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.

Description

   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)

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.