Emergence of coherent oscillations in stochastic models for circadian rhythms

Most living organisms have developed the capability of generating autonomously sustained oscillations with a period close to 24h. The mechanism responsible for these circadian rhythms relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models for circadian rhythms account for the occurrence of autonomous oscillations of the limit cycle type, for their entrainment by light–dark cycles, and for their phase shifting by light pulses. Such models, however, do not take into consideration the molecular fluctuations which arise when the number of molecules involved in the regulatory mechanism is low. Here we resort to a stochastic description of a core model for circadian rhythms to study the emergence of coherent oscillations in gene expression in the presence of molecular noise. We show that despite the “bar code” pattern of gene activation, robust circadian oscillations can be observed. Simulations of the deterministic, fully developed version of the circadian model indicate, however, that sustained oscillations only emerge above a critical value of the rate constants characterizing the reversible binding of repressor to the gene, while below this value the system evolves towards an excitable steady state. This explains why, depending on whether or not the critical value of these rate constants is exceeded, stochastic simulations of the model produce coherent oscillations or very noisy oscillations with a highly variable period.