American Institute of Mathematical Sciences

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doi: 10.3934/dcdss.2020346

Orthogonality of fluxes in general nonlinear reaction networks

D. R. Michiel Renger 1, and  Johannes Zimmer 2,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
2. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Dedicated, in gratitude, to Alex Mielke on the occasion of his 60th birthday

Received  July 2019 Revised  November 2019 Published  May 2020

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

Keywords: Chemical reactions, large deviations, orthogonality.
Mathematics Subject Classification: Primary: 82C35; Secondary: 82C31, 60J27.
Citation: D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020346
References:
[1]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 17pp. doi: 10.1098/rsta.2012.0341.  Google Scholar

[2]

D. F. Anderson and T. G. Kurtz, Continuous time {M}arkov chain models for chemical reaction networks, in Design and Analysis of Biomolecular Circuits, Springer, NY, 2011, 3–42. doi: 10.1007/978-1-4419-6766-4_1.  Google Scholar

[3]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory, Rev. Modern Phys., 87 (2015), 593-636.  doi: 10.1103/RevModPhys.87.593.  Google Scholar

[4]

B. Hilder, M. A. Peletier, U. Sharma and O. Tse, An inequality connecting entropy distance, {F}isher Information and large deviations, Stochastic Process. Appl., 130 (2020), 2596-2638.  doi: 10.1016/j.spa.2019.07.012.  Google Scholar

[5]

M. Kaiser, R. L. Jack and J. Zimmer, Canonical structure and orthogonality of forces and currents in irreversible Markov chains, J. Stat. Phys., 170 (2018), 1019-1050.  doi: 10.1007/s10955-018-1986-0.  Google Scholar

[6]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar

[7]

C. Maes, Frenetic bounds on the entropy production, Phys. Rev. Lett., 119 (2017). doi: 10.1103/PhysRevLett.119.160601.  Google Scholar

[8]

C. Maes and K. Netočný, Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhys. Lett. EPL, 82 (2008), 6pp. doi: 10.1209/0295-5075/82/30003.  Google Scholar

[9]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM J. Appl. Math., 77 (2017), 1562-1585.  doi: 10.1137/16M1102240.  Google Scholar

[10]

A. Mielke, M. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

[11]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91 (1953), 1505-1512.  doi: 10.1103/PhysRev.91.1505.  Google Scholar

[12]

R. I. A. Patterson and D. R. M. Renger, Large deviations of jump process fluxes, Math. Phys. Anal. Geom., 22 (2019), 32pp. doi: 10.1007/s11040-019-9318-4.  Google Scholar

[13]

D. R. M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, J. Stat. Phys., 172 (2018), 1291-1326.  doi: 10.1007/s10955-018-2083-0.  Google Scholar

[14]

D. R. M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy, 20 (2018). doi: 10.3390/e20080596.  Google Scholar

[15]

J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Modern Phys., 48 (1976), 571-585.  doi: 10.1103/RevModPhys.48.571.  Google Scholar

show all references

References:
[1]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 17pp. doi: 10.1098/rsta.2012.0341.  Google Scholar

[2]

D. F. Anderson and T. G. Kurtz, Continuous time {M}arkov chain models for chemical reaction networks, in Design and Analysis of Biomolecular Circuits, Springer, NY, 2011, 3–42. doi: 10.1007/978-1-4419-6766-4_1.  Google Scholar

[3]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory, Rev. Modern Phys., 87 (2015), 593-636.  doi: 10.1103/RevModPhys.87.593.  Google Scholar

[4]

B. Hilder, M. A. Peletier, U. Sharma and O. Tse, An inequality connecting entropy distance, {F}isher Information and large deviations, Stochastic Process. Appl., 130 (2020), 2596-2638.  doi: 10.1016/j.spa.2019.07.012.  Google Scholar

[5]

M. Kaiser, R. L. Jack and J. Zimmer, Canonical structure and orthogonality of forces and currents in irreversible Markov chains, J. Stat. Phys., 170 (2018), 1019-1050.  doi: 10.1007/s10955-018-1986-0.  Google Scholar

[6]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar

[7]

C. Maes, Frenetic bounds on the entropy production, Phys. Rev. Lett., 119 (2017). doi: 10.1103/PhysRevLett.119.160601.  Google Scholar

[8]

C. Maes and K. Netočný, Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhys. Lett. EPL, 82 (2008), 6pp. doi: 10.1209/0295-5075/82/30003.  Google Scholar

[9]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM J. Appl. Math., 77 (2017), 1562-1585.  doi: 10.1137/16M1102240.  Google Scholar

[10]

A. Mielke, M. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

[11]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91 (1953), 1505-1512.  doi: 10.1103/PhysRev.91.1505.  Google Scholar

[12]

R. I. A. Patterson and D. R. M. Renger, Large deviations of jump process fluxes, Math. Phys. Anal. Geom., 22 (2019), 32pp. doi: 10.1007/s11040-019-9318-4.  Google Scholar

[13]

D. R. M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, J. Stat. Phys., 172 (2018), 1291-1326.  doi: 10.1007/s10955-018-2083-0.  Google Scholar

[14]

D. R. M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy, 20 (2018). doi: 10.3390/e20080596.  Google Scholar

[15]

J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Modern Phys., 48 (1976), 571-585.  doi: 10.1103/RevModPhys.48.571.  Google Scholar

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