the inﬁnitesimal model - university of...

15The Infinitesimal Model

Normal theory is clearly the most powerful and problematic hypothesis inthe present analysis. — Chevalet (1988)

What, me normal? — Turelli and Barton (1994)

Draft Version 14 May 1998, c©Dec. 2000, B. Walsh and M. Lynch

Please email any comments/corrections to: [email protected]

The joint assumptions of normality and linear parent-offspring regressions under-lie most simple models of selection response, as in such cases the single-generationresponse can be predicted from knowledge of the appropriate variance compo-nents (Chapters 4-8). This is in sharp contrast to response under the one- andtwo-locus models examined in the previous chapter wherein prediction requiresdetailed knowledge of the underlying genotype frequencies and effects.

The infinitesimal model, which assumes a very large (effectively infinite)number of loci each with infinitesimal effect, satisfies both normality and linear-ity. Hence, most models of selection response are based on the (either explicit orimplicit) assumption that the infinitesimal model adequately describes the un-derlying dynamics. Of course, the infinitesimal model is not taken as an exactdescription of biological reality. It does, however, represent one extreme of as-sumptions about the underlying loci, allowing us to ignore the effects of allelefrequency changes. When a large number of loci, each of small effect, underlie acharacter the infinitesimal model provides a very satisfactory treatment of short-term response. However, given sufficient time, even with very weak selectionacting on each underlying locus, allele frequencies changes start to become con-siderable, and we can generally no longer describe the changes in variance withjust knowledge of the variance components in the base population.

Under the infinitesimal model, the amount of selection acting on any givenlocus is expected to be very small, and hence the expected change in allele fre-quencies over a few generations is also very small. When summed over a largenumber of loci, these very small allele frequency changes nonetheless allow forsignificant changes in the mean with little changes in the other moments, a pointwe demonstrate below. However, while allele frequency changes do not alter thevariance and higher order moments, as we saw in Chapter 5, selection-inducedchanges in gametic-phase (or linkage) disequilibrium can have profound effect on

235

236 CHAPTER 15

the genetic variance. Further, it can change higher-order moments as well, driv-ing the genotypic distribution away from normality and hence potentially causingparent-offspring (and more general relative pair) regressions to deviate from lin-earity. The focus of this chapter is thus rather technical, examining the robustnessof the infinitesimal model for predicting short-term response in some detail. Theresulting change in the variance (and higher order moments of the genotypicdistribution as well) are assumed to be due almost entirely to gametic-phasedisequilibrium. However, with a small number of underlying loci, the short-termchanges in the higher-order moments of the genotypic distribution are largely dueto changes in allele frequencies (Turelli and Barton 1994). Since our concern in thischapter is largely with the effects of gametic-phase disequilibrium, we are assum-ing models with a large number of underlying loci. Models assuming a smallernumber of underlying loci generally tend to ignore the effects of gametic-phasedisequilibrium, focusing instead on allele frequency changes. These models areexamined in Chapters 11 and 12 (which examine infinite- and finite-populationmodels, respectively), while Chapter 13 examines the very technical subject of theamount of selection maintained by mutation-selection balance.

THE INFINITESIMAL MODEL

Under the classic infinitesimal model, introduced by Fisher (1918), the charac-ter is determined by an infinite number of unlinked and nonepistatic loci, eachwith an infinitesimal effect. Under this model, allele frequencies are unchangedby selection: large changes in the mean occur by summing infinitesimal allelefrequency changes at a large number of loci. To see this, consider a characterdetermined by n completely additive diallelic loci. Further suppose that all lociare interchangeable, with each locus having the same effects and frequencies. Inparticular, assume each locus has two alleles, Q and q, with the genotypes QQ, Qq,and qq contributing 2a, a, and 0 (respectively) to the genotypic value, so that alleleQ has effect a. Further, assume the frequency of allele Q has the same value (p)at each locus. The resulting the mean is 2nap and the additive variance (ignoringthe contribution from gametic-phase disequilibrium) is σ2A = σ2a = 2na2p(1− p).For σ2A to remained bounded as the number of loci increase, a must be of or-der n−1/2. The change in mean due to a single generation of selection is easilyfound to be ∆µ = 2na∆p. Assuming the frequency of Q changes by the sameamount at each locus, ∆p = ∆µ/(2na). Since a is of order n−1/2, ∆p is of order1/(n · n−1/2) = n−1/2, approaching zero as the number of loci becomes infinite.Thus the infinitesimal model allows for arbitrary changes in the mean with (es-sentially) no change in the allele frequencies at underlying loci.

What effect does this amount of allele frequency change have on the variance?Letting p′ = p+∆pdenote the frequency after selection, the change in the additive

THE INFINITESIMAL MODEL 237

genic variance is∆σ2a = 2na

2p′(1− p′)− 2na2p(1− p)= 2na2∆p(1− 2p−∆p)≈ a (1− 2p) ∆µ

Sincea is of ordern−1/2, the change in variance due to changes in allele frequenciesis roughly 1/√n the change in mean. With a large number of loci, very largechanges in the mean can occur without any significant change in the variance. Inthe limit of an infinite number of loci, there is no change in the genic variance(∆σ2a = 0), while arbitary changes in the mean can occur. However, as discussed inChapter 5, the additive variance σ2A = σ2a+d, is the sum of the genic variance (σ2a)plus the effects of gametic-phase disequilibrium (d). When the latter is taken intoaccount, large changes in the overall additive variance σ2A can occur (Chapter 5).The reason for this can been seen from Equation 5.1. Changes in the covariancesCijbetween loci i and j (for i 6= j) are roughly of order n−2 (Bulmer 1980, Turelli andBarton 1990). Since there are n2 terms contributing to d, the total disequilibriumis of order one (n2 ·n−2) and does not necessarily approach zero as the number ofloci becomes infinite. The same reasoning holds for changes in the higher-ordermoments, which are caused by higher-order associations between groups of loci(Turelli and Barton 1990).

