A review of the deterministic and diffusion approximations for stochastic chemical reaction networks

Abstract

This work reviews deterministic and diffusion approximations of the stochastic chemical reaction networks and explains their applications. We discuss the added value the diffusion approximation provides for systems with different phenomena, such as a deficiency and a bistability. It is advocated that the diffusion approximation can be considered as an alternative theoretical approach to study the reaction networks rather than a simulation shortcut. We discuss two examples in which the diffusion approximation is able to catch qualitative properties of reaction networks that the deterministic model misses. We provide an explicit construction of the original process and the diffusion approximation such that the distance between their trajectories is controlled and demonstrate this construction for the examples. We also discuss the limitations and potential directions of the developments.

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Funding

P. Mozgunov and T. Jaki have received funding from the European Union‘s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No 633567.

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Correspondence to Enrico Bibbona.

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Mozgunov, P., Beccuti, M., Horvath, A. et al. A review of the deterministic and diffusion approximations for stochastic chemical reaction networks. Reac Kinet Mech Cat 123, 289–312 (2018). https://doi.org/10.1007/s11144-018-1351-y

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Keywords

  • Bistable systems
  • Deficiency
  • Diffusion approximation
  • Hungarian construction
  • Reaction networks
  • Stochastic differential equations