A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced

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June 2015, 20(4): 1077-1105. doi: 10.3934/dcdsb.2015.20.1077

A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced

Badal Joshi ^1,

1.	Dept. of Mathematics, CSU San Marcos, 333 S. Twin Oaks Valley, San Marcos, CA 92096, United States

Received December 2013 Revised September 2014 Published February 2015

Abstract
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Certain chemical reaction networks (CRNs) when modeled as a deterministic dynamical system taken with mass-action kinetics have the property of reaction network detailed balance (RNDB) which is achieved by imposing network-related constraints on the reaction rate constants. Markov chains (whether arising as models of CRNs or otherwise) have their own notion of detailed balance, imposed by the network structure of the graph of the transition matrix of the Markov chain. When considering Markov chains arising from chemical reaction networks with mass-action kinetics, we will refer to this property as Markov chain detailed balance (MCDB). Finally, we refer to the stochastic analog of RNDB as Whittle stochastic detailed balance (WSDB). It is known that RNDB and WSDB are equivalent. We prove that WSDB and MCDB are also intimately related but are not equivalent. While RNDB implies MCDB, the converse is not true. The conditions on rate constants that result in networks with MCDB but without RNDB are stringent, and thus examples of this phenomenon are rare, a notable exception is a network whose Markov chain is a birth and death process. We give a new algorithm to find conditions on the rate constants that are required for MCDB.

Keywords: stationary distribution., Detailed balance, stochastic models, chemical reaction network theory.

Mathematics Subject Classification: Primary: 37C25, 34A34, 60J2.

Citation: Badal Joshi. A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1077-1105. doi: 10.3934/dcdsb.2015.20.1077

References:

[1]	D. F. Anderson, G. Craciun and T. G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, 72 (2010), 1947-1970. doi: 10.1007/s11538-010-9517-4.
[2]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archive for Rational Mechanics and Analysis, 19 (1965), 81-99. doi: 10.1007/BF00282276.
[3]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions II. some addenda, Archive for Rational Mechanics and Analysis, 27 (1968), 356-364. doi: 10.1007/BF00251438.
[4]	H. Casimir, Some aspects of Onsager's theory of reciprocal relations in irreversible processes, Il Nuovo Cimento, 6 (1949), 227-231. doi: 10.1007/BF02780985.
[5]	A. Dickenstein and M. P. Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828. doi: 10.1007/s11538-010-9611-7.
[6]	R. Durrett, Probability: Theory and examples, Cambridge University Press, 2010. doi: 10.1017/CBO9780511779398.
[7]	M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827. doi: 10.1016/0009-2509(89)85124-3.
[8]	M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.
[9]	M. Feinberg, Lectures on Chemical Reaction Networks, Notes of lectures given at the Mathematics Research Center of the University of Wisconsin in 1979, available at http://www.crnt.osu.edu/LecturesOnReactionNetworks, 1979.
[10]	K. Gatermann, M. Eiswirth and A. Sensse, Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, Journal of Symbolic Computation, 40 (2005), 1361-1382. doi: 10.1016/j.jsc.2005.07.002.
[11]	J. Gunawardena, Chemical reaction network theory for in-silico biologists, Notes available at http://vcp.med.harvard.edu/papers/crnt.pdf, 2003.
[12]	F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.
[13]	F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.
[14]	B. Joshi, Complete characterization by multistationarity of fully open networks with one non-flow reaction, Applied Mathematics and Computation, 219 (2013), 6931-6945. doi: 10.1016/j.amc.2013.01.027.
[15]	B. Joshi and A. Shiu, Simplifying the Jacobian Criterion for precluding multistationarity in chemical reaction networks, SIAM Journal on Applied Mathematics, 72 (2012), 857-876. doi: 10.1137/110837206.
[16]	B. Joshi and A. Shiu, Atoms of multistationarity in chemical reaction networks, Journal of Mathematical Chemistry, 51 (2013), 153-178.
[17]	B. Joshi and A. Shiu, A survey of methods for deciding whether a reaction network is multistationary,, arXiv preprint, ().
[18]	F. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979.
[19]	G. Lewis, A new principle of equilibrium, Proceedings of the National Academy of Sciences of the United States of America, 11 (1925), 179-183. doi: 10.1073/pnas.11.3.179.
[20]	L. Onsager, Reciprocal relations in irreversible processes. I, Physical Review, 37 (1931), 405-426. doi: 10.1103/PhysRev.37.405.
[21]	L. Paulevé, G. Craciun and H. Koeppl, Dynamical properties of discrete reaction networks, Journal of mathematical biology, 69 (2014), 55-72. doi: 10.1007/s00285-013-0686-2.
[22]	P. Whittle, Systems in Stochastic Equilibrium, John Wiley & Sons, Inc., 1986.
[23]	E. Wigner, Derivations of Onsager's reciprocal relations, The Collected Works of Eugene Paul Wigner, A/4 (1997), 215-218. doi: 10.1007/978-3-642-59033-7_22.