MODIFICATIONS OF THE INFINITESIMAL MODEL

While in Chapter 5 we relaxed the assumption that selection does not changethe variance by considering the effects of selection-induced gametic-phase dise-quilibrium, the resulting models allowed the change in variance to be predictedentirely from the variance components of the base population. These models re-quired that the base population is in gametic-phase equilibrium (or that we canestimate the base-population value of d and σ2a). Further, they relied critically onthe assumption that the genotypic and phenotypic distribution remain normalduring selection.

We now relax some of the assumptions of the classic infinitesimal model. Sincewe assume that the reader is familiar with the results and notation from Chapter5, it may help the reader to review this material before proceeding. Throughoutthis section we continue to assume that certain normality assumptions hold (thesewill be stated below). Under these assumptions, we generalized the infinitesimalmodel by considering a finite number of loci, genetic drift, and the effects oflinkage. The last part of this chapter considers the more difficult issue of short-term response when the genotypic distribution is not normal. Chapters 11-13 dealwith the even more complicated subject of long-term response, where significantallele frequencies changes occur. In this case, the variance components of the basepopulation provide essentially no information for predicting long-term response.

238 CHAPTER 15

Gaussian Approximations Allowing for a Finite Number of Loci and DriftSeveral simulation studies (Bulmer 1974, 1976a, Sorensen and Hill 1983, Muellerand James 1983, Chevalet 1988) have shown that the infinitesimal model gives areasonably good fit to the change in variance over a few generations of selectionwhen the number of loci is finite. However, with a finite number of loci, allele fre-quency changes occur and after a sufficient number of generations the cumulativeeffects of these changes become so large that they cannot be ignored. Likewise,when the population is finite, genetic drift also changes allele frequencies. Thuswhen either the number of loci n or the population sizeN is finite, we must incor-porate changes in the genic variance σ2a into our model. As we will see at the endof this chapter and in Chapters 11 and 12, there generally is no simple expressionfor predicting changes in σ2A (recall from Chapter 5 that changes in the genic vari-ance σ2a require changes in the allele frequencies, while changes in the additivevariance σ2A can occur either through changes in gametic-phase disequilibrium orby changes in the genic variance).

If we are willing to assume that the distribution of allelic effects at each locus isnormal, so that the vector of contributions for all underlying loci is multivariate-normal, then fairly simple expressions for predicting the joint change in both σ2aand d can be obtained. This assumption is often refered to as the continuum-of-alleles model, and is also only an approximation, as it requires an infinite numberof alleles at each locus, an assumption clearly violated in finite populations. Thehistorical roots of this model trace back to the classic paper of Crow and Kimura(1964), which represents the first serious treatment of molecular evolution. Thefirst application of this model in quantitative genetics was Kimura (1965), whoused this approach to examine the amount additive variance maintained underthe balance between mutation and selection.

Modifications of the Bulmer equations for the change in variance (Equation5.7) allowing for a finite number of loci (n) were introduced by Lande (1975) andFelsenstein (1977, 1979), while Keightley and Hill (1987) allow for finite populationsize (Ne). The most general result is due to Chevalet (1988, 1994), who considersthe general case where both Ne and n are finite. Assuming that the phenotypicvariance after selection has the formσ2z∗ = κσ2z (Equation 5.10), then the equationsfor change in additive genic variance σ2a and the gametic-phase disequilibrium dbecome

∆σ2a(t) = −[σ2a(t)2Ne

+(

1− 1Ne

)κh2(t)σ2A(t)

2n

](15.1a)

∆ d(t) = −12

[(1 +

1Ne

)d(t) +

(1− 1

n

)(1− 1

Ne

)κh2(t)σ2A(t)

](15.1b)

The resulting response in the mean is given by the breeders’ equation, R(t) =h2(t)S(t), where the current values of h2(T ) (and perhaps S) are obtained fromEquation 15.1. When both N and n are infinite, these reduce (as expected) to


Bulmer’s model

∆σ2a(t) = 0, ∆ d(t) = −d(t) + κh2(t)σ2A(t)

2

Provided we are willing to accept the assumption that the distribution of effects ateach locus remains normally-distribution, we can simply iterate these expressionsto obtain the current values of σ2a and d. We return shortly to the validity of thisapproximation.

Linkage: An Approximate TreatmentA second modification of the standard version of the infinitesimal model is toallow for linkage between loci. As loci become more tightly linked, we expect |d|to increase relative to the standard infinitesimal model which assumes all loci areunlinked. Recall from Equation 5.1b that d(t) = 4

∑j

240 CHAPTER 15

exact treatment). Ignoring this for now, Equation 15.2c implies at equilibriumthat rij C̃ij = δ̃Cij . Using Equation 15.2b gives the equilibrium covariance as

C̃ij =h̃4 δ̃(σ2z)

4n (n− 1)1rij

(15.2d)

thusd̃ = 4

∑j


σ2z = 100, h2 = 0.5, κ = γ = 0.213, and the asymptotic rate of response ofR̃ = 1.4 h̃ 2 σ̃z . Substutiting into Equation 15.4 to obtainθ and recalling Equations5.13a-c gives

H θ d̃ σ̃ 2A h̃2 R̃

0.5 0.37 −12.60 37.40 0.43 5.600.4 0.35 −14.50 35.50 0.42 5.370.3 0.33 −17.11 32.89 0.40 5.060.2 0.29 −20.97 29.03 0.37 4.570.1 0.23 −27.49 22.51 0.31 3.70

H = 0.5 corresponds to free recombination, while H = 0.1 might be expectedin Drosophila melanogaster. Hence, with strong linkage (h = 0.1), the response isreduced 33% relative to unlinked loci.

