Neutra l Evolution of Multiple Qua ntitative Characters: A

Copyright ! 2007 by the Genetics Society of AmericaDOI: 10.1534/genetics.106.069658

Neutral Evolution of Multiple Quantitative Characters:A Genealogical Approach

Cortland K. Griswold,* Benjamin Logsdon†,1 and Richard Gomulkiewicz*,‡,2

*School of Biological Sciences, †School of Molecular Biosciences and ‡Department of Mathematics,Washington State University, Pullman, Washington 99164

Manuscript received December 13, 2006Accepted for publication February 19, 2007

ABSTRACTThe G matrix measures the components of phenotypic variation that are genetically heritable. The

structure of G, that is, its principal components and their associated variances, determines, in part, thedirection and speed of multivariate trait evolution. In this article we present a framework and results thatgive the structure of G under the assumption of neutrality. We suggest that a neutral expectation of thestructure of G is important because it gives a null expectation for the structure of G from which theunique consequences of selection can be determined. We demonstrate how the processes of mutation,recombination, and drift shape the structure of G. Furthermore, we demonstrate how shared commonancestry between segregating alleles shapes the structure of G. Our results show that shared commonancestry, which manifests itself in the form of a gene genealogy, causes the structure of G to be non-uniform in that the variances associated with the principal components of G decline at an approximatelyexponential rate. Furthermore we show that the extent of the nonuniformity in the structure of G isenhanced with declines in mutation rates, recombination rates, and numbers of loci and is dependent onthe pattern and modality of mutation.

PHENOTYPIC evolution is channeled through pat-terns of heritable variation. This is exemplified by

the breeder’s equation, R ¼ h2S , which for a single quan-titative trait states that the evolutionary response (R) toselection (S) of the trait is proportional to its herita-bility, h2 (see, e.g., Falconer and MacKay 1996; Lynchand Walsh 1998). Likewise, the simultaneous evolu-tionary responses of several potentially correlated quan-titative characters are predicted by the multivariateextension of the breeder’s equation, Dz ¼ Gb, whereDz is a vector describing the responses of each characterto selection, b—the selection gradient vector—describesthe pattern of directional selection applied to the traits,and G is the additive-genetic covariance matrix describ-ing the heritable patterns of variation and covariationof the traits (Lande 1979).

Quantitative geneticists and evolutionary biologistsalike are particularly interested in the principal compo-nents of theGmatrix, since these indicate independentpaths along which evolution can occur, and the varianceassociated with each principal component, since thesemodulate the speed of evolution along the directionof each principal component. For instance, the ‘‘evolu-

tionary line of least resistance’’ for adaptive radiation(Schluter 1996) is defined by the principal compo-nent ofG with the largest variance, i.e., with the greatestamount of heritable variation. At the other extreme,principal components of G with the smallest variancesrepresent evolutionary avenues with the greatest de-grees of genetic resistance (Kirkpatrick and Lofsvold1992; Hine and Blows 2006).Since the G matrix itself can evolve, it is natural to

consider the influences of evolutionary forces (such asselection, mutation, random genetic drift, and recom-bination) on the patterns of variation associated withthe principal components of G. For example, principalcomponents with the smallest eigenvalues might reflectthe strongest patterns of multivariate selection (e.g.,Blows et al. 2004; Hine et al. 2004). Indeed, this is ex-pected on general theoretical grounds (Lande 1980;Lande and Arnold 1983; Phillips and Arnold 1989)and has been verified by computer simulations (Joneset al. 2003).A second force that causes the G matrix to evolve is

random genetic drift. Phillips et al. (2001) showed thatsevere inbreeding led to an average reduction in totalvariation of the G matrix that was proportional to thetheoretical expectation 1 " F, where F is the inbreedingcoefficient. Furthermore, the principal components ofthe G matrices differed (in an apparently random way)among replicated inbred lines, although, on average,the G matrices across replicated inbred lines did not

1Present address:Department of Biological Statistics and ComputationalBiology, Cornell University, Ithaca, NY 14583.

2Corresponding author: School of Biological Sciences, P.O. Box 644236,Washington State University, Pullman, WA 99164.E-mail: [email protected]

Genetics 176: 455–466 (May 2007)

differ from those of the outbred control population.Subsequent work byWhitlock et al. (2002) on the sameset of inbred lines showed that the G matrix in somelines continued to change over 20 generations of ran-dom mating, but there was no evidence that the G ma-trices were returning to the original state in the outbredcontrol line. On the basis of work by Avery and Hill(1977) and Zeng and Cockerham (1991), Whitlocket al. (2002) hypothesized that the subsequent changesin G matrices in the generations following inbreedingwere caused, in part, by the decay of linkage disequi-libria that were generated during the inbreeding epi-sode. Besides the force of recombination, theoreticalwork by Lande (1980) shows that mutational pressurecan also cause the G matrix to evolve.

At equilibrium the proximate causes of the structureof a Gmatrix are linkage disequilibria and the pleiotro-pic and nonpleiotropic effects of segregatingmutations.The ultimate forces that contribute to the structure oftheGmatrix are recombination, mutation, genetic drift,and selection (e.g., Phillips and McGuigan 2006). Todate, no theoretical work presents a neutral expectationfor the structure of the G matrix when the only forcesshaping its evolution are mutation, recombination, anddrift, although Lynch and Hill (1986) presented aneutral model of the evolution of genetic variance for asingle character. In the case of multiple charactersLynch (1994) and Phillips and McGuigan (2006)postulate that G will evolve to be proportional to M, themutational variance–covariance matrix.

The focus of this article is the question of how mu-tation, recombination, and random genetic drift deter-mine the eigenstructure of theGmatrix at equilibrium,where the eigenstructure of the G matrix is measuredaccording to its eigenvectors and eigenvalues. The ei-genvectors of G are its principal components and theeigenvalues of G are the variances associated with eacheigenvector. Naıvely, for the case of free recombinationamong loci, one might expect that the eigenvalues of Gwould be proportional to those ofM, with drift-inducedlinkage disequilibrium and stochastic sampling from Mcausing nondirectional variation around this expecta-tion. Accordingly, if the eigenvalues of M were all simi-lar in size, then on average so would the eigenvalues ofG. We explore the validity of this intuitively reasonableexpectation.

One of the main insights of this article is that thisintuitively reasonable expectation is not valid. A primaryreason is that segregating alleles in a population arerelated to one another due to shared common ancestryor, to put it another way, alleles are related genealogi-cally through shared common descent via reproduction.Patterns of shared common ancestry can be describedfor a sample from a population using Kingman’s (1982)coalescent andmultilocus extensions such as Griffithsand Marjoram’s (1997) ancestral-recombination graph.In this article we use the coalescent and the ancestral-

recombination graph as a framework to investigateneutral evolution of G.

The neutral expectations developed here aid the in-terpretation and analysis of recent empirical work thatshowed significant additive genetic variance for 20 wing-shape characters inDrosophilamelanogaster and an approx-imately exponential decline in the ordered eigenvaluesof theGmatrix for these characters (Mezey and Houle2005). The exponential decline in ordered eigenvaluesmay be interpreted as signatures of either variable se-lection across characters and/or variable mutationalheritabilities across characters. Our analyses show that,under the neutral model, the ordered eigenvalues of Gwill evolve toward a regular pattern. Surprisingly, even ifthe ordered eigenvalues ofM are perfectly uniform andselection is absent, the expected eigenvalues of G in asample from a population are predicted to decline at anapproximately exponential rate.

METHODS

The model: We consider the evolution of a multivar-iate quantitative trait affected by multiple loci in adiploid population of sizeN withWright–Fisher randommating. (For nonideal populations, N ¼ Ne, the variance-effective population size.) Evolution is neutral as weassume the trait experiences no selection. Allelesaffecting the trait act additively within and across loci(i.e., no dominance, epistasis, or genotype–environmentinteractions). For example, if xi and x9i are multivariateeffects of maternally and paternally inherited alleles atlocus i (e.g., the effects of each allele on m traits), thenthe genotypic value g of a typical individual isg ¼

P‘i¼1ðxi 1 x9iÞ, where ‘ is the number of loci. The

phenotype is z ¼ g1 e, where e is an independent,mean-zero random variable. The G matrix is the m3 mcovariance matrix of additive values, g. We hereafter ig-nore e because, in the absence of selection, it plays norole in the evolution of G. It is important to note,however, that e can affect estimates of G, which may af-fect predictions of the eigenstructure of any particularG, especially for small samples from a population. Wedo not address this issue in this article and leave it forfurther study.

Under neutrality and additivity, three main processesinfluence G-matrix evolution: spontaneous mutation,recombination, and randomgenetic drift via stochastic re-production. In this article we focus on the recombination–mutation–drift equilibrium characteristics ofG. For spon-taneous mutation we examined two infinite-allele muta-tion models: the continuum-of-alleles model (Crow andKimura 1964; Kimura 1965) and the house-of-cardsmodel (Kingman 1977, 1978). The continuum-of-allelesmodel assumes that each mutation generates a newmultivariate effect that is added to the current effect ofan allele at a locus, where a new effect is sampled from a

456 C. K. Griswold, B. Logsdon and R. Gomulkiewicz

mutational distribution. In our simulations we used azero-mean multivariate normal distribution with covari-ance matrix A for the distribution of mutational effects.The diagonal elements of A are the variances in muta-tional effects and the off diagonals are the covariances ofmutational effects on different traits. The house-of-cardsmodel (Kingman 1978) assumes that each new muta-tion completely replaces the current allelic effect at amutated locus.

Coalescent approaches: Because the alleles contrib-uting to g are neutral and every mutation is novel, thealleles at individual loci should follow the distributionexpected at equilibrium for the infinite-alleles neutralmodel (see, e.g., Ewens 2004). Rather than sample fromthis distribution directly, we used a different but equiv-alent approach for n , N, where n is the samplesize—the coalescent. The coalescent is useful in that itretains information about the underlying genealogicalhistory of alleles. Thus it can be used to model howvariation in allelic values is determined by patterns ofshared common ancestry.

Coalescent with independent loci: Figure 1 illustrates theapproach that we used to generate the genotypic valuesg of individuals in a sample of size n based on the neutralcoalescent (Kingman 1982). Loci were treated inde-pendently. For each locus we generated a tree with 2ntips and placedmutations on the tree using the standardneutral coalescent approach. (We employed the algo-rithm in the Appendix of Hudson 1991.) When 1 unitof coalescent time is scaled to be 2N generations, co-alescent events occur at a rate kðk " 1Þ=2 per coalescenttime unit using the exponential approximation of ageometric process, where k is the number of lineagespresent at a particular time back in history. Once a genetree is drawn, branch segments in the tree have timesassociated with them as a result of the coalescent pro-cess. Mutations were randomly placed along branchsegments such that the number of mutations along abranch segment of the coalescent tree was Poisson dis-tributed with mean 2Nmti , where m is the haploid per lo-cus mutation rate and ti is the length (in coalescent time

units) of branch segment i. With each mutation, wethen associated a vector of effects drawn from amultivariate normal distribution with mean 0 andcovariance A. A population was generated by randomlyassigning two alleles from each locus (without replace-ment) to each of the n individuals. Note that thiscoalescent approach to generating individual geno-typic values imposes the strict rule that loci arecompletely independent; i.e., there are no transientlinkage disequilibria.The overall process is illustrated in Figure 1 for a

