Some dynamic properties of linear, hyperbolic and sigmoidal multi-enzyme systems with feedback control

Abstract

Numerical solutions of equations (3) (see scheme I) have been obtained for a wide range of parameters. For cases wherein limit cycles arise, the relationships between the amplitude, the period, the nature of the rate laws for the enzyme-catalyzed reactions (e.g. linear, hyperbolic, or sigmoidal), the number of kinetically important steps, the metabolic flux into the system, and the stoichiometry of the feedback were determined. Periods and amplitudes for systems with hyperbolic rate laws usually exceed periods and amplitudes for systems with corresponding linear rate laws, and periods and amplitudes for systems with sigmoidal rate laws are usually less than the periods and amplitudes for corresponding linear or hyperbolic systems. Furthermore, limit cycles can arise in hyperbolic systems which, if linear, would possess global asymptotic stability; limit cycles can arise in linear systems which, if sigmoidal, would possess global asymptotic stability. These results have been observed for systems wherein all the rate laws have been changed in the indicated manner, and in systems where some of the rate laws have been changed. The effect is usually (but not always) a monotonic function of the number of altered rate laws. The rate law for the last step is usually the most important, and the rate law for the first step can be much more important than the rate law for the steps between the first and the last. The rate laws for all these intermediate steps are usually of approximately equal importance. A striking feature of these results is that there are exceptions to all the “rules”. The waveforms of the oscillations can be “smooth” or “squared”, depending upon the parameters (including the nature of the rate laws) used. The period and the amplitude usually increase or decrease together, but in certain cases the amplitude increases as a function of a parametric variation that causes the period to decrease. Even the individual relationships between the amplitude or the period and certain parameters is not always the same. For example, the amplitude is usually an asymptotic function of the flux into the system, but in some systems there is an optimum amplitude at an intermediate flux. The period can be nearly independent of the flux, or it can depend nearly linearly on the flux. One important aspect of this result is that the period can be in the range of a day or two, or in the range of seconds, but the period is most likely to be insensitive to the flux when the flux is high and the period long. At the end of this paper I have included some speculative remarks about how these results might relate to biological timing mechanisms or to cellular differentiation.