The statistical mechanics of a polygenic character under stabilizing selection, mutation and drift

Journal of the Royal Society Interface

Volumn 8, Issue 58, 2011, Pages 720-739

  De Vladar, Harold P ^a   Barton, Nick H ^b  

^a INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA   (Austria)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Entropy; Epistasis; Evolution; Genetic variance; Quantitative genetics; Statistical mechanics

Indexed keywords

CHROMOSOMES; ENTROPY; FUNCTION EVALUATION; GENE EXPRESSION; POPULATION STATISTICS; STATISTICAL MECHANICS;

APPROXIMATION PROCEDURE; CUMULANTS; DIRECTIONAL SELECTION; ENTROPY MEASURE; EPISTASIS; EVOLUTION; GENERATING FUNCTIONS; GENETIC DRIFTS; GENETIC VARIANCE; GRADUAL CHANGES; HETEROZYGOSITY; INFINITE SERIES; MACROSCOPIC VARIABLES; POPULATION GENETICS; QUANTITATIVE GENETICS; STABILIZING SELECTION;

BIOLOGY;

ACCURACY; ARTICLE; ENTROPY; EVOLUTION; GENE FREQUENCY; GENE LOCUS; GENE MUTATION; GENETIC DRIFT; GENETIC SELECTION; GENETIC TRAIT; GENETIC VARIABILITY; HETEROZYGOSITY; HYSTERESIS; POPULATION GENETICS; STATISTICAL ANALYSIS;

EID: 79953245458     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2010.0438     Document Type: Article

References (62)

1. Lande, R. 1979 Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry. Evolution 33, 402-416. (doi:10.2307/2407630)
2. Turelli, M. 1988 Phenotypic evolution, constant covariances, and the maintenance of additive variance. Evolution 42, 1342-1347. (doi:10.2307/2409017)
3. Lynch, M. & Walsh, B. 1998 Genetics and analysis of quantitative traits. Sunderland, MA: Sinauer Associates.
4. Barton, N. H. & Turelli, M. 1987 Adaptive landscapes, genetic distance and the evolution of quantitative characters. Genet. Res. 49, 157-173. (doi:10.1017/S0016672300026951)
5. Bürger, R. 2000 The mathematical theory of selection, recombination and mutation. Oxford, UK: Wiley & Son.
6. Hill, W. G. 1982 Predictions of response to artificial selection from new mutations. Genet. Res. 40, 255-278. (doi:10.1017/S0016672300019145)
7. Zhang, X.-S. & Hill, W. G. 2010 Change and maintenance of variation in quantitative traits in the context of the price equation. Theor. Popul. Biol. 77, 14-22. (doi:10.1016/j.tpb.2009.10.004)
8. Wright, S. 1935 The analysis of variance and the correlations between relatives with respect to deviations from an optimum. J. Genet. 30, 243-256. (doi:10.1007/BF02982239)
9. Bulmer, M. 1972 The genetic variability of polygenic characters under optimizing selection, mutation and drift. Genet. Res. 19, 17-25. (doi:10.1017/S0016672300014221)
10. Barton, N. H. 1986 The maintenance of polygenic variation through a balance between mutation and stabilizing selection. Genet. Res. (Camb.) 49, 157-174. (doi:10.1017/S0016672300026951)
11. Wright, S. 1935Evolution in populations in approximate equilibrium. J. Genet. 30, 257-266. (doi:10.1007/BF02982240)
12. Bürger, R. 1991 Moments, cumulants, and polygenic dynamics. J. Math. Biol. 30, 199-213. (doi:10.1007/ BF00160336)
13. Bürger, R. 1993 Predictions of the dynamics of a polygenic character under directional selection. J. Theor. Biol. 162, 487-512. (doi:10.1006/jtbi. 1993.1101)
14. Turelli, M. & Barton, N. H. 1994 Genetic and statistical analyses of strong selection on polygenic traits: what, me normal? Genetics 138, 913-941. (Pubitemid 24337815)
15. Prügel-Bennett, A. & Shapiro, J. 1994 Analysis of genetic algorithms using statistical mechanics. Phys. Rev. Lett. 72, 1305-1309. (doi:10.1103/PhysRevLett.72.1305)
16. Prügel-Bennett, A. & Shapiro, J. 1997 The dynamics of a genetic algorithm for simple randomising systems. Physica D 104, 75-114. (doi:10.1016/S0167-2789(96)00163-7)
17. Rattray, M. 1995 The dynamics of a genetic algorithm under stabilizing selection. Complex Syst. 9, 213-234.
18. Prügel-Bennett, A. 1997 Modelling evolving populations. J. Theor. Biol. 185, 81-95. (doi:10.1006/jtbi.1996.0295)
19. Rogers, A. & Prügel-Bennett, A. 2000 Evolving populations with overlapping generations. Theor. Popul. Biol. 57, 121-129. (doi:10.1006/tpbi. 1999.1446)
20. Shannon, C. E. 1948 A mathematical theory of communication. Bell Syst. Tech. J. 27, 379-423.
21. Barton, N. H. & de Vladar, H. P. 2009 Statistical mechanics and the evolution of polygenic quantitative traits. Genetics 181, 997-1011. (doi:10.1534/genetics.108.099309)
22. Barton, N. H. & Coe, J. 2009 On the application of statistical physics to evolutionary biology. J. Theor. Biol. 259, 317-324. (doi:10.1016/j.jtbi.2009.03.019)
23. Barton, N. H. 1989 The divergence of a polygenic system under stabilizing selection, mutation and drift. Genet. Res. (Camb.) 54, 59-77. (doi:10.1017/S0016672300028378)
24. Aita, T., Morinaga, S. & Husimi, Y. 2004 Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in a evolution reactor, I. Bull. Math. Biol. 66, 1371-1403. (doi:10.1016/j.bulm.2004.01.004)
25. Aita, T., Morinaga, S. & Husimi, Y. 2005 Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in a evolution reactor, II. Bull. Math. Biol. 67, 613-635. (Pdoi:10.1016/j.bulm.2004.09.004)
26. Sella, G. & Hirsh, A. 2005 The application of statistical physics to evolutionary biology. Proc. Natl Acad. Sci. USA 102, 9541-9546. (doi:10.1073/pnas.0501865102)
27. Iwasa, Y. 1988 Free fitness that always increases in evolution. J. Theor. Biol. 135, 265-281. (doi:10.1016/S0022-5193(88)80243-1)
28. Wright, S. 1932 The roles of mutation, inbreeding, cross-breeding and selection in evolution. Proc. Sixth Int. Cong. Genet. 1, 356-366.
29. Barton, N. H. & Turelli, M. 1990 Dynamics of polygenic characters under selection. Theor. Popul. Biol. 38, 1-57. (doi:10.1016/0040-5809(90)90002- D)
30. Rattray, M. & Shapiro, J. 2001 Cumulant dynamics of a population under multiplicative selection, mutation, and drift. Theor. Popul. Biol. 60, 17-32. (doi:10.1006/tpbi.2001.1531)
