Population genetics of multiple loci.

*(English)*Zbl 0941.92019
Wiley Series in Mathematical and Computational Biology. Chichester: Wiley. xiv, 365 p. (1999).

This book presents the results of a study of the consequences of recombination and segregation for the evolution of genetic variation not subject to selection. A population of diploid individuals with nonoverlapping generations is considered. Many aspects of the population genetics of multiple loci, however, may be deduced without detailed knowledge of the linkage relationships among the loci, and therefore the assumption of a linkage order emerges as an unnecessary complication of multi-locus models. This statement is the thesis of this book. It is hoped to convince the validity of this thesis, and that the unordered set indexation is a fruitful and simple approach to the formulation and analysis of multi-locus models. We consider the convergence to linkage equilibrium among X-linked loci in an organism with \(XY\) sex determination. This also provides the principles of multi-locus population genetics for haplo-diploid species. We only consider the simplest case of inbreeding, where individuals are allowed to mate with themselves. This occurs in a large proportion of plant species, where fertilization may occur by pollen from the same flower or by pollen from another flower on the same individual (Fryxell 1957, Brown 1990). We will therefore assume that there is crossbreeding in addition to selfing, and we analyze a model where some of the ovules are pollinated by selfing, a fraction \(\alpha\), and some are pollinated by pollen chosen at random in the population, a fraction \(1-\alpha\), and we shall assume that \(\alpha<1\). This is the model of partial selfing and random outcrossing, or the mixed mating model, and the consideration of this model should form a part of any comprehensive theory of the population genetics of multi-locus systems. The following theorem is important for understanding the author’s thesis.

Theorem 4.1: Suppose partial selfing at the frequency \(\alpha\) and random outcrossing at the frequency \(1-\alpha\) prevails and that no selection occurs on the multi-allelic variation at n autosomal loci. If recombination occurs between any pair among the n loci, then the gamet frequencies in the population will converge to the Robbins proportions, where the alleles at the different loci occur independently in the gametes.

Corollary 1. With linkage equilibrium in the source populations, the linkage disequilibria at migration-recombination balance equilibrium in the island model are given by \[ \hat{D}_M=\psi(M)\prod_{a\in M} \Delta_a,\quad M \in S(N)/\{ 0 \}, \] where \(\psi(M), M \in {S(N)/{\{ 0 \}}}\), is given by the proposed algorithm. With linkage equilibrium in the source populations, the linkage disequilibria at migration-recombination balance equilibrium in the symmetric island model are given by \(\hat{D}_M=0\) when \(\sharp M\) is odd, and for \(\sharp M\) even the linkage disequilibria are given by \[ \hat{D}_M =\breve{D}_M\prod_{a\in M} \Delta_a, \quad M \in {S(N)/{\{ 0 \}}}, \] where the linkage disequilibrium ratio \(\breve{D}_X\) for \(\sharp X\) even is given by \[ (1-2R_X(X)(1-2m)) \breve{D}_X = 2m + (1-2m) \sum_{Y \in S(X)/{\{ 0,X \}}} 2^{-1} (1+\gamma (Y)) R_X (Y) \breve{D}_Y \breve{D}_{X\setminus Y}. \] For an even set M of absolute loci \(\breve{D}_M = 1\). The description of quantitative variation assumes Hardy-Weinberg proportions among the genotypes in the population. In general, without this assumption, the description becomes rather more complicated and tedious. Thus, as expected, the dominance deviation is undefined and the dominance variance vanishes in a fully inbred population where \(F_{\{ a \}}=1\) for all \(a \in N\). When the deviation from Hardy-Weinberg proportions is entirely due to inbreeding then \(F_{\{ a \}}=F\) for all \(a \in N\), whereas \(F_{\{ a,b\}}\) depends on the frequency \(r_ab\) of recombination between the two loci.

