Bulletin of Mathematical Biology

, Volume 58, Issue 1, pp 43–63 | Cite as

Extending the quasi-steady state approximation by changing variables

  • José A. M. Borghans
  • Rob J. de Boer
  • Lee A. Segel


The parameter domain for which the quasi-steady state assumption is valid can be considerably extended merely by a simple change of variable. This is demonstrated for a variety of biologically significant examples taken from enzyme kinetics, immunology and ecology.


Parameter Domain Slow Time Scale Fast Time Scale Replication Model Total Substrate 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arditi, R. and L. R. Ginzberg. 1989. Coupling in predator-prey dynamics: ratio-dependence.J. Theor. Biol. 139, 311–326.Google Scholar
  2. Baker, G. A. and P. R. Graves-Morris. 1984. Padé approximants.Encyclopedia of Mathematics and its Applications. New York: Cambridge University Press.Google Scholar
  3. Beddington, J. R. 1975. Mutual interference between parasites or predators and its effects on searching efficiency.J. Anim. Ecol. 51, 597–624.Google Scholar
  4. Borghans, J. A. M. and R. J. De Boer. 1995. A minimal model for T-cell vaccination.Proc. R. Soc. London Ser. B 259, 173–178.Google Scholar
  5. Cha, S. 1970. Kinetic behavior at high enzyme concentrations.J. Biol. Chem. 245, 4814–4818.Google Scholar
  6. Cha, S. and C.-J. M. Cha. 1965. Kinetics of cyclic enzyme systems.Mol. Pharmacol. 1, 178–189.Google Scholar
  7. DeAngelis, D. L., R. A. Goldstein and R. V. O'Neil. 1975. A model for trophic interaction.Ecology 6, 881–892.CrossRefGoogle Scholar
  8. De Boer, R. J. and A. S. Perelson. 1994. T cell repertoires and competitive exclusion.J. Theor. Biol. 169, 375–390.CrossRefGoogle Scholar
  9. De Boer, R. J. and A. S. Perelson. 1995. Towards a general function for T cell proliferation.J. Theor. Biol. in press.Google Scholar
  10. Eigen, M. and P. Schuster. 1979.The Hypercycle: A Principle of Natural Self-Organization. Berlin: Springer.Google Scholar
  11. Goldstein, A. 1944. The mechanism of enzyme-inhibitor-substrate reactions.J. Gen. Physiol. 27, 529–580.CrossRefGoogle Scholar
  12. Lim, H. C. 1973. On kinetic behavior at high enzyme concentrations. AIChEJ19, 659–661.CrossRefGoogle Scholar
  13. Palsson, B. O. 1987. On the dynamics of the reversible Michaelis-Menten reaction mechanisms.Chem. Eng. Sci. 42, 447–458.CrossRefGoogle Scholar
  14. Reiner, J. M. 1969.The Behavior of Enzyme Systems. New York: Van Nostrand Reinhold.Google Scholar
  15. Segel, L. A. 1984.Modeling Dynamic Phenomena in Molecular and Cellular Biology. New York: Cambridge University Press.Google Scholar
  16. Segel, L. A. 1988. On the validity of the steady state assumption of enzyme kinetics.Bull. Math. Biol. 6, 579–593.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Segel, L. A. and M. Slemrod. 1989. The quasi-steady state assumption: a case study in perturbation.SIAM Rev. 31 446–477.zbMATHMathSciNetCrossRefGoogle Scholar
  18. Sols, A. and R. Marco. 1970. Concentration of metabolites and binding sites. Implications in metabolic regulation. InCurrent Topics in Cellular Regulation, Vol. 2. New York: Academic Press.Google Scholar
  19. Straus, O. H. and A. Goldstein. 1943. Zone behavior of enzymes.J. Gen. Physiol 26, 559–585.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1996

Authors and Affiliations

  • José A. M. Borghans
    • 1
  • Rob J. de Boer
    • 1
  • Lee A. Segel
    • 2
  1. 1.Theoretical BiologyUtrecht UniversityUtrechtThe Netherlands
  2. 2.Dept. of Applied Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations