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.
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.
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. 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.
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.
The conductive curli module has different dynamics on different length scales:
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.
The behavior of the system on the cell level, that is the curli production of each cell in time.
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.
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.
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.
Click in the figure to move to the corresponding page.
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.
The goal of the modeling of the curli module is to prove that our system works as expected and to capture the dynamics of our system. The product we aim for is a chip where two parallel electrodes are a distance w apart. Between the electrodes, cells will grow and start building curli in the presence of DNT/TNT. Then, we will measure the conductivity of the resulting biofilm, which is related to the amount of DNT/TNT. Since even with bound gold nanoparticles the conductivity of the curli is very low, the chip is designed such that the electrodes are as long as possible.
The first question we are interested in is: can we prove that our system works as expected? So, does a conductive path between the two electrodes arise at a certain point in time and at which time does this happen? We do this by modeling the curli growth on the colony level; each cell is now visualized and has curli growth. First we have to make some approximations. Since the cells are grown on a chip, we assume that the cells and curli grow on a surface. This reduces our problem from 3D to 2D. This saves much computational time and memory. For this model, we take a chip of 500 by 500 µm. The electrodes are placed parallel to the y axis on x = 20 µm and x = 480 µm. The next approximation is that the cells are already present when they are induced by DNT/TNT, we neglect cell growth. In our model, E. coli are present with a density of \(\rho_{cell}\). Furthermore, we assume there is no spatial correlation between the cells; hence we place them at random on our chip. The cell density we use is \( 2 \cdot 10^4 \) cells \( mm^2 \). We'd like to model higher cell densities and larger chips. However, the memory cost of the solution increases with the amount of cells squared and, even when the code is neatly vectorized, the computational time increases drastically more.
We have come up with two different approaches.
The first is that we let the cells increase their conductive radius in time, according with our findings on the cellular level (figure 12). 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. However, we not only want to answer the question if our system works as expected with only a yes or no answer, but we also want to make predictions about the resistivity 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 resistivity between the two electrodes. The conductivity between the cells is computed from an integral that we have set up starting with formula 13.
Percolation
So, we now have designed our chip as a 500 by 500 µm square with an electrode on the left and right side. On this chip, we place cells randomly with a density of \(\rho_{cell}\). Subsequently, we increase the conductive radius of each cell in time, corresponding with our findings on the cellular level, see figure 12. A connection is created from one electrode to the other electrode when there is a conductive path between them, so when there is percolation. Conductive paths consists of cells that have a connection with each other, cells have a connection with each other when there is an overlap between their conductive radii. In practice we have programmed this by comparing the distances of all cells with all other cells with the conductive radius. A simulation of our resulting model is shown in figure 14. Percolation is computed by applying an algorithm that can find clusters of connected cells. When one of the clusters connects both electrodes, we have percolation.
Figure 14: NorthWest: A visual representation of our cells on the plate. The circels represent the cells with a conductive radius of 4.05 µm. In this simulation there are 500 cells present on a chip of 500µmx500µm. NorthEast: A spy matrix of 5000x5000 where the blue dots represent connections between the individual cells. A blue dot on position x,y means that cell x is connected with y. Each cell is connected to itself (diagonal). At the point of percolation, \( \approx 0.1 \% \) of the matrix is connected, meaning that each cell is on average connected to 5 others. SouthWest: Each square of nxn represents a cluster of n connected cells. The squares are sorted from small to large. SouthEast: This figure shows the largest cluster of cells in different colours.
We have stochasticity in our model, as we place the cells randomly with a density of \(\rho_{cell}\) on the chip. Therefore, we simulated our model 100 times and for each point in time we checked if there was percolation. We will only get a yes (1) or no (0) response. This enables us to find the chance of percolation at each time point, shown in figure 15 as the yellow line. The yellow line shows a sharp transition between 1.5 and 2 hours. Since this is a Bernoulli process, [reference], the variance is exactly equal to p(1-p). The variance must be as low as possible to get trustworthy measurement results, as in that case the transition from no percolation to percolation is as sharp as possible.
