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The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions.

Abstract

BACKGROUND

It has been established that stochastic effects play an important role in spatio-temporal biochemical networks. A popular method of representing such stochastic systems is the Reaction Diffusion Master Equation (RDME). However, simulating sample paths from the RDME can be computationally expensive, particularly at large populations. Here we investigate an uncommon, but much faster alternative: the Spatial Chemical Langevin Equation (SCLE).

METHODS

We investigate moment equations and correlation functions analytically, then we compare sample paths and moments of the SCLE to the RDME and associated deterministic solutions. Sample paths are generated computationally by the Next Subvolume method (RDME) and the Euler-Maruyama method (SCLE), while a deterministic solution is obtained with an Euler method. We consider the Gray-Scott model, a well-known pattern generating system, and a predator-prey system with spatially inhomogeneous parameters as sample applications.

RESULTS

For linear reaction networks, it is well known that the first order moments of all three approaches match, that the RDME and SCLE match to the second moment, and that all approaches diverge at third order moments. For non-linear reaction networks, differential equations governing moments do not form a closed system, but a general moment equation can be compared term wise. All approaches match at the leading order, and the RDME and SCLE match at the second leading order. As expected, the SCLE captures many dynamics of the RDME where deterministic methods fail to represent them. However, areas of the parameter space in the Gray-Scott model exist where either the SCLE and RDME give qualitatively different predictions, or the RDME predicts patterns, while the SCLE does not.

CONCLUSIONS

The SCLE provides a fast alternative to existing methods for simulation of spatial stochastic biochemical networks, capturing many aspects of dynamics represented by the RDME. This becomes very useful in search of quantitative parameters yielding desired qualitative solutions. However, there exist parameter sets where both the qualitative and quantitative behaviour of the SCLE can differ when compared to the RDME, so care should be taken in its use for applications demanding greater accuracy.

Authors+Show Affiliations

Integrative Systems Biology Unit Okinawa Institute of Science and Technology, Okinawa, Japan. atiyo.ghosh@oist.jp.Okinawa Institute of Science and Technology, Okinawa, Japan. andre.leier@oist.jp.Integrative Systems Biology Unit Okinawa Institute of Science and Technology, Okinawa, Japan. tatiana.marquez@oist.jp.

Pub Type(s)

Journal Article
Research Support, Non-U.S. Gov't

Language

eng

PubMed ID

25888773

Citation

Ghosh, Atiyo, et al. "The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: Moments and Qualitative Solutions." Theoretical Biology & Medical Modelling, vol. 12, 2015, p. 5.
Ghosh A, Leier A, Marquez-Lago TT. The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions. Theor Biol Med Model. 2015;12:5.
Ghosh, A., Leier, A., & Marquez-Lago, T. T. (2015). The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions. Theoretical Biology & Medical Modelling, 12, p. 5. doi:10.1186/s12976-015-0001-6.
Ghosh A, Leier A, Marquez-Lago TT. The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: Moments and Qualitative Solutions. Theor Biol Med Model. 2015 Feb 27;12:5. PubMed PMID: 25888773.
* Article titles in AMA citation format should be in sentence-case
TY - JOUR T1 - The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions. AU - Ghosh,Atiyo, AU - Leier,Andre, AU - Marquez-Lago,Tatiana T, Y1 - 2015/02/27/ PY - 2014/12/09/received PY - 2015/01/28/accepted PY - 2015/4/19/entrez PY - 2015/4/19/pubmed PY - 2015/9/17/medline SP - 5 EP - 5 JF - Theoretical biology & medical modelling JO - Theor Biol Med Model VL - 12 N2 - BACKGROUND: It has been established that stochastic effects play an important role in spatio-temporal biochemical networks. A popular method of representing such stochastic systems is the Reaction Diffusion Master Equation (RDME). However, simulating sample paths from the RDME can be computationally expensive, particularly at large populations. Here we investigate an uncommon, but much faster alternative: the Spatial Chemical Langevin Equation (SCLE). METHODS: We investigate moment equations and correlation functions analytically, then we compare sample paths and moments of the SCLE to the RDME and associated deterministic solutions. Sample paths are generated computationally by the Next Subvolume method (RDME) and the Euler-Maruyama method (SCLE), while a deterministic solution is obtained with an Euler method. We consider the Gray-Scott model, a well-known pattern generating system, and a predator-prey system with spatially inhomogeneous parameters as sample applications. RESULTS: For linear reaction networks, it is well known that the first order moments of all three approaches match, that the RDME and SCLE match to the second moment, and that all approaches diverge at third order moments. For non-linear reaction networks, differential equations governing moments do not form a closed system, but a general moment equation can be compared term wise. All approaches match at the leading order, and the RDME and SCLE match at the second leading order. As expected, the SCLE captures many dynamics of the RDME where deterministic methods fail to represent them. However, areas of the parameter space in the Gray-Scott model exist where either the SCLE and RDME give qualitatively different predictions, or the RDME predicts patterns, while the SCLE does not. CONCLUSIONS: The SCLE provides a fast alternative to existing methods for simulation of spatial stochastic biochemical networks, capturing many aspects of dynamics represented by the RDME. This becomes very useful in search of quantitative parameters yielding desired qualitative solutions. However, there exist parameter sets where both the qualitative and quantitative behaviour of the SCLE can differ when compared to the RDME, so care should be taken in its use for applications demanding greater accuracy. SN - 1742-4682 UR - https://www.unboundmedicine.com/medline/citation/25888773/The_Spatial_Chemical_Langevin_Equation_and_Reaction_Diffusion_Master_Equations:_moments_and_qualitative_solutions_ L2 - https://tbiomed.biomedcentral.com/articles/10.1186/s12976-015-0001-6 DB - PRIME DP - Unbound Medicine ER -