The creation of protein from DNA is a dynamic process consisting of numerous components that include transcription, translation and protein folding. Each of these components is further comprised of hundreds or thousands of sub-steps that must be completed before a fully mature protein is formed. 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. Instead of modeling each of these steps in detail, one way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. The stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed, complicating simulation and analysis. We examine this problem using different examples and approaches. First, we describe how queueing theory can be used 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 variance in protein production delay while holding the mean fixed increases signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. Next 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. We show that this behavior can be explained using a non-Markovian, analytically tractable reduced model. Finally, we explore the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. Simulations demonstrate that, if done correctly, a delay chemical Langevin approximation is accurate even at moderate system sizes. Together, these results provide a foundation for the implementation of detailed stochastic simulation algorithms in the study of the delay stochastic processes that model biochemical networks.Mathematics, Department o
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