quantitative genetic variability maintained by mutation

Genet. Res., Camb. (1989), 54, pp. 45-57 With 7 text-figures Printed in Great Britain 45

Quantitative genetic variability maintained by mutation-stabilizing selection balance: sampling variation andresponse to subsequent directional selection

PETER D. KEIGHTLEYf AND WILLIAM G. HILLDepartment of Genetics. University of Edinburgh. West Mains Road. Edinburgh EH9 3JN

(Received 10 October 1988 and in revised form 22 December 1988)

Summary

A model of genetic variation of a quantitative character subject to the simultaneous effects ofmutation, selection and drift is investigated. Predictions are obtained for the variance of thegenetic variance among independent lines at equilibrium with stabilizing selection. These indicatethat the coefficient of variation of the genetic variance among lines is relatively insensitive to thestrength of stabilizing selection on the character. The effects on the genetic variance of a change ofmode of selection from stabilizing to directional selection are investigated. This is intended tomodel directional selection of a character in a sample of individuals from a natural or long-established cage population. The pattern of change of variance from directional selection isstrongly influenced by the strengths of selection at individual loci in relation to effective populationsize before and after the change of regime. Patterns of change of variance and selection responsesfrom Monte Carlo simulation are compared to selection responses observed in experiments. Theseindicate that changes in variance with directional selection are not very different from those due todrift alone in the experiments, and do not necessarily give information on the presence ofstabilizing selection or its strength.

1. Introduction

Many selection experiments use as their sourcematerial samples from caged insect populations (oftenDrosophila) which have been in the cage environmentfor many generations. Samples are taken from thecage, separate selection lines started, and responsepatterns of individual lines and the variation inresponse among lines are obtained. Similarly, selectionof artificial populations is usually with species whichhave been under domestication for many generationsand there are often independent selection lines. Suchexperiments should provide information on theunderlying genetic basis of quantitative variation.

Many characters in natural (and perhaps in arti-ficial) populations are thought to have intermediateoptima. A popular model of selection with anintermediate optimum is stabilizing selection. This hasintuitive appeal, there is some evidence for itsoperation in nature and the model is amenable toanalysis. Previous analyses of the consequences ofstabilizing selection in natural populations have beenconcerned with the genetic variance maintained when

t Corresponding author.

the character is at or near the optimum, in which caseit is necessary to invoke mutation to maintain geneticvariation. Kimura (1965) analysed a 'continuum ofalleles' model, which involves loci at which the effectof mutant alleles differs only slightly from the previousallelic state. Kimura derived a formula for theequilibrium genetic variance of a locus at a mutation-stabilizing selection equilibrium, and showed that theequilibrium distribution of allelic effects segregatingat the locus is normal. The model was further analysedby Lande (1976) who argued that it predicts thatsubstantial variation can be maintained even withstrong stabilizing selection.

The second type of analysis differs from the firstbecause the effects of mutant alleles can be large.Turelli (1984) analysed a 'House of Cards' approxi-mation of the continuum of alleles model, which wasoriginally proposed by Kingman (1978). The criticalassumption which differed from that in Kimura's(1965) analysis above was that the effect of a mutantallele swamps existing variation at a locus controllinggenetic variation in the trait. This gives differentqualitative predictions of the variance maintained atthe locus at equilibrium and agrees with two alleleanalyses of Latter (1960) and Bulmer (1972). Turelli

P. D. Keightley and W. G. Hill 46

argued that a 'House of Cards' approximation of themutation process is appropriate for per-locus mutationrates likely to be found in nature.

With the exception of Bulmer (1972) the analysesdescribed above have been of infinite populations.This has an important consequence for the probabilitydistribution at equilibrium of allele frequencies at alocus influencing the trait. With stabilizing selection inpopulations near equilibrium, mutations are uncondi-tionally deleterious (Robertson, 1956). In an infinitepopulation, if there are few alleles segregating at thelocus the equilibrium probability distribution of allelefrequencies is therefore highly leptokurtic, i.e. mutantalleles are almost always very rare, and intermediateallele frequencies are absent. Such a distribution isvery 'U-shaped'. This affects the consequences of ashift in the optimum. Barton &Turelli (1987) analysedthe dynamics of the population mean and varianceafter a change of the optimum and showed anaccelerating rise in the mean, slowing down as itapproached the new optimum, and a rise in thegenetic variance because some previously deleteriousmutant alleles became advantageous and were selectedto intermediate frequencies where they contributedmore substantially to the genetic variance. In somecases, the variance fell again close to its original value(i.e. alleles became fixed) and in others a newequilibrium was reached with a higher variance. Theexistence of such multiple equilibria was predicted byBarton (1986). A change of optimum is similar toimposing directional selection on a character pre-viously subject to stabilizing selection.

In a finite population, alleles are able to drift infrequency, so the equilibrium probability distributionof allele frequencies becomes less leptokurtic thandescribed above. Bulmer (1972) derived an expressionfor the probability distribution of allele frequency at amutation-stabilizing selection-drift balance for thecase of up to two alleles per locus. This was used byKeightley & Hill (1988) and Burger, Wagner &Stettinger (1988) to investigate finite populationmodels of genetic variance maintained at a mutation-stabilizing selection balance for various distributionsof mutant gene effects.

