Simulating the Time Dependent Behaviour of Multienzyme Systems

Before we begin our matrix approach to modelling, it is worthwhile re-examining the approach we have used so far. This is best illustrated with the following simple example.

Q: Determine the time dependence of the concentrations of reactants in the two-reaction sequence,

where v1 and v2 are described by simple reversible Michaelis-Menten rate equations (Section 2.3.3) with all Fmax values being 1 mmol L-1 h-1 and all Km values being 1 mmol L-1. Suppose that the initial concentrations are S1 [0] = 1 mmol L-1, S2 [0] = 0, and S3 [0] = 0.

A: The analysis begins by defining the rate expressions and assigning values to the various parameters. Then the differential equations are solved numerically with NDSolve yielding a result that is stored as an InterpolatingFunction (Section 1.4.6).

S1[0] == 1, S2[0] == 0, S3[0] == 0}, {S1, S2, S3}, {t, 0, 10}]

{{s1^InterpolatingFunction[{{0., 10.}}, <>] , s2 ^InterpolatingFunction[{{ 0., 10.}}, <>] , s3 ^InterpolatingFunction[{{ 0., 10.}}, <>]}}

Hence, plotting the time course of all three metabolites over the entire 10 h of simulated time we obtain the following:

Plot[Evaluate[{s1[t] , s2[t], s3[t]} /. timecourSe], {t, 0, 10}, AxesLabel -> {"Time (h)", "Concentration (mM)"}, PlotRange -> {0, 1}] ;

Concentration (mM)

Concentration (mM)

Figure 4.1. Time course of the reaction scheme shown in Eqn [4.1]. Upper curve is Si, middle curve S2, and lower curve S3.

Recall that we can evaluate the concentration of the three metabolites with the following command:

{si[t], s2[t] , s3[t]} / . timecourse / . t -> 1

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