Advertisement

Bulletin of Mathematical Biology

, Volume 49, Issue 5, pp 539–595 | Cite as

A theoretical approach to the evolution and structural design of enzymatic networks; Linear enzymatic chains, branched pathways and glycolysis of erythrocytes

  • Reinhart Heinrich 
  • Hermann-Georg Holzhütter 
  • Stefan Schuster 
Article
  • 85 Downloads

Abstract

A theoretical approach to the explanation of the structural design of metabolic pathway is presented. It is based on the hypothesis that due to natural selection during evolution the cellular metabolism of present-day organisms may be characterized by optimal properties. Two cardinal terms enter the theory: (i) the efficiency of a metabolic pathway and (ii) the evolutionary effort for the change of the kinetic parameters of enzymes by mutations of the corresponding genes. For both quantities simple mathematical expressions are proposed. While the efficiency is related to the reaction rates of the enzymes constituting the metabolic pathway, the evolutionary effort is considered to be a monotonically increasing function of the parameter values. By maximizing the efficiency under the constraint of a fixed evolutionary effort the theory allows the calculation of the optimal parameter distribution as the outcome of evolution processes.

The methods developed are applied to the following systems: (a) linear reaction sequences with very low affinities of the enzymes towards substrates, (b) linear sequences consisting of saturable enzymatic reactions, (c) branched metabolic pathways consisting of segments of linear chains and (d) glycolysis of erythrocytes. The conclusion is derived that the optimal distribution of kinetic constants depends strongly on the equilibrium constants of the reactions as well as on the total osmolarity of the metabolic intermediates. Without osmotic constraints the evolutionary effort is mainly spent on the enzymes at the beginning of the chain. Using Michaelis-Menten equations the optimal state is characterized by a decrease of the maximal activities of the enzymes towards the end of the chain. These results are modified if osmotic constraints are taken into account. At the investigation of branched pathways the following results were obtained: firstly, if a certain end product may be synthesized along different pathways those which are thermodynamically more unfavourable (e.g. characterized by a small change of free energy) are eliminated in the course of evolution; secondly, if a branched pathway leads to several important end products those reaction segments which are thermodynamically unfavourable are characterized by a higher evolutionary effort.

The application of the theory to a realistic model of glycolysis of erythrocytes leads to a correct description of various functionally important properties of the system, such as the ratio between fluxes through different branches and the ATP/ADP ratio, whereas the theory cannot predict the strong separation of time constants observed in the real glycolytic system. It is concluded that the improvement of the predictive power of the theory necessitates the use of more complex functionals for the efficiency which take into account not only the fluxes but also other system properties such as the stability of the pathway or homoeostatic effects.

