Research article Special Issues

Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics


  • Received: 19 February 2022 Revised: 14 April 2022 Accepted: 18 April 2022 Published: 24 May 2022
  • 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.

    Citation: Daniele Cappelletti, Badal Joshi. Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7649-7668. doi: 10.3934/mbe.2022359

    Related Papers:

  • 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.



    加载中


    [1] D. F. Anderson, T. G. Kurtz, Continuous time markov chain models for chemical reaction networks, in Design and analysis of biomolecular circuits, Springer, (2011), 3–42. https://doi.org/10.1007/978-1-4419-6766-4_1
    [2] D. Schnoerr, G. Sanguinetti, R. Grima, Approximation and inference methods for stochastic biochemical kineticsa tutorial review, J. Phys. A: Math. Theor., 50 (2017), 093001. https://doi.org/10.1088/1751-8121/aa54d9 doi: 10.1088/1751-8121/aa54d9
    [3] S. Aoki, G. Lillacci, A. Gupta, A. Baumschlager, D. Schweingruber, M. Khammash, A universal biomolecular integral feedback controller for robust perfect adaptation, Nature, 570 (2019), 533–537. https://doi.org/10.1038/s41586-019-1321-1 doi: 10.1038/s41586-019-1321-1
    [4] T. Plesa, G. Stan, T. Ouldridge, W. Bae, Quasi-robust control of biochemical reaction networks via stochastic morphing, J. R. Soc. Interface, 18 (2021), 20200985. https://doi.org/10.1098/rsif.2020.0985 doi: 10.1098/rsif.2020.0985
    [5] J. Kim, G. Enciso, Absolutely robust controllers for chemical reaction networks, J. R. Soc. Interface, 17 (2020), 20200031. https://doi.org/10.1098/rsif.2020.0031 doi: 10.1098/rsif.2020.0031
    [6] H. Kang, T. Kurtz, Separation of time-scales and model reduction for stochastic reaction networks, Ann. Appl. Probab., 23 (2013), 529–583. https://doi.org/10.1214/12-AAP841 doi: 10.1214/12-AAP841
    [7] B. Brook, S. Waters, Mathematical challenges in integrative physiology, J. Math. Biol., 56 (2008), 893–896.
    [8] L. Preziosi, Hybrid and multiscale modelling, J. Math. Biol., 53 (2006), 977–978. https://doi.org/10.1007/s00285-006-0042-x doi: 10.1007/s00285-006-0042-x
    [9] A. Gupta, J. Mikelson, M. Khammash, A finite state projection algorithm for the stationary solution of the chemical master equation, J. Chem. Phys., 147 (2017), 154101. https://doi.org/10.1063/1.5006484 doi: 10.1063/1.5006484
    [10] J. Kuntz, P. Thomas, G. Stan, M. Barahona, Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations, SIAM Rev., 63 (2021), 3–64. https://doi.org/10.1137/19M1289625 doi: 10.1137/19M1289625
    [11] D. Anderson, G. Craciun, T. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bull. Math. Biol., 72 (2010), 1947–1970. https://doi.org/10.1007/s11538-010-9517-4 doi: 10.1007/s11538-010-9517-4
    [12] F. Horn, R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81–116.
    [13] M. Feinberg, Complex balancing in general kinetic systems, Arch. Ration. Mech. Anal., 49 (1972), 187–194. https://doi.org/10.1007/BF00255665 doi: 10.1007/BF00255665
    [14] G. Craciun, A. Dickenstein, A. Shiu, B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551–1565. https://doi.org/10.1016/j.jsc.2008.08.006 doi: 10.1016/j.jsc.2008.08.006
    [15] M. Gopalkrishnan, E. Miller, A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn., 13 (2014), 758–797. https://doi.org/10.1137/130928170 doi: 10.1137/130928170
    [16] B. Joshi, A. Shiu, Atoms of multistationarity in chemical reaction networks, J. Math. Chem., 51 (2013), 153–178. https://doi.org/10.1007/s10910-012-0072-0 doi: 10.1007/s10910-012-0072-0
    [17] B. Joshi, G. Craciun, Reaction network motifs for static and dynamic absolute concentration robustness, preprint, arXiv: 2201.08428.