Here, expressions are derived for the variance of thegenetic variance among independent lines at a muta-tion-selection-drift balance. The results of previousinvestigations of the problem using neutral models(Bulmer, 1976; Avery & Hill, 1977; Lynch & Hill,1986) are compared. The consequences of a changefrom stabilizing selection to directional selection onthe genetic variance of a character in a finitepopulation are investigated. These results togetherwith patterns of response to directional selection fromMonte Carlo simulation of populations previouslyunder stabilizing selection are compared to responsepatterns obtained from experiments published in theliterature.

2. Model

The assumptions of.the model are essentially thcsame.as described by Keightley & Hill (1988). The popu-lation consists of N random mating diploid individualswith constant population size and non-overlappinggenerations. The phenotypic value of an individual isX, the sum of contributions from the alleles affectingthe trait plus an independent normally distributedenvironmental effect of variance VE. New mutationsappear independently at random and effects are addedto the value at the locus at which they occur. Amutant effect, a, is the difference in value between thehomozygotes and q is the frequency of the highervalued allele. Mutant effects are sampled from agamma distribution,

f(a) = a11 e-*a a1'-1 /T(JS) (0 < a < oo).

The distribution was reflected about zero by randomlyassigning the sign of each mutant effect with proba-bility \ of being positive. The resulting distributionis termed a 'reflected gamma distribution'. Theparameter /? determines the shape of the distribution(fl-*0 gives a highly leptokurtic (geometric) distri-bution; /?-> oo is the limiting case of equal absolutevalues of effects), and the parameter a. is the scale ofthe distribution. The root mean square of thedistribution of mutant effects is defined as

e = E(a2/Vj = [/?(/?+ l)/(a2K,,)]i

The parameter Nu /<, where /* is the mutation rate atthe locus, is assumed to be sufficiently small, or theparameter \NL.s\, where .? is the selection coefficient ofthe mutant allele, is assumed to be sufficiently largethat no more than two alleles segregate at any time ateach locus. The genomic mutation rate is A = n/i,where n is the total number of loci affecting the trait.The expected genetic variation arising per generationin the population is KM = AE(a2)/2 (Hill, 1982 a).

The genie variance of the character is the sum ofvariance contributions from each locus, I.a2q(\ —q)/2.The additive variance is the variance of genotypicvalues and includes terms due to departures fromHardy-Weinberg and linkage equilibrium.

Where stabilizing selection is operating, it isassumed that it has acted for sufficient time that thepopulation is close to equilibrium. The fitness understabilizing selection is defined by a 'nor-optimal'model with optimum fixed at zero. The relative fitnessis

W(X) = exp (— X2/2w2), (1)

where vv is an inverse measure of the strength ofselection. The change in gene frequency is approxi-mated by

A? = 0% -1) <?( 1 - <?)/[4(V + <r2)] (2)

Selection and quantitative variation 47

(Robertson, 1956), where a is the phenotypic standarddeviation. New mutants are unconditionally deleteri-ous and there is a meta-stable equilibrium at q = \.Monte Carlo simulation has shown that this ap-proximation is accurate over a wide range of para-meters (Keightley & Hill, 1988). The term w2 + cr2 isoften referred to as Vs, the strength of stabilizingselection.

The model is used to investigate the effect on thegenetic variance of a change from stabilizing selectionto directional selection. There are therefore two phasesof selection, a stabilizing selection phase followed bya directional selection phase. From diffusion theory,in the first phase the steady state probability dis-tribution of allele frequency with recurrent mutationis a function of N1sl, where Nt is the effectivepopulation number in the first phase and Sj is theselective value of the mutant allele. With stabilizingselection st is frequency dependent (Robertson, 1956)and is approximated by

ors* = -

when the mutant allele is rare. The steady statedistribution of allele frequency with recurrent mu-tation and stabilizing selection is therefore a functionof the parameter

, E(s*) = - /v\e2 2 + o-2)].

In the second phase of directional selection, thedistribution of gene frequency at generation / is afunction of t/N2 and N2s2, where N2 is the effectivepopulation size in this phase and s2 is the selectioncoefficient of the favourable allele. For example, thepattern of change of allele frequency over 10 gener-ations in a population of 100 individuals with aselection coefficient of 01 is the same as that over 20generations in a population of 200 individuals with aselection coefficient of 005. With truncation selectionthe selection coefficient is approximately s2 = ia/a,where / is the intensity of selection. Since thedistribution of a is symmetrical in the stabilizingselection phase, the distribution of initial genefrequencies in the directional selection phase is alsosymmetrical, so the directional selection phase isparameterized by 7V2 E(s2) = N2 iE(\a\)/cr.

