Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1261–1267 | Cite as

Embracing Noise in Chemical Reaction Networks

  • German EncisoEmail author
  • Jinsu Kim


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.


Stochasticity Chemical reaction network Noise Scaling limit Systems biology 



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.


  1. Anderson DF, Cappelletti D (2018) Discrepancies between extinction events and boundary equilibria in reaction networks. arXiv:1809.04613 (Submitted)
  2. Anderson DF, Craciun G, Kurtz TG (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bull Math Biol 72(8):1947–1970MathSciNetCrossRefzbMATHGoogle Scholar
  3. Anderson DF, Enciso GA, Johnston MD (2014) Stochastic analysis of biochemical reaction networks with absolute concentration robustness. R Soc Interface 11:20130943CrossRefGoogle Scholar
  4. Bartholomay AF (1958) Stochastic models for chemical reactions. I. Theory of the unimolecular reaction process. Bull Math Biophys 20:175–190MathSciNetCrossRefGoogle Scholar
  5. Bartholomay AF (1959) Stochastic models for chemical reactions. II. The unimolecular rate constant. Bull Math Biophys 21:363–373MathSciNetCrossRefGoogle Scholar
  6. Benzi R, Sutera A, Vulpiani A (1999) The mechanism of stochastic resonance. J Phys A 14:L45301MathSciNetGoogle Scholar
  7. Delbrück M (1940) Statistical fluctuations in autocatalytic reactions. J Chem Phys 8(1):120–124CrossRefGoogle Scholar
  8. Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338CrossRefGoogle Scholar
  9. Etienne-Manneville S (2004) Cdc42: the centre of polarity. J Cell Sci 117(8):1291–1300CrossRefGoogle Scholar
  10. Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187–194MathSciNetCrossRefGoogle Scholar
  11. Gagniuc PA (2017) Markov chains: from theory to implementation and experimentation. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  12. Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287CrossRefGoogle Scholar
  13. Gang H, Ditzinger T, Ning CZ, Haken H (1993) Stochastic resonance without external periodic force. Phys Rev Lett 71:807–810CrossRefGoogle Scholar
  14. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
  15. 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 7:157CrossRefGoogle Scholar
  16. Horn FJM (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch Ration Mech Anal 49(3):172–186MathSciNetCrossRefGoogle Scholar
  17. Ingalls BP (2012) Mathematical modeling in systems biology: an introduction. MIT Press, CambridgezbMATHGoogle Scholar
  18. 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. Google Scholar
  19. Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978CrossRefGoogle Scholar
  20. Lin C-C, Segel L (1988) Mathematics applied to deterministic problems in the natural sciences. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  21. McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Probab 4:413–478MathSciNetCrossRefzbMATHGoogle Scholar
  22. Paulsson J (2005) Models of stochastic gene expression. Phys Life Rev 2(2):157–176CrossRefGoogle Scholar
  23. Paulsson J, Berg OG, Ehrenberg M (2000) Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc Natl Acad Sci USA 97(13):7148–7153CrossRefGoogle Scholar
  24. Pikovsky AS, Kurths J (1997) Coherence resonance in a noise-driven excitable system. Phys Rev Lett 78:775–778MathSciNetCrossRefzbMATHGoogle Scholar
  25. Potvin-Trottier L, Lord ND, Vinnicombe G, Paulsson J (2016) Synchronous long-term oscillations in a synthetic gene circuit. Nature 538:514–517CrossRefGoogle Scholar
  26. 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):2310–2315CrossRefGoogle Scholar
  27. Segel L, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446–477MathSciNetCrossRefzbMATHGoogle Scholar
  28. Székely T Jr, Burrage K (2014) Stochastic simulation in systems biology. Comput Struct Biotechnol J 12(20–21):14–25CrossRefGoogle Scholar
  29. 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):e1005307CrossRefGoogle Scholar
  30. Xu B, Jilkine A (2018) Modeling Cdc-42 oscillation in fission yeast. Biophys J 114(3):711–722CrossRefGoogle Scholar
  31. 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. Google Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California, IrvineIrvineUSA

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