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

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Cell Level Modeling

Now that the growth rate of curli and production of CsgB protein as function of time is obtained, 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.

Using the growth rate of curli and production of CsgB protein as function of time obtained from the gene level model, we obtained an analytical expression for $\rho_{curli}$, which represents the density of curli fibrils around the cell. We have fitted the function $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}}$$ to our curli density curves at each time $ n $, see figure 6. 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.
We also calculated the conductive radius of the cell as a function of the radius, see figure 8. We use the conductive radius in the colony level to determine when a conductive path between the two electrodes of our system arises.

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 figures 1 Section Gene Level Modeling.

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 [5] [6]. In reality, the distribution of CsgB on the cell surface is not uniformly distributed [4]. 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 [source].
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. 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. 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.make gif

Figure 2: 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.

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 radius. 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 $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}}$$ to our curli density curves (see figure 6) at each time $ n $. 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. A weighted fitting method is used, where the weights are inversely proportional to the variance of the density (green lines). At first we tried to fit our data to only the first term. 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.

@@ Line 28: / Line 28: @@
 <p>
-Using the growth rate of curli and production of CsgB protein as function of time obtained from the gene level model, we obtained an analytical expression for \(\rho_{curli}\). We have therefore fitted the function
+Using the growth rate of curli and production of CsgB protein as function of time obtained from the gene level model, we obtained an analytical expression for \(\rho_{curli}\), which represents the density of curli fibrils around the cell. We have fitted the function
 $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}}$$
   to our curli density curves at each time \( n \), see figure 6. 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. <br>