    [18] M. Pérez Millán, A. Dickenstein, The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 1650–1682. https://doi.org/10.1137/17M1113722 doi: 10.1137/17M1113722
    [19] D. Cappelletti, A. Gupta, M. Khammash, A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances, J. R. Soc. Interface, 17 (2020), 20200437. https://doi.org/10.1098/rsif.2020.0437 doi: 10.1098/rsif.2020.0437
    [20] G. Craciun, B. Joshi, C. Pantea, I. Tan, Multistationarity in cyclic sequestration-transmutation networks, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01021-7 doi: 10.1007/s11538-022-01021-7
    [21] D. Cappelletti, C. Wiuf, Product-form Poisson-like distributions and complex balanced reaction systems, SIAM J. Appl. Math., 76 (2016), 411–432. https://doi.org/10.1137/15M1029916 doi: 10.1137/15M1029916
    [22] B. Joshi, A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1077–1105. https://doi.org/10.3934/dcdsb.2015.20.1077 doi: 10.3934/dcdsb.2015.20.1077
    [23] D. Cappelletti, B. Joshi, Graphically balanced equilibria and stationary measures of reaction networks, SIAM J. Appl. Dyn. Syst., 17 (2018), 2146–2175. https://doi.org/10.1137/17M1153315 doi: 10.1137/17M1153315
    [24] L. Hoessly, C. Mazza, Stationary distributions and condensation in autocatalytic reaction networks, SIAM J. Appl. Math., 79 (2019), 1173–1196. https://doi.org/10.1137/18M1220340 doi: 10.1137/18M1220340
    [25] D. F. Anderson, S. L. Cotter, Product-form stationary distributions for deficiency zero networks with non-mass action kinetics, Bull. Math. Biol., 78 (2016), 2390–2407. https://doi.org/10.1007/s11538-016-0220-y doi: 10.1007/s11538-016-0220-y
    [26] D. Anderson, T. Nguyen, Results on stochastic reaction networks with non-mass action kinetics, Math. Biosci. Eng., 16 (2019), 2118–2140. https://www.aimspress.com/article/doi/10.3934/mbe.2019103
    [27] F. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979.
    [28] P. Whittle, Systems in Stochastic Equilibrium, John Wiley & Sons, Inc., 1986.
    [29] T. Kurtz, Limit theorems and diffusion approximations for density dependent markov chains, in Stochastic Systems: Modeling, Identification and Optimization, I, Springer, 1976, 67–78.
    [30] T. Kurtz, Strong approximation theorems for density dependent Markov chains, Stoch. Proc. Appl., 6 (1978), 223–240. https://doi.org/10.1016/0304-4149(78)90020-0 doi: 10.1016/0304-4149(78)90020-0
    [31] S. Leite, R. Williams, A constrained Langevin approximation for chemical reaction network, Ann. Appl. Probab., 29 (2019), 1541–1608.
    [32] A. Angius, G. Balbo, M. Beccuti, E. Bibbona, A. Horvath, R. Sirovich, Approximate analysis of biological systems by hybrid switching jump diffusion, Theor. Comput. Sci., 587 (2015), 49–72. https://doi.org/10.1016/j.tcs.2015.03.015 doi: 10.1016/j.tcs.2015.03.015
    [33] J. Norris, Markov Chains, Cambridge university press, 1998.
    [34] D. Anderson, D. Cappelletti, M. Koyama, T. Kurtz, Non-explosivity of stochastically modeled reaction networks that are complex balanced, Bull. Math. Biol., 80 (2018), 2561–2579. https://doi.org/10.1007/s11538-018-0473-8 doi: 10.1007/s11538-018-0473-8
    [35] G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, preprint, arXiv: 1501.02860.
    [36] L. Paulevé, G. Craciun, H. Koeppl, Dynamical properties of discrete reaction networks, J. Math. Biol., 69 (2014), 55–72. https://doi.org/10.1007/s00285-013-0686-2 doi: 10.1007/s00285-013-0686-2
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1822) PDF downloads(73) Cited by(0)

Article outline

Figures and Tables

Figures(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog