American Institute of Mathematical Sciences

Advanced
January  2021, 14(1): 205-217. 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, 2021, 14 (1) : 205-217. 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 Markov 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, Fisher 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 Markov 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, Fisher 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

[1]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[2]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[3]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[4]

Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
[5]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (80)
  • HTML views (284)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences