The Evolutionary Reduction Principle for Linear Variation in Genetic Transmission

Original Article

Abstract

The evolution of genetic systems has been analyzed through the use of modifier gene models, in which a neutral gene is posited to control the transmission of other genes under selection. Analysis of modifier gene models has found the manifestations of an “evolutionary reduction principle”: in a population near equilibrium, a new modifier allele that scales equally all transition probabilities between different genotypes under selection can invade if and only if it reduces the transition probabilities. Analytical results on the reduction principle have always required some set of constraints for tractability: limitations to one or two selected loci, two alleles per locus, specific selection regimes or weak selection, specific genetic processes being modified, extreme or infinitesimal effects of the modifier allele, or tight linkage between modifier and selected loci. Here, I prove the reduction principle in the absence of any of these constraints, confirming a twenty-year-old conjecture. The proof is obtained by a wider application of Karlin’s Theorem 5.2 (Karlin in Evolutionary biology, vol. 14, pp. 61–204, Plenum, New York, 1982) and its extension to ML-matrices, substochastic matrices, and reducible matrices.

Keywords

Evolution Evolutionary theory Modifier gene Recombination rate Mutation rate Spectral analysis Reduction principle Karlin’s theorem ML-matrix Essentially non-negative matrix 

References

  1. 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
  2. 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
  3. Balkau, B., Feldman, M.W., 1973. Selection for migration modification. Genetics 74, 171–174. MathSciNetGoogle Scholar
  4. Barton, N.H., 1995. A general model for the evolution of recombination. Genet. Res. (Camb.) 65, 123–144. Google Scholar
  5. Cavalli-Sforza, L.L., Feldman, M.W., 1973. Models for cultural inheritance. I. Group mean and within group variation. Theor. Popul. Biol. 4, 42–55. CrossRefGoogle Scholar
  6. Charlesworth, B., 1976. Recombination modification in a fluctuating environment. Genetics 83, 181–195. MathSciNetGoogle Scholar
  7. Charlesworth, B., 1993. Directional selection and the evolution of sex and recombination. Genet. Res. (Camb.) 61, 205–224. Google Scholar
  8. Cohen, J.E., 1978. Derivatives of the spectral radius as a function of non-negative matrix elements. Math. Proc. Camb. Philos. Soc. 83, 183–190. MATHCrossRefGoogle Scholar
  9. Cohen, J.E., 1979. Random evolutions and the spectral radius of a non-negative matrix. Math. Proc. Camb. Philos. Soc. 86, 345–350. MATHCrossRefGoogle Scholar
  10. Cohen, J.E., 1981. Convexity of the dominant eigenvalue of an essentially nonnegative matrix. Proc. Am. Math. Soc. 81, 657–658. MATHCrossRefGoogle 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. MATHCrossRefMathSciNetGoogle Scholar
  12. Deutsch, E., Neumann, M., 1985. On the first and second order derivatives of the Perron vector. Linear Algebra Appl. 71, 57–76. MATHCrossRefMathSciNetGoogle Scholar
  13. Feldman, M.W., 1972. Selection for linkage modification: I. Random mating populations. Theor. Popul. Biol. 3, 324–346. CrossRefGoogle Scholar
  14. Feldman, M.W., Balkau, B., 1973. Selection for linkage modification: II. A recombination balance for neutral modifiers. Genetics 74, 713–726. MathSciNetGoogle Scholar
  15. Feldman, M.W., Krakauer, J., 1976. Genetic modification and modifier polymorphisms. In: Karlin, S., Nevo, E. (Eds.), Population Genetics and Ecology, pp. 547–583. Academic Press, New York. Google Scholar
  16. Feldman, M.W., Liberman, U., 1986. An evolutionary reduction principle for genetic modifiers. Proc. Natl. Acad. Sci. USA 83, 4824–4827. MATHCrossRefMathSciNetGoogle Scholar
