Bulletin of Mathematical Biology

, Volume 73, Issue 6, pp 1227–1270 | Cite as

An Evolutionary Reduction Principle for Mutation Rates at Multiple Loci

Original Article

Abstract

A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (Evolutionary Biology, vol. 14, pp. 61–204, 1982) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models.

A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface—a generalization of the hyperplane found by Zhivotovsky et al. (Proc. Natl. Acad. Sci. USA 91, 1079–1083, 1994) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance.

Other results include the nonexistence of ‘viability analogous, Hardy–Weinberg’ modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the presence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains.

Keywords

Evolution Evolutionary theory Modifier gene Mutation rate Spectral analysis Reduction principle Karlin’s theorem Reversible Markov chain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ababneh, F., Jermiin, L. S., & Robinson, J. (2006). Generation of the exact distribution and simulation of matched nucleotide sequences on a phylogenetic tree. J. Math. Model. Algorithms, 5, 291–308. MathSciNetMATHCrossRefGoogle Scholar
  2. Altenberg, L. (1984). A generalization of theory on the evolution of modifier genes. Ph.D. thesis, Stanford University. Searchable online and available from University Microfilms, Ann Arbor, MI. Google Scholar
  3. Altenberg, L. (2009). The evolutionary reduction principle for linear variation in genetic transmission. Bull. Math. Biol., 71, 1264–1284. MathSciNetMATHCrossRefGoogle Scholar
  4. Altenberg, L., & Feldman, M. W. (1987). Selection, generalized transmission, and the evolution of modifier genes. I. The reduction principle. Genetics, 117, 559–572. Google Scholar
  5. Baer, C. F., Miyamoto, M. M., & Denver, D. R. (2007). Mutation rate variation in multicellular eukaryotes: causes and consequences. Nat. Rev. Genet., 8, 619–631. CrossRefGoogle Scholar
  6. Balkau, B., & Feldman, M. W. (1973). Selection for migration modification. Genetics, 74, 171–174. MathSciNetGoogle Scholar
  7. Brandon, R. N. (1982). The levels of selection. In P. Asquith & T. Nickles (Eds.), PSA 1982 (Vol. 1, pp. 315–323). East Lansing: Philosophy of Science Association. Google Scholar
  8. Charlesworth, B. (1990). Mutation-selection balance and the evolutionary advantage of sex and recombination. Genet. Res., 55, 199–221. CrossRefGoogle Scholar
  9. Charlesworth, B., & Charlesworth, D. (1979). Selection on recombination in clines. Genetics, 91, 581–589. MathSciNetGoogle Scholar
  10. Charlesworth, B., Charlesworth, D., & Strobeck, C. (1979). Selection for recombination in partially self-fertilizing populations. Genetics, 93, 237–244. MathSciNetGoogle Scholar
  11. Deutsch, E., & Neumann, M. (1984). Derivatives of the Perron root at an essentially nonnegative matrix and the group inverse of an M-matrix. J. Math. Anal. Appl., 102, 1–29. MathSciNetMATHCrossRefGoogle Scholar
  12. Duistermaat, J. J., & Kolk, J. A. C. (2004). Cambridge studies in advanced mathematics: Vol. 86. Multidimensional real analysis I: Differentiation. Cambridge: Cambridge University Press. ISBN 9780521551144. MATHCrossRefGoogle Scholar
  13. Eyre-Walker, A., & Keightley, P. D. (2007). The distribution of fitness effects of new mutations. Nat. Rev. Genet., 8, 610–618. CrossRefGoogle Scholar
  14. Feldman, M. W. (1972). Selection for linkage modification: I. Random mating populations. Theor. Popul. Biol., 3, 324–346. CrossRefGoogle Scholar
  15. Feldman, M. W., & Balkau, B. (1973). Selection for linkage modification II. A recombination balance for neutral modifiers. Genetics, 74, 713–726. MathSciNetGoogle Scholar
  16. Feldman, M. W., & Krakauer, J. (1976). Genetic modification and modifier polymorphisms. In S. Karlin & E. Nevo (Eds.), Population genetics and ecology (pp. 547–583). New York: Academic Press. Google Scholar
  17. Feldman, M. W., & Liberman, U. (1986). An evolutionary reduction principle for genetic modifiers. Proc. Natl. Acad. Sci. USA, 83, 4824–4827. MathSciNetMATHCrossRefGoogle Scholar
  18. Feldman, M. W., Christiansen, F. B., & Brooks, L. D. (1980). Evolution of recombination in a constant environment. Proc. Natl. Acad. Sci. USA, 77, 4838–4841. MathSciNetCrossRefGoogle Scholar
  19. Feller, W. (1971). An introduction to probability theory and its applications, Vol. I (3rd ed.). New York: Wiley. MATHGoogle Scholar
  20. Fox, A., Tuch, B., & Chuang, J. (2008). Measuring the prevalence of regional mutation rates: an analysis of silent substitutions in mammals, fungi, and insects. BMC Evol. Biol., 8, 186. CrossRefGoogle Scholar
  21. 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. Google Scholar
  22. Guillemin, V., & Pollack, A. (1974). Differential topology. Prentice-Hall: Englewood Cliffs. MATHGoogle Scholar
  23. Hirsch, M. W. (1976). Differential topology. New York: Springer. MATHGoogle Scholar
  24. Hoede, C., Denamur, E., & Tenaillon, O. (2006). Selection acts on DNA secondary structures to decrease transcriptional mutagenesis. PLoS Genet., 2, e176. http://dx.plos.org/10.1371%2Fjournal.pgen.0020176. CrossRefGoogle Scholar
  25. Holsinger, K., Feldman, M. W., & Altenberg, L. (1986). Selection for increased mutation rates with fertility differences between matings. Genetics, 112, 909–922. Google Scholar
  26. Holsinger, K. E., & Feldman, M. W. (1983a). Linkage modification with mixed random mating and selfing: a numerical study. Genetics, 103, 323–333. Google Scholar
  27. Holsinger, K. E., & Feldman, M. W. (1983b). Modifiers of mutation rate: evolutionary optimum with complete selfing. Proc. Natl. Acad. Sci. USA, 80, 6732–6734. MathSciNetMATHCrossRefGoogle Scholar
  28. Horn, R. A., & Johnson, C. R. (1985). Matrix analysis. Cambridge: Cambridge University Press. MATHGoogle Scholar
  29. Iosifescu, M. (1980). Finite Markov processes and their applications. Bucharest: Wiley. MATHGoogle Scholar
  30. Jayaswal, V., Jermiin, L. S., & Robinson, J. (2005). Estimation of phylogeny using a general Markov model. Evol. Bioinform. Online, 1, 62–80. Google Scholar
  31. Karlin, S. (1976). Population subdivision and selection migration interaction. In S. Karlin & E. Nevo (Eds.), Population genetics and ecology (pp. 616–657). New York: Academic Press. Google Scholar
  32. Karlin, S. (1982). Classification of selection-migration structures and conditions for a protected polymorphism. In M. K. Hecht, B. Wallace, & G. T. Prance (Eds.), Evolutionary biology (Vol. 14, pp. 61–204). New York: Plenum. Google Scholar
  33. Karlin, S., & McGregor, J. (1972a). Application of method of small parameters to multi-niche population genetic models. Theor. Popul. Biol., 3, 186–209. MathSciNetCrossRefGoogle Scholar
  34. Karlin, S., & McGregor, J. (1972b). The evolutionary development of modifier genes. Proc. Natl. Acad. Sci. USA, 69, 3611–3614. CrossRefGoogle Scholar
  35. Karlin, S., & McGregor, J. (1974). Towards a theory of the evolution of modifier genes. Theor. Popul. Biol., 5, 59–103. MathSciNetCrossRefGoogle Scholar
  36. Keilson, J. (1979). Markov chain models: rarity and exponentiality. New York: Springer. MATHGoogle Scholar
  37. King, D. G., & Kashi, Y. (2007). Mutation rate variation in eukaryotes: evolutionary implications of site-specific mechanisms. Nat. Rev. Genet., 8. Google Scholar
  38. Kingman, J. F. C. (1978). A simple model for the balance between selection and mutation. J. Appl. Probab., 15, 1–12. MathSciNetMATHCrossRefGoogle Scholar
  39. Kingman, J. F. C. (1980). Mathematics of genetic diversity. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-166-5. Google Scholar
  40. Kondrashov, A. S. (1982). Selection against harmful mutations in large sexual and asexual populations. Genet. Res., 40, 325–332. CrossRefGoogle Scholar
  41. Kondrashov, A. S. (1984). Deleterious mutations as an evolutionary factor. I. The advantage of recombination. Genet. Res., 44, 199–217. CrossRefGoogle Scholar
  42. Kondrashov, A. S. (1995). Modifiers of mutation-selection balance: general approach and the evolution of mutation rates. Genet. Res., 66, 53–69. CrossRefGoogle Scholar
  43. Kondrashov, F. A., & Kondrashov, A. S. (2010). Measurements of spontaneous rates of mutations in the recent past and the near future. Philos. Trans. R. Soc. B, 365, 1169–1176. CrossRefGoogle Scholar
  44. Lewontin, R. C. (1974). The genetic basis of evolutionary change. New York: Columbia University Press. Google Scholar
  45. Liberman, U., & Feldman, M. W. (1986a). A general reduction principle for genetic modifiers of recombination. Theor. Popul. Biol., 30, 341–371. MathSciNetMATHCrossRefGoogle Scholar
  46. Liberman, U., & Feldman, M. W. (1986b). Modifiers of mutation rate: A general reduction principle. Theor. Popul. Biol., 30, 125–142. MathSciNetMATHCrossRefGoogle Scholar
  47. Lynch, M. (2010). Rate molecular spectrum, and consequences of human mutation. Proc. Natl. Acad. Sci. USA, 107, 961–968. CrossRefGoogle Scholar
  48. Lynch, M., Sung, W., Morris, K., Coffey, N., Landry, C. R., Dopman, E. B., Dickinson, W. J., Okamoto, K., Kulkarni, S., Hartl, D. L., & Thomas, W. K. (2008). A genome-wide view of the spectrum of spontaneous mutations in yeast. Proc. Natl. Acad. Sci., 105, 9272–9277. CrossRefGoogle Scholar
  49. Munkres, J. R. (1975). Topology: a first course. Prentice-Hall: Englewood Cliffs. ISBN 0-13-925495-1. MATHGoogle Scholar
  50. Otto, S. P., & Feldman, M. W. (1997). Deleterious mutations, variable epistatic interactions, and the evolution of recombination. Theor. Popul. Biol., 51, 34–47. CrossRefGoogle Scholar
  51. Pylkov, K. V., Zhivotovsky, L. A., & Feldman, M. W. (1998). Migration versus mutation in the evolution of recombination under multilocus selection. Genet. Res., 71, 247–256. CrossRefGoogle Scholar
  52. Roach, J. C., Glusman, G., Smit, A. F. A., Huff, C. D., Hubley, R., Shannon, P. T., Rowen, L., Pant, K. P., Goodman, N., Bamshad, M., Shendure, J., Drmanac, R., Jorde, L. B., Hood, L., & Galas, D. J. (2010). Analysis of genetic inheritance in a family quartet by whole-genome sequencing. Science. http://dx.doi.org/10.1126/science.1186802.
  53. Rodríguez, F., Oliver, J., Marín, A., & Medina, J. (1990). The general stochastic model of nucleotide substitution. J. Theor. Biol., 142, 485–501. CrossRefGoogle Scholar
  54. Salmon, W. C. (1971). Statistical explanation and statistical relevance. Pittsburgh: University of Pittsburgh Press. MATHGoogle Scholar
  55. Salmon, W. C. (1984). Scientific explanation and the causal structure of the world. Princeton: Princeton University Press. Google Scholar
  56. Singer, I. M., & Thorpe, J. A. (1967). Lecture notes on elementary topology and geometry. New York: Springer. ISBN 0-387-90202-3. MATHGoogle Scholar
  57. Squartini, F., & Arndt, P. F. (2008). Quantifying the stationarity and time reversibility of the nucleotide substitution process. Mol. Biol. Evol., 25, 2525–2535. CrossRefGoogle Scholar
  58. Teague, R. (1977). A model of migration modification. Theor. Popul. Biol., 12, 86–94. CrossRefGoogle Scholar
  59. Whelan, S., & Goldman, N. (2004). Estimating the frequency of events that cause multiple-nucleotide changes. Genetics, 167, 2027–2043. CrossRefGoogle Scholar
  60. Wilkinson, J. H. (1965). The algebraic eigenvalue problem. Oxford: Clarendon Press. MATHGoogle Scholar
  61. Yang, Z. (1995). On the general reversible Markov process model of nucleotide substitution: a reply to Saccone et al. J. Mol. Evol., 41, 254–255. Google Scholar
  62. Yang, Z., & Nielsen, R. (2002). Codon-substitution models for detecting molecular adaptation at individual sites along specific lineages. Mol. Biol. Evol., 19, 908–917. Google Scholar
  63. Zhivotovsky, L. A., & Feldman, M. W. (1995). The reduction principle for recombination under density-dependent selection. Theor. Popul. Biol., 47, 244–256. CrossRefGoogle Scholar
  64. Zhivotovsky, L. A., Feldman, M. W., & Christiansen, F. B. (1994). Evolution of recombination among multiple selected loci: A generalized reduction principle. Proc. Natl. Acad. Sci. USA, 91, 1079–1083. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  1. 1.University of Hawai‘i at ManoaHonoluluUSA

Personalised recommendations