sample of n ¼ 2 individuals, each with two loci. Muta-tions in the coalescent tree aremarked with hatchmarksand the associated vector of effects is represented by~d.For locus A, alleles a1 and a2 are unmutated relative tothe ancestral allele and, accordingly, have the ancestralvalue ~0. Allele a3 is differentiated from the ancestralallele by one mutation and has the value ~01~d3 ¼ ~d3.The value of allele a4 depends on the mutation model.For the continuum-of-alleles model, the value of a4 is~01~d1 1~d2 ¼~d1 1 ~d2, but for the house-of-cards model,the value of a4 is~01~d2 ¼~d2, since the later mutationaleffect ~d2 completely replaces the earlier effect ~d1. Theallelic effects at locus B are determined similarly.Next, the alleles are randomly assigned two at a time,

without replacement, to an individual. In Figure 1,alleles a1 and a4 at locus A and alleles b3 and b4 at locusB were assigned to the first individual. On the basis ofthe alleles sampled and the mutations in the gene tree,the genotypic value of the first individual is g ¼ ~d1 1~d2 1 2~d4 for the continuum-of-alleles model and g ¼~d2 1 2~d4 assuming the house-of-cards.

Ancestral recombination graph: As stated at the outset,linkage disequilibria are also a proximate cause ofadditive-genetic covariances. Strictly applying indepen-dence among loci may therefore eliminate a factorpromoting genetic covariances even when there is freerecombination among loci. To allow for physical linkageamong loci and for the possibility of linkage disequi-libria we used the ancestral recombination graph co-alescent approach (Griffiths 1981, 1991; Griffiths

Figure 1.—Demonstration of how individualgenotypic values were generated using the neu-tral coalescent in which the evolutionary historiesof loci are completely independent. It is assumedthat there are two individuals in a sample andeach individual is composed of two loci. The genetree on the left is that for locus A and the genetree on the right is that for locus B. For each in-dividual two alleles from each locus are chosenwithout replacement. Genotypic values underthe continuum-of-alleles (COA) and house-of-cards (HOC) mutation models are given. Seemain text for further details.

Neutral Theory of G Evolution 457

and Marjoram 1996, 1997; see Nordborg 2001, pp.195–200, for a useful introduction).

Figure 2 illustrates how a sample of two individualseach consisting of two loci is generated using theancestral recombination graph. Instead of having twoindependent loci, the loci are linked. In Figure 2, therectangles at the top each represent a haplotype in thecurrent generation with the open section correspond-ing to locus A and the solid section corresponding tolocus B. In an ancestral recombination graph, two pro-cesses occur when forming a gene tree, namely recom-bination and coalescent events. Scaling time to be 2Ngenerations, coalescent events occur as before at ratekðk " 1Þ=2, where k is the number of lineages at a giventime. Recombination events occur at the rate 4Nrk/2,where r is the per haplotype recombination rate pergeneration. The total rate of events is kðk " 1Þ=214Nrk=2, and, given an event, the probability it is a co-alescent is ðk " 1Þ=ðk " 11 4Nr Þ. Although the rate ofrecombination is typically much faster than the rateof coalescence, a common ancestor is guaranteed(Griffiths and Marjoram 1997).

Proceeding backward in time from the present, a re-combination event is represented by the splitting of asingle haplotype lineage into two lineages. For a two-locus model, one locus is retained in one ancestral line-

age, whereas the other locus is retained in the secondancestral lineage. In Figure 2, shading corresponds to alocus that is recombined out in the descendant lineagefollowing a recombination event. Often recombinationevents occur in an ancestral lineage that consists of onelocus that is present in the current generation and onethat is subsequently recombined out and not present inthe current generation. With respect to the coalescenceof alleles in the sample, these recombination events arerelevant only with respect to incrementing time back-ward in the gene tree, but do not add additional in-formation. Consequently, the splitting of these types oflineages is not recorded in the gene tree, although thetime of the event is.

Once an ancestral recombination graph is drawn,there are times associated with each branch segment(in units of 2N generations) and numbers of mutationsare modeled to occur along each branch segmentaccording to a Poisson distribution with mean 2N‘mti ,where ti is the time associated with branch segment i.The per locus mutation rate is assumed to be equalacross loci, such that for ‘ loci and given a mutationalevent, the probability the mutation occurs at a partic-ular locus is 1=‘. In Figure 2, a hatch mark represents amutation, the box next to it represents the locus atwhich the mutation occurred, and ~di is the vector of

Figure 2.—Demonstration of how individualgenotypic values were generated using the ances-tral recombination graph. Loci are linked suchthat individuals inherit haplotypes as opposedto independent alleles. Haplotypes are presentedas rectangles and each haplotype consists of twoloci: locus A (open rectangles) and locus B (solidrectangles). Points at which a lineage bifurcatesinto two lineages (back in time) correspond torecombination events. Where lineages join arecoalescent events. In this example, each individ-ual consists of a random sample of two haplotypes(without replacement). Genotypic values underthe continuum-of-alleles (COA) and house-of-cards(HOC) mutation models are given. See main textfor further details.


effects of mutation i. To get an allele’s value, simplytrace its history back in the gene tree, being careful tofollow the correct ancestral lineage at a recombinationevent.

Because loci can be physically linked, individual ge-notypes are formed by randomly sampling haplotypes asopposed to alleles at independent loci. With a sampleconsisting of two individuals there are four haplotypes,each individual consisting of two haplotypes. In theexample (Figure 2), individual 1 consists of haplotypes 1and 4. Following the history of the allele a1 back in time,only the mutation with effect~d7 occurs before reachingthe common ancestor of the population so the additiveeffect of a1 is~01~d7 ¼~d7. Note that the mutation witheffect ~d3 occurred in an ancestral allele not present inthe current generation. As with the independent-locicoalescent approach described above, an allele’s addi-tive value is the sum of the effects of mutations in alineage for the continuum-of-alleles mutation modelwhereas, for the house-of-cards model, the additiveeffect is equal to themost recent mutation for the allele.

Given the relatively large sample sizes examined inthis study combined with a large number of loci and amoderate to fast recombination rate, there were toomany possible ancestral haplotypes for computers with1 GB of memory to simulate the ancestral recombina-tion graph algorithm; hence, there are significantcomputational limitations when considering recombina-tion while utilizing the coalescent method.

Parameters: Parameters underlying the results pre-sented in this article are cast in terms of 4Nm, 2Nr , thenumber of loci, andM (see below for a definition ofM).The expression 4Nm=2 gives the expected number ofnew mutations that enter a population each generationat each locus. In this article, 4Nm ranges from 0.1 to10.0. The expression 2Nr=2 gives the expected numberof recombination events per generation during themeiotic phase of the life cycle for an entire populationper adjacent pair of loci and it ranges from 0.0 to1000.0. Numbers of loci vary from 1 to 10 in the results,and it is assumed that each locus has the same valueof 4Nm.

In this article, mutations at a locus act either pleio-tropically or nonpleiotropically. When mutations actpleiotropically, a single mutation potentially affects allcharacters and the covariance matrix of their effects isgiven by A. For simplicity, we concentrate on diagonalcovariance matrices such that A ¼ 0:05I, where I is theidentity matrix. When mutations at a locus act non-pleiotropically, the mutations affect only a single char-acter and the variance of their effects is 0.05.

Throughout this article it is assumed that mutationrates and mutational effects are uniform across charac-ters. Thus, the expected eigenvalues of the mutationalcovariance matrix, M, are uniform, where M ¼ UA andU is the zygotic mutation rate. Wemake these simplifyingassumptions so that any nonuniformity in the eigenval-

ues of G arises via factors other than nonuniformity inthe eigenvalues of the mutational covariance matrix. Inreality, characters of organisms will likely have non-uniform mutational variance and this may be an im-portant factor in determining the ordered eigenvaluesof G. But, here we are particularly interested in dem-onstrating that genealogical history per se is important inshaping the ordered eigenvalues ofG. Furthermore, wedo not explore cases in which there is a nonorthogonaltransformation between the underlying mutationalprocess and the measured characters of an organism.If the transformation is not orthogonal then this willgenerally affect the eigenvalues of G.

RESULTS

All cases we examined were consistent with the pre-diction that, at equilibrium, the average G matrix isequal to 2NM underneutrality (Phillips andMcGuigan2006). However, our results revealed two new, consis-tent features of neutral evolution of G matrices atrecombination–mutation–drift balance. First, the eigen-values of equilibriumGmatrices are expected to behighlynonuniform—declining in anapproximately exponentialmanner—even when the mutation covariance matrix Mhas perfectly uniform eigenvalues (i.e., when M has asingle repeated eigenvalue). Second, we found that ge-nealogical ancestry of alleles in a finite population isparticularly important in determining the intensity ofdecay of the ranked eigenvalues of G matrices underneutrality. In the absence of selection, two forces, namelyrecombination and random genetic drift, generate gene-alogies of alleles, and the genotypic values of alleles in aparticular genealogy are determined by the pattern ofmutation. In what follows we show in turn how mutation,drift, and recombination shape the ordered eigenvaluesof G matrices.Genealogical effect on G: Figure 3 demonstrates how

genealogical ancestry cangeneratenonuniformity in theeigenvalues of G assuming allelic variation at a singlelocus. In Figure 3a, the top left corner shows a genealogyof a sample of six alleles. On the genealogy are 10 mu-tations that give rise to five allelic types out of the sixsampled alleles. The bivariate effects of these 10 muta-tions are plotted. The principal components of muta-tional variation are of similar magnitude, suggesting thatvariation is evenly distributed across the two dimen-sions. Figure 3b shows the values of the alleles at the tipsof the tree, plotted as solid dots, using the continuum-of-alleles model. What is immediately apparent in thisexample is that allelic variation, instead of being evenlyspread across two dimensions (as in Figure 3a), is pri-marily concentrated in one dimension. The reason forthis is that allelic values are dependent due to their sharedcommon ancestry whereas the mutational effects areindependent.


Figure 4 extends the principle illustrated in Figure 3by replicating the genealogical history of a single locusmany times to determine if, on average, there is a greaterrate of decline in the eigenvalues associated with breed-ing values vs.mutational variation. Furthermore, a largersample is taken fromeachpopulation and there aremoretrait dimensions. The boxes in Figure 4 give the averageordered eigenvalues of mutational variation for a singlelocus (as was done in Figure 3a, except replicated 500times). Note that variation is nonuniform across theprincipal components even thoughmutation is sampledfrom a distribution with uniform variance across itsprincipal components. The nonuniformity is the resultof stochasticity, in that there are a finite number of mu-tations segregating at a locus and by chance the effectsof these mutations are nonuniformly distributed acrosscharacters. The triangles in Figure 4 suggest that theeigenvalues associated with the breeding values ofindividuals decline at a greater rate than the eigenvaluesassociated with mutational variation. Two processeswithin the genealogy appear to bring about the en-hanced decline in eigenvalues. One is that genealogybuilds up statistical dependence among allelic valuesdue to the pattern of shared common ancestry (branch-ing). The second is that along a single branch multiplemutations may occur, which reduces the number ofindependently acting mutations. Figure 4 suggests thatthe point illustrated in Figure 3, namely that genealogypotentially enhances the nonuniformity in the eigen-values associated with variation in breeding values, isgeneral because, on average, there is a faster decline ineigenvalues associated with variation in breeding valuesthan in mutational effects. On occasion, genealogicalancestry may have no effect on eigenvalues, but, on av-erage, Figure 4 suggests that it enhances the nonuni-formity in their values.

