Team:TU Delft-Leiden/Modeling/Curli/Cell

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Latest revision as of 22:56, 17 October 2014

Cell Level Modeling

Now that the growth rate of curli and production of CsgB protein as function of time is obtained from the Gene Level Model, the conductance as a function of time can be computed for the cell. The approach we used for this is relatively simple:

We discretize the amount of curli subunits ($CsgA_{curli}$ in the gene level model) and CsgB proteins that have to be added for each time step.
At each time step, we add more curli subunits to growing curli fibrils. Also, we add more new curli fibrils to the model.
From the density of the curli fibrils around the cell as a function of the radius, we calculate the conductive radius of the cell. The conductive radius is the largest radius where $\rho_{curli}$, which represents the density of curli fibrils around the cell, is bigger than a certain threshold of curli density.

Curli Module

Discretization of Gene Level Model

We have discretized equations 6.2 and 12 of the Gene Level Model in 1000 discrete times between 0 hour and 10 hours, so each time step is equal to 36 seconds. Throughout the model, we use the same rates as shown in Table 1 of the Gene Level Model (unless otherwise specified). These equations give the expected number of new CsgB proteins and curli subunits for each time step, as we plotted the solution of these two equations in figures 1 in Section Gene Level Modeling. From these figures, we determine the expected number of new CsgB proteins and curli subunits for each time step. However, a fundamental assumption in deterministic modeling is that the concentration is continuous. In reality, the amount of added curli subunits is discrete, since we cannot add half a curli subunit.
Furthermore, in the gene level model we did not take into account the statistical variation of gene transcription and adding of curli subunits; sometimes less and some times more curli subunits are added with respect to the expected value. To include this in the cell level model, we drew the amount of new curli subunits from a Poisson distribution where λ equals the expected amount of added subunits.
So, for each time step we now have $B_n$ new CsgB proteins and $C_n$ new curli subunits, where $C_n$ varies for each time step, as it is drawn from a Poisson distribution. An assumption of this distribution is that the time at which a new curli subunit is added, is uncorrelated to the time at which the previous curli subunit was added, we think this is a fair assumption. Note that the cell level model we made, accounts for the stochasticity of adding curli subunits, but not for the stochasticity of gene expression, so for the production of CsgB protein. The value $B_n$ and the Poisson distribution are determined from figure 1 in the Gene Level Modeling section.

Building the Curli Fibrils

Firstly, $B_n$ CsgB proteins are added to our model that mark the starting points for new curli fibrils. These new curli fibrils are located at random points on a sphere with radius r, which represents the cell. The radius r is chosen such that the volume of the cell is$\ \sim 1.1 \ \mu m^3$ [1]. A CsgB protein is modeled by a line of length 4 nm that points radially outward, perpendicular to the cell surface. In reality, the distribution of CsgB on the cell surface is not uniformly distributed [4] and cells are not perfectly round. However, we assumed uniformly distributed CsgB to keep our model prehensile. This is a point that may be used to further improve the model.

Next, $C_n$, which is drawn from the Poisson distribution, where λ equals the expected amount of added curli subunits, new curli subunits are added to curli fibrils by repeating the following process $C_n$ times:

Firstly, a random curli fibril is selected, e.g. curli number k. A curli fibril is represented by a 3 (the x, y and z coordinates) by l+1 matrix, where l is the amount of curli subunits of the curli fibril and the origin is chosen to be the center of the sphere. Thus, by storing the ending coordinates of each curli subunit, we know the starting and end coordinates of each curli subunit. The curli subunits are modeled by a line of length 4 nm.
Secondly, the polar angle in spherical coordinates of the last curli subunit is computed, $\theta_{1}$.
Thirdly, the new curli subunit has a small angular deviation with respect to the previous one. This polar angle $\theta_{2}$ is chosen from a Gaussian distribution with parameters N(0,σ). σ is chosen such that the persistence length, the distance over which a fibril has bend by $90^{\circ}$ and has ‘lost’ its directional information, is 4 µm [5]. The azimuthal angle ϕ is completely random between 0 and 2π radians, and chosen from an uniform distribution.
Fourthly, for the new curli subunit for which we determined $\theta_{2}$ and ϕ, the polar angle is determined to be $\theta_{1} + \theta_{2}$. We now know the length of the new curli subunit (4 nm), its polar angle and its azimuthal angle. Subsequently, we add it to the previous curli subunit of the fibril and calculate the ending coordinate of the added curli subunit from its length, polar angle and azimuthal angle and the ending coordinate of the previous curli subunit. This calculated ending coordinate of the added curli subunit is stored in the matrix that represents the curli fibril.

The angular deviation σ is a critical parameter in our model. Increasing this value increases the flexibility of our curli, where decreasing this value increases the stiffness of the curli. This is shown in figure 1. If the length of one subunit is 4 nm and the total persistence length is 4 µm, then $\sigma = \ 3.47^{\circ}$. Furthermore, we think that it is justified to add the curli subunits one at a time to a random curli. We expect no discrimination of the CsgA proteins for binding to a large or small curli or one that has recently gotten a new curli subunit.

Figure 1: The persistence length in number of units of a curli fibril as function of the angular deviation per subunit in degrees.

An illustrative view of what our cell looks like during the adding of curli subunits is shown in figure 2.

meh — Figure 2: Schematic view of our cell (black sphere centred at x=y=z=0) with growing curli fibrils. The wires represent the curli fibrils. **Click to play!**

meh — Figure 2: Schematic view of our cell (black sphere centred at x=y=z=0) with growing curli fibrils. The wires represent the curli fibrils. **Click to play!**

Now that we have a model of a cell with growing curli, we want to extract relevant data for the colony level modeling. Ideally, the resistance as function of radius and time would be calculated by looking at connections between the curli fibrils. However, this requires insight of the behavior of the curli on the nanoscopic scale. For instance, what is the conductance of a single curli fibril with gold nanoparticles and what is the critical distance between the fibrils that make them connect? After an extensive literature study, we have decided to simplify this model. Furthermore, when interactions between the curli fibrils have to be taken into account, the model gets too computationally expensive.

To have a reasonable computational time, we decided to extract our parameters for the colony level modeling from the curli density around the cell. Figure 3 shows the length of all curli after 10 hours. Curli fibrils that are created first (low numbers) are much longer than the ones that are created last (high number). The steep drop in curli fibril length for the first couple of hundred fibrils is a consequence of the peak in curli production between 0 hour and 2 hours. After that, the curli length is linear with the time it has existed, precisely what you expect from the model.

Figure 3: The length of the curli fibrils in number of subunits on the y-axis at t=10 hours. On the x-axis is the time. A dot at height 1000 at 1 hour means that the curli fibril that was started at t=1 hour had length 1000 at t=10 hours.

