- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Fluctuation-response design rules for nonequilibrium flows

Ying-Jen Yang*

Ken A. Dill

  • *Contact author: ying-jen.yang@stonybrook.edu
PDF

Phys. Rev. Research 8, 043039 – Published 13 October, 2026

DOI: https://doi.org/10.1103/g8rc-615y

Abstract

Biomolecular machines operate as stochastic flows on networks. Here, we develop a principled approach to network design where local transition rates are systematically tuned to achieve global dynamical objectives. Based on the fluctuation-response duality in caliber force theory, our formalism unifies existing nonequilibrium response relations under a single conjugate structure. Topologically, a force-rate Jacobian yields response symmetries that establish a local orthogonal control protocol for network flows. Analytically, it parses covariance to diagnose noise mechanisms, revealing the transition from timing to branching fluctuations in a kinesin motor. With algebraic closure streamlining multiobjective gradient evaluations, this framework provides a unified blueprint for designing stochastic dynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (55)

  1. A. B. Kolomeisky and M. E. Fisher, Molecular motors: A theorist’s perspective, Annu. Rev. Phys. Chem. 58, 675 (2007).
  2. S. Liepelt and R. Lipowsky, Kinesin’s network of chemomechanical motor cycles, Phys. Rev. Lett. 98, 258102 (2007).
  3. J. A. Wagoner and K. A. Dill, Mechanisms for achieving high speed and efficiency in biomolecular machines, Proc. Natl. Acad. Sci. USA 116, 5902 (2019).
  4. T. A. J. Duke, N. Le Novère, and D. Bray, Conformational spread in a ring of proteins: A stochastic approach to allostery, J. Mol. Biol. 308, 541 (2001).
  5. Y. Tu, The nonequilibrium mechanism for ultrasensitivity in a biological switch: Sensing by Maxwell’s demons, Proc. Natl. Acad. Sci. USA 105, 11737 (2008).
  6. J. Monod, J. Wyman, and J.-P. Changeux, On the nature of allosteric transitions: A plausible model, J. Mol. Biol. 12, 88 (1965).
  7. J. J. Hopfield, Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity, Proc. Natl. Acad. Sci. USA 71, 4135 (1974).
  8. J. Ninio, Kinetic amplification of enzyme discrimination, Biochimie 57, 587 (1975).
  9. H. Qian and E. L. Elson, Single-molecule enzymology: Stochastic Michaelis–Menten kinetics, Biophys. Chem. 101–102, 565 (2002).
  10. A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, Proc. Natl. Acad. Sci. USA 78, 6840 (1981).
  11. H. Qian, Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Annu. Rev. Phys. Chem. 58, 113 (2007).
  12. T. L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics (Springer, New York, 1989).
  13. A. I. Brown and D. A. Sivak, Theory of nonequilibrium free energy transduction by molecular machines, Chem. Rev. 120, 434 (2020).
  14. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  15. M. Baiesi, C. Maes, and B. Wynants, Fluctuations and response of nonequilibrium states, Phys. Rev. Lett. 103, 010602 (2009).
  16. U. Seifert and T. Speck, Fluctuation-dissipation theorem in nonequilibrium steady states, Europhys. Lett. 89, 10007 (2010).
  17. A. C. Barato and R. Chetrite, A formal view on level 2.5 large deviations and fluctuation relations, J. Stat. Phys. 160, 1154 (2015).
  18. B. Altaner, M. Polettini, and M. Esposito, Fluctuation-dissipation relations far from equilibrium, Phys. Rev. Lett. 117, 180601 (2016).
  19. J. D. Mallory, A. B. Kolomeisky, and O. A. Igoshin, Kinetic control of stationary flux ratios for a wide range of biochemical processes, Proc. Natl. Acad. Sci. USA 117, 8884 (2020).
  20. J. A. Owen, T. R. Gingrich, and J. M. Horowitz, Universal thermodynamic bounds on nonequilibrium response with biochemical applications, Phys. Rev. X 10, 011066 (2020).
  21. G. Fernandes Martins and J. M. Horowitz, Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response, Phys. Rev. E 108, 044113 (2023).
  22. J. A. Owen and J. M. Horowitz, Size limits the sensitivity of kinetic schemes, Nat. Commun. 14, 1280 (2023).
  23. K. Ptaszyński, T. Aslyamov, and M. Esposito, Dissipation bounds precision of current response to kinetic perturbations, Phys. Rev. Lett. 133, 227101 (2024).
  24. T. Aslyamov and M. Esposito, General theory of static response for Markov jump processes, Phys. Rev. Lett. 133, 107103 (2024).
  25. T. Aslyamov and M. Esposito, Nonequilibrium response for Markov jump processes: Exact results and tight bounds, Phys. Rev. Lett. 132, 037101 (2024).
  26. T. Aslyamov, K. Ptaszyński, and M. Esposito, Nonequilibrium fluctuation-response relations: From identities to bounds, Phys. Rev. Lett. 134, 157101 (2025).
