Materials and methods: Building the model

We built a model of our propane pathway based on Michaelis-Menten enzyme kinetics. It is a way to model enzyme reactions which assumes that the change the enzyme causes is faster than the binding or release of the substrate by the enzyme.

Not all enzyme reactions in our pathway happen the same way, and thus they need to be modeled in various different ways. For more specific information about how each enzyme reaction is modeled and about the constants involved see our page of enzyme kinetics.

Because of time restrictions we couldn't measure how much enzymes there are in our cells. This is why in modeling we had to use estimated values, namely 1e-6 mol/l, for all the enzyme concentrations.

For modeling this propane pathway we used Copasi and Matlab.

Results and implications

Our files for download: Copasi file with Hbd, copasi file with FadB2 and our Matlab file.

CAR-activation

One of the enzymes in our pathway, CAR, needs activation before it can function. To further understand how this affects the function of this enzyme we modeled the reactions governing the activation. To summarize the results: in most scenarios CAR is mostly in its active form. This is why we have assumed that it is all activated in the following calculations.

Bottlenecks: Comparing enzyme rates

To find out what the rate limiting steps in our pathway are, we compared the rates of the enzyme reactions. This was done by calculating the reaction speeds with different substrate concentrations. The reactions are explained in depth here and the estimated Michaelis-Menten rate equations tell us directly the reaction speeds. We implemented the code to plot these with Matlab.

FadB2 reaction is reversible in our model but for this we approximated it as irreversible. This yields better results for it than in reality.

The results shown in figure 2 tell us that FadB2 is a really inefficient enzyme and one of the largest bottlenecks in our pathway. This caused us to change it to Hbd; an enzyme with same function and reportedly better performance.

The plot also shows us that ADO isn't a good one either. To ease the ADO bottleneck, we put the construct containing its gene to the backbone that had a higher copy number, to increase its relative concentration in the cell.

CAR isn’t the best enzyme in our pathway, and unfortunately we couldn’t do anything to make it’s performance better because it was in a different construct than ADO. We had ordered our constructs before we knew the bottleneck results and because of time restrictions we had to cope with what we had, but based on the obtained results we can deduce a better ordering of constructs than we now have. We should put as many of the slowest enzymes as possible to the higher copy number backbone.

We came to the same conclusions by checking the reaction fluxes and running parameter scans for different enzyme amounts with Copasi. Thus we identified and ranked the different bottlenecks in our pathway.

After getting these results we performed the bottleneck analysis again out of curiosity with relative enzyme amounts. When before we had all the enzyme concentrations to be 1e-6 mol/l, now we scaled them to correspond to the different copy numbers of different backbones. We had put CAR-construct into pSB6A1 (ORI: pMB1, copynumber: 15-20) and ADO-construct into pCDFDuet-1 (ORI: CloDF13, copynumber: 20-40). Based on this we approximated that there is about 1.5 times more of those enzymes that are in ADO construct; see figure 4 for results. It is good to remember that we don’t have accurate information on how much enzymes there are in the cell so the actual values might not be right. Despite that this approach gives us a good idea of how one could improve the pathway in the future.

Sensitivity analysis

We performed sensitivity analysis of our pathway model to see the robustness of different parameters. We performed this analysis with the aid of Copasi, which has a ready task for it. Further, we performed this analysis based on both parameters and initial concentrations. We did the analysis with different enzyme concentrations based on the backbone copy numbers.

Relative sensitivities are defined as \[\frac{p}{[S]}\frac{d[S]}{dp},\] where \([S]\) is substrate concentration and \(p\) is the parameter by which we wish to calculate the sensitivities. A small sensitivity coefficient with respect to a parameter tells us that the behaviour is robust to the perturbations. If the parameter is large, we have gained knowledge of a control point. There interventions will have significant effects.

When we calculated the scaled sensitivities in regarding to initial concentrations, butyryl-CoA was the most sensitive species of our reaction pathway. Hbd and NADPH influenced it positively (5.14 and 1.89) and YciA had a big negative influence (5.1). This doesn't really matter for us, since this doesn't have any effect on propane production. From the results we could see that propane was positively sensitive to ADO, CAR, Hbd and NADPH. This confirms the main bottlenecks and also suggests that NADPH affects our propane production. To improve the pathway in the future one could add mechanisms to create higher amounts of it in the cell.

Here you can download the sensitivity results for initial concentrations or our whole copasi file with Hbd.

The relative sensitivities when calculated based on the parameters (i.e kinetic constants) tell us the familiar tale of Hbd, CAR and ADO being the most influential enzymes in our pathway. The greatest single sensitivity (5.14) was again for Butyryl-Coa and caused by Hbd's \(K_{cat}\). YciA's \(K_{cat}\) has the smallest sensitivity value (-5.1). The results suggest that we could improve propane production by finding substitutive enzymes with better kinetic constants and performances.

Here you can download the sensitivity results for parameters or our whole copasi file with Hbd.

Time course

From time course analysis we can have some kind of idea of how much propane our system is able to produce. We performed this analysis with Copasi. Since we knew from the bottleneck calculations that FadB2 should be changed to Hbd, we considered only the latter. Also we had different amounts of enzymes based on which backbone they were.

In our copasi file we don't have the competing enzymes that also eat butyraldehyde, the last substrate before propane. This means that the values obtained here might be higher than in reality.

With step length of 0.1 minutes and total time 100 min, we got for propane concentration 6.8e-07 mol/l.

It is good to remember that this doesn’t tell us the exact value of propane produced, rather just an idea of the order of magnitude. We don’t know the amount of enzymes in our cell so we had to estimate it. Also, our model has some simplifications that very possibly affect the outcome.

The predicted amount of propane produced isn't very high. To produce propane more efficiently in the future, we would need to greatly improve the performance of our pathway. This could be done by taking a look into the bottlenecks of our pathway and other limiting factors such as NADPH.

Discussion

If we had had time we could have measured the kinetic constants ourselves. This might have given us a bit more accurate results, but isn't really necessary since we have reliable sources for most of our constants. Perhaps the only part of our model that would have gained a lot from this would have been our mutated ADO; a recently found enzyme that isn't yet thoroughly researched. Right now we had to approximate its performance very crudely.

We could have gained a bit more accuracy for modeling if we knew the enzyme concentrations in the cell. Unfortunately we couldn't measure these due to time restrictions, but this doesn't affect our modeling results very much. The bottleneck results are based on the properties of the enzymes, not the amounts of them.

We have made some simplifying assumptions that might have some minor effects on the results, mainly assuming some reversible reactions irreversible. This shouldn't however be too much of a problem since those reactions are strongly forward-favored according to their kinetic properties.

In the future, our model could be improved by taking other reactions that happen in the cell into account. In our model we concentrated only on properties of our pathway, but for example NADPH and acetyl-CoA production are something that happen outside of the pathway and has a strong influence on the propane production.