3. Methods

(i) Transition matrix iteration

Using this method, the expectation and variance ofheterozygosity maintained at a locus with recurrentmutation and the expected steady state allele frequencydistribution with recurrent mutation were computed.Details of the method are given by Keightley & Hill(1988). To model a change from stabilizing todirectional selection, the expected steady state allelefrequency distribution with recurrent mutation in the

stabilizing selection phase was computed using thetransition matrix with the expected change of genefrequency from (2). The expected variance was thencomputed for each generation of directional selectionby iterating a transition matrix with change of genefrequency given by

Aq = s2q(\-q)/[2(l+s2q)].

The transition matrices were of dimension N = 80. Togenerate expected responses for a gamma distributionof mutant effects, numerical integration was used (seeKeightley & Hill, 1988 for details).

(ii) Monte Carlo simulation

Using this method, the effects of simultaneouslysegregating mutants are assessed. The simulationprocedure has been described in detail elsewhere(Keightley & Hill, 1983). Here, only the case of freerecombination was considered. Essentially, there wasan infinite number of freely recombining sites and newmutants, with effects sampled from a reflected gammadistribution, arose at unique sites. Selection wasperformed by assigning fitnesses or probabilities ofproducing progeny (i.e. fertilities) to each individual.With stabilizing selection, the phenotypic value, X, ofan individual was the sum of genotypic contributionsplus an environmental effect of variance VE = 1, andthe relative fitness given by (1). Where the equilibriumbehaviour was needed, the population was allowed toapproach equilibrium by initiating the populationfrom an isogenic state and allowing mutations toaccumulate for 67V generations. The variances of genicand additive variances at equilibrium were computedfrom independent runs. After a change of selectionmode to directional selection, the fitness was simplythe genotypic value, because in this case it wasassumed that i/cr = 1, so the selection coefficient of anallele was the same as its effect.

4. Results

The first part of the results considers the variationamong independent lines of the genetic variancemaintained with mutation and stabilizing selection.Subsequently, the consequences of a change fromstabilizing selection to directional selection are con-sidered.

(i) Single locus analysis

Using diffusion theory (Kimura, 1969), Bulmer (1972)derived the density function of gene frequency, qi; ata locus (0 under stabilizing selection with equalforward and backward mutation rates, ft, betweentwo possible alleles

= Cexp[(-44X7,(1 - (3)


where 0 = 4A>>, d> = Nea2/[S(w2 + o-2)], Ne is the

effective population size, and C is a normalizingconstant. It follows that the expected heterozygosityat the locus is

•o:a:cxp [ -

This was shown to reduce to

(4)

3), (5)

where I(x, y) is a function of the complete betafunction and the confluent hypergeometric function(Bulmer, 1972). Assuming 0 is small (i.e. ignoringback-mutation) (5) simplifies to

(Keightley & Hill, 1988). Integrating over the dis-tribution of gene effects,/(a), gives the expected genievariance

E(Vg) = a2f(a)da.(6)

Bulmer's (1972) analysis can be extended to derive aformula, with similar assumptions, for the variance ofthe genie variance at a locus with recurrent mutationamong independent lines. It follows from (3), (4) and(5) that

= /(4O, 0 + 2)//(4O, (7)

Assuming 0 ^ 0 , by similar analysis the variance ofthe genie variance among lines is

Table 1. Comparison of predictions of variance ofgenie variance derived from (8) (diffusion theory ofindependent genes) and Monte Carlo simulation

NTheoryV(Vg)x 104

Simulation. xlO4

10152030

1-942-923-895-83

1 95 ±0082-85 + 0173-68 + 0135-65 + 018

The parameters of the simulation were A = 0-2, e = 0-1, anda reflected gamma distribution of mutant effects with shapeparameter /? = \. There was no selection.

Strong selection, d>-*oo. By simplifying (6),Keightley & Hill (1988) show that the expected genievariance is E{ Vg) = 4A(w2 + a-2) which equals thatobtained by Latter (1960) and Turelli (1984) whichassumed infinite population size. The variance of thegenie variance among lines by similar analysis from(8) is

Both the expectation and the variance of the genievariance with strong stabilizing selection (large Nos)are therefore independent of the magnitude of theeffects of mutant alleles. The coefficient of variation ofFgis

CV(Vg)=\/(4N0A)>. (12)

Comparison of (10) and (12) shows that for newmutant alleles of equal effect, the coefficient ofvariation varies by only a factor of l / \ /3 betweencases of weak and strong selection. The shape of the

I ) 2 - 1)) \/e0\a4f(a)da2 3O'/(/!(4(/

Equations (6) and (8) can be evaluated easily byiteration on a computer and converge readily. Theywere checked against results obtained from a transitionmatrix and were found to agree almost exactly.Equation (8) also agrees with results from the MonteCarlo simulation (Table 1). Two limiting cases are ofparticular interest.

Neutrality, <J>^-0. From (6) the expected genievariance is

(8)

The variance of the genie variance from (8) is

in agreement with Lynch & Hill (1986) who used adifferent derivation. The coefficient of variation of Vg

is therefore

CV(Vg) = [(E{a')/E\a2))/(\2NeA)t (10)

distribution of effects of mutant alleles becomesimportant as selection becomes weak. Fig. 1 shows thecoefficient of variation of the genie variance amonglines expressed as a fraction of that expected forstrong stabilizing selection (cj>-s- co) as a function of <J>= A'oe

2K1../[8(w2 + a-2)]. The curves are for a range ofvalues of /?, the shape parameter of the gammadistribution. All curves converge with increasing O asthe CV becomes independent of the shape of thedistribution but the shape parameter has increasinginfluence as <&-i-0.

