A Linear Steady-State Treatment of Enzymatic Chains. A Mathematical Model of Glycolysis of Human Erythrocytes

Abstract

A mathematical model of glycolysis in the steady state of human erythrocytes is proposed based on an analytical treatment. Assuming a linear dependence of the enzyme velocities on the substrate concentrations, which is justified for all non-equilibrium enzymes, simple analytical expressions for all metabolites of the glycolysis could be obtained. From a consideration of the NADH/NAD-coupling between the lactate dehydrogenase and the glyceraldehyde-phosphate dehydrogenase reactions a conservation quantity for the concentrations of the oxidized metabolites is derived. The glycolytic flux was shown to be controlled only by the hexokinase and phosphofructokinase. The control strengths for the hexokinase and phosphofructokinase were calculated to be 0.69–0.73 and 0.31–0.27 at pH 7.2 and 0.87–0.9 and 0.13–0.1 at pH 8.2, respectively for various experimental conditions. The metabolites glucose 6-phosphate and fructose 6-phosphate are only influenced by the characteristic times of the hexokinase and phosphofructokinase. All other metabolites and the NADH/NAD ratio depend additionally on the characteristic time the pyruvate kinase. The level of 2,3-bisphosphoglycerate is influenced furthermore by the characteristic time of the bisphosphoglycerate mutase and by the capacity of the 2,3-bisphosphoglycerate phosphatase. The effects of changes in the kinetic properties of enzymes on the concentrations of metabolites may be expressed in form of a control matrix. The enzymes which interact with an outer effector may be identified by means of three guide parameters. The best combination is based on the determination of changes of glucose 6-phosphate, phosphoenol-pyruvate and the total flux. This method is much to be preferred to the usual interpretation of crossover-plots. It is shown that pseudo-crossovers occuring at the glyceraldehyde-phosphate dehydrogenase, phosphoglycerate kinase and pyruvate kinase are not indicative for an interaction site with an effector. The model was tested in a number of experimental conditions including variation of the pH, Pi concentration, NH4+-ion concentration and temperature for which crossovers have been described. It explains quantitatively the complicated pattern of changes. The conclusions on the properties of the enzymes drawn from the metabolite concentrations are consistent with their behaviour under isolated conditions. It is further shown that under conditions in vitro with lactate accumulation the levels of fructose bisphosphate, triosephosphate and the NADH/NAD ratio are time-dependent.