The infinitesimal model predicts that d̃ increases under disruptive selectionas linkage becomes tighter, while Sorensen and Hill (1983) found in their simula-tions that d̃ decreases as linkage tightens. They reasoned that this discrepancy arisesdue to the interaction between a finite number of loci and the finite populationsizes used in the simulations. To see this, consider complete linkage. In a finitepopulation, the most extreme gamete observed (and hence the ultimate level ofd̃ ) is affected by sampling as selection can generate no gamete more extreme thanthose found in the initial sample. If the number of loci is small, the probability ofsampling the most extreme possible gamete is high, but this probability decreasesas the number of loci increases. Countering this, as recombination (measured byH) and/or the population size increases, the probability increases that recombi-nation can regenerate the most extreme possible gametes before the revelant lociare fixed by drift and/or selection. When population size becomes large enoughthat drift effects are no longer important, d̃ once again decreases with increasinglinkage. Interactions of this sort between drift, selection and recombination areconsidered in some detail in Chapter 12.

Linkage: A More Careful TreatmentA more rigorous treatment of how selection changes the within-gamete covari-ancesCij requires consideration of the between-gamete covarianceCi,j as well ashigher-order covariance terms that measure the amount of gametic-phase disequi-librium between groups of loci. A careful derivation highlights the importance ofthe normality assumptions we have liberally used above. The consequences of re-laxing normality are considered in detail at the end of this chapter and this sectionintroduces some of the notation needed to examine non-Gaussian distributionsof genotypic and phenotypic values.

242 CHAPTER 15

We start by defining the between-gamete covariance,

Ci,j = σ(a

(i)fa, a

(j)mo

)(15.5)

which is the covariance between the effect of an allele at the ith locus in thepaternal gamete and an allele at the jth locus in the maternal gamete. Underrandom mating, gametes unite at random and Ci,j = 0 at the start of each gen-eration. Selection generates correlations between gametes in much the same waythat it generates correlations among loci within gametes. For example, considera particular chromosome containing multiple loci influencing a character understabilizing selection. Initially, there is no correlation between the genetic valuesof the two copies of this chromosome in an offspring from randomly-mated par-ents. Stabilizing selection changes this initial distribution, favoring adults with anintermediate genotypic value. Thus surviving adults with a large genetic valueon one chromosome are expected to have a small value on the other and vice-versa, generating negative Ci,j . Likewise, assortative mating generates positiveCi,j , while disassortative mating generates negative Ci,j .

We assume random mating, so that Ci,j(t) = 0 at the start of each gen-eration. Letting C∗ denote the covariance after selection, C∗ij = Cij + δCij andC∗i,j = Ci,j+δCi,j = δCi,j . Assuming recombination follows selection, with prob-ability 1 − rij no recombination occurs between i and j and the within-gametecovariance is unchanged, while with probability rij recombination occurs and thenew covariance depends on the covariance between gametes, giving the result ofLande (1975) and Bulmer (1980),

Cij(t+ 1) = (1− rij)C∗ij(t) + rijC∗i,j(t) (15.6a)

Substituting for C∗ gives

Cij(t+ 1) = (1− rij) [ δCij(t) + Cij(t) ] + rij δCi,j(t)= (1− rij)Cij(t) + δCij(t)− rij [ δCij(t)− δCi,j(t) ] (15.6b)

Note that we recover Equation 15.2c only if δCij = δCi,j (selection changes thewithin-gamete and between-gamete covariances by the same amount). Turelli andBarton (1990) show that this occurs if there is either global gametic-phase equilib-rium (all groups of loci are in gametic-phase equilibrium) or if the distribution ofallelic effects is multivariate normal. This later assumption is a very strong one,as it requires that the distribution of allelic effects at each individual locus is normal,which strictly speaking requires an infinite number of alleles per locus.

General expressions for δCij and δCi,j have been obtained by Turelli andBarton (1990) for the case of no dominance or epistasis. Their expressions involve(i) generalizations of measures of selection to higher moments of a distribution and(ii) generalizations of disequilibrium measures to groups of k loci. Starting with


(i) first, recall (Equation 4.7a) that we defined the directional selection gradient,which measures how selection acts on the phenotypic mean, as ∂ lnw /∂µz . Wecan extend this notion to higher moments by considering ∂ lnw /∂µk,z , whereµk,z = E[ (z−µz)k ] is the kth central moment of the phenotypic distribution (fork ≥ 2). If selection is primarily on the mean and variance, selection gradients forthe skew and higher moments (k ≥ 3) are generally negligible. For the case wherephenotypes are normally distributed,

∂ ln w∂µz

=S

σ2z(15.7a)

∂ ln w∂σ2z

=δ(σ2z) + S

2

2σ4z(15.7b)

(Lande 1976, Lande and Arnold 1983). As shown in Chapters 14 and 16, whenselection acts only on the mean, the within-generation change in the phenotypevariance is δ(σ2z) = −S2, so that δ(σ2z) + S2 is the change in variance over thatexpected due to selection on the mean.

Using these extended selection gradients and ignoring selection acting on theskew and higher moments (by assuming that gradients for k ≥ 3 are negligible),Turelli and Barton (1990) found that

δCij =∂ ln w∂µz

∑k

Cijk +∂ ln w∂σ2z

∑k

∑l

(Cijkl − CijCkl ) + · · · (15.8a)

δCi,j =∂ ln w∂σ2z

2∑k

Cik∑l

Cjl + · · · (15.8b)

where Cijk refers to the third-order covariance between the alleles at loci i, j, andk. IfXi is the additive value of a randomly chosen allele at locus i and µi = E(Xi)is the average value for this locus, then Cijk = E[ (Xi − µi)(Xj − µj)(Xk − µk) ].Higher-order covariances are defined similarly. The covariances in Equation 15.8ameasure the amount of third (Cijk) and fourth (Cijkl) order gametic-phase dise-quilibrium (the departures from random assortment for triplets and quadrupletsof loci). If selection on the third (skew) or higher-order moments is significant,then Equation 15.8 includes covariance terms of order five and higher.

The key point about these equations is that changes in covariances depend crit-ically on very fine details of the genotypic distribution, details that are essentially im-possible to estimate empirically in realistic situations, and simplifying assump-tions are required to proceed further. For example, if the distribution of geno-typic values is multivariate normal (which, as previously mentioned, involvesthe rather strong assumption that allelic effects at each locus are normally dis-tributed), Equation 15.8 simplifies greatly as Cijk = 0 and Cijkl can be expressedin terms of second-order covariances (Cijlk = Cij Ckl + CikCjl + CilCjk). In thiscase, δCij = δCi,j , and combining Equations 15.7b and 15.8b gives

δCij 'δ(σ2z) + S

2

σ4zRiRj (15.9a)

244 CHAPTER 15

where Ri =∑j Cij . Thus when allelic effects are multivariate normal (normal at

each locus and multivariate normal for any subset of loci), the change in covarianceis given by

∆Cij(t+ 1) =δ(σ2z) + S

2

σ4zRi(t)Rj(t)− rijCij(t) (15.9b)

a result due to Lande (1975, 1977c). Since 2∑iRi = 2

∑ij Cij = σ

2A, then assum-

ing all theRi are equivalent, it follows for n loci thatRi = σ2A/(2n), and Equation15.9a reduces to

δCij 'σ4A

4n2σ4z

(δ(σ2z) + S

2

)=

h4

4n2

(δ(σ2z) + S

2

)When S2


effect underlie the trait. With a small number of loci, to an initial approximation,one can ignore effects of gametic-phase disequilibrium and instead focus on thechanges in the higher genotypic moments caused by allele frequency changes atthe underlying loci. The key results for such models is that even single-generationpredictions require extensive information about the underlying genetics. In con-trast, with a very large number of loci, the (short-term) effects of allele frequencychanges can be essentially ignored, and changes from gametic-phase disequilib-rium become critical. The nice (and somewhat surprising) result for this latterclass of models is that the breeders’ equation (and Bulmer’s extension to changesin the additive variance) are quite accurate for both directional selection and forstrong disruptive selection (Turelli and Barton, 1994).

Describing the Genotypic Distribution: MomentsWe now proceed to the general theory of response under arbitarty distributionsof genotypes. For starters, a few comments on the genotypic and phenotypic dis-tributions are in order. Under our assumption that genotypic and environmentalvalues are additive and independent, z = G + e. Thus, if environmental valuese are normally distributed, phenotypes are normally distributed if and only ifgenotypic values are Gaussian. However, the converse is not true — an approx-imately normal distribution of phenotypes does not imply that genotypic valuesare Gaussian. While we can test to see if phenotypes are normally distributed, thistells us little about the distribution of genotypes. While in theory we can estimatethe distribution of breeding values by computing the breeding values for a sam-ple of individuals (LW Chapter 26), this is impractical in most studies of naturalpopulations.

Since we assume no genotype-environment interactions, if the environmentremains constant over time, changes in the phenotypic distribution are entirelydue to changes in the genotypic distribution. The moments of a distribution pro-vide a convenient measure to describe its shape, and hence changes in the mo-ments provide descriptions of changes in the shape of the distribution. To seethe connection between the moments of the phenotypic and genotypic distribu-tions, note that the phenotypic mean, variance, and skew can be decomposed asµz = µG, σ2z = σ2G + σ2e , and µ3,z = µ3,G + µ3,e, hence changes in any of thefirst three phenotypic moments exactly equals the change in the correspondinggenotypic moments. The fourth phenotypic moment is slightly more complicated,

µ4,z = µ4,G + µ4,e + 6σ2G σ2e (15.10)

so that changes in the fourth moment of the phenotypic distribution can be dueto either changes in the second (variance) and/or fourth moments of the geno-typic distribution. When e is normal, µ3,e = 0 and µ4,e = 3σ4e , simplying theseexpressions.

How do the moments ofG depend on the distribution of effects at individual

246 CHAPTER 15

loci? If n loci control the character, our assumption of complete additivity implies

G =n∑i

(Xfa,i +Xmo,i) (15.11)

where Xfa,i (Xmo,i) is the value of the paternal (maternal) allele at the ith locus.Assuming both sexes have the same distribution of allelic effects, the moments ofG can be related to moments of the distribution of allelic effects at individual lociby expanding

µk,G = E(

[G− µG]k)

= E( [ n∑

i

Xp,i +Xm,i − 2E(Xi )]k )

for k ≥ 2 (15.12)

Finally, assume of random mating so thatXfa,i and Xmo,i are independent at thestart of each generation. Since we assume that the distribution of allelic effects isthe same in both sexes, we drop the subscript referring to parental origin.

When considering a particular moment of G, it will be important to dis-tinguish between contributions to that moment from individual loci (within-locus moments) and from gametic-phase disequilibrium (between-locus con-tributions). This distinction was used earlier with the additive genetic variance(Equation 5.2) and here extend this partitioning to the third and higher genotypicmoments. To describe the distribution of effects at locus i, let µ1,i = E(Xi ) = mibe the average value of an allele at locus i and define the kth moment for thislocus by µk,i = E( [Xi −mi]k ) for k ≥ 2. Summing over loci, define

M1 = 2∑i

µ1,i (15.13a)

M2 = 2∑i

µ2,i (15.13b)

M3 = 2∑i

µ3,i (15.13c)

as the contribution to the mean, variance, and skewness of the genotypic distri-bution due to the mean, variance, and skew at individual loci. Finally, define thewithin-locus kurtosis (LW Chapter 2) as

M4 = 2∑i

(µ4,i − 3µ22,i

)(15.13d)

While this may at first seem odd, recall for a normal that the fourth and secondmoments are related by µ4 = 3µ22. Hence, if the distribution of allelic effects at


each locus is normal, M4 = 0 and likewise M3 = 0 (as µ3,i = 0 ). On the otherhand, nonzero values of M3 and/or M4 cause G to be non-Gaussian.

