Team:KU Leuven/Modeling/Toulouse

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Toulouse FBA Model

We cooperated with Toulouse on the modeling. Here we describe the Flux-Balance-Analysis the Toulouse team generously performed for us.

Flux balance analysis is a widely used approach for studying the flow trough metabolic networks. In our case we are interested in the Leucine and AHL production rates of the type A cells and their effect on the growth rate of the bacteria. This is very important because if cell A has a big metabolic burden and a very low growth rate, cell B would overgrow it and our pattern formation system would not work. To check the effect of our system on the growth rate iGEM Toulouse ran a FB analysis. When a FBA is set up. The metabolic network of the organism in question is represented as a matrix $\mathbf{S}$ of size $m \times n$ is filled with the stoichiometric constants of each reaction. Each of the $m$ matrix rows represents a unique compound. Similarly each of the n columns represents one unique reaction. Next a vector $\mathbf{v}$ of length $n$ is defined which contains the flux trough each reaction. Finally the vector $\mathbf{x}$ is defined to contain the concentrations of each metabolite. The steady state solution in the insteresting one therefore: $$ \frac{dx}{dt} = \mathbf{Sv} = 0 $$ Which is the nullspace of $\mathbf{S}$. In this set of solutions a maximal or minimal value can be identified using numerical optimization. In order to run the optimization algorithm a cost function has to be defined. $$ f(x) = \mathbf{c}^{T}\mathbf{v} $$ The equation above shows such a cost function. Here the vector $\mathbf{c}$ represents a weight vector. In practice it is used to choose the metabolite of interested by setting the corresponding entry to one and all others to zero. From an optimization perspective the equation $\mathbf{Sv} = 0$ represents constraints, which guide the numerical solver to the right solution.
In our case the $\mathbf{S}$ matrix comes from the in silico E. Coli model K12MG1665. The model is contained in an XML file. We told the Toulouse modelers our two reaction of interest: $$ \text{glutamate} + \text{aKIC} \rightarrow \text{aKG} + \text{Leucine} $$ This reaction is interesting as Leucine serves as repellent in our scheme. The second reaction of interest is: $$ \text{acylACP} + \text{SAM} \rightarrow \text{ACP} + \text{MTA} + \text{AHL} $$ This reaction is important because AHL serves as the key to the motility system of cell B. Without AHL, cell B is unable to move away and form patterns. Given this information the Toulouse team was able to locate the reactions of interest in the XML model file, and simulate the cell metabolism. The results are:

Figure 1
AHL and Biomass production. Click to enlarge

Figure 2
AHL and Biomass production with maximal Leucine production rate.. Click to enlarge

Figure 3
Leucine and Biomass production. Click to enlarge

Figure 4
Leucine and Biomass production at AHL maximum. Click to enlarge

All of the graphs shown above are in $mmol \cdot gDW^{-1} \cdot h^{-1}$. The maximum value for AHL is:

$$ 150.71391 mmol \cdot gDW^{-1} \cdot h^{-1} $$

Similarly for Leucine toulouse found:

$$ 528.17774 mmol \cdot gDW^{-1} \cdot h^{-1}$$

In generally it follows from the analysis that the more AHL or Leucine is produced the smaller the Biomass output becomes. It also shows that Cell A can not operate at the maximum production rate, because the growth rate would then be so slow that Cell B would overgrow it. This is something to keep in mind when we choose our promoters for the system.