http://2014.igem.org/wiki/index.php?title=Special:Contributions&feed=atom&limit=20&target=Alejovigno&year=&month=2014.igem.org - User contributions [en]2024-03-29T00:45:56ZFrom 2014.igem.orgMediaWiki 1.16.5http://2014.igem.org/Team:Valencia_UPV/VUPV_PartsTeam:Valencia UPV/VUPV Parts2014-11-13T18:15:49Z<p>Alejovigno: </p>
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<p><h3 class="hook" align="left"><a>Achievements</a> > <a>Parts</a></h3></p><br/><br />
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<div align="center"><span class="coda"><roja>P</roja>arts</span> </div><br/><br />
<p>Our team sent 7 BioBrick Parts to the <a class="normal-link-page" href="http://parts.igem.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2014&group=Valencia_UPV">Registry</a>.</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554000">BBa_K1554000</a> – Ta29 Promoter</b></p><br />
<p>The Ta29 promoter is anther-specific. Anthers are the male organs of the flower, and their function is to produce pollen. A cytotoxic gene (barnase, ipt, RIP...) is usually used under the control of this specific promoter leading to a plant unable to produce pollen, inducing male-sterility. <a href=https://2014.igem.org/Team:Valencia_UPV/Project/modules/biosafety> See Biosafety</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554001">BBa_K1554001</a> – AtrΔ11</b></p><br />
<p>The AtrΔ11 protein is a delta-11-desaturase from Amyelois transitella that introduces an unsaturation between C11 and C12 in long-chain fatty acids. This DNA sequence underwent codon usage optimization for Nicotiana benthamiana.<br />
Acyl-CoA + Reduced acceptor + O2 = Delta11-acyl-CoA + Acceptor + 2 H2O<br />
This part was used in the Insect sexual pheromone production pathway. <a href=https://2014.igem.org/Team:Valencia_UPV/Project/modules/biosynthesis> See Biosynthesis</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554002">BBa_K1554002</a> – HarFAR</b></p><br />
<p>The HarFAR protein is a fatty acid reductase that catalyses the conversion of a long-chain fatty acid carboxyl group to an alcohol group. The DNA sequence was obtained from Helicoverpa armigera HarFAR-3 protein, including an Endoplasmic Reticulum retention signal (KKYR) in the C-terminal end. In addition, codon usage optimization was performed for Nicotiana benthamiana.<br />
This part was used in the Insect sexual pheromone production pathway. <a href=https://2014.igem.org/Team:Valencia_UPV/Project/modules/biosynthesis> See Biosynthesis</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554003">BBa_K1554003</a> – EaDAcT</b></p><br />
<p>The EaDAcT protein is a diacylglycerol acetyltransferase coming from Euonymus alatus. This enzyme can transform alcohol gropus from fatty acid to aldehyde gropus. This DNA sequence underwent codon usage optimization for Nicotiana benthamiana.<br />
This part was used in the Insect sexual pheromone production pathway. <a href=https://2014.igem.org/Team:Valencia_UPV/Project/modules/biosynthesis> See Biosynthesis </a></p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554004">BBa_K1554004</a> – Yellow biosafety module for plants</b></p><br />
<p>This biosafety device for plants consists of two submodules: a male sterility submodule and an identity preservation submodule. The male sterility submodule consists of a barnase specifically expressed in anthers under the regulation of the TA29 tapetum-specific promoter. As result, pollen from these plants is not fertile. The identity preservation submodule expresses the yellow chromoprotein AmilGFP, so that plants can be visually differentiated from non-transgenic plants. See Results: Constructs-Biosafety</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554005">BBa_K1554005</a> - Blue biosafety module for plants</b></p><br />
<p>This biosafety device is composed of the same parts as the previously explained one, BBa_K1554004. This device uses AmilCP as chromoprotein instead of AmilGFP. See Results: Constructs-Biosafety</p><br/><br />
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<p> <b><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1554006">BBa_K1554006</a> - Omega Undercover</b></p><br />
<p>The Omega undercover vector allows the conversión of GoldenBraid standard parts to BioBrick’s. <a href=https://2014.igem.org/Team:Valencia_UPV/Project/modules/methodology/parts_construction> See Methodology: Parts construction </a></p><br/><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/policy/outreachTeam:Valencia UPV/policy/outreach2014-11-13T18:13:11Z<p>Alejovigno: </p>
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<div align="center"><span class="coda"><roja>O</roja>utreach</span> </div><br/><br/><br />
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<h3>Workshop announcement</h3><br />
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<h3><i>“Generación Espontánea UPV” </i></h3><br />
<p><br />
Valencia UPV iGEM team participated in the communication and diffusion activity called <i>Generación Espontánea</i> (translated as Spontaneous Generation). Twenty-four teams of students showed their projects related to different areas: engineering, social, computing, architecture and culture. We presented our <span class="red-bold">Sexy Plant</span>, and made diffusion of Synthetic Biology. <i>Generación Espontánea</i> was organized by UPV and Dr. Larisa Dunai, who was awarded by MIT Technology Review with the <b>MIT Young Innovator under 35 Award</b> of this year. </p><br />
<p align="right"> Valencia, October 2014</p><br />
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<h3><i>Summer 2014 School courses </i></h3><br />
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Valencia UPV iGEM team introduced The Sexy Plant Project to students between 10 -15 years old, during this summer. Besides, we organised an activity for students to learn the pH scale using a natural indicator based on anthocyanins obtained from plants. Around one hundred students extrated anthocyanins from a red cabagge, and created a natural pH indicator. </p> <br />
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<p align="right"> Valencia, from June to July 2014 </p><br />
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<h3>See our Lipdub at IBMCP labs!</h3><br />
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<p>We have organized and participated in a Lipdub. Our team wanted to involve the different groups that are currently working on the institute that is hosting us (IBMCP) in a different manner. We celebrated the 10th anniversary of the iGEM competition and the 20th anniversary of the IBMCP, too. We spend a great time with our colleagues and we promoted social relationships between us. As a result we did not only achieve an awesome video, but found a magnificent way to popularize science in society.</p><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/policy/outreachTeam:Valencia UPV/policy/outreach2014-11-13T18:12:27Z<p>Alejovigno: </p>
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<h3><i>“Generación Espontánea UPV” </i></h3><br />
<p><br />
Valencia UPV iGEM team participated in the communication and diffusion activity called <i>Generación Espontánea</i> (translated as Spontaneous Generation). Twenty-four teams of students showed their projects related to different areas: engineering, social, computing, architecture and culture. We presented our <span class="red-bold">Sexy Plant</span>, and made diffusion of Synthetic Biology. <i>Generación Espontánea</i> was organized by UPV and Dr. Larisa Dunai, who was awarded by MIT Technology Review with the <b>MIT Young Innovator under 35 Award</b> of this year. </p><br />
<p align="right"> Valencia, October 2014</p><br />
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<h3><i>Summer 2014 School courses </i></h3><br />
<p><br />
Valencia UPV iGEM team introduced The Sexy Plant Project to students between 10 -15 years old, during this summer. Besides, we organised an activity for students to learn the pH scale using a natural indicator based on anthocyanins obtained from plants. Around one hundred students extrated anthocyanins from a red cabagge, and created a natural pH indicator. </p> <br />
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<p align="right"> Valencia, from June to July 2014 </p><br />
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<h3>See our Lipdub at IBMCP labs!</h3><br />
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<p>We have organized and participated in a Lipdub. Our team wanted to involve the different groups that are currently working on the institute that is hosting us (IBMCP) in a different manner. We celebrated the 10th anniversary of the iGEM competition and the 20th anniversary of the IBMCP, too. We spend a great time with our colleagues and we promoted social relationships between us. As a result we did not only achieve an awesome video, but found a magnificent way to popularize science in society.</p><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/policy/outreachTeam:Valencia UPV/policy/outreach2014-11-13T18:10:38Z<p>Alejovigno: </p>
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<div align="center"><span class="coda"><roja>O</roja>utreach</span> </div><br/><br/><br />
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<h3>Workshop announcement</h3><br />
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<h3><i>“Generación Espontánea UPV” </i></h3><br />
<p><br />
Valencia UPV iGEM team participated in the communication and diffusion activity called <i>Generación Espontánea</i> (translated as Spontaneous Generation). Twenty-four teams of students showed their projects related to different areas: engineering, social, computing, architecture and culture. We presented our <span class="red-bold">Sexy Plant</span>, and made diffusion of Synthetic Biology. <i>Generación Espontánea</i> was organized by UPV and Dr. Larisa Dunai, who was awarded by MIT Technology Review with the <b>MIT Young Innovator under 35 Award</b> of this year. </p><br />
<p align="right"> Valencia, October 2014</p><br />
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</html><br />
[[Image:estiu.jpg|500px|right]] <br />
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<h3><i>Summer 2014 School courses </i></h3><br />
<p><br />
Valencia UPV iGEM team introduced The Sexy Plant Project to students between 10 -15 years old, during this summer. Besides, we organised an activity for students to learn the pH scale using a natural indicator based on anthocyanins obtained from plants. Around one hundred students extrated anthocyanins from a red cabagge, and created a natural pH indicator. </p> <br />
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<p align="right"> Valencia, from June to July 2014 </p><br />
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<div align="left"></div><br />
</br></br></br></br></br></br><br />
<h3>See our Lipdub at IBMCP labs!</h3><br />
<br/><br />
<p>We have organized and participated in a Lipdub. Our team wanted to involve the different groups that are currently working on the institute that is hosting us (IBMCP) in a different manner. We celebrated the 10th anniversary of the iGEM competition and the 20th anniversary of the IBMCP, too. We spend a great time with our colleagues and we promoted social relationships between us. As a result we did not only achieve an awesome video, but found a magnificent way to popularize science in society.</p><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:59:00Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
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<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
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<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
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<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
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<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
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<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
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<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
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<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
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<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
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<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
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<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
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<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
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<p align="left"><strong>Approximation</strong></p><br/><br />
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The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
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$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
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where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
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If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
