Long delay times in reaction rates increase intrinsic fluctuations

Resonant noise amplification in a predator-prey model with quasi-discrete generations

Predator-prey models have been shown to exhibit resonance-like behaviour, in which random fluctuations in the number of organisms (demographic noise) are amplified when their frequency is close to the natural oscillatory frequency of the system. This behaviour has been traditionally studied in models with exponentially distributed replication and death times. Here we consider a biologically more realistic model, in which organisms replicate quasi-synchronously such that the distribution of replication times has a narrow maximum at some T > 0 corresponding to the mean doubling time. We show that when the frequency of replication f = 1 / T is tuned to the natural oscillatory frequency of the predator-prey model, the system exhibits oscillations that are much stronger than in the model with Poissonian (non-synchronous) replication and death. These oscillations lead to population instability and the extinction of one of the species much sooner than in the case of Poissonian replication. The effect can be explained by resonant amplification of coloured noise generated by quasi-synchronous replication events.

Modeling delay in genetic networks: from delay birth-death processes to delay stochastic differential equations

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner.

Stochastic delay accelerates signaling in gene networks

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.