Linkage: Linkage causes the genealogies of alleles atdifferent loci to be correlated. Our results show thattighter linkage between two loci tends to enhance thenonuniformity in the expected eigenvalues ofG (Figure5). When the loci are completely linked (2Nr¼ 0) thereis greater nonuniformity in eigenvalues than with ahigher rate of recombination (2Nr ¼ 1000).

Figure 3.—The effect of genealogy at a single lo-cus on the eigenstructure of allelic variation. (aand b) In the top left corner is an illustrationof a genealogy of six alleles with 10 mutations.(a) The bivariate effects of 10 random mutationsare plotted assuming normality, where the aver-age effect of a mutation per character is zeroand effects are uncorrelated. The shaded arrowsrepresent the principal components of the varia-tion in mutational effects with the length of eacharrow proportional to the variance explained byeach direction. The arrows are similar in length,suggesting that mutational variation is evenlydistributed across the two dimensions. (b) The ef-fects of the six alleles at the tips of the genealogyare plotted as solid dots. There are only five dots

because two alleles are identical in state. Lines of descent, shaded to aid comparison with the genealogy, connect allelic values. Asin a, the shaded arrows indicate the principal components of allelic variation. Comparing a and b suggests that shared commonancestry leads to a reduction in the dimensionality of allelic variation relative to mutational variation. Allelic variation is con-centrated along one principal component axis, whereas mutational variation is distributed relatively evenly over two principalcomponent axes.

Figure 4.—The effects of random mutation and genealogyon the eigenvalues of G. Shown are results for a single locusthat pleiotropically affects five characters with 4Nm ¼ 1:0and a sample size of 300. To generate this figure 500 replicatecoalescent trees were generated and random mutation wasplaced on the tree according to a Poisson process. Mutationswere drawn from a multivariate normal distribution with zeromean and uncorrelated effects. To determine the eigenvaluesof mutational effects, for each tree a covariance matrix wasestimated on the basis of the effects of mutations on the tree(as in Figure 3a). Allelic effects were calculated assuming thecontinuum-of-alleles model. Individual breeding values werecalculated by sampling without replacement two alleles from acoalescent tree. Eigenvalues are on a log scale and error barsrepresent 95% confidence intervals of the mean eigenvalue.The internal hatch mark within an error bar is the meaneigenvalue.


Other factors influencing eigenstructure: Besidesfinitemutation, genealogy, and linkage, our results showthat several other factors contribute to the pattern ofeigenvalues of G.

Number of loci: Alleles at each locus vary in how theycontribute to patterns of additive genetic variance. Asmore loci are added, the relative contribution of eachto the variability in G is reduced. Accordingly, there ismore nonuniformity in the eigenvalues ofG when thereare few vs. many loci (Figure 6).

Mutation rate and effective population size: The rate ofdecline in eigenvalues is greater with smaller values of

4Nm (Figure 7). In fact, there is a limit to the expectednumber of principal components that will have nonzerovariance associated with it (Table 1). In Table 1, the Mmatrix has 25 dimensions with equal variances acrossdimensions and no covariance, yet beyond 12 principalcomponents for a sample size of 75 and 15 principalcomponents for a sample size of 300, principal compo-nents are expected to have zero variance associated withthem. As the mutation rate or effective population sizeincreases, the number of principal components withnonzero variance will increase (data not shown). Fig-ure 7 illustrates another important point, namely thatthe decline in eigenvalues is not log linear for high-dimensional cases.Mutational matrices with variable eigenvalues or nonzero

correlations: Until now, we have discussed only cases inwhich the expected mutational variance–covariancematrix has perfectly uniform eigenvalues with com-pletely uncorrelated mutational effects on differenttraits. This is clearly a hypothetical extreme. In reality,we expect that the eigenvalues of M itself will vary andthat mutational effects on different traits will be corre-lated (e.g., see Ajie et al. 2005; Estes et al. 2005). Resultssuggest that nonuniformity in the eigenvalues of Mincreases nonuniformity in the eigenvalues of G (datanot shown).We also found that introducing nonuniformity in the

eigenvalues of M had an important influence on theeigenvectors of G. Simulations for two traits show that,whenM has uniform eigenvalues, the leading eigenvec-tor ofG is uniformly distributed between 0" and 360". In

Figure 5.—The effects of recombination on the eigenval-ues of G. A sample of 300 individuals was drawn from a pop-ulation in which 4Nm ¼ 1:0 per locus and four loci weremodeled using the ancestral recombination graph. 2Nr isgiven as the rate of recombination per adjacent pair of loci,and thus the total genomic recombination rate is 6Nr. Muta-tions pleiotropically affected five characters. A total of 500replicate recombination graphs were simulated. Eigenvaluesare on a log scale and error bars represent 95% confidenceintervals of the mean eigenvalue. The internal hatch markwithin an error bar is the mean eigenvalue.

Figure 6.—The effects of varying the number of loci on theeigenvalues of G. A sample of 300 individuals was drawn froma population in which 4Nm ¼ 1:0 per locus. For the 10- and100-loci cases, independent gene trees for each locus weresimulated using Hudson’s (1991) algorithm. Mean eigenval-ues and their confidence intervals are based on 500 repli-cates. Eigenvalues are on a log scale and error bars represent95% confidence intervals of the mean eigenvalue. The inter-nal hatch mark within an error bar is the mean eigenvalue.

