Team:TU Delft-Leiden/Modeling/Curli

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Curli Module

The goal of our project for the conductive curli module is to produce a biosensor that consists of E. coli 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, see the gadget section of our wiki and the extracellular electron transport (EET) module. The biofilm consists of curli containing His-tags that can connect to gold nanoparticles, see the conductive curli module. 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.
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 resistance between the two electrodes of our system in time.

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:

At the gene level, we calculate the curli subunits production rates and curli subunit growth that will be used in the cell level.
At the cell level, we use these production and growth rates to calculate the curli growth in time, which we will use at the colony level.
At the colony level layer, we 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. We also determine the change of the resistance between the two electrodes of our system in time.

A figure of our three-layered model is displayed below.

Click in the figure to move to the corresponding page.

Figure 1: A schematic view of our model, which is a three-layered model. Each layer determines characteristic parameters for the layer above it. At the gene level, we calculate the curli subunits production rates. At the cell level, we use these production rates to calculate the curli growth in time. At the colony level, we use the curli growth in time to determine the change of the resistance between the two electrodes of our system in time.

We start with the modeling of the gene expression of proteins involved in the curli formation pathway at the gene level. In the constructs we made in the wet lab, CsgA is continuously being produced and the CsgB gene is placed under the control of a landmine promoter, activated by either TNT or DNT, see the Landmine Detection Module. So, when the cells get induced by TNT or DNT, CsgB protein production will get started and CsgA will already be present in the system, as CsgA is continuously being produced. We first modeled this system by constructing an extensive gene expression model of the curli formation pathway. Subsequently, we simplified this model, so less parameters were needed.
Based on the simplified model, we made a plot of the curli growth as function of time for different initial concentrations of $CsgA_{free}$, see figure 2. We conclude the following from this figure:

Firstly, as expected, curli growth stabilizes to a rate equal to $p_{A}$ after approximately 2 hours, independent of the initial concentration of $CsgA_{free}$. The width of this peak is determined by the product $ k p_B$, where k is the production rate of curli and $ p_B$ is the production rate of CsgB proteins.
Secondly, increasing the initial concentration of $CsgA_{free}$, increases the height of the peak. Even with zero initial $CsgA_{free}$ concentration, a small peak can be found at one hour. This is a consequence of $CsgA_{free}$ build-up when the CsgB concentration is still very small.
Thirdly, during the first two hours, few CsgB proteins are present in the system. We therefore expect that the length of the curli fibrils that started in the first few hours are much longer than the fibrils that started at later times.

Figure 2: The curli subunit growth in units per second for various initial concentrations $ A_0 $ of CsgA as function of time. Initial concentrations that equal 0, 5, 10 or 15 hours of CsgA production are shown.

Using the growth rate of curli and production of CsgB protein as function of time obtained from the gene level model, the conductance as a function of time can be computed for the cell. We obtained an analytical expression for $\rho_{curli}$, which represents the density of curli fibrils around the cell. We have fitted the function $$ \rho_n = C_{1_n} e^{-\frac{r}{C_{2_n}}} + C_{3_n} e^{-\frac{r}{C_{4_n}}} \tag{1}$$ to our curli density curves at each time $ n $, see figure 3 the red line. At first, we tried to fit our data to only one exponential term (green line). It can clearly be seen in the figure that this does not adequately capture the dynamics of the curve. However, equation 1 gives a very good fit to the curli density curves at each time $ n $. 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 4. The conductive radius is the largest radius where $\rho_{curli}$ is bigger than a certain threshold of curli density. We use the conductive radius in the colony level to determine when a conductive path between the two electrodes of our system arises.

Figure 3: 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.

Figure 4: 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.

An illustrative view of what our cell looks like during the adding of curli subunits is shown in figure 5.

meh — Figure 5: 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 5: 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 determined values for $\rho_{curli}$ and $r_{cond}$ at the cell level, we can finally predict if our system works as expected and capture the dynamics of our system. Firstly, we want to prove that our system works as expected. So, we want to predict if a conductive path between the two electrodes arise at a certain point in time and at which time this happens. Secondly, we not only want to answer the question if our system works as expected with a yes or no answer, but we also want to make quantitative predictions about the resistance between the two electrodes of our system in time. We do this by modeling the curli growth on the colony level; each cell is now visualized and has curli growth. Now, we have come up with two different approaches:

Firstly, we let the cells increase their conductive radius in time, according with our findings on the cellular level (figure 4). A connection is created from one electrode to the other electrode when there is a conductive path between them. Conductive paths consists of cells that have a connection between each other; cells connect when there is an overlap between their conductive radius. This problem is very similar to problems in percolation theory. From this, we can make conclusions about how our system works in an experimental setting.
Secondly, we also want to make quantitative predictions about the resistance between the two electrodes of our system in time. Therefore, we used graph theory to translate the cells on the chip to a graph and used an algorithm from graph theory to calculate the resistance between the two electrodes. The conductance between the cells is computed from an integral and is equal to: $$ \sigma (y) = \ 2 \pi \left( C_{1}^2 C_2 e^{-\frac{y}{C_2}} + C_{3}^2 C_4 e^{-\frac{y}{C_4}} + \frac{4 C_{1}C_3 C_2^2 C_4^2}{y \left( C_2^2 - C_4^2 \right)} \left( e^{-\frac{y}{C_2}} -e^{-\frac{y}{C_4}} \right) \right) \tag{1}$$

Curli Module