Team:Valencia Biocampus/Modeling

From 2014.igem.org

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         <li><a href="#OrthogonalityTab" data-toggle="tab">Orthogonality</a></li>
         <li><a href="#OrthogonalityTab" data-toggle="tab">Orthogonality</a></li>
         <li><a href="#OpenLicenseTab" data-toggle="tab">Open License</a></li>
         <li><a href="#OpenLicenseTab" data-toggle="tab">Open License</a></li>
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         <li><a href="#OtherTab" data-toggle="tab">Other</a></li>
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         <li><a href="#OtherTab" data-toggle="tab">Other projects</a></li>
         <li><a href="#ReferencesTab" data-toggle="tab">References</a></li>
         <li><a href="#ReferencesTab" data-toggle="tab">References</a></li>
     </ul>
     </ul>
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             <h2>Presentation</h2>
             <h2>Presentation</h2>
<p>
<p>
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     We derive here some basic formulae needed to model the dependence of bio-bricks expression with temperature, pH or UV radiation. Our observable is the
+
     In this section, we present our models to describe the four legs of the ST$^2$OOL. We have closely followed all the experimental activities, so our models do not pretend to
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    relative change of the intensity of fluorescence per cell due to the variation of a given external factor.
+
completely characterize a single dynamical process of a cell, but instead we intend to capture the dependences of our systems on the variables explored
 +
by our experiments.
</p>
</p>
<p>
<p>
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    Many processes are involved in protein synthesis. Our goal is to identify the main processes that capture the response to changes and study the dynamics of
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<a href="Modeling#StabilityTab"><strong>Stability</strong></a> We have modeled the variation of the activity of cells transformed with Biobricks, when subjected to different temperatures or pH, by considering changes in the Biobrick transcription
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    the model. With this plan in mind, we simplify our problem by assuming that a) expression is linearly proportional to the number of cells, and b) mean
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and translation rates.  
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    number plasmids per cell remains constant.
+
</p>
 +
<p>
 +
<a href="Modeling#StandardizationTab"><strong>Standardization</strong></a> We have modeled the expression in different strains by comparing the transcription rates. We have built an app that computes the standardization-stability index and generates the promoter sequence for a given expression rate.
 +
</p>
 +
<p>
 +
<a href="Modeling#OrthogonalityTab"><strong>Orthogonality</strong></a>
 +
We have studied deterministic and stochastic models of the plasmid growth in order to learn about the unspecific interaction between two plasmids in a cell.
 +
</p>
 +
<p>
 +
<b><a href="Modeling#OpenLicenseTab"><strong>Open License</strong></a></b> We have modeled the susceptibility to patent a human creation and define the factors that influence it .
</p>
</p>
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</div>
</div>
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           <h2>Stability</h2>
           <h2>Stability</h2>
<p>
<p>
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     Living organisms have genetic and biochemical tools to respond to stress. These tools normally up-regulate some stress-specific genes, and, directly or
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     Living organisms have genetic and biochemical tools that allow them to respond to stress. These tools normally up-regulate some stress-specific genes, and, directly or
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     indirectly, down-regulate the rest of normal genes (most of them housekeeping genes). On the other hand, every E. coli component works worse under
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     indirectly, down-regulate the rest of normal genes (most of them housekeeping genes). On the other hand, every <em>Escherichia coli</em> component works worse under non-optimal conditions
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     conditions different from their optimum (for example, nutrients uptake or ATP production), so that, every E. coli function will be reduced.
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     (for example, nutrients uptake or ATP production), so that, every <em>E. coli</em> function will be reduced.
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     We model the relative expression of biobricks by variations of temperature, pH and salt concentration.  
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     We model the relative expression of biobricks when subjected to variations of temperature, pH and salt concentration.  
</p>
</p>
</html>
</html>
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<p>
<p>
     The expression level of our heterologous gene can be calculated by equations 1 and 2. Our main goal is to find which of the terms will change most
     The expression level of our heterologous gene can be calculated by equations 1 and 2. Our main goal is to find which of the terms will change most
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     significantly with the different stress factors. We will be interested in the activity relative to a reference one, which we take as the activity of E.
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     significantly with the different stress factors. We are interested in the activity in comparison to a reference one, which we take as the activity of <em>E. coli</em> bacteria grown on the LB medium at <u>38.7 <sup>o</sup>C, pH 7, 1% of NaCl, no UV irradiation and 1 atm of pressure.</u>
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    coli bacteria grown on the LB medium at <u>38.7 <sup>o</sup>C, pH 7 and 1% of NaCl.</u>
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</p>
</p>
<p>
<p>
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</p>
</p>
</html>
</html>
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=== Transcription rate ===
=== Transcription rate ===
<html>
<html>
<p>
<p>
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     Among several factors, we first discuss the transcription rate. In the range of temperatures (or pH) that we have studied, the polymerase is
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     Among several factors, we will first discuss the transcription rate. In the range of temperatures (or pH) that we have studied, the polymerase (RNAP) is
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     sufficiently stable to assume that the elongation rate is not affected and the interaction energies do not change significantly (for example, the promoter
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     sufficiently stable to assume that the elongation rate is not affected and the interaction energies RNAP-promoter do not change significantly (for example, the promoter
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     specific interaction do only change 16% from 33 <sup>o</sup>C to 44 <sup>o</sup>C). We did not find any significant difference between the nucleotide
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     specific interaction only changes 16%, from 33 <sup>o</sup>C to 44 <sup>o</sup>C). We did not find any significant difference in the nucleotide
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     compositions of E. coli under stress [2]. Therefore, main variations in the transcription rate are due to the probability of RNAP-promoter bounds  
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     compositions of <em>E. coli</em> under stress [2]. Therefore, main variations in the transcription rate are due to the probability of RNAP-promoter bounds  
$$
$$
\begin{equation}
\begin{equation}
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\end{equation}
\end{equation}
$$
$$
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     On the other hand, the down-regulation effect affects the expression level of our Biobrick, because it works as a housekeeping gene (with a normal -35 and
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     On the other hand, down-regulation affects the expression level of our Biobrick, because it works as a housekeeping gene (with a normal constitutive promoter for the RNAP). For example, <em>E. coli</em> has a common σ factor for housekeeping genes (σ70), and, another one (σ32) for temperature
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    -10 promoter for the RNA polymerase). For example, E. coli has a common σ factor for housekeeping genes (σ70), and, another one (σ32) for temperature
+
     stress. In the case of temperature stress, the bacteria will synthetize σ32, which will bind RNA polymerase (RNAP) cores that won’t be free for σ70, and so
     stress. In the case of temperature stress, the bacteria will synthetize σ32, which will bind RNA polymerase (RNAP) cores that won’t be free for σ70, and so
     they will not be able to synthetize mRNA of the normal housekeeping genes.
     they will not be able to synthetize mRNA of the normal housekeeping genes.
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<p>
<p>
The RNAP is bound to a promoter (unspecific DNA) with $\epsilon_{S}$ ($\epsilon_{NS}$) free energy. There are  
The RNAP is bound to a promoter (unspecific DNA) with $\epsilon_{S}$ ($\epsilon_{NS}$) free energy. There are  
-
$N_{NS}$ nonspecific sites on the E. coli genome (sequences of 41bp) and R free E$\sigma$70, with $R \ll N_{NS}$ .
+
$N_{NS}$ nonspecific sites on the <em>E. coli</em> genome (sequences of 41bp) and R free E$\sigma$70, with $R \ll N_{NS}$ .
The combinatory number of unbound states, with energy R$\epsilon_{NS}$, is  
The combinatory number of unbound states, with energy R$\epsilon_{NS}$, is  
$$
$$
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</p>
</p>
</html>
</html>
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=== Synthesis rate ===
=== Synthesis rate ===
<html>
<html>
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Modeling synthesis rate follows the lines discussed in transcription rate, but the working regime is quite different.
Modeling synthesis rate follows the lines discussed in transcription rate, but the working regime is quite different.
The energies of the mRNA-ribosome sequences interactions are much larger than in the RNAP-promoter ones.  
The energies of the mRNA-ribosome sequences interactions are much larger than in the RNAP-promoter ones.  
-
The average difference of free energies (computed with the RBS calculator for random sequences of E. coli genes) is  
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The average difference of free energies (computed with the RBS calculator for random sequences of <em>E. coli</em> genes) is  
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$\Delta\epsilon = -20.79 kT$, which leads to minor variations of the ribosomal protein with the temperature [5].
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$\Delta\epsilon = -20.79 kT$, which means that, in the equation:
 +
$$
 +
\begin{equation}
 +
\frac{r_T}{r^0_T}  = \frac{Prob_B}{Prob^0_B}  = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}} ,
 +
\end{equation}
 +
$$
 +
The term $\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}>>>\frac {R^0}{R}$. With this, and knowing that there is no effect on the ribosomal protein composition with temperature changes [5], we conclude that the most signifficant factor should be the elongation rate.
</p>
</p>
Therefore,  
Therefore,  
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$$
$$
where $r_E$ ($r_I$) is the elongation (initiation) rate and the superscript denotes our reference case.
where $r_E$ ($r_I$) is the elongation (initiation) rate and the superscript denotes our reference case.
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In E. coli, most of the ribosomes are translating (4 active every one free) [6], thus $\frac{r^0_E}{r^0_I} \simeq\frac{1}{4}$ and  
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In <em>E. coli</em>, most of the ribosomes are translating (4 active every one free) [6], thus $\frac{r^0_E}{r^0_I} \simeq\frac{1}{4}$ and  
$$
$$
\begin{equation}
\begin{equation}
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</p>
</p>
<p>
<p>
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We discuss next the dependence of elongation rates with temperature. We use the relation of amino acid concentration with the
+
We discuss next the dependence of elongation rates with temperature. We use the relation of amino acid concentration with
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temperature [7]. Using data from [8], the elongation rate does not depend on the ternary complex concentration. The fastest steps are:  
+
temperature [7]. Using data from [8], we know that small variations on the ternary complex does not affects the elongation rate. The slowests steps are:  
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1) arrival of the ternary complex (0.01 s), 2) dipeptide bond formation (0.03s) and 3) translocation (0.022s).  
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1) GTP hydrolysis on ternary complex (0.01 s), 2) dipeptide bond formation (0.03s) and 3) translocation (0.022s).  
$$  
$$  
\begin{equation}
\begin{equation}
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\end{equation}
\end{equation}
$$
$$
-
where $E^i_a$ is (32, 75.2, 29260) kJ/mol and $A^{-1}_i$ is (2.5 $\cdot 10^7$, 1.6 $\cdot 10^{14}$, 3.9 $\cdot 10^6$).   
+
where $E^i_a$ is (32, 75.2, 29.260) kJ/mol and $A^{-1}_i$ is (2.5 $\cdot 10^7$, 1.6 $\cdot 10^{14}$, 3.9 $\cdot 10^6$).   
</p>
</p>
</html>
</html>
 +
=== Summary ===
=== Summary ===
<html>
<html>
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</html>
</html>
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=== pH & SALT CONCENTRATION ===
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=== 2) pH ===
<html>
<html>
<p>
<p>
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In contrast with temperature variations, E. coli internal pH and salt concentration hardly vary. Therefore, we focus on the intrinsic E. coli element that may be affected by these stresses, and influence Biobrick expression. The main factor is the number of free RNAP, due to the synthesis of stress response genes.  
+
Contrary to the temperature situations, internal pH hardly varies in <em>E. coli</em> when the external pH does. Therefore, we focus on the intrinsic <em>E. coli</em> element that may be affected by this type of stress, subsequently influencing Biobrick expression. The main factor is the number of free RNAP, which will be reduced due to the synthesis of stress response genes. We estimate the ratio of free RNAP under different conditions by the ratio of their growth rates
-
We estimate the ratio of free RNAP under different conditions by the ratio of their growth rates  
+
$$
$$
\begin{equation}
\begin{equation}
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</html>
</html>
 +
=== Comparison with wet-lab data ===
=== Comparison with wet-lab data ===
<html>
<html>
<p>
<p>
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Our model captures the measured shape of the fluorescence per cell as a function of temperature. We have found that transcription rate goes up with temperature  
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<b><a href="Results#StabilitySection">Our model captures the shape of the observed temperature profile and predicts that additional variations at lower temperatures are independent of the chosen Biobrick</a></b>. We have found that transcription rate decreases with temperature as a consequence of the reduction of free RNAP; and synthesis rate grows above the optimal temperature. Their combination leads to the observed variation of fluorescence with temperature. Unfortunately, this is not the whole story. Our model fails to fit the amount of variation, which is larger experimentally than the predicted one. By comparing our theoretical predictions and experimental results of XL1 Blue strain transformed with Bb1 or Bb3, we find that the ratio between theoretical prediction and experimental results is the same at under-optimal temperatures (in fact, accidentally too close given the error bars). This fact suggests that the temperature dependence not included in our model should come from cell mechanisms; independent from Biobrick parameters (promoter strength, ...). On the contrary, the comparison of theoretical predictions and experimental results at higher temperatures show that our model might require more dynamics which depends on Biobrick parameters.
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by the increase of free $\sigma$ factors and synthesis rate goes down above the optimal temperature. Unfortunately, this is not all the story. Our model dramatically
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</p>
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fails to fit the amount of variation, which is much larger experimentally than the predicted one. Let us compare theoretical predictions and experimental results.
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<p>
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In both studies, strain XL1 Blue transformed with Biobricks Bb1 or Bb3, we find that the ratio between theoretical prediction and experimental result is the same at  
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<b><a href="Results#StabilitySection">Our model captures the shape of the observed pH profile</a></b>. We have found that the lowest fluorescence happens at the pH with lowest growth, while it is mildly dependent on the pH from 6 to 8.
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temperatures lower than the optimal (in fact, accidentally too close given the error bars). This fact suggests that the extra temperature dependence not included in  
+
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our model should come from cell mechanisms independent on parameters of the Biobricks (promoter strength, ...). Quite on the contrary, the comparison of  
+
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theoretical predictions and experimental results at larger temperatures show that our model has not yet capture the dependence of the magnitude of the decrease in
+
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activity with the Biobrick parameters.  
+
</p>
</p>
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         <div class="tab-pane fade" id="StandardizationTab">
         <div class="tab-pane fade" id="StandardizationTab">
             <h2>Standardization</h2>
             <h2>Standardization</h2>
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<div class="col-sm-9 center-block">
 