Keywords

Structural Design Evolutionary Effort Kinetic Constant Metabolite Concentration Linear Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson. 1983Molecular Biology of the Cell. New York, Garland.Google Scholar
  2. Atkinson, D. E. 1969. “Limitation of Metabolite Concentrations and the Conservation of Solvent Capacity in the Living Cell.”Curr. Topics Cell. Regul. 1, 29–43.Google Scholar
  3. Ataullakhanov, F. I., V. M. Vitvitsky, A. M. Zhabotinsky. 1981a. “Regulation of Glycolysis in Human Erythrocytes. The Mechanism of ATP-concentration Stabilization.”Acta biol. med. germ. 40, 991–997.Google Scholar
  4. ———, A. V. Pichugin, O. V. Platonova, B. N. Kholodenko and L. I. Ehrlich. 1981b. “The Regulation of Glycolysis in Human Erythrocytes. The Response of the Glycolytic Flux on the ATP-concentration.”Eur. J. Biochem. 115, 359–365.CrossRefGoogle Scholar
  5. Brumen, M. and R. Heinrich. 1984. “A Metabolic Osmotic Model of Human Erythrocytes.”BioSystems 17, 155–169.CrossRefGoogle Scholar
  6. Chernavsky, D. S., E. I. Volkov and E. V. Presnov. 1976. “Extremality Principle as a Method for Selecting the Form of a Dynamic Model.”Biofizika XXI, 334–340.Google Scholar
  7. Decroly, O. and A. Goldbeter. 1982. “Birhythmicity, Chaos and other Patterns of Temporal Self-organization in a Multiply Regulated Biochemical System.”Proc. natl. Acad. Sci. U.S.A. 79, 6917–6921.zbMATHMathSciNetCrossRefGoogle Scholar
  8. Ebeling, W. and R. Feistel. 1982.Physik der Selbstorganisation und Evolution. Berlin: Akademie.Google Scholar
  9. Eigen, M. 1971. “Self-organization of Matter and the Evolution of Biological Macromolecules.”Naturwissenschaften 58, 465–523.CrossRefGoogle Scholar
  10. — and P. Schuster. 1978. “The Hypercycle. A Principle of Natural Self-organization.”Naturwissenschaften 65, 341–369.CrossRefGoogle Scholar
  11. Garfinkel, D. and B. Hess. 1964. “Metabolic Control Mechanisms. A Detailed Computer Model of the Glycolytic Pathway in Ascites Cells.”J. biol. Chem. 239, 971–983.Google Scholar
  12. Goldbeter, A. and R. Lefever. 1972. “Dissipative Structures for an Allosteric Model. Application to Glycolytic Oscillations.”Biophys. J. 12, 1302–1315.CrossRefGoogle Scholar
  13. Heinrich, R. and H. G. Holzhütter. 1985. “Efficiency and Design of Simple Metabolic Systems.”Biomed. biochim. Acta 44, 959–969.Google Scholar
  14. — and T. A. Rapoport. 1973. “A Linear Theory of Enzymatic Chains. Its Application for the Analysis of the Crossover Theorem and of the Glycolysis of Human erythrocytes.”Acta biol. med. germ. 31, 479–494.Google Scholar
  15. — and —. 1974. “A Linear Steady State Treatment of Enzymatic Chains.”Eur. J. Biochem. 42, 89–95.CrossRefGoogle Scholar
  16. —, S. M. Rapoport and T. A. Rapoport. 1977. “Metabolic Regulation and Mathematical Models.”Progr. Biophys. Mol. Biol. 32, 1–82.MathSciNetCrossRefGoogle Scholar
  17. — and I. Sonntag. 1981. “Analysis of Selection Equations for a Multivariable Population Model.”J. theor. Biol. 93, 325–361.MathSciNetCrossRefGoogle Scholar
  18. — and —. 1982. “Dynamics of Nonlinear Biochemical Systems and the Evolutionary Significance of Time Hierarchy.”BioSystems 15, 301–316.CrossRefGoogle Scholar
  19. Hengartner, W. and R. Theodorescu. 1978.Einführung in die Monte-Carlo-Methode. Berlin: Deutscher Verlag der Wissenschaften.zbMATHGoogle Scholar
  20. Higgins, J. 1964. “A Chemical Mechanism for Oscillation of Glycolytic Intermediates in Yeast Cells.”Proc. natl. Acad. Sci. U.S.A. 51, 989–994.CrossRefGoogle Scholar
