Team:TU Delft-Leiden/Modeling

Modeling Overview

We developed models for each of the three different modules of our project: the conductive curli module, the extracellular electron transport (EET) module and the landmine detection module.
For the conductive curli module, we wanted to know if a conductive path between two electrodes of a chip filled with curli growing E. coli arise at a certain point in time. We also wanted to make quantitative predictions about the resistance between the two electrodes of our system in time.
For the EET module, our goal was to investigate the carbon metabolism providing the electrons for the EET module. Also, we want the EET pathway used by the cells in order to have a measurable electrical signal for our biosensor, see the gadget section of our wiki. Furthermore, in our modeling of the assembly of the EET complex, we wanted to predict how many EET complexes are formed under different initial conditions. We focused, in addition to the assembly mechanism, also on the apparent reduced cell viability.
For the landmine module, we tried to find a model which would be able to reproduce the response curves of both the landmine promoters, as found in [1].
For the EET and landmine modules, we used deterministic modeling. For the curli module, we used a stochastic modeling approach, and considered the system at the gene, cell and colony level. At the colony levvel, we employed percolation theory in order to predict if a conductive path between the two electrodes arise at a certain point in time and to predict at which time this happens. Our application of percolation theory to describe the formation of a conductive biological network represents a novel approach that has not been used in the literature before.

We used Matlab for most of the calculations; the scripts we made can be found in the code repository. We had great interactions with the Life Science and Microfluidics departments, which for the conductive curli module can be read here, for the EET module can be read here and for the landmine detection module can be read here.

Curli Module

For more information, see our entire section about the conductive 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, 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.

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.

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.

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

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: $$ \sigma (y) = \ 2 \pi \left( C_{1}^2 C_2 e^{-\frac{y}{C_2}} + C_{3}^2 C_4 e^{-\frac{y}{C_4}} + \frac{4 C_{1}C_3 C_2^2 C_4^2}{y \left( C_2^2 - C_4^2 \right)} \left( e^{-\frac{y}{C_2}} -e^{-\frac{y}{C_4}} \right) \right) \tag{1}$$

Using the first approach, a simulation of our resulting model is shown in figure 6. Percolation is computed by applying an algorithm that can find clusters of connected cells. When one of the clusters connects both electrodes, there is percolation.

We simulated our resulting model 100 times and for each point in time we checked the chance of percolation without variation in \(r_{cond}\) (yellow line in figure 7) and with cellular variation in \(r_{cond}\) (blue line in figure 7). Fortunately, the blue and yellow lines are very similar. 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.

As we want our system to be usable as a biosensor, it has to be strongly dependent on the analyte concentration, and therefore the CsgB production rate. From figure 8, we see that there are very distinguishable difference for different CsgB production rates. First of all, the moment of percolation differs a lot. Equally important, the transition from no percolation to percolation is much less sharper.

EET Module

For more information, see our entire section about the EET Module. This module consists of two separate sections, Flux Balance Analysis of the EET Module and Deterministic Model of EET Complex Assembly.

Flux Balance Analysis of the EET Module

For more information, see our entire section about Flux Balance Analysis of the EET Module.

In the wet lab, we integrated the Extracellular Electron Transport (EET) module of S. oneidensis into E. coli, see the extracellular electron transport (EET) module. For the modeling of the EET module, we wanted at first to gain insight in the consequences of the integration of the EET module into E. coli. To achieve this, we simulated the cell metabolism of E. coli including the EET module using the Flux Balance Analysis (FBA) method. Our goal was to investigate the carbon metabolism providing the electrons for the EET module, see Carbon Metabolism and Electron Transport. Also, we want the EET pathway used by the cells in order to have a measurable electrical signal for our biosensor, see the gadget section of our wiki.