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also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
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$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
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where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
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If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
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$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
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where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
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For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
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$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
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In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
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Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
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The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
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$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
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Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
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1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
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2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
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$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
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3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
<br/><br/><br/><br />
<div align="center"><br />
<iframe width="600" height="350"<br />
src="http://www.youtube.com/embed/URZgjbfEUwc"><br />
</iframe><br/><br/><br />
</div><br />
<br/><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
</div><br />
<br />
</div><br />
</div><br />
<br/><br/><br/><br />
<a class="button-content" id="goto-left" align="center" href="https://2014.igem.org/Team:Valencia_UPV/Modeling/overview"><strong>Go to Modeling Overview</strong></a><br />
<a class="button-content" id="goto-right" align="center" href="https://2014.igem.org/Team:Valencia_UPV/Modeling/fba"><strong>Go to Pheromone Production</strong></a></br></br></br><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:58:28Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
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<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
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<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<br/><br/><br/><br />
<div align="center"><br />
<iframe width="600" height="350"<br />
src="http://www.youtube.com/embed/URZgjbfEUwc"><br />
</iframe><br/><br/><br />
</div><br />
<br/><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
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<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
</div><br />
<br />
</div><br />
</div><br />
<br/><br/><br/><br />
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<a class="button-content" id="goto-right" align="center" href="https://2014.igem.org/Team:Valencia_UPV/Modeling/fba"><strong>Go to Pheromone Production</strong></a></br></br></br><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogoTeam:Valencia UPV/Modeling/sexyplants.nlogo2014-10-18T03:57:32Z<p>Alejovigno: </p>
<hr />
<div>breed [ moths moth ]<br />
breed [ females female ]<br />
breed [ standers stander ] <br />
<br />
<br />
moths-own<br />
[<br />
number-encounter<br />
count-down<br />
ocupado ;1 mating _ 0 free<br />
flockmates<br />
nearest-neighbor<br />
]<br />
<br />
females-own<br />
[<br />
count-down<br />
ocupado ;1 mating _ 0 free<br />
wait-time ;after mating, females don't emit pheromone and shouldn't mate a male<br />
]<br />
<br />
<br />
patches-own<br />
[<br />
old-pheromonelevel ; the pheromonelevel of the patch the last time thru go<br />
pheromonelevel ; the current pheromonelevel of the patch<br />
]<br />
<br />
globals<br />
[<br />
plate-size ; the size of the plate on which pheromone is diffusing <br />
min-pherolevel ; the minimum pheromonelevel <br />
max-pherolevel ; the maximum pheromonelevel <br />
old-number-encounter<br />
]<br />
<br />
<br />
;;;;;;;;;;;;;;;;;;;;;;;;<br />
;;; Setup Procedures ;;;<br />
;;;;;;;;;;;;;;;;;;;;;;;;<br />
<br />
to setup<br />
clear-all<br />
set plate-size max-pxcor <br />
<br />
<br />
;TO SEE STANDARS JUST REMOVE ; FROM THE FIVE LINES BELOW <br />
;create-standers 20000<br />
;[<br />
; setxy random-xcor random-ycor<br />
; set color gray<br />
;set size 0.5<br />
; ]<br />
<br />
ask patches<br />
[<br />
set pcolor white<br />
set-initial-pheromonelevels<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
]<br />
set min-pherolevel min [old-pheromonelevel] of patches<br />
set max-pherolevel max [old-pheromonelevel] of patches<br />
ask patches [ draw-plate ]<br />
<br />
<br />
set-default-shape moths "butterfly"<br />
make-moths number-moths<br />
<br />
set-default-shape females "butterfly"<br />
make-females number-females<br />
<br />
<br />
<br />
;SEXYPLANTS<br />
crt 1 [set shape "flower" set size 5 set xcor 0 set ycor 0 set color 86 ]<br />
crt 1 [set shape "flower" set size 5 set xcor 20 set ycor 20 set color 86 ]<br />
crt 1 [set shape "flower" set size 5 set xcor -20 set ycor 20 set color 86]<br />
crt 1 [set shape "flower" set size 5 set xcor 20 set ycor -20 set color 86]<br />
crt 1 [set shape "flower" set size 5 set xcor -20 set ycor -20 set color 86]<br />
<br />
;CROP <br />
crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 20 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -20 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 10 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -10 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 40 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -40 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 60 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -60 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 60 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -60 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 20 set color 44] <br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 40 set color 44]<br />
<br />
reset-ticks<br />
end<br />
<br />
to make-moths [ number ]<br />
create-moths number [<br />
set color brown<br />
jump random-float max-pxcor<br />
set size 2<br />
set number-encounter 0<br />
set count-down 40<br />
set ocupado 0<br />
]<br />
end<br />
<br />
to make-females [ number ]<br />
create-females number [<br />
set color 125<br />
jump random-float max-pxcor<br />
set size 2<br />
set count-down 40<br />
set ocupado 0<br />
set wait-time 0<br />
]<br />
end<br />
<br />
; Sets the pheromonelevel for inside of the plate<br />
to set-initial-pheromonelevels ;; Patch Procedure<br />
if ((abs pycor) < plate-size) and ((abs pxcor) < plate-size)<br />
[set pheromonelevel initial-plate-pherolevel] ;we consider no pheromone concentration in the field for t=0<br />
<br />
;Sexyplants<br />
if ( (pycor)= 0 ) and ( (pxcor) = 0)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= 20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= 20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= -20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= -20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
<br />
end<br />
<br />
<br />
<br />
;; Sets the pheromonelevels of the plate edges and corners<br />
;In this simulation this command is not activated, since the we consider an opened volume.<br />
to set-edge-pheromonelevels ;; patch procedure<br />
; set the pheromonelevels of the edges<br />
if (pxcor >= plate-size) and ((abs pycor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and ((abs pycor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pycor >= plate-size) and ((abs pxcor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pycor <= (- plate-size)) and ((abs pxcor) < plate-size )<br />
[set pheromonelevel 0]<br />
<br />
; set the pheromonelevels of the corners<br />
if (pxcor >= plate-size) and (pycor >= plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor >= plate-size) and (pycor <= (- plate-size))<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and (pycor >= plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and (pycor <= (- plate-size))<br />
[set pheromonelevel 0]<br />
<br />
<br />
end<br />
<br />
<br />
;;;;;;;;;;;;;;;;;;;;;;;;;;<br />
;;; Runtime Procedures ;;;<br />
;;;;;;;;;;;;;;;;;;;;;;;;;;<br />
<br />
;; Runs the simulation through a loop<br />
to go<br />
<br />
set max-pherolevel max [old-pheromonelevel] of patches<br />
set min-pherolevel min [old-pheromonelevel] of patches<br />
<br />
ask females [set color 125]<br />
ask standers [set color gray]<br />
<br />
move-thru-field ;males<br />
<br />
move-thru-field-females<br />
<br />
ask patches [<br />
;Diffusion process with advection (wind)<br />
set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4) - wind-forcey * ([old-pheromonelevel] of patch-at-heading-and-distance 0 1) - wind-forcex * ([old-pheromonelevel] of patch-at-heading-and-distance 90 1)) + ((1 - ( 4 * pheromone-diffusivity - 1 * wind-forcey - 1 * wind-forcex )) * old-pheromonelevel)<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
draw-plate<br />
]<br />
<br />
<br />
if switch = TRUE [<br />
ask patches<br />
[<br />
;; diffuse the pheromone of a patch with its neighbors<br />
if ( (pycor)= 0 ) and ( (pxcor) = 0)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= 20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= 20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= -20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= -20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
; set the edges back to their constant pheromone<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
draw-plate<br />
]<br />
<br />
] <br />
tick<br />
end<br />
<br />
<br />
to move-thru-field ; turtle procedure for male moths behavior<br />
<br />
ask moths<br />
[<br />
;MATING PROCEDURE<br />
ifelse (ocupado = 1) ;If i am mating<br />
[<br />
stay<br />
]<br />
[<br />
ifelse any? (females-on neighbors) <br />
[ <br />
set flockmates other females-on neighbors<br />
set nearest-neighbor min-one-of flockmates [distance myself]<br />
ifelse (([ocupado] of nearest-neighbor) = 0 and ([wait-time] of nearest-neighbor) = 0 )<br />
[<br />
set ocupado 1<br />
reset-count-down<br />
stay<br />
] ;If she is free and can<br />
[<br />
continue<br />
]<br />
][continue] <br />
]<br />
] <br />
;;;;;;;;;;; <br />
<br />
end<br />
<br />
to stay <br />
<br />
ifelse count-down = 0 <br />
[ <br />
set number-encounter number-encounter + 1<br />
set label number-encounter <br />
reset-count-down<br />
set ocupado 0<br />
] <br />
[<br />
set count-down count-down - 1 ;decrement-timer<br />
set label count-down <br />
] <br />
<br />
end<br />
<br />
to continue ;MALE MOTH RESPONSE <br />
<br />
<br />
ifelse ( pheromonelevel <= detectionthreshold) <br />
[ <br />
; if there is no detectable pheromone move randomly<br />
; flight by non-responding male moths involves short, fast movements in random directions (J.N. Perry and C.Wall, 1984)<br />
rt flutter-amount 45 ;RANDOM FLIGHT <br />
fd 1<br />
] <br />
[ <br />
ifelse (random 25 = 0) <br />
; add some additional randomness to the moth's movement, this allows some small<br />
; probability that the moth might "escape"<br />
[<br />
rt flutter-amount 60<br />
fd 1<br />
]<br />
[<br />
ifelse (pheromonelevel <= saturationlevel) <br />
[ <br />
maximize <br />
if (xcor = 0) and (ycor = 0) ;flying around our sexy plants<br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1<br />
]<br />
if (xcor = 20) and (ycor = 20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = -20) and (ycor = -20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = 20) and (ycor = -20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = -20) and (ycor = 20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
]<br />
<br />
[<br />
rt flutter-amount 60<br />
fd 1<br />
] <br />
]<br />
] <br />
if not can-move? 1<br />
[ maximize ]<br />
<br />
end<br />
<br />
to move-thru-field-females ; turtle procedure for female moths behavior<br />
<br />
ask females <br />
[<br />
<br />
;MATING PROCEDURE <br />
ifelse (ocupado = 1)<br />
[<br />
stayfemale<br />
]<br />
[ ;If i am free<br />
ifelse (any? moths-on neighbors) ;There are neighbors<br />
[ <br />
let flockmates-fem moths-on neighbors<br />
let nearest-neighbor-fem min-one-of flockmates-fem [distance myself]<br />
ifelse (([nearest-neighbor] of nearest-neighbor-fem) != 0) [<br />
ifelse ( ([who] of self) = ( [who] of ([nearest-neighbor] of nearest-neighbor-fem) ) and (wait-time = 0) ) ;ask if the neighbor of my neighbor is me: who-of myself<br />
[ <br />
set ocupado 1<br />
set color 85 ;change its colour in order to differentiate it by the observer<br />
stayfemale<br />
] [ continuefemale ]][continuefemale]<br />
]<br />
[<br />
continuefemale<br />
] <br />
]<br />
]<br />
end <br />
<br />
to stayfemale <br />
ifelse (count-down = 0)<br />
[ <br />
reset-count-down<br />
set ocupado 0<br />
set wait-time 100<br />
continuefemale<br />
]<br />
[ set count-down count-down - 1 ;decrement-timer<br />
set label count-down <br />
]<br />
<br />
end<br />
<br />
to continuefemale <br />
<br />
ifelse wait-time > 0 <br />
[set wait-time wait-time - 1] ;decrement timer for females after mating<br />
[pheromone-emission] <br />
rt flutter-amount 45 ;RANDOM FLIGHT <br />
if not can-move? 1<br />
[rt flutter-amount 60]<br />
fd 1 <br />
end<br />
<br />
to pheromone-emission <br />
ask patches in-radius 1[ <br />
set pheromonelevel releaserate<br />
set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate)<br />
set old-pheromonelevel pheromonelevel <br />
draw-plate ] <br />
end <br />
<br />
to-report flutter-amount [limit]<br />
;; This routine takes a number as an input and returns a random value between<br />
;; (+1 * input value) and (-1 * input value).<br />
;; It is used to add a random flutter to the moth's movements<br />
report random-float (2 * limit) - limit<br />
end<br />
<br />
;CHEMOATTRACTION<br />
to maximize ;; turtle procedure male moth<br />
move-to patch-here <br />
let p max-one-of neighbors [pheromonelevel] <br />
let gradient_p [pheromonelevel] of p - pheromonelevel<br />
ifelse (gradient_p > delta) [<br />
face p <br />
rt random-normal 0 (0.2 * (5 / gradient_p)) ;; RANDOM TURNING ANGLE INVERSELY RELATED TO THE GRADIENT OF PHEROMONE CONCENTRATION (model project approximation)<br />
fd 1<br />
]<br />
[rt flutter-amount 60<br />
fd 1] <br />
end<br />
<br />
;; Draws the patches that are within the plate<br />
to draw-plate ;; Patch Procedure<br />
if ((abs pycor) <= plate-size) and ((abs pxcor) <= plate-size)<br />
[color-patch]<br />
end<br />
<br />
;; color the patch based on its pheromonelevel<br />
to color-patch ;; Patch Procedure<br />
set pcolor scale-color 123 pheromonelevel 50 0<br />
end<br />
<br />
to-report releaserate<br />
report 3 * release-rate <br />
end <br />
<br />
<br />
to-report releaserate-sexyplant<br />
report 3 * release-rate-sexyplant ;Primary sex pheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
end ;for trap catch in the field at component emission rates similar to that used by the insect. <br />
;(W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386.<br />
<br />
;; report the pheromone diffusivity constant that we use for the calculations of pheromone diffusion<br />
to-report pheromone-diffusivity<br />
report number-diffusivity <br />
end <br />
<br />
to-report wind-forcex<br />
report wind-force-x<br />
end <br />
<br />
to-report wind-forcey<br />
report wind-force-y<br />
end <br />
<br />
to-report delta<br />
report 5 - moth-sensitivity<br />
<br />
end <br />
<br />
<br />
to-report saturationlevel<br />
report saturation-level-threshold <br />
end <br />
<br />
to-report detectionthreshold<br />
report detection-threshold<br />
end <br />
<br />
<br />
to reset-count-down <br />
set count-down 40 <br />
end<br />
<br />
<br />
<br />
<br />
; Copyright 2014 Valencia_UPV iGEM team<br />
; See Info tab for full copyright and license.<br />
@#$#@#$#@<br />
GRAPHICS-WINDOW<br />
976<br />
12<br />
1581<br />
627<br />
50<br />
49<br />
5.9<br />
1<br />
2<br />
1<br />
1<br />
1<br />
0<br />
1<br />
1<br />
1<br />
-50<br />
50<br />
-49<br />
49<br />
1<br />
1<br />
1<br />
ticks<br />
20.0<br />
<br />
SLIDER<br />
978<br />
637<br />
1178<br />
670<br />
initial-plate-pherolevel<br />
initial-plate-pherolevel<br />
0<br />
25<br />
0<br />
1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
BUTTON<br />
1480<br />
640<br />
1545<br />
673<br />
Go<br />
go<br />
T<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
BUTTON<br />
1273<br />
639<br />
1389<br />
672<br />
Setup<br />
setup<br />
NIL<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
BUTTON<br />
1398<br />
640<br />
1471<br />
673<br />
Go Once<br />
go<br />
NIL<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
SWITCH<br />
1216<br />
12<br />
1317<br />
45<br />
switch<br />
switch<br />
1<br />
1<br />
-1000<br />
<br />
SLIDER<br />
32<br />
33<br />
204<br />
66<br />
number-moths<br />
number-moths<br />
0<br />
100<br />
5<br />
1<br />
1<br />
u<br />
HORIZONTAL<br />
<br />
SLIDER<br />
33<br />
78<br />
205<br />
111<br />
number-females<br />
number-females<br />
0<br />
50<br />
5<br />
1<br />
1<br />
u<br />
HORIZONTAL<br />
<br />
SLIDER<br />
32<br />
443<br />
245<br />
476<br />
number-diffusivity<br />
number-diffusivity<br />
0.01<br />
0.2<br />
0.177<br />
0.001<br />
1<br />
cm^2/s<br />
HORIZONTAL<br />
<br />
PLOT<br />
273<br />
337<br />
957<br />
640<br />
Meetings in time<br />
Time<br />
Number of meeting<br />
0.0<br />
1000.0<br />
0.0<br />
10.0<br />
true<br />
false<br />
"" ""<br />
PENS<br />
"males mating" 1.0 0 -2674135 true "" "plot sum [ocupado] of moths"<br />
<br />
SLIDER<br />
27<br />
304<br />
199<br />
337<br />
moth-sensitivity<br />
moth-sensitivity<br />
0<br />
5<br />
4.6<br />
0.001<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
26<br />
262<br />
214<br />
295<br />
saturation-level-threshold<br />
saturation-level-threshold<br />
1<br />
100<br />
8<br />
1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
MONITOR<br />
306<br />
369<br />
575<br />
422<br />
number of encounters at this instant<br />
sum [ocupado] of moths<br />
17<br />
1<br />
13<br />
<br />
PLOT<br />
275<br />
12<br />
955<br />
329<br />
Total number of encounters in time<br />
Time<br />
Number of encounters<br />
0.0<br />
1000.0<br />
0.0<br />
100.0<br />
true<br />
true<br />
"" ""<br />
PENS<br />
"total number of encounters up to now" 1.0 0 -14070903 true "" "plot sum [number-encounter] of moths"<br />
"males mating at this moment" 1.0 0 -2674135 true "" "plot sum [ocupado] of moths"<br />
<br />
MONITOR<br />
314<br />
39<br />
507<br />
93<br />
total number of encounters<br />
sum [number-encounter] of moths<br />
17<br />
1<br />
13<br />
<br />
SLIDER<br />
27<br />
217<br />
198<br />
250<br />
detection-threshold<br />
detection-threshold<br />
0.5<br />
5<br />
1.2<br />
0.1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
32<br />
488<br />
176<br />
521<br />
wind-force-x<br />
wind-force-x<br />
-0.1<br />
0.1<br />
0<br />
0.01<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
33<br />
534<br />
172<br />
567<br />
wind-force-y<br />
wind-force-y<br />
-0.1<br />
0.1<br />
0<br />
0.01<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
27<br />
178<br />
198<br />
211<br />
release-rate<br />
release-rate<br />
0<br />
1<br />
0.999<br />
0.001<br />
1<br />
µg/h<br />
HORIZONTAL<br />
<br />
SLIDER<br />
27<br />
346<br />
246<br />
379<br />
release-rate-sexyplant<br />
release-rate-sexyplant<br />
0<br />
1<br />
0.998<br />
0.001<br />
1<br />
µg/h<br />
HORIZONTAL<br />
<br />
@#$#@#$#@<br />
## WHAT IS IT?<br />
<br />
This model simulates pheromone diffusion processes and its influence in moths response.<br />
https://2014.igem.org/Team:Valencia_UPV<br />
<br />
## HOW IT WORKS<br />
Visit our wiki!<br />
<br />
## HOW TO USE IT<br />
Visit our wiki!<br />
<br />
## THINGS TO TRY<br />
Visit our wiki!<br />
<br />
## THINGS TO NOTICE<br />
Visit our wiki!<br />
<br />
## EXTENDING THE MODEL<br />
<br />
This model simulates a classic partial differential equation problem (that of heat diffusion with advection). <br />
<br />
<br />
## HOW TO CITE<br />
<br />
If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:<br />
<br />
* Valencia_UPV iGEM, Alejandra González Boscá (2014) Pheromone diffusion and moths response with Sexyplants<br />
<br />
<br />
## COPYRIGHT AND LICENSE<br />
<br />
Valencia_UPV<br />
iGEM 2014<br />
<br />
Alejandra González Boscá<br />
<br />
<br />
@#$#@#$#@<br />
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@#$#@#$#@</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:54:00Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
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<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<br />
<div align="center"><br />
<iframe width="600" height="450"<br />
src="http://www.youtube.com/embed/URZgjbfEUwc"><br />
</iframe><br/><br/><br />
</div><br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
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</div><br />
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<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
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<br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:53:21Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<br />
<iframe width="600" height="450"<br />
src="http://www.youtube.com/embed/URZgjbfEUwc"><br />
</iframe><br/><br/><br />
<br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
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</div><br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:52:57Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
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<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<br />
<iframe width="800" height="450"<br />
src="http://www.youtube.com/embed/URZgjbfEUwc"><br />
</iframe><br/><br/><br />
<br />
<div align="center"><br />
<embed align="center" width="600" height="450"<br />
src="http://www.youtube.com/watch?v=URZgjbfEUwc"><br />
</div><br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
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<br />
</div><br />
</div><br />
<br/><br/><br/><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogoTeam:Valencia UPV/Modeling/sexyplants.nlogo2014-10-18T03:50:38Z<p>Alejovigno: </p>
<hr />
<div>breed [ moths moth ]<br />
breed [ females female ]<br />
breed [ standers stander ] <br />
<br />
<br />
moths-own<br />
[<br />
number-encounter<br />
count-down<br />
ocupado ;1 mating _ 0 free<br />
flockmates<br />
nearest-neighbor<br />
]<br />
<br />
females-own<br />
[<br />
count-down<br />
ocupado ;1 mating _ 0 free<br />
wait-time ;after mating, females don't emit pheromone and shouldn't mate a male<br />
]<br />
<br />
<br />
patches-own<br />
[<br />
old-pheromonelevel ; the pheromonelevel of the patch the last time thru go<br />
pheromonelevel ; the current pheromonelevel of the patch<br />
]<br />
<br />
globals<br />
[<br />
plate-size ; the size of the plate on which pheromone is diffusing <br />
min-pherolevel ; the minimum pheromonelevel <br />
max-pherolevel ; the maximum pheromonelevel <br />
old-number-encounter<br />
]<br />
<br />
<br />
;;;;;;;;;;;;;;;;;;;;;;;;<br />
;;; Setup Procedures ;;;<br />
;;;;;;;;;;;;;;;;;;;;;;;;<br />
<br />
to setup<br />
clear-all<br />
set plate-size max-pxcor <br />
<br />
<br />
;TO SEE STANDARS JUST REMOVE ; FROM THE FIVE LINES BELOW <br />
;create-standers 20000<br />
;[<br />
; setxy random-xcor random-ycor<br />
; set color gray<br />
;set size 0.5<br />
; ]<br />
<br />
ask patches<br />
[<br />
set pcolor white<br />
set-initial-pheromonelevels<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
]<br />
set min-pherolevel min [old-pheromonelevel] of patches<br />
set max-pherolevel max [old-pheromonelevel] of patches<br />
ask patches [ draw-plate ]<br />
<br />
<br />
set-default-shape moths "butterfly"<br />
make-moths number-moths<br />
<br />
set-default-shape females "butterfly"<br />
make-females number-females<br />
<br />
<br />
<br />
;SEXYPLANTS<br />
crt 1 [set shape "flower" set size 5 set xcor 0 set ycor 0 set color 86 ]<br />
crt 1 [set shape "flower" set size 5 set xcor 20 set ycor 20 set color 86 ]<br />
crt 1 [set shape "flower" set size 5 set xcor -20 set ycor 20 set color 86]<br />
crt 1 [set shape "flower" set size 5 set xcor 20 set ycor -20 set color 86]<br />
crt 1 [set shape "flower" set size 5 set xcor -20 set ycor -20 set color 86]<br />
<br />
;CROP <br />
crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 20 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -20 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -10 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 10 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor 0 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor 20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor 30 set ycor -20 set color 44]<br />
crt 1 [set shape "plant" set size 2 set xcor -30 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 10 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -10 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 40 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -40 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor 60 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 0 set ycor -60 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 60 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -60 set ycor 0 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor -20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 20 set color 44]<br />
; crt 1 [set shape "plant" set size 2 set xcor 40 set ycor 20 set color 44] <br />
; crt 1 [set shape "plant" set size 2 set xcor -40 set ycor 40 set color 44]<br />
<br />
reset-ticks<br />
end<br />
<br />
to make-moths [ number ]<br />
create-moths number [<br />
set color brown<br />
jump random-float max-pxcor<br />
set size 2<br />
set number-encounter 0<br />
set count-down 40<br />
set ocupado 0<br />
]<br />
end<br />
<br />
to make-females [ number ]<br />
create-females number [<br />
set color 125<br />
jump random-float max-pxcor<br />
set size 2<br />
set count-down 40<br />
set ocupado 0<br />
set wait-time 0<br />
]<br />
end<br />
<br />
; Sets the pheromonelevel for inside of the plate<br />
to set-initial-pheromonelevels ;; Patch Procedure<br />
if ((abs pycor) < plate-size) and ((abs pxcor) < plate-size)<br />
[set pheromonelevel initial-plate-pherolevel] ;we consider no pheromone concentration in the field for t=0<br />
<br />
;Sexyplants<br />
if ( (pycor)= 0 ) and ( (pxcor) = 0)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= 20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= 20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= -20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
if ( (pycor)= -20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel releaserate-sexyplant]<br />
<br />
end<br />
<br />
<br />
<br />
;; Sets the pheromonelevels of the plate edges and corners<br />
;In this simulation this command is not activated, since the we consider an opened volume.<br />
to set-edge-pheromonelevels ;; patch procedure<br />
; set the pheromonelevels of the edges<br />
if (pxcor >= plate-size) and ((abs pycor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and ((abs pycor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pycor >= plate-size) and ((abs pxcor) < plate-size)<br />
[set pheromonelevel 0]<br />
if (pycor <= (- plate-size)) and ((abs pxcor) < plate-size )<br />
[set pheromonelevel 0]<br />
<br />
; set the pheromonelevels of the corners<br />
if (pxcor >= plate-size) and (pycor >= plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor >= plate-size) and (pycor <= (- plate-size))<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and (pycor >= plate-size)<br />
[set pheromonelevel 0]<br />
if (pxcor <= (- plate-size)) and (pycor <= (- plate-size))<br />
[set pheromonelevel 0]<br />
<br />
<br />
end<br />
<br />
<br />
;;;;;;;;;;;;;;;;;;;;;;;;;;<br />
;;; Runtime Procedures ;;;<br />
;;;;;;;;;;;;;;;;;;;;;;;;;;<br />
<br />
;; Runs the simulation through a loop<br />
to go<br />
<br />
set max-pherolevel max [old-pheromonelevel] of patches<br />
set min-pherolevel min [old-pheromonelevel] of patches<br />
<br />
ask females [set color 125]<br />
ask standers [set color gray]<br />
<br />
move-thru-field ;males<br />
<br />
move-thru-field-females<br />
<br />
ask patches [<br />
;Diffusion process with advection (wind)<br />
set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4) - wind-forcey * ([old-pheromonelevel] of patch-at-heading-and-distance 0 1) - wind-forcex * ([old-pheromonelevel] of patch-at-heading-and-distance 90 1)) + ((1 - ( 4 * pheromone-diffusivity - 1 * wind-forcey - 1 * wind-forcex )) * old-pheromonelevel)<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
draw-plate<br />
]<br />
<br />
<br />
if switch = TRUE [<br />
ask patches<br />
[<br />
;; diffuse the pheromone of a patch with its neighbors<br />
if ( (pycor)= 0 ) and ( (pxcor) = 0)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= 20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= 20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= -20 ) and ( (pxcor) = 20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
if ( (pycor)= -20 ) and ( (pxcor) = -20)<br />
[set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate-sexyplant)]<br />
; set the edges back to their constant pheromone<br />
;set-edge-pheromonelevels<br />
set old-pheromonelevel pheromonelevel<br />
draw-plate<br />
]<br />
<br />
] <br />
tick<br />
end<br />
<br />
<br />
to move-thru-field ; turtle procedure for male moths behavior<br />
<br />
ask moths<br />
[<br />
;MATING PROCEDURE<br />
ifelse (ocupado = 1) ;If i am mating<br />
[<br />
stay<br />
]<br />
[<br />
ifelse any? (females-on neighbors) <br />
[ <br />
set flockmates other females-on neighbors<br />
set nearest-neighbor min-one-of flockmates [distance myself]<br />
ifelse (([ocupado] of nearest-neighbor) = 0 and ([wait-time] of nearest-neighbor) = 0 )<br />
[<br />
set ocupado 1<br />
reset-count-down<br />
stay<br />
] ;If she is free and can<br />
[<br />
continue<br />
]<br />
][continue] <br />
]<br />
] <br />
;;;;;;;;;;; <br />
<br />
end<br />
<br />
to stay <br />
<br />
ifelse count-down = 0 <br />
[ <br />
set number-encounter number-encounter + 1<br />
set label number-encounter <br />
reset-count-down<br />
set ocupado 0<br />
] <br />
[<br />
set count-down count-down - 1 ;decrement-timer<br />
set label count-down <br />
] <br />
<br />
end<br />
<br />
to continue ;MALE MOTH RESPONSE <br />
<br />
<br />
ifelse ( pheromonelevel <= detectionthreshold) <br />
[ <br />
; if there is no detectable pheromone move randomly<br />
; flight by non-responding male moths involves short, fast movements in random directions (J.N. Perry and C.Wall, 1984)<br />
rt flutter-amount 45 ;RANDOM FLIGHT <br />
fd 1<br />
] <br />
[ <br />
ifelse (random 25 = 0) <br />
; add some additional randomness to the moth's movement, this allows some small<br />
; probability that the moth might "escape"<br />
[<br />
rt flutter-amount 60<br />
fd 1<br />
]<br />
[<br />
ifelse (pheromonelevel <= saturationlevel) <br />
[ <br />
maximize <br />
if (xcor = 0) and (ycor = 0) ;flying around our sexy plants<br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1<br />
]<br />
if (xcor = 20) and (ycor = 20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = -20) and (ycor = -20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = 20) and (ycor = -20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
if (xcor = -20) and (ycor = 20) <br />
[jump 2<br />
rt flutter-amount 70<br />
fd 1]<br />
]<br />
<br />
[<br />
rt flutter-amount 60<br />
fd 1<br />
] <br />
]<br />
] <br />
if not can-move? 1<br />
[ maximize ]<br />
<br />
end<br />
<br />
to move-thru-field-females ; turtle procedure for female moths behavior<br />
<br />
ask females <br />
[<br />
<br />
;MATING PROCEDURE <br />
ifelse (ocupado = 1)<br />
[<br />
stayfemale<br />
]<br />
[ ;If i am free<br />
ifelse (any? moths-on neighbors) ;There are neighbors<br />
[ <br />
let flockmates-fem moths-on neighbors<br />
let nearest-neighbor-fem min-one-of flockmates-fem [distance myself]<br />
ifelse (([nearest-neighbor] of nearest-neighbor-fem) != 0) [<br />
ifelse ( ([who] of self) = ( [who] of ([nearest-neighbor] of nearest-neighbor-fem) ) and (wait-time = 0) ) ;ask if the neighbor of my neighbor is me: who-of myself<br />
[ <br />
set ocupado 1<br />
set color 85 ;change its colour in order to differentiate it by the observer<br />
stayfemale<br />
] [ continuefemale ]][continuefemale]<br />
]<br />
[<br />
continuefemale<br />
] <br />
]<br />
]<br />
end <br />
<br />
to stayfemale <br />
ifelse (count-down = 0)<br />
[ <br />
reset-count-down<br />
set ocupado 0<br />
set wait-time 100<br />
continuefemale<br />
]<br />
[ set count-down count-down - 1 ;decrement-timer<br />
set label count-down <br />
]<br />
<br />
end<br />
<br />
to continuefemale <br />
<br />
ifelse wait-time > 0 <br />
[set wait-time wait-time - 1] ;decrement timer for females after mating<br />
[pheromone-emission] <br />
rt flutter-amount 45 ;RANDOM FLIGHT <br />
if not can-move? 1<br />
[rt flutter-amount 60]<br />
fd 1 <br />
end<br />
<br />
to pheromone-emission <br />
ask patches in-radius 1[ <br />
set pheromonelevel releaserate<br />
set pheromonelevel (pheromone-diffusivity * (sum [old-pheromonelevel] of neighbors4)) + ((1 - ( 4 * pheromone-diffusivity )) * old-pheromonelevel + releaserate)<br />
set old-pheromonelevel pheromonelevel <br />
draw-plate ] <br />
end <br />
<br />
to-report flutter-amount [limit]<br />
;; This routine takes a number as an input and returns a random value between<br />
;; (+1 * input value) and (-1 * input value).<br />
;; It is used to add a random flutter to the moth's movements<br />
report random-float (2 * limit) - limit<br />
end<br />
<br />
;CHEMOATTRACTION<br />
to maximize ;; turtle procedure male moth<br />
move-to patch-here <br />
let p max-one-of neighbors [pheromonelevel] <br />
let gradient_p [pheromonelevel] of p - pheromonelevel<br />
ifelse (gradient_p > delta) [<br />
face p <br />
rt random-normal 0 (0.2 * (5 / gradient_p)) ;; RANDOM TURNING ANGLE INVERSELY RELATED TO THE GRADIENT OF PHEROMONE CONCENTRATION (model project approximation)<br />
fd 1<br />
]<br />
[rt flutter-amount 60<br />
fd 1] <br />
end<br />
<br />
;; Draws the patches that are within the plate<br />
to draw-plate ;; Patch Procedure<br />
if ((abs pycor) <= plate-size) and ((abs pxcor) <= plate-size)<br />
[color-patch]<br />
end<br />
<br />
;; color the patch based on its pheromonelevel<br />
to color-patch ;; Patch Procedure<br />
set pcolor scale-color 123 pheromonelevel 50 0<br />
end<br />
<br />
to-report releaserate<br />
report 3 * release-rate <br />
end <br />
<br />
<br />
to-report releaserate-sexyplant<br />
report 3 * release-rate-sexyplant ;Primary sex pheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
end ;for trap catch in the field at component emission rates similar to that used by the insect. <br />
;(W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386.<br />
<br />
;; report the pheromone diffusivity constant that we use for the calculations of pheromone diffusion<br />
to-report pheromone-diffusivity<br />
report number-diffusivity <br />
end <br />
<br />
to-report wind-forcex<br />
report wind-force-x<br />
end <br />
<br />
to-report wind-forcey<br />
report wind-force-y<br />
end <br />
<br />
to-report delta<br />
report 5 - moth-sensitivity<br />
<br />
end <br />
<br />
<br />
to-report saturationlevel<br />
report saturation-level-threshold <br />
end <br />
<br />
to-report detectionthreshold<br />
report detection-threshold<br />
end <br />
<br />
<br />
to reset-count-down <br />
set count-down 40 <br />
end<br />
<br />
<br />
<br />
<br />
; Copyright 2014 Valencia_UPV iGEM team<br />
; See Info tab for full copyright and license.<br />
@#$#@#$#@<br />
GRAPHICS-WINDOW<br />
976<br />
12<br />
1581<br />
627<br />
50<br />
49<br />
5.9<br />
1<br />
2<br />
1<br />
1<br />
1<br />
0<br />
1<br />
1<br />
1<br />
-50<br />
50<br />
-49<br />
49<br />
1<br />
1<br />
1<br />
ticks<br />
20.0<br />
<br />
SLIDER<br />
978<br />
637<br />
1178<br />
670<br />
initial-plate-pherolevel<br />
initial-plate-pherolevel<br />
0<br />
25<br />
0<br />
1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
BUTTON<br />
1480<br />
640<br />
1545<br />
673<br />
Go<br />
go<br />
T<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
BUTTON<br />
1273<br />
639<br />
1389<br />
672<br />
Setup<br />
setup<br />
NIL<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
BUTTON<br />
1398<br />
640<br />
1471<br />
673<br />
Go Once<br />
go<br />
NIL<br />
1<br />
T<br />
OBSERVER<br />
NIL<br />
NIL<br />
NIL<br />
NIL<br />
1<br />
<br />
SWITCH<br />
1216<br />
12<br />
1317<br />
45<br />
switch<br />
switch<br />
1<br />
1<br />
-1000<br />
<br />
SLIDER<br />
32<br />
33<br />
204<br />
66<br />
number-moths<br />
number-moths<br />
0<br />
100<br />
5<br />
1<br />
1<br />
u<br />
HORIZONTAL<br />
<br />
SLIDER<br />
33<br />
78<br />
205<br />
111<br />
number-females<br />
number-females<br />
0<br />
50<br />
5<br />
1<br />
1<br />
u<br />
HORIZONTAL<br />
<br />
SLIDER<br />
32<br />
443<br />
245<br />
476<br />
number-diffusivity<br />
number-diffusivity<br />
0.01<br />
0.2<br />
0.177<br />
0.001<br />
1<br />
cm^2/s<br />
HORIZONTAL<br />
<br />
PLOT<br />
273<br />
337<br />
957<br />
640<br />
Meetings in time<br />
Time<br />
Number of meeting<br />
0.0<br />
1000.0<br />
0.0<br />
10.0<br />
true<br />
false<br />
"" ""<br />
PENS<br />
"males mating" 1.0 0 -2674135 true "" "plot sum [ocupado] of moths"<br />
<br />
SLIDER<br />
27<br />
304<br />
199<br />
337<br />
moth-sensitivity<br />
moth-sensitivity<br />
0<br />
5<br />
4.6<br />
0.001<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
26<br />
262<br />
214<br />
295<br />
saturation-level-threshold<br />
saturation-level-threshold<br />
1<br />
100<br />
8<br />
1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
MONITOR<br />
306<br />
369<br />
575<br />
422<br />
number of encounters at this instant<br />
sum [ocupado] of moths<br />
17<br />
1<br />
13<br />
<br />
PLOT<br />
275<br />
12<br />
955<br />
329<br />
Total number of encounters in time<br />
Time<br />
Number of encounters<br />
0.0<br />
1000.0<br />
0.0<br />
100.0<br />
true<br />
true<br />
"" ""<br />
PENS<br />
"total number of encounters up to now" 1.0 0 -14070903 true "" "plot sum [number-encounter] of moths"<br />
"males mating at this moment" 1.0 0 -2674135 true "" "plot sum [ocupado] of moths"<br />
<br />
MONITOR<br />
314<br />
39<br />
507<br />
93<br />
total number of encounters<br />
sum [number-encounter] of moths<br />
17<br />
1<br />
13<br />
<br />
SLIDER<br />
27<br />
217<br />
198<br />
250<br />
detection-threshold<br />
detection-threshold<br />
0.5<br />
5<br />
1.2<br />