Looking back at figures 1 and 2 in Section Gene Level Modeling, the fact that, as can be seen in figure 3, the first curli are much longer that the later ones, can be explained by the fact that there is relatively large curli growth in the beginning, because few CsgB have been produced and therefore, only a few curli fibrils are available for CsgA proteins. After a couple of hours there are more CsgB proteins, thus more curli fibrils, but CsgA protein production does not increase. Therefore, the ratio [CsgA]/[curli fibrils] is much smaller than in the beginning and each curli will grow much slower. A consequence of this is that the ‘newer’ curli fibrils are much shorter.

Fitting the Curli Density

We think that a reasonable first approximation of the conductance is the density of the curli around the cell as a function of the distance from the membrane. When the density is higher, there are more gold particles, thus higher conductance. In our simplest approach we say that there is a critical density $\rho_{crit}$ of curli that is needed to have conductance. The density $\rho_{curli}$ decreases as function of the radius. The largest radius where $\rho_{curli} > \rho_{crit}$, we call the conductive radius $r_{cond}$. With only this simple approximation, we can calculate some interesting properties of our system at the colony level: the time at which we expect percolation to happen and the resistance of our system. Though this approximation seems to be rather arbitrary, we do have some reasoning for this:

First of all, the goal of this parameter is to get information about our system that will be calculated in colony level modeling. We use this parameter in colony level modeling to find connections between cells. To have a continuous path from one electrode to the other electrode, we must have a lot of cells that are connected to each other. In order to know when cells are connected to each other, we have to assume that everything at a certain radius from the cell is conductive; for this radius we use the critical density $\rho_{crit}$. However, for this to be true the fibrils on one side of the cell must be connected to the fibrils on the other side. The Percolation Theory prescribes that this is a sharp transition as a function of the density, so we can choose $\rho_{crit}$ in such a way that we are very sure that everything at $\rho_{crit}$ from the cell is conductive.
While the precise value of $\rho_{crit}$ may be unknown and should be measured, we think that we can still get plenty of information about the qualitative behavior of our system in advance, for instance we can investigate the sharp transition at which the conductance increases at the colony level.
Due to the simplifications that we made in order to be able to model our system, we cannot include interactions or cluster forming between the curli themselves. Using $\rho_{crit}$, we have an elegant way to filter out modeling errors.

As the building of the curli fibrils is a stochastic processes, we repeated our simulations on the cell level many times in order to get statistically valid results for the mean and standard deviation of $\rho_{curli}$ and $r_{cond}$.
When we ran our simulation 100 times, we got the results displayed in figure 4. Figure 4 displays the curli density at $\ t= \ 2 \ hours$ for all cells in the left figure. This should give us insight in the variation we might expect. In the right figure, the orange line represents the mean curli density, and the green lines represent the standard deviation. From figure 4, we conclude that the intercellular variation is relatively small. This makes sense, since the relative deviation of stochastic processes decreases with the sample size.

Figure 4: Left) The curli density in curli units $ \mu m ^{-3} $ as function of radial distance from the centre of the cell in $ \mu m$ for 100 different simulations at t=2 hr. The orange line represents the mean of all densities. Right) The orange line represents the mean curli density, and the green lines represent the standard deviation.

It is also interesting to study curli density as function of time at different times, shown in figure 5. This figure shows that, corresponding with what we have seen previously, $\rho_{curli}$ decreases as a function of the radius. Also, it decreases faster as a function of the radius in the first two hours. After two hours, we can see that the curli density increases only for small r, as mainly short curli are added to the system. This agrees with our previous results.

Figure 5: The mean curli density in curli units $ \mu m ^{-3} $ as function of radial distance from the centre of the cell in $ \mu m$, plotted at different times (.5 hr, 1hr, 2hr, 5hr and 10hr).

In order to be able to say something about the resistance of our system at the colony level, we need an analytical expression for $\rho_{curli}$. We have therefore fitted the function $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}} \tag{1}$$ to our curli density curves at each time $ n $,see figure 6 the red line. Here, $C_{1_n} $, $C_{2_n} $, $C_{3_n} $ and $ C_{4_n} $ are parameters that have to be fitted, and $ r $ is the distance from the centre of the cell. At first we tried to fit our data to only the first term (green line). It can clearly be seen in the figure that this does not adequately capture the dynamics of the curve. Either the approximation is bad at short distances or at large distances.

Figure 6: Blue line: Right behind the red line, at t=5 hr the mean of all density curves. Green line: a weighted fit of $ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} $. Red line: A fit $ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}} $ to the blue line.

It can be seen that the fit is certainly not perfect, but it is a reasonable approximation to the characteristics. The reason for fitting such a simple function is that, in the colony level, we need to quantify the conductance between the cells. The integral for this rather complicated and we need an analytical function for $\rho_{curli}$ to analytically solve this integral. In further research, we could improve our fit by fitting a set of decaying exponents.

Conductive Radius of the Cell

Different values of $\rho_{crit}$ result in different characteristic curves for $r_{cond}$, see figure 7. In this figure, we set $\rho_{crit}$ equal to a fraction of the maximum $ \rho_{curli} $ ($ 1.2 \cdot 10^5 \ \# \ \mu m^{-3} $ ) as observed in figure 5. So, we set $ \rho_{crit} = \max{ (\rho) } /K $, for the $ K $ shown in the legend.

Figure 7: The conductive radius in $ \mu m $ versus the time from t=0 to 10 hour for different values of $ \rho_{crit} $. The thick lines represent the mean conductive radius of 100 cells with a $ \rho_{crit} $ equal to to a fraction of the maximum ( $ 1.2 \cdot 10^5 \# \mu m^{-3} $ ) corresponding with the legend. The thinner lines of the same color are the mean $ \pm $ the standard deviation.

From figure 7, we conclude that low values of $\rho_{crit}$ result in a sharp increase of $r_{cond}$ followed by a steady, slow increase of $r_{cond}$ in time. During the steady, slow increase of $r_{cond}$ in time, the cellular variation is relatively large. For high values of $\rho_{crit}$, there is a delayed sharp increase of $r_{cond}$ and less cellular variation.

From figure 7, we conclude that low values of $\rho_{crit}$ result in a sharp increase of $r_{cond}$ followed by a steady, slow increase of $r_{cond}$ in time. During the steady, slow increase of $r_{cond}$ in time, the cellular variation is relatively large. For high values of $\rho_{crit}$, there is a delayed sharp increase of $r_{cond}$ and less cellular variation. Unfortunately we have no wetlab data to fit this parameter. However, we can argue what kind of behavior we would expect from $r_{cond}$.
A conductive radius of more than 5 $ \mu m $ seems unlikely to us, for the cell's diameter is only a micron. We set the $\rho_{crit}$ equal to $ 1.2 \cdot 10^5 \ \# \ \mu m^{-3} $. Even though this value might be off by a factor, we argue that this will change little in what we try to achieve in our model, namely to investigate the sharp transition at which the conductance increases at the colony level.

Figure 8: The green lines are the conductive radius plotted versus the time for 100 cells with a critical density of $ \rho_{crit}=1204 $ curli subuntis $ \mu m ^{-3} $. The orange red represents the mean conductive radius and the dark blue lines represent two standard deviations from the mean.

We conclude from figure 8 that a sharp increase in the conductive radius can be observed for $t < 1 \ hour$, and after $t = \ 1 \ hour$ the conductive radius increases slowly. The cellular variation in the second regime is relatively large, as is shown by the dark blue lines that represent two standard deviations from the mean. Note how the conductive radius increases in discrete steps. This is a result of the fact that density is a parameter that only exists over a certain volume. We have divided the volume around the cell in hollow spheres with thickness $ dr=0.08 \mu m $. Increasing the thickness would increase the accuracy over the mean, but would decrease the spatial volume. Decreasing the thickness would increase the variation between the conductive radii, but would increase the spatial volume.

References

[1] H.E. Kubitschek & J.A. Friske, "Determination of bacterial cell volume with the Coulter Counter", J. Bacteriol. 168, 3, 1986.

[2] calctoolo.org, (2014). Calctool. [online]
Available at: www.calctool.org/CALC/prof/bio/protein_size [Accessed 16 Oct. 2014].

[3] Q. Shu, C. Frieden et al. , "The E. coli CsgB nucleator of curli assembles to β-sheet oligomers that alter the CsgA fibrillization mechanism", Proc. Natl. Acad. Sci. 109, 6502-6507, 2012.

[4] E.A. Epstein, M.A. Reizian & M.R. Chapman, "Spatial clustering of the curlin secretion lipoprotein requires curli fiber assembly", J. Bacteriol. 191, 2, 2009.

[5] Gijsje H.Koenderink. Corianne C. ,Morphology and Persistence Length of Amyloid Fibrils areCorrelated to Peptide Molecular Structure, 14 Oct 2011 Journal of the American Chemical Society

@@ Line 8: / Line 8: @@
 <div class='grid_12'>
-<h2> Curli Module</h2>
+<h2>Cell Level Modeling</h2>
 <p>
-The goal of our project for the conductive curli module is to produce a biosensor that consists of <i>E. coli</i> that are able to build a conductive biofilm, induced by any promoter, in our case a promoter that gets activated in the presence of DNT/TNT. The biofilm consists of curli containing His-tags that can connect to gold nanoparticles. When the curli density is sufficiently high, a dense network of connected curli fibrils is present around the cells. Further increasing the amount of curli results in a conductive pathway connecting the cells, thereby forming conductive clusters. Increasing the amount of curli even further, sufficiently curli fibrils are present to have a cluster that connects the two electrodes and thus have a conducting system. <br>
+Now that the growth rate of curli and production of CsgB protein as function of time is obtained from the <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">Gene Level Model</a>, the conductance as a function of time can be computed for the cell. The approach we used for this is relatively simple: <br>
-The goal of the modeling of the curli module is to prove that our biosensor system works as expected and to capture the dynamics of our system. So, we want to answer the question: "Does a conductive path between the two electrodes arise at a certain point in time and at which time does this happen?" However, we not only want to answer the question if our system works as expected qualitatively, but we also want to make quantitative predictions about the resistivity between the two electrodes of our system in time.
-</p>
-<br>
-<p>
-The conductive curli module has different dynamics on different length scales:
 <ul>
 <li>
-The behavior of the system on the gene level, that is the dynamics of the activation of the promoter and the dynamics of the production of proteins needed for curli growth.
+We discretize the amount of curli subunits (\(CsgA_{curli}\) in the gene level model) and CsgB proteins that have to be added for each time step.
 </li>
 <li>
-The behavior of the system on the cell level, that is the curli production of each cell in time.
+At each time step, we add more curli subunits to growing curli fibrils. Also, we add more new curli fibrils to the model.
 </li>
 <li>
-The behavior of the system on the colony level, that is the change of the resistivity between the two electrodes of our system in time.
+From the density of the curli fibrils around the cell as a function of the radius, we calculate the conductive radius of the cell. The conductive radius is the largest radius where \(\rho_{curli}\), which represents the density of curli fibrils around the cell, is bigger than a certain threshold of curli density.
 </li>
 </ul>
 </p>
-<br>
-<p>
+<div class="tableofcontents">
-To capture the dynamics of our system, we have implemented a three-layered model, consisting of the gene level layer, the cell level layer and the colony level layer. <br>
-The gene level layer is used to determine characteristic parameters that will be used in the cell level layer. Subsequently, the cell level layer is used to determine characteristic parameters that will be used in the colony level layer. Lastly, the colony level layer is used to determine if our system works as expected, ie. determine if a conductive path between the two electrodes arises at a certain point in time and at which time this happens, and to determine the change of the resistivity between the two electrodes of our system in time. A figure of our three-layered model is displayed below.
-</p>
-<br>
+<ul>  <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli">
+                    <p>Curli Module</p>
+                    </a>
-<h2>Click in the figure to move to the corresponding page.</h2>
-<figure>
-<img src="https://static.igem.org/mediawiki/2014/f/fa/TU_Delft_2014_ModellingFigureCurli_small.png" width="800" height="920" usemap="#ModelMap">
-<map name="ModelMap">
-  <area shape="rect" coords="0,0,800,296" href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene" alt="Gene level">
-  <area shape="rect" coords="0,296,800,592" href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Cell" alt="Cell level">
-  <area shape="rect" coords="0,592,800,920" href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Colony" alt="Colony level">
-</map>
-<figcaption>
-Figure 0: A schematic view of our model. We aim to have a three layered model. Each level brings information to the next level. In the gene level, we calculate the curli production rates. In the cell level, we use this to calculate the curli growth over time. In the colony level, we use the curli growth to make predictions for the conductivity as function of time.
-</figcaption>
-</figure>
-<br>
-<p>
-<font color="red">summary of the conclusions</font>
-</p>
-<div class="tableofcontents">
-<center> <h3> Contents </h3> </center>
-<ul>
-   <li>
-        <a href="/Team:TU_Delft-Leiden/Modeling/Curli">
-        <p>Curli Module</p>
-        </a>
                 <ul>
                      <li>
-                     <a href="/Team:TU_Delft-Leiden/Modeling/Curli#Gene Level">
+                     <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">
                      <p>Gene Level Modeling</p>
                      </a>
                            <ul>
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#extendedgenelevel">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene#extendedgenelevel">
                                 <p>Extensive Gene Level Modeling</p>
                                 </a>
@@ Line 89: / Line 47: @@
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#simplifiedgenelevel">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene#simplifiedgenelevel">
                                 <p>Simplified Gene Level Modeling</p>
                                 </a>
@@ Line 97: / Line 55: @@
                      <li>
-                    <a href="/Team:TU_Delft-Leiden/Modeling/Curli#Cell Level">
                      <p>Cell Level Modeling</p>
-                    </a>
                            <ul>
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#discretization">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Cell#discretization">
                                 <p>Discretization of Gene Level Model</p>
                                 </a>
@@ Line 108: / Line 66: @@
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#building">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Cell#building">
                                 <p>Building the Curli Fibrils</p>
                                 </a>
@@ Line 114: / Line 72: @@
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#conductiveradius">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Cell#FittingCurliDensity">
+                               <p>Fitting the Curli Density</p>
+                               </a>
+                               </li>
+                               <li>
+                               <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Cell#conductiveradius">
                                 <p>Conductive Radius of the Cell</p>
                                 </a>
@@ Line 122: / Line 86: @@
                      <li>
-                     <a href="/Team:TU_Delft-Leiden/Modeling/Curli#Colony Level">
+                     <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Colony">
                      <p>Colony Level Modeling</p>
                      </a>
                            <ul>
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#percolation">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Colony#percolation">
                                 <p>Percolation</p>
                                 </a>
@@ Line 133: / Line 97: @@
                                 <li>
-                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli#resistivity">
+                                <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Colony#resistivity">
-                                <p>Resistivity</p>
+                                <p>Resistance</p>
+                               </a>
+                               </li>
+                               <li>
+                               <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Colony#recommendations">
+                               <p>Recommendations for product design and wet lab</p>
                                 </a>
                                 </li>
                            </ul>
+                                <li>
+                               <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Reflection">
+                               <p>Critical Reflection on our Model</p>
+                               </a>
+                               </li>
                      </li>
                 </ul>
-   </li>
+</ul>
-</ul>
 </div>
-<a name="Cell Level"></a>