  27. J. Zheng and Z. Lu, Unified linear fluctuation-response theory arbitrarily far from equilibrium, Phys. Rev. E 112, 064103 (2025).
  28. K. Ptaszyński, T. Aslyamov, and M. Esposito, Nonequilibrium fluctuation-response relations for state observables, Phys. Rev. E 113, 024130 (2026).
  29. K. Ptaszyński, T. Aslyamov, and M. Esposito, Nonequilibrium fluctuation-response relations for state-current correlations, Phys. Rev. E 113, 024131 (2026).
  30. L. T. Stutzer, C. Dieball, and A. Godec, Stochastic calculus for pathwise observables of Markov-jump processes: Unification of diffusion and jump dynamics, Phys. Rev. X 16, 021038 (2026).
  31. Y.-J. Yang and K. A. Dill, A principled basis for nonequilibrium network flows, Nat. Commun. 17, 4980 (2026).
  32. E. T. Jaynes, The minimum entropy production principle, Annu. Rev. Phys. Chem. 31, 579 (1980).
  33. S. Pressé, K. Ghosh, J. Lee, and K. A. Dill, Principles of maximum entropy and maximum caliber in statistical physics, Rev. Mod. Phys. 85, 1115 (2013).
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/g8rc-615y for more details, which includes Refs. [53, 54].
  35. We use standard asymptotic analysis notation a∼b to represent asymptotic equivalency, limt→∞a(t)/b(t)=1.
  36. Net flux must be cyclic at steady state because total flow into a state must equal the outward flow. Fundamental cycle is a graph-theoretic method to find a linearly independent set of cycle fluxes, by choosing a spanning tree and adding back edges (chords) one by one to define the cycles. See Sec. I B in the Supplemental Material [34] for an example and Refs. [12, 14, 31] for more details.
  37. We use the suffix “-like” as a formal analogy: Mathematically, this is the Hessian symmetry of the caliber in CFT conjugate coordinates, generally distinct from Onsager reciprocity among experimentally tunable parameters. In the special case of cycle forces under local detailed balance, however, Fcycle coincides with the reservoir free-energy drive, so this symmetry does reduce to a genuine far-from-equilibrium generalization of Onsager reciprocity—restorable, though, only via the coordinated global tuning demonstrated in Sec. II C in the Supplemental Material [34], not by adjusting the reservoir alone.
  38. This can be seen by evaluating k(F) following Ref. [31].
  39. E. Kwon and J. S. Lee, A unified framework for classical and quantum uncertainty relations using stochastic representations, Commun. Phys. 8, 444 (2025).
  40. S. E. Harvey, S. Lahiri, and S. Ganguli, Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing, Phys. Rev. E 108, 014403 (2023).
  41. R. Bao and S. Liang, Nonlinear response identities and bounds for nonequilibrium steady states, arXiv:2412.19602.
  42. E. Kwon, H.-M. Chun, H. Park, and J. S. Lee, Fluctuation-response inequalities for kinetic and entropic perturbations, Phys. Rev. Lett. 135, 097101 (2025).
  43. R. Bebon and T. Speck, Mutual linearity is a generic property of steady-state Markov networks, Phys. Rev. Lett. 136, 137401 (2026).
  44. K. Visscher, M. J. Schnitzer, and S. M. Block, Single kinesin molecules studied with a molecular force clamp, Nature (London) 400, 184 (1999).
  45. We derived this using the edge reciprocity shown later in Eq. (9) (see the Supplemental Material [34]), consistent with the mutual linearity results independently reported by others [43].
  46. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  47. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  48. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  49. H. Kacser and J. A. Burns, The control of flux, Biochem. Soc. Trans. 23, 341 (1995).
  50. R. Heinrich and T. A. Rapoport, A linear steady-state treatment of enzymatic chains, Eur. J. Biochem. 42, 89 (1974).
  51. While such localized tuning can be constructed algebraically from the master equation, CFT naturally extracts this protocol from the Jacobian identity.
  52. While AI-assisted coding tools lower the barrier to deploying query-specific analytical gradients, a unified mathematical structure remains advantageous as it eliminates repetitive derivations and simplifies debugging.
  53. Y.-J. Yang and H. Qian, Unified formalism for entropy production and fluctuation relations, Phys. Rev. E 101, 022129 (2020).
  54. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985).
  55. Y.-J. Yang, Codes for “Fluctuation-response design rules for nonequilibrium flows”, Zenodo, 2026, https://doi.org/10.5281/zenodo.18611573.

Outline

Information

Phys. Rev. Research 8, 043039– Published 13 October, 2026
  • Received 21 February 2026
  • Accepted 18 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation