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.
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28 May 2014
Research Article|
May 28 2014
Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations
Chinmaya Gupta;
Chinmaya Gupta
1Department of Mathematics,
University of Houston
, Houston, Texas 77004, USA
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José Manuel López;
José Manuel López
1Department of Mathematics,
University of Houston
, Houston, Texas 77004, USA
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Robert Azencott;
Robert Azencott
1Department of Mathematics,
University of Houston
, Houston, Texas 77004, USA
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Matthew R. Bennett;
Matthew R. Bennett
2Department of Biochemistry and Cell Biology,
Rice University
, Houston, Texas 77204, USA
and Institute of Biosciences and Bioengineering, Rice University
, Houston, Texas 77005, USA
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Krešimir Josić;
Krešimir Josić
1Department of Mathematics,
University of Houston
, Houston, Texas 77004, USA
3Department of Biology and Biochemistry,
University of Houston
, Houston, Texas 77204, USA
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William Ott
William Ott
1Department of Mathematics,
University of Houston
, Houston, Texas 77004, USA
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J. Chem. Phys. 140, 204108 (2014)
Article history
Received:
January 13 2014
Accepted:
May 07 2014
Citation
Chinmaya Gupta, José Manuel López, Robert Azencott, Matthew R. Bennett, Krešimir Josić, William Ott; Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations. J. Chem. Phys. 28 May 2014; 140 (20): 204108. https://doi.org/10.1063/1.4878662
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