At first we assumed in our model that \(r_{cond}\) is the same for each cell at each point in time (figure 12 red line). However, figure 12 shows that there clearly is some cellular variation in \(r_{cond}\). Therefore, we also added a feature to our model; the conductive radius of each cell can now deviate from the mean \(r_{cond}\) with the standard deviation as found in figure 12. We simulated our resulting model again 100 times and for each point in time we checked the chance of percolation, see figure 15 as the blue line. Fortunately, the resulting curve is very similar to the curve without cellular variation in \(r_{cond}\) (yellow line). This means that cellular variation has little influence on the chance of percolation at each point in time. Therefore, the results of our model are robust to cellular variation and it is likely that many factors that could increase the cellular variation, e.g. different CsgA or CsgB protein production rates, are relatively unimportant.
Figure 15: The chance of percolation with 5000 cells on a 500x500 \(\mu m \) chip. as function of time. The results are from 100 simulations. The yellow line represents the chance of percolation where all the cells have the same conductive radius. The blue line is the same simulation, but all cells have slightly different conductive radii. Note how there is no notable difference between the two.
Influence of the chip geometry on the point of percolation
To further investigate the point of percolation we have varied the shape of our chip. We have decreased the relative distance between the electrodes by making our chip 250 µm x 500 µm, where the electrodes are 250 µm apart. From the result, shown in figure 15.5, it can be seen that our system behaves as a percolation problem. The system is smaller, therefore the transition toward percolation is less sharp. This suggests that we want to increase the area size of our system. A larger area results in a shaper transition, thus a lower uncertainty.
Figure 15.5: The chance of percolation with 2500 cells on a 250x500 \(\mu m \) chip. as function of time. The results are from 100 simulations. The yellow line represents the chance of percolation where all the cells have the same conductive radius. The blue line is the same simulation, but all cells have slightly different conductive radii
Resistivity
To calculate the conductivity as function of time we repeat the following steps:
Place our cells on our chip.
Compute the conductivity between the cells.
Compute the conductivity between the electrodes.
Compute the conductivity between the cells
First, we have to get a quantitative measure for the conductivity between two cells. To do this, we will quantify the overlap of two conducting spheres, where we assumed that the conducting spheres represent cells surrounded by curli filaments. We subdivide the overlapping region in infinitesimal volumes \(dV\). The infinitesimal conductivity of such an infinitesimal volume is given by:
$$ d \sigma (y) = \ \frac{\rho_1}{r_1} dV \frac{\rho_2}{r_2} dV \tag{}$$
The factor \( 1/r \) is introduced to account for the conductivity of the wires itself, which is inversely proportional to the length of the conducting wire. [source: Narinder Kumar (2003). Comprehensive Physics XII. Laxmi Publications. pp. 282–. ISBN 978-81-7008-592-8.] Further away from the cell, the wires need a longer distance to go to the cell. Since we want to know the strength of the connection between the cells, we have to include this factor. For a straight line this is inversely proportional to the distance. For a single curli fibril, this relation does not hold. However, we assume that the curli density is high, thus there are many connections between the curli. Then there is a pathway from the origin to \( r \) roughly proportional to the distance from the cell. To find the total conductivity, we integrate on both sides. To account for the fact that both volume elements \(dV\) are the same, we make use of the Dirac-delta function \(\delta_3\) [source]. This gives us the following:
The Dirac delta allows us to remove the \(\vec{r_2}\) dependence by expressing these in \(\vec{r_1}\). The still undetermined relation between \(\vec{r_1}\) and \(\vec{r_2}\) is given by \(\vec{r_2} = f(\vec{r_1})\). Applying this removes one of the two volume integrations. Using spherical coordinates, the resulting single volume integration can be written as:
Here we have made use of the fact that the density \(\rho\) is only dependent on \(r\) and not on \(\phi\) and \(\theta
\). The integral over \(\phi_1\) is trivial and gives us a multiplication factor of \(2 \pi\):
Now that we have reduced our integration to two dimensions, we will work out \(f(\vec{r_1})\). To do this, we introduce the vector from the origin of cell 1 to the origin of cell 2, \(\vec{y}\). This allows us to express \(\vec{r_2}\) in terms of \(\vec{y}\) and \(\vec{r_1}\):
We will now have a closer look at the boundary values for \(r_1\) and \(\theta_1\). We want to integrate over the entire space. Therefore, \( \theta(max) = \pi \) and \( r_{max}=\infty \). By introducing no cut-off radius, we are able to take into account the possibility of having by chance a very large conductive radius. Here we have approximated our cells as points in space. Hence \( r_0 =0 \).
