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Title: State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation

Abstract

The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of anmore » arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.« less

Authors:
ORCiD logo [1];  [2];  [1]
  1. Univ. of Illinois, Chicago, IL (United States). Dept. of Bioengineering; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Univ. of Illinois, Chicago, IL (United States). Dept. of Bioengineering
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
National Institutes of Health (NIH); National Science Foundation (NSF)
OSTI Identifier:
1412851
Report Number(s):
LA-UR-16-27505
Journal ID: ISSN 0092-8240; TRN: US1800372
Grant/Contract Number:  
AC52-06NA25396; MCB1415589; GM079804
Resource Type:
Accepted Manuscript
Journal Name:
Bulletin of Mathematical Biology
Additional Journal Information:
Journal Volume: 78; Journal Issue: 4; Journal ID: ISSN 0092-8240
Publisher:
Society for Mathematical Biology - Springer
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 97 MATHEMATICS AND COMPUTING; Computer Science; Mathematics; Stochastic biological networks, discrete chemical master equation; state space truncation

Citation Formats

Cao, Youfang, Terebus, Anna, and Liang, Jie. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation. United States: N. p., 2016. Web. doi:10.1007/s11538-016-0149-1.
Cao, Youfang, Terebus, Anna, & Liang, Jie. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation. United States. doi:10.1007/s11538-016-0149-1.
Cao, Youfang, Terebus, Anna, and Liang, Jie. Fri . "State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation". United States. doi:10.1007/s11538-016-0149-1. https://www.osti.gov/servlets/purl/1412851.
@article{osti_1412851,
title = {State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation},
author = {Cao, Youfang and Terebus, Anna and Liang, Jie},
abstractNote = {The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.},
doi = {10.1007/s11538-016-0149-1},
journal = {Bulletin of Mathematical Biology},
number = 4,
volume = 78,
place = {United States},
year = {2016},
month = {4}
}

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