title: Modeling delay in genetic networks: from delay birth-death processes to delay stochastic differential equations.
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 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.
authors.affiliation: Department of Mathematics, University of Houston, Houston, Texas 77004, USA.
authors.forename: Chinmaya
authors.initials: C
authors.lastname: Gupta
authors.affiliation: Department of Mathematics, University of Houston, Houston, Texas 77004, USA.
authors.forename: José Manuel
authors.initials: JM
authors.lastname: López
authors.affiliation: Department of Mathematics, University of Houston, Houston, Texas 77004, USA.
authors.forename: Robert
authors.initials: R
authors.lastname: Azencott
authors.affiliation: Department of Biochemistry and Cell Biology, Rice University, Houston, Texas 77204, USA and Institute of Biosciences and Bioengineering, Rice University, Houston, Texas 77005, USA.
authors.forename: Matthew R
authors.initials: MR
authors.lastname: Bennett
authors.affiliation: Department of Mathematics, University of Houston, Houston, Texas 77004, USA.
authors.forename: Krešimir
authors.initials: K
authors.lastname: Josić
authors.affiliation: Department of Mathematics, University of Houston, Houston, Texas 77004, USA.
authors.forename: William
authors.initials: W
authors.lastname: Ott
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D053263:Gene Regulatory Networks; D011506:Proteins / Q000737:chemistry; D012333:RNA, Messenger / Q000737:chemistry*; D015398:Signal Transduction; D013269:Stochastic Processes
chemical_list: D011506:Proteins; D012333:RNA, Messenger