Flux model

Introduction

In this part of our modeling work we focus on genome scale analysis of the enterobactin production pathway. A major concern about our system is its non-orthogonality with the natural metabolic network of the cells : E. coli already possesses the genes for producing enterobactins. Therefore we wanted to assess the possible interactions between our system and the bacterial metabolism.

Observations

This model stems from the following observations :

• Enterobactin production pathway is a metabolic process;
• Any of the involved metabolites may limit the rate of the reactions.

The Figure 1 presents the biochemical pathway for producing the enterobactin compound in E. coli.

As can be seen in the Figure 1 the pathway is 4 steps long and composed of six different enzymes. Hence there exists 4 possible metabolites which concentration may be limiting :

• Chorismate
• Isochorismate
• 2,3-dihydroxy-2,3-dihydrobenzoate
• 2,3-dihydroxybenzoat

Goals

We highlighted two main kind of interactions between the bacterial and our system :

1. The synthetic system interacts with the bacterial metabolism. Leading to a depletion of the metabolites involved in the pathway for the other (possibly essential) metabolic reactions of the cell.
2. The other way round, the metabolic reactions could prevent our synthetic system to work as expected by limiting the quantity of the involved metabolites available.

From these assumptions we formulated the following questions :

• Is the metabolic model of E. coli able to provide any information about the possible interactions between our system and the metabolism?
• Is the concentration of any metabolite limiting ?
• In the latter case, what is the quantity of precursor to should be added in the medium?

Materials and Methods

Model

We used the metabolic model E. coli iJR904 downloaded from the BiGG model database[1]. We chose this model because it contains all the metabolites involved in the enterobactin production pathway.

This model contains 4037 reactions and 625 metabolites but lacks the enterobactin syntase (the last reaction of the enterobactin production pathway). Thus we extended it, adding this reactions and an enterobactin output reaction to be able to consider enterobactin as a component of the model. The inserted reactions are the following:

Name	Formula
ENTSYNTH	6 ATP + 3 2,3-Dihydroxybenzoate + 3 L-Serine <=> Enterobactin + 6 AMP + 6 Diphosphate
ENTOUT	=> -1 Enterobactin

The modified E. coli iJR904 model containing these two new reactions can be found in the download section at the bottom of the page.

Network Reduction

As an analysis of the whole network is too complex, we focused our analysis on the subnetwork presented in Figure 2. We considered the CHORISMIC ACID (hereafter called CHORISMATE) as the first step of the anterobactin production pathway. This is justified by the fact that chorismate can be bought from any compound supplier (for example sigma).

We also found that it would also be possible to by the 2-3-DIHYDROXYBENZOATE compound from the same provider. This compound may be very interesting to test our constructions later as the last precursor of the pathway. For the same reason, and because it does participate in any other reaction in the used model, this compound has no interest in our metabolic modeling.

Figure 2:

The considered subnetwork of metabolic reactions of E. coli. Red arrows are reaction consuming compounds used in the ENTEROBACTIN production pathway. Arrows : reactions, circles : enzymes.

Flux Balance Analysis

The metabolic model of E. Coli is represented as a stoichiometry matrix S representing the metabolic network. Our modified version has the size 4039 (reactions) * 625 (compounds). The unknown is the flux distribution vector v, a 4039-vector representing the flow of matter (mmol/gDW/h) going through each reactions.

The Flux Balance Analysis method[2] is about finding this flux repartition vector v given an objective function to optimize (usually the growth rate) and a set of constraints of fluxes.

Assumptions of the model

The Two assumptions at the heart of the method are the following :

1. steady state: The fluxes are considered to have attained a static equilibrium value and do not change through time.
2. No enzyme saturation: The enzymes are supposed to be not saturated, the number of enzymes is always greater than the number of the corresponding reactions happening.

Formalism

The FBA method use a representation of the metabolic reaction network in the form of a stoichiometry matrix S where :

• Each row corresponds to a reaction R_i
• Each column corresponds to a metabolite C_j

The definition of S is :

The FBA problem is then formulated as a maximisation problem under some constraints :

where :

• v is the vector of unknown reaction fluxes
• c is a vector of constants defining the objective function
• S is the stoichiometry matrix
• lowerbound and upperbound are vector of constraints (minimal and maximal flux values for each reactions)

The values for the lower and upper bounds on the flux of each reactions are either deduced from experiments or put to a very high value when unknown (most of the time) :

Tunning these boundaries allows to represent different experimental conditions, for example reducing an upper bound to a low value may represent a loss of reaction flux due to the scarcity of a certain compound.