The between-locus contributions from gametic-phase disequilibrium are de-scribed by Cij , Cijk and Cijkl, the covariances between groups of two, three, andfour loci as defined previously (e.g., Equation 15.8). Note that with this notationCii = µ2,i, Ciii = µ3,i and Ciiii = µ4,i. If loci are independent (in gametic-phaseequilibrium), then all other combinations involving four (or fewer) loci are zeroexpect Ciijj , which equals Cii · Cjj = µ2,i · µ2,j .

Following Turelli and Barton (1990), we are now in position to decomposegenotypic moments into within-locus effects (Mi) due to the moments at individ-ual loci and between-locus effects due to covariances generated by gametic-phasedisequilibrium. Remember that we are assuming the simplest case, complete ad-ditivity (no dominance or epistatis), so that the genotypic distribution G is thedistribution of additive genetic values (A). Expanding Equation 15.12 and takingexpectations gives the familiar expressions for the mean and variance,

µG = 2∑i

µ1,i = M1 (15.14a)

σ2G = σ2A = 2

∑i,j

Cij = M2 + 2∑i

∑j 6=i

Cij (15.14b)

Similarly, the skew can be partitioned as

µ3,G = 2∑i,j,k

Cijk = M3 + 2∑i

∑j,k 6=i

Cijk (15.14c)

All covariances in the second sums of Equations 15.14b and 14c are zero whenall groups of two and three loci (respectively) are in gametic-phase equilibrium.Partitioning the kurtosis requires a little more care. After some simplification(Turelli and Barton 1990),

µ4,G = 3σ4A +M4 + 2∑i

∑j,k,l 6=i

(Cijlk − Cij Ckl − CikCjl − CilCjk) (15.14d)

Again, the covariance terms are zero when all groups of four loci are in gametic-phase equilibrium. Since 3σ4A is the value expected when genotypic values areGaussian, the last two terms partition any kurtosis in G into the contributionfrom kurtosis at individual loci (M4) and the contribution generated by gametic-phase disequilibrium between groups of four loci. If the distribution of alleliceffects is multivariate normal, then M4 = 0 and each term within the covariancesum is zero as Cijlk = Cij Ckl + CikCjl + CilCjk.

A few remarks on the implications of Equations 15.14a-d for deviations fromnormality are in order at this point. Higher-order moments can depart from their

248 CHAPTER 15

expectations under normality by the presence of skewness and/or kurtosis at theindividual loci (generating non-zeroM3 and/orM4), which can result from allelefrequency changes. Alternatively, even if the within-locus moments are normal(M3 = M4 = 0), gametic-phase disequilibrium (nonzeroCijk and/orCijk`) can in-troduce skewness and/or kurtosis. When the number of loci is small, skew and/orkurtosis at individual loci can be significant, giving nonzeroM3 and/orM4, withthe resulting genotypic distribution deviating from normality. The motivation forassuming that the distribution of G is normal is the central limit theorem, whichstates that sums of random variables converge to a normal. Thus, as larger andlarger numbers of loci underlie the character, the sum should approach a normaldistribution as the contribution from each locus becomes smaller, reducing theeffects of individual deviations from normality. The problem for strict converge toa Gaussian in our cases is that the central limit theorem assumes that the variablesare independent (or at least only very weakly correlated). Selection, by introduc-ing gametic-phase disequilibrium, generates just such a lack of independence.

To see these points, first consider the changes due to within-locus moments.If n is the number of loci, then as we have seen earlier the effects (a) of allelesat individual loci must scale as 1/√n in order for the genetic variance to remainbounded, henceCii terms scale asa2 orn−1. Summing over alln loci,M2 is of ordern · n−1 = 1 and, as required, remains bounded as the number of loci increases.What happens to the skew and kurtosis as n increases? Since a is of order n−1/2,µ3,i is of order a3 orn−3/2, implying thatM3 is of ordern·n−3/2 = n−1/2. Hence, asthe number of loci becomes very large, the contribution from skew at individualloci becomes negligible. Likewise, µ4,i is of order n−4/2, implying M4 is of ordern · n−2 = n−1. As with skew, changes in kurtosis generated by within-locus (i.e.,allele frequency) changes become negligible as the number of loci becomes (very)large.

The behavior of the between-locus contributions, however, is quite different(Turelli and Barton 1990). Under weak selection, Turelli and Barton show thatCijk is proportional to CiiCjjCkk and is thus of order n−3. However, there aren(n − 1)(n − 2) ' n3 terms involving Cijk in the covariance contribution toskew. Thus, this contribution is of order one and does not necessarily convergeto zero even as the number of loci approaches infinity. The same argument holdsfor the kurtosis and higher moments (Turelli and Barton 1990). Thus, when thenumber of loci is very large, the distribution of genotypic values can depart fromGaussian due to selection generating third and higher-order covariances betweenloci, which in turn creates skew and kurtosis in the genotypic distribution. Evenif the distribution of genotypes is originally Gaussian, selection generates thesehigher-order disequilibria, driving the distribution away from a Gaussian (Bulmer1980; Zeng 1987; Turelli and Barton 1990, 1994).

Describing the Genotypic Distribution: Cumulants and Gram-Charlier SeriesWhile most readers are familiar with moments, an alternate approach to describ-


ing the shape of a distribution, and in particular how it departs from a Gaussian,is to examine the cumulants of that distribution. The first use of cumulants inexamining selection response appears to be O’Donald (1972) and Bulmer (1980).Sophisticated (and highly technical) treatments using cumulants have been de-veloped by Bürger (1991, 1993) and Turelli and Barton (1994). Our aim here is toboth give the fearless reader sufficient background to access this literature and toshow the connection between results using moments and those using cumulants.

