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Single-enzyme systems can undergo weakly damped concentration oscillations if the enzyme can be converted to an inactive form which only slowly reverts to the active form. Inactivation can take many forms, including a simple (hysteretic) conformational change of the enzyme. This observation may provide an explanation for the damped oscillations seen in photosynthetic activity following a sudden disturbance since rubisco, the enzyme responsible for fixing carbon, is a hysteretic enzyme with appropriate kinetic constants for oscillation. Copyright 1998 Academic Press

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.

In a recent experimental study, Ouyang et al. (1998, Proc. Math. Acad. Sci. U.S.A. 95 , 8660–8664) have shown that, in direct competition, cyanobacterial strains whose circadian clocks have free-running periods (FRPs) which match the period of an imposed light/dark (LD) cycle exclude strains whose FRPs are out of resonance with the LD cycle. These differences in competitive fitness are observed despite the lack of measurable differences in monoculture growth rates between the strains. Here we show that the experimental results are consistent with a mathematical model in which cells rhythmically produce a metabolic inhibitor to which they display a sensitivity modulated by their circadian rhythm. We argue that models in which there is a circadian modulation of nutrient uptake kinetics cannot account for the results of these experiments. We discuss possible experiments to further characterize this phenomenon. The experimental protocol we propose can be used to distinguish between mutual inhibition and substrate depletion as underlying causes of the competitive advantage of circadian resonance.