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Stochastic Models of Biological Processes

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Many processes in cell biology, such as those that carry out metabolism, the cell cycle, and various types of signaling, are comprised ofbiochemical reaction networks. It has proven useful to study these networks using computer simulations because they allow us to quantitatively investigatehypotheses about the networks. Deterministic simulations are sufficient to predict average behaviors at the population level, but they cannot addressquestions about noise, random switching between stable states of the system, or the behaviors of systems with very few molecules of key species. Thesetopics are investigated with stochastic simulations. In this article, we review the dominant types of stochastic simulation methods that are used toinvestigate biochemical reaction networks, as well as some of the results that have been found with them. As new biological experiments continue to revealmore detail about biological systems, and as computers continue to become more...

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Abbreviations

Brownian dynamics:

A level of detail in which each molecule is represented by a point-like particle and molecules move in response to diffusion and collisions

Chemical Fokker–Planck equation (CFPE):

Master equation for well-mixed systems that corresponds to the chemical Langevin equation

Chemical master equation (CME):

Master equation for the probability that the system has specific integer copy numbers for each type of chemical species; it is exact for a well-mixed system

Chemical Langevin equation (CLE):

Approximate stochastic differential equation for well-mixed systems which is based on continuous Gaussian statistics

Direct method:

An implementation of the Gillespie algorithm

Extrinsic noise:

In genetic noise studies, expression fluctuations of a gene that arise from upstream genes or global fluctuations

First‐reaction method:

An implementation of the Gillespie algorithm

Gillespie algorithm:

Exact algorithm for simulating individual trajectories of the CME

Hybrid algorithms:

Algorithms that are designed to efficiently simulate systems that have multiple timescales

Individual‐based spatial models:

Models that track individual molecules as they diffuse or react

Intrinsic noise:

Expression fluctuations of a gene that arise from that particular gene

Jump process:

A process in which the system abruptly changes from one state to another

Optimized direct method:

A computationally efficient implementation of the Gillespie algorithm

Population‐based spatial models:

Models that track how many molecules of each chemical species are in various spatial compartments

Reaction channel:

A possible reaction between specified reactant and product chemical species (the terminology distinguishes this meaning from an individual reaction event between single molecules)

Reaction‐diffusion equation:

Deterministic partial differential equation that combines mass action reaction kinetics and normal chemical diffusion

Reaction‐diffusion master equation (RDME):

Chemical master equation that accounts for diffusion as well as reactions

Reaction rate equation (RRE):

Deterministic ordinary differential equation for the net production rate of each chemical species from chemical reactions

Spatial chemical Langevin equation:

Chemical Langevin equation that accounts for diffusion as well as reactions

Stochastic simulation algorithm (SSA):

Alternative term for the Gillespie algorithm

Stoichiometric matrix (ν):

Matrix that gives the net production of each chemical species, for each chemical reaction

Tau‐leaping method:

Approximate simulation method for well-mixed systems in which molecule numbers are updated using discrete Poisson statistics

Well‐mixed hypothesis:

Assumption that mixing processes occur faster than the relevant reaction processes

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Acknowledgments

This work was funded by the US Department of Energy.

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Authors and Affiliations

  1. Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, USA

    Steven S. Andrews, Tuan Dinh & Adam P. Arkin

  2. Department of Bioengineering and Howard Hughes Medical Institute, University of California, Berkeley, USA

    Adam P. Arkin

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Andrews, S.S., Dinh, T., Arkin, A.P. (2009). Stochastic Models of Biological Processes. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_524

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