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Anderson DF, Howells AS. Stochastic Reaction Networks Within Interacting Compartments. Bull Math Biol 2023; 85:87. [PMID: 37624445 DOI: 10.1007/s11538-023-01185-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 06/30/2023] [Indexed: 08/26/2023]
Abstract
Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on [Formula: see text], and simulated via a version of the "Gillespie algorithm," have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in Duso and Zechner (Proc Nat Acad Sci 117(37):22674-22683 , Duso and Zechner (2020)), where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in Duso and Zechner (Proc Nat Acad Sci 117(37):22674-22683 , Duso and Zechner (2020)).
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Affiliation(s)
- David F Anderson
- Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
| | - Aidan S Howells
- Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA.
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Fan WTL, Yang YJ, Yuan C. Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:4322-4352. [PMID: 36896502 DOI: 10.3934/mbe.2023201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
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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Affiliation(s)
- Wai-Tong Louis Fan
- Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
| | - Yifan Johnny Yang
- Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
| | - Chaojie Yuan
- Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
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Cappelletti D, Joshi B. Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:7649-7668. [PMID: 35801439 DOI: 10.3934/mbe.2022359] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.
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Affiliation(s)
- Daniele Cappelletti
- DISMA-Dipartimento di Scienze Matematiche "G.L. Lagrange", Politecnico di Torino, Torino, Italy
| | - Badal Joshi
- Department of Mathematics, California State University San Marcos, San Marcos, USA
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Li YI, Turk G, Rohrbach PB, Pietzonka P, Kappler J, Singh R, Dolezal J, Ekeh T, Kikuchi L, Peterson JD, Bolitho A, Kobayashi H, Cates ME, Adhikari R, Jack RL. Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19. ROYAL SOCIETY OPEN SCIENCE 2021; 8:211065. [PMID: 34430050 PMCID: PMC8355677 DOI: 10.1098/rsos.211065] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/21/2021] [Accepted: 07/23/2021] [Indexed: 06/13/2023]
Abstract
Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.
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Affiliation(s)
- Yuting I. Li
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Günther Turk
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Paul B. Rohrbach
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Patrick Pietzonka
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Julian Kappler
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Rajesh Singh
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Jakub Dolezal
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Timothy Ekeh
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Lukas Kikuchi
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Joseph D. Peterson
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Austen Bolitho
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Hideki Kobayashi
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
| | - Michael E. Cates
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - R. Adhikari
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Robert L. Jack
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
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