Control Analysis of Periodic Phenomena in Biological Systems

Principles of the control and regulation of steady-state metabolic systems have been identified in terms of the concepts and laws of metabolic control analysis (MCA). With respect to the control of periodic phenomena MCA has not been equally successful. This paper shows why in case of autonomous (self-sustained) oscillations for the concentrations and reaction rates, time-dependent control coefficients are not useful to characterize the system: they are neither constant nor periodic and diverge as time progresses. This is because a controlling parameter tends to change the frequency and causes a phase shift that continuously increases with time. This recognition is important in the extension of MCA for periodic phenomena. For oscillations that are enforced with an externally determined frequency, the time-dependent control coefficients over metabolite concentration and fluxes (reaction rates) are shown to have a complete meaning. Two such time-dependent control coefficients are defined for forced oscillations. One, the so-called periodic control coefficient, measures how the stationary periodic movement depends on the activities of one of the enzymes. The other, the so-called transient control coefficient, measures the control over the transition of the system between two stationary oscillations, as induced by a change in one of the enzyme activities. For forced oscillations, the two control coefficients become equal as time tends to infinity. Neither in the case of forced oscillations nor in the case of autonomous oscillations is the sum of the time-dependent control coefficients time-independent, not even in the limit of infinite time. The sums of either type of control coefficients with respect to time-independent characteristics of the oscillations, such as amplitudes and time averages, do fulfill simple laws. These summation laws differ between forced oscillations and autonomous oscillations. The difference in control aspects between autonomous and forced oscillations is illustrated by examples.