Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations
arXiv:1401.2682 · doi:10.1063/1.4878662
Abstract
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 that the distance between these two descriptions, as measured by expectations of functionals of the processes, converges to zero with increasing system size. Further, we prove that the delay birth-death process converges to the thermodynamic limit as system size tends to infinity. 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 spatial and temporal distributions of transition pathways in metastable systems, oscillatory behavior in negative feedback circuits, and cross-correlations between nodes in a network. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.
References in corpus (7)
- Noise-induced dynamics in bistable systems with delay
- Epigenetics as a first exit problem
- Transcriptional delay stabilizes bistable gene networks
- Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation
- Intrinsic fluctuations in stochastic delay systems: theoretical description and application to a simple model of gene regulation
- Role of delay in the stochastic creation process
- Long delay times in reaction rates increase intrinsic fluctuations