Cumulants (the n-th of which we denoteKn) arise naturally in series approx-imations of probability distributions and are related to the central moments (µn).For example, the first five central moments can be expressed as functions of thecumulants as follows:

µ1 = K1, µ2 = K2, µ3 = K3, µ4 = K4 + 3K22 , µ5 = K5 + 10K2K3

Hence, the first three cumulants are equal to the mean, variance, and skew, whilethe fourth and fifth cumulants are

K4 = µ4 − 3µ22, K5 = µ5 − 10µ2 µ3 (15.15)

The major advantage of cumulants over moments is that they are additive,so that the n-th cumulant of a sum of random variables is just the sum of thecumulants for each, i.e.,Kn(x+ y) = Kn(x) +Kx(y). This linearity property doesnot hold for higher order moments, which are highly nonlinear functions of themoments of the individual distributions.

Example 2. Use cumulants to compute the fourth and fifth central moments ofthe phenotypic distribution. Here, z = G+ e, so that

µ4,z = K4,z + 3K22,z= K4,G +K4,e + 3 (K2,G +K2,e)

2

=[µ4,G − 3µ22,G

]+[µ4,e − 3µ22,e

]+ 3 [µ2,G + µ2,e ]

2

= µ4,G + µ4,e + 6σ2Gσ2e

where the second and third steps follow from the additivity propery of cumulants(Kn,z = Kn,G + Kn,e) and Equation 15.15, respectively. Simplifying recoversEquation 15.10. Likewise,

µ5,z = K5,z + 10K2,zK3,z= K5,g +K5,e + 10(K2,g +K2,e)(K3,g +K3,e)= [µ5,g − 10µ2,gµ3,g ] + [µ5,e − 10µ2,eµ3,e ]

+ 10(µ2,g + µ2,e)(µ3,g + µ3,e)= µ5,g + µ5,e + 10 (µ2,gµ3,e + µ2,eµ3,g)

250 CHAPTER 15

These nonlinear expressions for the higher moments of a sum of variables is insharp contrast to the expressions for cumulants, wherein Kn,z = Kn,g +Kn,e.

Thus, if the underlying genes are additive, then-th cumulant of the genotypicdistribution is the sum of the appropriate cumulants for each of the underlyingloci. To see the advantage of working with cumulants, consider the fourth cumu-lant of the genotypic distribution. Turelli and Barton (1994) show that this can bewritten as a sum of the fourth-order cumulants at the underlying loci by

K4,g =∑i,j,k,l

Kijkl =∑i

Kiiii +∑i

∑j,k,l 6=i

Kijkl

the sum over Kiiii represents the within-locus contributions to the fourth cu-mulant, while the sum over the other indicies are the contributions to K4 fromfourth-order disequilibrium generation associations between loci. This reduces toEquation 15.14d by noting that

Kijkl = Cijkl − Cij Ckl − CikCjl − CilCjk and M4 =∑i

Kiiii

A second advantage of working with cumulants is that the third and highercumulants for a normal random variable are zero, so nonzero values for thesehigher cumulants provide a convient measure of departures from normality. Bul-mer (1980) showed for the infinitesimal model (with unlinked loci) that, follow-ing the relaxation of selection, the j-th cumulant is decreased each generation by(1/2)j−1, so that the distribution rapidly returns to a normal.

Finally, cumulants appear in series approximations of arbitary probabilitydistributions. Consider a standardized random variable y = (z − µ)/σ, whichhas mean zero and variance one. If the true density function for y is φ(y), we canapproximate it as a unit normality density function ϕ(y) plus correction terms. Inparticular, the Gram-Charlier series approximation (here to order five) is givenby:

φ(y) ' ϕ(y)[1 +

K36H3(y) +

K424

H4(y) +K5120

H5(y)]

(15.16)

where Hx denotes the Chebsyshev-Hermite polynominal of order x, with

H3(y) = y3 − 3yH4(y) = y4 − 6x2 + 3H5(y) = y5 − 10x3 + 15x

Bulmer (1980), Zeng (1987), and Turelli and Barton (1994) have used Gram-Charlier series to examine departures from normality under selection. Further


properties of cumulants and Gram-Charlier (and other) series approximationsare discussed in Johnson and Kotz (1970) and Kendall and Stuart (1977).

Short-term Response Ignoring Linkage EffectsWith the above machinery in hand, we are now ready to examine the responseto selection under non-Gaussian genotypic distributions. We first consider thesituation where a small to modest number of loci underlie the character so thatmost of the changes in the higher-order moments are due to changes in allelefrequencies, rather than through generation of gametic-phase disequilibrium. Ourtreatment follows that of Barton and Turelli (1987).

If we are willing to assume additivity across loci and gametic-phase equilib-rium, then genetic changes in the character can be completely described by thedynamics of allele frequency changes at each locus. The complete dynamics for alocus with k alleles is described by the k−1 allele frequency change equations. Al-ternatively, we could completely describe the dynamics by using equations basedon any set of k− 1 independent new variables that can be expressed as functionsof allele frequencies (this is the standard multivariate transformation problem ofvector calculus and requires that the determinant of the Jacobian transformationmatrix is nonzero). Barton and Turelli show that one such set of new variables arethe first k − 1 moments of the allelic distribution. This is the motivation behindtheir approach, which focuses on allelic moments rather than allelic frequencies.If we ignore gametic-phase disequilibrium, then for n loci with k alleles each, wecan completely describe the dynamics by using the first n(k − 1) moments of thegenotypic distribution. This same approach can be used when linkage is consid-ered, but the number of equations increases dramatically. While this approach ofusing a new set of variables is exact, it is also as hopeless to solve as the original setof allele-frequency change equations. The hope, however, is that by consideringthe first few moments we can gain considerable insight into the actual dynamics.

To briefly sketch the approach used by Barton and Turelli, we first expressWright’s formula for multiple alleles (Equation 9.5b) as

∆pi =∑j

gij∂ lnw∂pj

(15.17a)

where gii = pi(1 − pi)/2 and gij = −pi pj/2 (for i 6= j). Wright’s formulaholds as long as fitnesses are frequency-independent. Now consider a functionf(p1, p2, . . . pk−1) that depends on the allele frequencies at this locus, such as aparticular moment of the allelic distribution. The change in f due to changes inallele frequencies can be approximated by a Taylor series to give

∆f =∑i

∂f

∂pi∆pi +

12

∑i

∑j

∂2f

∂pj ∂pi∆pi ∆pj + · · · (15.17b)

252 CHAPTER 15

where we have ignored higher terms of ∆pi. Substituting for ∆pi using Equation15.17a, and using the identity

∂ lnw∂pj

=∂ lnw∂f

∂f

∂pj

from the chain rule of differentiation gives (to first order)

∆f '∑i

∂f

∂pi

∑j

gij∂ lnw∂pj

=∂ lnw∂f

∑i

∑j

∂f

∂pigij

∂f

∂pj(15.17c)

This is a weak-selection approximation, as it assumes terms of order ∆pi ∆pjand higher can be ignored (if drift is considered, these second-order terms mustbe included even if selection is weak, see Turelli 1988). Using this expression toconsider changes in the allelic moments yields a set of equations where changesin a certain moment depend on higher order moments. After considerable algebra(see Barton and Turelli 1987 for details), the changes in genotypic moments (underthe assumptions of complete additivity and gametic-phase equilibrium) can beexpressed in matrix form as

∆µG 'M (∇ lnw) (15.18)

where

∆µG =

∆µ1,G∆µ2,G∆µ3,G

...

, ∇ lnw =∂ lnw /∂ µ1,z∂ lnw /∂ µ2,z∂ lnw /∂ µ3,z

...

and M =

2∑i

µ2,i µ3,i (µ4,i − 3µ22,i) · · ·µ3,i (µ4,i − µ22,i) (µ5,i − 4µ3,i µ2,i) · · ·

(µ4,i − 3µ22,i) (µ5,i − 4µ3,i µ2,i) (µ6,i − µ23,i − 6µ2,i µ4,i + 9µ22,i) · · ·...

...... . . .

The elements of M corresponding to selection on the fourth and higher momentsare more complicated than may be suggested by the simple dots in the matrix dueto the nonadditive mater of higher moments. Expressions based on ∂ lnw /∂ Ki,z(the partial derviative of fitness with respect to the i-th cumulant of the phenotypicdistribution) have a simpler form due to the additive nature of cumulants (Bürger1991, 1993; Turelli and Barton 1994), but these still have the undesirable feature


that the response on the i-th cumulant depends on cumulants of higher order. Thuspredicting changes in even the simplest genotypic moment, the mean, requiresa detailed knowledge of both higher order allelic moments (µk,i) and the natureof selection on these higher order moments (∂ lnw /∂ µk,z). In order to proceedfurther, we have to make additional assumptions about the distribution of alleliceffects at individual loci.

Example 3. Suppose the distribution of allelic effects at each locus is normal,in which case all odd central moments at each locus are zero (µ2k+1 = 0) andall even moments are related to the second moment by µ2k = µk2 (2k)!/(2kk!)(Kendall and Stewart 1977). For example, µ4 = 3µ22 so that µ4 − µ22 = 2µ22.Assuming that most of selection is on the mean and variance, we can neglect thethird and higher-order selection gradients. In this case, M becomes

M =

2

n∑i=1

µ2,i 0

0 4n∑i=1

µ22,i

=σ

2A 0

0σ4Ane

where

ne =σ4A

4∑i µ

22,i

can be thought of as the effective number of loci. To see this, suppose all loci con-tribute the same additive variance, so that µ2,i ≡ µ2. Thus, (2nµ2)2 = (σ2A)2,while 4

∑µ22 = 4nµ

22 = (2nµ2)

2/n = σ4A/n. This implies n = σ4A/( 4∑µ22 ),

which notivates the above definition of the effective number of loci when thecontribution to additive variances differs across loci. The expected response inthe genotypic mean and variance becomes

∆µ∆σ2A

'σ2A 0

0 σ4A/ne

∂ lnw /∂ µz∂ lnw /∂ σ2z

= σ

2A

∂ lnw∂ µz

σ4Ane

∂ lnw∂ σ2z

If the phenotypic distribution is exactly normal, since all moments can be ex-pressed in terms of the mean and variance, only gradients measuring selectionon the mean and variance appear and these equations exact. Recalling Equation15.7 gives

∆µ ' h2Sand

∆σ2A 'h4

ne

(δ(σ2z) + S

2

)

254 CHAPTER 15

Thus the expected change in the mean follows the breeders’ equation and short-term changes in variance are expected to be small if there are many loci of smalleffect (so that ne is large). We remind the reader that this ignores the effects ofgametic-phase disequilibrium.

Note that since µ2,i changes as allele frequencies change, predicting changes invariance over several generations even under the extreme simplifying assump-tions leading to these equations still requires a detailed knowledge about thedistribution of allelic effects at individual loci. Thus, while short-term changes inthe mean can be predicted without detailed knowledge of the underlying genet-ics (only σ2A is required, which can be estimated from phenotypic resemblancebetween relatives), changes in variance cannot (unless an estimate of

∑µ22,i can

be obtained).

If phenotypes are approximately normally distributed (but allelic effects atindividual loci are not necessarily Gaussian), the mean and variance terms ofthe selection gradient vector generally dominate. Considering only the first threegenotypic moments, Equation 15.18 reduces to

∆µG ' h2S +(δ(σ2z) + S

2

2σ4z

)M3 (15.19a)

∆σ2A 'S

σ2zM3 +

(δ(σ2z) + S

2

σ4z

) ∑i

(µ4,i − µ22,i

)(15.19b)

∆µ3,G 'S

σ2zM4 +

(δ(σ2z) + S

2

σ4z

) ∑i

(µ5,i − 4µ3,i µ2,i ) (15.19c)

where M3 and M4 are as defined by Equations 15.13c and 13d. As is discussed inChapters 14 and 16, when selection acts only on the mean, δ(σ2z) = −S2, so thatthe first term in each of these three equations accounts for the effect of selectionto change the mean and the second term accounts for the effect of selection onthe variance. Note that we obtained Equation 15.21a previously by an alterna-tive method using the secondary theorem (Equation 9.35b). When the genotypicdistribution is skewed (M3 6= 0), the single-generation change in the mean alsodepends on the nature of selection on the variance (O’Donald 1968, 1972; Bulmer1980; Gillespie 1984; Barton and Turelli 1987; Mitchell-Olds and Shaw 1987). Fur-ther, even if skew is initially absent, Equation 15.21c shows that if the kurtosisof the genotypic distribution differs from that expected for a Gaussian (M4 6= 0),selection strictly on the mean generates skew. Thus, even ignoring the effects ofgametic-phase disequilibrium, selection on the mean generates skew when thegenotypic distribution displays kurtosis.

Which factor, allele frequency changes at the individual loci or the generationof gametic-phase disequilibrium, is more important at producing departures from


normality depends on the number of underlying loci. Turelli and Barton (1990)simulated a character controlled by eight diallelic loci and found that most ofthe skew and kurtosis generated by selection was generated by allele frequencychanges, while the contribution from third and fourth-order disequilibrium wasquite small. Thus, when the number of loci is small, the relative contribution to thethird and higher moments from linkage effects may be largely negligible, and theerror by using Equation 15.18 (which assumes gametic-phase equilibrium) shouldbe small. As the number of (equivalent) loci increases, within-locus effects makea smaller and smaller contribution, with departures from normality caused bydisequilibrium eventually dominating as the number of loci becomes sufficientlylarge.

Response When Gametic-phase Disequilibrium is ConsideredTurelli and Barton (1990, 1994) extended basic moment analysis (Equation 15.18)to allow for gametic-phase disequilibrium, by considering both within-locus mo-ment changes due to allele frequency changes and between-locus contributionsgenerated by disequilibrium. The 1994 paper is the more general of the two, withthe analysis based on the cumulants of the distribution. While the mean, variance,and skew are equivalent to the first three cumulants, cumulants of order four andhigher provide much more compact expressions than those involving momentsof order four and higher, due to the additivity of cumulants versus the non-linearnature of higher order moments.

In parallel with their moments analysis, they define the gradients of selectionassociated with the i-th cumulant of the phenotypic distribution by

Li =∂ ln(W )∂Kz,i

(15.20a)

L1 and L2 correspond to selection on the mean and variance, while Li for i ≥ 3represents selection that drives the distribution away from normality. Turelli andBarton present general expressions for the change in all the cumulants of thedistribution. In particular, for a large number of loci, they show that if the majorityof selection is on the first four cumulants of the distribution, that the change inthe mean and variance are given by

∆µ = σ2A L1 +Kg,3 L2 +Kg,4 L3 +Kg,5 L4 (15.20b)

∆σ2A =σ2a − σ2A

2− ∆µ

2

2+Kg,3

2L1 +

(σ4A +

Kg,42

)L2

+(

3σ2AKg,3 +Kg,5

2

)L3 +

(3K2g,3 + 4σ

2AKg,4 +

Kg,62

)L4 (15.20c)

whereKg,i denotes the i-th moment of the genotypic distribution. If some cumu-lants of the genotype distribution of order three or higher are nonzero, selection

256 CHAPTER 15

to alter the higher-order culumants of the distribution (and hence drive the dis-tribution away from a Gaussian) also results in a change in the mean.

Example 4. If phenotypes are normally distributed, since the first two cumu-lants equal the mean and the variance, recalling Equation 15.7a and b gives

L1 =S

σ2z, L2 =

δ(σ2z) + S2

2σ4z(15.21)

If the genotypes follow a normal distribution, then Kg,i = 0 for i ≥ 3. In thiscase, Equation 15.20b reduces to

∆µ = σ2AS

σ2z= h2S

recovering the breeders’ equation. Recalling that σ2A = σ2a + d, Equation 15.20creduces to

∆σ2A =σ2a − σ2A

2− (h

2S)2

2+ σ4A

(δ(σ2z) + S

2

2σ4z

)= − d

2+h4

2δ(σ2z)

and we recover Bulmer’s formula for the short-term change in the variance (Equa-tion 5.7b).

Turelli and Barton (1994) examined the effects of both strong truncation (di-rectional) selection and strong disruptive selection on the Gaussian predictionsof Bulmer (Chapter 5). For strong truncation selection, they found that while se-lection does indeed generate nonzero culumants of order three and higher (andhence departures from normality), these are generally quite small. As a result,the breeder’s equation with the variance changes predicted from Bulmers modelgive quite accurate results for the predicted change in the mean and variance.Hence, the effects of disequilibrium in this case are essentially all accounted forby considering only the second-order disequilibrium, which is done in the basicBulmer model. Barton and Turelli found that the distribution of genotypic valuesis highly non-normal under strong disruptive selection, which a significant fourthcumulant (kurtosis) being generated by significant fourth-order disequilibrium(generating correlations between groups of four loci). However, even in this casethe change in variance is well predicted by the basic Bulmer model.

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