Figure 7.—The consequence of varying 4Nm (per locus)with a fixed number of loci (10). To aid direct comparisonbetween the different 4Nm values with respect to how rankeigenvalues decline, the points for 4Nm ¼ 10:0 were all shifteddown an equal amount such that the average leading eigen-values for 4Nm ¼ 1:0 and 4Nm ¼ 10:0 coincide. The figuredemonstrates that different per locus mutation rates (oreffective population sizes) lead to different rates of decreasein the eigenvalues, with smaller 4Nm values showing a fasterdecline. The independent locus coalescent approach wasused and n ¼ 300. Eigenvalues are on a log scale and errorbars represent 95% confidence intervals of the mean eigen-value. The internal hatch mark within an error bar is themean eigenvalue.


contrast, when the eigenvalues of M differ, the leadingeigenvector of G follows a unimodal, symmetric distri-bution approximately centered at the leading eigenvec-tor of M (results not shown). Thus, uneven eigenvaluesof M affect the entire expected eigenstructure (i.e., ei-genvalues and eigenvectors) of G under neutrality.

As for nonzero mutational correlations, their impactson the eigenstructure ofG are channeled through theireffects on the eigenstructure of M. For example, con-sider a pair of traits each with mutation variance Vm andmutational correlation r (as in Jones et al. 2003). That is,

M ¼ Vm1 rr 1

! ", which has eigenvalues Vmð16 rÞ. The

expected eigenvalues of G given this M are exactly thesame as those produced by the mutational covariance

matrix M ¼ Vm11 r 00 1" r

! ", which has the same ei-

genvalues asM but with uncorrelated mutational effects(results not shown). This suggests that the eigenvaluesof M—and not the specific patterns of mutational var-iances and covariances—determine the expected eigen-value distribution of G at equilibrium. Note, however,that the distributions of expected eigenvectors of Gwould differ for the two mutational matrices since theeigenvectors of M and M differ.

Mutational history: We examined two models ofmutation: the continuum-of-alleles model (Crow and

Kimura 1964; Kimura 1965) and the house-of-cardsmodel (Kingman 1977, 1978). The two models repre-sent opposing extremes of how to model the effects ofmutation. With respect to a genealogy of alleles, underthe house-of-cardsmodel the genealogy is important onlyin shaping allele frequencies, whereas in the continuum-of-alleles model the genealogy shapes both allele fre-quencies as well as the genotypic values of alleles.Thus, by comparing results from house-of-cards andcontinuum-of-alleles models, it is possible to disentan-gle, in part, the effects of variable allele frequency fromthe historical sequence of mutations on the eigenstruc-ture of G.

Both the house-of-cards and the continuum-of-allelesmodels yield, on average, nonuniform eigenvalues atequilibrium (Figure 8). Therefore, it appears that vari-able allele frequencies contribute to the nonuniformityof eigenvalues ofG and the mutational history of allelesfurther increases this nonuniformity.

Pleiotropic vs. nonpleiotropic mutation: All of the resultsdiscussed thus far have assumed universal pleiotropy.For example, if five characters are modeled then eachrandom mutation could potentially affect all five char-acters.Under universal pleiotropy variation at each char-acter is determined by the same set of gene trees. At theother extreme, when mutations at a locus are not pleio-tropic, each character has a unique set of gene treesunderlying its variation, provided that there is somerecombination between loci. Figure 9 shows that pleio-tropicmutation enhances nonuniformity in eigenvaluesofG (particularly for lower mutation rates), presumablybecause the allelic ancestries that underlie variation ofdifferent characters overlap. If mutations act nonpleio-tropically, then, to the extent that loci freely recombine,the allelic ancestries underlying each character’s varia-tion will be independent.

Whole-population properties of G: The results have fo-cused on the properties of G based on a sample of sizen from a population of sizeN, where n,N. An importantquestion is the extent to which the nonuniformity in Gis the result of taking a sample from a population. In theappendix, we describe results for the eigenstructure ofG for an entire population based on individual-basedsimulations. The results suggest that the eigenvalues ofG decline at an approximately exponential rate and thatthe decline is slightly less than the decline based ontaking a subsample from the population to estimate theproperties of G.

DISCUSSION

Our results are consistent with the equilibrium expec-tation of G, EðGÞ ¼ 2NM, predicted by Phillips andMcGuigan (2006) under neutral evolution. However,our results demonstrate that the expected eigenvaluesof G at equilibrium and the eigenvalues of E(G) are

TABLE 1

Limits to multivariate trait variation

Sample size

Eigenvalue rank 75 300

1 6.41e-1 6 3.20e-2 6.22e-1 6 2.80e-22 3.09e-1 6 1.86e-2 3.08e-1 6 1.74e-23 1.56e-1 6 1.25e-2 1.56e-1 6 1.17e-24 7.37e-2 6 7.32e-3 7.67e-2 6 7.22e-35 3.63e-2 6 4.69e-3 3.87e-2 6 4.62e-36 1.72e-2 6 2.71e-3 1.92e-2 6 2.78e-37 7.85e-3 6 1.60e-3 9.48e-3 6 1.69e-38 3.31e-3 6 8.07e-4 4.38e-3 6 9.05e-49 1.27e-3 6 4.42e-4 1.91e-3 6 5.04e-410 5.23e-4 6 2.44e-4 8.12e-4 6 2.95e-411 2.02e-4 6 1.34e-4 3.33e-4 6 1.74e-412 9.08e-6 6 1.78e-5 1.27e-4 6 9.90e-513 0.0 6.02e-5 6 5.00e-514 0.0 1.49e-5 6 1.60e-515 0.0 1.19e-6 6 2.34e-616 0.0 0.0

When the effective population size or genomic mutationrate for a set of characters is low enough, some principal com-ponents are expected to have zero variance associated withthem. For instance, the eigenvalues are given ranked in orderfrom largest to smallest when 4Nm ¼ 0:1 per locus and thenumber of loci is 10. The M matrix had dimensionality of25 with the same variance for each principal component.


highly discordant. For instance, when M has uniformeigenvalues associated with it, then E(G) has uniformeigenvalues, but the expected eigenvalues ofG are non-uniform. The practical consequence of this is that al-though E(G) may imply that variation in breeding valuesis evenly distributed across characters, in fact due tostochasticity, variation in breeding values is never evenlydistributed. Some characters by chance will have lessgenetic variation than others and furthermore, due tostochasticity and the pleiotropic nature ofmutation and/or linkage, breeding values will typically covary amongcharacters even if the expectation is independence. Ourresults suggest that the nonuniformity in the eigenvaluesof G is not trivial because the nonuniformity is approx-imately exponential. This approximate exponentialdecline in eigenvalues may lead to potentially strongconstraints on the freedom of a multivariate phenotypeto respond to selection.

