For the modeling component, the Fatty-Acid-Synthesis pathway of Synechococcus sp. PCC 7002 was modeled in R. Michaelis-Menten enzyme kinetics were used to compute the rate at which different steps in the reactions occurred, and information for calculating physical constants such as the turnover number came from several sources as listed below.
Minimal Impact of Concentration of FabD, FabZ and FabI on overall production: As is shown in the above plots, one can lower the concentrations of the various enzymes from the control value of 1 uM quite significantly (especially in the case of FabD) without significantly lowering the production rate of the system.
FabH findings: FabH was found to be the single most important enzyme in the pathway for optimizing production. The figure above shows the production of C16 fatty-acyl-ACP as a function of time for 3 different concentrations of fabH in the presence of all other enzymes at a control value concentration of 1 uM. As is shown above, FabH has some optimal value above and below which production slows. This is likely due to the interplay between FabF and FabH as they both consume the same reagent, Malonyl-ACP, which is needed once per cycle of elongation per molecule as well as once during initiation. As is shown below, FabH is clearly the rate limiting step in this model; upon lowering all enzyme concentrations to their minimum values at which the production rate does not decrease, we notice that the concentration of fabH is unchanged.
If we once again vary the concentration values for fabH at the adjusted values of concentration for all other enzymes(FabD = 0.01 uM,FabG = 0.004 uM, FabZ = 0.5 uM, FabI = 0.8 uM, FabF = 0.4 uM), we find that fabH clearly has an optimal value as is shown above.
Improved Ratio: As is shown above, tweaking the values of the different enzyme concentrations in this model can drastically impact the rate at which C16 Fatty-Acyl-ACP molecules are produced. One can lower all of the initial enzyme concentrations but slightly increase that of fabH and this actually increases the rate of production of C16 Fatty Acids. It turns out that a concentration for fabH of 1.06 uM is optimal under these conditions and yields a moderate increase in production rate.
In the process of modelling the Fatty-Acid-Synthesis (FAS) pathway of Synechococcus sp. PCC 7002, numerous assumptions were made and are detailed below. Constant Enzyme Concentration: In real cells, the enzyme concentration fluctuates with denaturation of the enzymes and the expression of genes encoding for those enzymes. Because the rate of denaturation as well as the rate of expression are not quantitatively understood, the underlying assumption in this model is that these two rates are equivalent and thereby the enzyme concentration is constant in the cell. Behavior of Dual Substrate Enzymes: In this model, it is assumed that there is an excess of one of the two reagents for all enzymes except FabF. Dual substrate enzyme kinetics behave like Michaelis-Menten single substrate enzyme kinetics when one reagent is in excess.
Ping-Pong Enzyme Kinetics with One Substrate in Excess For a dual substrate ping pong Enzyme, the rate in terms of kinetic constants is as follows for two substrates A and B:
Assuming that substrate B is in excess (has nonzero concentration), one can rearrange equation 1 by dividing both the numerator and denominator by the concentration of substrate B and take the limit as the concentration of substrate B approaches infinity to approximate the rate as only depending on the concentration substrate A.
Excess of Acyl-Carrier-Protein(ACP), Acetyl CoA, and NADPH: The assumption that these quantities are in excess does two things for this model. First, it acknowledges that it would be difficult to properly track the production of these compounds quantitatively in the cell, so these quantities are assumed to effectively have little influence over the rate of catalytic activity. Secondly, it allows the ping-pong enzyme rate assumption to hold true such that the catalytic rate is negligibly influenced by the concentration of these reactants so that their concentrations can be neglected. K50 for Fatty-Acyl-ACP and Malonyl-ACP is the same: Based on the limited data from (Kuo & Khosla, 2014), it was assumed that the k50 for Malonyl-ACP and Fatty-Acyl-ACP were the same quantity for the purpose of computing the catalytic rate of FabF. Kcat and k50 values: The k50 values and rates are assumed to be the same regardless of the size of the molecule acted on for all enzymes in the elongation cycle. Kcat was derived as shown below from Kuo and Khosla, 2014. Kcat*[E] was taken as Vmax in the equations for rate and the standard michaelis-menten equations were taken to yield the equations of rate for all enzymes except FabF. Fab F was taken to be a dual substrate enzyme and was calculated using the equation of rate from (Splittgerber, 1983).
Equations for kcatTo determine kcat from the enzyme data given in (Kuo & Khosla, 2014), the following equations were used.
In this equation, let the index n denote which enzyme in the initiation and elongation cycle for FAS (FabD->FabH->FabG->FabZ->FabI->FabF->FabG->FabZ->FabI-FabF->FabG...). Because the constant k50 describes the concentration at which the enzyme operates at half of its maximal rate, the above equation takes double this value to simulate the maximal rate. The maximal rate of the next enzyme is completely limited by the maximal rate of the previous enzyme. This equation thereby relies on this assumption to determine the value of kcat. Constant pH: Two protons are released per elongation cycle per molecule. The assumption of constant pH is one which relies on the fact that these protons don’t simply build up in the system during elongation. Thioesterase conversion of C16 Fatty-Acyl-ACP Only: This model assumes that only C16 Fatty-Acyl ACP molecules are operated on by the thioesterase to produce C16 fatty acids. This model also assumes that no further elongation beyond C16 occurs. Since data for the thioesterase was not used in this model, this model measures the buildup of C16 Fatty-Acyl-ACP as a benchmark for the production of long chain alkanes. All intermediates have nonzero initial concentrations: Due to the means by which the rates of production were calculated, it is necessary that there exists a nonzero initial concentration for some of the intermediates. These initial values are extremely small and simply serve the purpose of preventing a divide by zero error in the code.