0.1<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
32<br />
488<br />
176<br />
521<br />
wind-force-x<br />
wind-force-x<br />
-0.1<br />
0.1<br />
0<br />
0.01<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
33<br />
534<br />
172<br />
567<br />
wind-force-y<br />
wind-force-y<br />
-0.1<br />
0.1<br />
0<br />
0.01<br />
1<br />
NIL<br />
HORIZONTAL<br />
<br />
SLIDER<br />
27<br />
178<br />
198<br />
211<br />
release-rate<br />
release-rate<br />
0<br />
1<br />
0.999<br />
0.001<br />
1<br />
µg/h<br />
HORIZONTAL<br />
<br />
SLIDER<br />
27<br />
346<br />
246<br />
379<br />
release-rate-sexyplant<br />
release-rate-sexyplant<br />
0<br />
1<br />
0.998<br />
0.001<br />
1<br />
µg/h<br />
HORIZONTAL<br />
<br />
@#$#@#$#@<br />
## WHAT IS IT?<br />
<br />
This model simulates pheromone diffusion processes and its influence in moths response.<br />
<br />
<br />
<br />
<br />
The View shows a square thin plate as viewed from above. The plate is thermally isolated on the two faces parallel to the view such that heat can flow only in and out from the perimeter of the plate and not into or out of the world. Heat is kept constant at the edges. As the simulation runs, heat is transmitted from warmer parts of the plate to cooler parts of the plate as shown by the varying color of the plate. Therefore, the temperature of the plate begins to change immediately and possibly differently at different locations, gradually converging to a stable state. Overall, the temperature distribution over the plate is a function of time and location. In addition to this simple use of the model, you are encouraged to control various paramaters, such as the temperature of each edge edge of the plate and of the center of the plate before--and even while--the model is running.<br />
<br />
Heat diffuses ("spreads") at different rates through different media. These rates can be determined and are called the Thermal Diffusivity of the material. The Greek letter alpha is often associated with this value. The diffusivity of a material does not change based on how much of the material there is. It is always the same. Below is a table containing several different materials with different diffusivity rates. See that wood (bottom row) has a lower heat diffusivity than, say, iron. This means that it takes a longer for heat to spread through a wooden object than an iron one. That is one reason why the handles of iron saucepans are wooden, and not the other way round. Also, think of a marble table with iron legs that has just been put out in the sun in a street-side cafe. Which material part of the table do you expect will warm up faster? The model allows you to change thermal diffusivity of the plate in two ways. You can directly change the value of ALPHA to any value you like, or you can indirectly change ALPHA by selecting a material.<br />
<br />
### Thermal diffusivity of selected materials<br />
<br />
<table border><br />
<tr><th>Material<th>Thermal diffusivity<br>(alpha cm*cm/s)<br />
<tr><td>Wood (Maple)<td>0.00128<br />
<tr><td>Stone (Marble)<td>0.0120<br />
<tr><td>Iron<td>0.2034<br />
<tr><td>Aluminum<td>0.8418<br />
<tr><td>Silver<td>1.7004<br />
</table><br />
<br />
## HOW IT WORKS<br />
<br />
Initialize the plate and edges to have temperatures that equal their respective slider values. Each time through the GO procedure, diffuse the heat on each patch in the following way. Have each patch set its current temperature to the sum of the 4 neighbors' old temperature times a constant based on alpha plus a weighted version of the patch's old temperature. (For those interested, the updated temperature is calculated by using a Forward Euler Method.) Then the edges are set back to the specified values and the old temperature is updated to the current temperature. Then the plate is redrawn.<br />
<br />
## HOW TO USE IT<br />
<br />
There are five temperature sliders which enable users to set four fixed edge temperatures and one initial plate temperature: <br />
-- TOP-TEMP - Top edge temperature <br />
-- BOTTOM-TEMP - Bottom edge temperature <br />
-- IN-PLATE-TEMP - Initial plate temperature <br />
-- LEFT-TEMP - Left edge temperature <br />
-- RIGHT-TEMP - Right edge temperature<br />
<br />
There are two sliders that govern the thermal diffusivity of the plate: <br />
-- MATERIAL-TYPE - The value of the chooser is that of the above chart. You must press UPDATE ALPHA for this to change the value of ALPHA. <br />
-- ALPHA - The alpha constant of thermal diffusivity<br />
<br />
There are four buttons with the following functions: <br />
-- SETUP - Initializes the model <br />
-- GO - Runs the simulation indefinitely <br />
-- GO ONCE - Runs the simulation for 1 time step <br />
-- UPDATE ALPHA - press this if you want to set ALPHA to a preset value based on a material selected by the MATERIAL-TYPE chooser<br />
<br />
The TIME monitor shows how many time steps the model has gone through.<br />
<br />
## THINGS TO TRY<br />
<br />
Set the paramters on the temperature sliders. Pick a value for ALPHA (or pick MATERIAL-TYPE and press UPDATE ALPHA). After you have changed all the sliders to values you like, press Setup followed by GO or GO ONCE.<br />
<br />
## THINGS TO NOTICE<br />
<br />
How does the equilibrium temperature distribution vary for different edge temperature settings?<br />
<br />
Notice how an equilibrium (the steady-state condition) is reached.<br />
<br />
Keep track of the units:<br />
<br />
<table border><br />
<tr><th>Variables<th>Units<br />
<tr><td>time<td>0.1 second<br />
<tr><td>temperature<td>degrees Celsius<br />
<tr><td>length<td>centimeters<br />
<tr><td>diffusivity<td>square centimeters per second<br />
</table><br />
<br />
## THINGS TO TRY<br />
<br />
Try different materials to observe the heat transfer speed. How does this compare to physical experiments?<br />
<br />
Try the following sample settings:<br />
- Top:100, Bottom:0, Left:0, Right:0<br />
- Top:0, Bottom:100, Left:100, Right:100<br />
- Top:0, Bottom:66, Left:99, Right:33<br />
- Top:25, Bottom:25, Left:100, Right:0<br />
<br />
## EXTENDING THE MODEL<br />
<br />
This model simulates a classic partial differential equation problem (that of heat diffusion). The thin square plate is a typical example, and the simplest model of the behavior. Try changing the shape or thickness of the plate (e.g. a circular or elliptical plate), or adding a hole in the center (the plate would then be a slice of a torus, a doughnut-shaped geometric object).<br />
<br />
Add a slider to alter this thickness.<br />
<br />
Try modeling derivative or combined boundary conditions.<br />
<br />
<br />
## HOW TO CITE<br />
<br />
If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:<br />
<br />
*<br />
<br />
* Wilensky, U. (1998). NetLogo Heat Diffusion model. http://ccl.northwestern.edu/netlogo/models/HeatDiffusion. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.<br />
* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.<br />
<br />
## COPYRIGHT AND LICENSE<br />
<br />
Valencia_UPV<br />
iGEM 2014<br />
<br />
Alejandra González Boscá<br />
<br />
<br />
Copyright 1998 Uri Wilensky.<br />
<br />
![CC BY-NC-SA 3.0](http://i.creativecommons.org/l/by-nc-sa/3.0/88x31.png)<br />
<br />
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.<br />
<br />
Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.<br />
<br />
This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.<br />
<br />
This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2001.<br />
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@#$#@#$#@</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:45:20Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<br />
<br />
<div align="center"><br />
<embed align="center" width="600" height="450"<br />
src="http://www.youtube.com/watch?v=URZgjbfEUwc"><br />
</div><br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
</div><br />
<br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:42:23Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
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<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
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<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
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</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
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</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
<br/><br />
<div align="center"><br />
<embed align="center" width="600" height="450"<br />
src="http://www.youtube.com/v/x_URZgjbfEUwc"><br />
</div><br />
<br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
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<br/><br />
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</div><br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:40:50Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
<br />
<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
<div align="center"><br />
<embed align="center" width="600" height="450"<br />
src="http://www.youtube.com/v/x_URZgjbfEUwc"><br />
<br />
</div><br />
<br />
<br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
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</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
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<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:38:41Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
<div align="center"><br />
<embed align="center" width="600" height="450"<br />
src="http://www.youtube.com/watch?v=URZgjbfEUwc"><br />
</div><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br/><br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. The parameters ranges we found in the literature are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of physical search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of physical search: 0.02-1 µg/h <br/><br />
References: [4], [5], [8]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: Primary sexpheromone components are approximately defined as those emitted by the calling insect that are obligatory<br />
for trap catch in the field at component emission rates similar to that used by the insect [4].</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of physical search: 1000 molecules/ cm3<br/><br />
References: [4], [5], [8]</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd> References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4].<br/><br />
Range of physical search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br />
</dd><br />
<dt>Wind force</dt><br />
<dd>Range: 0 - 10 m/s <br/><br />
References: [7] </dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
<li>Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I.<br />
Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air.<br />
J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
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<br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:25:41Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:Upv_simu3.png|450px|center|Figure 3. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. Doing a model parameters swept, we simulated many possible scenarios, and then we came up with values of parameters corresponding to our desired goal: insects get confused. This will be useful to know the limitations of our system and to help to decide the final distribution of our plants in the crop field. The parameters we found are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of search: 0.02-1 µg/h <br/><br />
References: [4], [5]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4]. This emission rate above which male start to get confused could be the release rate from females.</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of search: 0.001-1 [Mass]/[Distance]^2</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd>Range of search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br/><br />
Range: 0-0.0009 <br/><br />
(The maximum level of moth sensitivity has to be less than the minimum level of release rate of females, since this parameter is obtained from the difference)</dd><br />
<dt>Wind force</dt><br />
<dd>Range: -0.1 – 0.1 cm/sec <br/><br />
References: [7] (700cm/sec!! we did not rely on this figure)</dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
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[[File:VUPV_difu_tabla6.png|600px|center]]<br />
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<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
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</p><br />
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[[File:VUPV_difu_tabla7.png|600px|center]]<br />
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<p align="left"><strong>Conclusions</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/File:VUPV_difu_orito2.pngFile:VUPV difu orito2.png2014-10-18T03:18:39Z<p>Alejovigno: </p>
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<div></div>Alejovignohttp://2014.igem.org/File:VUPV_difu_orito1.pngFile:VUPV difu orito1.png2014-10-18T03:17:16Z<p>Alejovigno: </p>
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<div></div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T03:15:51Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
