selection response in finite populationsa closed and finite population with a nested mating...

Copyright 0 1996 by the Genetics Society of America

Selection Response in 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. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), 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 ( C 8 ) can be approximated as

where N, is the effective population size, = Ne& is the genetic variance within 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.

G ENETIC response to one cycle of selection for a quantitative character is solely a function of selec-

tion accuracy (the correlation between the selection criterion and breeding values of individuals), selection intensity and additive genetic variance in the population (FALCONER and MACKAY 1996). However, for long term response to selection in a finite population, other factors such as genetic drift, gametic linkage disequilib rium, mutational variance and effective population size are all variables depending on character, population structure and selection strategy, which should be incor- porated to predict the cumulative response to selection. While long term response is also a function of the distribution of the effects and frequencies of individual loci, such information is not available, so it is necessary to consider only known, albeit restricted, parameters.

ROBERTSON (1960) developed the theory of selection limits under mass selection. He did not consider, however, that additive variance is reduced by selection due to gametic linkage disequilibrium, the “Bulmer effect” (PEARSON 1903; BULMER 1971). To predict long term response, the effects of gametic linkage disequilibrium and genetic drift on additive variance need to be considered simultaneously.

A selection index (LUSH 1947) and animal model best linear unbiased prediction (BLUP) (HENDERSON

Curresponding authur: William G. Hill, Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd., Edin- burgh EH9 3JT, United Kingdom. E-mail: [email protected]

Genetics 144: 1961-1974 (December, 1996)

1975) using all information available are formally the methods to achieve the maximum response for one generation of selection. Assuming an infinite population, WRAY and HILL (1989) computed the asymptotic response (the constant rate of response) after accounting for the Bulmer effect for various index selection schemes. DEKKERS (1992) and VILLANUEVA et aZ. (1993) extended the theory to BLUP selection. These works, however, did not consider the reduction in selection response due to the finite size of the populations (HILL 1985), which can be accounted for by means of the effective population size. In populations under selection, the effective size can be predicted depending on selection methods and mating schemes. ROBERTSON (1961), WRAY and THOMPSON (1990), WOOLLIAMS et aZ. (1993), and SANTIAGO and CABALLERO (1995) developed methods to predict the rate of inbreeding when a population is undergoing mass selection. WRAY et al. (1994) extended the theory to handle the situation of index selection. For a review of the principles of the above methods see CABALLERO (1994).

Selection intensity is also a factor that depends on population size. For the same proportion selected, the selection intensity is reduced when there are a small number of families and by the increased correlation between individuals’ estimated breeding values, particularly with selection using family information (HILL 1976; RAWLINGS 1976; MEUWISSEN 1990).

Different methods of selection (mass selection, index

1962 M. Wei, A. Caballero and W. G. Hill

selection, etc.) differ in the accuracywith which individual breeding values are estimated. A high selection accuracy gives a high response in the short term but because of coselection of family members it also gives a high rate of inbreeding, so that it reduces the selection response in the long run (ROBERTSON 1960, 1961). Thus methods achieving maximum short term response do not necessarily optimize long term response. Some simulation studies of long term selection response have been conducted (e.&, BELONSKY and KENNEDY 1988; VEWER et al. 1993) that give some insight into the comparison of short and long term selection response.

Spontaneous mutation has been found to be an important source of new variation (CLAETON and ROBERT- SON 1955; FRANKHAM 1980; LYNCH 1988). HILL (1982) developed the theory for genetic response from new mutations, but the model including both initial segregating and mutational variances to predict selection response has not been developed, except under simple assumptions (HILL 1985; KEIGHTLEY and HILL 1992).

The aim of this study is to develop the analytical theory to predict cumulative response to selection under various selection schemes (mass, family, within-family, index and BLUP selection) taking into account genetic drift, gametic linkage disequilibrium, initial and mutational variances, inbreeding depression, common environmental variance and reduced selection intensity due to correlation of family members. The main objec- tive is not to give the most accurate predictions of response to selection but to incorporate all the factors that affect short, medium and long-term response in relatively simple equations as functions of known parameters, which allow the effect of such factors and parameter values to be evaluated. Results are discussed in relation to previous theory and the results of long term selection experiments.

THEORY

Model and selection schemes assumed: A single quantitative character is considered that is controlled by an infinitesimal model of gene effects ( i e . , the character is controlled by genes at many unlinked loci, each of small additive effect). A closed and finite population with a nested mating structure (full-sib families nested within half-sib families), discrete generations, random mating, constant population structure and size is assumed. Each full-sib family is assumed to have a constant number of individuals (n), n/2 of each sex. At generation t, the total phenotypic variance (a&) consists of additive genetic variance (&), environmental variance (a:) and common environmental variance between full-sib family members (a:). The residual variance (a:) is defined as a: = a: + 0% and is assumed to be constant over generations. The constant input of new genetic variance per generation due to mutation is assumed to be aL. Subscripts for generations are

omitted for simplicity if a formula or parameter applies for any generation.

In this study, mass selection refers to truncation selection on individual phenotypic values. Family selection is defined as the selection of the best families based on phenotypic family means. Within-family selection is the selection of the best progeny within each family based on individual phenotype. Index selection refers to the selection of individuals based on an index of individual, full-sib and/or half-sib family information, optimally weighted every generation (FALCONER and MACKAY 1996). BLUP selection is carried out by the selection of individuals with highest breeding values estimated by the “pseudo-animal model BLUP”, i e . , an index of individual, full- and half-sib information plus estimated breeding values of the dam, sire and mates of the sire (WRAY and HILL 1989). Pseudo-BLUP is almost identi- cal to true BLUP when there are no fixed effects other than the overall mean (WRAY and GODDARD 1994), and we will simply call it BLUP henceforth.

