1747

q 2003 The Society for the Study of Evolution. All rights reserved.

Evolution, 57(8), 2003, pp. 1747–1760

STABILITY OF THE G-MATRIX IN A POPULATION EXPERIENCING PLEIOTROPICMUTATION, STABILIZING SELECTION, AND GENETIC DRIFT

ADAM G. JONES,1,2 STEVAN J. ARNOLD,3 AND REINHARD BURGER4

1School of Biology, 310 Ferst Drive, Georgia Institute of Technology, Atlanta, Georgia 303322E-mail: adam.jones@biology.gatech.edu

3Department of Zoology, 3029 Cordley Hall, Oregon State University, Corvallis, Oregon 973314Institut fur Mathematik, Universitat Wien, A-1090 Wien, Austria

Abstract. Quantitative genetics theory provides a framework that predicts the effects of selection on a phenotypeconsisting of a suite of complex traits. However, the ability of existing theory to reconstruct the history of selectionor to predict the future trajectory of evolution depends upon the evolutionary dynamics of the genetic variance-covariance matrix (G-matrix). Thus, the central focus of the emerging field of comparative quantitative genetics isthe evolution of the G-matrix. Existing analytical theory reveals little about the dynamics of G, because the problemis too complex to be mathematically tractable. As a first step toward a predictive theory of G-matrix evolution, ourgoal was to use stochastic computer models to investigate factors that might contribute to the stability of G overevolutionary time. We were concerned with the relatively simple case of two quantitative traits in a populationexperiencing stabilizing selection, pleiotropic mutation, and random genetic drift. Our results show that G-matrixstability is enhanced by strong correlational selection and large effective population size. In addition, the nature ofmutations at pleiotropic loci can dramatically influence stability of G. In particular, when a mutation at a single locussimultaneously changes the value of the two traits (due to pleiotropy) and these effects are correlated, mutation cangenerate extreme stability of G. Thus, the central message of our study is that the empirical question regarding G-matrix stability is not necessarily a general question of whether G is stable across various taxonomic levels. Rather,we should expect the G-matrix to be extremely stable for some suites of characters and unstable for others over similarspans of evolutionary time.

Key words. Genetic correlation, genetic covariance, genetic variance, pleiotropy, response to selection, quantitativegenetics.

Received October 23, 2002. Accepted March 2, 2003.

Modern quantitative genetics theory provides points ofconnection between microevolution and macroevolution (Ar-nold et al. 2001). For a phenotype comprising multiple traits,the single-generation response to selection is given by themultivariate version of the breeder’s equation (Lande 1979),Dz 5 Gb, where z is a vector of population trait means, bis a vector of directional selection gradients, and G is thegenetic variance-covariance matrix (the G-matrix). Hence,the response to selection depends upon the intensity and di-rection of selection, as well as upon the amount of geneticvariation and the nature of genetic correlations among traits.This equation for the change in the mean phenotype can beextrapolated over multiple generations to reconstruct the his-tory of selection or to predict the future trajectory of thephenotype as a consequence of selection. This potential forextrapolation provides a connection between microevolu-tionary processes and macroevolutionary patterns (Lande1979; but see Zeng 1988). However, such an extrapolationis possible only if the G-matrix remains relatively constantover long spans of evolutionary time. An extremely unstableG-matrix would render the goal of understanding selectionover evolutionary time unachievable within the existingquantitative genetics theory framework.

Because of the central role of the G-matrix in quantitativegenetics and the implications of its evolution, G-matrix sta-bility has been a major focus of recent studies. In fact, thisenterprise has grown so much in size that it is fair to saythat a new field of comparative quantitative genetics has aris-en (Steppan et al. 2002), whose primary purview is the evo-lution of the G-matrix itself. However, despite several de-cades of work, how the G-matrix changes over evolutionary

time remains a major unresolved issue. Neither empirical northeoretical investigations have led to a consensus with respectto the expected stability of the G-matrix over evolutionarytime.

On the empirical side, the study of G-matrix evolution hasrelied upon comparisons of G-matrices across distinct pop-ulations of organisms. Clearly, over extremely long spans ofevolutionary time the G-matrix must change appreciably,since among very divergent taxa, dramatic changes in bau-plans result in many structures that cannot be equated withone another in a quantitative genetic framework. Over shorterperiods of evolutionary time, however, empirical results dem-onstrate that the G-matrix often remains stable. For example,numerous comparisons of G between populations within spe-cies have revealed G-matrix equality (Billington et al. 1988;Shaw and Billington 1991; Spitze et al. 1991; Platenkampand Shaw 1992; Brodie 1993; Podolsky et al. 1997; Service2000). Other studies have shown that G-matrices may varysomewhat between populations while still retaining evolu-tionarily important aspects of their structure (Arnold andPhillips 1999; Roff and Mousseau 1999). Stability or con-servation of structure has even been demonstrated for somecomparisons between species within genera (Roff et al. 1999;Begin and Roff 2001). However, enough studies have dem-onstrated G-matrix inequality that we cannot tacitly assumethat the G-matrix remains constant over long spans of time.Instability of G is particularly obvious in comparisons amongspecies or among genera (Kohn and Atchley 1988; Paulsen1996; Waldmann 2000), but G has also been shown to varyin response to genetic drift and environmental perturbations

1748 ADAM G. JONES ET AL.

(Shaw et al. 1995; Guntrip et al. 1997; Begin and Roff 2001;Phillips et al. 2001).

Existing theory on the evolution of the G-matrix producesconclusions similar to those that can be drawn from the em-pirical studies: some conditions are expected to produce sta-bility, while others are not. Turelli (1988) outlined severalconditions that may promote G-matrix constancy, including:(1) no genotype-by-environment interaction with respect tothe values of genetic variances and covariances; (2) no directchange in the nature of environmental effects on the phe-notype over time; (3) a constant curvature and orientation ofthe adaptive landscape (even though the optimum canchange); and (4) no change in the nature of genetic covari-ances associated with new mutations. In addition, the numberof loci, number of alleles per locus, and the distribution ofallelic effects can be important to the stability of G (Turelli1988). Furthermore, linkage disequilibrium during periods ofstrong, fluctuating selection can induce transient changes inthe G-matrix, a phenomenon referred to as the ‘‘Bulmer ef-fect’’ (Bulmer 1980; Turelli 1988; Shaw et al. 1995). Overall,these theoretical considerations have fostered a grim outlookon the prospects of long-term stability of G, with some au-thors suggesting that the G-matrix is not useful over evo-lutionary time scales. Nevertheless, the current state of an-alytical theory on G-matrix evolution is that existing theorycannot guarantee stability of the G-matrix (Turelli 1988), butalso does not guarantee instability. In fact, existing theoryprovides very little information with respect to how much Gwill change as a result of violations of the criteria for stabilitylaid out by Turelli (1988). Consequently, we are left to won-der if the changes in G-matrix structure that seem likely basedon theoretical considerations are of sufficient magnitude toaffect evolutionary inferences.

Due to the complexity of the problem, analytical theoryappears to have reached an impasse with respect to furtherprogress on the issue of G-matrix stability, but a hithertounexplored approach is to use stochastic simulations to in-vestigate this important topic. Such a modeling approach hasled to useful insights in quantitative genetic studies involvingsingle traits. For example, stochastic models have been usedto address the maintenance of genetic variance for quanti-tative traits in finite populations (Bulmer 1972; Barton 1989;Burger et al. 1989; Keightley and Hill 1989; Foley 1992;Burger and Lande 1994) and the risk of extinction due toquantitative trait evolution in response to environmentalchange (Burger and Lynch 1995). So far, few studies haveattempted to extend these stochastic models to problems in-volving a phenotype comprising multiple traits (Wagner1989; Baatz and Wagner 1997; Wagner et al. 1997; Reeve2000), and none has investigated the stability of G in detail.Hence, our aim in the present study is to extend these sto-chastic models to investigate the factors that influence G-matrix stability.

