Preliminary Simulations & Model Development

The BISEN toolset generated a pH-dependent model of hepatic glycolytic flux represented as a system of ordinary differential equations (ODEs) (Table 3-3), that are pH-dependent with respect to enzyme kinetics.

d[GLC] /dt = - JGLK + JG6PASE - JFD + JGLUT2 d[ATP] /dt = - JGLK - JPFK + 2 JPGK + 2 JPYK d[ADP] /dt = JGLK + JPFK - 2 JPGK - 2 JPYK d[G6P] /dt = JGLK - JG6PASE - JPGI d[F6P] /dt = JPGI - JPFK + JFBP1 d[Pi] /dt = 0 d[F16P] /dt = JPFK - JFBP1 - JALD d[BPG] /dt = JGAPDH - JPGK d[F26P] /dt = 0

d[DHAP] /dt = JALD - JTPI

d[GHAP] /dt = JALD + JTPI - JGAPDH

d[NAD] /dt = - JGAPDH + JLDH

d[NADH] /dt = JGAPDH - JLDH

d[PG2] /dt = JPGK - JPGYM

d[PG3] /dt = JPGYM - JENO

d[PEP] /dt = JENO - JPYK

d[PYR] /dt = JPYK - JLDH

d[LAC] /dt = JLDH - JLACT

d[GLC_e] /dt = - JGLUT2

d[LAC_e] /dt = JLACT

Variable time course and reaction flux simulations were generated by integration of model equations using the variable-order stiff solver ode15s in Matlab 2015b, coupled with relative integration and absolute integration tolerances of 1.0 ×10-4 _and

1.0 ×10-10 _{respectively. An additional solver option setting the maximum solver step}

size “MaxStep” was used, set to 5.0 ×10-2_{. For time course simulations, steady state}

solutions are reached when changes in all variable concentrations do not exceed the size of the absolute integration tolerance for a time interval of ≥2000 seconds. Rather than derive a steady state mathematically, this definition of a steady state is better suited to represent a large scale dynamic system. The initial preliminary simulation was conducted over a 300 minute time period. The primary aim of the initial simulations was to establish whether or not the BISEN toolset could successfully construct a model that could reproduce the well-defined states previously achieved in the literature.

Initial model simulations over a duration of 300 minutes yielded dynamic variable curves that deviate from initial conditions only slightly for the majority of the variables, predominantly remaining within the same order of magnitude, shown in Figure 3-4. A dynamic pH time course was simulated accounting for differences in pH in both cytoplasm and extracellular compartment. This was accomplished using the method described in chapter 1 and is a function of total proton stoichiometry within the model. The initial decrease in intracellular pH suggest an increase of intracellular proton concentration caused by an imbalance in the intracellular proton stoichiometry. Extracellular pH also decreased as a result of the MCT1 transporter successfully pumping protons into the extracellular environment.

Variable time course solutions failed to equilibrate by the end of the simulation. The reactions that influence the variables F16P, GAP, and DHAP were examined and altered in order to correctly reflect the homeostasis conditions in the unperturbed system.

to these variables, which may be causing instability. Figure 3-5 shows the total flux, J, of each reaction within the model during the simulation. The majority of the reaction fluxes were constant over the simulation, however, FBP and PYK were still in a dynamic state as the simulation concluded. Unsteady flux through the FBP reaction shown in the FPB flux plot in Figure 3-5 presented a possible explanation for instability of the F16P variable and therefore this variable and its associated reactions were examined first.

Plotting the individual reaction fluxes shows discrepancies in production and consumption fluxes specific to the variables of concern. Figure 3-6 highlights the reaction fluxes responsible for three variables of concern and one flux balanced variable BPG. The most important part of these simulations are the black cross and red circle plots that are the total production and consumption fluxes respectively. If the model were at steady state, these two plots would be the same shown in the F6P plot. The GAP plot shows only slight imbalance between the production and consumption with the two plots almost aligned. The DHAP flux plot clearly shows an imbalance of production and consumption fluxes, with the red consumption flux approximately 0.5 mM min-1 _lower.

The F16P flux terms are of higher concern as the production and consumption terms are neither close nor appear to be converging. In fact, towards the latter stages of the simulation they appear to diverge. From this qualitative flux balance analysis, it was decided that F16P is at this point the variable of highest concern, provoking investigation of the reactions responsible for its consumption and production.

Figure 3-6: Variable specific flux analysis: Reaction fluxes responsible for production and consumption of specific variables were separated by showing production

flux simulations in black and consumption fluxes in red. The sum of all production and consumption fluxes are plotted with crosses and circles respectively. F16P, DHAP and GAP are variables of concern, with F6P plotted as an example of balanced production and consumption

As previously discussed, there are significant pathway omissions in the BISEN generated pH-dependent model compared to the glucose metabolism model from which the kinetic terms were selected. For example, the pH-dependent model excludes glycogenolysis and the mitochondria along with its components. Focusing on differences between the two models around the F16P variable, two enzyme reactions were omitted directly linked to the production and consumption of F16P, phosphofructokinase (PFK2) 2 and fructose-2,6-bisphosphatase 2 (F26P) (Figure 3-7).

Omission of these enzymes may be responsible for the discrepancy highlighted in the flux balance analysis of F16P either by decreasing production flux of F16P via the FBP reaction from F26P, or increasing the consumption flux via the PFK2 reaction. Since F26P is held constant within the pH-dependent model used only as an allosteric reactant, it was decided to implement a removal term in the model for F16P in the form of a linear decay within the FBP reaction. The new kinetic term for FBP now includes a mass-action-based decay term with a new parameter kfbp1 which was manually adjusted to 2.0 ×10-3_. 𝐽𝐹𝐵𝑃 = 𝑣_𝑚𝑎𝑥 1 +[𝐹26𝑃]_𝑘 𝑖𝐹26𝑃 ( [𝐹16𝑃] [𝐹16𝑃] + 𝑘𝑚𝐹16𝑃)+ 𝑘𝑓𝑏𝑝1[𝐹16𝑃]. (3-2) Figure 3-7: Enzyme mediated removal and production omitted from the pH-dependent

The initial simulation was repeated to include the new FBP kinetic term, as well as setting any variable that is devoid of a sink to constant. This included extracellular pH, extracellular lactate and extracellular glucose. Furthermore, the simulation duration was extended to 3.0×104 _{minutes to allow the model to reach a steady state.}

A steady state of the model was shown by this extended variable time course simulation and flux balance analysis simulation in Figure 3-8 and Figure 3-9 respectively. Using both variable and flux simulations allows the model to be viewed on a component and system scale, as the fluxes are often described as the variables of the system. The variable concentrations must conform to the demands of the system in order to balance its fluxes in a steady state. Figure 3-8 and Figure 3-9 illustrate the models capability of having a steady state that satisfies both a component and system level.

Figure 3-9: Plots showing variable specific flux balance analysis time courses for the hepatic

glycolytic model. Reaction fluxes responsible for production and consumption of specific variables were separated by showing production flux simulations in black and consumption fluxes in red. The sum of all production and consumption fluxes are plotted with crosses and circles respectively.