  21. Holzhütter, H. G., G. Jacobasch and A. Bisdorf. 1985. “Mathematical Modelling of Metabolic Pathways Affected by an Enzyme Deficiency.”Eur. J. Biochem. 149, 101–111.CrossRefGoogle Scholar
  22. Kacser, H. and R. Beeby. 1984. “Evolution of Catalytic Proteins.”J. molec. Evol. 20, 38–51.CrossRefGoogle Scholar
  23. — and J. A. Burns. 1973. “The Control of Flux.”Symp. Soc. exp. Biol. 27, 65–104.Google Scholar
  24. Nicolis, G. and I. Prigogine. 1977.Self Organization in Non-equilibrium Systems. New York: Wiley & Sons.Google Scholar
  25. Rapoport, T. A. and R. Heinrich. 1975. “Mathematical Modelling of Multienzyme Systems. I. Modelling of the Glycolysis of Human Erythrocytes.”Biosystems 7, 120–129.CrossRefGoogle Scholar
  26. —, R. Heinrich, G. Jacobasch and S. M. Rapoport. 1974. “A Linear Steady State Treatment of Enzymatic Chains. A Mathematical Model of Glycolysis of Human Erythrocytes.”Eur. J. Biochem. 42, 107–120.CrossRefGoogle Scholar
  27. —, R. Heinrich and S. M. Rapoport. 1976. “The Regulatory Properties of Glycolysis in Erythrocytesin vivo andin vitro.”Biochem. J. 154, 449–469.Google Scholar
  28. Reich, J. G. 1984a. “Zur Ökonomie im Proteinhaushalt der lebenden Zelle.”Biomed. biochim. Acta 42, 839–848.Google Scholar
  29. — 1984b. “Über Selektionskriterien in metabolischen Systemen.”Biomed. biochim. Acta 43, 47–55.Google Scholar
  30. — 1985. “On Optimum Properties in the Design of Metabolic and Epigenetic Systems.”Biomed. biochim. Acta 44, 845–852.Google Scholar
  31. — and E. E. Selkov. 1981.Energy Metabolism of the Cell. London: Academic Press.Google Scholar
  32. Savageau, M. A. 1975. “Optimal Design of Feedback Control by Inhibition.”J. molec. Evol. 5, 199–222.CrossRefGoogle Scholar
  33. — 1976.Biochemical Systems Analysis. London: Addison-Wesley.zbMATHGoogle Scholar
  34. Schauer, M., R. Heinrich and S. Rapoport. 1981. “Mathematische Modellierung der Glykolyse und des Adeninnukleotidstoffwechsels menschlicher Erythrozyten.”Acta biol. med. germ. 40, 1659–1682.Google Scholar
  35. Schulmeister, T. and E. E. Selkov. 1978. “Folded Limit Cycles and Quasi-stochastic Self-oscillations in a Third Order Model of Open Biochemical Systems.”Stud. Biophys. 72, 111–112.Google Scholar
  36. Selkov, E. E. 1968. “Self Oscillations in Glycolysis.”Eur. J. Biochem. 4, 79–86.CrossRefGoogle Scholar
  37. — 1975. “Stabilization of Energy Charge, Generation of Oscillations and Multiple Steady States in Energy Metabolism as a Result of Purely Stoichiometric Regulation.”Eur. J. Biochem. 59, 151–157.CrossRefGoogle Scholar
  38. Umbarger, H. E. 1956. “Evidence for a Negative Feedback Mechanism in the Biosynthesis of Isoleucine.”Science,123, 848.Google Scholar
  39. Waley, S. G. 1964. “A Note on the Kinetics of Multi-enzyme Systems.”Biochem. J. 91, 514–517.Google Scholar
  40. Werner, A. and R. Heinrich. 1985. “A Kinetic Model for the Interaction of Energy Metabolism and Osmotic States of Human Erythrocytes.”Biomed. biochim. Acta 44, 185–212.Google Scholar
  41. Westerhoff, H. V., A. K. Groen and R. J. A. Wanders. 1984. “Modern Theories of Metabolic Control and Their Applications.”Bioscience Rep. 4, 1–22.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1987

Authors and Affiliations

  • Reinhart Heinrich 
    • 1
  • Hermann-Georg Holzhütter 
    • 2
  • Stefan Schuster 
    • 1
  1. 1.Humboldt-Universität zu Berlin, Sektion Biologie, Bereich BiophysikBerlinGDR
  2. 2.Humboldt-Universität zu Berlin, Institut für BiochemieBerlinGDR

Personalised recommendations