-
 
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<p>
<p>
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We discuss now Biobrick expression in different strains of E. coli. Changes on the number of free RNAP affect the mRNA synthesis rate depending on the energy of the promoter. The relative transcription rate in different strains can be written as
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We will now discuss Biobrick expression in different <em>E. coli</em> strains. The effect of the changes in the number of free RNAP on mRNA synthesis rate depend on the energy of the promoter. The relative transcription rate in different strains can be written as
$$
$$
\begin{equation}
\begin{equation}
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\end{equation}
\end{equation}
$$
$$
-
where we can see that the dominance of the second term over the first one, controlled by the promoter strength, would lead to similar rate independently on the particular strain chosen. Comparing this formula with the relative transcription rate in stability, we find that promoters whose  binding energy makes them more standard also implies more stable promoters. Using the energy matrix derived in [3], promoter sequences with large promoter strengths lead to synthesis rates that go down from a maximum to almost zero with increasing strength. We use these results to create an application that  is able to compute promoter sequences for a given relative (to the maximum) expression rate and associated standard and stable rate. The physical parameters behind are  the promoter strength (using the data of Anderson’ promoters) and the binding energy and the tool create a sequence of a constitutive promoter with the appropriated trade-off between strength and energy, therefore, with the appropriated trade-off between stability & standardization and rate of expression. We define the factor of stability & standardization as $$ F_{st} = (F - 0.012)/0.51 $$ with $$F = \frac{R'}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}, $$ where $F_{st}$ has been scaled to run from 1 to 0. We have used a fitting formula for the rate of expression relative to the maximum rate [3] as a function of the binding energy $\Delta\epsilon$,  $$ r = m - n \frac{\Delta\epsilon}{kT}. $$  With this approach, we can infer the stability and standardization level and the promoter sequence for a given expression rate. We show here our app in javascript:
+
where we can see that the dominance of the second term over the first one, controlled by the promoter strength, would lead to similar rate independently on the particular strain chosen. Comparing this formula with the relative transcription rate in stability, we find that promoters whose  binding energy makes them more standard also implies more stable promoters. Strong promoter sequences experience a decrease in synthesis rates when subjected to an increase of promoter energy. We use these results and the energy matrix derived in [3] to create an application that  is able to compute promoter sequences for a given relative (to the maximum) expression rate and associated standard and stable rate. The physical parameters behind are  the promoter strength (using the data of Anderson’ promoters) and the binding energy and the tool create a sequence of a constitutive promoter with the appropriated trade-off between strength and energy, therefore, with the appropriated trade-off between stability & standardization (ST$^2$)and rate of expression. We define the ST$^2$ index as $$ F_{st2} = (F - 0.012)/0.51 $$ with $$F = \frac{R'}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}, $$ where $F_{sty}$ has been scaled to run from 1 to 0. We have used a fitting formula for the rate of expression relative to the maximum rate [3] as a function of the binding energy $\Delta\epsilon$,  $$ r_T = 2.29 + 0.27 \frac{\Delta\epsilon}{kT}. $$  With this approach, we can infer the ST$^2$ level and the promoter sequence for a given expression rate. We show below our app in javascript.
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<p>Move the cursor to a given expression rate (fraction of the maximum shown in the red box) and generate a chain</p>
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<div class="col-sm-9 center-block" style="padding:20px;">
 +
<p><b>Generate a promoter sequence and compute the ST$^2$ (Stability & Standardization) index for a given expression rate</b></p>
 +
<p>Move the cursor to a given expression rate (as a fraction of its maximum) and press Generate a chain</p>
<p style="text-align:center">
<p style="text-align:center">
       <input id="sliderApp" type="text" data-slider-min="0" data-slider-max="100" data-slider-step="1" data-slider-value="25" />
       <input id="sliderApp" type="text" data-slider-min="0" data-slider-max="100" data-slider-step="1" data-slider-value="25" />
       <span class="label label-success" id="percent1">25%</span>
       <span class="label label-success" id="percent1">25%</span>
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      <span class="label label-danger" id="percent2">75%</span>
 
       <input type="hidden" name="textbox1" id="textbox1" value="25"/>  
       <input type="hidden" name="textbox1" id="textbox1" value="25"/>  
<br/><br/>
<br/><br/>
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       <input type="submit" name="button" id="button1" class="btn btn-default" onclick="CE()" value="Generate chain" />
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       <button type="button" name="button" id="button1" data-loading-text="Generating..." class="btn btn-default">Generate a chain</button>
</p>
</p>
<br/>
<br/>
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<p>Promoter sequence (from -41 to -1 nt) is:</p> <p id="output"></p>
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<p class="text-centered">Promoter sequence (from -41 to -1 nt) is:</p>
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<script type="text/javascript" src="https://2014.igem.org/wiki/index.php?title=Template:Team:Valencia_Biocampus/Templates/js/ModelingApp&amp;action=raw&amp;ctype=text/javascript"></script>
+
                <p id="output"></p>
 +
                <p class="text-centered">ST$^2$ index:
 +
                    <span class="label label-danger" id="percent2"></span>
 +
                </p>
 +
<script type="text/javascript" src="https://2014.igem.org/wiki/index.php?title=Template:Team:Valencia_Biocampus/Templates/js/NewModelingApp&amp;action=raw&amp;ctype=text/javascript"></script>
<script type="text/javascript">
<script type="text/javascript">
-
// With JQuery
 
$("#sliderApp").slider({
$("#sliderApp").slider({
formatter: function(value) {
formatter: function(value) {
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     $("#textbox1").val(slideEvt.value);     
     $("#textbox1").val(slideEvt.value);     
     $("#percent1").text(slideEvt.value+'%');
     $("#percent1").text(slideEvt.value+'%');
-
    $("#percent2").text(100-slideEvt.value+'%');
 
});
});
 +
  $('#button1').click(function () {
 +
    $("#input").empty();
 +
    $(this).prop("disabled",true);
 +
    $(this).text("GENERATING ...");
 +
 +
    CE();
 +
  });
</script>
</script>
</div>
</div>
</html>
</html>
 +
=== Comparison with wet-lab data ===
=== Comparison with wet-lab data ===
<html>
<html>
<p>
<p>
-
Our wet-lab results show that de Biobrick 5 is the more standard one. Our model predicts that systems with less modules or factors will be the more stable and standard. Biobrick 5 has a tetracycline depending promoter, whose limitating step is the binding of tetracycline to TetR. In this scenario, the intracellular concentration of aTc and the binding to the TetR repressor do not depend on the strain, so there is only one factor in which the strain could have an important effect: the TetR intracellular concentration, which has a constitutive expression. Analyzing the other Biobricks, we found that the strains with less expression in the constitutive promoter are the ones with higher expression when we use the Ptet promoter (but JM109 and BB1), which agree with our prediction.
+
<b><a href="Results#StandardizationSection">Our wet-lab results show that Biobrick 5 is the most standard one.</a></b>Our model predicts that systems with less modules or factors will be more stable and standard. Biobrick 5 has a tetracycline depending promoter, whose limitating step is the binding of tetracycline to TetR. In this scenario, the intracellular concentration of aTc and the binding to the TetR repressor do not depend on the strain, so there is only one factor in which the strain could have an important effect: the TetR intracellular concentration, which has a constitutive expression. Analyzing the other Biobricks, we found that the strains with less expression of the constitutive promoter are those ones with higher expression when we use the Ptet promoter (but JM109 and BB1), which agree with our prediction.
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<p>
<p>
 +
Plasmid dynamics have been studied exhaustively over these years using mathematical models: from how plasmid growth is regulated to plasmid partitioning after cell division. However, little is known about how two identical plasmids interact with each other. This part of our modelling project aims to shed light upon this issue using previous models of ColE1 plasmids and the properties of two different plasmids with same origin of replication when these are introduced in a cell. We employed both deterministic and stochastic models to try to answer whether two plasmids behave independently, and developed some approximations to highlight important features of the system.
 +
<h3>
<h3>
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Why modelling plasmid dynamics?
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Deterministic model
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</h3><br><br>
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</h3>
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Synthetic biology aims to engineer a living cell to perform well-defined tasks. In order to achieve these aims, biologists introduce genes into cells using self-replicating DNA carriers called plasmids. Therefore, the levels of these molecules must be tightly regulated for our system to behave correctly in a predictable fashion.
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-
Plasmid dynamics have been studied exhaustively over these years using mathematical models: from how plasmid growth is regulated to plasmid partitioning after cell division. However, little is known about how two identical plasmids interact each other.
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<br>
<br>
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In this part of our modelling project aims to shed light upon this issue using previous models of ColE1 plasmids and the properties when two different plasmids with same origin of replication are introduced in a cell: Do they behave independently, that is, in an orthogonal way? To address this question, we employed both deterministic and stochastic models to create a theoretical framework on this field. Also, we developed some approximations to discover some important features of the system.
+
<p>
-
<br><br><br>
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This model is taken from [16]. Here, plasmid levels are regulated both by an RNA molecule that inhibits plasmid replication and by the dilution of plasmids as a result of cell growth and division:
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<h3>
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-
Deterministic model
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-
</h3><br><br>
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-
This model is taken from (Paulsson, 2001). Here, the plasmid levels are regulated both by a RNA molecule that inhibits plasmid replication and the dilution of plasmids as a result of cell growth and division:<br><br>
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-
$$ \frac{d P}{dt} = r \cdot P \cdot R(s) - \frac{\log 2}{\tau} \cdot P $$
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$$ \frac{d P}{dt} = r \cdot P \cdot R(s) - \frac{\log 2}{\tau} \cdot P, $$
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$$ \frac{d s}{dt} = \alpha \cdot P - \beta \cdot s $$
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$$ \frac{d s}{dt} = \alpha \cdot P - \beta \cdot s, $$
-
where:
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where $P$ is the number of plasmids, $s$ is the number of RNA inhibitors, $r$ is the plasmid growth rate, $\tau$ is the cell growth rate, $\alpha$ is the RNA inhibitor synthesis rate
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<br>$P$ is the number of plasmids
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, $\beta$ is the RNA inhibitor decay rate and $R(s)$ is the repression effect of RNA inhibitor with
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<br>$s$ is the number of RNA inhibitors
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$$ R(s) = \frac{1}{(1+\frac{s}{s_0})^n},$$
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<br>$r$ is the plasmid growth rate
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where $s_0$ is the value of RNA inhibitor necessary to repress half of the replication events and $n$ is the cooperativity. Plasmid levels are regulated both by a negative feedback and by cell growth. If we consider several plasmids, we get<br><br>
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<br>$\tau$ is the cell growth rate
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-
<br>$\alpha$ is the RNA inhibitor synthesis rate
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<br>$\beta$ is the RNA inhibitor decay rate
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<br>$R(s)$ is the repression effect of RNA inhibitor:
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-
$$ R(s) = \frac{1}{(1+\frac{s}{s_0})^n} $$
+
$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(s) - \frac{\log 2}{\tau} \cdot P_i,$$
-
where $s_0$ is the value of RNA inhibitor to repress half of the replication events and $n$ is the cooperativity
+
$$ \frac{d s}{dt} = \alpha \cdot \sum \limits_{i=1}^n{P_i} - \beta \cdot s,$$
 +
with a steady state solution<br><br>
-
<br><br><br>
+
$$ s_{ss} = R^{-1}(\frac{\log 2}{\tau \cdot r_i}),$$
-
Plasmids levels are regulated both by a negative feedback and cell growth. In order to take into account the presence of several plasmids, it has been modified to obtain this set of equations:<br><br>
+
-
$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(s) - \frac{\log 2}{\tau} \cdot P_i $$
+
$$ \sum \limits_{i=1}^n{P_{i,ss}} = (\frac{ \beta \cdot s_{ss}}{\alpha}).$$
-
$$ \frac{d s}{dt} = \alpha \cdot \sum \limits_{i=1}^n{P_i} - \beta \cdot s $$
+
<br>In a simplified way, with two identical plasmids, we get:
 +
$$ P_{1,ss} + P_{2,ss} = K.$$
-
<br><br>
 
-
The steady state solution is then:<br><br>
 
-
$$ s_{ss} = R^{-1}(\frac{\log 2}{\tau \cdot r_i})$$
+
It is noticeable that plasmid levels are not independent. Small perturbations can increase the relative level of one plasmid, favoring its growth. As they are repressed as a function of the total number of plasmids but they grow depending on their number, the system is prone to introduce asymmetries
-
$$ \sum \limits_{i=1}^n{P_{i,ss}} = (\frac{ \beta \cdot s_{ss}}{\alpha})$$
+
$$ \frac{d P_1}{d P_2} = \frac{r_1 \cdot P_1}{r_2 \cdot P_2} .$$
-
<br>In a simplified way, if we had two identical plasmids:
 
-
$$ P_{1,ss} + P_{2,ss} = K $$
 
-
<br><br>
+
If growth rates are constant, the slope is determined by the initial conditions
 +
$$ m = \frac{P_{01}}{P_{02}} .$$
-
It is noticeable that plasmid levels are not independent and has a determined value: they depend on each other. However, the value for a system is not known directly using this formula. A hypothesis to explain this solution is that small perturbations can increase the relative level of one plasmid, favouring its growth. As they are repressed as a function of the total number of plasmids but they growth depending on their number, the system is prone to enhance one over another.
+
<h4>Quasi-steady state approach</h4>
 +
To simplify our equations, we assume inhibitor dynamics are faster than plasmid growth, so RNA levels are a direct representation of plasmid levels
-
$$ \frac{d P_1}{d P_2} = \frac{r_1 \cdot P_1}{r_2 \cdot P_2} $$
+
$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(P_i) - \frac{\log 2}{\tau} \cdot P_i .$$
-
<br>
+
<h4>Dependence on initial conditions:</h4>
-
If both growth rate are constant, both plasmids are going to growth only depending of its level, then, the slope could be calculated using the initial conditions:
+
To test our hypothesis, we integrate our differential equations using numerical methods with several initial conditions. As expected, small differences in initial proportions are amplified by the dynamics.
-
<br>
+
-
$$ m = \frac{P_{01}}{P_{02}} $$
+
-
<br>
+
-
<br><br>
+
<div class="thumbnail col-sm-11 center-block">
-
<h4>Quasi-steady state approach</h4><br>
+
     <a href="https://static.igem.org/mediawiki/2014/d/d6/VB2014_Det_FinalDistribution.jpg" rel="lightbox">
-
To simplify this set of equations, inhibitor dynamics are thought to be faster than plasmid growth, so RNA levels are a direct representation of plasmid levels.
+
-
<br><br>
+
-
 
+
-
$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(P_i) - \frac{\log 2}{\tau} \cdot P_i $$
+
-
 
+
-
<br><br>
+
-
 
+
-
<h4>Dependence on initial conditions:</h4><br>
+
-
To test the previous hypothesis, we integrated the set of two equations using numerical methods as we changed the initial number of each plasmid. As expected, small differences in proportions were amplified.
+
-
<br><br>
+
-
<div style="text-align:center;">
+
-
     <a href="https://static.igem.org/mediawiki/2014/d/d6/VB2014_Det_FinalDistribution.jpg">
+
         <img src="https://static.igem.org/mediawiki/2014/d/d6/VB2014_Det_FinalDistribution.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/d/d6/VB2014_Det_FinalDistribution.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/7/73/VB2014_Det_PhaseMap.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/7/73/VB2014_Det_PhaseMap.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/7/73/VB2014_Det_PhaseMap.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/7/73/VB2014_Det_PhaseMap.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 1. Dependence on initial conditions</strong> A) Steady state distribution of plasmids from different initial conditions. As seen, small differences in initial conditions are amplified. B) Phase space map of plasmid growth.
 +
  </div>
</div>
</div>
<br>
<br>
-
 
-
 
-
<br><br><br>
 
<h3>
<h3>
Stochastic model
Stochastic model
-
</h3><br><br>
+
</h3>
-
The previous model is suitable when the system is not noisy. However, this is not the case and small variations in plasmid levels could be highly amplified. Hence, we modeled the effect of noise in plasmid growth and how it affects. In order to obtain we used the <b>Stochastic Simulation Algorithm (SSA)</b> developed by Gillespie (Erbran et al, 2007 is a good paper about the topic). In order not to enter into details and giving a general overview of the process, SSA simulates by iterations which processes of a given set of reactions occur at which time using a random number generator.<br>
+
<br>
<br>
-
First of all, we developed a code using this algorithm, considering the growth of two plasmids and cell division as independent process. The steps are the following:<br>
+
The deterministic model assumes negligible noise. However, this might not be the case and small variations in plasmid levels could be highly amplified. Hence, we now discuss the importance of noise in plasmid growth. We used the <b>Stochastic Simulation Algorithm (SSA)</b> developed by Gillespie (see [13]). In short, SSA simulates by iteration which processes occur at any time using a random number generator.<br>
-
<br>
+
 
-
<li>Calculation of probabilities:
+
First of all, we used the algorithm to develop a code assuming growth of the plasmids and cell division as independent processes. The steps in the algorithm are:<br>
-
$$ \alpha_{P_i \rightarrow P_i+1} = \frac{1}{r_i \cdot P_i \cdot R(P_i)}$$
+
 
-
<li>Calculation of the timestep when the next reaction occurs:
+
<li>Compute the propensity function of each reaction and the overall propensity function:
-
$$ t = \sum \limits_{i=1}^{n}{\alpha_{P_i \rightarrow P_i+1}} \cdot \log(\frac{1}{\theta_1})$$
+
$$ \alpha_{P_i \rightarrow P_i+1} = r_i \cdot P_i \cdot R(P_i), $$
-
where $\theta_1$ is a random number from a uniform distribution U(0,1). <br><br>
+
$$ \alpha_{0} = \sum \limits_{i=1}^{n}\alpha_{P_i \rightarrow P_{i+1}}. $$
-
<li>Calculation the next reaction. The next reaction taking place will be the one which:
+
<li>Compute when the next chemical reaction takes place :
-
$$ \sum \limits_{i=1}^{j}{\alpha_{P_i \rightarrow P_i+1}} < \theta_2 \leq \sum \limits_{i=1}^{j+1}{\alpha_{P_i \rightarrow P_i+1}} $$
+
$$ t = \frac{1}{\alpha_{0}} \cdot \log(\frac{1}{\theta_1}),$$
 +
where $\theta_1$ is a random number from a uniform distribution U(0,1) we make use of the random distribution for exponential reactions.<br><br>
 +
<li>Compute which reaction happens:
 +
$$ \frac{1}{\alpha_{0}} \sum \limits_{i=1}^{j-1}{\alpha_{P_i \rightarrow P_i+1}} < \theta_2 \leq \frac{1}{\alpha_{0}} \sum \limits_{i=1}^{j}{\alpha_{P_i \rightarrow P_i+1}},$$
 +
where $\theta_2$ is a random number given by the uniform distribution U(0,1).
<br>
<br>
<li>Plasmid partition after cell division at t = $\tau$ <br>
<li>Plasmid partition after cell division at t = $\tau$ <br>
-
<br>
+
We show below the evolution of a two plasmids system for two different initial conditions: $P_1=P_2=10$ and $P_1 = 10$ and $P_2 = 50$. <br><br>
-
Here is a couple of examples of what can be obtained for two different conditions: the first one takes places when $P_1=P_2=10$ and the second one when $P_1 = 10$ and $P_2 = 50$  <br><br>
+
 
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-11 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Example.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Example.jpg" rel="lightbox">
-
         <img src="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Example.jpg" width="512" height="384" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
+
         <img src="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Example.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/c/cd/VB2014_Sto_Example2.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/c/cd/VB2014_Sto_Example2.jpg" rel="lightbox">
-
         <img src="https://static.igem.org/mediawiki/2014/c/cd/VB2014_Sto_Example2.jpg" width="512" height="384" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
+
         <img src="https://static.igem.org/mediawiki/2014/c/cd/VB2014_Sto_Example2.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 2. Stochastic simulation (solid lines) versus deterministic solution (solid + dots lines) for different initial conditions (left $P_1=P_2=10$, right $P_1 = 10$ and $P_2 = 50$)</strong> This figure depicts the differences between these two modeling approaches. When several cells are taken into account, the dynamics are similar to the deterministic solution. However, the intracellular levels of plasmids can vary from the deterministic solution as shown here. Each plasmid is represented by a color and the total level of plasmids of the deterministic solution is in black.
 +
  </div>
</div>
</div>
<br>
<br>
-
As expected, the deterministic model gives a mean version of the dynamics. However these are not the single cell dynamics, which show oscillatory behaviour. Although the levels of each plasmid can vary, their sum remains unchaged (black line)
+
As expected, the deterministic model gives a mean version of the dynamics. Single cell dynamics shows oscillatory behaviour. The levels of each plasmid can vary but their sum remains unchaged (black line)
-
<br><br><br>
 
<h4>
<h4>
A lineage model
A lineage model
-
</h4><br>
+
</h4>
-
Aiming to obtain a better description of our system and perform a more exhaustive analysis, we developed a modeling approach to study the plasmids level in a lineage from a single cell. <br>
+
 
-
As tracking bacteria and their plasmid levels is an important issue, we implemented a tree plot in our code using <a href="http://www.mathworks.co.uk/matlabcentral/fileexchange/35623-tree-data-structure-as-a-matlab-class"><b>a matlab package</b></a><br>
+
Aiming to obtain a better description, we studied plasmid levels in a lineage from a single cell. <br>
 +
As tracking bacteria and their plasmid levels is an important issue, we implemented a tree plot in our code using <a href="http://www.mathworks.co.uk/matlabcentral/fileexchange/35623-tree-data-structure-as-a-matlab-class"><b>a matlab package</b></a>.
Plasmids were given to each daughter cell following a gaussian distribution with a standard deviation of $\frac{1}{\sqrt{n}}$
Plasmids were given to each daughter cell following a gaussian distribution with a standard deviation of $\frac{1}{\sqrt{n}}$
-
<br><br>
+
 
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-11 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/6/65/VB2014_Lin_PlasmidExample.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/6/65/VB2014_Lin_PlasmidExample.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/6/65/VB2014_Lin_PlasmidExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/6/65/VB2014_Lin_PlasmidExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/d/de/VB2014_Lin_TreeExample.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/d/de/VB2014_Lin_TreeExample.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/d/de/VB2014_Lin_TreeExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/d/de/VB2014_Lin_TreeExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 3. Plasmid dynamics over a lineage (I)</strong> As cells divide, they can suffer from variations in their number of copies of a plasmid, influencing the levels of daughter cells. Left picture shows the evolution in the levels of each plasmid (in different colors) and right picture is a lineage tree to track each cell.
 +
  </div>
</div>
</div>
<br>
<br>
-
Also we are able to track the plasmids levels of each bacteria at the time of their division.
+
Also we were able to track the plasmid levels of each bacteria at the time of their division.
<br>
<br>
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-11 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/4/4e/VB2014_Lin_TreeExampleP1.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/4/4e/VB2014_Lin_TreeExampleP1.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/4/4e/VB2014_Lin_TreeExampleP1.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/4/4e/VB2014_Lin_TreeExampleP1.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/0/08/VB2014_Lin_TreeExampleP2.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/0/08/VB2014_Lin_TreeExampleP2.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/0/08/VB2014_Lin_TreeExampleP2.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/0/08/VB2014_Lin_TreeExampleP2.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 4. Plasmid dynamics over a lineage (II)</strong> Variations of plasmid copy number in each lineage can be track using a lineage tree. Each tree shows the level of its associated plasmid.
 +
  </div>
</div>
</div>
<br>
<br>
-
The distribution will be the following for the begining and for the ending of the simulation:
+
The distribution at the beginning and at the end of the simulation is:
-
<br><br>
+
<div class="thumbnail col-sm-11 center-block">
-
<div style="text-align:center;">
+
     <a href="https://static.igem.org/mediawiki/2014/e/ea/VB2014_Sto_Dist0-1000.jpg" rel="lightbox">
-
     <a href="https://static.igem.org/mediawiki/2014/e/ea/VB2014_Sto_Dist0-1000.jpg">
+
         <img src="https://static.igem.org/mediawiki/2014/e/ea/VB2014_Sto_Dist0-1000.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/e/ea/VB2014_Sto_Dist0-1000.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Dist2000-2500.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Dist2000-2500.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Dist2000-2500.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/6/64/VB2014_Sto_Dist2000-2500.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 5. Heat maps of time evolution of plasmid distribution in a lineage</strong> Here we show the distributions of the plasmid levels in a lineage for the first 500 simulations (left picture) and the last 500 simulations (right picture), from a single cell with the initial conditions $P_1 = P_2 = 75$. Notice the
 +
broadening of the distribution with the lineage growth.
 +
  </div>
</div>
</div>
<br>
<br>
-
If compared with a plasmid distribution for orthogonal plasmids, a notable feature can be see: both distributions are symmetric using as reference the straight line $P_1 = P_2$, but the previous one is broader.  
+
Comparing with a distribution of orthogonal plasmids, we find that while both are symmetric (respect to the $P_1 = P_2$ line),  
-
<div style="text-align:center;">
+
non orthogonal plasmids lead to broader heat maps.  
-
     <a href="https://static.igem.org/mediawiki/2014/0/07/VB2014_NoDependence.jpg">
+
 
 +
<div class="thumbnail col-sm-5 center-block">
 +
     <a href="https://static.igem.org/mediawiki/2014/0/07/VB2014_NoDependence.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/0/07/VB2014_NoDependence.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/0/07/VB2014_NoDependence.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 6. Plasmid distribution in a system with no influence between different plasmids</strong> When each plasmid represses itself, the distribution gets narrower and the time-dependent evolution is lost.
 +
  </div>
</div>
</div>
-
<br><br><br>
 
<h4>
<h4>
A step beyond: Resistance
A step beyond: Resistance
-
</h4><br>
+
</h4>
-
As a proof of concept, we modelled the bacterial time of division as a function of both plasmids, being minimal when both plasmid levels are similar. However, this is naive approach and further work should be done in this issue. Anyway, for this simple kind of regulation, we obtained interesting results: the average levels of plasmids in a population are the same for each DNA molecule.  
+
As a proof of concept, we modelled the bacterial time of division as a function of both plasmids, being minimal when both plasmid levels are similar. For this simple regulation, we obtained that the average level of plasmids in a population is the same for each DNA molecule.  
<br><br>
<br><br>
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-11 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/a/af/VB2014_Res_PlasmidExample.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/a/af/VB2014_Res_PlasmidExample.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/a/af/VB2014_Res_PlasmidExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/a/af/VB2014_Res_PlasmidExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/b/b4/VB2014_Res_TreeExample.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/b/b4/VB2014_Res_TreeExample.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/b/b4/VB2014_Res_TreeExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/b/b4/VB2014_Res_TreeExample.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
    <div class="caption"><strong>Figure 7. Plasmid dynamics over a lineage in presence of resistance and plasmid-dependent growth </strong> Introduction of a new feature to the system, a plasmid-dependent growth, cells shows loss of synchrony in growth, favouring cells keeping similar levels of both plasmids. Left picture shows the plasmid copy number over time and right picture shows the lineage tree as a chronogram.
 +
</div>
</div>
</div>
<br>
<br>
-
This way to tune the plasmid levels was shown to work correctly, altough it is unable to make them equal, as depicted in the following picture. Also, there are variation in growth rate.
+
This way to tune the plasmid levels is unable to make them equal, as depicted in the following picture. Also, there are variations in their growth rates.
<br><br>
<br><br>
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-11 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/9/91/VB2014_Res_Evol.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/9/91/VB2014_Res_Evol.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/9/91/VB2014_Res_Evol.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/9/91/VB2014_Res_Evol.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
-
     <a href="https://static.igem.org/mediawiki/2014/b/b5/VB2014_Res_HistogramTau.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/b/b5/VB2014_Res_HistogramTau.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/b/b5/VB2014_Res_HistogramTau.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/b/b5/VB2014_Res_HistogramTau.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
      <div class="caption">
 +
<strong>Figure 8. Resistance-induced plasmid levels modulation (left) and doubling cell time (right)</strong> Resistance is able to equal the levels of both plasmids, even there was a difference in initial conditions, although there may be a small difference. This is achieved by decreasing the doubling time of cells with similar levels of each plasmid, obtaining a distribution in these times as shown in the right picture.
 +
</div>
</div>
</div>
<br>
<br>
-
 
-
<br><br><br>
 
<h4>
<h4>
Approaches for less computational expensive
Approaches for less computational expensive
-
</h4><br>
+
</h4>
-
Although these simulations are powerful and helpful and yield informative results, they are computationally expensive. This could become an important pitfall when dealing with a huge number of cells or long times. We could bypass the problem using an approach if we are working with populations as explained in (Paulsson, 2001)In order to do so, we added a degradation process for each plasmids following this equation: <br>
+
<br>
 +
Although these simulations are powerful and helpful and yield informative results, they are computationally expensive. This could become an important pitfall when dealing with a huge number of cells or long times. We tried to bypass this problem by using an approach if we are working with populations as explained in [16]We added a degradation process for each plasmid following the equations: <br>
$$ \alpha_{P_i \rightarrow P_i-1} = \frac{log(2)}{\tau} \cdot P_i$$
$$ \alpha_{P_i \rightarrow P_i-1} = \frac{log(2)}{\tau} \cdot P_i$$
-
In this new system we are observing the global dynamics for a cell population. If there is no resistance, we could obtain the following dynamics, where a plasmid is lost:
+
This approach can be used to study cell dynamics in a population. With the system of equations, we capture the global dynamics for a cell population.  
 +
If there is no resistance, we find dynamical examples where a plasmid is lost.
<br><br>
<br><br>
-
<div style="text-align:center;">
+
<div class="thumbnail col-sm-5 center-block">
-
     <a href="https://static.igem.org/mediawiki/2014/4/4b/VB2014_Approach_Example.jpg">
+
     <a href="https://static.igem.org/mediawiki/2014/4/4b/VB2014_Approach_Example.jpg" rel="lightbox">
         <img src="https://static.igem.org/mediawiki/2014/4/4b/VB2014_Approach_Example.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
         <img src="https://static.igem.org/mediawiki/2014/4/4b/VB2014_Approach_Example.jpg" width="340" height="255" alt="Simple Linear Regression Model" class="thumbnail" style="display: inline; float: none;"/>
     </a>
     </a>
 +
  <div class="caption">
 +
    <strong>Figure 9. Plasmid levels in cell lineages without resistance modulation using a less computational approach.</strong> Simulation of all features are time-consuming and computation expensive. However, simpler approaches can lead to similar results. In this particular example, the drift of a plasmid is shown.
 +
  </div>
</div>
</div>
<br>
<br>
-
This new approach could lead to study cell dynamics in a population.
 
-
<br><br><br>
+
 
-
<h3>
+
</html>
-
Conclusions
+
 
-
</h3><br>
+
=== Comparison with wet-lab data ===
-
In this work we aimed to bring the attention to a relevant issue in synthetic biology: plasmid dynamics. Yet they are widely used, we lack of great knowledge about the interaction between two plasmids in the same cell. Plasmids are used because they can perform the tasks we want. However, are their behaviour the expected one? A relevant example is <i>Marguet </i>et al.<i>, 2010</i> where they obtained an emergent behaviour as result of an unexpected relation between the bacterial physiology and plasmid dynamics. <br>
+
<html>
-
In first place, we look for studies with two plasmids in a cell but we were unable to find, although there are interesting work in single plasmid dynamics. Using these models we switch between deterministic and stochastic models to learn about the unspecific interaction between two plasmids in a cell. Therefore, we have developed a new framework to work on this topic. Moreover, we have studied this trying to shed light on this issue. <br>
+
<p>
-
As we could have learnt, we found that this system is unstable and the only way to keep the plasmid levels is introducing a resistance. However, it is not a good way to maintain those levels: bacteria which have non-desired levels of plasmids are killed. This leads to a lower yield. Also, there is an important variability, which can drive the system away from the expected behaviour.<br>
+
The interaction between two plasmids in the same cell is still an uncharted territory. A relevant example is shown in [15], where the authors obtained an emergent behaviour as result of an unexpected relation between the bacterial physiology and plasmid dynamics. Using models in single plasmid dynamics, we switch between deterministic and stochastic models to learn about the unspecific interaction between two plasmids in a cell. <a href="Results#OrthogonalitySection"><b>There is an important variability, which drives the system away from orthogonality. Plasmids with identical origin of replication are not orthogonal</b> <i>per se</i></a>. Although we can introduce different resistances to tune the system, it  
-
In conclusion, plasmids with identical origin of replication are not orthogonal <i>per se</i>. Although we can introduce use different resistances to tune the system, it could not behave as expected due to a great variability. As a solution, we could use plasmids with different mechanisms of replication, but they could show different dynamics, making the system even mor difficult to control. Therefore, new plasmids must be designed to make engineered cells more reliable and predictable to make progress in synthetic biology.
+
is hard to stabilize the system due to the large variability. Plasmids with different mechanisms of replication might solve it but it is not guaranteed. Certainly, new plasmids must be designed to make engineered cells more reliable and predictable.
 +
</p>
Line 539: Line 556:
               history: false,
               history: false,
               highlightOffset: -40
               highlightOffset: -40
-
             }).data("toc-tocify");
+
             }).data("toc-tocify2");
             $(".optionName").popover({ trigger: "hover" });
             $(".optionName").popover({ trigger: "hover" });
Line 549: Line 566:
         <div class="tab-pane fade" id="OpenLicenseTab">
         <div class="tab-pane fade" id="OpenLicenseTab">
             <h2>Open License</h2>
             <h2>Open License</h2>
-
         
