- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Geometric phase in the Crow-Kimura model of molecular evolution on dynamic environments

Vladimir Suvorov1, Ricard Solé2,3, and David B. Saakian4,5,*

  • *Contact author: saakian@yerphi.am
PDF

Phys. Rev. E 112, 054409 – Published 17 November, 2025

DOI: https://doi.org/10.1103/rv4q-l8cq

Abstract

When taking place under fluctuating environments, some classical results of evolutionary dynamics in fitness landscapes need to be reconsidered. Under such nonequilibrium conditions, the properties of adaptive evolution might escape from the expectations grounded in equilibrium systems. Here, an important contribution to this nonequilibrium dynamics results from the presence of a geometry (Berry) phase in the Crow-Kimura model of molecular evolution with asymmetric mutations. By considering changes in fitness alone as well as changes in both fitness and mutation, analytical expressions for the Berry phase are derived, showing strong singularities at bulk transition points. Periodically varying parameters are also analyed for the two-dimensional case. The potential implications for evolutionary and prebiotic scenarios are discussed.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (39)

  1. J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory (Harper Row, New York, 1970).
  2. H. Woodcock and P. G. Higgs, J. Theor. Biol. 179, 61 (1996).
  3. E. Baake, M. Baake, and H. Wagner, Phys. Rev. Lett. 78, 559 (1997).
  4. E. Baake and H. Wagner, Genet. Res. 78, 93 (2001).
  5. M. Eigen, J McCaskill, and P Schuster, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice (John Wiley & Sons, Inc., 19891), Vol. 75, pp. 149–263.
  6. D. B. Saakian, O. Rozanova, and A. Akmetzhanov, Phys. Rev. E 78, 041908 (2008).
  7. D. B. Saakian and C.-K. Hu, Phys. Rev. E 69, 046121 (2004).
  8. M. Ancliff and J. M. Park, Phys. Rev. E 80, 061910 (2009).
  9. A. G. M. Neves, Phys. Rev. E 82, 031915 (2010).
  10. M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000).
  11. V. Mustonen and M. Lässig, Trends Genet. 25, 111 (2009).
  12. E. Kussell and S. Leibler, Science 309, 2075 (2005).
  13. A. Skanata and E. Kussell, Phys. Rev. Lett. 117, 038104 (2016).
  14. M. Assaf, M. Mobilia, and E. Roberts, Phys. Rev. Lett. 111, 238101 (2013).
  15. P. G. Hufton, Y. T. Lin, T. Galla, and A. J. McKane, Phys. Rev. E 93, 052119 (2016).
  16. R. Yuan, X. Zhu, G. Wang, S. Li, and P. Ao, Rep. Prog. Phys. 80, 042701 (2017).
  17. O Rivoire, J. Stat. Phys. 162, 1324 (2016).
  18. A. Mayer, T. Mora, O. Rivoire, and A. M. Walczak, Phys. Rev. E 96, 032412 (2017).
  19. D. B. Saakian, Phys. Rev. E 97, 012409 (2018).
  20. R. Rao and S. Leibler, Proc. Natl. Acad. Sci. USA 119, e2112083119 (2022).
  21. M. Sireci and M. A. Muñoz, Phys. Rev. Res. 6, 023070 (2024).
  22. Y. J. Yang, C. D. Kocher, and K. A. Dill, arXiv:2506.19018 (2025).
  23. N. A. Sinitsyn and I. Nemenman, Phys. Rev. Lett. 99, 220408 (2007).
  24. S. Tănase-Nicola and I. Nemenman, J. R. Soc. Interface 9, 1354 (2012).
  25. M. M. Tanaka, C. T. Bergstrom, and B. R. Levin, Genetics 164, 843(2003).
  26. Y. Sughiyama and T. J. Kobayashi, Phys. Rev. E 95, 012131 (2017).
  27. S. Kwon and J.-M. Park, Phys. Rev. E 103, 062405 (2021).
  28. A. Melikyan, Generalized Characteristics of First Order PDEs (Birkhäuser, Boston, 1998).
  29. L. C. Evans, Partial Differential Equations (AMS, Providence, RI, 2002).
  30. D. B. Saakian, C. K. Biebricher, and C. K. Hu, Phys. Rev. E 79, 041905 (2009).
  31. J.-Y. Zhang, X. Yin, M.-Y. liu, J. Zhao, Y. Ding, and J. Chang, Phys. Scr. 98, 055927 (2023).
  32. S. J. Maynard and E. Szathmáry, The Major Transitions in Evolution (W. H. Freeman, Oxford, UK, 1995).
  33. N. Goldenfeld, and C. Woese, Annu. Rev. Condens. Matter Phys. 2, 375 (2011).
  34. J. W. Biddle and J. Gunawardena, Phys. Rev. E 101, 062125 (2020).
  35. P. Pietzonka, J. Guioth, and R. L. Jack, Phys. Rev. E 104, 064137 (2021).
  36. X. Fang, K. Kruse, T. Lu, and J. Wang,Rev. Mod. Phys. 91, 045004 (2019).
  37. J. L. Bada and J. Korenaga, Life 8, 55 (2016).
  38. B. Damer and D. Deamer, Astrobiology 20, 429 (2020).
  39. P. G. Higgs, Life 6, 24 (2016).

Outline

Information

Phys. Rev. E 112, 054409– Published 17 November, 2025
  • Received 16 June 2025
  • Accepted 6 October 2025
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