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Results on stochastic reaction networks with non-mass action kinetics

  • Received: 12 December 2018 Accepted: 18 February 2019 Published: 12 March 2019
  • In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons [1]. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in [1], and proved it is a well known Lyapunov function [2]. In 2016, Cappelletti and Wiuf showed the converse of the main result in [1]: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced [3]. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in [1] are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.

    Citation: David F. Anderson, Tung D. Nguyen. Results on stochastic reaction networks with non-mass action kinetics[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2118-2140. doi: 10.3934/mbe.2019103

    Related Papers:

  • In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons [1]. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in [1], and proved it is a well known Lyapunov function [2]. In 2016, Cappelletti and Wiuf showed the converse of the main result in [1]: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced [3]. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in [1] are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.
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    [1] D. F. Anderson, G. Craciun and T. G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, B. Math. Biol., 72 (2010), 1947–1970.
    [2] D. F. Anderson, G. Craciun, M. Gopalkrishnan, et al., Lyapunov functions, stationary distributions, and non-equilibrium potential for reaction networks, B. Math. Biol., 77 (2015), 1744–1767.
    [3] D. Cappelletti and C.Wiuf, Product-form poisson-like distributions and complex balanced reaction systems, SIAM . Appl. Math., 76 (2016), 411–432.
    [4] D. F. Anderson and S. L. Cotter, Product form stationary distributions for deficiency zero networks with non-mass action kinetics, B. Math. Biol., 78(2016), 2390–2407.
    [5] D. F. Anderson, D. Cappelletti, M. Koyama, et al., Non-explosivity of stochastically modeled reaction networks that are complex balanced, B. Math. Biol., 80 (2018), 2561–2579.
    [6] A. Agazzi, A. Dembo and J. P. Eckmann, Large deviations theory for Markov jump models of chemical reaction networks, Ann. Appl. Probab., 28 (2018), 1821–1855.
    [7] H. Ge and H. Qian, Mathematical formalism of nonequilibrium thermodynamics for nonlinear chemical reaction systems with general rate law, J. Stat. Phys., 166 (2017), 190–209.
    [8] T. G. Kurtz, Representations of markov processes as multiparameter time changes, Ann. Prob., 8 (1980), 682–715.
    [9] C. Chan, X. Liu, L. Wang, et al., Protein scaffolds can enhance the bistability of multisite phosphorylation systems, PLoS Comput. Biol., 8 (2012), 1–9.
    [10] G. Gnacadja, Univalent positive polynomial maps and the equilibrium state of chemical networks of reversible binding reactions, Adv. Appl. Math., 43 (2009), 394–414.
    [11] H. W. Kang, L. Zheng and H. G. Othmer, A new method for choosing the computational cell in stochastic reaction–diffusion systems, J. Mathe. Biol., 65 (2012), 1017–1099.
    [12] E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading of t-cell receptor signal transduction, IEEE Trans. Auto. Cont., 46 (2001), 1028– 1047.
    [13] F. J. M. Horn and R. Jackson, General mass action kinetics, Arch. Rat. Mech. Anal, 47 (1972), 81–116.
    [14] M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. the deficiency zero and deficiency one theorems, review article 25, Chem. Eng. Sci., 42 (1987), 2229–2268.
    [15] M. Feinberg, Lectures on chemical reaction networks, Delivered at the Mathematics Research Center, Univ. Wisc.-Madison, (1979). Available from http://www.che.eng.ohio-state. edu/~feinberg/LecturesOnReactionNetworks.
    [16] J. Gunawardena, Chemical reaction network theory for in-silico biologists. Available from http: //vcp.med.harvard.edu/papers/crnt.pdf, (2003).
    [17] F. P. Kelly, Reversibility and stochastic networks, J. Wiley, 1979.
    [18] P. Whittle, Systems in stochastic equilibrium, J. Wiley, 1986.
    [19] H. G. Othmer, Y. Kim and M. A. Stolarska, The role of the microenvironment in tumor growth and invasion, Prog. Biophys. Mol. Bio., 106 (2011), 353–379.
    [20] D. F. Anderson and T. G. Kurtz, Continuous time Markov chain models for chemical reaction networks, in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology, Springer, (2011), 3–42.
    [21] D. F. Anderson and T. G. Kurtz, Stochastic analysis of biochemical systems, Springer, 2015.
    [22] T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stoch. Proc. Appl., 6 (1977/78), 223–240.
    [23] R. B. Paris and A. D. Wood, Asymptotics of high order differential equations, Pitman Research Notes in Mathematics Series, 1986.
    [24] G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, preprint, arXiv:1501.02860.
    [25] F. J. M. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rat. Mech. Anal, 49 (1972), 172–186.
    [26] V. Kazeev, M. Khammash, M. Nip, et al., Direct solution of the chemical master equation using quantized tensor trains, PLoS Comput. Biol., 10 (2014) ,e1003359. Available from: https:// doi.org/10.1371/journal.pcbi.1003359.

    © 2019 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)
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