Such optimization is then realized through a linear programming algorithm finding an approximated solution of the real optimal flux distribution (the distribution maximizing the objective function).

Software used

To perform the simulations we decided to use the cobrapy software[3] in conjunction with the GLPK (GNU Linear Progamming Kit) linear programming solver.

All the simulations of model modifications were made through python scripts, available at the end of the document.

Results

Question 1: Interractions between the artifitial system and the metabolism of E. coli

Model : E. coli iJR904 with addition of ENTSYNTH and ENTOUT reactions.

In order to test whether the system interacts with the metabolism of the bacteria we applied the FBA with a modification to the objective function:

By varying i between 0 and 1 we can vary the contributions of enterobacting production and biomass growth.

Simulation

The Figure 3 presents the results of the simulation following the previous settings.

Figure 3:

Fluxes as function of the i coefficient of the modified objective function. Dotted lines : Important fluxes; Solid lines : Reactions belonging to the ENTEROBACTIN biosynthesis pathway; Dashed lines : Other reactions consuming important metabolites.

• Setup : i varied from 0 to 1 with a step of 0.01.
• The code for this model can be found at the bottom of the page

Interpretation

The graph is divided in two parts :

i < 0.303

In this part, no ENTOUT flux is present, hence there is no production of enterobactin. In the other hand there are two non-zero constant fluxes, the CHORISMATE MUTASE (at 0.28 mmol/gDW/h) and DEOXYCHORISMATE SYNTHASE (at 0.04 mmol/gDW/h). The objective function is linearly decreasing with i meaning that the system does not change in this part.

i = 0.303

For this value of i there is a brutal change in the different fluxes :

• CHORMISMATE MUTASE flux goes to 0 mmol/gDW/h
• DEOXYCHORISMATE SYNTHASE flux goes to 0 mmol/gDW/h
• ISOCHORISMATE SYNTHASE, ISOCHORISMATASE, ENTOUT fluxes goes to their maximal values 2.1 mmol/gDW/h

The objective function f stops decreasing.

i > 0.303

After this value of i the system stays the same as proved by the linear increase of the objective function with i.

Conclusion

The production of enterobactin modifies the flux distribution for only two reactions out of 5 (the others have a nul flux) : CHORMISMATE MUTASE and DEOXYCHORISMATE SYNTHASE.
Thus there is an interaction between the enterobactin production system and the E. coli metabolism that happen at the level of these reactions.

Studying the perturbation

model: E. coli iJR904 wild type

In order to understand the type of perturbation happening two new simulations were created in which the upper bound of the fluxes going through those reactions are progressively reduced until attaining 0. The response of the system (the biomass function) is plotted against these changes in upper bound flux value.

Upper bound modification for the CHORMISMATE MUTASE reaction

Simulation

The results of the simulation are presented in Figure 4.

Figure 4:

CHORISTMATE MUTASE reaction flux and growth rate as a function of the upper bound of the CHORISMATE MUTASE flux.

• Setup: upperbound from 1 mmol/gDW/h to 0 with a step of 0.01
• The code for this model can be found at the bottom of the page

Interpretation

As can be seen on Figure 4, the relationship between the CHORISMATE MUTASE flux and the growth rate is linear, hence this reaction is central to the metabolism of E. coli.

It appears that the CHORISMATE MUTASE reaction takes CHORISMATE and produces PREPHENATE, a compound participating the production of tryphosphate, an amino acid. Hence one must be very careful no to perturb this reaction, note that when the flux is not perturbated, the growth rate value is the maximal predicted one.

Upper bound modification for the DEOXYCHORISMATE SYNTHASE

Simulation

The results of the simulation are presented in Figure 5.

Figure 5:

DEOXYCHORISTMATE SYNTHASE reaction flux and growth rate as a function of the upper bound of the CHORISMATE MUTASE flux.

• Setup: upperbound from 1 mmol/gDW/h to 0 with a step of 0.01
• The code for this model can be found at the bottom of the page

Interpretation

The dynamic of the curve in Figure 5 is the same as the one in Figure 4, except that the dependency relation between the growth rate and the DEOXYCHORISMATE SYNTHASE is higher (as seen by the slope of the growth rate curve in its ascending part).

Again, this may be explained by the fact that the DEOXYCHORISMATE SYNTHASE reaction produces L-GLUTAMATE whichi is a central metabolite of the bacteria.

