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Stochastically modeled weakly reversible reaction networks with a single linkage class

Published online by Cambridge University Press:  04 September 2020

David F. Anderson
Affiliation:
University of Wisconsin-Madison
Daniele Cappelletti
Affiliation:
ETH Zurich
Jinsu Kim
Affiliation:
University of California at Irvine
Corresponding
E-mail address:

Abstract

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete-time Markov chain.

Type
Research Papers
Copyright
© Applied Probability Trust 2020

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References

Agazzi, A., Dembo, A. and Eckmann, J.-P. (2018). Large deviations theory for Markov jump models of chemical reaction networks. Ann. Appl. Prob. 28, 18211855.CrossRefGoogle Scholar
Anderson, D. F. (2011). Boundedness of trajectories for weakly reversible, single linkage class reaction systems. J. Math. Chem. 49, 22752290.CrossRefGoogle Scholar
Anderson, D. F. (2011). A proof of the global attractor conjecture in the single linkage class case. SIAM J. Appl. Math 71, 14871508.CrossRefGoogle Scholar
Anderson, D. F., Cappelletti, D., Kim, J. and Nguyen, T. (2018). Tier structure of strongly endotactic reaction networks. Preprint, arXiv:1808.05328.Google Scholar
Anderson, D. F., Cappelletti, D., Koyama, M. and Kurtz, T. G. (2018). Non-explosivity of stochastically modeled reaction networks that are complex balanced. Bull. Math. Biol. 80, 25612579.CrossRefGoogle ScholarPubMed
Anderson, D. F. and Cotter, S. L. (2016). Product-form stationary distributions for deficiency zero networks with non-mass action kinetics. Bull. Math. Biol. 78, 23902407.CrossRefGoogle ScholarPubMed
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, 17441767.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, 19471970.CrossRefGoogle ScholarPubMed
Anderson, D. F. and Ehlert, K. W. (2019). Conditional Monte Carlo for reaction networks. Preprint, arXiv:1906.05353.Google Scholar
Anderson, D. F. and Kim, J. (2018). Some network conditions for positive recurrence of stochastically modeled reaction networks. SIAM J. Appl. Math 78, 26922713.CrossRefGoogle Scholar
Ball, K., Kurtz, T. G., Popovic, L. and Rempala, G. A. (2006). Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Prob. 16, 19251961.CrossRefGoogle Scholar
Briat, C., Gupta, A. and Khammash, M. (2016). Antithetic integral feedback ensures robust perfect adaptation in noisy biomolecular networks. Cell Syst. 2, 1526.CrossRefGoogle ScholarPubMed
Briat, C., Gupta, A., andKhammash, M. (2018). Antithetic proportional-integral feedback for reduced variance and improved control performance of stochastic reaction networks. J. R. Soc. Interface 15, 20180079.CrossRefGoogle ScholarPubMed
Cappelletti, D. and Wiuf, C. (2016). Product-form Poisson-like distributions and complex balanced reaction systems. SIAM J. Appl. Math. 76, 411432.CrossRefGoogle Scholar
Cappelletti, D. and Wiuf, C. (2017). Uniform approximation of solutions by elimination of intermediate species in deterministic reaction networks. SIAM J. Appl. Dynam. Syst. 16, 22592286.CrossRefGoogle Scholar
Craciun, G., Dickenstein, A., Shiu, A. and Sturmfels, B. (2009). Toric dynamical systems. J. Symbolic Comput. 44, 15511565.CrossRefGoogle Scholar
Craciun, G. Tang, Y. andFeinberg, M. (2006). Understanding bistability in complex enzyme-driven networks. Proc. Nat. Acad. Sci. 103, 86978702.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley & Sons, New York.CrossRefGoogle Scholar
Feinberg, M. (1972). Complex balancing in general kinetic systems. Arch. Rational Mech. Anal. 49, 187194.CrossRefGoogle Scholar
Gopalkrishnan, M., Miller, E. and Shiu, A. (2013). A projection argument for differential inclusions, with applications to mass-action kinetics. SIGMA 9, 25.Google Scholar
Gupta, A., Mikelson, J. and Khammash, M. (2017). A finite state projection algorithm for the stationary solution of the chemical master equation. J. Chem. Phys. 147, 154101.CrossRefGoogle ScholarPubMed
Horn, F. J. M. (1972). Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Rat. Mech. Anal. 49, 172186.CrossRefGoogle Scholar
Horn, F. J. M. and Jackson, R. (1972). General mass action kinetics. Arch. Rat. Mech. Anal. 47, 81116.CrossRefGoogle Scholar
Jahnke, T. and Huisinga, W. (2007). Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54, 126.CrossRefGoogle ScholarPubMed
Kang, H.-W. and Kurtz, T. G. (2013). Separation of time-scales and model reduction for stochastic reaction networks. Ann. Appl. Prob. 23, 529583.CrossRefGoogle Scholar
Karp, R. L., Pérez Millán, M., Dasgupta, T., Dickenstein, A. and Gunawardena, J. (2012). Complex-linear invariants of biochemical networks. J. Theor. Biol. 311, 130138.CrossRefGoogle ScholarPubMed
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian Processes III: Foster–Lyapunov criteria for Continuous-Time Processes. Adv. Appl. Prob. 25, 518548.CrossRefGoogle Scholar
Munsky, B. and Khammash, M. (2006). The finite state projection algorithm for the solution of the chemical master equation. J. Chem Phys. 124, 044104.CrossRefGoogle ScholarPubMed
Norris, J. R. (1997). Markov Chains. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Paulevé, L., Craciun, G. and Koeppl, H. (2014). Dynamical properties of discrete reaction networks. J. Math. Biol. 69, 5572.CrossRefGoogle ScholarPubMed
Pfaffelhuber, P. and Popovic, L. (2015). Scaling limits of spatial compartment models for chemical reaction networks. Ann. Appl. Prob. 25, 31623208.CrossRefGoogle Scholar

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