The bulletin of mathematical biophysics

, Volume 31, Issue 4, pp 775–787 | Cite as

Some conditions for sustained oscillations in biochemical chains with feedback inhibition

  • H. D. Landahl


A chain ofn reactions is considered in which the last substance inhibits the production of the first with degreep, p being the order of the inhibition. Maintained oscillations are possible for certain values of the parameters under the following conditions: (1) If there is no time delay, then there must be at least three compartments (n=3) and either the degree of inhibition is sufficiently large (p>8 forn=3) or there must be enzymatic removal from the first compartment, in which casep≥1. (2) If there are time delays, but there is no enzymatic removal, the degree of the inhibition must be greater than or equal to 2 for any value ofn. (3) If there is a time delay in addition to enzymatic removal, one compartment with simple first order inhibition is sufficient. Conditions on the parameters necessary for maintained oscillations are given for many of the cases discussed.


Time Delay Vortex Point Biochemical System Constant Input Sustained Oscillation 
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  1. Danziger, L. and G. L. Elmergreen. 1954. “Mathematical Theory of Periodic Relapsing Catatonia”.Bull. Math. Biophysics,16, 15–21.CrossRefGoogle Scholar
  2. Goodwin, B. C. 1963.Temporal Organization in Cells. New York: Academic Press Inc.Google Scholar
  3. — 1965. “Oscillatory Behavior in Enzymatic Control Processes”.Advances in Enzyme Regulation,3, 425–438.CrossRefGoogle Scholar
  4. Griffith, J. S. 1968. “Mathematics of Cellular Control Processes. I. Negative Feedback to One Gene”.J. Theoretical Biology,20, 202–208.CrossRefGoogle Scholar
  5. Higgins, J. 1966. “The Theory of Oscillating Reactions”.Ind. Eng. Chem.,59, No. 5, 18–69.Google Scholar
  6. Morales, M. F. and D. McKay. 1967. “Biochemical Oscillations in ‘Controlled’ Mechanisms”.Biophys. J.,7, 621–625.CrossRefGoogle Scholar
  7. Rashevsky, N. 1964.Some Medical Aspects of Mathematical Biology. Springfield, Illinois: Charles C. Thomas, Publisher.Google Scholar
  8. —. 1968. “Mathematical Theory of Biological Periodicities: Formulation of then-Body Case”.Bull. Math. Biophysics,30, 735–749.MATHCrossRefGoogle Scholar
  9. —. 1968. “A Suggestion of a New Approach to the Theory of Some Biological Periodicities” —Ibid.,30, 751–760.MATHCrossRefGoogle Scholar
  10. Selkov, E. E. 1967. InOscillation Processes in Biological and Chemical Systems. Moscow, Academy of Sciences.Google Scholar
  11. Smith, J. M. 1968.Mathematical Ideas in Biology. New York and London: Cambridge University Press.Google Scholar
  12. Spangler, R. A. and F. M. Snell. 1961. “Sustained Oscillations in a Catalytic Chemical System”.Nature,191, 457–458.CrossRefGoogle Scholar
  13. Viniegra-Gonzales, G. and H. M. Martinez. 1969. “Stability of Biochemical Feedback Systems”.Biophys. Soc. Abstracts,9, 210.Google Scholar
  14. Walters, C. F. 1968. “A Model for Sustained Rhythmic Binary Logic in Biochemical Systems”. InQuantitative Biology of Metabolism, ed. A. Locker. Berlin: Springer-Verlag.Google Scholar
  15. Walters, C. F. 1969. “Oscillations in Controlled Biochemical Systems”.J. Theoretical Biology (in press).Google Scholar

Copyright information

© N. Rashevsky 1968

Authors and Affiliations

  • H. D. Landahl
    • 1
  1. 1.Department of Biochemistry and BiophysicsUniversity of California at San FranciscoSan FranciscoUSA

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