+
<p>
 +
We model the susceptibility to patent a human creation by
 +
$$
 +
P.I. = M \cdot N \cdot A ^{log(\frac{A+I}{2})},
 +
$$
 +
where P.I. is the patentability index, M is the morality, which is a non-legal term accepted in the E.U., and N refers to novelty. Both factors are described by a discrete variable which can be valued 0 or 1 depending on the results of a questionnaire. A is the inventive step and I is the industrial application which can be valued from 1 to 10 depending of the
 +
results of a test.
 +
</p>
 +
<p>
 +
The Inventive Step is the most relevant variable but is regulated by the industrial application. The highest inventive step can be reduced by almost a factor 2 with the lowest industrial application. On the contrary, the lowest inventive step is not affected by the industrial application. After computing all the discussed factors, the patentability index can run from 0 to 10. The reader can find a detailed discussion of the questionnaires, tests, analysis of the index and references in the <b><a href="HumanPractices"><strong>Open License section</strong></a></b>.
 +
</p>
         </div>
         </div>
         <div class="tab-pane fade" id="OtherTab">
         <div class="tab-pane fade" id="OtherTab">
             <h2>Applications to other projects</h2>
             <h2>Applications to other projects</h2>
           </html>
           </html>
-
=== Enzymatic activity ===
 
<html>
<html>
<p>
<p>
-
External agents may modify the protein folding efficiency, and therefore change the fluorescence emission. The fraction of proteins correctly folded (active) in equilibrium at a given temperature can be calculated from the folding-unfolding equilibrium reactions:
+
    In our project, we have developed a tool that allows us to predict the stability of <em>E. coli </em>carrying a reporter gene (in particular, the stability when subjected to different temperatures). The variation of activity with temperature was described by:
 +
</p>
 +
$$
 +
\begin{equation}
 +
\frac{A_{c}}{A^0_{c}}  = \prod_i \frac{V_i}{V^0_i} ,
 +
\end{equation}
 +
$$
 +
where $V_i$ is a temperature dependent factor, and A<sub>c</sub><sup>0</sup> is the activity in reference conditions (in this case, 37 ºC).
</p>
</p>
<p>
<p>
 +
We found (<a href="#StabilityTab">see Stability</a>):
$$
$$
\begin{equation}
\begin{equation}
-
K_{eq}  = \frac{N^{eq}_A}{N_{Pr}- N^{eq}_A} = e^{-\frac{\Delta G_F}{RT } }  \nonumber,
+
\frac{Ac}{Ac^0}  = \frac{N_Pr}{N^0_Pr} =\frac{r_T}{r^0_T} \frac{r_P}{r^0_P},
\end{equation}
\end{equation}
$$
$$
-
where R is the gas constant. This equation leads to
+
with
-
 
+
$$
$$
\begin{equation}
\begin{equation}
-
N^{eq}_A = \frac{N_{Pr}}{1+e^{\frac{\Delta G_F}{RT } } } \nonumber.
+
\frac{r_T}{r^0_T}  = \frac{Prob_B}{Prob^0_B} = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}},
\end{equation}
\end{equation}
$$
$$
 +
where
 +
$$
 +
\begin{equation}
 +
\frac{R^0}{R}  = \frac {0.027 (T^0)^2 - 2.12 T^0 + 42.2} {0.027 T^2 - 2.12 T + 42.2} ,
 +
\end{equation}
 +
$$
 +
and with $$
 +
\begin{equation}
 +
\frac{r_P}{r^0_P}  = \frac{\frac{5}{4}}{\frac{r^0_E}{r_E}+\frac{1}{4}},
 +
\end{equation}
 +
$$
 +
where
 +
$$
 +
\begin{equation}
 +
r_E  = \frac{1}{\frac{1}{A_1}e^{\frac{E^1_a}{RT}}+\frac{1}{A_2}e^{\frac{E^2_a}{RT}}+\frac{1}{A_3}e^{\frac{E^3_a}{RT}}}.
 +
\end{equation}
 +
$$
 +
In more complex systems, one expects more temperature dependent factors, and varying more significantly. For an enzyme, the activity is:
 +
$$
 +
\begin{equation}
 +
A_{c}  = k  e^{-\frac{E_a}{RT } }  \frac{N_{Pr}}{1+e^{\frac{\Delta G_F}{RT } } } .
 +
\end{equation}
 +
$$
 +
where N<sub>Pr</sub> is the number of total proteins, ΔG is the free energy of the folding process, k is the Arrhenius constant of the chemical reaction
 +
    and E<sub>a</sub> is the activation energy of the catalyzed reaction.
</p>
</p>
<p>
<p>
-
The free energy of folding $\Delta G_F$ can be estimated as a function of length and temperature by [1]
+
A crude estimation can be obtained by assuming: a) all genes are controlled by the same promoter, b) same folding energy for all proteins, and c) same activation energy
 +
for all reactions. The activation energy for each reaction (mean activation energy is from [1]) is:
 +
$$
 +
\begin{equation}
 +
\frac{A_{c}}{A^0_{c}}  = {(\frac{r_P}{r^0_P} )^n}{({\frac{r_P}{r^0_P}}{\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^m},
 +
\end{equation}
 +
$$
 +
where n is the number of different mRNAs, m is the number of proteins, E$_ a$ is the activation energy of the catalyzed reaction and $\Delta$G is the protein folding energy. Using:
 +
$$
\begin{equation}
\begin{equation}
\Delta G_F (L,T)  = \Delta H(L) + \Delta C_p (L)(T-T_h) - T\Delta S(L) - \Delta C_p (L)T\ln\frac{T}{T_S} \nonumber,
\Delta G_F (L,T)  = \Delta H(L) + \Delta C_p (L)(T-T_h) - T\Delta S(L) - \Delta C_p (L)T\ln\frac{T}{T_S} \nonumber,
\end{equation}
\end{equation}
-
with $T_h=373.5$K, $T_s=385$K and
+
$$
 +
with
 +
$$
\begin{eqnarray}
\begin{eqnarray}
\Delta H(L) &=& -5.03 L - 41.6 (kJ/mol)\nonumber, \\
\Delta H(L) &=& -5.03 L - 41.6 (kJ/mol)\nonumber, \\
Line 584: Line 652:
\Delta S(L) &=& -16.8 L -85 (J/mol)\nonumber.
\Delta S(L) &=& -16.8 L -85 (J/mol)\nonumber.
\end{eqnarray}
\end{eqnarray}
 +
$$
 +
and T<sub>h</sub> = 373.5 K, T<sub>s</sub> = 385 K and L = 300 aa is the <em>E. coli</em> mean protein length is 300 aa, we computed the
 +
relative activity and relative protein number for n mRNAs and m proteins. The results are shown below:
 +
</p>
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/5/59/Vbt_modeling_other_figure1.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/5/59/Vbt_modeling_other_figure1.png" alt="Figure 1" /> 
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 1:</strong>
 +
    Variation of activity with temperature for a system that has 1, 2, 3, 4 and 5 genes.
 +
  </div>
 +
</div>
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/d/d9/Vbt_modeling_other_figure2.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/d/d9/Vbt_modeling_other_figure2.png" alt="Figure 2" /> 
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 2:</strong>
 +
    Variation of synthetized protein with temperature for a system that has 1, 2, 3, 4 and 5 genes.
 +
  </div>
 +
</div>
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/b/b6/Vbt_modeling_other_figure3.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/b/b6/Vbt_modeling_other_figure3.png" alt="Figure 3" /> 
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 3:</strong>
 +
    Variation of activity with temperature for a system that has 1, 2, 3, 4 and 5 enzymes, but only one mRNA.
 +
  </div>
 +
</div>
 +
<h3>
 +
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/b/b2/VBT_graf7.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/b/b2/VBT_graf7.png" />
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 4:</strong>
 +
    Variation of activity with temperature of a system that has 5 proteins,
 +
using 1, 2, 3, 4 and 5 mRNA.
 +
  </div>
 +
</div>
 +
 +
</html>
 +
=== Summary ===
 +
<html>
 +
<p>
 +
    - Stability decreases with increasing modules number. In case of many modules, stability is larger when one of the modules is the limiting step.
</p>
</p>
<p>
<p>
-
The enzymatic activity would be
+
    - Stability grows with decreasing number of mRNA at lower temperatures.  On the other hand, higher temperatures and lower number of mRNA show higher activity.
 +
</p>
 +
 