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References:

[1]	D. F. Anderson, G. Craciun and T. G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, 72 (2010), 1947-1970. doi: 10.1007/s11538-010-9517-4.
[2]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archive for Rational Mechanics and Analysis, 19 (1965), 81-99. doi: 10.1007/BF00282276.
[3]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions II. some addenda, Archive for Rational Mechanics and Analysis, 27 (1968), 356-364. doi: 10.1007/BF00251438.
[4]	H. Casimir, Some aspects of Onsager's theory of reciprocal relations in irreversible processes, Il Nuovo Cimento, 6 (1949), 227-231. doi: 10.1007/BF02780985.
[5]	A. Dickenstein and M. P. Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828. doi: 10.1007/s11538-010-9611-7.
[6]	R. Durrett, Probability: Theory and examples, Cambridge University Press, 2010. doi: 10.1017/CBO9780511779398.
[7]	M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827. doi: 10.1016/0009-2509(89)85124-3.
[8]	M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.
[9]	M. Feinberg, Lectures on Chemical Reaction Networks, Notes of lectures given at the Mathematics Research Center of the University of Wisconsin in 1979, available at http://www.crnt.osu.edu/LecturesOnReactionNetworks, 1979.
[10]	K. Gatermann, M. Eiswirth and A. Sensse, Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, Journal of Symbolic Computation, 40 (2005), 1361-1382. doi: 10.1016/j.jsc.2005.07.002.
[11]	J. Gunawardena, Chemical reaction network theory for in-silico biologists, Notes available at http://vcp.med.harvard.edu/papers/crnt.pdf, 2003.
[12]	F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.
[13]	F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.
[14]	B. Joshi, Complete characterization by multistationarity of fully open networks with one non-flow reaction, Applied Mathematics and Computation, 219 (2013), 6931-6945. doi: 10.1016/j.amc.2013.01.027.
[15]	B. Joshi and A. Shiu, Simplifying the Jacobian Criterion for precluding multistationarity in chemical reaction networks, SIAM Journal on Applied Mathematics, 72 (2012), 857-876. doi: 10.1137/110837206.
[16]	B. Joshi and A. Shiu, Atoms of multistationarity in chemical reaction networks, Journal of Mathematical Chemistry, 51 (2013), 153-178.
[17]	B. Joshi and A. Shiu, A survey of methods for deciding whether a reaction network is multistationary,, arXiv preprint, ().
[18]	F. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979.
[19]	G. Lewis, A new principle of equilibrium, Proceedings of the National Academy of Sciences of the United States of America, 11 (1925), 179-183. doi: 10.1073/pnas.11.3.179.
[20]	L. Onsager, Reciprocal relations in irreversible processes. I, Physical Review, 37 (1931), 405-426. doi: 10.1103/PhysRev.37.405.
[21]	L. Paulevé, G. Craciun and H. Koeppl, Dynamical properties of discrete reaction networks, Journal of mathematical biology, 69 (2014), 55-72. doi: 10.1007/s00285-013-0686-2.
[22]	P. Whittle, Systems in Stochastic Equilibrium, John Wiley & Sons, Inc., 1986.
[23]	E. Wigner, Derivations of Onsager's reciprocal relations, The Collected Works of Eugene Paul Wigner, A/4 (1997), 215-218. doi: 10.1007/978-3-642-59033-7_22.

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