Bulletin of Mathematical Biology

, Volume 65, Issue 6, pp 1111–1129 | Cite as

Michaelis-Menten kinetics at high enzyme concentrations

  • A. R. Tzafriri
Article

Abstract

The total quasi-steady state approximation (tQSSA) for the irreversible Michaelis-Menten scheme is derived in a consistent manner. It is found that self-consistency of the initial transient guarantees the uniform validity of the tQSSA, but does not guarantee the validity of the linearization in the original derivation of Borghans et al. (1996, Bull. Math. Biol., 58, 43–63). Moreover, the present rederivation yielded the noteworthy result that the tQSSA is at least roughly valid for any substrate and enzyme concentrations. This reinforces and extends the original assertion that the parameter domain for which the tQSSA is valid overlaps the domain of validity of the standard quasi-steady state approximation and includes the limit of high enzyme concentrations. The criteria for the uniform validity of the original (linearized) tQSSA are corrected, and are used to derive approximate solutions that are uniformly valid in time. These approximations overlap and extend the domains of validity of the standard and reverse quasi-steady state approximations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atkinson, D. E. (1977). Cellular Energy Metabolism and its Regulation, New York: Academic Press.Google Scholar
  2. Borghans, J. A. M., R. J. De Boer and L. A. Segel (1996). Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43–63.CrossRefMATHGoogle Scholar
  3. Briggs, G. E. and J. B. S. Haldane (1925). A note on the kinetics of enzyme action. Biochem. J. 19, 338–339.Google Scholar
  4. Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth (1996). On the Lambert W function. Adv. Comput. Math. 5, 329–359.MathSciNetCrossRefMATHGoogle Scholar
  5. Gutfreund, H. and B. R. Hammond (1959). Steps in the reaction of chymotrypsin with tyrosine derivatives. Biochem. J. 73, 526–530.Google Scholar
  6. Kruskal, M. D. (1963). Asymptotology, in Mathematical Models in Physical Science, S. Drobot (Ed.), New Jersey: Prentice-Hall, pp. 17–48.Google Scholar
  7. Li, G., A. S. Tomlin, H. Rabitz and J. Toth (1993). Determination of approximate lumping schemes by a singular perturbation method. J. Chem. Phys. 99, 3562–3574.CrossRefGoogle Scholar
  8. Lim, H. C. (1973). On kinetic behavior at high enzyme concentrations. AIChE J. 19, 659–661.CrossRefGoogle Scholar
  9. Lin, C. C. and L. A. Segel (1988). Mathematics Applied to Deterministic Problems in the Natural Sciences, Philadelphia: Society for Industrial and Applied Mathematics (SIAM), pp. 303–320.MATHGoogle Scholar
  10. Michaelis, L. and M. L. Menten (1913). Die kinetik der invertinwirkung. Biochem. Z. 49, 333–369.Google Scholar
  11. Schnell, S. and P. K. Maini (2000). Enzyme kinetics at high enzyme concentrations. Bull. Math. Biol. 62, 483–499.CrossRefGoogle Scholar
  12. Schnell, S. and P. K. Maini (2002). Enzyme kinetics far from the standard quasi-steady-state and equilibrium approximations. Math. Comput. Model 35, 137–144.MathSciNetCrossRefMATHGoogle Scholar
  13. Schnell, S. and C. Mendoza (1997). Closed form solution for time-dependent enzyme kinetics. J. Theor. Biol 187, 207–212.CrossRefGoogle Scholar
  14. Segel, L. A. (1988). On the validity of the steady-state assumption of enzyme kinetics. Bull. Math. Biol. 50, 579–593.MATHMathSciNetCrossRefGoogle Scholar
  15. Segel, L. A. and M. Slemrod (1989). The quasi-steady-state assumption: a case study in perturbation. SIAM Rev. 31, 446–477.MathSciNetCrossRefMATHGoogle Scholar
  16. Sols, A. and R. Marco (1970). Concentrations of metabolites and binding sites. Implications in metabolic regulation, in Current Topics in Cellular Regulation, Vol. 2, B. Horecker and E. Stadtman (Eds), New York: Academic Press, pp. 227–273.Google Scholar
  17. van Slyke, D. D. and G. E. Cullen (1914). The mode of action of urease and of enzymes in general. J. Biol. Chem. 19, 141–180.Google Scholar
  18. Tzafriri, A. R., M. Bercovier and H. Parnas (2002). Reaction diffusion modeling of the enzymatic erosion of insoluble fibrillar matrices. Biophys. J. 83, 776–793.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2003

Authors and Affiliations

  • A. R. Tzafriri
    • 1
  1. 1.Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations