Abstract
A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature
Albery, W. J. and J. R. Knowles. 1976. Evolution of enzyme function and the development of catalytic efficiency.Biochemistry 15, 5631–5639.
Belfiore, F. 1980.Enzyme Regulation and Metabolic Disease. Basel: S. Karger.
Brocklehurst, K. 1977. Evolution of enzyme catalytic power: Characteristics of optimal catalysis evaluated for the simplest plausible kinetic model.Biochem. J. 163, 111–116.
Burbaum, J. J. and J. R. Knowles. 1989. Internal thermodynamics of enzymes determined by equilibrium quench: Values ofK int for enolase and creatine kinase.Biochemistry 28, 9306–9317.
Burbaum, J. J., R. T. Raines, W. J. Albery and J. R. Knowles. 1989. Evolutionary optimization of the catalytic effectiveness of an enzyme.Biochemistry 28, 9293–9305.
Chin, J. 1983. Perfect enzymes: Is the equilibrium constant between the enzyme's bound species unity?J. Am. Chem. Soc. 105, 6502–6503.
Cornish-Bowden, A. 1976. The effect of natural selection on enzymic catalysis.J. Mol. Biol. 101, 1–9.
Crowley, P. H. 1975. Natural selection and the Michaelis constant.J. theor. Biol. 50, 461–475.
Eigen, M. and G. G. Hammes. 1963. Elementary steps in enzyme reactions.Adv. Enzymol. 25, 1–38.
Ellington, A. D. and S. A. Benner. 1987. Free energy differences between enzyme bound states.J. theor. Biol. 127, 491–506.
Fersht, A. R. 1974. Catalysis, binding and enzyme-substrate complementary.Proc. R. Soc. 187, 397–407.
Fersht, A. R. 1985.Enzyme Structure and Mechanism. New York: Freeman.
Heinrich, R. 1993. Mathematical Models of enzymic systems: Simulation, control analysis and optimization.Proceedings of the First World Congress of Nonlinear Analysts. Tampa, U.S.A. (in press).
Heinrich, R. and E. Hoffmann. 1991. Kinetic parameters of enzymatic reactions in states of maximal activity; an evolutionary approach.J. theor. Biol. 151, 249–283.
Heinrich, R., H.-G. Holzhütter and S. Schuster. 1987. A theoretical approach to the evolution and structural design of enzymatic networks; linear enzymatic chains, branched pathways and glycolysis of erythrocytes.Bull. math. Biol. 49, 539–595.
Heinrich, R., S. Schuster and H.-G. Holzhütter. 1991. Mathematical analysis of enzymic reaction systems using optimization principles.Eur. J. Biochem. 201, 1–21.
Lindskog, S., L. E. Henderson, K. K. Kannan, A. Liljas, P. O. Nyman and B. Strandberg. 1971. Carbonic anhydrase. InThe Enzymes. P. D. Boyer (Ed.). New York: Academic Press.
Lowry, O. H. and J. V. Passonneau. 1964. The relationships between substrates and enzymes of glycolysis in brain.J. biol. Chem. 239(1), 31–42.
Meléndez-Hevia, E. and N. V. Torres. 1988. Economy of design in metabolic pathways: Further remarks in the game of the pentose phosphate cycle.J. theor. Biol. 132, 97–111.
Meléndez-Hevia, E., T. G. Waddell and F. Montero. 1993. Selection of enzymatic mechanisms which account for simplicity in the evolution of metabolic pathways.Proceedings of the First World Congress of Nonlinear Analysts. Tampa, U.S.A. (in press).
Peller, L. and R. A. Alberty. 1959. Multiple intermediates in steady state enzyme kinetics. I. The mechanism involving a single substrate and product.J. Am. Chem. Soc. 81, 5907–5914.
Pettersson, G. 1989. Effect of evolution on the kinetic properties of enzymes.Eur. J. Biochem. 184, 561–566.
Pettersson, G. 1991. Why do many Michaelian enzymes exhibit an equilibrium constant close to unity for the interconversion of enzyme-bound substrate and product?Eur. J. Biochem. 195, 663–670.
Pettersson, G. 1992. Evolutionary optimization of the catalytic efficiency of enzymes.Eur. J. Biochem. 206, 289–295.
Schuster, S. and R. Heinrich. 1991. Minimization of intermediate concentrations as a suggested optimality principle for biochemical networks. I. Theoretical analysis.J. math. Biol. 29, 425–442.
Sel'kov, E. E., I. I. Goryagin, N. P. Kaimatchnikow, E. L. Shevelev and I. A. Yunus. 1989. Factographic data bank on enzymes and metabolic pathways.Stud. Biophys. 129, 155–164.
Silverman, D. N. and S. Lindskog. 1988. The catalytic mechanism of carbonic anhydrase: Implications of a rate-limiting proteolysis of water.Acc. Chem. Res. 21, 30–36.
Somogyi, B. and S. Damjanovich. 1975. Relationship between the lifetime of an enzyme-substrate complex and the properties of the molecular environment.J. theor. Biol. 48, 393–401.
Srivastava, D. K. 1991. Physiological constraints on evolution of enzymes for cellular metabolic pathways.J. theor. Biol. 152, 93–100.
Stackhouse, J., K. P. Nambiar, J. J. Burbaum, D. M. Stauffer and S. A. Benner. 1985. Dynamic transduction of energy and internal equilibria in enzymes: A reexamination of pyruvate kinase.J. Am. Chem. Soc. 107, 2757–2763.
Stucki, J. W. 1988. Thermodynamics of energy conversion in the cell. InFrom Chemical to Biological Organization, M. Markus, S. C. Müller and G. Nicolis (Eds). Berlin: Springer.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wilhelm, T., Hoffmann-Klipp, E. & Heinrich, R. An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms. Bltn Mathcal Biology 56, 65–106 (1994). https://doi.org/10.1007/BF02458290
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02458290