A semi-symmetric two-locus model

Puniyani, Amit; Feldman, Marcus W.

doi:10.1016/j.tpb.2005.07.004

Cited by 2 publications

(2 citation statements)

References 9 publications

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“…For instance, Hastings and Hom ( 1990 ) determined the equilibrium structure of a model with stabilizing selection toward an optimum with arbitrary position by assuming absence of linkage disequilibrium. More recently, Feldman and Puniyani ( 2006 ) examined an extension of the Lewontin–Kojima version of the full two-locus symmetric viability model, in which substitutions at locus A do not depend on the background alleles at locus B , but substitutions at locus B depend on the background locus A . The boundary flow classes that arise in this framework are , and …”

Section: Discussionmentioning

confidence: 99%

Evolutionary dynamics in the two-locus two-allele model with weak selection

2017

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Two-locus two-allele models are among the most studied models in population genetics. The reason is that they are the simplest models to explore the role of epistasis for a variety of important evolutionary problems, including the maintenance of polymorphism and the evolution of genetic incompatibilities. Many specific types of models have been explored. However, due to the mathematical complexity arising from the fact that epistasis generates linkage disequilibrium, few general insights have emerged. Here, we study a simpler problem by assuming that linkage disequilibrium can be ignored. This is a valid approximation if selection is sufficiently weak relative to recombination. The goal of our paper is to characterize all possible equilibrium structures, or more precisely and general, all robust phase portraits or evolutionary flows arising from this weak-selection dynamics. For general fitness matrices, we have not fully accomplished this goal, because some cases remain undecided. However, for many specific classes of fitness schemes, including additive fitnesses, purely additive-by-additive epistasis, haploid selection, multilinear epistasis, marginal overdominance or underdominance, and the symmetric viability model, we obtain complete characterizations of the possible equilibrium structures and, in several cases, even of all possible phase portraits. A central point in our analysis is the inference of the number and stability of fully polymorphic equilibria from the boundary flow, i.e., from the dynamics at the four marginal single-locus subsystems. The key mathematical ingredient for this is index theory. The specific form of epistasis has both a big influence on the possible boundary flows as well as on the internal equilibrium structure admitted by a given boundary flow.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-017-1140-7) contains supplementary material, which is available to authorized users.

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Section: Discussionmentioning

confidence: 99%

Evolutionary dynamics in the two-locus two-allele model with weak selection

2017

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“…The present paper does not touch at all upon the question of the stability of the equilibria. Studies on the stability of equilibria include the papers cited previously, and, as a selected additional list, Feldman et al (1974), Karlin (1975), Karlin and Liberman (1976), Feldman and Liberman (1979), Karlin (1979), Karlin (1980), Hastings (1981, Hastings (1985), Franklin and Feldman (2000), and Puniyani and Feldman (2006).…”

Section: Discussionmentioning

confidence: 99%

Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system

Altenberg

2010

Theoretical Population Biology

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Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2 n − 1. Here a proof is provided. The upper bound of 3 n−1 obtained by Lyubich et al. (2001) using Bézout's Theorem (1779) is reduced here to 2 n through a change of representation that reduces the third-order polynomials to second order. A further reduction to 2 n − 1 is obtained using the homogeneous representation of the system, which yields always one solution 'at infinity'. While the original conjecture was made for systems of viability selection and recombination, the results here generalize to viability selection with any arbitrary system of bi-parental transmission, which includes recombination and mutation as special cases. An example is constructed of a mutation-selection system that has 2 n − 1 fixed points given any n, which shows that 2 n − 1 is the sharpest possible upper bound that can be found for the general space of selection and transmission coefficients.

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A semi-symmetric two-locus model

Cited by 2 publications

References 9 publications

Evolutionary dynamics in the two-locus two-allele model with weak selection

Evolutionary dynamics in the two-locus two-allele model with weak selection

Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system

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