A key factor shaping the nonuniformity in the eigen-values ofG is genealogy. Genealogy causes the effects ofalleles at a locus to be dependent. The principle that ge-nealogy causes dependencies is not new and has been akey basis of analyses in phylogenetics (e.g., Piazza andCavalli-Sforza 1974 and Felsenstein 1985). That ge-nealogy is important in shaping the eigenstructure ofG has been until now unrecognized. Shared commondescent enhances the nonuniformity in the eigenvaluesof G due to dependencies generated by branching pat-terns within the genealogy. In addition, multiple muta-tions may occur within a single branch of a genealogy,and this too can enhance the nonuniformity in the ei-genvalues of G because there are fewer independentlyacting mutations. The genealogical ancestry of alleles isintrinsic to biology. The only way to partially overcomethe covariance in allelic values that is induced by gene-alogy is for the value of an allele to be decoupled fromits ancestry. The house-of-cards mutation model is onewhere the value of an allele is decoupled from its an-cestry and our results show that this causes the decline ineigenvalues to be less than if the value of an allele is cou-pled to its ancestry (such as in the continuum-of-allelesmodel). Tighter linkage enhances nonuniformity in theeigenvalues of G because alleles at different loci do notsegregate independently, thus decreasing the numberof independent gene genealogies contributing to thestructure of G.In general, then, the nonuniformity in the eigenval-

ues of G is exaggerated with more dependency amongallelic effects. We found that these dependencies can beaffected by a variety of other factors as well. For instance,when there are fewer loci or the per locus mutation rateis lower the observed nonuniformity in the eigenvaluesof G is increased. The degree of pleiotropy of new mu-tations affects the dependency of allelic effects. Underuniversal pleiotropy all characters are influenced by thesame set of gene trees and this overlap increases theobserved nonuniformity in the eigenvalues of G. Incontrast, in the absence of pleiotropy, each characterhas a unique set of gene trees (as long as there is somerecombination), and this independence decreases cor-relations between genotypic values and reduces non-uniformity in the expected eigenvalues of G.

Figure 8.—The effect of differentmutational models: house-of-cards vs.continuum-of-alleles. Three hundred in-dividuals were sampled and 4Nm ¼ 1:0for each locus. The independent locus co-alescent approach was used. Estimates arebased on 500 replicates. Eigenvalues areon a log scale and error bars represent95% confidence intervals of the meaneigenvalue. The internal hatch markwithinanerrorbar is themeaneigenvalue.

Figure 9.—The effects of pleiotropy on the nonuniformityin G. Three characters and loci were modeled. In the pleiot-ropy cases, mutations at the three loci affect all characters,whereas in the no pleiotropy case a single locus affects onlyone of the characters. The top set of points corresponds toa per character 4Nm ¼ 3:0 and the bottom set of points cor-responds to a per character 4Nm ¼ 0:3. Mutation rate wasmodeled on a per character basis so that a direct comparisonbetween the pleiotropy and the no pleiotropy case can bemade. The ancestral-recombination graph was simulated inwhich the genomic recombination rate was 4Nr ¼ 2000. Ei-genvalues are on a log scale and error bars represent 95% con-fidence intervals of the mean eigenvalue. The internal hatchmark within an error bar is the mean eigenvalue.


What leads to the discrepancy between the expectedeigenvalues of G and the eigenvalues of E(G)? Mathe-matically, the discordance traces to the fact that eigen-values are nonlinear functions of the elements ofG, andhence the expectations of the eigenvalues of G will notgenerally be equal to the eigenvalues of the expectationof G. This can be seen more directly in the followingtwo-trait case. Suppose the expected G matrix is di-

agonal with the repeated eigenvalue l: EðGÞ ¼ l 00 l

! ",

where l. 0. A realized G under neutral evolution canbe viewed as a perturbation of E(G), say, G ¼l1 e1 e2e2 l1 e3

! ", where the ei’s are random variables with

zero means and variances much smaller than l. If weimagine that the perturbations are drawn, say, from azero-mean normal distribution with variance s2>lthen it is not hard to show that the expected eigenvaluesof G are not uniform (i.e., not repeated). Indeed, theexpected leading eigenvalue of G exceeds l by at leasts

ffiffiffiffiffiffiffiffiffi2=p

pwhereas the smaller eigenvalue ofG is expected

to fall below l by at least the same amount. This sug-gests that nonuniformity among the expected eigen-values of G is driven by the variability of perturbationsfrom E(G). This variability arises from the assortedsources of stochasticity inherent to evolution in finitepopulations.

That the eigenvalues of random covariance matrices(such as aGmatrix) are nonuniform is not a new result(e.g., see a review by Edelman and Rao 2005 andWagner’s 1984 random pleiotropy model). Addition-ally, similar patterns found in this article for the averageof ranked eigenvalues are observed if it is assumed thatthe variances for characters are independently drawnfrom a chi square for a given degrees of freedom. Inparticular, a deviation from a purely exponential de-cline occurs as was observed in Figure 7. Broadly, sto-chasticity causes the nonuniformity in eigenvalues, butthere are biologically unique processes that contributeto this stochasticity. This article suggests that genealog-ical ancestry is one important process.