There are equations available to approximate effective population size under the selection schemes under- lined above (WRAY et al. 1994). In this paper, however, we use values obtained by simulation, for simplicity. For a description of the simulation procedure and examples of predictions using the equation of WRAY et al. (1994), see CABALLERO et al. (1996b). The reduced selection intensity due to the correlation between index of family members is calculated by using the formulae of HILL (1976) and MEUWISSEN (1990) to correct for finite populations.

Prediction of genetic variance: Based on the infinitesimal model, the additive genetic variance at generation t (&) can be predicted if gametic linkage disequilibrium is ignored. It is then equal to the genic variance (&), Le., ait = a:, = aio [(l - 1/(2N,)lt = aioe-1’/2N~, where a:o is the initial genetic variance and Ne is the effective population size. When gametic linkage disequilibrium is accounted for, the prediction of ail is not straightforward. However, the within-family additive variance (aiwl) is not affected by the linkage disequilibrium (BULMER 1971) and therefore can still be predicted, i e . , aiw = uiuoe-t/2Ne, so we will use it as the reference point in all predictions. We now add to this last equation the contribution of mutation. The within- family additive variance due to mutations accumulated until generation tis equal to NgL(1 - Lie., half the total additive variance, given by CLA~TON and ROB- ERTSON (1955) and HILL (1982)l. Thus, the total within-family genetic variance at generation t is

o A W l 2 = a i w m + (oAWO - a ~ w m ) e - t / 2 ~ , (1) 2

where

CAW^ = Ne& 2 (2)

is the steady state within-family additive genetic variance, regardless of the value of ~ W O .

Selection Response 1963

Prediction of response: The response to selection at generation t(&) is

R,= ZPptffAt, ' (3)

where i is the selection intensity and pt is the accuracy of selection at generation t. The cumulative response to selection until generation t (CR,) is

CR, = R, = s' ipraArdT. (4)

R, and CR, can, therefore, be evaluated if pt and ait are expressed in terms of the within-family additive variance at generation t, which is predicted from (1). Equations 3 and 4 can be tedious to derive in terms of aiw, except under simple assumptions, and numerical solutions can be obtained using a mathematical package (e.g., MAPLE; see HECK 1993). However, a first order Taylor series approximation can always be used providing t/N, is sufficiently small (say, t/N, < 2),

r=l 0

where & = ipoaAo is the selection response at generation 0 and the derivative

d log R d log aiw P =

(log indicates natural logarithm) gives the relative rate of change in response ( R ) and aiw No subscript generation is given in the derivative, because it can be evaluated at any generation, but here we evaluate it using the genetic parameters at the reference generation, i.e., generation zero. Substituting (1) into the above expression,

or approximately

assuming t/N, < 1. The cumulative response to selection until generation t (CR,) can be obtained by integrating functions (5a) or (5b) over generations,

X [ t - 2N,(1 - e-"'y)] , (6a) I

Note that, if mutation is ignored (a" = 0 ) , then from (2) aiwm = 0 in (6, a and b).

The general equations (Equation 6, a and b) allow the cumulative response to selection to be approximated for a range of selection methods and parameter values. We now describe how gametic linkage disequilibrium is specifically accounted for. In an infinite p o p ulation, a limiting value of genetic variance is closely approached after the Bulmer effect occurs in a few cycles of directional selection. The response (variance) predicted with this limiting value was called the asymp totic response (variance) by WRAY and HILL (1989). In this study, the response (variance) is continuously increased by mutation and erased by drift and, therefore, it does not necessarily asymptote. We shall still call it the asymptotic response (variance) for simplicity, however.

The approach that we will follow for the derivation of parameters in generation 0 (p and &) consists of assuming that the population has already experienced similar selection (which is a realistic assumption in breeding programs) so that the initial response (&) will actually be the asymptotic response. Even if the scheme of preselection is different, the approximation of response would not be significantly influenced because the asymptotic response is closely approached within a few generations. We assume that the variance within families at generation zero (aim), which is the result of previous selection, mutation and genetic drift, is known. Then we calculate aio as the asymptotic variance assuming that the initial genetic variance was 2aio. Analogously at any generation t we predict the genetic variance within families (aiwt) accounting for mutation and drift using (1), and ait as the asymptotic variance assuming that the initial genetic variance was 2aiW. Explicit formulae to calculate the asymptotic variance are derived in APPENDICES A.1 (Equation Al) and A.2 (Equation A3) for mass and family selection, respectively. Those for index and BLUP were obtained numerically.

Thus, the genetic variance at generation t (ai,) is always predicted using as a reference the within-family variance at that generation. The response to selection at generation t is then R, = ZppAt, where both p, and ai, account for initial asymptotic variance, genetic drift, mutation and Bulmer effect for a further t generations.

To find solutions for the derivative p = d log R / d log o ~ W needed in (6, a and b), this can be expressed as

d log R d log a: P = p 1 P 2 = d l o p d l o g a ~ w '

or where PI indicates the relative proportional rate of change of R and 02. This can be derived for mass selec-


tion by setting h2 = &‘(ai + a:) and R = ihoA = i&ap to give

p d log R d ( b A ) ai 1 dlog da; h o A 2

1 - - - 1 - - h2. (7)

Equation 7 also applies for family (subscript f) and within-family (subscript w ) selection, replacing h2 by h;, the heritability of family means, and ?& the heritability of within-family deviations, respectively (see Table 1). p2 reflects the relative rate of change of ai and aiw. Derivations of p2 for mass selection and family selection are given in APPENDICES B.l and B.2 (Equations A4 and A5). Within-family selection is not influenced by the Bulmer effect since it utilizes only the within-family additive variance and, therefore, pw = pwl. For index and BLUP selection, derivatives of ,8 are more complicated (APPENDICES B.3 and B.4). Note that, for BLUP selection (subscript B ) , p B = P e l , because the response including the Bulmer effect is proportional to that without including it (DEKKERS 1992; see APPENDIX B.4).

