Bulletin of Mathematical Biology

, Volume 74, Issue 11, pp 2650–2675 | Cite as

Mutation Rate Evolution in Replicator Dynamics

Original Article
  • 450 Downloads

Abstract

The mutation rate of an organism is itself evolvable. In stable environments, if faithful replication is costless, theory predicts that mutation rates will evolve to zero. However, positive mutation rates can evolve in novel or fluctuating environments, as analytical and empirical studies have shown. Previous work on this question has focused on environments that fluctuate independently of the evolving population. Here we consider fluctuations that arise from frequency-dependent selection in the evolving population itself. We investigate how the dynamics of competing traits can induce selective pressure on the rates of mutation between these traits. To address this question, we introduce a theoretical framework combining replicator dynamics and adaptive dynamics. We suppose that changes in mutation rates are rare, compared to changes in the traits under direct selection, so that the expected evolutionary trajectories of mutation rates can be obtained from analysis of pairwise competition between strains of different rates. Depending on the nature of frequency-dependent trait dynamics, we demonstrate three possible outcomes of this competition. First, if trait frequencies are at a mutation–selection equilibrium, lower mutation rates can displace higher ones. Second, if trait dynamics converge to a heteroclinic cycle—arising, for example, from “rock-paper-scissors” interactions—mutator strains succeed against non-mutators. Third, in cases where selection alone maintains all traits at positive frequencies, zero and nonzero mutation rates can coexist indefinitely. Our second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics.

Keywords

Mutation rate Evolution Replicator dynamics Adaptive dynamics Evolvability 

References

  1. Aharoni, A., Gaidukov, L., Khersonsky, O., Gould, S. M. Q., Roodveldt, C., & Tawfik, D. S. (2005). The ‘evolvability’ of promiscuous protein functions. Nat. Genet., 37(1), 73–76. Google Scholar
  2. André, J.-B., & Godelle, B. (2006). The evolution of mutation rate in finite asexual populations. Genetics, 172(1), 611–626. CrossRefGoogle Scholar
  3. Bonneuil, N. (1992). Attractors and confiners in demography. Ann. Oper. Res., 37, 17–32. MathSciNetMATHCrossRefGoogle Scholar
  4. Brandstrom, M., & Ellegren, H. (2007). The genomic landscape of short insertion and deletion polymorphisms in the chicken (Gallus gallus) genome: a high frequency of deletions in tandem duplicates. Genetics, 176, 1691–1701. CrossRefGoogle Scholar
  5. Buss, L. W., & Jackson, J. B. C. (1979). Competitive networks: nontransitive competitive relationships in cryptic coral reef environments. Am. Nat., 113(2), 223–234. CrossRefGoogle Scholar
  6. Chen, J. Q., Wu, Y., Yang, H., Bergelson, J., Kreitman, M., & Tian, D. (2009). Variation in the ratio of nucleotide substitution and indel rates across genomes in mammals and bacteria. Mol. Biol. Evol., 26, 1523–1531. CrossRefGoogle Scholar
  7. Chen, F., Liu, W.-Q., Eisenstark, A., Johnston, R., Liu, G.-R., & Liu, S.-L. (2010). Multiple genetic switches spontaneously modulating bacterial mutability. BMC Evol. Biol., 10(1), 277. CrossRefGoogle Scholar
  8. Chicone, C. C. (2006). Ordinary differential equations with applications. Berlin: Springer. MATHGoogle Scholar
  9. Cortez, M. H., & Ellner, S. P. (2010). Understanding rapid evolution in predator–prey interactions using the theory of fast–slow dynamical systems. Am. Nat., 176(5), e109–e127. CrossRefGoogle Scholar
  10. Dercole, F. (2002). Evolutionary dynamics through bifurcation analysis: methods and applications. Ph.D. thesis, Department of Electronics and Information, Politecnico di Milano, Milano, Italy. Google Scholar
  11. Dercole, F., & Rinaldi, S. (2008). Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton: Princeton University Press. MATHGoogle Scholar
  12. Dercole, F., Ferrière, R., Gragnani, A., & Rinaldi, S. (2006). Coevolution of slow–fast populations: evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics. Proc. R. Soc. B, Biol. Sci., 273, 983–990. 1589. CrossRefGoogle Scholar
  13. Desai, M. M., & Fisher, D. S. (2011). The balance between mutators and nonmutators in asexual populations. Genetics, 188(4), 997–1014. CrossRefGoogle Scholar
  14. Dieckmann, U., & Law, R. (1996). The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34(5), 579–612. MathSciNetMATHCrossRefGoogle Scholar
  15. Dieckmann, U., Marrow, P., & Law, R. (1995). Evolutionary cycling in predator–prey interactions: population dynamics and the Red Queen. J. Theor. Biol., 176(1), 91–102. CrossRefGoogle Scholar
  16. Doebeli, M. (1995). Evolutionary predictions from invariant physical measures of dynamic processes. J. Theor. Biol., 173(4), 377–387. MathSciNetCrossRefGoogle Scholar
  17. Doebeli, M., & Koella, J. C. (1995). Evolution of simple population dynamics. Proc. R. Soc. B, Biol. Sci., 260(1358), 119–125. CrossRefGoogle Scholar
  18. Duret, L. (2009). Mutation patterns in the human genome: more variable than expected. PLoS Biol., 7, e1000028. CrossRefGoogle Scholar
  19. Earl, D. J., & Deem, M. W. (2004). Evolvability is a selectable trait. Proc. Natl. Acad. Sci., 101(32), 11531–11536. CrossRefGoogle Scholar
  20. Eckmann, J. P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57(3), 617–656. MathSciNetCrossRefGoogle Scholar
  21. Elango, N., Kim, S. H., Program, N. C. S., Vigoda, E., & Soojin, V. Y. (2008). Mutations of different molecular origins exhibit contrasting patterns of regional substitution rate variation. PLoS Comput. Biol., 4, e1000015. CrossRefGoogle Scholar
  22. Fryxell, K. J., & Moon, W. J. (2005). CpG mutation rates in the human genome are highly dependent on local GC content. Mol. Biol. Evol., 22, 650–658. CrossRefGoogle Scholar
  23. Gaunersdorfer, A. (1992). Time averages for heteroclinic attractors. SIAM J. Appl. Math., 52(5), 1476–1489. MathSciNetMATHCrossRefGoogle Scholar
  24. Geritz, S. A. H. (2005). Resident–invader dynamics and the coexistence of similar strategies. J. Math. Biol., 50(1), 67–82. MathSciNetMATHCrossRefGoogle Scholar
