Summary
For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.
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References
Allwright, D.: A global stability criterion for simple control loops. In press J. Math Biol. (1977).
Bünning, E.: The Physiological Clock. Circadian Rhythms and Biological Chronometry. London: The English Universities Press Ltd. (1973)
Cesari, L.: Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations. New York: Academic Press (1963)
Dieudonné, J.: Foundations of Modern Analysis. New York: Academic Press (1960)
Golenhofen, K.: Slow rhythms in smooth muscle (Minute-rhythms). Smooth Muscle, pp. 316–342 E. Bülbring, ed. (1970)
Hartman, P.: Ordinary Differential Equations. New York: Wiley and Sons (1964)
Hastings, S., Tyson, J., Webster, D.: Existence of periodic solutions for negative feedback cellular control systems. J. Diff. Eqn., 25, 39–64 (1977).
Hirsch, M. W., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press (1974)
Hopf, E.: Bifurcation of a periodic solution from a stationary solution of a system of differential equations. Berichte der Mathematisch-Physikalischen Klassen der Sächsischen Acad. Wissenschaften, Leipzig, XCIV. 3–22 (1942)
Hsu, J., Meyer, A.: Modern Control Principles and Applications. New York: McGraw-Hill (1968)
Johnsson, A.: Oscillatory water regulation in plants. Bul. Inst. Maths. Applics. 12, 22–26 (1976)
Johnsson, A., Karlsson, H. G.: A feedback model for biological rhythms, I. Mathematical description and basic properties of the model. J. theor. Biol. 36, 153–174 (1972)
Junge, D., Stephens, C. L.: Cyclic variation of potassium conductance in a burst-generating neurone in Aplysia. J. Physiol. 235, 155–181 (1973)
Karlsson, H. G., Johnsson, A.: A feedback model for biological rhythms, II. Comparisons with experimental results, especially on the petal rhythm of Kalanchoë, J. theor. Biol. 36, 175–194 (1972)
Kelley, A.: Note on an Invariant Manifold Theorem. Appendix C in: Transversal Mappings and Flows. (R. Abraham and J. Robbin) New York: Benjamin (1967)
King-Smith, E. A., Morley, A.: Computer simulation of granulopoiesis: Normal and im-paired granulopoiesis. Blood 36, 254–262 (1970)
MacDonald, N.: Bifurcation theory applied to a simple model of a biochemical oscillator. J. theor. Biol. 65, 727–734 (1977)
Mandelstam, J.: Co-ordination: Induction, Repression and Feedback Inhibition. In: Bichemistry of Bacterial Growth. 1st Edition, pp. 414–461. Edited by J. Mandelstam and K. McQuillen, Oxford and Edinburgh: Blackwell Scientific Publications (1968)
Marsden, J., McCracken, M.: The Hopf Bifurcation Theorem and its Applications. Lectures in Applied Mathematics. Berlin Heidelberg New York: Springer (1976)
Noble, D.: The Initiation of the Heartbeat. Oxford: Clarendon Press (1975)
Othmer, H. G.: The qualitative dynamics of a class of biochemical control circuits. J. math. Biology 3, 53–78 (1976)
Pavlidis, T.: Populations of biochemical oscillators as circadian clocks. J. theor. Biol. 33, 319–338 (1971)
Pittendrigh, C. S., Bruce, V. C.: An oscillator model for biological clocks. In: Rhythmic and Synthetic Processes in Growth, pp. 75–109. D. Rudnick ed. Princeton: Princeton University Press (1957)
Pontryagin, L. S.: Ordinary Differential Equations. (An English translation of the 1960 original) Reading, Mass.: Addison-Wesley (1962)
Poore, A. B.: Bifurcation of periodic orbits in a chemical reaction problem. Math. Biosciences 26, 99–107 (1975)
Rapp, P. E.: A theoretical investigation of a large class of biochemical oscillators. Math. Biosciences 25, 165–188 (1975)
Rapp, P. E.: Mathematical techniques for the study of oscillations in biochemical control loops. Bul. Inst. Maths. Applics. 12, 11–21 (1976)
Rapp, P. E., Berridge, M. J.: Oscillations in calcium-cyclic AMP control loops form the basis of pacemaker activity and other high frequency biological rhythms. J. theor. Biol. 66, 497–525 (1977)
Toates, F. M.: A model of an autonomic effector control loop. Measurement and Control 5, 354–357 (1972)
Tyson, J. J., Othmer, H. G.: The dynamics of feedback control circuits in biochemical path ways. In press. Progress in Theoretical Biology
Walter, C.: The absolute stability of certain types of controlled biological systems. J. theor. Biol. 23, 39–52 (1969)
Walter, C.: Kinetic and thermodynamic aspects of biological and biochemical control mechanisms. In: Biochemical Regularity Mechanisms in Eukaryotic Cells. Edited by F. Kun and S. Grisolia. pp. 355–489. New York: Wiley Interscience (1971)
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Mees, A.I., Rapp, P.E. Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks. J. Math. Biology 5, 99–114 (1978). https://doi.org/10.1007/BF00275893
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DOI: https://doi.org/10.1007/BF00275893