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Bounding the stationary distributions of the chemical master equation via mathematical programming

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Title: Bounding the stationary distributions of the chemical master equation via mathematical programming
Authors: Kuntz Nussio, J
Thomas, P
Stan, GB
Barahona, M
Item Type: Journal Article
Abstract: The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schl¨ogl’s model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerisation model with multiple stationary distributions.
Issue Date: 21-Jul-2019
Date of Acceptance: 24-Jun-2019
URI: http://hdl.handle.net/10044/1/70872
DOI: 10.1063/1.5100670
ISSN: 0021-9606
Publisher: AIP Publishing
Journal / Book Title: Journal of Chemical Physics
Volume: 151
Issue: 3
Replaces: 10044/1/62777
http://hdl.handle.net/10044/1/62777
Copyright Statement: ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license(http://creativecommons.org/licenses/by/4.0/)
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/N014529/1
EP/M002187/1
Keywords: Science & Technology
Physical Sciences
Chemistry, Physical
Physics, Atomic, Molecular & Chemical
Chemistry
Physics
AUGMENTED TRUNCATION APPROXIMATIONS
MARKOV-CHAINS
EQUILIBRIUM DISTRIBUTION
ERROR-BOUNDS
SQUARES
IDENTIFICATION
OPTIMIZATION
STABILITY
SYSTEMS
MODELS
Cell Biology
Mathematical Computing
Models, Biological
Models, Chemical
Stochastic Processes
Stochastic Processes
Mathematical Computing
Models, Biological
Models, Chemical
Cell Biology
math.PR
math.PR
math.OC
q-bio.MN
q-bio.PE
q-bio.QM
Chemical Physics
02 Physical Sciences
03 Chemical Sciences
09 Engineering
Publication Status: Published
Article Number: ARTN 034109
Online Publication Date: 2019-07-18
Appears in Collections:Faculty of Engineering
Bioengineering
Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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