-<h3>Cell Level Modeling</h3>
-<p>
-Now that the growth rate of curli and production of CsgB protein as function of time is obtained, the conductivity as a function of time can be computed. The relevant length scale is the cell length, or the micrometre scale. The approach we used for this is relatively simple: <br>
-<ul>
-<li>
-We discretize the amount of curli subunits (\(CsgA_{curli}\) in the gene level model) and CsgB proteins that have to be added for each time step.
-</li>
-<li>
-At each time step, we add more curli subunits to growing curli fibrils. Also, we add more new curli fibrils to the model.
-</li>
-<li>
-From the density of the curli fibrils around the cell as a function of the radius, we calculate the conductive radius of the cell.
-</li>
-</ul>
-</p>
-<br>
 <a name="discretization"></a>
-<h4>Discretization of Gene Level Model</h4>
+<h3>Discretization of Gene Level Model</h3>
 <p>
-We have discretized equations 5.2 and 12 in N time steps. These give the expected number of new CsgB proteins and curli subunits for each time step, as we plotted the solution of these two equations in figures 1 and 2. From these figures we determine the expected number of new CsgB proteins and curli subunits for each time step. However, a fundamental assumption in deterministic modeling is that the concentration is continuous. In reality, the amount of added curli subunits is discrete, since we cannot add half a curli subunit. <br>
+We have discretized equations 6.2 and 12 of the <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">Gene Level Model</a> in 1000 discrete times between 0 hour and 10 hours, so each time step is equal to 36 seconds. Throughout the model, we use the same rates as shown in Table 1 of the Gene Level Model (unless otherwise specified). These equations give the expected number of new CsgB proteins and curli subunits for each time step, as we plotted the solution of these two equations in figures 1 in Section <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">Gene Level Modeling</a>. From these figures, we determine the expected number of new CsgB proteins and curli subunits for each time step. However, a fundamental assumption in deterministic modeling is that the concentration is continuous. In reality, the amount of added curli subunits is discrete, since we cannot add half a curli subunit. <br>
 Furthermore, in the gene level model we did not take into account the statistical variation of gene transcription and adding of curli subunits; sometimes less and some times more curli subunits are added with respect to the expected value. To include this in the cell level model, we drew the amount of new curli subunits from a Poisson distribution where λ equals the expected amount of added subunits. <br>
-So, for each time step we now have \(B_n\) new CsgB proteins and \(C_n\) new curli subunits, where \(C_n\) varies for each time step, as it is drawn from a Poisson distribution. An assumption of this distribution is that the time at which a new curli subunit is added, is uncorrelated to the time at which the previous curli subunit was added, we think this is a fair assumption. Note that the cell level model we made, accounts for the stochasticity of adding curli subunits, but not for the stochasticity of gene expression, so for the production of CsgB protein. The value \(B_n\) and the Poisson distribution are determined from figures 1 and 2. We have used 1000 discrete times between 0 hr and 10 hr.
+So, for each time step we now have \(B_n\) new CsgB proteins and \(C_n\) new curli subunits, where \(C_n\) varies for each time step, as it is drawn from a Poisson distribution. An assumption of this distribution is that the time at which a new curli subunit is added, is uncorrelated to the time at which the previous curli subunit was added, we think this is a fair assumption. Note that the cell level model we made, accounts for the stochasticity of adding curli subunits, but not for the stochasticity of gene expression, so for the production of CsgB protein. The value \(B_n\) and the Poisson distribution are determined from figure 1 in the <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">Gene Level Modeling</a> section.
-<font color="red">say something about the time steps, how much time represents each step and determine Bn and Cn from figures</font>
 </p>
@@ Line 181: / Line 130: @@
 <a name="building"></a>
-<h4>Building the Curli Fibrils</h4>
+<h3>Building the Curli Fibrils</h3>
 <p>
-Firstly, \(B_n\)  CsgB proteins are added to our model that mark the starting points for new curli fibrils. These new curli fibrils are located at random points on a sphere with radius r, which represents the cell. The radius r is chosen such that the volume of the cell is\(\ \sim 1.1 \ \mu m^3\) [5]. A CsgB protein is modeled by a line of length 4 nm that points radially outward, perpendicular to the cell surface <font color="red">[source]</font>. In reality, the distribution of CsgB on the cell surface is not uniformly distributed [6]. However, we assumed uniformly distributed CsgB to keep our model prehensile. This is a point that may be used to further improve the model.
+Firstly, \(B_n\)  CsgB proteins are added to our model that mark the starting points for new curli fibrils. These new curli fibrils are located at random points on a sphere with radius r, which represents the cell. The radius r is chosen such that the volume of the cell is\(\ \sim 1.1 \ \mu m^3\) [1]. A CsgB protein is modeled by a line of length 4 nm that points radially outward, perpendicular to the cell surface. In reality, the distribution of CsgB on the cell surface is not uniformly distributed [4] and cells are not perfectly round. However, we assumed uniformly distributed CsgB to keep our model prehensile. This is a point that may be used to further improve the model.
 </p>
@@ Line 193: / Line 142: @@
 <ul>
 <li>
-Firstly, a random curli fibril is selected, e.g. curli number k. A curli fibril is represented by a 3 (the x, y and z coordinates) by l+1 matrix, where l is the amount of curli subunits of the curli fibril and the origin is chosen to be the center of the sphere. Thus, by storing the ending coordinates of each curli subunit, we know the starting and end coordinates of each curli subunit. The curli subunits are modeled by a line of length 4 nm <font color="red">[source]</font>.
+Firstly, a random curli fibril is selected, e.g. curli number k. A curli fibril is represented by a 3 (the x, y and z coordinates) by l+1 matrix, where l is the amount of curli subunits of the curli fibril and the origin is chosen to be the center of the sphere. Thus, by storing the ending coordinates of each curli subunit, we know the starting and end coordinates of each curli subunit. The curli subunits are modeled by a line of length 4 nm.
 </li>
 <li>
@@ Line 199: / Line 148: @@
 </li>
 <li>
-Thirdly, the new curli subunit has a small angular deviation with respect to the previous one. This polar angle \(\theta_{2}\) is chosen from a Gaussian distribution with parameters N(0,σ). σ is chosen such that the persistence length, the distance over which a fibril has bend by \(90^{\circ}\) and has ‘lost’ its directional information, is 4 µm. The azimuthal angle ϕ is completely random between 0 and 2π radians, and chosen from an uniform distribution.
+Thirdly, the new curli subunit has a small angular deviation with respect to the previous one. This polar angle \(\theta_{2}\) is chosen from a Gaussian distribution with parameters N(0,σ). σ is chosen such that the persistence length, the distance over which a fibril has bend by \(90^{\circ}\) and has ‘lost’ its directional information, is 4 µm [5]. The azimuthal angle ϕ is completely random between 0 and 2π radians, and chosen from an uniform distribution.
 </li>
 <li>
@@ Line 210: / Line 159: @@
 <p>
-The angular deviation σ is a critical parameter in our model. Increasing this value increases the flexibility of our curli, where decreasing this value increases the stiffness of the curli. This is shown in figure 3. If the length of one subunit is 4 nm and the total persistence length is 4 µm, then \(\sigma = \ 3.47^{\circ}\). Furthermore, we think that it is justified to add the curli subunits one at a time to a random curli. We expect no discrimination of the CsgA proteins for binding to a large or small curli or one that has recently gotten a new curli subunit.
+The angular deviation σ is a critical parameter in our model. Increasing this value increases the flexibility of our curli, where decreasing this value increases the stiffness of the curli. This is shown in figure 1. If the length of one subunit is 4 nm and the total persistence length is 4 µm, then \(\sigma = \ 3.47^{\circ}\). Furthermore, we think that it is justified to add the curli subunits one at a time to a random curli. We expect no discrimination of the CsgA proteins for binding to a large or small curli or one that has recently gotten a new curli subunit.
 </p>
@@ Line 216: / Line 165: @@