As a first step in developing stochastic models of the G-matrix, we address the situation in which a finite populationis subject to stabilizing selection and mutation. Our specificgoals are to investigate the relative roles of effective popu-lation size, the shape of the adaptive landscape, and the natureof mutational parameters on the G-matrix over thousands ofgenerations of evolution. The essential features of the G-

matrix can be illustrated by using a phenotype composed oftwo quantitative traits, so we restrict our attention in thisinitial study to a situation in which two traits are determinedby a suite of pleiotropic loci. The results of this analysis arerelevant to debates regarding the stability of the G-matrix,and they pinpoint certain parameters of fundamental impor-tance to the genetic architecture of the multivariate pheno-type.

METHODS

The Simulation Model

The simulation model used in this study is a direct exten-sion of that employed by Burger et al. (1989) and Burgerand Lande (1994). We used Monte Carlo simulations, withan additive genetic model in which all loci in all individualswere explicitly modeled. The extension of the univariatemodel to two traits resulted in numerous additional param-eters, so for those parameters that were investigated in thesingle-trait simulations, we chose parameter values thatseemed reasonable from those earlier studies (Burger andLande 1994). The two traits in our model were determinedby n unlinked loci, all of which were pleiotropic. This sit-uation can result in a genetic correlation between the twotraits, but it need not do so, depending on the nature ofmutational effects and selection. A mutation at a locus re-sulted in a new allele with new effects on both traits. Theseeffects were drawn from a bivariate Gaussian distributionwith means of zero, variances of and , and a correlation2 2a a1 2of rm. They then were added to the existing effects of theallele in accord with the continuum of alleles model (Crowand Kimura 1964). We determined an individual’s phenotypeat each trait by summing across loci and adding environ-mental variation drawn randomly from a normal distributionwith a mean of zero and a variance of one. Environmentaleffects on multiple traits were uncorrelated.

We simulated a diploid, sexually reproducing populationwith a constant population size of N. The life cycle consistedof: (1) production of progeny from the previous generationof adults (including mutation); (2) viability selection; and (3)random choice of the new generation of adults from the sur-vivors of selection. We used a monogamous mating systemfor these simulations, and generations did not overlap. Forthe parameter combinations that we investigated, selectionwas relatively weak, such that at least N progeny alwayssurvived the viability selection phase of the life cycle. Thefitness of each individual was determined by an individualselection surface with the shape of a multivariate Gaussiandistribution, such that the fitness of phenotype z, W(z), wasgiven by

1 T 21W(z) 5 exp 2 (z 2 u) v (z 2 u) , (1)[ ]2

where z is a column vector of trait values, u is a columnvector of trait optima, and the superscript T denotes matrixtransposition (Lande 1979). The matrix v describes the cur-vature and orientation of the individual selection surface. Forthe two-character case, v contains the elements v11, v22, andv12. The diagonal elements, v11 and v22, represent thestrength of stabilizing selection and are analogous to the var-

1749EVOLUTION OF THE G-MATRIX

FIG. 1. The combined influence of selectional and mutational cor-relation on the average between-generation change of the orienta-tion of the G-matrix (Dw). As indicated, each of the five linesdisplays Dw as a function of the selectional correlation rv for agiven value of rm. The following parameters are the same for allpanels: Ne 5 342, v11 5 v22 5 49, 5 5 0.05. See text for2 2a a1 2other parameter values used in the model.

iance of the bivariate normal distribution. Thus, larger valuesof v11 and v22 result in weaker stabilizing selection. Theorientation of the selection surface is determined by v12,which is analogous to the covariance of a bivariate normaldistribution.

Each simulation run was preceded by 10,000 generationsof stabilizing selection, during which a genetically uniformstarting population reached a mutation-selection-drift equi-librium. The following 2000 to 4000 generations were theexperimental generations, during which population-level val-ues were calculated every generation. Most notably, we cal-culated the elements of the G-matrix, as well as the geneticand phenotypic means. We also calculated the single-gen-eration change in each population-level variable. In this anal-ysis, we restrict attention to the absolute values of these sin-gle-generation changes, because under a stabilizing selectionequilibrium there is no net directionality to the changes onaverage.

Because the mating system was monogamous in our model,with no variance in the number of offspring produced byeach female, our effective population size was actually some-what larger than the census population size. Each breedingpair in our simulation produced exactly 2B offspring, and weheld B constant at 2. Under these circumstances, the effectivepopulation size is Ne 5 4N/(Vk 1 2), where the variance infamily size, Vk, is given by Vk 5 2(1 2 1/B)[1 2 (2B 2 1)/(BN 2 1)] (Burger and Lande 1994). We used census pop-ulation sizes of 256, 512, 1028, and 2056, which correspond-ed to effective population sizes of 342, 683, 1366, and 2731,respectively.

Throughout this study, we used the following standard pa-rameters (unless otherwise mentioned): the number of locicontributing to the trait, n, is 50, the mutation rate is m 50.0002 per haploid locus, the variances of mutational effects( and ) are 0.05 for both traits, and the loci are freely2 2a a1 2recombining. Therefore, the genomic mutation rate is 2nm 50.02 and for each trait the input of genetic variance due tonew mutations per generation is 1023. With these parametercombinations, Vm/ is also 1023, a value close to those ob-2se

served in empirical studies of mutational variance (Lande1975; Lynch 1988; Lynch and Walsh 1998). Our choice ofparameters was governed mainly by the previous results ofBurger and Lande (1994), so the parameter values that weused in our simulations were similar to those that formed thenucleus of this previous simulation-based analysis. Futurestudies may well benefit from a greater departure from theparameters used by Burger and Lande (1994), but such anal-yses are outside the scope of this initial study.

Quantifying G-matrix Stability

The most obvious way to study the degree of stability ofthe size and shape of the G-matrix under a balance betweenmultivariate stabilizing selection, pleiotropic mutation, andrandom genetic drift would be to consider the genetic vari-ances (G11, G22) and the covariance (G12), or correlation (rg),and how they change during evolution. Another way is tolook at the eigenvalues (l1, l2) of the G-matrix, at their ratio(e, defined here as the smaller eigenvalue divided by thelarger) as a measure of the shape of the G-matrix (thus, e is

inversely related to the eccentricity), and at the angle (w)between the leading eigenvector and the axis along whichthe first trait is measured (i.e., the x-axis). This angle w ismeasured in degrees, ranging from 2908 to 1908, thus pro-viding a convenient measure for the orientation of the G-matrix. Small e means high eccentricity, or as we often shallcall it, a cigar-shaped G-matrix. As a measure for the overallsize of the G-matrix we use the total genetic variance, S 5G11 1 G22. Thus, a two-dimensional G-matrix is fully de-scribed by S, e, and w. Both perspectives, variances and co-variance versus size, shape, and orientation yield interestingand complementary insights.

In our tables, we present the average values of the geneticvariances of the two traits, their genetic covariance and theircorrelation. We also present average values of the eigenval-ues (l1, l2), total size (S), shape (e), and orientation (w) ofthe G-matrix. These average values were obtained by aver-aging over 20 replicate runs, each over 2000 generations(except in the case of Fig. 1, for which values were averagedover 4000 generations), measuring the quantities of interesteach generation. This method is a compromise between twoalternative methods for obtaining an estimator for the meanof a random variable distributed according to a stationarystochastic process: namely, averaging over a very long timeseries or averaging over single values from a large numberof replicate runs. Because the generation of an initial pop-ulation in quasi-equilibrium requires much computer time,and because of high temporal autocorrelations within singlereplicate runs, the adopted method seems to be appropriate(see also Burger et al. 1989; Burger and Lande 1994).

