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Effective Size of Populations with Heritable Variation in Fitness

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Effective Size of Populations with Heritable Variation in Fitness Heredity (2002) 89, 413–416 2002 Nature Publishing Group All rights reserved 0018-067X/02 $25.00 www.nature.com/hdy SHORT REVIEW Effective size of populations with heritable variation in fitness T Nomura Department of Biotechnology, Faculty of Engineering, Kyoto Sangyo University, Kyoto 603–8555, Japan The effective size of monogamous populations with heritable zygotes are produced by random union of gametes, each variation in fitness is formulated, and the expression from conceptual male and female gametic pools. A con- obtained is compared with a published equation. It is shown venient equation for practical use is proposed, and the appli- that the published equation for dioecious populations is inap- cation is illustrated with the estimation of the effective size propriate for most animal and human populations, because of a rural human community in Japan. the derivation is implicitly based on the assumption that Heredity (2002) 89, 413–416. doi:10.1038/sj.hdy.6800169 Keywords: effective population size; random genetic drift; fitness; heritability; human population; animal population Introduction situations have been developed (Santiago and Caballero, 1995; Nomura, 1996, 1997; Wang, 1998; Bijma et al, 2000, The effective size of a population is a parameter central 2001). Only one extension to directional selection on fit- to understanding evolution in small populations, because ness was given by Nei and Murata (1966). Based on the the magnitude of this parameter determines the genetic approach of Robertson (1961), they worked out an equ- effects of both inbreeding and genetic drift (Falconer, ation for monoecious populations. They also extended 1989; Caballero, 1994). This parameter is also important their derivation to dioecious populations. The equations for solving some practical issues; it has been used as a have been used to estimate the effective population size key parameter in designing strategies for conservation of of humans (Nei and Murata, 1966) and in wild animals endangered animal and plant species (eg, Lande and (Ryman et al, 1981; Kelly, 2001). Barrowclough, 1987; Santiago and Caballero, 2000; In this paper, I will first derive an expression of the Yonezawa et al, 2000) and breeding of domestic animals effective size of monogamous populations with heritable (eg, Caballero et al, 1996; Bijma et al, 2001). variation in fitness, and will compare it with the equa- The effective size of a population depends on various tions of Nei and Murata (1966). We will see that their demographic and ecological variables (Nunney, 1991, equation for dioecious populations is inappropriate for 1993). Among these, the variation in the number of pro- most animal and human populations. Finally, a more geny per parent (family size) is one with the most pro- useful equation for practical use is presented. The appli- nounced effect (Crow and Denniston, 1988; Caballero, cation will be illustrated with published data on a human 1994). When the variation is due to non-inherited causes, community in Japan. the effective population size is simply a function of the variance of family size, and predictions have been developed for a variety of cases (Caballero, 1994; Wang, Effective size of monogamous populations 1996; Wang and Caballero, 1999). However, when the dif- Suppose an unstructured population of monogamous ferences in family sizes are, at least partially, due to species, consisting of N/2 couples each generation. Ran- inherited causes, formulation of the effective size is com- dom mating and discrete generations are assumed. An plicated by the fact that the offspring and later descend- autosomal neutral gene unlinked to the genes affecting ants of a parent with a large (or small) family size inherit fitness is considered. Formulation of the effective size is the property. based on the amount of genetic drift in the frequency of The problem of heritable variation of family size in the the neutral allele. But as emphasized by Hill (1972), the formulation of the effective size was first addressed by same expression could be derived from the loss of hetero- Robertson (1961) in the context of artificial selection. He zygosity (or the increase in inbreeding) in the neutral introduced the idea of the accumulation of selective locus. For the effective size under selection, the identity advantages of individuals over generations. Following has been proved by Nomura (1999) and Bijma et al (2000). this approach, formulae applicable to