 +
</html>
 +
=== Application to other iGEM projects ===
 +
<html>
 +
<p>
 +
    <a href="https://2010.igem.org/Team:Slovenia/ABOUT_US" target="blank">Team Slovenia 2010 (DNA CODING BEYOND TRIPLETS)</a>
 +
developed a system to use a DNA sequence as a scaffold for enzymes, in order to optimize a metabolic pathway (<a href="https://2010.igem.org/Team:Slovenia" target="blank"> TS10 </a>). We apply our model as follows:
 +
</p>
 +
<p>
 +
$$
\begin{equation}
\begin{equation}
-
A_{c}  = e^{-\frac{E_a}{RT } } \frac{N_{Pr}}{1+e^{\frac{\Delta G_F}{RT } } } .
+
\frac{A_{c}}{A^0_{c}}  = {(\frac{r_P}{r^0_P} )^n}{({\frac{r_P}{r^0_P}}{\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^m}{({{\frac{\frac{1}{1+e^{\frac{\Delta G_Z}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_Z}{RT^0 } } }}}}{{\frac{\frac{1}{1+e^{\frac{\Delta G_I}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_I}{RT^0 } } }}}})^z}{{\frac{\frac{1}{1+e^{\frac{\Delta G_D}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_D}{RT^0 } } }}}}
\end{equation}
\end{equation}
-
where k is the Arrhenius constant and $E_a$ is the activation energy of the folding reaction.
+
$$
 +
where n is the number of required transcripts, m is the number of required translated proteins, z is the number of required zinc-finger—DNA interactions, E$_a$ is the activation energy, $\Delta$G$_F$ is the folding energy of any enzyme required for the pathway, $\Delta$G$_Z$ is the  folding energy of every zinc-finger domain [17], $\Delta$G$_I$ is the interaction energy of a normal zinc-finger-DNA interaction [17] and $\Delta$G$_D$ is the DNA melting energy.
 +
</p>
 +
<p>
 +
The goal was to look for a more efficient synthesis of violacein. Using the mean activation energy (50 kJ/mol) for any catalyzed reaction [1] we get
 +
</p>
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/8/8f/Vbt_modeling_other_figure4.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/8/8f/Vbt_modeling_other_figure4.png" alt="Figure 4" /> 
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 5:</strong>
 +
    Variation of activity with temperature of a metabolic pathway with 1, 2, 3, 4 and 5 enzymes.
 +
  </div>
 +
</div>
 +
We conclude that the new modules (zinc-finger and DNA) do not significantly affect the stability.
 +
<p>
 +
    <a href="https://2013.igem.org/Team:Heidelberg/Team" target="blank">Team Heidelberg 2013 (PHILOSOPHER’S STONE)</a>
 +
developed a system to synthesize non ribosomal peptides. One of the simplest mechanisms was the synthesis of indigoidine (<a href="https://2013.igem.org/Team:Heidelberg" target="blank">TH13 wiki</a>) :
 +
</p>
 +
<div class="thumbnail col-sm-7 center-block">
 +
    <img src="https://static.igem.org/mediawiki/2014/c/cc/Vbt_modeling_other_schema.png" alt="" /> 
 +
</div>
 +
    For this synthesis, we need: 1) acquisition of Glu (assuming that the bacteria acquires from the external medium with a specific transporter), 2) transform Glu in cGlu and 3) transform cGlu in Ind (assuming that there ir one lower step). We get:
 +
$$
 +
\begin{equation}
 +
\frac{A_{c}}{A^0_{c}}  = {(\frac{r_P}{r^0_P} )^3}{(\frac{r_P}{r^0_P})^3}{({\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^2}{\frac{\frac{1}{1+e^{\frac{\Delta G_T}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_T}{RT^0 } } }}}{\frac{e^{-\frac{E_aT}{RT } }}{e^{-\frac{E_aT}{RT^0 } }}}
 +
\end{equation}
 +
$$
 +
where E$_a$ is the activation energy of every chemical reaction, E$_{aT}$ is the activation energy of the transport process, $\Delta$G$_F$ is the folding energy of any enzyme required for the pathway and $\Delta$G$_T$ is the folding energy of the transporter.
 +
</p>
 +
<p>
 +
We use the mean activation energy (50 kJ/mol) for any catalyzed reaction [1] and the activation energy for the Lysine transport through membranes derived from [18],
 +
E$_{aT}$=33.244 kJ/mol, we get
 +
</p>
 +
<div class="thumbnail col-sm-7 center-block">
 +
  <a href="https://static.igem.org/mediawiki/2014/5/50/Vbt_modeling_other_figure5.png" rel="lightbox">
 +
    <img src="https://static.igem.org/mediawiki/2014/5/52/VBT_graf8.png" alt="Figure 5" /> 
 +
  </a>
 +
  <div class="caption">
 +
    <strong>Figure 6:</strong>
 +
    Variation of the Ind production with temperature compared to the variation of activity of the systems composed by 2 or 3 enzymes.
 +
  </div>
 +
</div>
 +
<p>
 +
In this case, we observe deviations to the expectation: the optimum temperature and the maximum value of Ac/Ac$^0$ are lower than in the n,m = 3 case. This implies a significant contribution of the Glu transport module.
</p>
</p>
Line 642: Line 815:
<li>Paulsson J and Ehrenberg M (2001) <i>Noise in a minimal regulatory network: plasmid copy number control</i> Cambridge Press <a href="http://journals.cambridge.org/action/displayFulltext?type=1&fid=75292&jid=QRB&volumeId=34&issueId=01&aid=75291"><b>link</b></a>
<li>Paulsson J and Ehrenberg M (2001) <i>Noise in a minimal regulatory network: plasmid copy number control</i> Cambridge Press <a href="http://journals.cambridge.org/action/displayFulltext?type=1&fid=75292&jid=QRB&volumeId=34&issueId=01&aid=75291"><b>link</b></a>
 +
 +
<li>Shiro I and Kuldell N (2007) <i>Zinc Finger proteins: from atomic contact to cellular function</i>. Landes Bioscience. <a href="http://www.landesbioscience.com/iu/Iuchi_9781587066429.pdf"><b>link</b></a>
 +
 +
<li>Berlin, R.D (1973) <i>Temperature dependence of nucleoside membrane transport in rabbit alveolar macrophages and polymorphonuclear leukocytes</i>. J. Biol. Chem., 248:4724-4730. <a href="http://www.ncbi.nlm.nih.gov/pubmed/4718740"><b>link</b></a>
</ol>
</ol>

Latest revision as of 23:35, 17 October 2014

Modeling

Presentation

In this section, we present our models to describe the four legs of the ST$^2$OOL. We have closely followed all the experimental activities, so our models do not pretend to completely characterize a single dynamical process of a cell, but instead we intend to capture the dependences of our systems on the variables explored by our experiments.

Stability We have modeled the variation of the activity of cells transformed with Biobricks, when subjected to different temperatures or pH, by considering changes in the Biobrick transcription and translation rates.

Standardization We have modeled the expression in different strains by comparing the transcription rates. We have built an app that computes the standardization-stability index and generates the promoter sequence for a given expression rate.

Orthogonality We have studied deterministic and stochastic models of the plasmid growth in order to learn about the unspecific interaction between two plasmids in a cell.

Open License We have modeled the susceptibility to patent a human creation and define the factors that influence it .

Stability

Living organisms have genetic and biochemical tools that allow them to respond to stress. These tools normally up-regulate some stress-specific genes, and, directly or indirectly, down-regulate the rest of normal genes (most of them housekeeping genes). On the other hand, every Escherichia coli component works worse under non-optimal conditions (for example, nutrients uptake or ATP production), so that, every E. coli function will be reduced. We model the relative expression of biobricks when subjected to variations of temperature, pH and salt concentration.

Protein production

The mRNA and protein production of a cell can be described by the kinetic equations:

$$ \begin{eqnarray} \frac{d N_{mRNA} }{dt}&=& P \cdot r_T - N_{mRNA} \cdot \lambda_{mRNA} \nonumber ,\\ \frac{d N_{Pr} }{dt}&=& N_{mRNA} \cdot r_P - N_{Pr} \cdot \lambda_{Pr}, \nonumber \end{eqnarray} $$

where $N_{\it{mRNA}} (N_{Pr} )$ is the number of biobrick mRNA (protein) molecules per cell, $\lambda_{mRNA} $ ($\lambda_{Pr}$) is the mRNA (protein) decay rate, $r_T$ ($r_P$) is the transcription (translation) rate, P is the mean number of plasmids per cell.

Assuming stationary messenger and protein content during the cell cycle, $\frac{d N_{mRNA} }{dt}=0$ and $\frac{d N_{Pr} }{dt}=0$, we get

$$ \begin{equation} N_{Pr} = \frac{ r_T \cdot r_P }{ \lambda_{mRNA}\cdot \lambda_{Pr} } P . \end{equation} $$

This equation describes the amount of total synthesized protein as a function of calculable rates and the number of free plasmids. It can be easily shown that, after many generations, the mean number of proteins per cell is 1.5 times the protein production during a cell cycle.

1) TEMPERATURE

The expression level of our heterologous gene can be calculated by equations 1 and 2. Our main goal is to find which of the terms will change most significantly with the different stress factors. We are interested in the activity in comparison to a reference one, which we take as the activity of E. coli bacteria grown on the LB medium at 38.7 oC, pH 7, 1% of NaCl, no UV irradiation and 1 atm of pressure.

$$ \begin{equation} \frac{A_{c}}{A^0_{c}} = \prod_i \frac{V_i}{V^0_i} . \end{equation} $$ where $V$ is any of the factors contributing to the activity and the index $0$ refers to the reference case.

Transcription rate

Among several factors, we will first discuss the transcription rate. In the range of temperatures (or pH) that we have studied, the polymerase (RNAP) is sufficiently stable to assume that the elongation rate is not affected and the interaction energies RNAP-promoter do not change significantly (for example, the promoter specific interaction only changes 16%, from 33 oC to 44 oC). We did not find any significant difference in the nucleotide compositions of E. coli under stress [2]. Therefore, main variations in the transcription rate are due to the probability of RNAP-promoter bounds $$ \begin{equation} r_T (T)=\frac{1}{\frac{1}{r_I(\Delta\epsilon)}+\frac{1}{r_E}}Prob_B (\Delta\epsilon, T) \propto Prob_B (T). \nonumber \end{equation} $$ On the other hand, down-regulation affects the expression level of our Biobrick, because it works as a housekeeping gene (with a normal constitutive promoter for the RNAP). For example, E. coli has a common σ factor for housekeeping genes (σ70), and, another one (σ32) for temperature stress. In the case of temperature stress, the bacteria will synthetize σ32, which will bind RNA polymerase (RNAP) cores that won’t be free for σ70, and so they will not be able to synthetize mRNA of the normal housekeeping genes.

Therefore, we just need to find a relation between the amount of σ factors and the expression level, which we derive next with thermodynamic arguments, and also the variation of σ factors with temperature, which can be determined experimentally.

The RNAP is bound to a promoter (unspecific DNA) with $\epsilon_{S}$ ($\epsilon_{NS}$) free energy. There are $N_{NS}$ nonspecific sites on the E. coli genome (sequences of 41bp) and R free E$\sigma$70, with $R \ll N_{NS}$ . The combinatory number of unbound states, with energy R$\epsilon_{NS}$, is $$ \begin{equation} \frac{N_{NS}!}{R!(N_{NS}-R)!} \simeq \frac{N^R_{NS}}{R!} . \nonumber \end{equation} $$ Similarly, the combinatory number with one bound state, with energy $\epsilon_{S}$+(R-1)$\epsilon_{NS}$, is $$ \begin{equation} \frac{N_{NS}!}{(R-1)!(N_{NS}-R-1)!} \simeq \frac{N^{R-1}_{NS}}{(R-1)!} . \nonumber \end{equation} $$ Then, the probability of finding a RNAP bound to a promoter is with the enter $$ \begin{equation} Prob_B = \frac {\frac{R}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}}{1+\frac{R}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}}, \end{equation} $$ with $\Delta\epsilon = \epsilon_{S} - \epsilon_{NS}$, calculated with the energy matrix developed in [3] and $k_B$ is the Bolztmann constant. R $\simeq$ 1000 and $N_{NS} \simeq 10^7$ for E. coli.
Using the measured variation of free E$\sigma$70, in arbitrary units, with temperature, in $^o$C, [4] $$ \begin{equation} E\sigma70 = 0.027 T^2 - 2.12 T + 42.2, \nonumber \end{equation} $$ we can determine the relative change of transcription rate $$ \begin{equation} \frac{r_T}{r^0_T} = \frac{Prob_B}{Prob^0_B} = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}} , \end{equation} $$ with $$ \begin{equation} \frac{R^0}{R} = \frac {0.027 (T^0)^2 - 2.12 T^0 + 42.2} {0.027 T^2 - 2.12 T + 42.2} . \end{equation} $$

Synthesis rate

Modeling synthesis rate follows the lines discussed in transcription rate, but the working regime is quite different. The energies of the mRNA-ribosome sequences interactions are much larger than in the RNAP-promoter ones. The average difference of free energies (computed with the RBS calculator for random sequences of E. coli genes) is $\Delta\epsilon = -20.79 kT$, which means that, in the equation: $$ \begin{equation} \frac{r_T}{r^0_T} = \frac{Prob_B}{Prob^0_B} = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}} , \end{equation} $$ The term $\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}>>>\frac {R^0}{R}$. With this, and knowing that there is no effect on the ribosomal protein composition with temperature changes [5], we conclude that the most signifficant factor should be the elongation rate.

Therefore, $$ \begin{equation} \frac{r_P}{r^0_P} = \frac{1+\frac{r^0_E}{r^0_I}}{\frac{r^0_E}{r_E}+\frac{r^0_E}{r^0_I}}, \end{equation} $$ where $r_E$ ($r_I$) is the elongation (initiation) rate and the superscript denotes our reference case. In E. coli, most of the ribosomes are translating (4 active every one free) [6], thus $\frac{r^0_E}{r^0_I} \simeq\frac{1}{4}$ and $$ \begin{equation} \frac{r_P}{r^0_P} = \frac{\frac{5}{4}}{\frac{r^0_E}{r_E}+\frac{1}{4}}. \end{equation} $$

We discuss next the dependence of elongation rates with temperature. We use the relation of amino acid concentration with temperature [7]. Using data from [8], we know that small variations on the ternary complex does not affects the elongation rate. The slowests steps are: 1) GTP hydrolysis on ternary complex (0.01 s), 2) dipeptide bond formation (0.03s) and 3) translocation (0.022s). $$ \begin{equation} r_E = \frac{1}{\frac{1}{A_1}e^{\frac{E^1_a}{RT}}+\frac{1}{A_2}e^{\frac{E^2_a}{RT}}+\frac{1}{A_3}e^{\frac{E^3_a}{RT}}}, \end{equation} $$ where $E^i_a$ is (32, 75.2, 29.260) kJ/mol and $A^{-1}_i$ is (2.5 $\cdot 10^7$, 1.6 $\cdot 10^{14}$, 3.9 $\cdot 10^6$).