 <img src="https://static.igem.org/mediawiki/2014/a/ab/TUDelft_2014_AnglePersistence.png" width="60%" height="60%">
 <figcaption>
-Figure 3: The persistence length in number of units of a curli fibril as function of the angular deviation per subunit in degrees.
+Figure 1: The persistence length in number of units of a curli fibril as function of the angular deviation per subunit in degrees.
 </figcaption>
 </figure>
@@ Line 223: / Line 172: @@
 <p>
-An illustrative view of what our cell looks like during the adding of curli subunits is shown in figure 4. This figure is created when just a few curli were added (\( \sim 1/2 \ hour\)). A similar figure after \(t = \ 10 \ hr\) would look like a fuzzy ball of curli.
+An illustrative view of what our cell looks like during the adding of curli subunits is shown in figure 2.
 </p>
 <figure>
-<img src="https://static.igem.org/mediawiki/2014/5/5b/TUDelft_2014_fuzzycurlicell.png" width="100%" height="100%">
+<img class="theimage" src="https://static.igem.org/mediawiki/2014/e/e3/TU_Delft_2014_Curli_picturestill.png" alt="meh" width="60%" height="60%"/>
+<img class="theimage2 hidden" src="https://static.igem.org/mediawiki/2014/3/37/TU_Delft_2014_Curli_picture.gif" alt="meh" width="60%" height="60%"/>
 <figcaption>
-Figure 4: Schematic view of our cell at t=1/2 hr after initiation (black sphere centred at x=y=z=0). The wires represent the curli fibrils. The labels on the axis are in meter.
+Figure 2: Schematic view of our cell (black sphere centred at x=y=z=0) with growing curli fibrils. The wires represent the curli fibrils. <b> Click to play!</b>
 </figcaption>
 </figure>
+<script>
+$(".theimage").click(function() {
+$(this).addClass('hidden');
+$(this).next().removeClass('hidden');
+});
+$(".theimage2").click(function() {
+ $(this).addClass('hidden');
+$(this).prev().removeClass('hidden');
+});
+</script>
 <br>
+<p>
+Now that we have a model of a cell with growing curli, we want to extract relevant data for the colony level modeling. Ideally, the resistance as function of radius and time would be calculated by looking at connections between the curli fibrils. However, this requires insight of the behavior of the curli on the nanoscopic scale. For instance, what is the conductance of a single curli fibril with gold nanoparticles and what is the critical distance between the fibrils that make them connect? After an extensive literature study, we have decided to simplify this model. Furthermore, when interactions between the curli fibrils have to be taken into account, the model gets too computationally expensive.
+</p>
 <br>
 <p>
-<font color="red">[write something about the part where we tried the percolation on this level], low priority</font>
+To have a reasonable computational time, we decided to extract our parameters for the colony level modeling from the curli density around the cell. Figure 3 shows the length of all curli after 10 hours. Curli fibrils that are created first (low numbers) are much longer than the ones that are created last (high number). The steep drop in curli fibril length for the first couple of hundred fibrils is a consequence of the peak in curli production between 0 hour and 2 hours. After that, the curli length is linear with the time it has existed, precisely what you expect from the model.
 </p>
-<br>
-One thing interesting thing to look at is the length of the curli fibrils at t=10 hr, shown in figure 6. Curli fibrils that are created first (low numbers) are much longer than ones that are created last (high number). The steep drop in curli fibril length for the first couple of hundred fibrils comes from the peak in curli production between 0 hr and 2 hr. After that, and the curli length is linear with the time it has existed, precisely what you'd expect from the model.
 <figure>
 <img src="https://static.igem.org/mediawiki/2014/f/f1/TUDelft_2014_Curli_Length.png" width="60%" height="60%">
 <figcaption>
-Figure 6: The length of the curli fibrils in number of subunits on the y-axis at t=10 hours. On the x-axis is the time. A dot at height 1000 at 1 hour means that the curli fibril that was started at t=1 hour had length 1000 at time=10 hours.
+Figure 3: The length of the curli fibrils in number of subunits on the y-axis at t=10 hours. On the x-axis is the time. A dot at height 1000 at 1 hour means that the curli fibril that was started at t=1 hour had length 1000 at t=10 hours.
 </figcaption>
 </figure>
@@ Line 256: / Line 218: @@
 <br>
+<p>
+Looking back at figures 1 and 2 in Section <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Curli/Gene">Gene Level Modeling</a>, the fact that, as can be seen in figure 3, the first curli are much longer that the later ones, can be explained by the fact that there is relatively large curli growth in the beginning, because few CsgB have been produced and therefore, only a few curli fibrils are available for CsgA proteins. After a couple of hours there are more CsgB proteins, thus more curli fibrils, but CsgA protein production does not increase. Therefore, the ratio [CsgA]/[curli fibrils] is much smaller than in the beginning and each curli will grow much slower. A consequence of this is that the ‘newer’ curli fibrils are much shorter.
-<p>
-Since adding curli on the colony level would result in unreasonable computation times, we decided to extract our parameters for the colony level modeling from the curli density around the cell. Figure 5 contains a histogram with the amount of curli subunits as a function of the cell radius after 10 hours. <font color="red">[add figure]</font> Note how no curli are found below the actual cell radius. It can be seen from the figure that there is a large peak, followed by a plateau. When this histogram is observed in time, you would notice that at first large curli are being created. Figure 6 shows the length of all curli after 10 hours.
 </p>
@@ Line 267: / Line 228: @@
-<a name="Extracting information for the colony level. "></a>
+<a name="FittingCurliDensity"></a>
-<h4> Extracting information for the colony level. </h4>
+<h3> Fitting the Curli Density </h3>
 <p>
-Now that we have a model of a cell with growing curli, we want to extract relevant data for the colony level modeling. Ideally, the resistance as function of radius and time would be calculated by looking at connections between the curli fibrils. However, this requires insight of the behavior of the curli on the nanoscopic scale. For instance, what is the conductivity of a single curli fibril with gold nanoparticles and what is the critical distance between the fibrils that make them connect? Furthermore, when interactions between the curli fibrils have to be taken into account, the model becomes computationally too expensive. After an extensive literature study, [] we have decided to simplify this model.  The simplest approach is by saying there is a critical density of curli that is needed to make connections. Also we tried to parametrize the curli density for more quantitative results.
+We think that a reasonable first approximation of the conductance is the density of the curli around the cell as a function of the distance from the membrane. When the density is higher, there are more gold particles, thus higher conductance. In our simplest approach we say that there is a critical density \(\rho_{crit}\) of curli that is needed to have conductance. The density \(\rho_{curli}\) decreases as function of the radius. The largest radius where \(\rho_{curli} > \rho_{crit}\), we call the conductive radius \(r_{cond}\). With only this simple approximation, we can calculate some interesting properties of our system at the colony level: the time at which we expect percolation to happen and the resistance of our system. Though this approximation seems to be rather arbitrary, we do have some reasoning for this:
+<ul>
+<li>
+First of all, the goal of this parameter is to get information about our system that will be calculated in colony level modeling. We use this parameter in colony level modeling to find connections between cells. To have a continuous path from one electrode to the other electrode, we must have a lot of cells that are connected to each other. In order to know when cells are connected to each other, we have to assume that everything at a certain radius from the cell is conductive; for this radius we use the critical density \(\rho_{crit}\). However, for this to be true the fibrils on one side of the cell must be connected to the fibrils on the other side. The <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Techniques#PercolationTheory">Percolation Theory</a> prescribes that this is a sharp transition as a function of the density, so we can choose \(\rho_{crit}\) in such a way that we are very sure that everything at \(\rho_{crit}\) from the cell is conductive.
+</li>
+<li>
+While the precise value of \(\rho_{crit}\) may be unknown and should be measured, we think that we can still get plenty of information about the qualitative behavior of our system in advance, for instance we can investigate the sharp transition at which the conductance increases at the colony level.
