Team:Aalto-Helsinki/Modeling propane

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Under construction

Propane is a commonly used, convenient and clean-burning fuel, currently produced from non-renewable sources. Our project is about producing propane in bacteria, paving way for its sustainable production from renewable biomass. Ultimately, the pathway could be transferred to cyanobacteria, producing propane from CO2 and solar energy.

In our mathematical model our goal is to grasp the important concepts underlying the experiments made in the lab, and to see how those concepts could help us produce more propane. By having a better understanding of the ideas that govern our project, we could see the influence of each compound in the reaction pathway and have a basis to make decisions that would have a long term impact in our results.

We separated our modeling in four modules:

Derministic modeling of the reaction pathway

Finding bottlenecks in our reactions, identifying which substrates could be overproduced, and comprehending better the role each component, as the substrates concentrations, plays in our pathway are a few of the reasons we decided to do a deterministic model. With the help of differential equations applied to each reaction, we could have simultaneously a specific and a broad view of our pathway.

Sensitivity analysis

We wanted to go further in our understanding of the main reaction pathway. By completing our deterministic model, it became easier for us to interpret how each substrate affects another one in our system. This is crucial for us to then invest more resources in those substrates that affect the most our propane production, the main goal of this project.

Relative sensitivity of concentration in steady state, $s^{ss}$, with respect to variable $p$, is defined as \[\frac{p}{s^{ss}}\frac{ds^{ss}}{dp}.\] It relates the size of a relative perturbation in $p$ to a relative change in $s^{ss}$. If a system shows a small sensitivity coefficient with respect to a parameter, then behaviour is robust with respect to perturbations of that parameter. Large values suggest 'control points' at which interventions will have significant effects.

The steady state concentrations can be calculated from the basic differential equations. \[\begin{align*} \text{Acetoacetyl-CoA}(t) &= \frac{k_{AtoB}\, \text{AcetylCoA}^2}{k_{Hbd}[NADPH]} \\ \text{3-hydroxybutyryl-CoA}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{Crt}} \\ \text{Crotonyl-CoA}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{Ter}[NADH]} \\ \text{Butyryl-CoA}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{YciA}[H_2O]} \\ \text{Buryric acid}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{CAR}[ATP][H_2O][NADPH]}\\ \text{Butyraldehyde}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{ADO}[NADPH]^2[H]^2[O_2]}\\ \text{Propane}(t) &= \frac{k_{AtoB}\,\text{AcetylCoA}^2}{k_{out}} \end{align*}\]

Now the relative sensitivities are as follows: \[ \begin{array} {|l | c|c|c|c|c|c|c|} \hline & b(t) & c(t) & d(t) & e(t) & f(t) & g(t) & \text{Propane}(t) \\ \hline k_{AtoB} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline k_{Hdb} & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline k_{Crt} & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ \hline k_{Ter} & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ \hline k_{YciA} & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ \hline k_{CAR} & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ \hline k_{ADO} & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ \hline k_{out} & 0 & -1 & 0 & 0 & 0 & 0 & -1 \\ \hline \text{AcetylCoA} & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline \text{NADPH} & -1 & 0 & 0 & 0 & -1 & -2 & 0\\ \hline \text{NADH} & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ \hline H_2O & 0 & 0 & 0 & -1 & -1 & 0 & 0\\ \hline \text{ATP} & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ \hline \text{H} & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ \hline O_2 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ \hline \end{array} \] where $b$ is Acetoacetyl-CoA, $c$ is 3-hydroxybutyryl-CoA and so on, until $g$ is Butyraldehyde.

The table tells us which concentrations or speed constants affect the most to the reaction. It seems that the system is robust with respect to many perturbations of the parameters, and that the propane production could be controlled mainly trough Acetyl-CoA (and the speed of the first reaction). Also -2 values in NADPH and H must be noted.

Stability analysis

We wanted to know whether our pathway could produce propane steadily. In order to understand if this would be plausible, we performed a stability analysis of our reaction. To conclude these calculations, we used again the ideas behind our deterministic modeling.

Modeling the whole cell

Although our models tell a lot about the reaction pathway, they are still lacking since not everything can be taken into account. This is why we did modeling with Cobra toolbox, for which there is a ready model of e. coli that could be modified to take into account our pathway.