RESULTS

Characteristics of equations and validation: In the above section we have derived explicit equations to predict cumulative response in terms of known parameters of the population. To check the validity of these equations, we can compare them with more exact recur- rence calculations. In these, gametic linkage disequilibrium is accounted for (see, e.g., VERRIER et al. 1991) and we include mutational variance (a:). Let aist and ahnt be the variance between half-sib families and full- sib families within sires at generation t , respectively, and a:, the genic variance at generation t , i.e., the additive variance which would result in the absence of selection. Thus

a:, = anl-l(l - 1/2Ne) + a:, (8a) 2

where pst-l is the accuracy of selection of sires at generation t - 1; Ns is the number of sires selected; (1 - l/Ns) is used to correct for the finite population size (KEIGHTLEY and HILL 1987; VERRIER et al. 1993), assuming sampling of parents without replacement; ks = is( is - xS), where is is the selection intensity and xs is the standardized truncation point; and corresponding quantities apply for dams (subscript D ) .

Note that, in Expressions 8, the mutational variance (a&) initially arises within families (Equation 8b) but it is later redistributed among and within families (Equa- tion 8a). In the absence of selection ai, = a:,, and for t -+ a, a:= -P 2NpL and oiwm ”* Nea:, as expected.

12

0 0.5 1 1.5 2 GenerationlN e

FIGURE 1.-Cumulative response (in upunits) to mass selection predicted by Formulas 6a (---) and 6b ( - - -) compared with that from recurrent equations (Equation 8) ( - ). Pa- rameters used were as follows: 20 males, 40 females, six progeny scored per couple, N, = 46, $ = 0.2 and uk = 0.

The accuracy of selection ( p s and pul) was obtained with the usual procedure (see FALCONER and MACKAY

1996, pp. 243-244). Response to selection every generation was calculated from (3) using the asymptotic variance as the initial value (aio).

The cumulative response (C&) predicted by (6 , a and b) is compared in Figure 1 with that obtained by the recurrent equations (Equation 8). Comparisons are shown only for mass selection, for simplicity, but the results are similar for other selection methods. Formula 6a fits well until about 2N, generations, after which it starts to overestimate response; (6b) fits well up to N, generations, after which response starts to be substantially underestimated.

The role of the different parameters determining cumulative response can be seen in Formula 6b. The initial response (6) depends mainly on the accuracy of selection and is the key factor in determining the early response. When t G Ne, the term [p(1 - 0 ~ ~ J a i ~ ) t ~ / ( 4 N , ) ] G t so that CR, = t&. With increasing t , however, the roles of N, and p become more important.

The value of p determines the relative rate of change in response and variance, i.e., how much the response is reduced due to the loss of variance by genetic drift and the Bulmer effect. As the response is proportional to the accuracy of selection ( p ) (Equation 3), p will be small if the accuracy is fairly insensitive to changes in additive variance (or heritability), and it will be large if the accuracy is highly dependent on the heritability. In general, p lies between 0.5 and 1, being smaller with a larger p, h2, and/or family size (n). This is shown in Figure 2, A-F, where the values of p and p are plotted against h2 for different selection schemes and population parameters. In general, 0 is small for selection

Selection Response 1 !Kl

A ’

0.8

0 h 0.6 E a 4 0.4 0 0

0.2

0

B 1

0.8

6 0.6 E a 3 0.4 0

0.2

0

c 1 0.9

0.8

0.7

0.6

5 0.5 2 0.4

0.3

0.2

0.1

0

m 0

0.9

.- g 0.8 c .- P 1 0 0.7

0.6

0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Heritability Heritability

E 1

0.9

.- $! 0.8 U

.- !2 1

d 0.7

0.6

0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


0.9

.- $! 0.8 m >

0 0.7

U

.- 1

0.6

0.5 4 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


FIGURE 2.-The accuracy ( p ) (A-C) and derivative, p = d log R / d log azw (D-F), of different selection methods plotted against initial heritability (R is the initial response and uiw is the within-family additive variance). Parameters used were as follows: 20 males, 20 females. A and D: n = six progeny scored per couple, u: = 0. B and E: n = 24, u:; = 0. C and F: n = 24, u:; = 0.2 u$ Both P and p are evaluated in the initial generation.


schemes using family information (family, index and BLUP selection), particularly when family size is large, because the change of accuracy is insensitive to change in heritability ( c f : Figure 2, A and B). For example, in the extreme case of family selection with large family size (Figure 2, B and E), for values of h2 < 0.1, the accuracy is a very steep function of h*, while for h2 > 0.1, it is nearly independent of h2 (Figure 2B). This means that a small reduction in additive variance when h2 < 0.1 has a large impact on the response, i.e., is large; but when h2 > 0.1 a reduction in additive variance has almost no impact on the response, i.e. p is small (see Figure 2E). For large family size (Figure 2E), the value of p under BLUP and index selection is around 0.65 and changes very little with h2 between 0.2 and 0.8.

For a given heritability, an increase in common environmental variance (a:) implies a reduction in a: and thus an increase in the selection accuracy for within- family selection, but a decrease for family, index and BLUP selection ( c f : Figure 2, B and C). Accordingly, the value of p increases with increasing a: for family, index and BLUP selection, but decreases for within- family selection, especially with high h2 ($ Figure 2, E and F).