Summary

Our model includes two main contributions to the changes of the enzymatic activity due to temperature variation: transcription and synthesis rates. In the case of the GFP, the folding free energy term is small and the relative enzymatic activity can be simplified to our master equation for stability $$ \begin{equation} \frac{Ac}{Ac^0} = \frac{r_T}{r^0_T} \frac{r_P}{r^0_P} . \end{equation} $$

2) pH

Contrary to the temperature situations, internal pH hardly varies in E. coli when the external pH does. Therefore, we focus on the intrinsic E. coli element that may be affected by this type of stress, subsequently influencing Biobrick expression. The main factor is the number of free RNAP, which will be reduced due to the synthesis of stress response genes. We estimate the ratio of free RNAP under different conditions by the ratio of their growth rates $$ \begin{equation} \frac{r_T}{r^0_T} = \frac{Prob_B}{Prob^0_B} = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}} , \end{equation} $$ with $$ \begin{equation} \frac{R^0}{R} = \frac{growth^0} {growth} . \end{equation} $$

Comparison with wet-lab data

Our model captures the shape of the observed temperature profile and predicts that additional variations at lower temperatures are independent of the chosen Biobrick. We have found that transcription rate decreases with temperature as a consequence of the reduction of free RNAP; and synthesis rate grows above the optimal temperature. Their combination leads to the observed variation of fluorescence with temperature. Unfortunately, this is not the whole story. Our model fails to fit the amount of variation, which is larger experimentally than the predicted one. By comparing our theoretical predictions and experimental results of XL1 Blue strain transformed with Bb1 or Bb3, we find that the ratio between theoretical prediction and experimental results is the same at under-optimal temperatures (in fact, accidentally too close given the error bars). This fact suggests that the temperature dependence not included in our model should come from cell mechanisms; independent from Biobrick parameters (promoter strength, ...). On the contrary, the comparison of theoretical predictions and experimental results at higher temperatures show that our model might require more dynamics which depends on Biobrick parameters.

Our model captures the shape of the observed pH profile. We have found that the lowest fluorescence happens at the pH with lowest growth, while it is mildly dependent on the pH from 6 to 8.

Standardization

We will now discuss Biobrick expression in different E. coli strains. The effect of the changes in the number of free RNAP on mRNA synthesis rate depend on the energy of the promoter. The relative transcription rate in different strains can be written as $$ \begin{equation} \frac{r_T}{r'_T} = \frac{Prob_B}{Prob'_B} = \frac { 1+\frac{R'}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R'}{R}+\frac{R'}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}} , \end{equation} $$ where we can see that the dominance of the second term over the first one, controlled by the promoter strength, would lead to similar rate independently on the particular strain chosen. Comparing this formula with the relative transcription rate in stability, we find that promoters whose binding energy makes them more standard also implies more stable promoters. Strong promoter sequences experience a decrease in synthesis rates when subjected to an increase of promoter energy. We use these results and the energy matrix derived in [3] to create an application that is able to compute promoter sequences for a given relative (to the maximum) expression rate and associated standard and stable rate. The physical parameters behind are the promoter strength (using the data of Anderson’ promoters) and the binding energy and the tool create a sequence of a constitutive promoter with the appropriated trade-off between strength and energy, therefore, with the appropriated trade-off between stability & standardization (ST$^2$)and rate of expression. We define the ST$^2$ index as $$ F_{st2} = (F - 0.012)/0.51 $$ with $$F = \frac{R'}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}, $$ where $F_{sty}$ has been scaled to run from 1 to 0. We have used a fitting formula for the rate of expression relative to the maximum rate [3] as a function of the binding energy $\Delta\epsilon$, $$ r_T = 2.29 + 0.27 \frac{\Delta\epsilon}{kT}. $$ With this approach, we can infer the ST$^2$ level and the promoter sequence for a given expression rate. We show below our app in javascript.

Generate a promoter sequence and compute the ST$^2$ (Stability & Standardization) index for a given expression rate

Move the cursor to a given expression rate (as a fraction of its maximum) and press Generate a chain

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Promoter sequence (from -41 to -1 nt) is:

ST$^2$ index:

Comparison with wet-lab data

Our wet-lab results show that Biobrick 5 is the most standard one.Our model predicts that systems with less modules or factors will be more stable and standard. Biobrick 5 has a tetracycline depending promoter, whose limitating step is the binding of tetracycline to TetR. In this scenario, the intracellular concentration of aTc and the binding to the TetR repressor do not depend on the strain, so there is only one factor in which the strain could have an important effect: the TetR intracellular concentration, which has a constitutive expression. Analyzing the other Biobricks, we found that the strains with less expression of the constitutive promoter are those ones with higher expression when we use the Ptet promoter (but JM109 and BB1), which agree with our prediction.

Orthogonality

Plasmid dynamics have been studied exhaustively over these years using mathematical models: from how plasmid growth is regulated to plasmid partitioning after cell division. However, little is known about how two identical plasmids interact with each other. This part of our modelling project aims to shed light upon this issue using previous models of ColE1 plasmids and the properties of two different plasmids with same origin of replication when these are introduced in a cell. We employed both deterministic and stochastic models to try to answer whether two plasmids behave independently, and developed some approximations to highlight important features of the system.

Deterministic model


This model is taken from [16]. Here, plasmid levels are regulated both by an RNA molecule that inhibits plasmid replication and by the dilution of plasmids as a result of cell growth and division: $$ \frac{d P}{dt} = r \cdot P \cdot R(s) - \frac{\log 2}{\tau} \cdot P, $$ $$ \frac{d s}{dt} = \alpha \cdot P - \beta \cdot s, $$ where $P$ is the number of plasmids, $s$ is the number of RNA inhibitors, $r$ is the plasmid growth rate, $\tau$ is the cell growth rate, $\alpha$ is the RNA inhibitor synthesis rate , $\beta$ is the RNA inhibitor decay rate and $R(s)$ is the repression effect of RNA inhibitor with $$ R(s) = \frac{1}{(1+\frac{s}{s_0})^n},$$ where $s_0$ is the value of RNA inhibitor necessary to repress half of the replication events and $n$ is the cooperativity. Plasmid levels are regulated both by a negative feedback and by cell growth. If we consider several plasmids, we get

$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(s) - \frac{\log 2}{\tau} \cdot P_i,$$ $$ \frac{d s}{dt} = \alpha \cdot \sum \limits_{i=1}^n{P_i} - \beta \cdot s,$$ with a steady state solution

$$ s_{ss} = R^{-1}(\frac{\log 2}{\tau \cdot r_i}),$$ $$ \sum \limits_{i=1}^n{P_{i,ss}} = (\frac{ \beta \cdot s_{ss}}{\alpha}).$$
In a simplified way, with two identical plasmids, we get: $$ P_{1,ss} + P_{2,ss} = K.$$ It is noticeable that plasmid levels are not independent. Small perturbations can increase the relative level of one plasmid, favoring its growth. As they are repressed as a function of the total number of plasmids but they grow depending on their number, the system is prone to introduce asymmetries $$ \frac{d P_1}{d P_2} = \frac{r_1 \cdot P_1}{r_2 \cdot P_2} .$$ If growth rates are constant, the slope is determined by the initial conditions $$ m = \frac{P_{01}}{P_{02}} .$$

Quasi-steady state approach

To simplify our equations, we assume inhibitor dynamics are faster than plasmid growth, so RNA levels are a direct representation of plasmid levels $$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(P_i) - \frac{\log 2}{\tau} \cdot P_i .$$

Dependence on initial conditions:

To test our hypothesis, we integrate our differential equations using numerical methods with several initial conditions. As expected, small differences in initial proportions are amplified by the dynamics.
Simple Linear Regression Model Simple Linear Regression Model
Figure 1. Dependence on initial conditions A) Steady state distribution of plasmids from different initial conditions. As seen, small differences in initial conditions are amplified. B) Phase space map of plasmid growth.

Stochastic model


The deterministic model assumes negligible noise. However, this might not be the case and small variations in plasmid levels could be highly amplified. Hence, we now discuss the importance of noise in plasmid growth. We used the Stochastic Simulation Algorithm (SSA) developed by Gillespie (see [13]). In short, SSA simulates by iteration which processes occur at any time using a random number generator.
First of all, we used the algorithm to develop a code assuming growth of the plasmids and cell division as independent processes. The steps in the algorithm are:
  • Compute the propensity function of each reaction and the overall propensity function: $$ \alpha_{P_i \rightarrow P_i+1} = r_i \cdot P_i \cdot R(P_i), $$ $$ \alpha_{0} = \sum \limits_{i=1}^{n}\alpha_{P_i \rightarrow P_{i+1}}. $$
  • Compute when the next chemical reaction takes place : $$ t = \frac{1}{\alpha_{0}} \cdot \log(\frac{1}{\theta_1}),$$ where $\theta_1$ is a random number from a uniform distribution U(0,1) we make use of the random distribution for exponential reactions.

  • Compute which reaction happens: $$ \frac{1}{\alpha_{0}} \sum \limits_{i=1}^{j-1}{\alpha_{P_i \rightarrow P_i+1}} < \theta_2 \leq \frac{1}{\alpha_{0}} \sum \limits_{i=1}^{j}{\alpha_{P_i \rightarrow P_i+1}},$$ where $\theta_2$ is a random number given by the uniform distribution U(0,1).
  • Plasmid partition after cell division at t = $\tau$
    We show below the evolution of a two plasmids system for two different initial conditions: $P_1=P_2=10$ and $P_1 = 10$ and $P_2 = 50$.

    Simple Linear Regression Model Simple Linear Regression Model
    Figure 2. Stochastic simulation (solid lines) versus deterministic solution (solid + dots lines) for different initial conditions (left $P_1=P_2=10$, right $P_1 = 10$ and $P_2 = 50$) This figure depicts the differences between these two modeling approaches. When several cells are taken into account, the dynamics are similar to the deterministic solution. However, the intracellular levels of plasmids can vary from the deterministic solution as shown here. Each plasmid is represented by a color and the total level of plasmids of the deterministic solution is in black.

    As expected, the deterministic model gives a mean version of the dynamics. Single cell dynamics shows oscillatory behaviour. The levels of each plasmid can vary but their sum remains unchaged (black line)

    A lineage model

    Aiming to obtain a better description, we studied plasmid levels in a lineage from a single cell.
    As tracking bacteria and their plasmid levels is an important issue, we implemented a tree plot in our code using a matlab package. Plasmids were given to each daughter cell following a gaussian distribution with a standard deviation of $\frac{1}{\sqrt{n}}$
    Simple Linear Regression Model Simple Linear Regression Model
    Figure 3. Plasmid dynamics over a lineage (I) As cells divide, they can suffer from variations in their number of copies of a plasmid, influencing the levels of daughter cells. Left picture shows the evolution in the levels of each plasmid (in different colors) and right picture is a lineage tree to track each cell.

    Also we were able to track the plasmid levels of each bacteria at the time of their division.
    Simple Linear Regression Model Simple Linear Regression Model
    Figure 4. Plasmid dynamics over a lineage (II) Variations of plasmid copy number in each lineage can be track using a lineage tree. Each tree shows the level of its associated plasmid.

    The distribution at the beginning and at the end of the simulation is:
    Simple Linear Regression Model Simple Linear Regression Model
    Figure 5. Heat maps of time evolution of plasmid distribution in a lineage Here we show the distributions of the plasmid levels in a lineage for the first 500 simulations (left picture) and the last 500 simulations (right picture), from a single cell with the initial conditions $P_1 = P_2 = 75$. Notice the broadening of the distribution with the lineage growth.

