Team:ULB-Brussels/Modeling

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Questions from the Wetlab part

Questions asked by the dihydrogen module were mainly general questions : how much the deletion of the whole, or of one part, of the genes proposed would increase the amount of dihydrogen produced ? To answer these questions, the metabolic pathway of mixed acid fermentation had to be modelled before we can, eliminating equations for some of the arms of the pathway, observe the advantages in term of dihydrogen production.

Initial equations and further enhancements

We posited the differential equations expressing the variation in the main intermediates of the metabolic pathway, vs the time. Generally speaking, these equations comprise a creation term (positive) representing the formation of this intermediate from a reactive which is before it in the pathway and from one or several destruction terms (negative) that stand for the degradation of this intermediate in the next product. We distinguish two types of terms (creation and destruction ones). First of all, they are linear terms, characteristic of non-catalyzed reactions that we assume being spontaneous. They are of the type coefficient.reagent concentration. There is, then, the big category of the non-linear terms, for more complex situations, in which enzymes act. In the H2 module, these non-linear terms are Michaëlis-Menten like, which is the typical cinetic of enzyme-catalyzed reactions. We'll then see, in the next section, other non-linear terms, more particulary used when we modelize systems with important genetic regulations. Here are, then, the first equations which represent the metabolic pathway of the mixed acid fermentation.

v0 is here a constant of production of the phosphoenolpyruvate by glycolysis. The two next terms are enzymatic terms of first order, characterizing the catalyzed reaction ADP + PEP <=> ATP + PYR + 2H+ (v1) and the catalyzed reaction PEP + CO2 + H2O <=> OXA + P + H+ (v2)

The first term is the term of production of the pyruvate, from phosphoenolpyruvate The second term (in v3) is a term of Michaëlis-Menten for the catalyzed reaction PYR+ CoA  FOR + AcCO. The third term (in v4) is once again a non-linear term traducing the catalyzation of pyruvate disappearance, which is transformed in lactate, following the reaction NADH + PYR + H+ <=> LAC + NAD+ )

The first term is a term of formation of the formate, from pyruvate. The second term (in v5) traduce the catalysation of the reaction FOR + H+ <=> CO2 + H2. The third term (in v6) represents facilitated diffusion of formate, by specific and non-specific permeases. It's again an enzymatic reaction, with a typical non-linear kinetic.

The first term is a term of formation of the acetyl-CoA, from pyruvate. The second (in v7) is a non-linear term of the second order in OXA and AcCO, traducing the reaction OXA + AcCO+ H2O <=> CIT + CoA + H+

The first term is a term of apparition of the oxaloacetate, from PEP The second term is a non-linear term of the second order in OXA and AcCO, traducing the reaction OXA + AcCO+ H2O <=> CIT + CoA + H+

The first term is a term of apparition of the citrate from acetyl-CoA and oxaloacétate The second term is a term of consumption of the citrate, in following reaction.

The first term is the term of apparition of the lactate from pyruvate. The second term (in v9) is the facilitate diffusion term for lactate.

We then realized the importance of an equilibrium term in the last reaction leading to the formation of hydrogen  (FOR + H+  <=> CO2 + H2.),this product can accumulate, which encourages the reverse reaction.
We also refined the kinetic of the reaction OXA + AcCO+ H2O <=> CIT + CoA + H+, using a non-linear term which considers the concentration and Michaëlis-Menten constant specific of each substrate.
Finally, and to more easily observe the benefit of each deletion, we added an equation modeling the output speed of H2 diffusing out of the bacteria.    
Here are the final equations :

Where the term in v5r is the equilibrium term characterizing the reverse reaction of the H2 production. This term has been added to account for the balance of the reaction, and avoid an irrealist of H2 in the cell.

Where the term in v7 is a modification of the preceding term in v7, modeling a more general kinetic, to better take into account the differences between the two substrates. Where the term in K7b is a term of utilisation of acetyl-CoA in later reactions.

With the same modification than above.

With the same modification than above.

Where the term in v5r is the equilibrium term characterizing the reverse reaction of the H2 production. This term has been added to account for the balance of the reaction, and avoid an irrealist of H2 in the cell.

Results

Such a differential equation system can't be analytically solved. We then had to use a specialized software able to numerically integrate the system. To do that, we used the XPP software, specifically designed to model biological systems. This software allows us to study the evolution of each variable depending on time. It also has an extension, Auto, able to draw stability diagrams of the systems. At first, we only vary the parameters, depending on the expected wetlab results. For example, v2=0 means the PPC gene has been Knock Out successfully (see table 1 for the corresponding between genes deletions and the parameters.). We can see, in the diagram 1, the theoretical effects of such a deletion on the intra-cellular concentration of dihydrogen at the steady-state: it's almost doubled (from 0.057 to 0.118). In the same way, we put each parameters to a nul value, alternately and in a combined way, to observe the effects of the deletions and their optimal combination.

We noticed that for double deletions, the results cluster in three increased levels of production (cf. Diagram 2). A deletion with particularly strong associated effect: FocA. LdhA and PPC are middle effect deletions, whereas the deletion of both hydrogenases seems to have a lesser effect on productivity. Indeed, the combinations including FocA are always in the two higher levels of production, while those including the hydrogenases are always in the two lowest levels.

In triple deletions, the above results are confirmed and show even more the importance of FocA in dihydrogen production, while the weakness of the hydrogenases deletion seems less important. Indeed, we notice here two very marked dihydrogen production levels, one being FocA and two other genes (no matter what ever they are) KO bacteria and the other one with a functional FocA.  

There is an important clarification to make here. The intracellular concentration in dihydrogen presented here are in no case absolute values, they are given in arbitrary units. Generally speaking, all of the numbers presented in this section are only for a comparative purpose. This is easily understood if we remember the largely empirical way to establish the parameters of the differential equations.