Abstract
We provide a short review of stochastic modeling in chemical reaction networks for mathematical and quantitative biologists. We use as case studies two publications appearing in this issue of the Bulletin, on the modeling of quasi-steady-state approximations and cell polarity. Reasons for the relevance of stochastic modeling are described along with some common differences between stochastic and deterministic models.
This is a preview of subscription content, log in via an institution to check access.
Images from Xu et al. (2019) (Color figure online)
Similar content being viewed by others
References
Anderson DF, Cappelletti D (2018) Discrepancies between extinction events and boundary equilibria in reaction networks. arXiv:1809.04613 (Submitted)
Anderson DF, Craciun G, Kurtz TG (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bull Math Biol 72(8):1947–1970
Anderson DF, Enciso GA, Johnston MD (2014) Stochastic analysis of biochemical reaction networks with absolute concentration robustness. R Soc Interface 11:20130943
Bartholomay AF (1958) Stochastic models for chemical reactions. I. Theory of the unimolecular reaction process. Bull Math Biophys 20:175–190
Bartholomay AF (1959) Stochastic models for chemical reactions. II. The unimolecular rate constant. Bull Math Biophys 21:363–373
Benzi R, Sutera A, Vulpiani A (1999) The mechanism of stochastic resonance. J Phys A 14:L45301
Delbrück M (1940) Statistical fluctuations in autocatalytic reactions. J Chem Phys 8(1):120–124
Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338
Etienne-Manneville S (2004) Cdc42: the centre of polarity. J Cell Sci 117(8):1291–1300
Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187–194
Gagniuc PA (2017) Markov chains: from theory to implementation and experimentation. Wiley, New York
Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287
Gang H, Ditzinger T, Ning CZ, Haken H (1993) Stochastic resonance without external periodic force. Phys Rev Lett 71:807–810
Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361
Hahl SK, Kremling A (2016) A comparison of deterministic and stochastic modeling approaches for biochemical reaction systems: on fixed points, means, and modes. Front Genet 7:157
Horn FJM (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch Ration Mech Anal 49(3):172–186
Ingalls BP (2012) Mathematical modeling in systems biology: an introduction. MIT Press, Cambridge
Kang H-W, KhudaBukhsh WR, Koeppl H, Rempala GA (2019) Quasi-steady-state approximations derived from the stochastic model of enzyme kinetics. Bull Math Biol. https://doi.org/10.1007/s11538-019-00574-4
Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978
Lin C-C, Segel L (1988) Mathematics applied to deterministic problems in the natural sciences. SIAM, Philadelphia
McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Probab 4:413–478
Paulsson J (2005) Models of stochastic gene expression. Phys Life Rev 2(2):157–176
Paulsson J, Berg OG, Ehrenberg M (2000) Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc Natl Acad Sci USA 97(13):7148–7153
Pikovsky AS, Kurths J (1997) Coherence resonance in a noise-driven excitable system. Phys Rev Lett 78:775–778
Potvin-Trottier L, Lord ND, Vinnicombe G, Paulsson J (2016) Synchronous long-term oscillations in a synthetic gene circuit. Nature 538:514–517
Samoilov M, Plyasunov S, Arkin AP (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. Proc Natl Acad Sci USA 102(7):2310–2315
Segel L, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446–477
Székely T Jr, Burrage K (2014) Stochastic simulation in systems biology. Comput Struct Biotechnol J 12(20–21):14–25
Wang Q, Holmes WR, Sosnik J, Schilling T, Nie Q (2017) Cell sorting and noise-induced cell plasticity coordinate to sharpen boundaries between gene expression domains. PLoS Comput Biol 13(1):e1005307
Xu B, Jilkine A (2018) Modeling Cdc-42 oscillation in fission yeast. Biophys J 114(3):711–722
Xu B, Kang H-W, Jilkine A (2019) Comparison of deterministic and stochastic regime in a model for Cdc42 oscillations in fission yeast. Bull Math Biol. https://doi.org/10.1007/s11538-019-00573-5
Acknowledgements
We would like to thank our anonymous reviewer, who contributed significantly to this manuscript, and especially for his tireless reviews of Kang et al. (2019). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1616233.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Enciso, G., Kim, J. Embracing Noise in Chemical Reaction Networks. Bull Math Biol 81, 1261–1267 (2019). https://doi.org/10.1007/s11538-019-00575-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-019-00575-3