    Comparing with a distribution of orthogonal plasmids, we find that while both are symmetric (respect to the $P_1 = P_2$ line), non orthogonal plasmids lead to broader heat maps.
    Simple Linear Regression Model
    Figure 6. Plasmid distribution in a system with no influence between different plasmids When each plasmid represses itself, the distribution gets narrower and the time-dependent evolution is lost.

    A step beyond: Resistance

    As a proof of concept, we modelled the bacterial time of division as a function of both plasmids, being minimal when both plasmid levels are similar. For this simple regulation, we obtained that the average level of plasmids in a population is the same for each DNA molecule.

    Simple Linear Regression Model Simple Linear Regression Model
    Figure 7. Plasmid dynamics over a lineage in presence of resistance and plasmid-dependent growth Introduction of a new feature to the system, a plasmid-dependent growth, cells shows loss of synchrony in growth, favouring cells keeping similar levels of both plasmids. Left picture shows the plasmid copy number over time and right picture shows the lineage tree as a chronogram.

    This way to tune the plasmid levels is unable to make them equal, as depicted in the following picture. Also, there are variations in their growth rates.

    Simple Linear Regression Model Simple Linear Regression Model
    Figure 8. Resistance-induced plasmid levels modulation (left) and doubling cell time (right) Resistance is able to equal the levels of both plasmids, even there was a difference in initial conditions, although there may be a small difference. This is achieved by decreasing the doubling time of cells with similar levels of each plasmid, obtaining a distribution in these times as shown in the right picture.

    Approaches for less computational expensive


    Although these simulations are powerful and helpful and yield informative results, they are computationally expensive. This could become an important pitfall when dealing with a huge number of cells or long times. We tried to bypass this problem by using an approach if we are working with populations as explained in [16]. We added a degradation process for each plasmid following the equations:
    $$ \alpha_{P_i \rightarrow P_i-1} = \frac{log(2)}{\tau} \cdot P_i$$ This approach can be used to study cell dynamics in a population. With the system of equations, we capture the global dynamics for a cell population. If there is no resistance, we find dynamical examples where a plasmid is lost.

    Simple Linear Regression Model
    Figure 9. Plasmid levels in cell lineages without resistance modulation using a less computational approach. Simulation of all features are time-consuming and computation expensive. However, simpler approaches can lead to similar results. In this particular example, the drift of a plasmid is shown.

    Comparison with wet-lab data

    The interaction between two plasmids in the same cell is still an uncharted territory. A relevant example is shown in [15], where the authors obtained an emergent behaviour as result of an unexpected relation between the bacterial physiology and plasmid dynamics. Using models in single plasmid dynamics, we switch between deterministic and stochastic models to learn about the unspecific interaction between two plasmids in a cell. There is an important variability, which drives the system away from orthogonality. Plasmids with identical origin of replication are not orthogonal per se. Although we can introduce different resistances to tune the system, it is hard to stabilize the system due to the large variability. Plasmids with different mechanisms of replication might solve it but it is not guaranteed. Certainly, new plasmids must be designed to make engineered cells more reliable and predictable.

  • Open License

    We model the susceptibility to patent a human creation by $$ P.I. = M \cdot N \cdot A ^{log(\frac{A+I}{2})}, $$ where P.I. is the patentability index, M is the morality, which is a non-legal term accepted in the E.U., and N refers to novelty. Both factors are described by a discrete variable which can be valued 0 or 1 depending on the results of a questionnaire. A is the inventive step and I is the industrial application which can be valued from 1 to 10 depending of the results of a test.

    The Inventive Step is the most relevant variable but is regulated by the industrial application. The highest inventive step can be reduced by almost a factor 2 with the lowest industrial application. On the contrary, the lowest inventive step is not affected by the industrial application. After computing all the discussed factors, the patentability index can run from 0 to 10. The reader can find a detailed discussion of the questionnaires, tests, analysis of the index and references in the Open License section.

    Applications to other projects

    In our project, we have developed a tool that allows us to predict the stability of E. coli carrying a reporter gene (in particular, the stability when subjected to different temperatures). The variation of activity with temperature was described by:

    $$ \begin{equation} \frac{A_{c}}{A^0_{c}} = \prod_i \frac{V_i}{V^0_i} , \end{equation} $$ where $V_i$ is a temperature dependent factor, and Ac0 is the activity in reference conditions (in this case, 37 ºC).

    We found (see Stability): $$ \begin{equation} \frac{Ac}{Ac^0} = \frac{N_Pr}{N^0_Pr} =\frac{r_T}{r^0_T} \frac{r_P}{r^0_P}, \end{equation} $$ with $$ \begin{equation} \frac{r_T}{r^0_T} = \frac{Prob_B}{Prob^0_B} = \frac { 1+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }} } { \frac {R^0}{R}+\frac{R^0}{N_{NS}} e^{-\frac{\Delta\epsilon}{k_BT }}}, \end{equation} $$ where $$ \begin{equation} \frac{R^0}{R} = \frac {0.027 (T^0)^2 - 2.12 T^0 + 42.2} {0.027 T^2 - 2.12 T + 42.2} , \end{equation} $$ and with $$ \begin{equation} \frac{r_P}{r^0_P} = \frac{\frac{5}{4}}{\frac{r^0_E}{r_E}+\frac{1}{4}}, \end{equation} $$ where $$ \begin{equation} r_E = \frac{1}{\frac{1}{A_1}e^{\frac{E^1_a}{RT}}+\frac{1}{A_2}e^{\frac{E^2_a}{RT}}+\frac{1}{A_3}e^{\frac{E^3_a}{RT}}}. \end{equation} $$ In more complex systems, one expects more temperature dependent factors, and varying more significantly. For an enzyme, the activity is: $$ \begin{equation} A_{c} = k e^{-\frac{E_a}{RT } } \frac{N_{Pr}}{1+e^{\frac{\Delta G_F}{RT } } } . \end{equation} $$ where NPr is the number of total proteins, ΔG is the free energy of the folding process, k is the Arrhenius constant of the chemical reaction and Ea is the activation energy of the catalyzed reaction.

    A crude estimation can be obtained by assuming: a) all genes are controlled by the same promoter, b) same folding energy for all proteins, and c) same activation energy for all reactions. The activation energy for each reaction (mean activation energy is from [1]) is: $$ \begin{equation} \frac{A_{c}}{A^0_{c}} = {(\frac{r_P}{r^0_P} )^n}{({\frac{r_P}{r^0_P}}{\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^m}, \end{equation} $$ where n is the number of different mRNAs, m is the number of proteins, E$_ a$ is the activation energy of the catalyzed reaction and $\Delta$G is the protein folding energy. Using: $$ \begin{equation} \Delta G_F (L,T) = \Delta H(L) + \Delta C_p (L)(T-T_h) - T\Delta S(L) - \Delta C_p (L)T\ln\frac{T}{T_S} \nonumber, \end{equation} $$ with $$ \begin{eqnarray} \Delta H(L) &=& -5.03 L - 41.6 (kJ/mol)\nonumber, \\ \Delta C_p(L) &=& -0.062 L + 0.53 (kJ/mol)\nonumber, \\ \Delta S(L) &=& -16.8 L -85 (J/mol)\nonumber. \end{eqnarray} $$ and Th = 373.5 K, Ts = 385 K and L = 300 aa is the E. coli mean protein length is 300 aa, we computed the relative activity and relative protein number for n mRNAs and m proteins. The results are shown below:

    Figure 1
    Figure 1: Variation of activity with temperature for a system that has 1, 2, 3, 4 and 5 genes.
    Figure 2
    Figure 2: Variation of synthetized protein with temperature for a system that has 1, 2, 3, 4 and 5 genes.
    Figure 3
    Figure 3: Variation of activity with temperature for a system that has 1, 2, 3, 4 and 5 enzymes, but only one mRNA.

    Figure 4: Variation of activity with temperature of a system that has 5 proteins, using 1, 2, 3, 4 and 5 mRNA.

    Summary

    - Stability decreases with increasing modules number. In case of many modules, stability is larger when one of the modules is the limiting step.

    - Stability grows with decreasing number of mRNA at lower temperatures. On the other hand, higher temperatures and lower number of mRNA show higher activity.

    Application to other iGEM projects

    Team Slovenia 2010 (DNA CODING BEYOND TRIPLETS) developed a system to use a DNA sequence as a scaffold for enzymes, in order to optimize a metabolic pathway ( TS10 ). We apply our model as follows:

    $$ \begin{equation} \frac{A_{c}}{A^0_{c}} = {(\frac{r_P}{r^0_P} )^n}{({\frac{r_P}{r^0_P}}{\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^m}{({{\frac{\frac{1}{1+e^{\frac{\Delta G_Z}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_Z}{RT^0 } } }}}}{{\frac{\frac{1}{1+e^{\frac{\Delta G_I}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_I}{RT^0 } } }}}})^z}{{\frac{\frac{1}{1+e^{\frac{\Delta G_D}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_D}{RT^0 } } }}}} \end{equation} $$ where n is the number of required transcripts, m is the number of required translated proteins, z is the number of required zinc-finger—DNA interactions, E$_a$ is the activation energy, $\Delta$G$_F$ is the folding energy of any enzyme required for the pathway, $\Delta$G$_Z$ is the folding energy of every zinc-finger domain [17], $\Delta$G$_I$ is the interaction energy of a normal zinc-finger-DNA interaction [17] and $\Delta$G$_D$ is the DNA melting energy.

    The goal was to look for a more efficient synthesis of violacein. Using the mean activation energy (50 kJ/mol) for any catalyzed reaction [1] we get

    Figure 4
    Figure 5: Variation of activity with temperature of a metabolic pathway with 1, 2, 3, 4 and 5 enzymes.
    We conclude that the new modules (zinc-finger and DNA) do not significantly affect the stability.

    Team Heidelberg 2013 (PHILOSOPHER’S STONE) developed a system to synthesize non ribosomal peptides. One of the simplest mechanisms was the synthesis of indigoidine (TH13 wiki) :

    For this synthesis, we need: 1) acquisition of Glu (assuming that the bacteria acquires from the external medium with a specific transporter), 2) transform Glu in cGlu and 3) transform cGlu in Ind (assuming that there ir one lower step). We get: $$ \begin{equation} \frac{A_{c}}{A^0_{c}} = {(\frac{r_P}{r^0_P} )^3}{(\frac{r_P}{r^0_P})^3}{({\frac{\frac{1}{1+e^{\frac{\Delta G_F}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_F}{RT^0 } } }}}{\frac{e^{-\frac{E_a}{RT } }}{e^{-\frac{E_a}{RT^0 } }}})^2}{\frac{\frac{1}{1+e^{\frac{\Delta G_T}{RT } } }}{\frac{1}{1+e^{\frac{\Delta G_T}{RT^0 } } }}}{\frac{e^{-\frac{E_aT}{RT } }}{e^{-\frac{E_aT}{RT^0 } }}} \end{equation} $$ where E$_a$ is the activation energy of every chemical reaction, E$_{aT}$ is the activation energy of the transport process, $\Delta$G$_F$ is the folding energy of any enzyme required for the pathway and $\Delta$G$_T$ is the folding energy of the transporter.

    We use the mean activation energy (50 kJ/mol) for any catalyzed reaction [1] and the activation energy for the Lysine transport through membranes derived from [18], E$_{aT}$=33.244 kJ/mol, we get

    Figure 5
    Figure 6: Variation of the Ind production with temperature compared to the variation of activity of the systems composed by 2 or 3 enzymes.

    In this case, we observe deviations to the expectation: the optimum temperature and the maximum value of Ac/Ac$^0$ are lower than in the n,m = 3 case. This implies a significant contribution of the Glu transport module.

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