  25. Geritz, S. A. H., Kisdi, E., Meszeńa, G., & Metz, J. A. J. (1997). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol., 12(1), 35–57. CrossRefGoogle Scholar
  26. Geritz, S. A. H., Gyllenberg, M., Jacobs, F. J. A., & Parvinen, K. (2002). Invasion dynamics and attractor inheritance. J. Math. Biol., 44, 548–560. MathSciNetMATHCrossRefGoogle Scholar
  27. Giraud, A., Matic, I., Tenaillon, O., Clara, A., Radman, M., Fons, M., & Taddei, F. (2001). Costs and benefits of high mutation rates: adaptive evolution of bacteria in the mouse gut. Science, 291(5513), 2606–2608. CrossRefGoogle Scholar
  28. Gore, J., Youk, H., & Van Oudenaarden, A. (2009). Snowdrift game dynamics and facultative cheating in yeast. Nature, 458(7244), 253–256. CrossRefGoogle Scholar
  29. Gyllenberg, M., Osipov, A., & Söderbacka, G. (1996). Bifurcation analysis of a metapopulation model with sources and sinks. J. Nonlinear Sci., 6, 329–366. MathSciNetMATHCrossRefGoogle Scholar
  30. Hadeler, K. P. (1981). Stable polymorphisms in a selection model with mutation. SIAM J. Appl. Math., 41(1), 1–7. MathSciNetCrossRefGoogle Scholar
  31. Heino, M., Metz, J. A. J., & Kaitala, V. (1998). The enigma of frequency-dependent selection. Trends Ecol. Evol., 13(9), 367–370. CrossRefGoogle Scholar
  32. Hodgkinson, A., Ladoukakis, E., & Eyre-Walker, A. (2009). Cryptic variation in the human mutation rate. PLoS Biol., 7, e1000027. CrossRefGoogle Scholar
  33. Hofbauer, J. (1985). The selection mutation equation. J. Math. Biol., 23(1), 41–53. MathSciNetMATHCrossRefGoogle Scholar
  34. Hofbauer, J. (1994). Heteroclinic cycles in ecological differential equations. Tatra Mt. Math. Publ., 4, 105–116. MathSciNetMATHGoogle Scholar
  35. Hofbauer, J., & Sigmund, K. (1990). Adaptive dynamics and evolutionary stability. Appl. Math. Lett., 3(4), 75–79. MathSciNetMATHCrossRefGoogle Scholar
  36. Hofbauer, J., & Sigmund, K. (1998). Evolutionary games and replicator dynamics. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  37. Hofbauer, J., & Sigmund, K. (2003). Evolutionary game dynamics. Bull. Am. Math. Soc., 40(4), 479–520. MathSciNetMATHCrossRefGoogle Scholar
  38. Hofbauer, J., Schuster, P., & Sigmund, K. (1979). A note on evolutionary stable strategies and game dynamics. J. Theor. Biol., 81(3), 609–612. MathSciNetCrossRefGoogle Scholar
  39. Horn, R. A., & Johnson, C. R. (1990). Matrix analysis. Cambridge: Cambridge University Press. MATHGoogle Scholar
  40. Ishii, K., Matsuda, H., Iwasa, Y., & Sasaki, A. (1989). Evolutionarily stable mutation rate in a periodically changing environment. Genetics, 121, 163–174. Google Scholar
  41. Johnson, T. (1999a). Beneficial mutations, hitchhiking and the evolution of mutation rates in sexual populations. Genetics, 151(4), 1621–1631. Google Scholar
  42. Johnson, T. (1999b). The approach to mutation–selection balance in an infinite asexual population, and the evolution of mutation rates. Proc. R. Soc. B, Biol. Sci., 266(1436), 2389–2397. CrossRefGoogle Scholar
  43. Kerr, B., Riley, M. A., Feldman, M. W., & Bohannan, B. J. M. (2002). Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature, 418(6894), 171–174. CrossRefGoogle Scholar
  44. Kessler, D. A., & Levine, H. (1998). Mutator dynamics on a smooth evolutionary landscape. Phys. Rev. Lett., 80, 2012–2015. CrossRefGoogle Scholar
  45. Khibnik, A. I., & Kondrashov, A. S. (1997). Three mechanisms of red queen dynamics. Proc. R. Soc. Lond. B, Biol. Sci., 264(1384), 1049–1056. CrossRefGoogle Scholar
  46. Kimura, M. (1967). On the evolutionary adjustment of spontaneous mutation rates. Genet. Res., 9, 23–34. CrossRefGoogle Scholar
  47. King, D. G., Soller, M., & Kashi, Y. (1997). Evolutionary tuning knobs. Endeavour, 21, 36–40. CrossRefGoogle Scholar
  48. Kirkup, B. C., & Riley, M. A. (2004). Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature, 428(6981), 412–414. CrossRefGoogle Scholar
  49. Kirschner, M., & Gerhart, J. (1998). Evolvability. Proc. Natl. Acad. Sci. USA, 95(15), 8420–8427. CrossRefGoogle Scholar
  50. Krivan, V., & Cressman, R. (2009). On evolutionary stability in prey–predator models with fast behavioral dynamics. Evol. Ecol. Res., 11, 227–251. Google Scholar
  51. Leigh, E. G. (1970). Natural selection and mutability. Am. Nat., 104, 301–305. CrossRefGoogle Scholar
  52. Leigh, E. G. (1973). The evolution of mutation rates. Genetics, 73, 1–18. MathSciNetGoogle Scholar
  53. Levinson, G., & Gutman, G. A. (1987a). High frequencies of short frameshifts in poly-CA/TG tandem repeats borne by bacteriophage M13 in Escherichia coli K-12. Nucleic Acids Res., 15, 5323–5338. CrossRefGoogle Scholar
  54. Levinson, G., & Gutman, G. A. (1987b). Slipped-strand mispairing: a major mechanism for DNA evolution. Mol. Biol. Evol., 4, 203–221. Google Scholar
  55. Lynch, M. (2007). The origins of genome architecture. Sunderland: Sinauer Associates. Google Scholar
  56. Magnus, J. R., & Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics. New York: Wiley. MATHGoogle Scholar
  57. Marrow, P., Law, R., & Cannings, C. (1992). The coevolution of predator–prey interactions: ESSs and red queen dynamics. Proc. R. Soc. Lond. B, Biol. Sci., 250(1328), 133–141. CrossRefGoogle Scholar
  58. May, R. M. (1972). Limit cycles in predator–prey communities. Science, 177(4052), 900–902. CrossRefGoogle Scholar
  59. May, R. M. (2001). Stability and complexity in model ecosystems. Princeton: Princeton University Press. MATHGoogle Scholar
  60. May, R. M., & Leonard, W. J. (1975). Nonlinear aspects of competition between three species. SIAM J. Appl. Math., 29(2), 243–253. MathSciNetMATHCrossRefGoogle Scholar
  61. Maynard Smith, J., & Price, G. R. (1973). The logic of animal conflict. Nature, 246(5427), 15–18. CrossRefGoogle Scholar
  62. Metz, J. A. J., Nisbet, R. M., & Geritz, S. A. H. (1992). How should we define ‘fitness’ for general ecological scenarios? Trends Ecol. Evol., 7(6), 198–202. CrossRefGoogle Scholar
  63. Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. A., & van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In S. J. van Strien & S. M. V. Lunel (Eds.), Stochastic and spatial structures of dynamical systems (pp. 183–231). Amsterdam: KNAW Verhandelingen, Afd. Google Scholar
  64. Milinski, M. (1987). Tit for tat in sticklebacks and the evolution of cooperation. Nature, 325(6103), 433–435. CrossRefGoogle Scholar
  65. Murphy, G. L., Connell, T. D., Barritt, D. S., Koomey, M., & Cannon, J. G. (1989). Phase variation of gonococcal protein II: regulation of gene expression by slipped-strand mispairing of a repetitive DNA sequence. Cell, 56, 539–547. CrossRefGoogle Scholar
  66. Nowak, M. A. (2006). Five rules for the evolution of cooperation. Science, 314(5805), 1560–1563. CrossRefGoogle Scholar
  67. Nowak, M. A., & Sigmund, K. (2004). Evolutionary dynamics of biological games. Science, 303(5659), 793–799. CrossRefGoogle Scholar
  68. Nowak, M. A., Komarova, N. L., & Niyogi, P. (2001). Evolution of universal grammar. Science, 291(5501), 114–118. MathSciNetMATHCrossRefGoogle Scholar
  69. Oliver, A., Cantón, R., Campo, P., Baquero, F., & Blázquez, J. (2000). High frequency of hypermutable Pseudomonas aeruginosa in cystic fibrosis lung infection. Science, 288(5469), 1251. CrossRefGoogle Scholar
  70. Orr, H. A. (2000). The rate of adaptation in asexuals. Genetics, 155(2), 961–968. Google Scholar
  71. Paquin, C. E., & Adams, J. (1983). Relative fitness can decrease in evolving asexual populations of S. cerevisiae. Nature, 368–371. Google Scholar
  72. Pigliucci, M., & Box, P. (2008). Is evolvability evolvable? Nat. Rev. Genet., 9, 75–82. CrossRefGoogle Scholar
  73. Radman, M., Matic, I., & Taddei, F. (1999). Evolution of evolvability. Ann. N.Y. Acad. Sci., 870(1), 146–155. CrossRefGoogle Scholar
  74. Rainey, P. B., & Rainey, K. (2003). Evolution of cooperation and conflict in experimental bacterial populations. Nature, 425(6953), 72–74. CrossRefGoogle Scholar
  75. Rand, D. A., Wilson, H. B., & McGlade, J. M. (1994). Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philos. Trans. R. Soc. B, Biol. Sci., 343(1305), 261–283. CrossRefGoogle Scholar
  76. Rosche, W., Foster, P., & Cairns, J. (1999). The role of transient hypermutators in adaptive mutation in Escherichia coli. Proc. Natl. Acad. Sci., 96, 6862–6867. CrossRefGoogle Scholar
  77. Ruelle, D. (1989). Chaotic evolution and strange attractors. Cambridge: Cambridge University Press. MATHCrossRefGoogle Scholar
  78. Schnabl, W., Stadler, P. F., Forst, C., & Schuster, P. (1991). Full characterization of a strange attractor. Physica D, 48(1), 65–90. MathSciNetMATHCrossRefGoogle Scholar
