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Selection, Subdivision and Extinction and Recolonization

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Selection, Subdivision and Extinction and Recolonization Copyright 2004 by the Genetics Society of America Selection, Subdivision and Extinction and Recolonization Joshua L. Cherry1 National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, Bethesda, Maryland 20894 Manuscript received April 14, 2003 Accepted for publication November 2, 2003 ABSTRACT In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for fitness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity Nese, which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations. OPULATION subdivision has many population- between the Ne that applies to fixation probabilities of Pgenetic consequences. The amount of neutral varia- selected alleles and the Ne describing the behavior of tion maintained in a population, the distribution of neutral alleles can in some cases be resolved with the coalescence times, expected times to fixation or loss of notion of the effective selection coefficient se (Cherry alleles, and fixation probabilities of selected alleles can and Wakeley 2003). Nese, rather than Nes, determines all be altered by subdivision. An important determinant fixation probability. Ne is raised by subdivision, but se is of the magnitudes, and even the directions, of these lowered such that Nese is unaltered by subdivision. This effects is whether gene flow among subpopulations in- framework is consistent with the fact that although fixa- cludes local extinction and subsequent recolonization. tion probabilities are unaffected by subdivision, times In the absence of extinction and recolonization, sub- to fixation are increased. division increases the amount of standing neutral varia- If the assumption of simple genic selection is re- tion and lengthens coalescence times. By these measures laxed—if there is dominance for fitness or frequency- subdivision therefore increases effective population size, dependent selection—fixation probabilities are altered Ne. Extinction and recolonization can diminish and by subdivision, even in the absence of extinction and even reverse these effects (Slatkin 1977; Maruyama recolonization (Slatkin 1981; Lande 1985; Spirito et and Kimura 1980), so that effective size can be either al. 1993; Cherry 2003a). Specifically, subdivision de- larger or smaller than actual size. creases the importance of dominance or frequency de- The simplest case involving natural selection is genic pendence. This effect can be described in terms of effec- selection in the absence of extinction and recoloniza- tive values of the additional parameters that describe tion. Under these conditions, so long as migration is the more complex selection regime (Cherry 2003a). symmetric, subdivision has no effect on the fixation Extinction and recolonization can alter fixation prob- probability of an allele under selection (Maruyama abilities, even with simple genic selection (Barton 1970, 1974). This would seem to suggest that for the 1993). The effect of this type of population structure purpose of understanding the behavior of selected al- is to decrease the importance of selection relative to leles Ne is unaffected by subdivision, since fixation prob- stochastic forces; extinction and recolonization is an ability is generally thought of as a function of Nes, where additional stochastic force that brings with it no addi- s is the selection coefficient. The apparent discrepancy tional directional change. The interaction between nat- ural selection and extinction and recolonization has been a subject of much recent theoretical work (Cherry 2003b; Roze and Rousset 2003; Whitlock 2003). 1Address for correspondence: National Center for Biotechnology Infor- mation, National Library of Medicine, National Institutes of Health, 45 Cherry (2003b) derived results for genic selection Center Dr., Bethesda, MD 20894. E-mail: [email protected] in a finite island model with extinction and recoloniza- Genetics 166: 1105–1114 ( February 2004) 1106 J. L. Cherry tion by a single founding allele. Here I extend these parameters of this equivalent panmictic population are, results in two ways. First, I allow for recolonization by by definition, the effective population size (Ne), the multiple founders, which may have a tendency to origi- effective selection coefficient (se), and, in the case of nate from the same deme. Second, I obtain results that dominance, the effective dominance parameter (he). apply when there is dominance for fitness or, what is Founders chosen independently: I first consider the formally equivalent, a form of frequency-dependent se- case of genic selection where the founders that recolo- lection. nize an extinct deme have no particular tendency to originate from a common source deme. This case corre- sponds to Slatkin’s (1977) “migrant pool” model. MODELS AND RESULTS The mean change in allele frequency in the popula- The model of population structure considered here tion as a whole is the mean of the within-deme mean is the finite island model. In this model D demes, each changes due to selection (because migration and recolo- consisting of N haploid or N/2 diploid individuals, ex- nization are symmetric they do not, on average, change change migrants among themselves and also serve as the allele frequency). For a deme whose allele frequency sources for recolonization of extinct demes. The ex- is x, the mean change due to selection is approximately pected fraction of migrant alleles entering a deme in a sx(1 Ϫ x), where s is the selection coefficient. Thus generation is m. The probability of extinction of any the expected value of the mean change in the entire sE[x(1 Ϫ x)], where the expectation isف given deme in any generation is ␭. Subsequent to extinc- population is tion, a deme is recolonized by k founding alleles. The taken across the quasi-equilibrium distribution of x. The subpopulation then immediately grows to full size. quasi-equilibrium value of E[x(1 Ϫ x)] can be obtained Every deme receives migrants and colonists from the from a recursion that gives this expected value in one population as a whole, in which the allele frequency is generation as a function of its value in the previous x. Suppose that x changes slowly compared to the allele generation and the previous value of E[x]. Let H ϭ frequency in any deme. From the point of view of any E[2x(1 Ϫ x)]. This quantity is the probability that two deme, the population as a whole looks, in the short copies of the gene sampled from the same deme, inde- term, much like a source population with constant allele pendently and with replacement, are in distinct allelic frequency x. The distribution of the within-deme allele states. Let Ht be the value of H in generation t. We seek frequencies will then attain a quasi-equilibrium, which an expression for Htϩ1 in terms or Ht and E[x]. corresponds to the equilibrium of a source-sink (conti- For two copies of the gene sampled at time t ϩ 1to nent-island) model. For x to change sufficiently slowly, be in different allelic states, it is necessary that the same two conditions must hold. First, the number of demes copy of the locus has not been sampled twice (probabil- must be large, so that the stochastic change in the overall ity 1 Ϫ 1/N). Assuming that this is the case there are allele frequency is small compared to the within-deme several possibilities to consider. There may have just stochastic change. Second, selection must be weak in been an extinction/recolonization event (probability the sense that the magnitudes of selective differences ␭), in which case the two copies of the gene may or are small compared to the reciprocal of the deme size may not be descended from distinct founders (probabil- N. This condition allows selection to have a significant ities 1 Ϫ 1/k and 1/k, respectively). The two can be effect on the behavior of an allele in the population as allelically distinct only if they are descended from dis- a whole because the size of the entire population is tinct founders, in which case they are different with much larger than N. This assumption has the additional probability 2x(1 Ϫ x). If there has not just been an ex- consequence that selection can be neglected in the deri- tinction/recolonization event (probability 1 Ϫ␭), zero, vation of the quasi-equilibrium distribution of within- one, or both of the sampled alleles may be migrants. If deme allele frequency. I make these two assumptions— neither is a migrant, the immediate ancestors of the that the number of demes is large and that selection is sampled pair are chosen independently and with re- weak compared to 1/N—in all that follows. placement from generation t. Thus the probability that The temporal trajectory of an allele’s frequency is they are allelically distinct is Ht. If exactly one is a mi- the outcome of the interaction between the directional grant, the probability is E[x](1 Ϫ x) ϩ (1 Ϫ E[x])x, effects of natural selection and the stochastic effects of where E[x] refers to the expectation in the previous genetic drift and extinction/recolonization.
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