An enzyme E is synthesized at a constant rate, and degraded in a first order process, -ke*E, only when it is unbound. Input of S is sinusoidal. The system amplifies to give an oscillation in product P. From Reich and Selkov 1981 and really from Selkov 1968.
Grisolia (1964) noted that enzyme degradation was a normal and apparently regulated process, and that the rates of degradation were much higher when the enzyme was not bound to substrate. The system for cleaning out damaged proteins takes place mostly in lyzosomes where the proteins are taken apart and the amino acids made available for new protein building. The process, autophagy, or eating of self, was identified by Ciechanover (2012 for his review) as an important feature of cell survival, keeping the house clean and tidy, for which he was awarded a Nobel Prize. In mice, He and colleagues found that rates of autophagy were increased by exercise, and that the mice lived longer. The model, taken from Reich and Sel'kov (1981), assumes a constant rate of synthesis, vE, of the enzyme E by transcription from the genome, and lysis of E by a first order hydrolysis, kE. Substrate S is supplied at a sinusoidally varying rate Sinp. Under the conditions, the enzyme is almost fully bound most of the time, i.e. ES/(ES+E) is close to 1.0 so there is little degradataion. When S gets used up, ES -> E and degradation occurs.
Figure: (Top) Concentration time course. (Bottom) Phase plane diagram of Concentration of E and S versus enzyme substrate complex (ES) concentration. Simulation run with low equilbrium dissociation constant (keq = 1e-8 mM)
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Ciechanover A. Intracellular Protein Degradation: From a Vague Idea through the Lysosome and the Ubiquitin- Proteasome System and onto Human Diseases and Drug Targeting. Ramban Maimonides Med J 3: 1-20, 2012. He C, Bassik MC, Moresi V, Sun K, Wei Y, Zou Z, An Z, Loh J, Fisher J, Sun Q, Korsmeyer S, Packer M, May HI, Hill JA, Virgin HW, Gilpin C, Xiao G, Bassel-Duby R, Scherer PE, and Levine B. Exercise-induced BCL2-regulated autophagy is required for muscle glucose homeostasis. Nature advance online publication: 2012/01/18/online, 2012/01/18/online. Grisolia S. The catalytic environment and its biological implications. Physiol Rev 44: 657-714, 1964. Reich JG and Sel'kov EE. Energy Metabolism of the Cell: A Theoretical Treatise. London: Academic Press, 1981. (p 40) Selkov EE. Self-oscillations in glycolysis. 1. A simple kinetic model. European J Biochem 4: 79-86, 1968.
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