  79. Schuster, P., & Sigmund, K. (1983). Replicator dynamics. J. Theor. Biol., 100(533), 8. MathSciNetGoogle Scholar
  80. Seger, J., & Antonovics, J. (1988). Dynamics of some simple host–parasite models with more than two genotypes in each species [and discussion]. Philos. Trans. R. Soc. B, 319(1196), 541–555. CrossRefGoogle Scholar
  81. Shaver, A. C., & Sniegowski, P. D. (2003). Spontaneously arising mutl mutators in evolving Escherichia coli populations are the result of changes in repeat length. J. Bacteriol., 185(20), 6076–6079. CrossRefGoogle Scholar
  82. Sigmund, K. (1992). Time averages for unpredictable orbits of deterministic systems. Ann. Oper. Res., 37, 217–228. doi:10.1007/BF02071057. MathSciNetMATHCrossRefGoogle Scholar
  83. Sinervo, B., & Calsbeek, R. (2006). The developmental, physiological, neural, and genetical causes and consequences of frequency-dependent selection in the wild. Annu. Rev. Ecol. Evol. Syst., 37, 581–610. CrossRefGoogle Scholar
  84. Sinervo, B., & Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature, 380, 240–243. CrossRefGoogle Scholar
  85. Sinervo, B., Miles, D. B., Frankino, W. A., Klukowski, M., & DeNardo, D. F. (1996). Testosterone, endurance, and Darwinian fitness: natural and sexual selection on the physiological bases of alternative male behaviors in side-blotched lizards. Horm. Behav., 38, 222–233. CrossRefGoogle Scholar
  86. Sniegowski, P. D., Gerrish, P. J., & Lenski, R. E. (1997). Evolution of high mutation rates in experimental populations of Escherichia coli. Nature, 387(6634), 703–705. CrossRefGoogle Scholar
  87. Sniegowski, P. D., Gerrish, P. J., Johnson, T., & Shaver, A. (2000). The evolution of mutation rates: separating causes from consequences. BioEssays, 22(12), 1057–1066. CrossRefGoogle Scholar
  88. Taddei, F., Radman, M., Maynard Smith, J., Toupance, B., Gouyon, P. H., & Godelle, B. (1997). Role of mutator alleles in adaptive evolution. Nature, 387(6634), 700–702. CrossRefGoogle Scholar
  89. Takens, F. (1985). On the numerical determination of the dimension of an attractor. In B. Braaksma, H. Broer, & F. Takens (Eds.), Lecture notes in mathematics: Vol. 1125. Dynamical systems and bifurcations (pp. 99–106). Berlin: Springer. CrossRefGoogle Scholar
  90. Taylor, P. D., & Jonker, L. B. (1978). Evolutionary stable strategies and game dynamics. Math. Biosci., 40(1–2), 145–156. MathSciNetMATHCrossRefGoogle Scholar
  91. Tenaillon, O., Toupance, B., Le Nagard, H., Taddei, F., & Godelle, B. (1999). Mutators, population size, adaptive landscape and the adaptation of asexual populations of bacteria. Genetics, 152(2), 485–493. Google Scholar
  92. Tian, D., Wang, Q., Zhang, P., Araki, H., Yang, S., Kreitman, M., Nagylaki, T., Hudson, R., Bergelson, J., & Chen, J. Q. (2008). Single-nucleotide mutation rate increases close to insertions/deletions in eukaryotes. Nature, 455, 105–108. CrossRefGoogle Scholar
  93. Travis, J. M. J., & Travis, E. R. (2002). Mutator dynamics in fluctuating environments. Proc. R. Soc. B, Biol. Sci., 269(1491), 591–597. CrossRefGoogle Scholar
  94. Wagner, A. (2008). Robustness and evolvability: a paradox resolved. Proc. R. Soc. B, Biol. Sci., 275, 91. 1630. CrossRefGoogle Scholar
  95. Weber, M. (1996). Evolutionary plasticity in prokaryotes: a Panglossian view. Biol. Philos., 11, 67–88. CrossRefGoogle Scholar
  96. Woods, R. J., Barrick, J. E., Cooper, T. F., Shrestha, U., Kauth, M. R., & Lenski, R. E. (2011). Second-order selection for evolvability in a large Escherichia coli population. Science, 331(6023), 1433–1436. CrossRefGoogle Scholar
  97. Wylie, C. S., Ghim, C., Kessler, D., & Levine, H. (2009). The fixation probability of rare mutators in finite asexual populations. Genetics, 181, 1595–1612. CrossRefGoogle Scholar
  98. Zhao, Z., & Jiang, C. (2007). Methylation-dependent transition rates are dependent on local sequence lengths and genomic regions. Mol. Biol. Evol., 24, 23–25. MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Benjamin Allen
    • 1
    • 2
  • Daniel I. Scholes Rosenbloom
    • 3
  1. 1.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsEmmanuel CollegeBostonUSA
  3. 3.Program for Evolutionary Dynamics, Department of Organismic and Evolutionary BiologyHarvard UniversityCambridgeUSA

Personalised recommendations