Simulating a Time Course

So far, we have examined the behavior of the erythrocyte model with the assumption that metabolites such as CO2, glucose, extracellular lactate, pyruvate, and phosphate remain at constant fixed concentrations; in other words, these metabolites are treated as so-called external metabolites. The assumption of this situation is reasonable because in situ the erythrocyte is exposed to approximately constant concentrations in the blood plasma. This enables the erythrocyte to establish a steady state of metabolite concentrations in this thermodynamically open system. However, in many important experimental conditions and many disease states, erythrocyte metabolism does not attain a steady state and the concentrations of these metabolites can no longer be treated as fixed in a simulation.

We finish this chapter with an example that reveals how the present model of erythrocyte metabolism(2-4) simulates time-dependent changes that occur over short times before the attainment of a global steady state of metabolite concentrations. This type of analysis is very useful for the interpretation of experimental data as well as for testing the behavior of the model under conditions that simulate inborn errors of enzyme function.

Q: Simulate the time course of glucose, total 2,3-BPG, and lactate concentrations that might occur in an experiment in which erythrocytes (hematocrit = 0.5) are incubated with an initial concentration of 10 mmol L-1 glucose and assuming constant CO2. Assume that during the time course the intracellular pH decreased linearly from 7.2 to 6.8 in 10 h.

A: The starting point for this simulation requires external parameters, initial conditions, and equations from the previous simulation. However, these are required to be modified to simulate the new time course.

First, the external parameters are identified and their concentrations specified. Four of the parameters that were defined as external ones in the previous model are now internal parameters. Hence we must Clear these values.

It is assumed that CO2 remains as an external parameter (which has already been assigned a value of 1.2 mmol L-1). In the model pH also remains an external parameter but its value will no longer be constant. Since pH is assumed to change linearly from 7.2 to 6.8 in 10 h, the following equation describes this change.

Glucose, external lactate, pyruvate, and phosphate are no longer deemed to be external parameters, so initial conditions are required for them; these are appended to the initial conditions list, ic1, which was defined previously. This is achieved as follows:

ic2 = Union[{Glc [0] == 10 x 10-3, Lace [0] == 1.82 x 10-3, Phose [0] == 1.92 x 10-3, Pyre[0] == 85 x 10-6} , ic1] ;

All that remains to be done is to define the matrices and vectors of the new model. To do this, use is made of the original equation list, eqns, but CO2 only is specified as an external parameter. Hence the new matrices and vectors are nVN = NMatrix[eqns, {CO2[t]}] ; nS = SMatrix[eqns, {CO2[t]}] ;

The reaction velocity vector remains unchanged from previously. Now simulate some selected time courses of concentrations.

sol2 = NDSolveMatrix[nS, nVN, V, ic2,

{t, 0, 36000}, AccuracyGoal^ 10, PrecisionGoal^ 10, WorkingPrecision^ 15, MaxSteps ^ 2000];

Plot [Evaluate [{Glc [t] , (Lac[t] Voli + Lac[t] Vole) / (Vole + Voli)} /. sol2], {t, 0, 30000}, AxesLabel -> {"Time", "Concentration (M)"}];

Concentration (M)

Concentration (M)

Figure 7.5. Simulated time course of glucose consumption (declining curve) and lactate production (increasing curve) for the model of human erythrocyte metabolism.

Plot[Evaluate[{B23PG[t] +Hb$B23PG[t] +Mg$B23PG[t]} /. sol2], {t, 0, 30000}, PlotRange -> {0., 8 x 10-3}, AxesLabel -> {"Time", "Concentration (M)"}];

Concentration (M)

Concentration (M)

Figure 7.6. Simulated time course of 2,3-BPG depletion in human erythrocytes.

These simulations demonstrate that in such an experiment the concentration of total 2,3-BPG would be expected to decline from an initial concentration of ~7 mmol L-1 to a concentration of ~0 mmol L-1 after 10 h. In fact, when NMR spectroscopy was used to follow the metabolite changes in such an experiment this is precisely what was observed.® The decline in 2,3-BPG was shown to be mainly due to the decrease in pH that occurs during the experiment and the fact that a low pH inhibits many enzymes of glycolysis and the 2,3-BPG shunt. The inhibition of glycolytic enzymes is reflected in the plot of glucose and lactate concentrations during the experiment; as the pH drops, the rate of glucose utilization (and lactate production) falls.

The role of pH in regulating erythrocyte metabolism is addressed in greater detail in the next chapter, but for now an easy way to demonstrate its importance is to perform Exercise 7.6.

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