Conclusion on the perturbations

We observed linear dependencies of the growth rate for the two perturbated reactions DEOXYCHORISMATE SYNTHASE and CHORISMATE MUTASE which means that reducing the flux of any of these reactions may have very important consequences on the growth rates of the bacterias.

Hence, the CHORISMATE metabolite can be considered as a limiting factor of the enterobacting bionsythesis pathway.

Questions 2 and 3 : How many chorismate to add ?

model: E. coli iJR904 with addition of ENTSYNTH, ENTOUT and CHORIN (intake of CHORISMATE) reactions.

This section is about finding the good quantity of CHORISMATE to give to the bacteria for having a maximal production of enterobactin.

To answer this question, we set up a simulation using the same objective function as the first simulation :

At the same time we simulate an addition of CHORISMATE into the medium by adding an input flux of CHORISMATE named CHORIN. The ENTOUT flux (production of ENTEROBACTIN) is then monitored as a function of both parameters in order to get the best quantity of CHORISMATE to add in all conditions.

Simulation

The results of the simulation are presented in Figure 6.

Figure 6:

ENTOUT flux (ENTEROBACTIN output flux) as a function of CHORIN (CHORISMATE input flux) and i coefficient of the modified objective function.

• Setup: upperbound from 1 mmol/gDW/h to 0 with a step of 0.01
• The code for this model can be found at the bottom of the page

Interpretation

As we can see on Figure 6, the relationship between the ENTOUT flux and the CHORIN flux is linear whatever the i choosen. A stable region is reached for the ENTOUT flux at a flux for CHORIN of about 17 mmol/gDW/h for all the i. Hence this result gives an idea of the quantity of CHORISMATE to add to the medium in order to maximize the production of ENTEROBACTIN.

Plus, the dynamic of the graph becomes more regular as more CHORISMATE is added, this means that adding a CHORISMATE input flux successfuly counter the effect of the consumption of CHORISMATE by the enterobacting production pathway.

Another interesting point to analyze is the presence of a plateau when saturating the medium with CHORISMATE precusor. It means that the presence of the this reagent is no longer the limiting step. As we can see on Figure 1, the pathway uses NAD+ for the production of the non ribosomal peptide. We can assume therefore that the origin of the limiting step is the necessity for the cell to regenerate its Red/Ox potential, which is coupled to other essential metabolic transformation, and therefore to growth. Unless we add a regenerative reaction, our production of enterobactin, may be theoretically limited to the flux we calculated when the chorismate is supplied.

Conclusion

This model gives two interesting results :

1. The dynamic of the ENTOUT flux as a function of CHORIN and i is very regular;
2. A flat region for the ENTOUT flux is reached for all values of i for about 17 mmol/gDW/h of CHORISMATE.
3. There is probably no way we can improve this theoric ration unless we change the red-ox potential of the cell or change host.

Conclusion

The metabolic models presented in this sections have been used to answer the general questions we had about our system.

First, we wanted to know if adding new synthetic constructs would perturb the metabolism of E. coli. This was answered by a first model plotting a modified objective function optimizing both ENTEROBACTIN production and growth rate. The model showed that the enterobactin pathway perturbs the CHORISMATE MUTASE and DEOXYCHORISMATE SYNTHASE reactions.

In two other models, we showed that the CHORISMATE MUTASE AND DEOXYCHORISMATE SYNTHASE were very crucial for the bacteria, as a reduction of their maximal upper flux lead to a direct (linear) decrease of the growth rate function. Resulting in the extreme case with a nul growth rate for a nul flux.

Finally, we analysed how we could find a workaround to this problem by adding some CHORISMATE to the medium. This situation was represented in a third model with a CHORISMATE input flux that we plotted against the modified objective function. This model clearly showed that the ENTEROBACTIN flux attained a maximal stable region for any value of the objective function starting from 17 mmol/gDW/h of CHORISMATE. Finally, the dynamic of the model is very smoother as more CHORISMATE is added, showing that this addition could paliate the effects of the addition of the ENTEROBACTIN pathway.

In the end, this FBA model shows that it is possible to paliate the effect of our construction on the metabolism of E. coli by adding some CHORISMATE to the medium. This justifies the other models developed that did the assumption that no metabolite was limiting.

Models and Scripts

Metabolic Models

The two metabolic models used are :

1. Wild-Type E. coli iJR904
2. ENTOUT(enterobactin outflux) + ENTSYNTH(enterobactin synthase) E. coli iJR904

These models can be found in this tarball.

Scripts

The scripts used to generates the curves on this page can be found on this archive. They are Python scripts with distinct names corresponding to the simulation they represent.

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