Stochastic Simulation of Delay-Induced Circadian Rhythms in Drosophila

Almost all living organisms, including animals, plants, fungi, and cyanobacteria, exhibit daily periodic oscillations in their biochemical or physiological behavior, which are known as circadian rhythms [1–7]. The mechanism of circadian oscillation has been an extensive research topic in the last three decades. It has been found that circadian rhythms in fact are determined by oscillatory expression of certain genes [89]. Specifically, circadian clocks consist of a network of interlocked transcriptional-translational feedback loops formed by a number of genes [2]. In Drosophila, transcription of per and tim genes is activated by a heterodimer consisting of two transcriptional activators dCLOCK and CYCLE [10–13]. The PER protein in turn binds to the dCLOCK-CYCLE heterodimer, which inhibits the DNA binding activity of the dimer, thereby repressing the transcription of per and tim [11–14]. While this forms a negative feedback loop, there is also a positive feedback loop, in which PER and TIM activate dCLOCK synthesis by binding dCLOCK and relieving dCLOCK's repression of dclock transcription [1516].Several mathematical models have been proposed for circadian oscillation in Drosophila [121417–22]. The models of Smolen et al. [1214] introduce time delays in the expression of dclock and per genes, while other models do not have such delays. Numerical simulations using ordinary differential equations (ODE) show that all these models can produce circadian oscillations. In particular, times delays were found to be essential for simulation of circadian oscillations with the model of Smolen et al. [1214].Since there is significant stochasticity in gene expression arising from fluctuations in transcription and translation [23–25], it is desirable to simulate circadian oscillations in the presence of noise. Toward this end, several stochastic models were proposed [426–29], and Gillespie's stochastic simulation algorithm (SSA) [3031] was employed to simulate circadian oscillation