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Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks

Part of: Mathematical biology in general Markov processes Limit theorems Chemistry Physiological, cellular and medical topics

Published online by Cambridge University Press:  08 October 2021

Chen Jia ,
Da-Quan Jiang  and
Youming Li
Chen Jia*
Affiliation:
Beijing Computational Science Research Center
Da-Quan Jiang*
Affiliation:
Peking University
Youming Li*
Affiliation:
Peking University
*
*Postal address: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China.
**Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China; Center for Statistical Science, Peking University, Beijing 100871, China. Email: [email protected]
***Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.
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Abstract

Detailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays a central role in the large deviation theory and the Freidlin–Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.

Keywords

Chemical reaction systemsmicroscopic reversibilityKolmogorov’s cycle conditionquasi-potentiallarge deviationmetastability

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J28: Applications of continuous-time Markov processes on discrete state spaces 60F10: Large deviations 92E20: Classical flows, reactions, etc.
Secondary: 92C40: Biochemistry, molecular biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 92B05: General biology and biomathematics
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 886 - 922
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Agazzi, A., Dembo, A. and Eckmann, J.-P. (2018). On the geometry of chemical reaction networks: Lyapunov function and large deviations. J. Statist. Phys. 172, 321–352.10.1007/s10955-018-2035-8CrossRefGoogle Scholar
Agazzi, A., Dembo, A., Eckmann, J.-P. (2018). Large deviations theory for Markov jump models of chemical reaction networks. Ann. Appl. Prob. 28, 1821–1855.CrossRefGoogle Scholar
Agazzi, A. and Mattingly, J. C. (2020). Seemingly stable chemical kinetics can be stable, marginally stable, or unstable. Commun. Math. Sci. 18, 1605–1642. Available at https://dx.doi.org/10.4310/ CMS.2020.v18.n6.a5.CrossRefGoogle Scholar
Alberty, R. A. (2004). Principle of detailed balance in kinetics. J. Chem. Educ. 81, 1206.CrossRefGoogle Scholar
Anderson, D. F., Cappelletti, D., Kim, J. and Nguyen, T. D. (2020). Tier structure of strongly endotactic reaction networks. Stoch. Process. Appl. 130, 7218–7259.10.1016/j.spa.2020.07.012CrossRefGoogle Scholar
Anderson, D. F., Craciun, G., Gopalkrishnan, M. and Wiuf, C. (2015). Lyapunov functions, stationary distributions, and non-equilibrium potential for reaction networks. Bull. Math. Biol. 77, 1744–1767.CrossRefGoogle ScholarPubMed
Anderson, D. F., Craciun, G. and Kurtz, T. G. (2010). Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol. 72, 1947–1970.CrossRefGoogle ScholarPubMed
Anderson, D. F. and Kurtz, T. G. (2015). Stochastic Analysis of Biochemical Systems. Springer, Cham.CrossRefGoogle Scholar
Chen, X. and Jia, C. (2020). Limit theorems for generalized density-dependent Markov chains and bursty stochastic gene regulatory networks. J. Math. Biol. 80, 959–994.10.1007/s00285-019-01445-1CrossRefGoogle ScholarPubMed
Conradi, C., Feliu, E., Mincheva, M. and Wiuf, C. (2017). Identifying parameter regions for multistationarity. PLoS Comput. Biol. 13, e1005751.10.1371/journal.pcbi.1005751CrossRefGoogle ScholarPubMed
Conradi, C. and Pantea, C. (2019). Multistationarity in biochemical networks: results, analysis, and examples. In Algebraic and Combinatorial Computational Biology, Elsevier, London, pp. 279–317.CrossRefGoogle Scholar
Cornish-Bowden, A. (2012). Fundamentals of Enzyme Kinetics, 4th edn. Wiley-Blackwell, Weinheim.Google Scholar
Delbrück, M. (1940). Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124.10.1063/1.1750549CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2009). Markov Processes: Characterization and Convergence. John Wiley, Hoboken, NJ.Google Scholar
Feinberg, M. (1989). Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44, 1819–1827.CrossRefGoogle Scholar
Freidlin, M. I. and Wentzell, A. D. (1998). Random perturbations. In Random Perturbations of Dynamical Systems, Springer, Berlin, Heidelberg, pp. 15–43.10.1007/978-1-4612-0611-8_2CrossRefGoogle Scholar
Gorban, A. N. and Yablonsky, G. S. (2011). Extended detailed balance for systems with irreversible reactions. Chem. Eng. Sci. 66, 5388–5399.CrossRefGoogle Scholar