Role of mutational variance: The role of mutational variance (a") in determining the cumulative response to selection is clearly shown in Formula 6b. If aiwo > aiwm, the term 1 - aiwm/aiwo is positive and selection response will become slower every generation, other- wise it will accelerate. Estimates of 0% from various experiments (LYNCH 1988) are in the range from 0.0001 to 0.005 of, depending on the character, species and population, although most estimates come from Dro- sophila. Estimates for body weight in mice range from 0.001 to 0.006 a: (CABALLERO et al. 1995). Figure 3 shows cumulative responses to mass and BLUP selection (using Equation 6b) for no mutation and two extreme values of aL (= 0.001 a; and 0.01 a;). Mutation can substantially increase the long term selection response.

The cumulative response to mass selection surpasses that to BLUP selection earlier when mutation is present, because selection schemes with smaller accuracy but larger Ne (e.g., mass selection) exploit the mutational variation more efficiently than selection schemes with larger accuracy but smaller Ne (e.g., BLUP). The reason is that the long term response will eventually be determined by the variance at the steady state, i.e., ai, = 2Np" (cf : Equation 2 ) . Different selection methods (say j and k ) can be compared in terms of the ratio of long term responses at the limit [R, from (3), assuming the same selection intensity],

1 Re = P m j f f A m j - pmj&q

h k P m k f f A m k P m k J N e k '

- (9)

where pm is the accuracy of selection at the steady state. In general, a selection method with low Ne is predicted

0 10 20 30 40 Generation

FIGURE 3.-Cumulative response (in apunits) to mass selection (---) and BLUP selection (-) in case of different mutational variance: ak = 0 (lines without symbols); a" = 0.001 a$ (lines with X); and 0% = 0.01 a$ (lines with A). Parameters used were as follows: 20 males, 40 females, 12 progeny scored per couple, @ = 0.2.

to give a lower asymptotic response than another with larger Ne, even if the first has a higher accuracy of selection for a given value of genetic variance. This is because the accuracy will eventually depend on the steady-state variance (ab) that is also a function of Ne. Thus, if Ne is low the accuracy will eventually be low at the steady state. Compare, for example, mass selection (subscript m) and within-family selection (subscript w ) .

In this case, (Nm/Nm)1/2 2 & in general, because N, = 2N irrespective of selection, while N, is usually smaller than N , particularly with high h2 and intense selection (see, e.g., SANTIAGO and CABALLERO 1995). The accuracies at the steady state are

p m w = JYLPL/~(N~PL + ( ~ N L P L ) and

pm, = 2 N , a ~ / d ( 2 N m a ~ + a:) (2N,aL).

Thus, again because N, 2 2N,, it follows that pmw/pm, 2 l/&. Therefore, substituting these inequalities into (9), ReW/km 2 1, and a larger response in the long term is predicted for within-family selection than mass selection, even though the accuracy for within-family selection is about one half that for mass selection, for a given genetic variance.

Short vs. long term selection response for different selection methods: Inbreeding depression needs to be taken into account in long term selection (FALCONER and MACKAY 1996). Assuming a linear relationship between inbreeding coefficient and genetic mean, the total reduction of genetic response at generation t is D( 1 - e-'lzNe) = Dt/ (2Ne) , where D is the inbreeding depression per unit of inbreeding (in am units). Thus from Equation 6b,


' " I .

0 10 20 30 40 50 60 Generation

FIGURE 4.-Cumulative response (in apunits) to mass selection (---) and BLUP selection (-) with @ = 0.2 in case of different population sizes and different proportions selected, i.e., NS = 20 males, N D = 40 females, n = six progeny scored per couple (lines without symbols); Ns = 20, N D = 40, n = 12 (lines with X); and = 40, ND = 80, n = 12 (lines with A).

C & = & t - P 1 - 7 - -D-. (10) [ ( 2;) 2N, t

The more general Equation 10 can now be used to compare different selection methods. For example, the generation when two different selection schemes (j and k) are expected to achieve the same cumulative response ( Tmss) is

/ *

4% 4N* defining a = 1 - a:wm/a:wo, for simplicity.

Selection intensity affects the long term response both directly and indirectly through its influence on Ne and the Bulmer effect, particularly for intense selection, low h2 and index or BLUP (HILL 1985; MEUWISSEN 1990). For example, with h2 = 0.2,20 sires and 40 dams selected every generation out of 240 progeny, i = 1.296 for an infinite population. If computed for a finite p o p ulation, i = 1.283 for mass selection and 1.257 for BLUP selection. The difference between i values would be larger with more intense selection.

Predicted responses from mass and BLUP selection for different population sizes and selection intensities

0 10 20 30 40 Generation

FIGURE 5.-Cumulative response (in cpunits) to mass selection (---) and BLUP selection (-) when common environmental variance (a;) is assumed to be 0 (lines without A) and 0.1 a; (lines with A). Parameters used were as follows: 20 males, 40 females, 12 progeny scored per couple, @ = 0.2.

are compared in Figure 4 where no inbreeding depression is assumed. The cumulative response to mass selection surpasses that to BLUP earlier with a small population size or a high selection intensity. If inbreeding depression were considered, mass selection would catch up BLUP even earlier, as Ne for BLUP is smaller and, therefore, it is more affected by inbreeding depression than is mass selection.

The effect of common environmental variance (a:) on selection response is illustrated for mass and BLUP selection in Figure 5. With an increase in a;, the effective population size is reduced for mass selection but increased for index and BLUP selection, for less weight is given to family information (see, e.g., CABALLERO 1994). However, because a common environmental variance substantially reduces the accuracy of BLUP selection but not that of mass selection for the same value of heritability, cumulative response to mass selection surpasses that to BLUP earlier when common environmental variance is present (Figure 5) .

Comparison of equations with previous results: If the Bulmer effect is ignored, Formulas 3 and 4 can easily be developed for illustration and as a reference for comparison to previous results. Let us first consider mass selection, for which the accuracy of selection at generation tis pf = ht, the square root of the heritability. Ignoring the Bulmer effect, the total genetic variance at generation tis azf = 2 0 ; ~ from (l), and the response from (3) is


TABLE 1

Parameters used to predict cumulative response to family and within-family selection, which substitute for the corresponding parameters in Formulas 3-7 for mass selection

Mass selection Within-family selection Family selection

where a& = ai, + af is the steady-state phenotypic variance.

The cumulative response to selection until generation t can be obtained by integrating (12),

X tanh" ( a m - g p l ) c J P m 2 a p P m - UP-

where tanh" is the hyperbolic arc tangent function. If there is no mutation, aL = aim = 0, and (13) reduces to

CR, = 4Nei(am - a&). (14)

Predictions for the cumulative response to family and within-family selection (ignoring the Bulmer effect) can also be obtained from the same formulae (Equations 12-14), but replacing Ne, i and other corresponding parameters as listed in Table 1. Expressions for within- family selection in the table are taken from HILL et al. (1996).

ROBERTSON (1960) derived a similar formula to predict cumulative response to mass selection without taking account of the Bulmer effect, i e . , CR, = 2Ne& (1 - giving a limit of C& = 2Ne& = 2Neia~0/am. In this formula, however, a& is assumed to be a constant over generations, while in our derivation of (14), both ait and a& are assumed to change, and only a: remains constant. Hence apm = a, and noting that aio = oPo - a: = (apo + a,) (apo - aT), the selection limit from (14) is C& = 4Ne2bio/ (apo + a?). Thus, the ratio of ROBERTSON'S limit to the limit derived in this study is

2

This ratio is near to one for small g, but if this is large, ROBERTSON'S formula predicts a substantially smaller response (down to one-half for = 1).

If the population is initially isogenic (aio = 0 ) , the cumulative response from mutational variance alone can be obtained from (13), substituting a& = &

(1 - e - ' lnNe) + of. The resulting equation differs from that derived by HILL (1982), i e . , CR, = 2Nei(aL/ap) [ t - 2N,(1 - e- t /2Ne)] , which also does not allow for change in a;. While in the short term (13) agrees well with HILL'S formula, the difference becomes significant in the long term, especially when aL and/or Ne are large. For example, with a" = 0.001 a& and N, = 100, the difference in predicted cumulative response is only -2% at generation 80, but with a value of 0% five times larger, the difference is 3% at generation 20, and 10% at generation 80.

A formula to predict cumulative response accounting for both initial and mutational variance has not been formally derived previously. KEIGHTLEY and HILL (1992) used an approximation by adding two independent terms from the formulae of ROBERTSON (1960) and HILL (1982), but this inappropriately applies two different values of selection accuracy to the same selection process. Equation 13, by contrast, uses a value of selection accuracy applicable to the initial and mutational variation together, so if evaluated with aio = 0 it would not simply be the difference between the results of Equations 13 (aio f 0, & # 0) and 14 (DL = 0).

DISCUSSION

We have derived expressions to predict selection response as a function of known parameters in the population, including a number of factors. These expressions are useful in the assessment of the impact of such factors and parameter values on the selection limits, the comparison of the response with different selection methods, and the interpretation of selection experiments. We discuss now some of the possible uses of the equations.

Selection limits and interpretation of results from long term selection experiments: WEBER and DIGGINS (1990) reviewed responses in long term selection experiments and found that the cumulative response predicted by ROBERTSON'S formula was always larger than that achieved after 50 generations of selection at various population sizes. One possible reason for this discrep- ancy would be the fact that ROBERTSON'S formula ig- nores the Bulmer effect. Generally, the predicted cumulative response from (6a) is -5-10% smaller than


TABLE 2

Predicted cumulative response to mass selection (CRJ compared with that achieved in the long term selection experiment of YOO

Predicted C& by different formulae

Generations Robertson's, Equation 14, Equation 6a, Equation 6a, selected Achieved no Bulmer no Bulmer Bulmer Bulmer (average) CR, (UL = 0 ) (UL = 0 ) ( U t = 0) (UL = 0.002 u;)

~~~~

76 22.4 24.9 26.0 23.1 26.7 88 28.8 27.6 28.9 25.8 29.2

In YOO'S (1980) experiment, six replicate lines were selected to increase abdominal bristle number in Drosophila (three lines selected for an average of 76 generations, another three lines for 88 generations). Every generation, 50 pairs of parents were selected at an intensity of 20% (Nc = 60). In the base population, = 0.5 and @ = 0.2.

ROBERTSON'S equation, and would give a better fit to the examples given by WEBER and DIGGINS (1990).

Let us consider the results of YOO'S (1980a) long term selection experiment to illustrate these effects. In Table 2 are shown the observed responses in this experiment and the predictions without accounting for Bulmer effect and mutation (Equation 14), accounting for Bulmer effect but no mutation (Equation 6a with o', = O), accounting for Bulmer effect and mutation (Equation 6a with o', = 0.002a;), and ROBERTSON'S (1960) prediction. We note that ROBERTSON'S prediction is slightly smaller than the corresponding prediction by (14), as shown by (15). The inclusion of the Bulmer effect (Equation 6a with g', = 0 ) reduces con- siderably the predictions, while the account for mutation (Equation 6a with o& > 0 ) increases them again. For the first group of three lines selected for 76 or so generations, the formula excluding mutation but accounting for the Bulmer effect (Equation 6a with o& = 0 ) fits best to the experimental results. For the second group of three lines selected for 88 or so generations, predictions with and without accounting for both mutation and the Bulmer effect (Equation 6a with a& > 0 and Equation 14, respectively) are closest to observa- tions. Interestingly, mutations of large effect were de- tected in two lines of the second group (Yo0 1980b). However, not too much emphasis can be given to these comparisons, because of the restrictions in the prediction model (see below).

Based on an infinitesimal model without mutation, ROBERTSON'S (1960) theory predicts a selection limit of 2NeR0. Such a limit is very unlikely to be achieved, but even if the infinitesimal model were true, it would be very difficult to be observed experimentally. First, the limit cannot be reached in a real experiment unless population size and generation interval are very small. Further, populations of small size are difficult to maintain for many generations because of reproduction and fitness problems (FALCONER and MACKAY 1996). Finally, mutational variance cannot be ignored in the long term, and additive genetic variance would never be ex- hausted in theory under the model used here.

DEMPFLE (1974) compared the selection limits to within-family ( w ) and mass selection (m) considering the effects of genetic drift and Bulmer effect, and concluded that within-family selection achieves a higher selection limit. We can generalize this result as follows. Assume first, for simplicity, that a& = 0, a: = 0 and ignore the Bulmer effect. Using (14),

" C L 4Nk,jk(apmo - 0,)

C L 4N,i,(ffm - 0,) -

1 - (1 - h2)1/2

= [z] (1 - /&2/2)1/2 - (1 - h2)1/2.

Because N, = 2Nm and, if n is sufficiently large, i, = i,, the above expression reduces to

C L 1 - (1 - h2)1/2

C L 2[(1 - h2/2)1/2 - (1 - h y ] -= 5 1.

The ratio has a maximum of 1 for h2 + 0, equals about 0.9 for h2 = 0.5, and a minimum of about 0.7 for h2 = 1. If the Bulmer effect were included, the ratio would be even smaller because this reduces effective size and genetic variance for mass selection, but not for within- family selection, so that N, 2 2N, in most cases. Analo- gously, an increase in common environmental variance (a:) decreases effective size for mass selection but not for within-family selection, so CR,,/ CR,, becomes even smaller with 0% > 0. Finally, the long term response from mutation will also be smaller for mass selection than for within-family selection, as was explained before.

The above argument refers to the theoretical limit to selection. In a feasible time span, however, within-family selection will give smaller responses than mass selection, as it utilizes only about half of the total genetic variance. For this reason, the use of such a scheme in breeding practice is not useful except, perhaps, for breed conservation purposes.

Comparison of short and long term response for different selection methods: Equation 14 nicely illustrates the antagonistic relation between effective size and se-

1970 M. Wei. A. Caballero and W. G. Hill

lection accuracy in the cumulative response. This latter is a function of two factors, Ne and (gA) - oR). On the one hand, the magnitude of the value of om - opt for a given tis mainly determined by the selection accuracy, which depends on the particular selection scheme: a scheme of higher accuracy generally results in a larger reduction of variance and, therefore, in a larger value of om - C T ~ . On the other hand, the larger Ne, the larger is the cumulative response resulting from the same reduction in variance. Because a selection method of higher accuracy gives a smaller Ne but a larger reduction of additive variance (i .e. , larger om - oR), the two factors have an antagonistic effect on the response. It is then expected that the quicker the genetic variance is reduced, the less efficiently this is exploited to change the population mean. Therefore, selection schemes with higher accuracy achieve less response in the long term, and it seems to be impossible to manipulate accuracy and effective size simultaneously to achieve maximum cumulative response for both the short and long term.

WRAY and HILL (1989) concluded, ignoring genetic drift, that the ranking of breeding schemes is not greatly altered when compared by predicted responses after one-generation rather than asymptotic responses, and DEKKERS (1992) showed something similar for BLUP selection. Even if genetic drift is considered, the predicted response after one generation of selection may be still sufficient to rank breeding schemes in the short term (say, up to five generations), because the accuracy of selection is still the major determinant of response.

Simulation studies on long term selection response in finite populations have been conducted by many authors, with conclusions depending on population size and time span. For example, BELONSKY and KENNEDY

(1988) concluded that BLUP achieves larger response than mass selection for 10 generations of selection, because the increased accuracy of BLUP selection more than counterbalances the reduction in effective size. VERRIER et al. (1993) used different population sizes and showed that mass selection achieves larger response than BLUP if population size is small. QUINTON et al. (1992) showed that mass selection can yield higher response than BLUP selection when the comparison is made at the same level of inbreeding, i.e., when the population size is assumed to be different for the two schemes, so that more intense mass selection yields the same inbreeding rate as BLUP selection. As shown by (11) and illustrated by Figures 3-5, the time for the ranking in response of selection methods to change depends mainly on effective size, but also on other parameters such as r&, i, &, etc.

Some systematic methods to control inbreeding for long term selection have been proposed (e.g., TORO and PEREZ-ENCISO 1990; VILLANUEVA et al. 1994), for example, the use of a biased upward heritability in

BLUP evaluation gives less weight to family information, reducing the rate of inbreeding more than selection accuracy (GRUNDY et al. 1994). WRAY and GODDARD (1994) and BRISBANE and GIBSON (1995) have proposed an iterative selection algorithm to maximize genetic response in a certain time horizon by balancing response and inbreeding, i.e., selecting individuals based on their estimated breeding values and relationship to other individuals. These methods enable BLUP selection schemes to achieve more response for a given time horizon, but mass selection will eventually produce a larger response because of its larger effective size.

Limitations of the study: The infinitesimal model is unrealistic because there can only be a finite number of loci that control the character and the gene effects may not be all small. Change of variance due to change of gene frequencies as well as dominance, linkage and epistatic effects are not accounted for under this model. Thus, an infinitesimal model is likely to be adequate for predicting short or medium term selection response, but to become less valid as the number of generations increases. Approximate formulae given in this paper apply up to about Ne generations for which the infinitesimal model assumptions may still hold.

In the model, a constant environmental variance (a:) is assumed, but its pattern of change is actually unknown. A phenomenon observed from selection experiments (e.g., BUNGER and HERRENDOWER 1994; HEATH et al. 1995) is that it increases with selection. However, little is known of the magnitude of this incre- ment and how it relates to genetic response or the reduction of additive variance.

In this study, mutational variance is assumed to be constant, and all mutants are neutral with respect to fitness and of small effect (CLAWON and ROBERTSON 1955; HILL 1982; LYNCH and HILL 1986). Mutants with large effects on quantitative traits and deleterious pleiotropic effects on fitness certainly occur, however ( U T - TER 1966; HILL and KEIGHTLFY 1988; CABALLERO and KEIGHTLEY 1994). The fate of mutations of large effect under artificial selection with different selection methods has been investigated by CABALLERO et al. (1996b). Natural selection on phenotype, ie., stabilizing selection, may also play a role, so that the actual response is smaller than predicted here (UTTER 1966; NICHOLAS and ROBERTSON 1980; ZENG and HILL 1986).

The model of mutation assumed in this paper is the random walk mutation model (e.g. , LYNCH and HILL 1986). A house-of-cards mutation model ( C O C K E M and TACHIDA 1987; ZENG and COCKERHAM 1993) was not considered but is not likely to affect the results of this paper as the difference between the random walk and house-of-cards mutation model is expected to be large only in a time scale far beyond that considered in this paper (see ZENG and COCKERHAM 1993).

Finally, random mating is assumed in the analysis,


but in practice mating can be controlled to reduce the rate of inbreeding and to maintain more genetic variation (e.g., TORO and PEREZ-ENCISO 1990; CABALLERO et al. 1996a).

CHEAVALET (1994) has developed a theory for selection response assuming a finite number of unlinked loci, and HILL and RASBASH (1986) proposed a nonin- finitesimal model allowing for different distributions of gene effects and frequencies, although nonadditive effects and linkage are not accounted for. As more information on distribution of gene effects and gene frequencies and on collateral effects on fitness for quantitative traits become available, more realistic models can be applied. In the meantime, however, an infinitesimal model continues to be a useful reference point to inter- pret selection experiments and to predict selection response.

We are grateful to B. VILLANUEVA and particularly J. DEKKERS for useful comments on the manuscript and to the Biotechnology and Biological Sciences Research Council for financial support.

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APPENDIX A

Derivation of additive genetic variance at the limit under mass selection and family selection: The expressions derived in this appendix are given as a function of ai and aio. To apply them for any generation t, replace a: and aio by ait and 2aiwt, respectively.

Mass selection: GOMEZ-RAYA and BURNSIDE (1990) gave a formula to calculate the genetic variance at the limit (ai) for mass selection assuming an infinite population. For a finite population, we still assume that the within-family variance (aio/2) does not change over generations, but consider a correction in the between- family variance for the number of sires and dams selected (N, and No) due to the sampling of parents with replacement (KEIGHTLEY and HILL 1987; VERRIER et al. 1993). Thus, from (8)

ai = 0.25(1 - l/N.) (1 - k.&)ai

+ 0.25 (1 - l/ND)(l - k,&)d + d0/2,

where p: and p; are the squared limiting accuracy of selection in sires and dams, respectively, and are equal to each other and to the limiting heritability h2 = aV(ai + a:). Rearranging the equation ai = aio/ (yml + y,h2), where Yrnl = 1 (I/& + 1/ND)/2, and ym2 = [(I - 1/Ns)k.$ f (1 - 1/ND)kD1/2, giving

+ h2,I2 + 47dh:(1 - ho)) 2 1/2

where @ = uio/ (aio + a:). Equation A1 reduces to that given by GOMEZ-RAYA and BURNSIDE (1990) when N7 and ND + w, i.e., yrnl = 1 and yd = ( k ~ s + kD)/2.

Family selection: The additive variance of family means at the limit (05~) can be derived as a function of initial additive variance of family means (a:p) and k, ( i e . , k for family selection). Analogously as before, from (8) the recurrent equation at the limit is

ai = 0.25 (1 - l /N.)( l - k,p;)a;

+ 0.25 (1 - l/ND) (1 - k&)a: + 0:0/2,

where k, = kD = k . under family selection. p; = a&/(aia$) is the limiting value of squared accuracy under family selection, where a$ = aif + 0% is the phenotypic variance of family means at the limit. The recurrent equation can be rearranged as ai = a i o / [ (2 - yf) + kfyp;] where yf = 1 - (l/Ns + 1/N1)/2. Substitut-

ing p; gives

[r,(k,- 1) + 2 1 4 + rpd,

1 u2 -

A f - 2[yXk,- 1) + 21

where

APPENDIX B

Rate of change in selection response with respect to rate of change in genetic variance for different selection schemes: All the derivations in this appendix are evaluated at generation 0, which is taken as a reference generation. The subscript 0 , however, has been omitted from all parameters, for simplicity.

Mass selection: The derivative pm = d log R,,,/d log aiw can be expressed as Pmlpm2 = ( d log R J d log ai) X ( d log o i / d log aiw). From (7), pml = 1 - h2/2 . Substituting (1) into (Al) and deriving

-1 + [h2 + (yml + 2y,)(l- h2)l

p,=1-h2 +([yrnl(l - h2) + h2I2 + 4y,h2(1 - h2))l/'

yrnl(l - h2) - h2 + {[yrnl(l - h2) + h2I2 ' + 4y,h2(1 - h2))l/'

(A4)


where yml and ym2 are given in Mass selection of APPEN- DIX A. The function Pm2 increases with increasing hz, and limh' + 1 Pm2 = 1. Equation A4 also applies to a full-sib ( y m l = 1 + 1/[2ND] and ym2 = [ l - l/ND]kD) or half-sib family structure ( yml = 1 + 1/ [2Ns], and

Family selection: Analogously to the above, Pp = (1 - h;/2), where hj is the limiting value of heritability of family means, and

ym2 = [1 - 1/Nslks).

- (1 - n ) y f - 2 n + 1 - 2Yf2]

(A5)

where yf, yfl and yf2 are given in Family selection of APPENDIX A. The function P f 2 decreases with increasing h; and family size, and limh2 + 1 Pf2 = 1.

Index selection: First, we just derive an expression (Pn) without accounting for the Bulmer effect. The index consists of individual performance (expressed as deviation from full-sib family mean), full-sib mean (as deviation from half-sib family mean) and half-sib family mean. The phenotypic variance-covariance matrix is diagonal with the following elements:

P I 1 = [ ( n - l)/nI(aS + ai /2) ,

PIZ = [(m - l ) / ( n m ) l [ d + d + (n + 2)d/41,

P I , = {a: + na2c + [n(m + 1) + 2Iai/4)/(nm),

where n is the full-sib family size, and m is the number of dams mated to a sire. The vector of covariance between genetic values and index information has the following elements:

wn = [(n - 1) / (2n) Ig i ,

WE = [(n + 2)(m - 1)/(4nm)]ai,

wn = [2 + n(m + 1)]oi/(4nm).

The variance of the index is a: = Z dj/po. Thus, Pn = d log Rl/d log aiw is

2 + n(m + 1) 2prs -

+ $3 "1 . (A6) ~3

When half-sib information is not available ( m = l ) , then the second term in these equations ( wR, p R ) drops out; and if there are only half-sibs (n = l ) , the first term drops out. Similarly, if n = 1 and m = 1, the index

reduces to mass selection and (A6) reduces to (7). For the extreme case where n + UJ and a: = 0, Pn + 0.5h2

In general, the value of Pn lies between 0.5 and 1, and decreases with increasing family size (n and m) and/or h2.

When the Bulmer effect is accounted for it is, however, very tedious to express ai as a function of aiw, a: and k, and to obtain its derivative. The formula for PI accounting for the Bulmer effect is not presented, and a numerical solution has been used (PRESS et al. 1992).

BLUP selection: BLUP selection is approximated by selection based on an index (pseudo-animal model BLUP) consisting of individual record (I), full-sib mean (FS), half-sib mean (HS), estimated breeding values of the dam (AD), the mean of the estimated breeding value of all dams mated to the sire (AD,) and the estimated breeding value of the sire (A,y) (WMY and HILL 1989). To facilitate the derivation, the index is set up using independent linear functions of the records so the variance-covariance matrix is diagonal, where the diagonals are for (1 - m), (n - HS - &/2 + Au/2), (HS - As/2 - Au/2), and (AD/2 + As/2). These diagonal elements are

+ (1 - h z ) / ( 2 - h').

P B l = [(n - 1 ) / n ] ( d + d / 2 ) ,

p B 2 = [(m - l ) / ( n m ) I [ d + 4 + (n + 2)ai/4] - [1/4 - 1/4m]p2ai,

p B 3 = [a: + na: + (n + nm + 2)ai/4]/nm

- [1/4 + 1/4m]p2ui,

p , = p2ai/2.

Here, p2 is the squared accuracy of BLUP selection. The vector of covariances between breeding values and information sources has elements

W B l = [(n- 1 ) / ( 2 n ) ] d ,

WBZ = [(n + 2)(m - l ) / ( n m ) l a i - [1/4 - 1/4m]p2ai,

WBS = [(n + 2 + nm)/(4nm)]ai - [1/4 + 1/4m]p2ai,

wB4 = p2&2.

The variance of the index is aiI = Z wij/ps, for j = 1, . . . , 4. Let 6 = d p2/d ai, then


HENDERSON (1982) found that under animal model BLUP the variance of prediction error is determined only by the amount of information on self and relatives and is not affected by selection. Hence, DEKKERS (1992) showed that the ratio of asymptotic to original response is equal to a constant, [2/ (2 + ks + k D ) ] 1’2. Therefore, in this case p B = pBl = d log R/d log cTiwand, therefore,

with 4 from (A7). If half-sib information is not available,

m = 1, and the second term in w and p in (A7) and (A8) is dropped. Further, when the pseudo-BLUP index includes only individual information and parental estimated breeding values, Le., the case that DEKKERS (1992) considered to provide lower limit to accuracy of

p2a;/2, pH = p2a?/2, and the first two terms in w and p are dropped. Then

BLUP, wBJ = (1 - p2/2)& WM = p2&2, = 0; -

and

selection response in finite populationsa closed and finite population with a nested mating...

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