On constrained Langevin equations and (bio)chemical reaction networks

David F. Anderson, Desmond J. Higham, Saul C. Leite, Ruth J. Williams

Research output: Contribution to journalArticle

Abstract

Stochastic effects play an important role in modeling the time evolution of chemical reaction systems in fields such as systems biology, where the concentrations of some constituent molecules can be low. The most common stochastic models for these systems are continuous time Markov chains, which track the molecular abundance of each chemical species. Often, these stochastic models are studied by computer simulations, which can quickly become computationally expensive. A common approach to reduce computational effort is to approximate the discrete valued Markov chain by a continuous valued diffusion process. However, existing diffusion approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (chemical Langevin equation).

In this paper, we propose (obliquely) reflected diffusions, which respect the non-negativity of chemical concentrations, as approximations for Markov chain models of chemical reaction networks. These reflected diffusions satisfy “constrained Langevin equations,” in that they behave like solutions of chemical Langevin equations in the interior of the positive orthant and are constrained to the orthant by instantaneous oblique reflection at the boundary. To motivate their form, we first illustrate our constrained Langevin approximations for two simple examples. We then describe the general form of our proposed approximation. We illustrate the performance of our approximations through comparison of their stationary
distributions for the two examples with those of the Markov chain model and through simulations of more complex examples.
LanguageEnglish
Pages1-30
Number of pages30
JournalMultiscale Modeling and Simulation: A SIAM Interdisciplinary Journal
Volume17
Issue number1
DOIs
Publication statusPublished - 3 Jan 2019

Fingerprint

Chemical Reaction Networks
Langevin Equation
chemical reaction
Chemical reactions
chemical reactions
Markov chains
Markov processes
Markov chain
Reflected Diffusion
approximation
Approximation
Stochastic models
Markov Chain Model
Stochastic Model
Continuous time systems
Diffusion Approximation
Continuous-time Markov Chain
Nonnegativity
Systems Biology
Oblique

Keywords

  • density dependent Markov chains
  • diffusion approximation
  • Langevin equation
  • linear noise approximation
  • chemical reaction networks
  • stochastic differential equation with reflection
  • systems biology

Cite this

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abstract = "Stochastic effects play an important role in modeling the time evolution of chemical reaction systems in fields such as systems biology, where the concentrations of some constituent molecules can be low. The most common stochastic models for these systems are continuous time Markov chains, which track the molecular abundance of each chemical species. Often, these stochastic models are studied by computer simulations, which can quickly become computationally expensive. A common approach to reduce computational effort is to approximate the discrete valued Markov chain by a continuous valued diffusion process. However, existing diffusion approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (chemical Langevin equation). In this paper, we propose (obliquely) reflected diffusions, which respect the non-negativity of chemical concentrations, as approximations for Markov chain models of chemical reaction networks. These reflected diffusions satisfy “constrained Langevin equations,” in that they behave like solutions of chemical Langevin equations in the interior of the positive orthant and are constrained to the orthant by instantaneous oblique reflection at the boundary. To motivate their form, we first illustrate our constrained Langevin approximations for two simple examples. We then describe the general form of our proposed approximation. We illustrate the performance of our approximations through comparison of their stationarydistributions for the two examples with those of the Markov chain model and through simulations of more complex examples.",
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On constrained Langevin equations and (bio)chemical reaction networks. / Anderson, David F.; Higham, Desmond J.; Leite, Saul C.; Williams, Ruth J.

In: Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal , Vol. 17, No. 1, 03.01.2019, p. 1-30.

Research output: Contribution to journalArticle

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