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

Cell Level Modeling

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:

• 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.

summary of the conclusions

Discretization of Gene Level Model

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.
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 and 2. We have used 1000 discrete times between 0 hr and 10 hr. say something about the time steps, how much time represents each step and determine Bn and Cn from figures

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\(\ \sim 1.1 \ \mu m^3\) [5]. A CsgB protein is modeled by a line of length 4 nm that points radially outward, perpendicular to the cell surface [source]. 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.

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 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.

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.

[write something about the part where we tried the percolation on this level], low priority

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.

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. [add figure] 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.

Extracting information for the colony level

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.

Our model is subject to stochastic processes. Therefore, to acquire enough in silico 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. insert caption

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.

Conductive Radius of the Cell

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:

• 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 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.
• 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.

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.

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.

Parametrization of the curli density

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 $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} \tag{13} $$ 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).

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.

References

still has to be made