various practical Environmental correlation between parental and off- spring fitness is assumed to be absent. Let k be the number of progeny (which reach repro- Correspondence: T Nomura, Department of Biotechnology, Faculty of i Engineering, Kyoto Sangyo University, Kyoto 603-8555, Japan. E-mail: ductive age) of couple i. Following the terminology in the [email protected] related works (eg, Caballero and Santiago, 1995; Santiago Received 25 October 2001; accepted 27 July 2002 and Caballero, 1995), we will refer to ki as the ‘fitness’ of Effective population size and fitness T Nomura 414 couple i. Assuming an additive model, the fitness of cou- + + 1/2(1/2 + 1/4 + 1/8 + ···)cov(p , f ) (3) ple i can be partitioned as: i m(f)i = + {1 + 1/2(1/2 + 1/4 + 1/8 + ···)}cov(pi, ff(f)i) ki gm(m)i + gf(f)i + ei , = ⌬ ⌬ ⌬ ⌬ Q1 pm(m) + Q2 pf(m) + Q2 pm(f) + Q1 pf(f) , where gs(s)i is the additive genetic component from parent = of sex s( m or f), and ei is the environmental (including where the nonadditive and sampling) component. The subscript = = Q1 1 + 1/2(1/2 + 1/4 + 1/8 + ···) 3/2 s in parentheses in gs(s)i is attached to emphasize the sex = = Q2 1/2(1/2 + 1/4 + 1/8 + ···) 1/2 of the parent. In general, gs(u)i is the genetic component of fitness of parent of sex u(= m or f), when the genotype and is expressed in sex s. Note that for s u, the genotype ⌬p = cov(p , f ). is not expressed in the fitness of the couple, but will be s(u) i s(u)i expressed when it is transmitted into descendants of sex Following the argument of Santiago and Caballero ⌬ s. The two genetic components of fitness (gm(m)i and gf(f)i) (19951), the variance of ps(u) can be written as may be partially controlled by different sets of genes, − ⌬ = p(1 p) because some components of fitness, such as prenatal and V( ps(u)) Vgs . postnatal maternal abilities, are sex-limited. Thus, we 8N ⌬ ⌬ treat the contributions of male and female parents as two Analogously, the covariance between pm(u) and pf(u) is different sex-limited traits with genetic correlation rg. The expressed as variance of ki is decomposed as − ⌬ ⌬ = p(1 p) ͱ = cov( pm(u), pf(u)) rg VgmVgf . Vk Vgm + Vgf + Ve , 8N where Vgs and Ve are the variances of gs(s)i and ei, respect- With these expressions, the variance of ⌺⌬p is obtained as ively. Putting p(1 − p) h2 = V /V , V(⌺⌬p) = ͫ(Q2 + Q2)V s gs k 8N 1 2 gm the heritability of fitness may be written as +4Q Q r ͱV V +(Q2 + Q2)V ͬ . V + V 1 2 g gm gf 1 2 gf 2 = gm gf = 2 2 h hm + hf . (1) Vk ⌺⌬ Since the contribution of V( p)to1/Ne is The effective population size (denoted by Ne) after one 1 generation is approximated by ͫ(Q2 + Q2)V +4Q Q r ͱV V +(Q2 + Q2)V ͬ 4N 1 2 gm 1 2 g gm gf 1 2 gf = 4N 1 5 5 Ne (2) = ͫ ͱ ͬ Vgm +3rg VgmVgf + Vgf . 2+Vk 4N 2 2 (Crow and Denniston, 1988; Caballero, 1994). The cumu- an expression of Ne is obtained from equation 2 as lative effect of selection can be incorporated into the above expression by considering the partial persistence = 4N Ne 2 2 . (4) of the random association between the relative fitness 2 + [1 + 3/2 (hm + hf +2rghmhf)]Vk and the frequency of a neutral allele for which the effec- If male and female components of fitness are controlled tive size is defined (Santiago and Caballero, 1995, 1998). 2 = 2 = = by the same set of genes, ie, hm hf and rg 1, equation Let pi ( 0,1/4, 1/2, 3/4 or 1) be the frequency of the = 4 reduces to neutral allele in couple i, and fs(u)i( gs(u)i/2) be the expected contribution of parent of sex u in couple i to the = 4N Ne 2 , (5) relative fitness of couple when the genotype is expressed 2+(1+3h )Vk in sex s. The frequency of the neutral allele in the popu- = ͸N/2 which is the same expression as the equation for a mon- lation is p pi/(N/2). Only the two relative fit- oecious population given by Nei and Murata (1966). i=1 ⌬ nesses, fm(m)i and ff(f)i, contribute to the change ( p) in the frequency of the neutral allele after one generation as Comparison with published equations ⌬ = p cov(pi, fm(m)i) + cov(pi, ff(f)i) Nei and Murata (1966) extended equation 5 to a dioecious (Santiago and Caballero, 1995, 1998). But in the following population with equal numbers of male and female par- ents. Their equation comparable to equation 4 can be generations, all the associations between pi and fs(u)i are relevant to the total cumulative change of the gene fre- written as quency. Noting that under weak selection the associ- 4N N = , (6) ations are halved each generation due to segregation and e 2 + [1 + 3/4 (h2 + h2 +2r h h )]V recombination (Santiago and Caballero, 1995, 1998), and m f g m f k half of the persisting association contributes to the which is different from equation 4.
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