Abstract
The paper is concerned with the conditions of dynamic (asymptotic) stability of steady states in unbranched metabolic pathways. The stationary flux in such pathways is generally determined by the concentration of the end product due to the effector action of this product on the reactions proceeding in its synthetic pathway. The delay in feedback circuits causes violation of dynamic stability at large static stabilization factors. A method permitting analytic estimation of the critical stabilization factor is suggested. Sufficient and necessary conditions for asymptotic stability of the steady state in the general case of the pathway with a single feedback loop have been established. Mechanisms for maintenance of the steady state asymptotic stability at large static stabilization factors are studied. It has been shown that the range of dynamic stability can be widened greatly, if the pathway contains one or two reactions (but not more) of relatively small effective rate constants. Short strong negative feedback is also found to extend considerably the range of dynamic stability of the pathway. The feedback is more effective if it acts on the reaction with small effective rate constant.
Similar content being viewed by others
References
Aizermann, M. A.: Automatic control theory. Publ. House Nauka, Moscow 1966
Ataullakhanov, F. I., Vitvitsky, V. M., Zhabotinsky, A. M., Pichughin, A. V., Platonova, O. V., Kholodenko, B. N., Ehrlich, L. I.: The regulation of glycolysis in human erythrocytes. The dependence of the glycolytic flux on the ATP concentration. Eur. J. Biochem. 115, 359–365 (1981)
Atkinson, D. E.: The energy charge of the adenylate pool as a regulatory parameter. Interaction with feedback modifiers. Biochem. J. 7, 4030–4034 (1968)
Dibrov, B. F., Zhabotinsky, A. M., Kholodenko, B. N.: Local stability of the metabolic pathway with end-product inhibition. Biofizika 26, 590–595 (1981)
Dibrov, B. F., Zhabotinsky, A. M., Kholodenko, B. N.: Steady state dynamic stability and parametric stabilization in unbranched metabolic pathways. Biofizika 26, 790–795 (1981)
Hunding, A.: Limit-cycles in enzyme system with nonlinear negative feedback. Biophys. Struct. Mechanism 1, 47–54 (1975)
Meerov, M. B.: Synthesis of precision automatic control systems. Publ. House Nauka, Moscow 1967
Nazarenko, V. G., Sel'kov, E. E.: Mechanism of contact suppression as a possible source of ultralow frequency biological rhythms. In: Frank, G. M., Zhabotinsky, A. M., Molchanov, A. M., Shnol, S. (eds.). Oscillatory processes in biological and chemical systems, Vol. 2, pp. 145–148. Publ. by the USSR Academy of Sciences, Pushino-Oka 1971
Othmer, H. G.: The qualitative dynamics of a class of biochemical control circuits. J. Math. Biol. 3, 53–78 (1976)
Savageau, M. A.: Optimal design of feedback control by inhibition. Dynamic consideration. J. Mol. Evol. 5, 199–222 (1975)
Tyson, J. J., Othmer, H. G.: The dynamics of feedback control circuits in biochemical pathways. Progr. Theor. Biol. 5, 1–60 (1978)
Viniegra-Gonzales, G.: Stability properties of metabolic pathways with feedback interactions. In: Chance, B., Kendall Pye, E., Ghosh, A. K., Hess, B. (eds). Biological and biochemical oscillations, pp. 41–59. New York-London: Academic Press 1973
Walter, C. F.: Stability of controlled biological systems. J. Theoret. Biol. 23, 23–38 (1969)
Walter, C. F.: The occurrence and the significance of limit cycle behavior in controlled biochemical systems. J. Theoret. Biol. 27, 259–272 (1970)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dibrov, B.F., Zhabotinsky, A.M. & Kholodenko, B.N. Dynamic stability of steady states and static stabilization in unbranched metabolic pathways. J. Math. Biology 15, 51–63 (1982). https://doi.org/10.1007/BF00275788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275788