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<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
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<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
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</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
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<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:Upv_simu3.png|450px|center|Figure 3. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. Doing a model parameters swept, we simulated many possible scenarios, and then we came up with values of parameters corresponding to our desired goal: insects get confused. This will be useful to know the limitations of our system and to help to decide the final distribution of our plants in the crop field. The parameters we found are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of search: 0.02-1 µg/h <br/><br />
References: [4], [5]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4]. This emission rate above which male start to get confused could be the release rate from females.</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of search: 0.001-1 [Mass]/[Distance]^2</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd>Range of search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br/><br />
Range: 0-0.0009 <br/><br />
(The maximum level of moth sensitivity has to be less than the minimum level of release rate of females, since this parameter is obtained from the difference)</dd><br />
<dt>Wind force</dt><br />
<dd>Range: -0.1 – 0.1 cm/sec <br/><br />
References: [7] (700cm/sec!! we did not rely on this figure)</dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexy plants are switched-off and males only interact with females.</li><br />
<li>When sexy plants are switched-on and have the effect of trapping males.</li><br />
<li>When sexy plants are switched-on and males get confused as the level of pheromone concentration is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? This gives an idea of the minimum number of male-female encounters that we should expect in a fully random scenario, with no pheromones at play.</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
If Sexy Plant works, the first scenario should give higher number of encounters than the second and third ones.<br />
</p><br />
<br/><br />
<p align="left"><strong>Scenarios</strong></p><br/> <br />
<br/><br />
<br />
<p><br />
With all values fixed excepting the number of males and females, we started the simulations. Each test was simulated more than once, in order to consider the stochastic nature of the process. Again, we considered different sub-scenarios for each one of the cases mentioned above. In particular, we considered the cases of having male and female subpopulations of equal size, or one larger than the other one.<br />
</p><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 1</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> The results show that the number of encounters during the time sexy plants are switched-on is almost the same, but in most cases lower than when sexy plants are switched-off.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla1.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>The time at which the insects start to get confused and move randomly is shorter as the population increases. Even for high numbers, males get confused before sexy plants are switched-on. That is because there is such amount of females that they saturate the field. This rarely happens in nature, so when this occurs in our simulation we should think that we are out of real scenarios, and then we should modify the rest of parameter values. In these experiments we see that at a population equal to 12 we start be on this limit (insects gets confused when the sexy plants are going to be switched-on). </li><br />
<li>An aspect that should also be considered is the time of the insects getting confused among experiments, (when the number of females is the same). One could think that this “saturation” time would depend on the number of encounters before it happens. Since females wouldn’t be emitting pheromones after mating, males should get confused later if the previous number of meetings is larger. However, results are not decisive in this matter.</li><br />
<br />
</ul><br />
<br/><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Experiment 2</strong></p><br/> <br />
<br/><br />
<p><br />
What does it happen when the number of females is equal to the number of males? (F=M)<br />
</p><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1000}: Switch-ON</li><br />
<li>T_{2000}: End</li><br />
</ul><br />
<br/><br />
<p> Based on the results of experiment 1, we fixed 10 as the top number of females for the next tests. The number of females is conserved in each test.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla2.png|600px|center]]<br />
<html><br />
<br/><br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>It is observed that the number of encounters is higher if the number of males increases (this makes sense). </li><br />
<li>In all cases it can be deduced that while the number of males increase against the number of females, the time required for them to get confused is larger. This possibly has its origin in the number of encounters, which is higher according to the first point. When males mate females, they give up emitting pheromones during a certain period of time, so the contribution to the field saturation decreases.</li><br />
</ul><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla3.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li><br />
In contrast with the Experiment 1, it is observed that while the number of males increases, the number of encounters is considerably higher when sexy plants are switched-off than when they are switched-on. This is seen with more clarity when the number of males is larger. We believe that with more experiments, this fact can be easily tested.</li><br />
</ul><br />
<br />
<br/><br/><br />
<p align="left"><strong>Comparing Experiments 1 and 2</strong></p><br/> <br />
<br/><br />
<p><br />
Experiment 1: F=10 M=10<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla4.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we did not see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) are very similar, so the effect is not achieved satisfactorily.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito1.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Experiment 2: F=10 M=30<br />
</p><br />
<br />
</html><br />
[[File:VUPV_difu_tabla5.png|600px|center]]<br />
<html><br />
<br/><br />
<p><br />
In this experiment we do see the result we are looking for. We are interested in obtaining a high proportion in the third column when sexy plants are working. We see that the graphs counting the number of encounters (purple for the Switch-OFF, green for the Switch-ON) differ visibly, so the effect is achieved.<br />
</p><br />
<br/><br />
</html><br />
[[File:VUPV_difu_orito2.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br />
<br/><br />
<p align="left"><strong>Experiment 3</strong></p><br/> <br />
<br/><br />
<p><br />
<b>Females don’t emit pheromones. Thus, males and females move randomly. How much would our results differ from the ones with females emitting?</b><br />
</p><br />
<br/><br />
<p><br />
<We decided to set out the end time according to the moment in which the pheromone level in the field is entirely over the male saturation threshold (in this case 8). We take as reference the top population female number: 10. For the rest of tests the pheromone concentration in the field will be lower.</p><br />
<br/><br />
<br />
<ul style="position: relative; left: 4%; width: 90%;"><br />
<li>T_{0} : Start</li><br />
<li>T_{1700}: End</li><br />
</ul><br />
<br/><br />
<br />
<p><br />
In almost every cases, the number of encounters is higher when females emit pheromones. It means that in our model, males can follow females being guided by pheromone concentration gradients. Moreover, it is seen in the interface during simulations. Results for “pheromone emission”. Showed below are an average of an amount of experiments.<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla6.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<p><br />
Also see the contribution of the pheromone supply to the environment depending on the number of females (directly related) and the number of meetings (inversely related)<br />
For population 1 to 1 and this time ending given, no more than 2 encounters have been observed. In contrast with the random movement, in which not encounters have been showed in the range of experiments we have checked.<br />
<br />
</p><br />
<br/><br />
<br />
</html><br />
[[File:VUPV_difu_tabla7.png|600px|center]]<br />
<html><br />
<br/><br />
<br />
<br/><br />
<p align="left"><strong>Conclusions/strong></p><br/> <br />
<br/><br />
<br />
<p><br />
We have used a methodology for the results comparison in which experiments have been repeated several times. The interpretation of the performances has based on the values obtained. Nevertheless an exhaustive replay of the same realizations would give us more accurate values. <br />
</p><br/><br />
<p><br />
The experiments with the same number of males than females give results we haven’t expected. Maybe changing the model parameter values one would obtain a different kind of performance. <br />
</p><br />
<br/><p><br />
Other aspect that we have taken into account is that some of the encounters during the time males are following pheromone traces from females may be also due to random coincidence.<br />
</p><br />
<br/><p><br />
We have used a procedure useful to discard scenarios and contrast different realizations. With this, logic conclusions can be derived. Thus, they are a way of leading a potential user of this application to widen the search of parameters and improve our model. And that could be useful to know the limitations of our system and helpful to decide the final distribution of our synthetic plants in the field.<br />
</p><br />
<br/><br />
<br />
<br />
<br/><br />
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<br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T02:43:52Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
<html><br/><br />
<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
<br />
<br />
With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
<html><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
<br />
</ol><br />
</div><br />
<br />
<br />
<br />
</div><br />
<br />
<div id="tab3" class="tab"><br />
<br/><br />
<h3>The Idea</h3><br/><br />
<br />
<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
<br />
<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
<br />
<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
<br />
<br />
<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
<br />
<br />
<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
<br />
<br />
</html><br />
[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
<br />
<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
<br />
<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
<br />
<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
<br />
<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
<br />
<br />
</html><br />
[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
<br />
<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
</ol><br />
</div><br />
<br />
</div><br />
<br />
<div id="tab4" class="tab"><br />
<br/><br />
<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
<br />
<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
<br />
<p align="left"><strong>Setup</strong></p><br/><br />
<br />
<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
<br />
<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
<br />
<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
<br />
<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
<br />
</html><br />
[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
<br />
<p align="left"><strong>Runs</strong></p><br/><br />
<br />
<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
<br />
<br />
<br />
</html><br />
[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
<br />
<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
<br />
</html><br />
[[File:Upv_simu3.png|450px|center|Figure 3. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.]]<br />
<html><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
<br />
<p align="left"><strong>Parameters</strong></p><br/><br />
<br />
<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. Doing a model parameters swept, we simulated many possible scenarios, and then we came up with values of parameters corresponding to our desired goal: insects get confused. This will be useful to know the limitations of our system and to help to decide the final distribution of our plants in the crop field. The parameters we found are: </p> <br/><br />
<br />
<br />
<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of search: 0.02-1 µg/h <br/><br />