,q\ (ii) Disequilibrium

The above analysis applies to the genie variance andits variance among independent lines. Such quantitiescannot easily be measured. The additive variance andits variance among lines, which can be estimated, is

1 influenced by departures from Hardy-Weinberg and


linkage equilibrium at different loci affecting the trait.Bulmer (1976) and Avery & Hill (1977) showed thatvariation in disequilibrium can be an importantcontributor to the variation in the additive variationamong lines. Here, Monte Carlo simulation is used toexamine previous results on a neutral model andexamine the effects of selection.

Neutrality. Using results obtained by Avery & Hill(1979), Lynch & Hill (1986) derived an expression forthe coefficient of variation of the additive variance

3-0

2-5 •

2 0 •

u10 -

0-5 -

00

= 0-25

12 16 20Ne

Fig. 1. The coefficient of variation of the genie varianceamong independent lines at equilibrium as a proportionof that predicted for very strong stabilizing selection,namely [l/(4yVeA)]5, as a function of NeE(s*) = <J> =Nee

tVK/[S(wt + a1)]. The curves were generated bynumerical integration of (6) and (8) using gammadistributions of mutant effects. The shape parameterranges from /?-> oo to /? = \ with intermediate values of/? = 8, 4, 2, l , i

among independent lines at an equilibrium betweendrift and mutation in the absence of linkage,

CV(VA) = [(£(a4)/£V))/(12/VeA) + 2/(3AQ]5 (13)

(note, the additional n~x term given by Lynch & Hill(1986) is inappropriate). The first term is the varianceof the genie variance [cf. (10)] and the second is thevariance of disequilibrium (note, although there is noselection and no net disequilibrium, the disequilibriumin each line varies stochastically about zero). Thevariance of the additive variance is therefore

V(VA) =

12

10 -

8-

oX

(14)

c20

15

10

5

0 40 80 120N'

160 200 240

Fig. 2 The variance of the additive variance amongindependent lines for various parental population sizes(JV) with variation in N', the number of individuals usedto estimate VA. The curves were generated by (a) MonteCarlo simulation ( ) in the absence of selection andmutation parameters such that many alleles segregate (A= 0-2, e = 0-1 sampled from a reflected gammadistribution with /? = ±); {b) equation (15) ( ) with

] and fi = \.

Table 2. Comparison of predictions of variances of genie and additivevariances among lines from Monte Carlo simulation and theory

Theory Simulation Theory SimulationNl N,. K(Ke)xl04 V(Vt)±\ s.E. xlO4 K(KA)xl04 K(KJ± 1 s.E. x 104

5102030

418-5

170260

0-751-452-583-54

0-78 + 0041-35 + 0-062-81 ± 0 1 33-62 + 0-20

0-861 662-97409

0-86 + 0021-53 + 0-03303 ± 0 0 6407 ± 0 0 7

The mutation parameters of the Monte Carlo simulation were the same as Table1 (A = 0-2, e = 0 1 , and a gamma distribution of mutant effects with shapeparameter ft = \ reflected about zero). The character was under stabilizingselection with w2 + cr2 = 2. The effective population size was measured in thesimulation by following the fates of independent neutral alleles. The effectivepopulation size is less than the actual population size because of selection. Thevalue of yv. computed by the computer program was used to compute thetheoretical values in the Table. The theoretical value of V(VJ is from (8). Thetheoretical value of K(KA) is ^((/

B) + 2£2(K|!)/(3Af(,), i.e. includes the disequilibriumcomponent from the neutral model [cf. (14)] and E(Vt) was computed from (6).

GRH 54

P. D. Keighlley and W. G. Hill 50

V/VB

l-6-i

1-2 -

0-8 -

0-4 -

-

(o) (b)

0 0

Fig. 3. Strength of directional selection, N2E(si) = 1.

10t/N2

2 0 0 0 10t/N,

2 0 0 0 10IIN2

20

2-8 -,

2-4 -

20 -

(a)

V/Vo

(b)

20 00

(c)

20 00 10UN,

20

Figs. 3-5. The expected genie variance as a proportion ofthat at t = 0 plotted against t/N2 for four strengths ofstabilizing selection, TV, E(s*). The solid line on eachfigure is the expected variance for any value of A E(s*)for no directional selection [N2 E(s2) = 0]. Curves forthree reflected gamma distributions of mutant effects areshown (a) fi = i; (b) fi = 1; (c) fi^ oo.