The second part of book is dedicated to the selection problem. Transmission of genes from a parent population to its offspring was the focus of Part I, and the ”physiological” aspects of population genetics were neglected. The key assumption was that the genetic variation under consideration is not subject to selection. The theory of weak selection assumes that the effect of recombination overrides the effect of selection, and the population therefore rapidly reaches a state where it is approximately in linkage equilibrium. The remaining part is known as intermediate complexity, even though this merely reflects how difficult it is to analyze. The simplest models of the action of selection assume that the genotypes have different probabilities of survival to maturity and that no other fitness variation is present. Some general results are presented in following corollaries.

Corollary 2. In the generalized multiplicative model, a necessary condition for the stability of an existing totally polymorphic Robbins equilibrium is marginal overdominance (or neutrality) at equilibrium for every locus \(a \in N\). For free recombination the equilibrium is stable when equilibrium marginal overdominance prevails for all loci \(a \in N\), and it is unstable when any locus shows marginal underdominance at equilibrium. The proof of this corollary uses the product structure of the equilibrium gamete frequencies to get a simple expression for the marginal fitnesses.

Corollary 3. In the generalized multiplicative model, a necessary condition for the stability away from the boundary of an existing \(N\setminus M\) Robbins equilibrium is that the average fitness, evaluated assuming Robbins proportions, of any rare heterozygote at the loci in \(M\), multiplied by the probability of segregating the parental gametes from that heterozygote, is less than or equal to the average fitness at equilibrium. For free recombination the strict inequality version of this condition is also sufficient for stability away from the boundary.

Corollary 4. Under diploid viability selection with mutation, the frequency of the marginal gamete A with respect to the loci M changes in one generation from \(\pi_M (A)\) to \(\pi'_M (A)\), which satisfies the bounding inequality \[ \pi_M' (A) \geq (1-\tilde{\mu})^{\sharp M}\left(1-(1-2R_M(0))(1 - \pi_M (A)) \right)V_M(A)W^{-1}\pi_M (A), \] where \(\tilde{\mu} =\max_{a \in N} \mu_a\) bounds mutation. The fitness effect of inbreeding may be due to mutation-selection balance equilibria in the population. If linkage is not too right, the deviation from gametic linkage equilibrium is at most of the order of the square of the gene frequencies of the mutant alleles, and we may therefore approximate the fitness of inbred individuals assuming Robbins proportions. Neglecting second-order terms in the mutant frequencies, an individual with the one-locus inbreeding coefficient \(F\) is expected to have the fitness \[ W_F = (1-\sum_{a \in N} (2(1-F)q_a + Fq_a))\omega(0,0) + \sum_{a \in N}\left(2(1 - F)q_a \omega(0,{a}) + Fq_a \omega({\{ a \}},{\{ a \}}) \right) + O({\mathbf q}\cdot{\mathbf q}). \] The inbreeding depression is therefore \[ W_0 - W_F \approx F \sum_{a \in N} q_a \delta_a, \] where \(\delta_a \approx {\partial}_a \approx -\omega(0,0)+2\omega(0,{\{ a \}}) - \omega({\{ a \}},{\{ a \}})\) is the dominance effect of fitness at locus a. Thus, inbreeding depression is the result at a mutation-selection balance equilibrium if the dominance effects are positive, on average. As expected, recessive deleterious alleles always contribute to inbreeding depression.

Then the problems of migration and selection are considered. This view of immigration as a perturbation of the population or as a source of new genetic variation is discussed in the next sections. The study of the dynamic effects of selection is therefore naturally extended to include the effects of geographical structure and recurrent migration within a population. The results of the modification theory seem to be at variance with the ubiquitous phenomenon of chromosomal crossover and genetic recombination. The reduction principle predicts that the frequency of recombination should decrease and eventually vanish. The widespread occurrence of positive interference in recombination and chiasma formation, increasing towards complete interference for closely situated chromosomal regions, is at variance with theoretical predictions in a similar way. If recombination by and large is advantageous in a finite population because of the variation-maintaining capability, will that solve the problem of evolution and maintenance of recombination? The way in which the problem has been formulated in relation to genetic algorithms suggests that recombination provides a way to improve the performance of the population. This indicates that group selection is at play, but a more interesting question is whether the effect can produce selection at the level of the individual. Evidently, in a population fixed at a rather limited number of gamete types, an individual which produces gametes by ameiosis that allows for recombination is expected to be able to produce offspring with a higher fitness. However, recombinant gametes may also contribute a detrimental effect to the offspring, and so we are stuck in the original dilemma of the evolutionary effect of recombination: Is bringing together on average more beneficial than breaking apart? Our deterministic analysis leans towards a resounding no to this question, but it is not unlikely that the stochastic effects in a finite population may change this answer. The experience from the application of genetic algorithms is that they do not always work, so even in a finite population we cannot expect a clear answer. A more dichotomous result, however, may be interesting, because it may allow some deductions on the actions and interactions of genes in a world in which recombination is widespread.

Theorem 4.1: Suppose partial selfing at the frequency \(\alpha\) and random outcrossing at the frequency \(1-\alpha\) prevails and that no selection occurs on the multi-allelic variation at n autosomal loci. If recombination occurs between any pair among the n loci, then the gamet frequencies in the population will converge to the Robbins proportions, where the alleles at the different loci occur independently in the gametes.

Corollary 1. With linkage equilibrium in the source populations, the linkage disequilibria at migration-recombination balance equilibrium in the island model are given by \[ \hat{D}_M=\psi(M)\prod_{a\in M} \Delta_a,\quad M \in S(N)/\{ 0 \}, \] where \(\psi(M), M \in {S(N)/{\{ 0 \}}}\), is given by the proposed algorithm. With linkage equilibrium in the source populations, the linkage disequilibria at migration-recombination balance equilibrium in the symmetric island model are given by \(\hat{D}_M=0\) when \(\sharp M\) is odd, and for \(\sharp M\) even the linkage disequilibria are given by \[ \hat{D}_M =\breve{D}_M\prod_{a\in M} \Delta_a, \quad M \in {S(N)/{\{ 0 \}}}, \] where the linkage disequilibrium ratio \(\breve{D}_X\) for \(\sharp X\) even is given by \[ (1-2R_X(X)(1-2m)) \breve{D}_X = 2m + (1-2m) \sum_{Y \in S(X)/{\{ 0,X \}}} 2^{-1} (1+\gamma (Y)) R_X (Y) \breve{D}_Y \breve{D}_{X\setminus Y}. \] For an even set M of absolute loci \(\breve{D}_M = 1\). The description of quantitative variation assumes Hardy-Weinberg proportions among the genotypes in the population. In general, without this assumption, the description becomes rather more complicated and tedious. Thus, as expected, the dominance deviation is undefined and the dominance variance vanishes in a fully inbred population where \(F_{\{ a \}}=1\) for all \(a \in N\). When the deviation from Hardy-Weinberg proportions is entirely due to inbreeding then \(F_{\{ a \}}=F\) for all \(a \in N\), whereas \(F_{\{ a,b\}}\) depends on the frequency \(r_ab\) of recombination between the two loci.

The second part of book is dedicated to the selection problem. Transmission of genes from a parent population to its offspring was the focus of Part I, and the ”physiological” aspects of population genetics were neglected. The key assumption was that the genetic variation under consideration is not subject to selection. The theory of weak selection assumes that the effect of recombination overrides the effect of selection, and the population therefore rapidly reaches a state where it is approximately in linkage equilibrium. The remaining part is known as intermediate complexity, even though this merely reflects how difficult it is to analyze. The simplest models of the action of selection assume that the genotypes have different probabilities of survival to maturity and that no other fitness variation is present. Some general results are presented in following corollaries.

Corollary 2. In the generalized multiplicative model, a necessary condition for the stability of an existing totally polymorphic Robbins equilibrium is marginal overdominance (or neutrality) at equilibrium for every locus \(a \in N\). For free recombination the equilibrium is stable when equilibrium marginal overdominance prevails for all loci \(a \in N\), and it is unstable when any locus shows marginal underdominance at equilibrium. The proof of this corollary uses the product structure of the equilibrium gamete frequencies to get a simple expression for the marginal fitnesses.