Our neutral theoretical predictions for the expectedstructure of G at recombination–mutation–drift bal-ance form an important set of null expectations for em-pirical studies of multivariate evolution. In particular,our finding that, under neutrality, the eigenvalues of Gtend to be unequal and decay approximately exponen-tially from largest to smallest is consistent with empiricalestimates ofG (e.g., Mezey and Houle 2005). However,our results also show that the rate of decay is sensitive inpredictable ways to numerous factors including pop-ulation size, the number and patterns of linkage amongloci, and the rate of and modality of mutation. If oneincludes selection as well (as in Jones et al. 2003),current theoretical considerations suggest that infer-ring underlying process from an equilibrium estimate ofG may be difficult because different underlying pro-

cesses have similar effects on the eigenvalues associatedwith G. For instance, it may be difficult to disentanglemutation rates from numbers of loci and the pattern ofrecombination among loci. Single-locus expectationsand statistics may be developed, but this ignores thelikely possibility that more than one locus contributesvariation to G. Rather, to infer processes that influencethe evolution of G may require direct approaches (seeEstes et al. 2004) or possibly the study of transientchanges in G (Phillips et al. 2001; Whitlock et al.2002).

Comments by Ruth Shaw, Joe Felsenstein, and an anonymous re-viewer improved the presentation and content of this article.We thankthe anonymous reviewer for suggesting that we examine the case inwhich the variance for each character is independent and chi-squaredistributed. We thank P. Phillips and K. McGuigan for allowing us tocite their then unpublished work and the National Science Founda-tion (NSF) (grants DEB 0337582, EF 0328594, and EF 0531870 to RG)for financial support. The Initiative for Bioinformatics and Evolution-ary Studies program at the University of Idaho kindly providedcomputer resources and they are supported by NSF grant EPS0080935 and National Institutes of Health grants P20 RR16454 andP20 RR16448 from the Center for Biomedical Research Excellenceand Idea Networks of Biomedical Research Excellence programsof the National Center for Research Resources.

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Communicating editor: M. Nordborg

APPENDIX

Whole-population predictions for G: To investigateneutral evolution ofG for an entire population we usedforward time, individual-based simulations. Our ap-proach is similar to that of Jones et al. (2003) except

for the absence of selection and our use of Wright–Fishermating (i.e., sampling parents with replacement).The individual-based simulations were founded on thesame model assumptions as the two coalescent-basedmethods. Events in the individual-based simulationswere ordered mutation / recombination / randommating within generations. At the beginning of a rep-licate simulation run, each individual was initializedwith a genotypic value of zero by assigning allelic valuesof zero to all loci. Population sizes of 140 and 1366 weresimulated. An inspection of simulations suggests thatpopulations reach quasi-recombination–mutation–driftbalance by 12,000 generations for populations consist-ing of 140 and 1366 individuals given our mutation rateand rates of recombination. The basis for this inferencewas that the average G matrix from simulations was notsignificantly different from2NM the expectedGmatrix.For each replicate run, we calculated the G matrix at

Figure A1.—The eigenvalues of G for an entire populationbased on individual-based simulations (squares) for (a) smalland (b) large population sizes. Mutations pleiotropically af-fected five characters at five loci. The continuum-of-allelesmutation model was used. In the individual-based simula-tions, the recombination rate (r) was 0.5 per locus pair. Forcomparison, results based on the independent locus coales-cent are provided for a sample of 10 (a) and 30 (b) individu-als from populations of size 140 and 1366, respectively. Theper locus mutation rate and number of loci were the samein the coalescent and the individual-based simulations. Re-sults are based on 200 replicates for both the individual-basedand 500 replicates for the coalescent simulations. Eigenvaluesare on a log scale and error bars represent 95% confidenceintervals of the mean eigenvalue. The internal hatch markwithin an error bar is the mean eigenvalue. Points usingthe coalescent approach are offset relative to points usingthe individual-based approach to aid comparison.


FigureA2.—The eigenvalues of G for an entire populationbased on individual-based simulations and when the recombi-nation rate varies between loci (squares). The population sizein all cases was N ¼ 1366. Mutations pleiotropically affectedfive characters at two loci. The continuum-of-alleles mutationmodel was used. For comparison, results based on the ances-tral recombination graph are provided (triangles) in whichthe same number of loci, population size, per locus mutationrate, and recombination rate is modeled, except now resultsare based on a sample of 30 individuals from the population.Results are based on 200 replicates for the individual-basedand 500 replicates for the coalescent simulations. Eigenvaluesare on a log scale and error bars represent 95% confidenceintervals of the mean eigenvalue. The internal hatch markwithin an error bar is the mean eigenvalue. Points using the an-cestral recombination graph approach are offset relative topoints using the individual-based approach to aid comparison.

generation 12,000 using the genotypic values of allindividuals in the population. We ran 20 replicates for aparticular parameter set when using 50 loci and 200replicates when using 2 and 5 loci.

Some of the parameters used in both the individual-based simulations were similar to those utilized by Joneset al. (2003). For example, the per haploid locus mu-tation rate utilized was 0.0002 and all traits were un-correlated with an expected per character variance inthe effects of mutations of 0.05.

Results: Figure A1 illustrates that when the wholepopulation is modeled, the eigenvalues of G decline atan approximately exponential rate and that the rate ofdecline is greater for smaller populations. Figure A2suggests that nonindependent segregation of allelesamong loci enhances the nonuniformity in the eigen-values of G when the whole population is modeled.Overall these whole-population simulations combinedwith the coalescent simulations suggest that the non-uniformity in the eigenvalues of G is a property of bothan entire population and subsamples from the popula-tion. Furthermore, a comparison of individual-basedand coalescent results in Figures A1 andA2 suggests thatestimating the eigenstructure ofG on the basis of a sub-sample from a population (as is done with the co-alescent) results in slightly enhanced nonuniformity inthe eigenvalues of G relative to the true value of thewhole population.


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Neutra l Evolution of Multiple Qua ntitative Characters: A