Bulmer (1980, ch. 12) points out that there is anambiguity in the interpretation of (13) and (14)because the variance of the additive variance dependson the number of individuals measured to estimate theadditive variance within each line, and this may bedifferent from the effective population size, A c. As a

starting point for resolving this difficulty, let Ne be theeffective number of parents in each line (as before) andN' be the number of individuals per line used toestimate the^ additive variance. Assuming a normaldistribution of observations, the variance in theestimate of the variance among lines due to sampling

Selection and quantitative variation

3-6 -,

3-2 •

24 \

51

VIV0

20 •

(c)

Fig. 5. N2E(s2)= 10.

is approximately 2V\/(N'- 1) ~ 87V,2 V\JN'. Thisadditional source of variation affects V(VA) so (14)can be rewritten taking this source into account,

(15)

This reduces to (14) as N' becomes large, in which casethe additional variance from estimating VA becomessmall. The results of Monte Carlo simulation of aneutral model with many alleles segregating arecompared to evaluation of (15) in Fig. 2. Lines ofvarious sizes were allowed to reach steady state andthe additive variance and its variance among lineswere computed using various numbers of progeny.The agreement between the models is very close.

There would be an additional source of variation inestimating V(VA) caused by error in estimating VA

within lines, using, for example, offspring-parentregression, correlation of sibs or other appropriatemethod. For example, if VA were estimated withinlines from the covariance of half-sibs, it can be shown(cf. Robertson, 1959) that the variance of the estimateof VA is approximately 2V\[\ + 4/(vh2)]2/N', where vis the number of progeny per half-sib family (assumedconstant), h2 is the heritability and N' in this casemeans the number of sires. Adding this additional

term to (14), V{VS) (where the estimation of VA isdone by half-sib covariance) becomes,

V(VA) = N0\E{ai)/\2 + {2Nc V»(16)

For small numbers of progeny and sires and traits oflow heritability at equilibrium, the last term in (16)can dominate. Equation (16) gives the variance of theadditive variance in the absence of linkage. Withlinkage, the term 2/(3^,) is inflated, although not bymuch for species with many chromosomes (Lynch &Hill, 1986).

Stabilizing selection. No formulae are available topredict the variance among sublines of the disequi-librium component at steady state with stabilizingselection, although formulae for the expected disequi-librium with selection and an 'infinitesimal model'have been derived elsewhere (Keightley & Hill, 1987).However, with free recombination, disequilibrium in apopulation has a very short 'memory', with onaverage half the previous disequilibrium lost due torecombination each generation. It is likely thereforethat the additional term in (14) for the neutral case isa good predictor of the variation in disequilibrium forthe case of stabilizing selection. Simulation runs(Table 2) show good agreement with eqn (8) forand (14) for V(VA).


Table 3. The mean, coefficient of variation (CV), and range ofcumulative response to selection to generations t = 10 and t = 20 from JOindependent populations initially at a mutation-stabilizing selection-driftequilibrium

N, E{s*)

0

0

0

0

2

2

8

8

e

0-2

0-2

0-8

0-8

0-2

0-2

0-8

0-8

fiCO

i2

0 0

12

CO

12

CO

I

t

10201020102010201020102010201020

Mean

2-213-622153-351-692131-781 491-362-460-661160-420-640-250-36

CV

0190170-360-340-440-410-510-660-320-220-360-411141061-641 67

Range

1-27-2-902-74-^-911-06-3-631-77-5-120-52-2-820-88-3-340-39-3-380-39-3-450-64-2-291-85-3-820-42-1-180-58-1 95

-006-1-50002-2-26

-0-02-1-31-002-1-94

Mutations occurred in the stabilizing selection phase only and were sampled froma reflected gamma distribution with shape parameter (i and scale parameter e givenin the Table. The mutation rate A was such that VM/ Vv = 10~3. The population sizein the stabilizing selection phase was /V, = 160 and in the directional selectionphase was N2 = 20. Response in units of i/cr.

(iii) Effect of change of selection mode on variance

The results of the previous section show that thetendency for stabilizing selection to generate anextremely U-shaped distribution of allele frequenciesinfluences the variance of the genetic variance betweensublines. This effect also influences the pattern ofresponse and change of variance of a character understabilizing selection subsequently subjected to di-rectional selection.

For a gamma distribution of mutant effects, theeffect of selection on the genetic variance of the charac-ter is a function of three parameters: (1) N, £(.?*),the expected selective value in the stabilizing selectionphase; (2) N2E(s2), the expected selective value in thedirectional selection phase; (3) /?, the shape parameterof the gamma distribution. Figs. 3-5 show expectedgenie variances (ignoring disequilibrium) in the gener-ations after a change in mode of selection for arange of N1E(sf) and N2E(s2). Mutants occurredonly in the stabilizing selection phase and effects weresampled from reflected gamma distributions. Curvesfor three different values of the shape parameter, /?,are shown: (1) fi = f, a highly leptokurtic distribution(see Keightley & Hill, 1988) ;{2)ft= 1, an exponentialdistribution; (3) /?^- co, all mutant effects have equalabsolute values. In all cases, the probability of amutant of positive or negative effect was assumed tobe the same. An implicit assumption of the analysis isthat there is no stabilizing selection operating in thedirectional selection phase or, equivalently, thatdirectional selection is strong relative to stabilizing

selection. Figs. 3-5 show a wide range of values of theparameters. The selective values in the stabilizingselection phase range from Nx E(s*) = 0 (neutrality)to N1 E(s*) = 40. The latter case would pertain, forexample, if [E(a2)f = 0-1, w2 + a2 = 20 and N1 =6-4 xlO5. The range of selective values during thedirectional selection phase is from N2 E(s2) = 0 (neu-t ra l i t y ) t o N2E(s2) = 10 [e .g. E(\a\)/a = 0-\, i=\,and N2 = 100].

