Journal of the Korean Physical Society

Korean Physical Society (한국물리학회)
권호 표지
DOI QR코드

DOI QR Code

Effects of Phenotypic Variation on Evolutionary Dynamics

  • Received : 2018.10.22
  • Accepted : 2018.10.30
  • Published : 2018.11.30
⟨ Previous Next ⟩

Abstract

Phenotypic variation among clones (individuals with identical genes, i.e. isogenic individuals) has been recognized both theoretically and experimentally. We investigate the effects of phenotypic variation on evolutionary dynamics of a population. In a population, the individuals are assumed to be haploid with two genotypes : one genotype shows phenotypic variation and the other does not. We use an individual-based Moran model in which the individuals reproduce according to their fitness values and die at random. The evolutionary dynamics of an individual-based model is formulated in terms of a master equation and is approximated as the Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find that there is a discrete phase transition in the population distribution when the probability of reproducing the fitter individual is equal to the critical value determined by the stability of the fixed points. Next, we take demographic stochasticity into account and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of simulations are consistent with the analytic predictions.

Keywords

Acknowledgement

Supported by : Hanshin University, National Research Foundation of Korea

References

  1. E. Korobkova, T. Emonet, J. M. Vilar, T. S. Shimidzu and P. Cluzel, Nature 428, 574 (2004). https://doi.org/10.1038/nature02404
  2. J. Paulsson, Nature 427, 415 (2004). https://doi.org/10.1038/nature02257
  3. M. Acar, J. T. Mettetal and A. V. Oudenaarden, Nat. Genet. 40, 471 (2008). https://doi.org/10.1038/ng.110
  4. N. Q. Balaban, J. Merrin, R. Chait, L. Kowalik and S. Leibler, Science 305, 1622 (2004). https://doi.org/10.1126/science.1099390
  5. K. Kaneko, PLoS ONE 2, e434 (2007). https://doi.org/10.1371/journal.pone.0000434
  6. A. Wagner, Nature 24, 355 (2000).
  7. T. Yomo, Y. Ito, K. Sato and K. Kaneko, Physica A 350, 1 (2005). https://doi.org/10.1016/j.physa.2004.11.025
  8. K. Sato, Y. Ito, T. Yomo and K. Kaneko, Proc. Natl. Acad. Sci. USA 100, 14086 (2003). https://doi.org/10.1073/pnas.2334996100
  9. M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Science 297, 1183 (2002). https://doi.org/10.1126/science.1070919
  10. A. P. Feinberg and R. A. Irizzary, Proc. Natl. Acad. Sci. USA 107, 1757 (2010). https://doi.org/10.1073/pnas.0906183107
  11. M. F. Wernet, E. O. Mazzoni, A. Celik, D. M. Duncan, I. Duncan and C. Desplan, Nature 440, 174 (2006). https://doi.org/10.1038/nature04615
  12. M. J.West-Eberhard, Developmental Plasticity and Evolution (Oxford University Press, Oxford, 2003).
  13. J. M. Pedraza and A. V. Oudenaarden, Science 307, 1965 (2005). https://doi.org/10.1126/science.1109090
  14. N. Rosenfeld, J. W. Young, U. Alon, P. S. Swain and M. B. Elowitz, Science 307, 1962 (2005). https://doi.org/10.1126/science.1106914
  15. E. M. Ozbudak, M. Thattai, I. Kurtser, A. D. Grossman and A. V. Oudenaarden, Nat. Genet. 31, 69 (2002). https://doi.org/10.1038/ng869
  16. K. Kaneko and C. Furusawa, J. Theor. Biol. 240, 78 (2006). https://doi.org/10.1016/j.jtbi.2005.08.029
  17. Y. Ito, H. Toyota, K. Kaneko and T. Yomo, Molecular Systems Biology 5, 264 (2009). https://doi.org/10.1038/msb.2009.23
  18. M. J. Baldwin, Am. Nat. 30, 441 (1896). https://doi.org/10.1086/276408
  19. G. G. Simpson, Evolution 7, 110 (1953). https://doi.org/10.1111/j.1558-5646.1953.tb00069.x
  20. G. E. Hinton and S. J. Nowlan, Complex Syst. 1, 495 (1987).
  21. H. Dopazo, M. B. Gordon, R. Perazzo and S. Riau-Gusman, Bull. Math. Biol. 63, 117 (2001). https://doi.org/10.1006/bulm.2000.0207
  22. L. W. Ancel, Theor. Pop. Biol. 58, 307 (2000). https://doi.org/10.1006/tpbi.2000.1484
  23. I. Paenke, B. Sendhoff and T. J. Kawecki, Am. Nat. 170, E47 (2007). https://doi.org/10.1086/518952
  24. E. Borenstein, I. Meilijson and E. Ruppin, J. Evol. biol. 19, 1555 (2006). https://doi.org/10.1111/j.1420-9101.2006.01125.x
  25. T. D. Price, A. Qvarnstrom and D. E. Irwin, Proc. R. Soc. London B 270, 1433 (2003). https://doi.org/10.1098/rspb.2003.2372
  26. N. Saito, S. Ishihara and K. Kaneko, Phys. Rev. E 87, 052701 (2013).
  27. C. W. Gardiner, Handbook of Stochastic Methods, 4th ed. (Springer, Berlin, 2009).
  28. Y-G. Kang and J-M. Park, J. Korean Phys. Soc. 71, 528 (2017). https://doi.org/10.3938/jkps.71.528
  29. G. W. A. Constable and A. J. McKane, Phys. Rev. E 89, 032141 (2014). https://doi.org/10.1103/PhysRevE.89.032141
  30. T. Funaki and H. Nagai, Stochastics 44, 1 (1993).
  31. G. S. Katzenberger, Ann. Probab. 19, 1587 (1991). https://doi.org/10.1214/aop/1176990225
  32. T. Biancalani, L. Dyson and A. J. McKane, Phys. Rev. Lett. 112, 038101 (2014). https://doi.org/10.1103/PhysRevLett.112.038101
  33. F. Jafarpour, T. Biancalani and N. Goldenfeld, Phys. Rev. Lett. 115, 158101 (2015). https://doi.org/10.1103/PhysRevLett.115.158101
  34. G. W. A. Constable, T. Rogers, A. J. McKane and C. E. Tarnita, Proc. Natl. Acad. Sci. U. S. A. 113, E4745 (2016). https://doi.org/10.1073/pnas.1603693113

Cited by

  1. Phase Transitions in the Evolution Model with Phenotypic Variation via Learning vol.75, pp.8, 2018, https://doi.org/10.3938/jkps.75.636