+</li>
+<li>
+Due to the simplifications that we made in order to be able to model our system, we cannot include interactions or cluster forming between the curli themselves. Using \(\rho_{crit}\), we have an elegant way to filter out modeling errors.
+</li>
+</ul>
 </p>
+<br>
 <p>
-Our model is subject to stochastic processes. Therefore, to acquire enough <i>in silico</i> results, we have repeat the script that builds the curli fibrils for 10 hours a hundred times. This should give us insight in the variation we might expect. Figure 9 displays the curli density at \(\ t= \ 2 \ hours\) for all cells. in the left figure. The orange line represents the average of the simulations. It can be concluded that the intercellular variation is relatively small. This makes sense, since the relative deviation of stochastic processes decreases with the sample size. In the right figure, the mean and standard deviation of the curli density as a function of the radius is shown. <font color="red">insert caption</font>
+As the building of the curli fibrils is a stochastic processes, we repeated our simulations on the cell level many times in order to get statistically valid results for the mean and standard deviation of \(\rho_{curli}\) and \(r_{cond}\). <br>
+When we ran our simulation 100 times, we got the results displayed in figure 4. Figure 4 displays the curli density at \(\ t= \ 2 \ hours\) for all cells in the left figure. This should give us insight in the variation we might expect. In the right figure, the orange line represents the mean curli density, and the green lines represent the standard deviation. From figure 4, we conclude that the intercellular variation is relatively small. This makes sense, since the relative deviation of stochastic processes decreases with the sample size.
 </p>
@@ Line 282: / Line 256: @@
 <img src="https://static.igem.org/mediawiki/2014/d/d5/TUDelft_2014_2h_curli_density.png" width="60%" height="60%">
 <figcaption>
-Figure 9: Left) The curli density in curli units \(  \mu m ^{-3} \) as function of radial distance from the centre of the cell in \( \mu m\) for 100 different simulations at t=2 hr. The orange line represents the mean of all densities. Right) The orange line represents the mean curli density, and the green lines represent the variation within the simulations.
+Figure 4: Left) The curli density in curli units \(  \mu m ^{-3} \) as function of radial distance from the centre of the cell in \( \mu m\) for 100 different simulations at t=2 hr. The orange line represents the mean of all densities. Right) The orange line represents the mean curli density, and the green lines represent the standard deviation.
 </figcaption>
 </figure>
@@ Line 289: / Line 263: @@
 <p>
-It is also interesting to study curli density as function of time at different times, shown in figure 10. This figure shows that, corresponding with what we have seen previously, \(\rho_{curli}\) decreases as a function of the radius. Also, it decreases faster as a function of the radius in the first two hours. After two hours, we can see that the curli density increases only for small r, as mainly short curli are added to the system. This agrees with our previous results.
+It is also interesting to study curli density as function of time at different times, shown in figure 5. This figure shows that, corresponding with what we have seen previously, \(\rho_{curli}\) decreases as a function of the radius. Also, it decreases faster as a function of the radius in the first two hours. After two hours, we can see that the curli density increases only for small r, as mainly short curli are added to the system. This agrees with our previous results.
 </p>
 <figure>
-<img src="https://static.igem.org/mediawiki/2014/c/c4/TU_Delft_2014_curli_density.png" width="70%" height="70%">
+<img src="https://static.igem.org/mediawiki/2014/c/c4/TU_Delft_2014_curli_density.png" width="60%" height="60%">
 <figcaption>
-Figure 10: The mean curli density in curli units \( \mu m ^{-3} \) as function of radial distance from the centre of the cell in \( \mu m\), plotted at different times (.5 hr, 1hr, 2hr, 5hr and 10hr).</figcaption>
+Figure 5: The mean curli density in curli units \( \mu m ^{-3} \) as function of radial distance from the centre of the cell in \( \mu m\), plotted at different times (.5 hr, 1hr, 2hr, 5hr and 10hr).</figcaption>
 </figure>
 <br>
-</p>
-<a name="conductiveradius"></a>
-<h5>Conductive Radius of the Cell</h5>
 <p>
-We think that a reasonable first approximation of the conductivity is the density of the curli around the cell as a function of the radius. When the density is higher, there are more gold particles, thus higher conductivity. In our simplest approach we say that there is a critical density \(\rho_{crit}\) of curli that is needed to have conductivity. The density \(\rho_{curli}\) decreases as function of the radius. The largest radius where \(\rho_{curli} > \rho_{crit}\), we call the conductive radius \(r_{cond}\). Let's take a look at what this would look like from figure 10. If the critical density would be \( 1 \cdot 10^3\), then at 30 min, the conductive radius would be \(\approx 2.5 \mu m \), at 1 hour it would be \( \approx 4.5 \mu m \) and at 2 hours it would be \( \approx 5 \mu m \). How this looks for 100 different cells is shown in figure 12. With only this simple approximation we can calculate some interesting properties of our system: the time at which we expect percolation to happen and the resistivity of our system. Though this approximation seems to be rather arbitrary, we do have some reasoning for this:
+In order to be able to say something about the resistance of our system at the colony level, we need an analytical expression for \(\rho_{curli}\). We have therefore fitted the function
-<ul>
+$$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}} \tag{1}$$
-<li>
+ to our curli density curves at each time \( n \),see figure 6 the red line. Here, \(C_{1_n} \), \(C_{2_n} \), \(C_{3_n} \) and \( C_{4_n} \) are parameters that have to be fitted, and \( r \) is the distance from the centre of the cell. At first we tried to fit our data to only the first term (green line). It can clearly be seen in the figure that this does not adequately capture the dynamics of the curve. Either the approximation is bad at short distances or at large distances.
-First of all, the goal of this parameter is to get information about our system that will be calculated in colony level modeling. We use this parameter in colony level modeling to find connections between cells. To have a continuous path from one electrode to the other electrode, we must have a lot of cells that are connected to each other. In order to know when cells are connected to each other, we have to assume that everything at a certain radius from the cell is conductive; for this radius we use the critical density \(\rho_{crit}\). However, for this to be true the fibrils on one side of the cell must be connected to the fibrils on the other side. The <a href="https://2014.igem.org/Team:TU_Delft-Leiden/Modeling/Techniques#PercolationTheory">Percolation Theory</a> prescribes that this is a sharp transition as a function of the density, so we can choose \(\rho_{crit}\) in such a way that we are very sure that everything at \(\rho_{crit}\) from the cell is conductive.
+</p>
-</li>
-<li>
-While the precise value of \(\rho_{crit}\) may be unknown and should be measured, we think that we can still get plenty of information about the qualitative behaviour of our system in advance. Figure 8 at the bottom shows the conductive radius \(r_{cond}\) as function of time using \(\rho_{crit}\) as shown in figure 8 in the middle as the red line. Increasing or decreasing \(\rho_{crit}\) would result in a similar \(r_{cond}\) as function of time. Hence, the qualitative behaviour is preserved.
-</li>
-<li>
-Due to the simplifications that we made in order to be able to model our system, we cannot include interactions or cluster forming between the curli themselves. Using \(\rho_{crit}\), we have an elegant way to filter out modeling errors.
-</li>
-</ul>
-</p>
-<br>
 <figure>
-<img src="https://static.igem.org/mediawiki/2014/e/e1/TUDelft_2014_cellradius.png" width="70%" height="70%">
+<img src="https://static.igem.org/mediawiki/2014/3/3f/TUDelft_2014_FittingCurli.png" width="60%" height="60%">
 <figcaption>
-Figure 12: The green lines are the conductive radius plotted versus the time for 100 cells with a critical density of \( \rho_{crit}=1204 \) curli subuntis \( \mu m ^{-3} \).  The orange red represents the mean conductive radius. A sharp increase in the conductive radius can be observed for \(t < 1 \ hour\), and after \(t = \ 1 \ hour\) the conductive radius increases slowly. The cellular variation in the second regime is relatively large, as is shown by the dark blue lines that represent two standard deviations from the mean. Note how the conductive radius increases in discrete steps. This is a result of the fact that density is a parameter that only exists over a certain volume. We have divided the volume around the cell in hollow spheres with thickness \( dr=0.08 \mu m \). Increasing would increase the accuracy over the mean, but would decrease the spatial volume. Decreasing this would increase the variation between the conductive radii, but would increase the spatial volume.
+Figure 6: Blue line: Right behind the red line, at t=5 hr the mean of all density curves. Green line: a weighted fit of \( \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} \). Red line: A fit \( \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}} \) to the blue line.
 </figcaption>
 </figure>
@@ Line 337: / Line 291: @@
 <br>
+<p>
+It can be seen that the fit is certainly not perfect, but it is a reasonable approximation to the characteristics. The reason for fitting such a simple function is that, in the colony level, we need to quantify the conductance between the cells. The integral for this rather complicated and we need an analytical function for \(\rho_{curli}\) to analytically solve this integral. In further research, we could improve our fit by fitting a set of decaying exponents.
+</p>
+<br>
+<a name="conductiveradius"></a>
+<h3>Conductive Radius of the Cell</h3>
 <p>
-Different values of \(\rho_{crit}\) result in different characteristic curves for \(r_{cond}\), see figure 11. In this figure, we set \(\rho_{crit}\) equal to a fraction of the maximum \( \rho_{curli} \) (\( 1.2 \cdot 10^5 \# \mu m^{-3} \) ) as observed in figure 10. So, we set \( \rho_{crit} = \max{ (\rho) } /K \) for the \( K \) shown in the legend.
+Different values of \(\rho_{crit}\) result in different characteristic curves for \(r_{cond}\), see figure 7. In this figure, we set \(\rho_{crit}\) equal to a fraction of the maximum \( \rho_{curli} \) (\( 1.2 \cdot 10^5  \ \# \ \mu m^{-3} \) ) as observed in figure 5. So, we set \( \rho_{crit} = \max{ (\rho) } /K \), for the \( K \) shown in the legend.
 </p>
-<br>
 <figure>
-<img src="https://static.igem.org/mediawiki/2014/e/ec/TUDelft_2014_Different_Critical_values.png" width="80%" height="80%">
+<img src="https://static.igem.org/mediawiki/2014/e/ec/TUDelft_2014_Different_Critical_values.png" width="60%" height="60%">
 <figcaption>
-Figure 11: The conductive radius in \( \mu m \) versus the time from t=0 to 10 hr for different values of \( \rho_{crit} \). The thick lines represent the mean conductive radius of 100 cells with a \( \rho_{crit} \) equal to to a fraction of the maximum ( \( 1.2 \cdot 10^5 \# \mu m^{-3} \) ) corresponding with the legend. The thinner lines of the same color are the mean \( \pm \) the standard deviation.
+Figure 7: The conductive radius in \( \mu m \) versus the time from t=0 to 10 hour for different values of \( \rho_{crit} \). The thick lines represent the mean conductive radius of 100 cells with a \( \rho_{crit} \) equal to to a fraction of the maximum ( \( 1.2 \cdot 10^5 \# \mu m^{-3} \) ) corresponding with the legend. The thinner lines of the same color are the mean \( \pm \) the standard deviation.
 </figcaption>
 </figure>
@@ Line 358: / Line 314: @@
 <p>
-From figure 11, we conclude that low values of \(\rho_{crit}\) result in a sharp increase of \(r_{cond}\) followed by a steady, slow increase of \(r_{cond}\) in time. During the steady, slow increase of \(r_{cond}\) in time, the cellular variation is relatively large. For high values of \(\rho_{crit}\), there is a delayed sharp increase of \(r_{cond}\) and less cellular variation. Unfortunately we have no wetlab data to fit this parameter. We can speculate however. A a conductive radius of more than 5 \( \mu m \) seems unlikely to us, for the cell diameter is only a micron. We set the critical value to \( 1.2 \cdot 10^5 \ \# \ \mu m^{-3} \). Even though this value might be off by a factor, we claim that this will change little in what we try to achieve in this approximation, namely that there is a sharp transition at which the conductivity increases.
+From figure 7, we conclude that low values of \(\rho_{crit}\) result in a sharp increase of \(r_{cond}\) followed by a steady, slow increase of \(r_{cond}\) in time. During the steady, slow increase of \(r_{cond}\) in time, the cellular variation is relatively large. For high values of \(\rho_{crit}\), there is a delayed sharp increase of \(r_{cond}\) and less cellular variation.
 </p>
+<br>
-<h5>Parametrization of the curli density</h5>
 <p>
-We aim to not only say something about the moment of percolation, but also predict the conductivity as function of time. Using a conductive radius captures only little information of our simulations. We have therefore fitted the function
+From figure 7, we conclude that low values of \(\rho_{crit}\) result in a sharp increase of \(r_{cond}\) followed by a steady, slow increase of \(r_{cond}\) in time. During the steady, slow increase of \(r_{cond}\) in time, the cellular variation is relatively large. For high values of \(\rho_{crit}\), there is a delayed sharp increase of \(r_{cond}\) and less cellular variation. Unfortunately we have no wetlab data to fit this parameter. However, we can argue what kind of behavior we would expect from \(r_{cond}\). <br>
-$$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} \tag{13} $$
+A conductive radius of more than 5 \( \mu m \) seems unlikely to us, for the cell's diameter is only a micron. We set the \(\rho_{crit}\) equal to \( 1.2 \cdot 10^5 \ \# \ \mu m^{-3} \). Even though this value might be off by a factor, we argue that this will change little in what we try to achieve in our model, namely to investigate the sharp transition at which the conductance increases at the colony level.
- to our curli density curves (see figure 13) at each time \( n \). Here \(C_{2_n} \) and \( C_{2_n} \) are parameters that have to be fitted, and \( r \) is the distance from the cell centre. A weighted fitting method is used, where the weights are inversely proportional to the variance of the density (green lines).
+</p>
-</p>
 <figure>
-<img src="https://static.igem.org/mediawiki/2014/3/3f/TUDelft_2014_FittingCurli.png" width="80%" height="80%">
+<img src="https://static.igem.org/mediawiki/2014/e/e1/TUDelft_2014_cellradius.png" width="60%" height="60%">
 <figcaption>
-Figure 13: Orange: The cell sensitivity as function of time with the standard deviation (green lines). The black line is a weighted fit of \( \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} \).
+Figure 8: The green lines are the conductive radius plotted versus the time for 100 cells with a critical density of \( \rho_{crit}=1204 \) curli subuntis \( \mu m ^{-3} \).  The orange red represents the mean conductive radius and the dark blue lines represent two standard deviations from the mean.
 </figcaption>
 </figure>
+<br>
 <p>
-It can be seen that the fit is certainly not perfect, but it a reasonable approximation of the characteristics. The reason for fitting such a simple function is that, in the colony level, we need to quantify the conductivity between the cells. The integral for this rather complicated. In further research, we could improve our fit by fitting a set of decaying exponents.
+We conclude from figure 8 that a sharp increase in the conductive radius can be observed for \(t < 1 \ hour\), and after \(t = \ 1 \ hour\) the conductive radius increases slowly. The cellular variation in the second regime is relatively large, as is shown by the dark blue lines that represent two standard deviations from the mean. Note how the conductive radius increases in discrete steps. This is a result of the fact that density is a parameter that only exists over a certain volume. We have divided the volume around the cell in hollow spheres with thickness \( dr=0.08 \mu m \). Increasing the thickness would increase the accuracy over the mean, but would decrease the spatial volume. Decreasing the thickness would increase the variation between the conductive radii, but would increase the spatial volume.
-<p>
+</p>
+<br>
 <h3> References </h3>
+<p>[1] H.E. Kubitschek & J.A. Friske, "Determination of bacterial cell volume with the Coulter Counter", J. Bacteriol. 168, 3, 1986.</p>
-<p>
+<p>[2] calctoolo.org, (2014). Calctool. [online] <br>
-<font color="red">still has to be made</font>
+Available at: <a href="www.calctool.org/CALC/prof/bio/protein_size">www.calctool.org/CALC/prof/bio/protein_size</a> [Accessed 16 Oct. 2014].
 </p>
+<p>[3] Q. Shu, C. Frieden <i> et al. </i>, "The E. coli CsgB nucleator of curli assembles to β-sheet oligomers that alter the CsgA fibrillization mechanism", Proc. Natl. Acad. Sci. 109, 6502-6507, 2012.</p>
+<p>[4] E.A. Epstein, M.A. Reizian & M.R. Chapman, "Spatial clustering of the curlin secretion lipoprotein requires curli fiber assembly", J. Bacteriol. 191, 2, 2009.</p>
+<p>[5] Gijsje H.Koenderink. Corianne C. ,Morphology and Persistence Length of Amyloid Fibrils areCorrelated to Peptide Molecular Structure, 14 Oct 2011 Journal of the American Chemical Society</p>
 </div>