Hiriart-Urruty, J.-B. and LemarÉchal, C. (2012). Fundamentals of Convex Analysis. Springer, Berlin, Heidelberg.Google Scholar
Horn, F. J. M. and Jackson, R. (1972). General mass action kinetics. Arch. Rat. Mech. Anal. 47, 81–116.10.1007/BF00251225CrossRefGoogle Scholar
Hornos, J. et al. (2005). Self-regulating gene: an exact solution. Phys. Rev. E 72, 051907.10.1103/PhysRevE.72.051907CrossRefGoogle Scholar
Jia, C. (2016). A solution to the reversible embedding problem for finite Markov chains. Statist. Prob. Lett. 116, 122–130.10.1016/j.spl.2016.04.020CrossRefGoogle Scholar
Jia, C. (2020). Kinetic foundation of the zero-inflated negative binomial model for single-cell RNA sequencing data. SIAM J. Appl. Math. 80, 1336–1355.10.1137/19M1253198CrossRefGoogle Scholar
Jia, C. and Grima, R. (2020). Small protein number effects in stochastic models of autoregulated bursty gene expression. J. Chem. Phys. 152, 084115.10.1063/1.5144578CrossRefGoogle ScholarPubMed
Jia, C., Qian, H., Chen, M. and Zhang, M. Q. (2018). Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks. J. Chem. Phys. 148, 095102.10.1063/1.5009749CrossRefGoogle Scholar
Jia, C., Wang, L. Y., Yin, G. G. and Zhang, M. Q. (2019). Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback. Phys. Rev. E 100, 052406.10.1103/PhysRevE.100.052406CrossRefGoogle Scholar
Jia, C., Zhang, M. Q. and Qian, H. (2017). Emergent LÉvy behavior in single-cell stochastic gene expression. Phys. Rev. E 96, 040402(R).10.1103/PhysRevE.96.040402CrossRefGoogle ScholarPubMed
Joshi, B. (2015). A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete Cont. Dynam. Systems B 20, 1077–1105.CrossRefGoogle Scholar
Joshi, B. and Shiu, A. (2013). Atoms of multistationarity in chemical reaction networks. J. Math. Chem. 51, 153–178.Google Scholar
Kolmogoroff, A. (1936). Zur Theorie der Markoffschen Ketten. Math. Ann. 112, 155–160.10.1007/BF01565412CrossRefGoogle Scholar
Kumar, N., Platini, T. and Kulkarni, R. V. (2014). Exact distributions for stochastic gene expression models with bursting and feedback. Phys. Rev. Lett. 113, 268105.CrossRefGoogle Scholar
Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes. J. Appl. Prob. 8, 344–356.10.2307/3211904CrossRefGoogle Scholar
Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57, 2976–2978.10.1063/1.1678692CrossRefGoogle Scholar
Kurtz, T. G (1978). Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223–240.10.1016/0304-4149(78)90020-0CrossRefGoogle Scholar
Lazarescu, A., Cossetto, T., Falasco, G. and Esposito, M. (2019). Large deviations and dynamical phase transitions in stochastic chemical networks. J. Chem. Phys. 151, 064117.CrossRefGoogle Scholar
Lee, J. M. (2013). Introduction to Smooth Manifolds, 2nd edn. Springer, New York.Google Scholar
Li, T. and Lin, F. (2017). Large deviations for two-scale chemical kinetic processes. Commun. Math. Sci. 15, 123–163.10.4310/CMS.2017.v15.n1.a6CrossRefGoogle Scholar
Niculescu, C. and Persson, L.-E. (2006). Convex Functions and Their Applications. Springer, Cham.10.1007/0-387-31077-0CrossRefGoogle Scholar
Olivieri, E. and Vares, M. E. (2004). Large Deviations and Metastability. Cambridge University Press.Google Scholar
Peccoud, J. and Ycart, B. (1995). Markovian modeling of gene-product synthesis. Theoret. Pop. Biol. 48, 222–234.10.1006/tpbi.1995.1027CrossRefGoogle Scholar
Qian, H. (2007). Phosphorylation energy hypothesis: open chemical systems and their biological functions. Annu. Rev. Phys. Chem. 58, 113–142.10.1146/annurev.physchem.58.032806.104550CrossRefGoogle ScholarPubMed
Sakmann, B. and Neher, E. (2009). Single-Channel Recording, 2nd edn. Springer, New York.Google Scholar
Shahrezaei, V. and Swain, P. S. (2008). Analytical distributions for stochastic gene expression. Proc. Nat. Acad. Sci. USA 105, 17256–17261.CrossRefGoogle Scholar
Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis: Queues, Communication and Computing. Chapman and Hall, New York.Google Scholar
Vellela, M. and Qian, H. (2008). Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J. R. Soc. Interface 6, 925–940.CrossRefGoogle ScholarPubMed
Walter, W. (1998). Ordinary Differential Equations. Springer, New York.CrossRefGoogle Scholar
Zhang, F. (2011). Matrix Theory: Basic Results and Techniques. Springer, New York.10.1007/978-1-4614-1099-7CrossRefGoogle Scholar
Zhang, X. J., Qian, H. and Qian, M. (2012). Stochastic theory of nonequilibrium steady states and its applications. Part I. Phys. Reports 510, 1–86.CrossRefGoogle Scholar