Abstract

The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale. master equation ͉ stochastic delay equations ͉ noise ͉ time delay ͉ systems biology T here is considerable experimental evidence that noise can play a major role in gene regulation (1-10). These fluctuations can arise from either intrinsic sources, which are related to the small numbers of reactant biomolecules, or extrinsic sources, which are attributable to a noisy cellular environment. Although the importance of fluctuations in gene regulation was stressed Ͼ30 years ago (11), recent experimental advances have renewed interest in the stochastic modeling of the biochemical reactions that underlie gene regulatory networks (12-16). The most typical approaches are the utilization of the Gillespie algorithms One major difficulty often encountered in the analysis of gene regulatory networks is the vast separation of time scales between what are typically the fast reactions (dimerization, protein-DNA binding͞unbinding) and the slow reactions (transcription, translation, degradation). There have been many studies devoted to the development of reduced descriptions of these systems using the idea of quasiequilibrium for the fast processes compared with the slow dynamics (cf. ref. 22 and references therein). These approaches have thus far implicitly assumed that all of the reactions (fast and slow) are Markovian processes obeying Poissonian statistics. In this regard, it is important to note that the transcriptional and translational processes are not just slow but also are compound multistage reactions involving the sequential assembly of long molecules. Thus, by virtue of the central limit theorem, these processes should obey Gaussian statistics with a certain characteristic mean delay time. When delays in biochemical reactions are small compared with other significant time scales characterizing the genetic system, one can safely ignore them in simulations. Furthermore, time delays usually do not affect the quasiequilibrium behavior of gene regulatory networks and mean values of corresponding observables, and the conventional stochastic models without delays work properly here (for a review, see ref. 23). However, if indeed the time delays are of the order of other processes or longer, and the feedback loops associated with these delays are strong, taking the delays into account can be crucial for description of transient processes. This finding implies that when delay times are significant, both analytical and numerical modeling should take into account the nonMarkovian nature of gene regulation. The fact that delayed-induced stochastic oscillations can occur during transcriptional regulation is supported by recent studies of circadian oscillations in Neurospora, Drosophila, and others. It is widely accepted now that these oscillations are caused by delays in certain elements of gene regulation networks [see recent experimental (24, 25) and modeling (26-29) studies]. It is plausible that the role of time delays in circadian rhythms has come to light because the delays in the corresponding reactions are particularly long (several hours) in comparison with other characteristic times of the system. It would be logical to suppose that shorter delays present in other systems also can have a significant impact on dynamics; however, they may be more difficult to detect with currently available experimental methods. The behavior of stochastic delay-differential equations (SDDEs) has been extensively studied, and various approximation techniques have been developed and utilized (30-35). For example, delayed differential equations have been reduced to coupled map lattices and perturbed by white noise, demonstrating how the phase space density reaches a limit cycle in the asymptotic regime (30). The stability of the moment equations for linear SDDEs has been explored to elucidate the oscillatory properties of the first and second moments (31), and the master equation approach has been applied to a delayed random walker in an effort to demonstrate the effects of delay in an analytically tractable system (32, 33). The limit of short delay time has been used to show how a univariate nondelayed stochastic differential equation can be used to approximate the original system (34), and, more recently, a noise-driven bistable system with delayed feedback was reduced to a two-state model with delayed transition rates to demonstrate the phenomenon of coherence resonance (35). These important studies have provided many valuable insights, yet little work has been directed toward realistic delay times coupled with intrinsic noise in the context of gene regulation. In the present work, we develop methodologies for the analysis and simulation of delayed biochemical reactions that describe gene regulation. Specifically, we establish a theoretical approach for reducing and solving master equations that describe gene expres-