With values of N2E(s2) at the high end of the rangeshown, the pattern of change of variance departssubstantially from that observed with neutrality. Insome cases, there is a marked rise in variance followedby a rapid fall. The rise in variance occurs during thefixation of beneficial alleles segregating initially at lowfrequency. Such a pattern is therefore observed whenthe following conditions pertain: (i) strong directionalselection, so such alleles have a high probability offixation; (ii) strong stabilizing selection because theprobability distribution of allele frequencies becomesincreasingly U-shaped with increasingly strong stabi-lizing selection, so the expected initial frequency ofbeneficial mutants of large effect is low. The pattern ofchange of variance becomes most extreme with aleptokurtic distribution of mutant effects (e.g. /? = \)because as the mutational distribution becomes moreleptokurtic, a higher proportion of the mutationalvariance is contributed by mutations of large effect.

The pattern of rapid rise followed by rapid fall invariance depends on the presence of beneficial mutantsin the directional selection phase, i.e. mutants ofpositive effect. In Figs. 3-5 there are equal probabilities


of mutants of positive and negative effect, but thepattern of rapid rise followed by a rapid fall invariance becomes more extreme with a higher pro-portion of mutants of positive effect (results notshown). In other cases, the variance falls off morequickly than for neutral genes. This occurs with weakstabilizing selection, but strong directional selection,in which case alleles initially segregating at inter-mediate frequencies become fixed at a high rate andthe genetic variance falls rapidly.

The curves show the expected genie variance andignore the consequences of disequilibrium. Selectiongenerates a negative disequilibrium component ofvariance which increases with increasingly tightlinkage (see Bulmer, 1980, ch. 9; Keightley & Hill,1987). In such circumstances, the additive variance isless than the genie variance and the pattern of increasein additive variance would be less extreme thanshown.

4 8 12 16 20 0 4 8 12 16 20t I

Fig. 6. There was no selection in the stabilizing phase.The sizes of gene effects were given by e = 02 and VM wasio-3.

(iv) Variation in response

Using similar methods to the above, Hill & Rasbash(1986) analysed the variation in response to directionalselection. Higher variation in response was noted withincreasingly leptokurtic distributions of effects ofsegregating alleles and with increasingly U-shapedprobability distributions of allele frequency. The vari-ation in the genie variance or response can be easilycomputed with a transition matrix. Using MonteCarlo simulation, however, it is possible to generatereplicates of responses for different NlE(s*) andN2E(s2) parameter combinations and the generalpattern of the response is perhaps easier to visualise(and compare to the results of experiments).

Table 3 shows cumulative responses and CVs ofcumulative responses to generations 10 and 20 among10 independent replicates sampled from independentpopulations initially at a mutation-stabilizing selectionbalance with a range of strengths of stabilizingselection and sizes of gene effects (examples ofresponses are plotted in Figs. 6 and 7). The Tablecompares results from a reflected gamma distributionof mutant effects with shape parameter (1 = \ andequal probabilities of positive and negative effects (/?-» oo). The population size in the stabilizing selectionphase was N1 = 160 and in the directional selectionphase was N2 = 20, and KM was 10"3. The main pointsto note from the table are: (i) In theory, the averageinitial response rate is equal to the standing additivevariance in the stabilizing selection phase. With nostabilizing selection [N1 E(s*) = 0], the theoreticalinitial rate is therefore 2NxVM=0-32VK, but thecumulative response to generation 10 was less than tentimes this because of the presence of disequilibriumgenerated by directional selection and a loss of geneticvariance due to changes in gene frequency; the averageresponse with stabilizing selection is lower than theaverage response from initially unselected populations

(b)

12 16 20 0 12 16 20

Fig. 7. The strength of stabilizing selection was such thatN, E(s2) = 8, and the value of e was 0-8 with VM = 10"3.

Figs. 6-7. Examples of selection responses generated byMonte Carlo simulation from directional selection ofsamples of N2 = 20 individuals from independentpopulations of size JV, = 160 individuals at mutation-stabilizing selection-drift equilibrium (a) Reflected gammadistribution with ft = \. {b) Equal probabilities of positiveand negative effects. Response in units of ;/<r.

because the expected steady state variance is lowered;(ii) the more leptokurtic distribution of mutant effects(/? = |) gives a higher coefficient of variation ofresponse than that for equal absolute values of mutanteffects; (iii) with stronger stabilizing selectionfjV, E(s*) = 8], because there is little standing variance,average response is generally small, but occasionallyan allele of large effect segregating at low frequencygives a rapid early response so the range of responseis large relative to the mean; (iv) many responsepatterns give similar means, variances and ranges of


response and in practice would not be distinguishablefrom one another. Some examples of responses, theresults of which are summarized in Table 3, are shownin Figs. 6 and 7. Fig. 6(a, b) shows results for areflected gamma distribution with shape parameters/?^oo and /? = f respectively, for the case of nostabilizing selection and small gene effects (e = 0-2).These response patterns are similar to those commonlyobserved in selection experiments, although it shouldbe emphasised that weak stabilizing selection wouldnot lead to much change in pattern. In contrast, Fig.7(a, b) which again is for cases of reflected gammadistributions with shape parameter /? = oo and ft = \respectively show much higher variation in response.In these cases, stabilizing selection was relativelystrong [Nl E(s*) = 8] and gene effects were large (e =0-8). Such responses are not typical of selectionexperiments.

The responses in Figs. 6 and 7 show the change inmean genotype, so there would be more variation inmean phenotype than shown because of the presenceof an environmental component of variation. How-ever, this would contribute little to the variation ofresponse unless N2 is very small or the heritability ofthe character is low. The simulated selection responseswere generated from independent populations. Inpractice, a caged population is often used to initiateindependent lines, so variation in genes segregatinginitially and hence variation in response would be dueto sampling from the base population rather than todifferent genes segregating in independent popula-tions. Simulation showed that responses generatedfrom sub-populations of a single population ratherthan a set of independent populations tend to vary lessthan the responses discussed above because of reducedvariance between lines in the initial genetic varianceand alleles of large effect which are not rare are likelyto be fixed in all replicate lines. If, however, themutational variance is generated by few mutants oflarge effects (e.g. e, = 08, cf. Fig. 7), under stabilizingselection, because genes segregate at very low fre-quency at equilibrium, the variation in response isvery similar to that obtained by sampling fromindependent populations.

5. Discussion

(i) Mutation-stabilizing selection balance

The two allele models of Latter (1960) and Bulmer(1972) and Turelli's (1984) 'House of Cards' ap-proximation of the continuum of alleles model showedthat the expected genie variance in an infinitepopulation is independent of the effects of alleles atthe loci controlling the trait. Similarly, our resultsshow that with strong stabilizing selection the varianceof the genie variance among independent lines is alsoindependent of the size of gene effects, and is only afunction of the effective population size, strength of

stabilizing selection, and the genomic mutation rate,A. As drift becomes more important relative toselection, the coefficient of variation of the genievariance becomes increasingly dependent on the shapeof the distribution of effects of new mutants, but wehave little if any information about this parameterand can only conjecture that distributions of mutanteffects are very leptokurtic (Robertson, 1967;Keightley & Hill, 1988; Shrimpton & Robertson,1988). Variation in estimates of genetic variance fromdifferent populations therefore does not necessarilytell us much about the selective forces operating in thepopulation.

Turelli's (1984) 'House of Cards' analysis ofmutation-stabilizing selection balance is multi-allele,but the formula for the expected genie variance atequilibrium is the same as obtained from the two alleleanalyses of Latter (1960) and Bulmer (1972). Whyis this so? In these models, the population sizeis assumed to be infinite or, equivalently, selection isassumed to be very strong. The probability dis-tribution of allele frequencies is therefore of extremelyU-shaped form with alleles at intermediate frequenciesabsent. Each new mutant allele almost always occursat a locus previously carrying the 'wild-type' allele.The fates of new mutant alleles are therefore almostindependent of any other mutant alleles segregating atthe same locus in the population, and a two alleleanalysis with the parameter n/i replaced with genomicmutation rate, A, is sufficient. Similarly, with weakselection the fates of different alleles at the same locusare essentially independent of one another, they canbe considered as occurring at separate loci, and thetwo allele treatment also applies. As shown previously(Keightley & Hill, 1988), the model of stabilizingselection is very similar to a model of unconditionallydeleterious genes with selection coefficient s pro-portional to a2 and independent of gene frequency, q.

The tendency for stabilizing selection to generatean extremely U-shaped probability distribution ofallele frequencies has important consequences forsubsequent changes of variance and hence responseswith directional selection. Under certain circum-stances, namely strong directional and strong stabi-lizing selection, large increases in variance occur inearly generations due to the fixation of alleles initiallyat low frequency; in other circumstances, namelystrong directional selection and weak stabilizingselection, a rapid fall in variance (much faster thanexpected from drift alone) can occur due to the rapidfixation of genes initially at intermediate frequencies.Seldom, if ever, are such response patterns seen inselection experiments. For example, the replicatedDrosophila abdominal bristle selection experiments ofClayton, Morris & Robertson (1957), Frankham,Jones & Barker (1968) and Yoo (1980), which wereinitiated from cage populations, showed little sign ofearly accelerated or rapidly falling responses. Similarregular patterns were observed in a Tribolium egg