We will now use the previously [link] found fact that the curli density can be described as:
This integral looks very complicated, but don't panic! It can algebraically be simplified with some substitutions. We can rewrite this integral by moving all terms independent of \( \theta \) out of the integral over \(\theta_1\). Furthermore, using that \( \sin^2 (\theta_1) + \cos^2(\theta_1) = 1 \) we get.
In the second integral we recognize something of the form \( \int \frac{e^{-\sqrt{a+bx}}}{C_2\sqrt{a+bx}} dx \) with \( a= \frac{y^2+r_1^2}{C^2_2} \) and \(b=-\frac{2yr_1}{C^2_2} \). Substituting \( h= \sqrt{a+bx} \) with \( dx= \frac{2h}{b} dh \) yields:
For future research, we could extend our models such that the cellular variation is included. If \( \rho_1(r) = \ C_{1}e^{-\frac{r}{C_{2}}} \) and \( \rho_2(r) = \ C_{3}e^{-\frac{r}{C_{4}}} \) then the conductivity between the two electrodes, using the approach as described above is.
Now, we use graph theory to translate the cells on the chip to a graph and use an algorithm from graph theory to calculate the resistivity between the two electrodes.
Results
We have calculated the conductivity as function of time for various different dimensions of our plate. The distances of the electrodes is varied from 250-500-1000x500 \( \mu m \). The result is shown in figure 16. The process is, for the smaller chips repeated 10 times to say something about the variance. For the 1000x100 \( \mu m \) plate it is only repeated thrice, for a single curve already takes over twelve hours to compute.
Figure 16: Orange: The conductivity (rbitrary units) as function of time (hours) for different dimensions of our plate. The first dimension is the distance between the two electrodes. The second dimension represents the length of the electrodes.
From figure 16 we can draw a couple of conclusions.
It seems that the conductivity increases exponentially over time. We expect that even after a long time, there is low conductivity [paper]. This means that before that time, it is hard to measure changes in conductivity.
The response-curve of the system is independent of the shape of our plate. The blue lines that have the distance between the electrodes doubled compared to the green line also has half the conductivity (and a quarter of the red lines). Thus the conductance is inversely proportional to the distance between the electrodes. This is precisely what you would expect if you see the system as a single resistor.
Increasing the chip size decreases the relative uncertainty of the response. The red lines are much further apart (also relatively) than the blue lines. This makes sense from a physical point of view, since we're dealing with larger samples. It is then more insensitive to the randomness due to the placement of the cells.
Our findings are in accordance to the behaviour expected from a random resistor network with percolation theory, where the conductivity increases exponentially after percolation.
It is impossible to observe a point of percolation in this. This is because we have made a continuous model.
Other simulations shows us that indeed, the conductivity scales linearly with the length of the electrodes. If we want to make a design of our system and have as high conductivity as possible, we want to decrease the distance between the electrodes as much as possible. At the same time, the total area of the chip should be reasonable large to reduce the effects of the stochastic behaviour of the system.
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