31. Wright, S. 1931 Evolution in Mendelian populations. Genetics 16, 97-159.
32. Kimura, M. 1965 A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl Acad. Sci. USA 54, 731-736. (doi:10.1073/pnas.54.3.731)
33. Crow, J. F. & Kimura, M. 1970 An introduction to population genetics theory. New York, NY: Harper & Row.
34. Kimura, M. 1962 On the probability of fixation of mutant genes in a population. Genetics 47, 713.
35. Wright, S. 1937 The distribution of gene frequencies in populations. Proc. Natl Acad. Sci. USA 23, 307-320. (doi:10.1073/pnas.23.6.307)
36. Ewens, W. J. 1979 Mathematical population genetics. Berlin, Germany: Springer.
37. Gardiner, C. 2004 Handbook of stochastic methods. Berlin, Germany: Springer.
38. Barton, N. H. & Turelli, M. 1989 Evolutionary quantitative genetics: how little do we know? Annu. Rev. Genet. 23, 337-370.
39. Higham, D. J. 2001 An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525-546. (doi:10.1137/ S0036144500378302)
40. Cox, D. R. & Hinkley, D. 1974 Theoretical statistics. London, UK: Chapman and Hall.
41. Fowler, K. & Whitlock, M. C. 2002 Environmental stress, inbreeding, and the nature of phenotypic and genetic variance in Drosophila melanogaster. Proc. R. Soc. Lond. B 269, 677. (doi:10.1098/rspb.2001.1931)
42. Turelli, M. & Barton, N. H. 1990 Dynamics of polygenic characters under selection. Theor. Popul. Biol. 38, 1-57. (doi:10.1016/0040-5809(90)90002- D)
43. Bulmer, M. G. 1980 The mathematical theory of quantitative genetics. Broadbridge, UK: Clarendon Press.
44. Bürger, R. 2005 A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait. J. Math. Biol. 50, 355-396. (doi:10.1007/s00285-004-0294-2)
45. Waxman, D. 2005 Dynamics of a quantitative character subject only to stabilising selection. Math. Biosci. 194, 81-93. (doi:10.1016/j.mbs.2004.11.009)
46. Kopp, M. & Hermisson, J. 2007 Adaptation of a quantitative trait to a moving optimum. Genetics 176, 715. (doi:10.1534/genetics.106.067215)
47. Sato, M. & Waxman, D. 2008 Adaptation to slow environmental change, with apparent anticipation of selection. J. Theor. Biol. 252, 166-172. (doi:10.1016/j.jtbi.2007.12.020)
48. Kopp, M. & Hermisson, J. 2009 The genetic basis of phenotypic adaptation. I. Fixation of beneficial mutations in the moving optimum model. Genetics 182, 233-249. (doi:10.1534/genetics.108.099820)
49. Lynch, M. & Lande, R. 1993 Evolution and extinction in response to environmental change. In Biotic interactions and global change (eds P. M. Kareiva, J. G. Kingsolver & R. B. Huey) pp 234-250. Sunderland MA: Sinauer Associates.
50. Falconer, D. S. 1981 Introduction to quantitative genetics, 2nd edn. London, UK: Longman.
51. Hoffmann, A. A. & Merilä, J. 1999 Heritable variation and evolution under favourable and unfavourable conditions. Trends Ecol. Evol. 14, 96-101. (doi:10.1016/S0169-5347(99)01595-5)
52. Barton, N. H. & Keightley, P. D. 2002 Understanding quantitative genetic variation. Nat. Rev. Genet. 3, 11-21. (doi:10.1038/nrg700)
53. Steppan, S., Phillips, P. & Houle, D. 2002 Comparative quantitative genetics: evolution of the G matrix. Trends Ecol. Evol. 17, 320-327. (doi:10.1016/S0169-5347(02)02505-3)
54. Blows, M. 2007 A tale of two matrices: multivariate approaches in evolutionary biology. J. Evol. Biol. 20, 1-8. (doi:10.1111/j.1420-9101.2006. 01164.x)
55. Turelli, M. 1985 Effects of pleiotropy on predictions concerning mutation-selection balance for polygenic traits. Genetics 111, 165.
56. Barton, N. H. 1990 Pleiotropic models of quantitative variation. Genetics 124, 773-782. (Pubitemid 20106347)
57. Wagner, G., Kenney-Hunt, J. P., Pavlicev, M., Peck, J. R., Waxman, D. & Cheverud, J. M. 2008 Pleiotropic scaling of gene effects and the 'cost of complexity'. Nature 452, 470-473. (doi:10.1038/nature06756)
58. Keightley, P. D. & Hill, W. G. 1990 Variation maintained in quantitative traits with mutation-selection balance: pleiotropic side-effects on fitness traits. Proc. R. Soc. Lond. B 242, 99-100. (doi:10.1098/rspb.1990.0110)
59. Kondrashov, A. S. & Turelli, M. 1992 Deleterious mutations, apparent stabilizing selection and the maintenance of quantitative variation. Genetics 132, 603-618.
60. Gavrilets, S. & de Jong, G. 1993 Pleiotropic models of polygenic variation, stabilizing selection, and epistasis. Genetics 134, 609-625. (Pubitemid 23156955)
61. Orr, H. A. 2000 Adaptation and the cost of complexity. Evolution 54, 13-20. (doi:10.1111/j.0014-3820.2000.tb00002.x)
62. Abramowitz, M. & Stegun, I. A. 1972 Handbook of mathematical functions. New York, NY: Dover Publications Inc.