As measures for the stability of the size, the shape, andthe orientation, we use the average of the absolute values ofchange between two successive generations and denote themby DG11, DG22, DG12, Dl1, Dl2, DS, De, and Dw. We stan-dardized DG11, DG22, Dl1, Dl2, DS, and De relative to their

1750 ADAM G. JONES ET AL.

average magnitudes by dividing them by their correspondingmean (i.e., DG11 is the average single-generation change inthe genetic variance for trait 1, divided by the mean geneticvariance for trait 1). However, the change of the angle (Dw)does not require this kind of normalization. In addition, themean covariance is close to zero for some parameter com-binations, so we did not standardize DG12. Thus, DG11, DG22,Dl1, Dl2, DS, and De represent the average between-gen-eration change relative to the mean, whereas Dw and DG12are simply the average between-generation changes.

The Expected Genetic Variance

For comparison of the dynamics of the genetic variance ofour traits determined by pleiotropic loci with those expectedof a single, completely independent trait experiencing sta-bilizing selection, genetic drift, and mutation, we also cal-culated the stochastic house-of-cards approximation for theexpected genetic variance of a single trait (Barton 1989; Burgeret al. 1989; Houle 1989; Keightley and Hill 1989). Under thismodel, the formula for the expected genetic variance is

24nma Ne2s (SHC) 5 , (2)g 21 1 (a N /V )e s

where n is the number of loci, m is the per-locus mutationrate, a2 is the variance of the distribution of mutational ef-fects, and Vs, the strength of stabilizing selection on breedingvalues, is equal to the v corresponding to the trait (i.e., v11for trait 1) plus the environmental variance (which we areholding constant at one; Burger and Lande 1994). This for-mula gives the expected genetic variance for a single traitevolving in complete isolation from other traits. We wereinterested in comparing these expectations with the equilib-rium levels of genetic variation in a trait experiencing sta-bilizing selection, while genetically associated, through plei-otropy, with a second trait also experiencing stabilizing se-lection.

RESULTS

Stabilizing Selection and the G-matrix

Table 1 summarizes the effects of variation in the strengthof stabilizing selection on the two traits and on their co-variance for a population of effective size Ne 5 342 in theabsence of correlation between the mutational effects (rm 50). Table 1 clearly shows that stronger correlational selection(increasing rv) decreases the genetic variance of both traitsas well as both eigenvalues, hence also decreasing the overallsize S of the matrix. In parallel, it increases the genetic co-variance and correlation between the traits. In addition, inthe absence of selectional correlation, asymmetric stabilizingselection (v11 ± v22) has the consequence that stronger se-lection on the second trait (v11 . v22) reduces the varianceof this trait to a greater extent than that of the first trait. Thevariance of the first trait is reduced, too, despite the absenceof genetic covariance, because each single mutation haspleiotropic effects on both traits, and only on average do theycancel. The shape measure e decreases with increasing se-lectional correlation, but not very much, that is, correlationalselection promotes eccentricity of the G-matrix to a moderate

extent. The average angle w does not differ significantly fromzero if mutational and selection correlations are both zero(rm 5 rv 5 0), but it is highly variable in this case. Withincreasing rv, it quickly increases to nearly 458 if the twotraits experience the same strength of stabilizing selection.For asymmetric stabilizing selection, this angle increases toa much smaller degree.

The average between-generation change of the genetic var-iances and of the overall size is almost unaffected by anychange in the selection pattern, reflecting the fact that fluc-tuations in the variances and the size of the G-matrix aremainly determined by random genetic drift (see Table 2,which shows that DG11, DG22, and DS decrease as Ne in-creases). However, the between-generation change of the ma-jor eigenvalue increases with increasing correlational selec-tion, whereas that of the smaller eigenvalue decreases cor-respondingly. Interestingly, the standardized between-gen-eration change of the shape, De, is nearly independent of rv,although e itself decreases as rv increases. In other words,regardless of the value of rv, the per-generation change in eis a constant fraction (about 10% for Ne 5 342) of its mean.However, De does depend on Ne (Table 2). By contrast, thebetween-generation change of the covariance (DG12), as wellas that of the orientation (Dw), decreases with increasing rv.These patterns reflect the fact that stronger correlational se-lection causes higher stability in the orientation of the G-matrix, though it has virtually no effect on the stability ofthe genetic variances, the total size, and the shape.

Interestingly, if one trait is under stronger stabilizing se-lection than the other, the between-generation change of theangle is reduced below that observed if both traits experiencethe same strength of selection, weak or strong. This is truefor any value of rv. The between-generation change of theshape parameter, De, is remarkably inert to any changes inthe strength of selection on the two characters, even to ex-treme asymmetries. Thus, like correlational selection, asym-metric stabilizing selection promotes orientation stability.Overall, Table 1 shows that the stability properties of thesize and shape of a G-matrix depend in a qualitatively similarway on the shape of the fitness landscape. Also, at least insuch small populations, the size, shape, and orientation ofthe G-matrix are rather unstable in the absence of mutationalcorrelation: The average between-generation change of thetotal size is between 5% and 6% of the mean, that of theshape parameter is about 10% of the mean, and the averagebetween-generation change of the orientation, Dw, is alwaysat least 38 and reaches nearly 108 in the absence of any cor-relation (see Fig. 2 for a time series showing the evolutionof w).

The Nature of Mutations and G-matrix Stability

In contrast to stabilizing selection alone, mutational cor-relation has a much stronger influence on the shape of theG-matrix and its orientation stability. Table 3 summarizesthe effects of increasing the mutational correlation for dif-ferent strengths of stabilizing selection on the two traits inthe absence of correlational selection. First, increasing rm hasonly a slight negative effect on the average genetic variances(or even none for G22 in the case of unequal selection

1751EVOLUTION OF THE G-MATRIX

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0.10

0.10

0.10

0.10

0.11

0.11

6.60

6.29

5.68

4.39

3.02

3.02

99 99 99 99 99

99 49 19 9 4

0 0 0 0 0

0.56

10.

504

0.45

50.

375

0.35

1

0.51

90.

463

0.35

40.

224

0.14

4

0.00

40.

015

20.

011

0.01

10.

000

0.00

70.

038

20.

028

0.03

50.

006

0.66

0.60

0.50

0.40

0.36

0.42

0.37

0.30

0.20

0.14

1.08

0.97

0.81

0.60

0.50

0.65

0.62

0.63

0.55

0.41

4.2

8.4

24.

63.

70.

8

0.07

10.

072

0.07

20.

072

0.07

3

0.07

20.

072

0.07

20.

074

0.07

5

0.02

70.

024

0.02

00.

015

0.01

1

0.05

70.

058

0.05

60.

058

0.05

0

0.09

10.

091

0.09

70.

100

0.13

4

0.05

20.

053

0.05

30.

055

0.05

7

0.10

0.10

0.10

0.10

0.10

9.55

9.03

8.79

6.68

3.75

1752 ADAM G. JONES ET AL.

TA

BL

E2.

The

infl

uenc

eof

popu

lati

onsi

zeon

G-m

atri

xst

abil

ity

duri

ngst

abil

izin

gse

lect

ion.

The

foll

owin

gpa

ram

eter

sar

efi

xed:

v1

15

v2

25

49,

55

0.05

.S

eeth

e2

2a

a1

2te

xtan

dT

able

1fo

rm

ore

info

rmat

ion

abou

tot

her

para

met

erva

lues

,m

etho

ds,

and

the

mea

ning

ofsy

mbo

ls.

