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Selection Response in Finite Populations

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Selection Response in Finite Populations Copyright 0 1996 by the Genetics Society of America Selection Responsein Finite Populations Ming Wei, Armando Caballero and William G. Hill Institute of Cell, Animal and Population Biology, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JT, United Kingdom Manuscript received November 28, 1995 Accepted for publication September 5, 1996 ABSTRACT Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Accountwas taken of genetic drift, gametic (linkage) disequilibrium (Bulmereffect), inbreeding depression, common environmental variance, and both initial segregating variance within families (a;wo)and mutational (&) variance. The cumulative response to selection until generation t(C8) can be approximated as where N, is the effective population size, = Ne& is the genetic variancewithin families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. & is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. P is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, ie., their relative rate of change. & is the major determinant of the short term selection response, but aL, Ne and p are also important for the long term. A selection method of high accuracy using family information gives a small N, and will lead to a larger- response in the short term and a smaller response in the long term, utilizing mutation less efficiently. ENETIC response to one cycle of selection for a 1975) using all information available are formally the G quantitative character is solely a function of selec- methods to achieve the maximum response for one tion accuracy (the correlation between the selection generation of selection. Assuming an infinite popula- criterion and breeding values of individuals), selection tion, WRAYand HILL (1989) computed the asymptotic intensity and additive genetic variance in the popula- response (the constant rate of response) after account- tion (FALCONERand MACKAY 1996). However, for long ing for the Bulmer effect for various index selection term response to selection in a finite population, other schemes. DEKKERS(1992) and VILLANUEVAet aZ. (1993) factors such as genetic drift, gametic linkage disequilib extended the theory to BLUP selection. These works, rium, mutationalvariance and effective population size however, did not consider the reduction in selection are all variables depending on character, population response due to the finite size of the populations (HILL structure and selection strategy, which should be incor- 1985), which can be accounted for by means of the porated to predict thecumulative response to selection. effective population size. In populations under selec- While long term response is also a function of the distri- tion, the effective size can be predicted depending on bution of the effects and frequencies of individual loci, selection methodsand mating schemes. ROBERTSON such information is not available, so it is necessary to (1961), WRAYand THOMPSON(1990), WOOLLIAMSet aZ. consider only known, albeit restricted, parameters. (1993), and SANTIAGOand CABALLERO (1995) devel- ROBERTSON(1960) developed the theory of selection oped methods to predict the rate of inbreeding when limits under mass selection. He did not consider, how- a population is undergoing mass selection. WRAYet al. ever, that additive variance is reduced by selection due (1994) extended the theory to handle the situation of to gametic linkage disequilibrium, the “Bulmer effect” index selection. For a review of the principles of the (PEARSON 1903; BULMER1971). To predict long term above methods see CABALLERO (1994). response, the effects of gametic linkage disequilibrium Selection intensity is also a factor that depends on and genetic drift on additive variance need to be consid- population size. For the same proportion selected, the ered simultaneously. selection intensity is reduced when there are a small A selection index (LUSH1947) and animal model number of families and by the increased correlation best linear unbiased prediction (BLUP) (HENDERSON between individuals’estimated breeding values, particu- larlywith selection using family information (HILL Curresponding authur: William G. Hill, Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd., Edin- 1976; RAWLINGS 1976; MEUWISSEN1990). burgh EH9 3JT, United Kingdom. E-mail: [email protected] Different methods of selection (mass selection, index Genetics 144: 1961-1974 (December, 1996) 1962 M. Wei, A. Caballero and W. G. Hill selection, etc.) differ in the accuracywith which individ- omitted for simplicity if a formula or parameter applies ual breeding values are estimated. A high selection ac- for any generation. curacy gives a high response in the short term but be- In this study, mass selection refers to truncation selec- cause of coselection of family members it also gives a tion on individual phenotypic values. Family selection high rate of inbreeding, so that it reduces the selection is defined as the selection of the best families based on response in thelong run (ROBERTSON1960, 1961). phenotypic family means. Within-family selection is the Thus methodsachieving maximum short term response selection of the best progeny within each family based do not necessarily optimize long term response. Some on individual phenotype. Index selection refers to the simulation studies of long term selection response have selection of individuals based on an indexof individual, beenconducted (e.&, BELONSKYand KENNEDY 1988; full-sib and/or half-sibfamily information, optimally VEWER et al. 1993) that give some insight into the weighted every generation (FALCONERand MACKAY comparison of short and longterm selection response. 1996). BLUP selection is carried out by the selection Spontaneous mutation has been found to be an im- of individuals with highest breeding values estimated portant source of new variation (CLAETONand ROBERT- by the “pseudo-animal model BLUP”, ie., an index of SON 1955; FRANKHAM1980; LYNCH1988). HILL (1982) individual, full- and half-sib information plus estimated developed the theory for genetic response from new breeding values of the dam, sire and mates of the sire mutations, but the model including both initial segre- (WRAYand HILL 1989).Pseudo-BLUP is almost identi- gating and mutational variances to predict selection re- cal to true BLUP when there are no fixed effects other sponse has not been developed, except under simple than the overall mean (WRAYand GODDARD1994), and assumptions (HILL1985; KEIGHTLEYand HILL 1992). we will simply call it BLUP henceforth. The aim of this study is to develop the analytical There are equations available to approximate effec- theory to predict cumulative response to selection un- tive population size under theselection schemes under- der various selection schemes (mass, family, within-fam- lined above (WRAYet al. 1994). In this paper, however, ily, index and BLUP selection) taking into account ge- we use values obtained by simulation, for simplicity. For netic drift, gametic linkage disequilibrium, initial and a description of the simulation procedure and examples mutational variances, inbreeding depression, common of predictions using the equation of WRAYet al. (1994), environmental variance and reduced selection intensity see CABALLERO et al. (1996b). The reduced selection due to correlation of family members. The main objec- intensity due to the correlationbetween index of family tive is not to give the most accurate predictions of re- members is calculated by using the formulae of HILL sponse to selection but to incorporate all the factors (1976) and MEUWISSEN(1990) to correct for finite pop- that affect short, medium and long-term response in ulations. relatively simple equations as functions of known pa- Prediction of genetic variance: Based on the infini- rameters, which allow the effect of such factors and tesimal model, the additive genetic variance at genera- parameter values to be evaluated. Results are discussed tion t (&) can be predicted if gametic linkage disequi- in relation to previous theory and the results of long librium is ignored. Itis then equal to the genic variance term selection experiments. (&), Le., ait = a:, = aio[(l - 1/(2N,)lt = aioe-1’/2N~, where a:o is the initial genetic variance and Ne is the THEORY effective population size. When gametic linkage dis- equilibrium is accounted for, the prediction of ail is Modeland selectionschemes assumed: A single not straightforward. However, the within-family additive quantitative character is considered that is controlled variance (aiwl)is not affected by the linkage disequilib- by an infinitesimal model of gene effects (ie., the char- rium (BULMER1971) and therefore can still be pre- acter is controlled by genes at many unlinked loci, each dicted, ie., aiw = uiuoe-t/2Ne,so wewill use it as the of small additive effect). A closed and finite population reference point in all predictions. We now add to this with a nested mating structure (full-sib families nested last equation the contributionof mutation. The within- within half-sib families), discrete generations, random family additive variance due to mutations accumulated mating, constant population structure and sizeis as- until generationtis equal to NgL(1 - Lie.,
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