Corollary 3. In the generalized multiplicative model, a necessary condition for the stability away from the boundary of an existing \(N\setminus M\) Robbins equilibrium is that the average fitness, evaluated assuming Robbins proportions, of any rare heterozygote at the loci in \(M\), multiplied by the probability of segregating the parental gametes from that heterozygote, is less than or equal to the average fitness at equilibrium. For free recombination the strict inequality version of this condition is also sufficient for stability away from the boundary.

Corollary 4. Under diploid viability selection with mutation, the frequency of the marginal gamete A with respect to the loci M changes in one generation from \(\pi_M (A)\) to \(\pi'_M (A)\), which satisfies the bounding inequality \[ \pi_M' (A) \geq (1-\tilde{\mu})^{\sharp M}\left(1-(1-2R_M(0))(1 - \pi_M (A)) \right)V_M(A)W^{-1}\pi_M (A), \] where \(\tilde{\mu} =\max_{a \in N} \mu_a\) bounds mutation. The fitness effect of inbreeding may be due to mutation-selection balance equilibria in the population. If linkage is not too right, the deviation from gametic linkage equilibrium is at most of the order of the square of the gene frequencies of the mutant alleles, and we may therefore approximate the fitness of inbred individuals assuming Robbins proportions. Neglecting second-order terms in the mutant frequencies, an individual with the one-locus inbreeding coefficient \(F\) is expected to have the fitness \[ W_F = (1-\sum_{a \in N} (2(1-F)q_a + Fq_a))\omega(0,0) + \sum_{a \in N}\left(2(1 - F)q_a \omega(0,{a}) + Fq_a \omega({\{ a \}},{\{ a \}}) \right) + O({\mathbf q}\cdot{\mathbf q}). \] The inbreeding depression is therefore \[ W_0 - W_F \approx F \sum_{a \in N} q_a \delta_a, \] where \(\delta_a \approx {\partial}_a \approx -\omega(0,0)+2\omega(0,{\{ a \}}) - \omega({\{ a \}},{\{ a \}})\) is the dominance effect of fitness at locus a. Thus, inbreeding depression is the result at a mutation-selection balance equilibrium if the dominance effects are positive, on average. As expected, recessive deleterious alleles always contribute to inbreeding depression.

Then the problems of migration and selection are considered. This view of immigration as a perturbation of the population or as a source of new genetic variation is discussed in the next sections. The study of the dynamic effects of selection is therefore naturally extended to include the effects of geographical structure and recurrent migration within a population. The results of the modification theory seem to be at variance with the ubiquitous phenomenon of chromosomal crossover and genetic recombination. The reduction principle predicts that the frequency of recombination should decrease and eventually vanish. The widespread occurrence of positive interference in recombination and chiasma formation, increasing towards complete interference for closely situated chromosomal regions, is at variance with theoretical predictions in a similar way. If recombination by and large is advantageous in a finite population because of the variation-maintaining capability, will that solve the problem of evolution and maintenance of recombination? The way in which the problem has been formulated in relation to genetic algorithms suggests that recombination provides a way to improve the performance of the population. This indicates that group selection is at play, but a more interesting question is whether the effect can produce selection at the level of the individual. Evidently, in a population fixed at a rather limited number of gamete types, an individual which produces gametes by ameiosis that allows for recombination is expected to be able to produce offspring with a higher fitness. However, recombinant gametes may also contribute a detrimental effect to the offspring, and so we are stuck in the original dilemma of the evolutionary effect of recombination: Is bringing together on average more beneficial than breaking apart? Our deterministic analysis leans towards a resounding no to this question, but it is not unlikely that the stochastic effects in a finite population may change this answer. The experience from the application of genetic algorithms is that they do not always work, so even in a finite population we cannot expect a clear answer. A more dichotomous result, however, may be interesting, because it may allow some deductions on the actions and interactions of genes in a world in which recombination is widespread.

Reviewer: F.T.Adylova (Tashkent)