  17. Feldman, M.W., Otto, S.P., 1991. A comparative approach to the population genetic theory of segregation distortion. Am. Nat. 137, 443–456. CrossRefGoogle 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. CrossRefMathSciNetGoogle Scholar
  19. Feldman, M.W., Otto, S.P., Christiansen, F.B., 1997. Population genetic perspectives on the evolution of recombination. Annu. Rev. Genet. 20, 261–295. Google Scholar
  20. Felsenstein, J., 1974. The evolutionary advantage of recombination. Genetics 78, 737–756. Google Scholar
  21. Felsenstein, J., Yokoyama, S., 1976. The evolutionary advantage of recombination. II. Individual selection for recombination. Genetics 83, 845–859. Google Scholar
  22. Fisher, R.A., 1922. On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–341. Google Scholar
  23. Fisher, R.A., 1930. The Genetical Theory of Natural Selection. Clarendon Press, Oxford. MATHGoogle Scholar
  24. Friedland, S., 1981. Convex spectral functions. Linear Multilinear Algebra 9, 299–316. MATHCrossRefMathSciNetGoogle Scholar
  25. Friedland, S., Karlin, S., 1975. Some inequalities for the spectral radius of non-negative matrices and applications. Duke Math. J. 42, 459–490. MATHCrossRefMathSciNetGoogle Scholar
  26. Gantmacher, F.R., 1959. The Theory of Matrices, vol. 2. Chelsea, New York. MATHGoogle Scholar
  27. Haldane, J.B.S., 1924. A mathematical theory of natural and artificial selection. Part I. Trans. Camb. Philos. Soc. 23, 19–41. Google Scholar
  28. Hamilton, W.D., 1980. Sex versus non-sex versus parasite. Oikos 35, 282–290. CrossRefGoogle Scholar
  29. Karlin, S., 1982. Classification of selection-migration structures and conditions for a protected polymorphism. In: Hecht, M.K., Wallace, B., Prance, G.T. (Eds.), Evolutionary Biology, vol. 14, pp. 61–204. Plenum, New York. Google Scholar
  30. Karlin, S., McGregor, J., 1972. The evolutionary development of modifier genes. Proc. Natl. Acad. Sci. USA 69, 3611–3614. CrossRefGoogle Scholar
  31. Karlin, S., McGregor, J., 1974. Towards a theory of the evolution of modifier genes. Theor. Popul. Biol. 5, 59–103. CrossRefMathSciNetGoogle Scholar
  32. Kimura, M., 1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278–287. CrossRefGoogle Scholar
  33. Liberman, U., Feldman, M.W., 1986a. Modifiers of mutation rate: A general reduction principle. Theor. Popul. Biol. 30, 125–142. MATHCrossRefMathSciNetGoogle Scholar
  34. Liberman, U., Feldman, M.W., 1986b. A general reduction principle for genetic modifiers of recombination. Theor. Popul. Biol. 30, 341–371. MATHCrossRefMathSciNetGoogle Scholar
  35. Liberman, U., Feldman, M.W., 1989. The reduction principle for genetic modifiers of the migration rate. In: Feldman, M.W. (Ed.), Mathematical Evolutionary Theory, pp. 111–137. Princeton University Press, Princeton. Google Scholar
  36. Maynard Smith, J., 1988. Selection for recombination in a polygenic model—the mechanism. Genet. Res. (Camb.) 51, 59–63. CrossRefGoogle Scholar
  37. Nei, M., 1967. Modification of linkage intensity by natural selection. Genetics 57, 625–641. MathSciNetGoogle Scholar
  38. Odling-Smee, J., 2007. Niche inheritance: A possible basis for classifying multiple inheritance systems in evolution. Biol. Theory 2, 276–289. CrossRefGoogle Scholar
  39. Schauber, E., Goodwin, B., Jones, C., Ostfeld, R., 2007. Spatial selection and inheritance: applying evolutionary concepts to population dynamics in heterogeneous space. Ecology 88, 1112–1118. CrossRefGoogle Scholar
  40. Seneta, E., 1981. Non-negative Matrices and Markov Chains. Springer, New York. MATHGoogle Scholar
  41. Wright, S., 1931. Evolution in Mendelian populations. Genetics 16, 97–159. Google Scholar
  42. 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 2009

Authors and Affiliations

  1. 1.University of Hawai‘i at ManoaHonoluluUSA

Personalised recommendations