Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions

Wai-Tong (Louis) Fan; Yifan (Johnny) Yang; Chaojie Yuan; Wai-Tong (Louis) Fan; Yifan (Johnny) Yang; Chaojie Yuan

doi:10.3934/mbe.2023201

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 4322-4352. doi: 10.3934/mbe.2023201

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Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Academic Editor: Ankit Gupta

Received: 29 August 2022 Revised: 05 December 2022 Accepted: 08 December 2022 Published: 22 December 2022

The Togashi Kaneko model (TK model) is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. This CLA, derived under the classical scaling, is an obliquely reflected diffusion process on the positive orthant and hence respects the constraint that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent and converges exponentially fast to the unique stationary distribution. We also characterize the stationary distribution and show that it has finite moments. In addition, we simulate both the TK model and its CLA in various dimensions. For example, we describe how the TK model switches between meta-stable patterns in dimension six. Our simulations suggest that, when the volume of the vessel in which all of the reactions that take place is large, the CLA is a good approximation of the TK model in terms of both the stationary distribution and the transition times between patterns.
- reaction network,
- Langevin approximation,
- reflected diffusion,
- Feller property,
- stochastic simulation,
- stationary distribution,
- Lyapunov function,
- Harris recurrent,
- exponential ergodicity
Citation: Wai-Tong (Louis) Fan, Yifan (Johnny) Yang, Chaojie Yuan. Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4322-4352. doi: 10.3934/mbe.2023201

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Abstract

The Togashi Kaneko model (TK model) is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. This CLA, derived under the classical scaling, is an obliquely reflected diffusion process on the positive orthant and hence respects the constraint that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent and converges exponentially fast to the unique stationary distribution. We also characterize the stationary distribution and show that it has finite moments. In addition, we simulate both the TK model and its CLA in various dimensions. For example, we describe how the TK model switches between meta-stable patterns in dimension six. Our simulations suggest that, when the volume of the vessel in which all of the reactions that take place is large, the CLA is a good approximation of the TK model in terms of both the stationary distribution and the transition times between patterns.

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