production selection experiment (i.e. a characterclosely related to fitness) initiated from a cagepopulation (Ruano, Orozco & Lopez-Fanjul, 1975)and in a selection experiment of cannon-bone lengthin Scottish blackface sheep (Atkins & Thompson,1986). In the latter case, Atkins & Thompson showedthat the response closely matched the predictedresponse of an 'infinitesimal model' which incor-porated the effect of disequilibrium on the additivevariance (Bulmer, 1980 ch. 9). The results of theexperiment of Frankham et al. (1968) are of particularinterest because selection on bristle score was per-formed using various population sizes and a range ofselection intensities. The expected response can beestimated if the 'infinitesimal model' is assumed andthe initial genetic variance is obtained from: (i) theheritability estimated from the base population; or (ii)the realised heritability estimated from the selectionresponse in the first one or two generations which intheory is almost independent of the magnitude of geneeffects. Such analysis shows substantially lower pre-dicted responses using the base population geneticvariance estimate than observed in the experiment.Using the genetic variance obtained from the realisedheritability in the first two generations, the agreementbetween the experimental results and the infinitesimalmodel is closer. With the strongest selection strength(10 %) and the biggest population (N = 80), some hintof an accelerating response was observed, but unfortu-nately this line was not replicated. The responses fromYoo's (1980) long-term experiment fit closely aninfinitesimal model if parameters derived from theinitial generations are assumed although the responsecontinued longer than predicted by the infinitesimalmodel. The results of Falconer's (1973) replicatedmouse body-weight selection experiment also give areasonable fit to the infinitesimal model if the geneticvariance derived from the response in the initialgenerations is assumed, but this realised heritability israther higher than Falconer's estimate of the heri-tability in the base population. In this case the lineswere derived from crosses of inbred lines andpresumably some alleles were initially at intermediatefrequencies.

The experimental selection response patterns do nottell us very much about the strength of stabilizingselection which might affect the characters in the basepopulation because, as iV2 s2 -*• 0, the expected varianceeach generation of directional selection becomes thesame irrespective of the distribution of allele fre-quencies. There is, however, a general absence ofobservations of either rapid rises or rapid falls invariance compared to those expected for drift alone. Atleast two explanations are possible for these observa-tions, although they are not mutually exclusive, (i)Selective values of directional selection [N2 E(s*)] arenot high, say greater than one, so drift dominates.This would also imply a very large number of loci ofsmall effect controlling variation in the trait, (ii) The

initial distribution of allele frequencies is not extreme(implying weak stabilizing selection or some othermechanism generating such a distribution). Variationfrom initially segregating alleles falls due to changes ingene frequency from directional selection, but mu-tation contributes sufficiently to variation to maintainresponses (Hill, 1982a, b).

Many of the responses generated by Monte Carlosimulation, using a wide range of parameters, are verysimilar to selection responses obtained experimentally.Some types of response patterns in Table 3 and Figs.6-7 are not, however, observed experimentally. Forexample, with e = 0-8 and a gamma distribution withshape parameter /? = \, responses are very variablebecause there are few genes segregating. This patternbecomes more extreme with strong stabilizing selectionN1E(s*) = 8, Fig. 1b) with some lines giving a rapidearly burst of response. Bursts of response have beenseen in selection lines, generally in long-term experi-ments (Thoday, Gibson & Spickett, 1964; Yoo, 1980),and are most likely the result of fixation of mutantsappearing since the start of the experiment (Hill,19826). The alternative hypothesis of segregation ofrare recessive alleles is also possible, though unlikely ifthe burst occurs late in the experiment as in the casescited above (Robertson, 1978). Breakdown of linkagedisequilibrium is also an unlikely explanation(Keightley & Hill, 1983).

6. Concluding remarks

The validity of the stabilizing selection model ofnatural selection has been discussed extensivelyelsewhere (Robertson, 1973; Turelli, 1984, 1985;Keightley & Hill, 1988; Hill, 1989). The most im-portant weakness is that the pure stabilizing selectionmodel ignores selection which might be acting at thelocus through pleiotropic effects on characters directlyrelated to fitness. Other models where the mutantallele is at a selective disadvantage (e.g. Hill &Keightley, 1988) have similar qualitative effects on theprobability distribution of allele frequencies. Theessential problem in explaining quantitative geneticvariation is not whether mutation is an adequate forceto explain observed variation, for in the absence ofselection it is a more than adequate force. Theproblem is the mode of action of natural selection andthe selective values of the genes affecting the character.

The analysis here is purely additive and ignoringdominance is a serious limitation. As shown byKacser & Burns (1981), the larger the absolute effectof a mutant allele, the more likely it is to behave asnearly recessive. The consequences of this could bededuced with specific models of the relationshipbetween mutant effect and dominance. It is likely thatthe tendency to give bursts or rapid falls in responsewould be reduced, however, because alleles of largeadditive effect would contribute little to the variance


in the stabilizing selection phase and would have littlechance of fixation from directional selection.

Other models of the mutation process might also beconsidered. For example, Cockerham & Tachida's(1987) model differs from the present step-wise modelbecause the effect of a new mutation replaces thecurrent value at the locus, not as in this case adding tothe value. This additional constraint does not affectthe equilibrium behaviour with stabilizing selection asthe model is formally the same as the 'House ofCards'. It can lead, however, to limits in the case ofdirectional selection. The present results, therefore,would only be applicable in the short term.

We are grateful to Michael Turelli for helpful discussions.We wish to thank the Agricultural and Food ResearchCouncil for financial support.

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