Not

eth

atth

epa

ram

eter

sus

edin

the

firs

tfi

vero

ws

ofth

ista

ble

corr

espo

ndto

thos

eus

edto

gene

rate

Fig

ures

2–

5.F

orth

ese

para

met

erva

lues

,th

eS

HC

-exp

ecte

dge

neti

cva

rian

ces

(see

text

)ar

e0.

51,

0.81

,1.

15,

and

1.46

for

Nes

of34

2,68

3,13

66,

and

2731

,re

spec

tive

ly.

Ne

r vr m

G11

G22

G12

r gl

1l

2S

ew

DG

11D

G22

DG

12D

l1

Dl

2D

SD

eD

w

342

342

342

342

342

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.43

20.

355

0.40

80.

410

0.46

5

0.44

30.

353

0.39

90.

399

0.46

7

0.01

20.

062

0.16

70.

226

0.41

9

0.02

80.

164

0.41

10.

548

0.89

5

0.54

0.46

0.59

0.64

0.89

0.33

0.25

0.22

0.17

0.05

0.88

0.71

0.81

0.81

0.93

0.63

0.58

0.44

0.29

0.05

5.9

28.6

42.4

44.7

45.0

0.07

20.

072

0.07

20.

072

0.07

1

0.07

20.

072

0.07

20.

072

0.07

2

0.02

20.

018

0.02

20.

024

0.03

2

0.05

90.

071

0.07

10.

072

0.07

2

0.09

10.

072

0.07

20.

072

0.07

1

0.05

0.05

0.06

0.06

0.07

0.10

0.10

0.10

0.10

0.10

9.10

7.48

3.64

2.34

0.71

683

683

683

683

683

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.65

80.

477

0.59

80.

628

0.66

3

0.66

00.

470

0.59

90.

603

0.63

9

0.01

30.

127

0.22

80.

350

0.58

0

0.02

20.

260

0.38

10.

563

0.89

0

0.77

0.62

0.84

0.97

1.23

0.55

0.33

0.36

0.26

0.07

1.32

0.95

1.20

1.23

1.30

0.71

0.56

0.44

0.28

0.06

2.8

42.1

45.0

43.8

44.4

0.05

10.

051

0.05

10.

051

0.05

1

0.05

10.

051

0.05

10.

051

0.05

1

0.02

30.

018

0.02

30.

025

0.03

1

0.04

40.

050

0.05

10.

051

0.05

1

0.05

90.

051

0.05

10.

051

0.05

0

0.04

0.04

0.04

0.04

0.05

0.07

0.07

0.07

0.07

0.07

8.80

4.24

2.65

1.51

0.52

1366

1366

1366

1366

1366

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.85

30.

591

0.79

60.

746

0.80

6

0.84

00.

587

0.79

00.

740

0.81

3

20.

004

0.21

10.

292

0.44

70.

722

20.

003

0.35

20.

366

0.59

70.

891

0.95

0.81

1.10

1.19

1.53

0.75

0.37

0.49

0.29

0.09

1.69

1.18

1.59

1.49

1.62

0.79

0.47

0.46

0.25

0.06

0.6

45.1

44.4

44.8

45.2

0.03

60.

036

0.03

60.

036

0.03

6

0.03

60.

036

0.03

60.

036

0.03

6

0.02

10.

016

0.02

10.

022

0.02

8

0.03

10.

036

0.03

60.

036

0.03

6

0.04

00.

036

0.03

60.

036

0.03

6

0.03

0.03

0.03

0.03

0.03

0.05

0.05

0.05

0.05

0.05

9.71

2.02

1.88

0.99

0.37

2731

2731

2731

2731

2731

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.93

70.

650

0.91

40.

820

0.95

0

0.91

10.

676

0.89

60.

832

0.93

1

20.

007

0.27

10.

269

0.52

00.

840

20.

007

0.40

60.

298

0.62

90.

893

1.02

0.94

1.18

1.35

1.78

0.83

0.39

0.63

0.30

0.10

1.85

1.33

1.81

1.65

1.88

0.82

0.42

0.54

0.23

0.06

23.

546

.544

.145

.344

.7

0.02

50.

025

0.02

50.

025

0.02

5

0.02

50.

025

0.02

50.

025

0.02

5

0.01

60.

013

0.01

70.

017

0.02

2

0.02

30.

025

0.02

50.

025

0.02

5

0.02

80.

026

0.02

50.

025

0.02

5

0.02

0.02

0.02

0.02

0.02

0.03

0.04

0.04

0.04

0.04

7.60

1.16

1.68

0.63

0.26

FIG. 2. Time series of the orientation, w, the angle between theleading eigenvector and the x-axis. Each figure displays the anglew from one replicate run near stochastic equilibrium for 2000 gen-erations. As in Figure 3, from top to bottom, the following selec-tional and mutational correlations are chosen: (a) rm 5 rv 5 0; (b)rm 5 0, rv 5 0.75; (c) rm 5 0.5, rv 5 0; (d) rm 5 0.5, rv 5 0.75;(e) rm 5 rv 5 0.90. This figure was produced using the same pa-rameter values as those used for Figure 3.

strength). However, it has larger effects on the average ei-genvalues. Increasing the mutational correlation rm alwaysincreases the leading eigenvalue but decreases the minor one.Accordingly, the eccentricity increases substantially, that is,the shape parameter e gets very small and the genetic co-variance and correlation increase strongly. In other words,mutational correlation strongly promotes a cigar-shaped G-matrix.

As in Table 1, the between-generation changes of the ge-netic variances, hence of the overall size, as well as of theshape, are virtually unaffected by varying the mutational cor-relation. However, because De (as well as DG11 and DG22)is measured relative to the mean and the mean e is very smallfor large rm, the absolute change in shape is very small ifmutational correlation is large. The between-generationchange of the eigenvalues is affected in a very similar wayby increasing rm as by increasing rv (cf. with Table 1). How-ever, an increasing mutational correlation leads to much high-er stability of the orientation, that is, small Dw. The averageper-generation change in the genetic covariance (DG12) in-creases slightly as rm increases. However, this increase isassociated with a dramatic increase in the average magnitudeof the genetic covariance. Thus, relative to its mean, DG12

1753EVOLUTION OF THE G-MATRIXT

AB

LE

3.T

hein

flue

nce

ofm

utat

iona

lco

rrel

atio

non

G-m

atri

xst

abil

ity

duri

ngst

abil

izin

gse

lect

ion.

The

foll

owin

gpa

ram

eter

sar

efi

xed:

Ne

534

2,r v

50,

55

0.05

.2

2a

a1

2S

eete

xtan

dT

able

1fo

rad

diti

onal

info

rmat

ion

rega

rdin

got

her

para

met

erva

lues

,m

etho

ds,

sym

bol

defi

niti

ons,

and

expe

cted

leve

lsof

gene

tic

vari

ance

acco

rdin

gto

the

SH

Cap

prox

imat

ion.

v11

v22

r mG

11G

22G

12r g

l1

l2

Se

wD

G11

DG

22D

G12

Dl

1D

l2

DS

De

Dw

9 9 9 9 9 9

9 9 9 9 9 9

0 0.25

0.50

0.75

0.85

0.90

0.18

80.

192

0.18

30.

178

0.16

60.

162

0.19

50.

197

0.17

80.

167

0.16

80.

165

0.00

30.

031

0.06

70.

097

0.11

70.

129

0.01

20.

161

0.36

10.

556

0.69

30.

700

0.24

0.25

0.26

0.27

0.29

0.29

0.15

0.14

0.11

0.07

0.05

0.03

0.38

0.39

0.36

0.34

0.33

0.33

0.65

0.59

0.40

0.18

0.12

0.08

2.0

27.5

42.9

43.0

45.5

45.4

0.07

50.

074

0.07

50.

074

0.07

50.

075

0.07

50.

075

0.07

50.

075

0.07

50.

075

0.01

00.

010

0.01

00.

010

0.01

10.

011

0.06

00.

068

0.07

50.

075

0.07

50.

075

0.09

60.

082

0.07

50.

073

0.07

30.

072

0.06

0.06

0.06

0.06

0.07

0.07

0.10

0.10

0.10

0.10

0.10

0.10

9.35

7.92

4.37

2.30

1.55

1.16

49 49 49 49 49 49

49 49 49 49 49 49

0 0.25

0.50

0.75

0.85

0.90

0.43

90.

452

0.40

80.

412

0.40

50.

409

0.45

30.

444

0.39

90.

408

0.41

80.

405

20.

019

0.06

80.

167

0.28

20.

327

0.34

8

20.

045

0.15

60.

411

0.68

60.

790

0.84

8

0.55

0.57

0.59

0.70

0.74

0.76

0.35

0.33

0.22

0.12

0.08

0.06

0.89

0.90

0.81

0.82

0.82

0.81

0.63

0.59

0.44

0.28

0.18

0.12

210

.025

.242

.444

.745

.744

.8

0.07

20.

072

0.07

20.

072

0.07

30.

073

0.07

20.

072

0.07

20.

072

0.07

30.

073

0.02

30.

023

0.02

20.

025

0.02

70.

028

0.05

50.

066

0.07

10.

072

0.07

30.

073

0.09

60.

079

0.07

20.

071

0.07

10.

070

0.05

0.05

0.06

0.06

0.07

0.07

0.10

0.10

0.10

0.10

0.10

0.10

9.90

7.88

3.64

1.54

1.12

0.90

49 49 49 49 49 49

9 9 9 9 9 9

0 0.25

0.50

0.75

0.85

0.90

0.34

20.

365

0.33

10.

282

0.27

50.

258

0.22

70.

236

0.22

40.

224

0.22

40.

227

0.01

20.

042

0.11

40.

153

0.18

10.

197

0.04

30.

162

0.41

30.

608

0.72

70.

809

0.37

0.40

0.41

0.41

0.43

0.44

0.20

0.20

0.14

0.09

0.06

0.04

0.57

0.60

0.56

0.51

0.50

0.48

0.58

0.51

0.37

0.24

0.16

0.10

5.1

18.1

32.8

39.9

40.9

42.6

0.07

20.

073

0.07

30.

073

0.07

30.

073

0.07

40.

073

0.07

30.

074

0.07

40.

074

0.01

40.

015

0.01

50.

015

0.01

60.

016

0.05

90.

065

0.07

30.

074

0.07

40.

074

0.09

60.

087

0.07

30.

073

0.07

20.

072

0.05

0.05

0.06

0.06

0.07

0.07

0.10

0.10

0.10

0.10

0.10

0.10

7.13

5.77

3.18

1.92

1.37

1.05

becomes very small as rm increases. Asymmetric selectionstrength increases stability of the orientation slightly if themutational correlation is low, but not otherwise. Thus, mu-tational correlation may be an important agent in stabilizingthe orientation of a G-matrix; it does not, however, stabilizeits size or shape.

To investigate more fully the role of the pattern of pleio-tropic mutations in determining the shape and orientation ofthe G-matrix, we also performed simulations in which themutational variances of the traits differed, but without anyselectional or mutational correlation. The results presentedin Table 4 clearly demonstrate that for symmetric selectionon the two traits, increasingly different mutational variancesincrease the eccentricity of the G-matrix and enhance sta-bility of the orientation. Clearly, the total size S of the G-matrix is strongly correlated with the total mutational vari-ance. Unequal mutational variances do not at all affect thestability in size or shape. The same is true for asymmetricstabilizing selection. However, instability of the orientationis maximized under asymmetric stabilizing selection if themutational variance of the trait under weaker selection issomewhat, but not much, higher than that of the other trait.

The Effect of Population Size on G-matrix Stability

Almost all the variation in the size of the G-matrix, asmeasured by the genetic variances and the total variance, isdue to random genetic drift. This is clearly demonstrated byTable 2. Also, the between-generation change of the eigen-values and all other measures of instability decrease withincreasing population size. Increasing population size has aslight increasing effect on the shape parameter e unless mu-tational and selectional correlation both are strong, but almostno effect on the orientation w. The between-generationchange De decreases with increasing population size. In-creasing population size has only a weak effect on the sta-bility in orientation, Dw, and increases stability most if cor-relational selection is strong or if there is substantial muta-tional correlation.

Mutation Rate Considerations

The per-locus mutation rate that we employed for the ma-jority of our analyses may be unrealistically high for singleloci affecting quantitative traits (Kondrashov 2003). Justifi-cation for our use of this high mutation rate comes from twosources. First, if several physically linked loci affect the sametrait, then they can behave as a single locus with a highermutation rate (Burger 2000). If this phenomenon is common,it would justify the use of a mutation rate considerably higherthan the empirically estimated single-locus mutation rate.Second, this mutation rate is necessary to maintain geneticvariation in the small populations under consideration in thisstudy, given the assumption that each trait is determined by50 loci. Small population size was a necessary constraint inthis study, because the Monte Carlo simulations become veryslow as population size increases.

To test the validity of our conclusions for loci with smallermutation rates, we performed some simulations with mutationrates of 1 3 1024 and 2 3 1025 per locus per generation.The results of these simulations are presented in Table 5.

1754 ADAM G. JONES ET AL.

TA

BL

E4.

The

infl

uenc

eof

mut

atio

nal

vari

ance

onG

-mat

rix

stab

ilit

ydu

ring

stab

iliz

ing

sele

ctio

n.T

hefo

llow

ing

para

met

ers

are

fixe

d:N

e5

342,

r v5

r m5

0,5

0.05

.2

a1

See

the

text

and

Tab

le1

for

mor

ein

form

atio

nab

out

othe

rpa

ram

eter

valu

es,

met

hods

,an

dth

em

eani

ngof

sym

bols

.In

noca

sedo

esth

ean

gle

mea

sure

,w

,di

ffer

sign

ifica

ntly

from

zero

,w

hich

isto

beex

pect

edbe

caus

ew

itho

utm

utat

iona

lor

sele

ctio

nal

corr

elat

ion

nom

echa

nism

ispr

esen

tto

prod

uce

ano

nzer

oan

gle.

Not

eth

atth

efo

urth

colu

mn

show

sth

eex

pect

edge

neti

cva

rian

cefo

rth

ese

cond

trai

tac

cord

ing

toth

eS

HC

appr

oxim

atio

n(s

eete

xt).

v11

v22

a2 2

G22

G11

G22

G12

r gl

1l

2S

ew

DG

11D

G22

DG

12D

l1

Dl

2D

SD

eD

w

9 9 9 9 9 9

9 9 9 9 9 9

0.01

0.02

0.03

0.05

0.10

0.20

0.10

0.16

0.20

0.25

0.31

0.35

0.22

30.

213

0.21

10.

188

0.16

40.

139

0.07

20.

112

0.14

60.

195

0.23

70.

286

0.00

00.

001

0.00

00.

003

0.00

42

0.00

3

20.

005

0.00

80.

006

0.01

20.

022

20.

019

0.23

0.22

0.23

0.24

0.26

0.30

0.07

0.01

0.13

0.15

0.14

0.13

0.30

0.33

0.36

0.38

0.40

0.43

0.32

0.49

0.59

0.65

0.57

0.45

20.

11.

30.

42.

05.

92

7.0

0.07

30.

074

0.07

40.

075

0.07

50.

077

0.07

30.

073

0.07

30.

075

0.07

70.

081

0.00

60.

008

0.00

90.

010

0.01

00.

011

0.04

60.

054

0.05

70.

060

0.06

10.

055

0.16

10.

116

0.10

20.

096

0.10

20.

137

0.06

0.06

0.06

0.06

0.06

0.06

0.10

0.10

0.10

0.10

0.10

0.10

2.59

5.39

7.84

9.35

7.66

4.83

49 49 49 49 49 49

49 49 49 49 49 49

0.01

0.02

0.03

0.05

0.10

0.20

0.13

0.24

0.34

0.51

0.81

1.16

0.48

40.

479

0.42

70.

439

0.42

80.

370

0.12

00.

207

0.27

90.

453

0.69

80.

995

0.00

22

0.00

32

0.00

22

0.01

90.

013

0.00

6

0.00

32

0.02

02

0.00

22

0.04

50.

025

0.00

7

0.49

0.49

0.46

0.55

0.74

1.01

0.11

0.10

0.25

0.35

0.39

0.35

0.60

0.69

0.71

0.89

1.13

1.36

0.25

0.41

0.57

0.63

0.54

0.36

20.

02

1.8

1.4

210

.0 5.0

4.2

0.07

20.

072

0.07

20.

072

0.07

20.

073

0.07

10.

072

0.07

30.

072

0.07

30.

074

0.01

20.

016

0.01

70.

023

0.02

80.

031

0.04

50.

049

0.05

40.

055

0.05

50.

051

0.18

40.

129

0.10

20.

096

0.10

30.

139

0.06

0.06

0.05

0.05

0.05

0.06

0.10

0.10

0.10

0.10

0.10

0.10

1.93

3.75

7.16

9.90

6.59

3.11

49 49 49 49 49 49

9 9 9 9 9 9

0.01

0.02

0.03

0.05

0.10

0.20

0.10

0.16

0.20

0.25

0.31

0.35

0.44

90.

440

0.39

50.

342

0.31

30.

255

0.09

30.

143

0.18

30.

227

0.28

60.

356

0.00

32

0.00

72

0.00

20.

012

0.00

62

0.00

8

0.01

22

0.03

32

0.00

40.

043

0.01

92

0.02

7

0.45

0.45

0.41

0.37

0.37

0.39

0.09

0.14

0.17

0.20

0.23

0.21

0.54

0.58

0.58

0.57

0.60

0.61

0.21

0.33

0.45

0.58

0.63

0.57

0.2

21.

92

0.2

5.1

4.5

26.

4

0.07

20.

072

0.07

20.

072

0.07

30.

074

0.07

20.

073

0.07

30.

074

0.07

60.

078

0.01

00.

013

0.01

40.

014

0.01

50.

016

0.04

60.

044

0.05

20.

059

0.06

10.

057

0.20

20.

165

0.12

10.

096

0.09

20.

109

0.06

0.06

0.06

0.05

0.05

0.06

0.10

0.10

0.10

0.10

0.10

0.10

1.72

2.66

4.21

7.13

9.47

7.59

9 9 9 9 9 9

49 49 49 49 49 49

0.01

0.02

0.03

0.05

0.10

0.20

0.13

0.24

0.34

0.51

0.81

1.16

0.24

50.

231

0.24

40.

227

0.21

40.

202

0.08

70.

165

0.24

00.

342

0.58

60.

818

0.00

10.

003

20.

009

0.01

22

0.00

62

0.01

7

0.00

50.

009

20.

031

0.04

32

0.01

72

0.03

4

0.25

0.25

0.30

0.37

0.60

0.83

0.08

0.14

0.19

0.20

0.20

0.19

0.33

0.40

0.48

0.57

0.80

1.02

0.36

0.58

0.65

0.58

0.36

0.24

20.

11.

32

8.6

5.1

28.

22

10.5

0.07

30.

073

0.07

40.

074

0.07

40.

074

0.07

20.

072

0.07

20.

072

0.07

30.

075

0.00

70.

010

0.01

20.

014

0.01

80.

021

0.04

90.

057

0.05

60.

059

0.04

60.

042

0.14

20.

097

0.09

60.

096

0.15

40.

215

0.06

0.05

0.05

0.05

0.06

0.06

0.10

0.10

0.10

0.10

0.10

0.10

2.97

7.58

9.79

7.13

3.16

2.01

1755EVOLUTION OF THE G-MATRIX

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mr v

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11G

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l1

l2

Se

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G11

DG

22D

G12

Dl

1D

l2

DS

De

Dw

23

102

4

23

102

4

23

102

4

23

102

4

23

102

4

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.85

30.

591

0.79

60.

746

0.80

6

0.84

00.

587

0.79

00.

740

0.81

3

20.

004

0.21

10.

292

0.44

70.

722

20.

003

0.35

20.

366

0.59

70.

891

0.95

0.81

1.10

1.19

1.53

0.75

0.37

0.49

0.29

0.09

1.69

1.18

1.59

1.49

1.62

0.79

0.47

0.46

0.25

0.06

0.6

45.1

44.4

44.8

45.2

0.03

60.

036

0.03

60.

036

0.03

6

0.03

60.

036

0.03

60.

036

0.03

6

0.02

10.

016

0.02

10.

022

0.02

8

0.03

10.

036

0.03

60.

036

0.03

6

0.04

00.

036

0.03

60.

036

0.03

6

0.03

0.03

0.03

0.03

0.03

0.05

0.05

0.05

0.05

0.05

9.71

2.02

1.88

0.99

0.37

13

102

4

13

102

4

13

102

4

13

102

4

13

102

4

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.43

80.

285

0.40

80.

401

0.44

3

0.40

90.

302

0.39

80.

399

0.45

7

0.00

60.

097

0.13

80.

246

0.40

4

0.01

40.

327

0.33

70.

613

0.89

8

0.48

0.40

0.55

0.65

0.86

0.36

0.19

0.26

0.15

0.04

0.85

0.59

0.81

0.80

0.90

0.76

0.49

0.49

0.24

0.05

3.3

46.9

43.0

44.9

45.5

0.03

60.

037

0.03

60.

036

0.03

6

0.03

60.

036

0.03

60.

036

0.03

6

0.01

10.

008

0.01

10.

012

0.01

5

0.03

50.

036

0.03

60.

036

0.03

6

0.03

60.

037

0.03

60.

037

0.03

6

0.03

0.03

0.03

0.03

0.03

0.05

0.05

0.05

0.05

0.05

8.18

2.19

2.48

0.94

0.35

23

102

5

23

102

5

23

102

5

23

102

5

23

102

5

0 0.75

0 0.75

0.90

0 0 0.50

0.50

0.90

0.07

70.

050

0.08

90.

065

0.09

3

0.08

10.

060

0.08

60.

067

0.09

6

20.

001

0.01

90.

032

0.04

00.

084

0.02

30.

344

0.37

10.

595

0.87

6

0.11

0.08

0.13

0.11

0.18

0.05

0.03

0.05

0.02

0.01

0.16

0.11

0.17

0.13

0.19

0.50

0.38

0.39

0.25

0.07

21.

440

.737

.345

.445

.9

0.04

00.

042

0.03

90.

041

0.03

9

0.03

90.

040

0.03

90.

040

0.03

9

0.00

20.

002

0.00

20.

002

0.00

3

0.03

90.

040

0.03

80.

040

0.03

9

0.04

20.

046

0.03

90.

042

0.03

9

0.03

0.03

0.03

0.03

0.04

0.06

0.06

0.05

0.06

0.05

3.54

2.09

2.06

1.25

0.42

The smaller mutation rates produce results that are very sim-ilar to the results with higher mutation rates. The major dif-ference is that the amount of standing genetic variance issubstantially reduced, resulting in a decrease in S. Eventhough these low mutation rates eliminate most of the geneticvariance in the population, resulting in trait heritabilities lessthan 0.10 in the most extreme case, the patterns of changein the G-matrix remain remarkably similar to the simulationswith much greater genetic variance. Thus, our main conclu-sions appear to be robust to large changes in the per locusmutation rate.

The Influence of Alignment on G-matrix Evolution

Another issue concerns the interplay of selectional andmutational correlations. It is to be expected that alignmentbetween the selection matrix, determining the shape of theadaptive surface, and the matrix of mutational effects en-hances stability. This is indeed borne out by our numericalresults. Figure 1 displays the change in orientation of the G-matrix as a function of the selectional correlation rv for fivedifferent values of rm. This figure clearly shows that for pos-itive mutational correlation, the between-generation changeof the angle, Dw, always decreases with increasing selectionalcorrelation unless the mutational correlation is very weak.Put otherwise, mutational and selectional correlations of dif-ferent signs lead to instability of the orientation. Withoutmutational correlation, any increase in selection correlation,positive or negative, clearly increases stability of the ori-entation. Figure 1 also demonstrates that mutational corre-lation is much more important in producing stability thancorrelational selection.

Visualizing the Evolution of the G-matrix

Figure 3 depicts the change of the size, shape, and ori-entation of the G-matrix within one replicate run under eachof five different scenarios by displaying snapshots of graph-ical representations of the G-matrix at intervals of 200 gen-erations. Whereas the G-matrix apparently is fluctuating andflipping around randomly if rm 5 rv 5 0, with no visibledifference in the case of rv 5 0.75, fluctuations of the shapeand orientation (but not size) are markedly restricted in thetwo cases where rm 5 0.5. Obviously, increasing selectionalcorrelation substantially aids their stability. The case of rm

5 rv 5 0.90 results in a highly stable G-matrix that is ex-tremely cigar-shaped. It still fluctuates in size, however.

Figure 2 displays the complete time series of the angle wbetween the leading eigenvector and the x-axis for the fiveruns shown in Figure 3. Because the angle is confined to theinterval 2908 to 1908, and, for example, 958 corresponds to2858, flipping from near 2908 to near 1908 does not implya great change in the G-matrix. It is interesting to observethe high degree of autocorrelation for the runs without mu-tational or selectional correlation. Often, the orientation staysaround the same value for hundreds of generations beforechanging substantially. Figure 2b also shows distinctivelythat with strong selectional correlation but not mutationalcorrelation, there are prolonged periods during which theorientation remains quite stable (e.g., between about gener-ations 1500 to 1700), whereas during other periods (between

1756 ADAM G. JONES ET AL.

FIG. 3. Time dependence of size, shape, and orientation of the G-matrix, displayed as ellipses with axis length proportional to thecorresponding eigenvalues. These are snapshots, taken every 200th generation, each from a single replicate run. From top to bottom,the following selectional and mutational correlations are chosen: (a) rm 5 rv 5 0; (b) rm 5 0, rv 5 0.75; (c) rm 5 0.5, rv 5 0; (d) rm 50.5, rv 5 0.75; (e) rm 5 rv 5 0.90. The following parameters are the same for all panels: Ne 5 342, v11 5 v22 5 49, 5 5 0.05.2 2a a1 2

generations 1000 and 1300) the orientation is flipping aroundwildly. Figures 2d and 2e demonstrate to what extent mod-erate to high selectional and mutational correlations can sta-bilize the orientation of the G-matrix provided they are ofthe same sign and of similar magnitude. Despite this amazingstability in orientation, there is still substantial variation inthe size (S) of these G-matrices (Fig. 4). Figure 5 shows thetime series for the eccentricity (e) of the G-matrix from thesame runs as those shown in Figures 2, 3, and 4. Even thoughthe change in eccentricity, when standardized to the mean,does not change as a consequence of mutational or selectionalcorrelation, the magnitude of e becomes so small as corre-lations stabilize the angle w that the G-matrix retains itsoverall shape for extremely long spans of time.

The Reduction of Genetic Variance Due to Pleiotropy

Stabilizing selection on multiple traits determined bypleiotropic loci is expected to reduce the genetic variance ofthese traits relative to what would be expected if each traitwere determined by a completely independent set of loci.Indeed, our simulations verify that pleiotropy does decreasethe equilibrium level of additive genetic variance, as pre-dicted by models developed by Turelli (1985) and Wagner(1989). This reduction of variance due to pleiotropy is ob-vious when we consider the SHC-approximation for the ex-pected genetic variance of a single trait. In all cases that weinvestigated (see Tables 1–5), the expected genetic varianceaccording to the SHC model was higher than the amount of

genetic variance that we observed in our simulated popula-tions.

DISCUSSION

Our simulations show that the G-matrix is stable undersome conditions and unstable under others. This result helpsdefine a central issue in evolutionary genetics. Following onthe heels of Lande’s (1979, 1980) pioneering papers, thiscentral issue was cast in terms of G-matrix constancy.Lande’s models assumed G-matrix constancy. Is that prop-osition literally true? Because the proposition of literal con-stancy is easily refuted from first principles (Turelli 1988;Shaw et al. 1995) and, sometimes, on empirical grounds (Ar-nold and Phillips 1999), some workers have considered theissue settled. Clearly the G-matrix is not constant. This char-acterization, however, trivializes the constancy issue and soachieves closure prematurely. The primary claim of this ar-ticle is that the constancy issue has deeper ramifications andcan be approached by asking three questions. First, underwhat conditions is the G-matrix stable and under what con-ditions is it unstable? Second, how common are conditionspromoting stability or instability in nature? And third, howmuch and what kind of stability is required to make mean-ingful extrapolations of equations for drift or response toselection? This article focuses on the first question. Previousstudies correctly identified several factors that contribute tostability but were unable to assess their relative importance(Turelli 1988). We found that stable orientation of the G-

1757EVOLUTION OF THE G-MATRIX

FIG. 4. Time series of the size of the G-matrix, S, the sum of thegenetic variances, which is equal to the sum of the eigenvalues. Asin Figure 3, from top to bottom, the following selectional and mu-tational correlations are chosen: (a) rm 5 rv 5 0; (b) rm 5 0, rv 50.75; (c) rm 5 0.5, rv 5 0; (d) rm 5 0.5, rv 5 0.75; (e) rm 5 rv 50.90. This figure was produced using the same parameter values asthose used for Figure 3.

FIG. 5. Time series of the eccentricity of the G-matrix, e, thesmaller eigenvalue divided by the larger eigenvalue. As in Figure3, from top to bottom, the following selectional and mutationalcorrelations are chosen: (a) rm 5 rv 5 0; (b) rm 5 0, rv 5 0.75; (c)rm 5 0.5, rv 5 0; (d) rm 5 0.5, rv 5 0.75; (e) rm 5 rv 5 0.90. Thisfigure was produced using the same parameter values as those usedfor Figure 3.

matrix is promoted by correlational selection, large popula-tion size, and especially by pleiotropic mutation. The highestlevels of instability prevailed in small populations with nocorrelational selection and no mutational correlation.

The recognition of different kinds of G-matrix stability isan important contribution of the present study. By charac-terizing the G-matrix in terms of its eigenvalues and eigen-vectors, we can recognize three varieties of stability that arerelated to the size, shape, and orientation of the matrix. Theimportance of recognizing these three kinds of stability isemphasized by our finding that they respond differently tovarious conditions. Thus, size and shape stability are pro-moted chiefly by large population size. In contrast, orienta-tion stability is promoted by pleiotropic mutation, correla-tional selection, and the alignment of these two processes (inaddition to large population size). This result of differentialsensitivity to conditions supports the view that comparativestudies of the G-matrix should focus on both eigenvalues andeigenvectors (Phillips and Arnold 1999). This type of focusis necessary, because a complete description of the G-matrixrequires knowledge of the eigenvectors as well as the eigen-values. In addition, this approach is appealing because ei-

genvalues and eigenvectors provide an intuitive descriptionof the G-matrix that lends itself to graphical depiction.

Turning to our second question—the prevalence of stabil-ity-promoting conditions in nature—it seems likely that somekinds of characters are more likely than others to have stableG-matrices. Thus, bilaterally symmetrical characters mayhave highly stable G-matrices because of strong correlationsbetween mutational effects, as well as strong correlationalselection between right and left sides. At the opposite ex-treme, we may expect fitness components to have unstableG-matrices because their adaptive landscapes lack curvatureas a consequence of strong, persistent directional selection.Hence, the G-matrix for these types of traits will not expe-rience the restraining effects of stabilizing selection. In ad-dition, fitness components are probably determined by a verylarge number of loci, only some of which may be expectedto pleiotropically affect multiple fitness components. Mostcharacters probably lie between these extremes of stabilityand instability. Although our results provide some expecta-tions about stability, the characterization of G-matrix vari-ability for different kinds of characters is fundamentally anempirical exercise.

Our third question—how much stability is necessary for

1758 ADAM G. JONES ET AL.

meaningful extrapolation—is only tangentially addressedby the results presented here. In cases of extreme stability(e.g., Fig. 3e), there can be little doubt that response toselection equations could be accurately extrapolated overmany generations. At the opposite extreme, when the G-matrix is prone to erratic fluctuation, extrapolations basedon the supposition of constancy are bound to be misleading.The parameter space for much of the natural world may liebetween these two extremes. One interesting pattern in ourresults is that the G-matrix tended to be unstable when itwas less cigar-shaped, whereas the more cigar-shaped G-matrices tended to be very stable. Because the more cigar-shaped G-matrices are the ones that constrain the evolu-tionary trajectory, the observed stability of these G-matricesmay justify extrapolations over many generations. Similar-ly, even though the less cigar-shaped G-matrices were un-stable, they constrain the evolutionary trajectory very littlein any case, so their instability is less consequential. How-ever, to definitively address this issue we need not onlyestimates of G-matrix stability, but also analyses of howmuch instability can be tolerated in extrapolations based onmatrix averages. Furthermore, Turelli (1988) has pointedout that calculation of the net selection gradient may becomplicated not only by fluctuation in G but by covariancesinduced by those fluctuations. One other important consid-eration is that the conditions for stability under stabilizingselection alone may differ in important ways from the con-ditions for stability when directional selection is operating.Simulation-based models may provide an expedient way toevaluate these issues.

Some of the present results were anticipated by simulationstudies of between-generation change in the genetic varianceof a single trait. High autocorrelation in the mean and geneticvariance were noted in studies of stabilizing and directionalselection on a single trait (Keightley and Hill 1989; Burgeret al. 1989; Burger and Lande 1994). Strong autocorrelationis a conspicuous feature of G-matrix dynamics. In simula-tions, a particular orientation of the G-matrix can be rela-tively stable for hundreds of generations, but can then changeto a new stable orientation or enter a period of chaotic fluc-tuation. The complex time series for G-matrix orientationillustrated in Figure 2 underscores the difficulty of achievinganalytical results.

The present results extend our understanding of G-matrixevolution based on past studies by computer simulation.Reeve’s (2000) main focus was the long-term accuracy ofLande’s (1979) equation for the evolution of the multivar-iate mean. He addressed the issues of G evolution and sta-bility only in passing. Although Reeve (2000) explored threedifferent models for the distribution of allelic effects(Gaussian, leptokurtic and biallelic), he did not allow cor-relational selection and did not report effects on geneticcovariance. Hence, Reeve’s (2000) analysis did not explic-itly address G-matrix stability at all, except to note thatsmall changes in the G-matrix had little effect on the ac-curacy of predictions of the response to selection in sim-ulated populations.

Our simulations do not explore some potentially im-portant contributions to G-matrix stability or instability.Most notably, we have not addressed the possible effects

on G-matrix stability of dominance, epistasis, linkage,number of loci, and distributions of allelic effects (in-cluding genes of major effect). Likewise, we have onlyexplored cases in which the position and shape of the adap-tive landscape and the pattern of mutation are stable. It islikely that some aspects of the dynamics of the G-matrixwould change if the adaptive landscape were changing,either with respect to the location of its optima, its ori-entation, or its curvature. These issues are beyond thescope of this initial study, and additional simulation stud-ies are needed to address them.

Our results help to inform the comparative study of G-matrices. Under some conditions the G-matrix is capableof pronounced, random fluctuation. Under those circum-stances, multiple independent samples are needed to makea convincing case for G-matrix stability. Thus, neutral orweakly selected characters with little or no genetic cor-relation in small populations will be especially prone toG-matrix fluctuation. Such characters may be poor choicesfor comparative work, because even considerable effortcould yield inconclusive results. In contrast, characters un-der strong stabilizing and correlational selection with pro-nounced genetic correlations in large populations are apriori likely to have stable G-matrices. Under these con-ditions, a modest number of samples would be required tomake a convincing case for matrix identity or proportion-ality. Comparative studies should examine multiple as-pects of stability because different aspects may be differ-entially stable. Thus, eigenvalues (size and eccentricity)can show different stability profiles than eigenvectors (ori-entation). Finally, systematic sampling of matrices on aphylogeny may provide especially revealing data on sta-bility. Under conditions for which the G-matrix is likelyto be stable, such phylogenetically planned sampling couldreveal a pattern of long-term stability in the matrix. Thisapproach may also be the most practical way to obtainconvincing evidence for long-term stability in the patternsof stabilizing selection and mutation.

Several unresolved empirical issues are highlighted by ourresults. Most notably, we need estimates of pleiotropic mu-tational covariance. Although estimates of mutational vari-ance are available for a number of characters (Lynch 1988;Lynch and Walsh 1998), mutational covariances have seldombeen estimated (Camara and Pigliucci 1999). Consequently,we have some basis for choosing parameters for the maindiagonal of the mutational matrix but virtually no empiricalbasis for specification of the off-diagonal elements. Our un-derstanding of multivariate selection is plagued by a similarlimitation. Estimates of stabilizing selection are seldom cor-rected for character correlation (Lande and Arnold 1983; En-dler 1986; Kingsolver et al. 2001), and corrected estimatesof correlational selection are even less common. Given theselimitations, it is not surprising that key issues of alignmentare an open field of investigation. Are the matrices describingselection and mutation commonly aligned? Our simulationsindicate that such alignment would enhance G-matrix sta-bility. Finally, is the adaptive landscape stable on a geologicaltime scale, producing alignment of mutation and inheritancewith selection? These are the overarching empirical issues

1759EVOLUTION OF THE G-MATRIX

that will determine how widely quantitative genetic theorycan be applied to the natural world.

ACKNOWLEDGMENTS

We are grateful to M. E. Pfrender for comments on themanuscript and to P. C. Phillips for helpful discussion. Thisresearch was partially supported by grants from the NationalInstitutes of Health (AGJ) and the National Science Foun-dation (SJA).

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Corresponding Editor: D. Houle