References: [4], [5]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4]. This emission rate above which male start to get confused could be the release rate from females.</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of search: 0.001-1 [Mass]/[Distance]^2</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd>Range of search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br/><br />
Range: 0-0.0009 <br/><br />
(The maximum level of moth sensitivity has to be less than the minimum level of release rate of females, since this parameter is obtained from the difference)</dd><br />
<dt>Wind force</dt><br />
<dd>Range: -0.1 – 0.1 cm/sec <br/><br />
References: [7] (700cm/sec!! we did not rely on this figure)</dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
</dl><br />
<br />
<br />
<br/><br />
<br/><br />
<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
<br/> <br />
<br />
<br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
</ol><br />
</div><br />
<br />
<br />
</div><br />
<br />
<br />
<div id="tab5" class="tab"><br />
<br/><br />
<br />
<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
</p><br />
<br />
<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexyplants are switched-off and males only interact with females.</li><br />
<li>When sexyplants are switched-on and have an effect of trapping males.</li><br />
<li>When sexyplants are swiched-on and males get confused when the concentration of pheromone level is higher than their saturation threshold.</li><br />
</ol><br />
<br />
<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? :</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
</ol><br />
<br />
<p><br />
What is important is that between the first and the third case, the number of meetings should be less in the latter than in the former. Then we are closer to our objective fulfillment.<br />
</p><br/><br />
</div><br />
<br />
</div><br />
</div><br />
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{{:Team:Valencia_UPV/footer_img}}</div>Alejovignohttp://2014.igem.org/Team:Valencia_UPV/Modeling/diffusionTeam:Valencia UPV/Modeling/diffusion2014-10-18T02:37:22Z<p>Alejovigno: </p>
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<h3 class="hook" align="left"><a href="#">Modeling</a> > <a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/diffusion">Pheromone Diffusion</a></h3></p></br><br />
<br />
<div align="center"><span class="coda"><roja>P</roja>heromone <roja>D</roja>iffusion <br/><br/> and <roja>M</roja>oths <roja>R</roja>esponse</span> </div><br />
</br></br><br />
<div class="tabs"><br />
<ul class="tab-links"><br />
<li class="active"><a href="#tab1">Introduction</a></li><br />
<li><a href="#tab2">Diffusion Equation</a></li><br />
<li><a href="#tab3">Moth Response</a></li><br />
<li><a href="#tab4">Simulation</a></li><br />
<li><a href="#tab5">Results</a></li><br />
</ul><br />
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<div class="tab-content"><br />
<div id="tab1" class="tab active"><br />
<p>Sexual communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" consists of <span class="black-bold">sexual pheromones</span>.</p><br/><br />
<br />
<div align="center"><img width="540px" src="https://static.igem.org/mediawiki/2014/9/9d/VUPVIntro_sexpheromone.png" alt="female_sex_pheromones" title="Female and Male Moths"></img></div><br/><br />
<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.8em; width: 700px;"><span class="black-bold">Figure 1</span>. Female moth releasing sex pheromones and male moth.</p></div><br/><br />
<br />
<br />
<br />
<p>Pheromones are molecules that easily diffuse in the air. During the diffusion process, the random movement of gas molecules transport the chemical away from its source [1]. Diffusion processes are complex ones, and modeling them analytically and with accuracy is difficult. Even more when the geometry is not simple. For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, the equation is solved using the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. </p><br/><br />
<br />
<p> Moths seem to respond to gradients of pheromone concentration to be attracted towards the source. Yet, there are other factors that lead moths to sexual pheromone sources, such as optomotor anemotaxis [2]. Moreover, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation [3]. </p><br/><br />
<br />
<p>Using a modeling environment called <a class="normal-link-page" href="https://ccl.northwestern.edu/netlogo/">Netlogo</a>, we simulated the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of <span class="red-bold">Sexy Plant</span>.</p><br/><br />
<br />
<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop, Phil. Trans. R. Soc. Lond. B 10 May 1984 vol. 306 no. 1125 19-48</li><br />
<li>W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues, Annual Review of Entomology<br />
Vol. 22: 377-405, 1977</li><br />
</ol><br />
<br />
<br />
</div><br />
<br />
<div id="tab2" class="tab"><br />
<p>Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.</p><br/><br />
<br />
<p>The motion of fluids can be described by the <span class="black-bold">Navier–Stokes equations</span>. But the typical nonlinearity of these equations when there may exist turbulences in the air flow, makes most problems difficult or impossible to solve. Thus, attending to the particles suspended in the fluid, a simpler effective option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior.<br />
That is, pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].</p><br/><br />
<br />
<p>There are two ways to introduce the notion of diffusion: either using a phenomenological approach starting with <span class="black-bold"> Fick's laws of diffusion</span> and their mathematical consequences, or a physical and atomistic one, by considering the <span class="black-bold"> random walk</span> of the diffusing particles [2].</p><br/><br />
<br />
<p>In our case, we decided to model our diffusion process using the <span class="black-bold">Fick's laws</span>. Thus, it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult. Even more when the geometry is not simple (e.g. consider the potential final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey the heat diffusion equation.</p><br/><br/><br />
<br />
<p align="left"><strong>Approximation</strong></p><br/><br />
<br />
</html> <br />
<br />
The diffusion equation is a partial differential equation that describes density dynamics<br />
in a material undergoing diffusion. It is also used to describe processes exhibiting<br />
diffusive-like behavior, like in our case. The equation is usually written as:<br />
<br />
$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$<br />
<br />
where $\phi(r, t)$ is the density of the diffusing material at location r and time t, and<br />
$D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and<br />
$\nabla$ represents the vector differential operator.<br />
<br />
If the diffusion coefficient does not depend on the density then the equation is linear and<br />
$D$ is constant. Thus, the equation reduces to the linear differential equation:<br />
$$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$<br />
<br />
also called the '''heat equation'''. Making use of this equation we can write the pheromones chemicals diffusion equation with no<br />
wind effect consideration as:<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$<br />
<br />
where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is<br />
the pheromone diffusion constant in the air.<br/><br />
<br />
If we consider the wind, we face a diffusion system with drift, and an advection term is<br />
added to the equation above.<br />
<br />
$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$<br />
<br />
where $\vec{v}$ is the average ''velocity''. Thus, $\vec{v}$<br />
would be the velocity of the air flow in or case.<br/><br />
<br />
For simplicity, we are not going to consider the third dimension. In $2D$ the equation<br />
would be:<br />
<br />
<br />
$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$<br />
<br />
<br />
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<div align="center"><img width="650px" src="https://static.igem.org/mediawiki/2014/1/11/VUPVDiffusion_purple.png" alt="modeling_equations_solving" title="Netlogo Screen"></img></div><br/><br/><br />
</html><br />
<br />
In order to determine a numeric solution for this partial differential equation, the so-called finite difference methods are used. <br />
With finite difference methods, partial differential equations are replaced by<br />
its approximations as finite differences, resulting in a system of algebraic equations. This is solved at each node<br />
$(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial<br />
distribution of the particles diffusing.<br/><br />
<br />
Although implicit methods are unconditionally stable, so time steps could be larger and<br />
make the calculus process faster, the tool we have used to solve our heat equation is the<br />
Euler explicit method, for it is the simplest option to approximate spatial derivatives.<br/><br/><br />
<br />
The equation gives the new value of the pheromone level in a given node in terms of initial values at that<br />
node and its immediate neighbors. Since all these values are known, the process is called<br />
explicit.<br />
<br />
$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$<br />
<br />
Now, applying this method for the first case (with no wind consideration) we followed the<br />
next steps:<br />
<br />
1. Split time $t$ into $n$ slices of equal length <i>dt</i>:<br />
$$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t<br />
\end{array} \right. $$<br />
<br />
2. Considering the backward difference for the Euler explicit method, the<br />
expression that gives the current pheromone level each time step is:<br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$<br />
<br />
3. And now considering the spatial dimension, central differences is applied to the Laplace operator $\Delta$, and backward differences are applied to the vector differential operator $\nabla$ (in 2D and assuming equal steps in x and y directions): <br />
<br />
$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$<br />
$$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$<br />
$$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{c_{i,j} – c_{i,j-1}}{h} $$<br />
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With respect to the boundary conditions, they are null since we are considering an open space. Attending to the implementation and simulation of this method, <i>dt</i> must be small enough to avoid instability.<br />
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<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Sol I. Rubinow, Mathematical Problems in the Biological Sciences, chap. 9, SIAM, 1973</li><br />
<li> J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2,1.1-1.10, 2005.</li><br />
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</ol><br />
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<div id="tab3" class="tab"><br />
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<h3>The Idea</h3><br/><br />
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<p>When one observes moths behavior, they apparently move with erratic flight paths. This is possibly to avoid predators. This random flight is modified by the presence of sex pheromones. Since these are pheromones released by females in order to attract an individual of the opposite sex, it makes sense that males respond to <span class="purple-bold">gradients of sex pheromone concentration</span>, being attracted towards the source. As soon as a flying male <span class="green-bold">randomly</span> enters into a conical pheromone-effective sphere of sex pheromone released by a virgin female, the male begins to seek the female following a zigzag way. The male approaches the female, and finally copulates with her [1].</p><br/><br/><br/><br />
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<p align="left"><strong>Approximation</strong></p><br/><br />
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<img width="150px" style="float:left; margin-right: 15px; margin-bottom: 15px;" src="https://static.igem.org/mediawiki/2014/1/17/VUPVPolillita_con_vectores_v1.png" alt="moth_array"></img><br />
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<p>In <span class="red-bold">Sexy Plant</span> we approximate the resulting moth movement as a vectorial combination of a <span class="purple-bold">gradient vector</span> and a <span class="green-bold">random vector</span>. The magnitude of the gradient vector depends on the change in the pheromone concentration level between points separated by a differential stretch in space. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The random vector is constrained in this ‘moth response’ model by a fixed angle upper bound, assuming that the turning movement is relatively continuous. For example, one can asume that the moth cannot turn 180 degrees from one time instant to the next.</p><br/><br />
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<p>Our synthetic plants are supposed to release enough sexual pheromone so as to be able to <span class="red-bold">saturate moth perception</span>. In this sense the resulting moth vector movement will depend ultimately on the pheromone concentration levels in the field and the moth ability to follow better or worse the gradient of sex pheromone concentration.</p><br/><br />
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<p>The three clases of male moth behavior we consider for the characterization of males moth behavior are described in Table 1.</p><br/><br />
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[[File:Table_behavior.png|600px|center|Male moths behaviour characterization.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Male moths behaviour characterization.</p></div><br />
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<p>This ensemble of behaviors can be translated into a sum of vectors in which the random vector has constant module and changing direction within a range, whereas the module of the gradient vector is a function of the gradient in the field.<br />
The question now is how do we include the saturation effect in the resulting moth shift vector. With this in mind, and focusing on the implementation process, our approach consists on the following:</p><br />
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<p>To model chemoattraction, the gradient vector will be always have fixed unit magnitude, and its direction is that of the greatest rate of increase of the pheromone concentration. </p><br/><br />
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<p>To model the random flight, instead of using a random direction vector with constant module, we consider a random turning angle starting from the gradient vector direction.</p><br/><br />
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<p>Thus, how do we include the saturation effect in the resulting moth shift vector? This is key to achieve sexual confusion. Our answer: the behaviour dependence on the moth saturation level --in turn related to the pheromone concentration in the field-- will be included in the random turning angle. </p><br/><br />
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[[File:Moth_vector.png|600px|center|Approximation of the male moths behaviour.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Table 1</span>. Approximation of the male moths behaviour.</p></div><br />
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<p>This random turning angle will not follow a uniform distribution, but a Poisson distribution in which the mean is zero (no angle detour from the gradient vector direction) and the standard-deviation will be inversely proportional to the intensity of the gradient of sex pheromone concentration in the field. This approach leads to ‘sexual confusion’ of the insect as the field homogeneity increases. This is because the direction of displacement of the moth will equal the gradient direction with certain probability which depends on how saturated it is.</p><br/><br />
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<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li> Yoshitoshi Hirooka and Masana Suwanai. Role of Insect Sex Pheromone in Mating Behavior I. Theoretical Consideration on Release and Diffusion of Sex Pheromone in the Air. J. Ethol, 4, 1986</li> <br />
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<div id="tab4" class="tab"><br />
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<p>Using a modeling environment called Netlogo, we simulate the approximate moth population behavior when the pheromone diffusion process take place.</p><br/><br />
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<p> The <a href="http://ccl.northwestern.edu/netlogo/">Netlogo</a> simulator can be found in its website at Northwestern University. To download the source file of our <span class="red-bold">Sexy plant</span> simulation in Netlogo click here: <br />
<a href="https://2014.igem.org/Team:Valencia_UPV/Modeling/sexyplants.nlogo" download>sexyplants.nlogo</a></p><br/><br />
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<p align="left"><strong>Setup</strong></p><br/><br />
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<ul style="list-style: disc; position: relative; left: 4%; width: 96%;"><br />
<li>We consider three <span class="black-bold">agents</span>: <span class="marron-bold">male</span> and <span class="fucsia-bold">female</span> moths, and <span class="red-bold">sexy plants</span>.</li><br />
<li>We have two kinds of sexual pheromone emission sources: <span class="fucsia-bold">female</span> moths and <span class="red-bold">sexyplants</span>. </li><br />
<li>Our scenario is an open crop field where <span class="red-bold">sexy plants</span> are intercropped, and moths fly following different patterns depending on its sex.</li><br />
</ul><br />
<p><span class="fucsia-bold">Females</span>, apart from emitting sexual pheromones, move following erratic random flight paths. After mating, females do not emit pheromones for a period of 2 hours.</p><br />
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<p><span class="marron-bold">Males</span> also move randomly while they are under its detection threshold. But when they detect a certain pheromone concentration, they start to follow the pheromone concentration gradients until its saturation threshold is reached. </p><br />
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<p> <span class="red-bold">Sexy plants</span> act as continuously- emitting sources, and their activity is regulated by a <span class="black-bold">Switch</span>.</p><br/><br />
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<p> The pheromone diffusion process, it is simulated in Netlogo by implementing the Euler explicit method. </p><br/><br />
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[[File:Upv_simu1.png|600px|center|Figure 1. NETLOGO Simulation environment.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 1</span>. NETLOGO Simulation environment.</p></div><br />
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<p align="left"><strong>Runs</strong></p><br/><br />
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<p>When <span class="red-bold">sexy plants</span> are switched-off, <span class="marron-bold">males</span> move randomly until they detect pheromone traces from <span class="fucsia-bold">females</span>. In that case they follow them. </p><br />
<p>When <span class="red-bold">sexy plants</span> are switched-on, the pheromone starts to diffuse from them, rising up the concentration levels in the field. At first, <span class="red-bold">sexy plants</span> have the effect of acting as pheromone traps on the <span class="marron-bold">male</span> moths.</p><br/><br />
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[[File:VUPV_Polillas.png|600px|center|Figure 2. On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 2</span>.On the left: sexy plants are switched-off and a male moth follows the pheromone trace from a female. On the right: sexy plants are switched on and a male moth go towards the static source as it happens with synthetic pheromone traps.</p></div><br />
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<p>As the concentration rises in the field, it becomes more homogeneous. Remember that the <span class="green-bold">random turning angle</span> of the insect follows a Poisson distribution, in which the standard-deviation is inversely proportional to the intensity of the <span class="purple-bold">gradient</span>. Thus, the probability of the insect to take a bigger detour from the faced gradient vector direction is higher. This means that it is less able to follow pheromone concentration gradients, so sexual confusion is induced.</p><br />
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[[File:Upv_simu3.png|450px|center|Figure 3. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.]]<br />
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<div align="center"><p style="text-align: justify; font-style: italic; font-size: 0.9em; width: 700px;"><span class="black-bold">Figure 3</span>. NETLOGO Simulation of the field: sexyplants, female moths, pheromone diffusion and male moths.</p></div><br />
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<p align="left"><strong>Parameters</strong></p><br/><br />
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<p>The parameters of this model are not as well-characterized as we expected at first. Finding the accurate values of these parameters is not a trivial task. In the literature it is difficult to find a number experimentally obtained. So we decided to take an inverse engineering approach. Doing a model parameters swept, we simulated many possible scenarios, and then we came up with values of parameters corresponding to our desired goal: insects get confused. This will be useful to know the limitations of our system and to help to decide the final distribution of our plants in the crop field. The parameters we found are: </p> <br/><br />
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<dl><br />
<dt>Diffusion coefficient</dt><br />
<dd>Range of search: 0.01-0.2 cm^2/s <br/><br />
References: [1], [2], [3], [5]</dd><br />
<dt>Release rate (female)</dt><br />
<dd>Range of search: 0.02-1 µg/h <br/><br />
References: [4], [5]</dd><br />
<dt>Release rate (Sexy Plant)</dt><br />
<dd>The range of search that we have considered is a little wider than the one for the release rate of females. <br/><br />
References: It generally has been found that pheromone dispensers releasing the chemicals above a certain emission rate will catch fewer males. The optimum release rate or dispenser load for trap catch varies greatly among species [4]. This emission rate above which male start to get confused could be the release rate from females.</dd><br />
<dt>Detection threshold</dt><br />
<dd>Range of search: 0.001-1 [Mass]/[Distance]^2</dd><br/><br />
<dt>Saturation threshold </dt><br />
<dd>Range of search: 1-5[Mass]/[ Distance]^2</dd><br/><br />
<dt>Moth sensitivity</dt><br />
<dd>This is a parameter referred to the capability of the insect to detect changes in pheromone concentration in the patch it is located and the neighbor patch. When the field becomes more homogeneous, an insect with higher sensitivity will be more able to follow the gradients.<br/><br />
Range: 0-0.0009 <br/><br />
(The maximum level of moth sensitivity has to be less than the minimum level of release rate of females, since this parameter is obtained from the difference)</dd><br />
<dt>Wind force</dt><br />
<dd>Range: -0.1 – 0.1 cm/sec <br/><br />
References: [7] (700cm/sec!! we did not rely on this figure)</dd><br />
<dt>Population</dt><br />
<dd>The number of males and females can be selected by the observer.</dd><br />
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<p align="left"><strong>Patches</strong></p><br/><br />
<p>One can modify the number of patches that conform the field so as to analyze its own case. In our case we used a field of 50x50 patches. </p><br />
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<p align="left"><strong>References</strong></p><br/><br />
<div style="position: relative; left: 3%; width: 96%;"><br />
<ol><br />
<li>Wilson et al.1969, Hirooka and Suwanai, 1976.</li><br />
<li>Monchich abd Mauson, 1961, Lugs, 1968.</li><br />
<li>G. A. Lugg. Diffusion Coefficients of Some Organic and Other Vapors in Air.</li><br />
<li>W. L. Roelofs and R. T. Carde. Responses of Lepidoptera to Synthetic Sex Pheromone Chemicals and their Analogues, Page 386. </li><br />
<li>R.W. Mankiny, K.W. Vick, M.S. Mayer, J.A. Coeffelt and P.S. Callahan (1980) Models For Dispersal Of Vapors in Open and Confined Spaces: Applications to Sex Pheromone Trapping in a Warehouse, Page 932, 940.</li><br />
<li> Tal Hadad, Ally Harari, Alex Liberzon, Roi Gurka (2013) On the correlation of moth flight to characteristics of a turbulent plume. </li><br />
<li> Average Weather For Valencia, Manises, Costa del Azahar, Spain. </li> <br />
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<div id="tab5" class="tab"><br />
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<p><br />
The aim consists of reducing the possibility of meeting among moths of opposite sex. Thus, we will analyze the number of meetings in the three following cases:<br />
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<ol style="position: relative; left: 4%; width: 90%;"><br />
<li>When sexyplants are switched-off and males only interact with females.</li><br />
<li>When sexyplants are switched-on and have an effect of trapping males.</li><br />
<li>When sexyplants are swiched-on and males get confused when the concentration of pheromone level is higher than their saturation threshold.</li><br />
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<p><br />
It is also interesting to analyze a fourth case, what does it happen if females wouldn’t emit pheromones and males just move randomly through the field? :</p><br />
<ol start="4" style="position: relative; left: 4%; width: 90%;"><br />
<li>Males and females move randomly. How much would our results differ from the rest of cases? </li><br />
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What is important is that between the first and the third case, the number of meetings should be less in the latter than in the former. Then we are closer to our objective fulfillment.<br />
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