From the FBA method, we conclude that in aerobic conditions the cell does not use the EET pathway, but oxygen gets reduced instead, as it is a stronger oxidizing agent. However, in anaerobic growth the cell does use the EET pathway to export electrons out of the cell. When the cell is grown on glucose, the growth rate will be higher than when the cell is grown on D-lactate.
We also conclude that in an experimental setting the EET pathway has a higher chance of being used when the cells are grown on D-lactate as the EET pathway is necessary in order for the cells to grow, while when grown on glucose and the EET pathway is turned off (represented by \(0 \ mmol \ (gDW)^{-1} \ hr^{-1}\) (per gram dry weight per hour) maximum EET flux) growth is still possible, see figure 2. From Flux Variability Analysis (FVA) we conclude that for maximum growth for each specific combination of carbon source uptake flux and maximum EET flux, only one possible EET flux is possible for both growth on glucose and growth on D-lactate, namely the EET flux shown in figure 2.

As performing the FBA method while maximizing the EET flux yielded no growth, we wondered if there are pathways possible that would yield growth. So, we performed FVA, the results can be found in figure 3. From this figure, we conclude that when maximizing the EET flux, there are pathways possible that yield growth, as the figure displays the difference between the optimized maximum value and the optimized minimum value of growth. We see that this value is not equal to zero everywhere in the figure, thus there are multiple pathways possible when maximizing EET flux that all yield different values for growth. Note that the EET flux is 2.8 times higher and 2 times higher in comparison to maximizing for growth rate, for glucose and D-lactate as a carbon source, respectively.

From figures 2 and 3 we conclude that there are different regions in which the cell can operate. In an experimental setting, it can be investigated in which region the cell actually operates and if it maximizes its growth rate or its EET flux. To be able to do this, the experimental observed pathway has to be compared to the possible pathways when maximizing the EET flux and to the pathway when maximizing the growth rate. From these regions, it can be deduced if the experimentally observed EET flux and growth rate are carbon source limited or limited by the maximum possible EET flux.
Finally, we investigated an extended model of E. coli metabolism. This model contains, in contrast to the previously used core model, L-lactate as a metabolite. Using the extended model, we found that for glucose and D-lactate as carbon sources, the maximized growth rate agreed quite well to the previous analysis's using the core model. Using L-lactate as a carbon source, we conclude that a steady state solution in which E. coli can grow on L-lactate and use the EET pathway is not possible. A possible way to obtain information about the EET flux when the cells are not in steady state as observed by Goldbeck et al. [2], would be by the use of dynamic flux balance analysis (dFBA), which can also model the dynamics of a system before it reaches steady state [3].

Deterministic Model of EET Complex Assembly

For more information, see our entire section about Deterministic Model of EET Complex Assembly.

The EET module consists of three proteins: MtrA, a cytochrome on the inside of the outer membrane, MtrB, a β-barrel protein located in the outer membrane, and MtrC, another cytochrome, located on the cell surface. This complex enables the cell to transport electrons from the cytoplasm of the cell to the extracellular environment, see the extracellular electron transport (EET) module.
The assembly of the trans-membrane EET complex depends on many factors other than transcriptional and translational control, as it requires a large amount of post-translational modifications. We set up a simplified model of this assembly process, largely based on section 1.3 of the thesis of Jensen [4]. With the use of deterministic modeling methods, our goal is to predict how many EET complexes are formed under different initial conditions.
In our modeling of the assembly of the EET complex, in addition to the assembly mechanism, we also focus on the apparent reduced cell viability. Jensen [4] proposes two possible explanations for this: the formation of cytosolic aggregates and reduced membrane integrity due to the high amount of trans-membrane protein complexes.

We constructed two different models, one based on an extensive model of EET complex assembly, which we based upon the work of Jensen [4], the other based on a very much simplified model of EET complex assembly, which includes only the most fundamental reactions of the assembly process.
From the extensive model, we concluded that the amount of δ-ALA (and therefore heme) is rate limiting and not the amount of available binding sites. We therefore predict that adding extra δ-ALA to the cells will increase the amount of EET complexes, see figure 4. This effect is also observed by Jensen [4].

Although the extensive model proved to be valuable in the investigation of the mechanism which assembles the EET pathway, it is not suitable for the quantitative prediction of the amount of EET complexes. The most important reason for this is the large number of unknown parameters. Therefore we decided not to aim at enhancing this model, and rather set up a more simplified model.

To sidestep the difficulties experienced with the extensive model of the assembly of the EET complex, we reduced the system to a bare minimum. For this simplified model, we only included the production of MtrCAB, the formation of cytosolic aggregates and the assembly of the EET complex.
The simplified model does not represent all the actual mechanisms of the EET complex assembly process that happens in nature, but it is able to match the experimental data of Goldbeck et al. [2], see figure 5. A maximum at low promoter strength is clearly visible. This corroborates the statement in [2] that maximum promoter strength does not result in maximum EET concentration due to reduced cell viability.

This is a confirmation that our assumptions concerning cell viability might be correct. Therefore, we conclude that reduced cell viability because of the implementation of the EET pathway is the consequence of three molecular processes, namely firstly, the amount of EET complexes reduces the transcription and translation of the MtrCAB proteins due to reduced membrane integrity, secondly, the forming of MtrCAB aggregates and thirdly, the clogging of the secretion system transporting the MtrCAB complexes.

Landmine Module

For more information, see our entire section about the landmine detection module.

An important part of our iGEM project is a promoter sensitive to DNT/TNT. We will use two promoters that are sensitive to DNT/TNT, namely ybiJ and ybiFB2A1, in our project, see the landmine detection module. Of these promoters, not much is known other than the fact that they have a DNT/TNT-dependent response curve. Our goal was to find a model which would be able to reproduce the response curves of both promoters. To achieve this, we constructed two different models, both using deterministic modeling methods. One model is based on a simple binding model of DNT to the promoter, the other is based on cooperative binding of DNT to the promoter.
When based on the simple binding model, fits of promoter activation with respect to DNT concentration to the experimental data of [1] did not yield good results. However, when the fits were based on the cooperative binding model, we were able to match the experimental data in [1] really well, see figure 1.
From the fits shown in figure 6, it is clear that both the standard and cooperative activation model fail to describe the experimental data we obtained. This might have several reasons. Firstly, the response ratio of our measurements is very small compared to the measurements in [1]. Secondly, our experimental data suggest a detrimental effect of induction with high DNT concentration. This effect is not described in [1] and we don't know its reason. It is therefore not included in our model. Thirdly, both the data set from [1] and our experimental data set are small compared to the amount of parameters. This makes finding the right model difficult.
To improve the promoter activation model, a larger and more consistent data set has to be obtained. We need to conduct fluorescence measurements over a wide range of DNT concentrations. Besides that, possible detrimental effects of its solvent (acytonitrile) have to be investigated. Also, the induction of the N genes with rhamnose might influence the measurements in an unexpected way. Doing all this is unfortunately not possible in the time span of this iGEM project.

References

[1] S. Yagur-Kroll, S. Belkin et al., “Escherichia Coli bioreporters for the detection of 2,4-dinitrotoluene and 2,4,6-trinitrotoluene”, Appl. Microbiol. Biotechnol. 98, 885-895, 2014.

[2] C.P. Goldbeck, H.M. Jensen et al., “Tuning Promoter Strengths for Improved Synthesis and Function of Electron Conduits in Escherichia coli”, ACS Synth. Biol. 2, 150-159, 2013.

[3] R. Mahadevan, J.S. Edwards & F.J. Doyle, “Dynamic Flux Balanace Analysis of Diauxic Growth in Escherichia coli”, Biophys. J. 83, 1331-1340, 2002.

[4] H.M. Jensen, “Engineering Escherichia coli for molecularly defined electron transfer to metal oxides and electrodes”, PhD Thesis Chemistry UC Berkeley, 2013.