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Fixation probability with multiple alleles and projected average allelic effect on selection

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  • Lessard, Sabin
  • Lahaie, Philippe

Abstract

The first-order effect of selection on the probability of fixation of an allele, with respect to an intensity of selection s>0 in a diploid population of fixed finite size N, undergoing discrete, non-overlapping generations, is shown to be given by the sum of the average effects of that allele on the coefficient of selection in the current generation and all future generations, given the population state in the current generation. This projected average allelic effect is a weighted sum of average allelic effects in allozygous and autozygous offspring in the initial generation, with weights given in terms of expected coalescence times, under neutrality, for the lineages of two or three gametes chosen at random in the same generation. This is shown in the framework of multiple alleles at one locus, with genotypic values determining either viability or fertility differences, and with either multinomial or exchangeable reproduction schemes. In the limit of weak selection in a large population such that Ns tends to zero, the initial average allelic effects in allozygous offspring and autozygous offspring have the same weight on the fixation probability only in the domain of application of the Kingman coalescent. With frequency-dependent selection in a linear-game-theoretic context with two phenotypes determined by additive gene action, the first-order effect on the fixation probability is a combination of two effects of frequency-independent selection, one in a haploid population, the other in a diploid population. In the domain of application of the Kingman coalescent as the population size goes to infinity and Ns to zero, the first effect is three times more important than the second effect. This explains the one-third law of evolutionary dynamics in this domain, and shows how this law can be extended beyond this domain.

Suggested Citation

  • Lessard, Sabin & Lahaie, Philippe, 2009. "Fixation probability with multiple alleles and projected average allelic effect on selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 266-277.
    • Handle: RePEc:eee:thpobi:v:75:y:2009:i:4:p:266-277
    DOI: 10.1016/j.tpb.2009.01.009
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    References listed on IDEAS

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    1. Hisashi Ohtsuki & Christoph Hauert & Erez Lieberman & Martin A. Nowak, 2006. "A simple rule for the evolution of cooperation on graphs and social networks," Nature, Nature, vol. 441(7092), pages 502-505, May.
    2. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
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    Cited by:

    1. Sabin Lessard, 2011. "On the Robustness of the Extension of the One-Third Law of Evolution to the Multi-Player Game," Dynamic Games and Applications, Springer, vol. 1(3), pages 408-418, September.
    2. Kermany, Amir R. & Lessard, Sabin, 2012. "Effect of epistasis and linkage on fixation probability in three-locus models: An ancestral recombination–selection graph approach," Theoretical Population Biology, Elsevier, vol. 82(2), pages 131-145.
    3. Dhaker Kroumi & Sabin Lessard, 2015. "Conditions for Cooperation to be More Abundant than Defection in a Hierarchically Structured Population," Dynamic Games and Applications, Springer, vol. 5(2), pages 239-262, June.

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