Download The evolution of the G matrix: selection or drift?

Heredity 84 (2000) 135±142
Received 11 October 1999, accepted 17 December 1999
Short Review
The evolution of the G matrix: selection or drift?
DEREK ROFF
Department of Biology, McGill University, 1205 Dr Pen®eld Ave., Montreal, Quebec, Canada H3A 1B1
The evolution of quantitative characters can be described by
the equation Dz ˆ GPÿ1 S where Dz is the vector of mean
responses, G is the matrix of additive genetic variances and
covariances, P is the matrix of phenotypic variances and
covariances and S is the vector of selection di€erentials. This
equation can be used to predict changes in trait values or to
retrospectively estimate the selection gradient and is thus
a central equation of evolutionary quantitative genetics.
Genetic variances and covariances will be reduced by
stabilizing selection, directional selection and drift, and
increased by mutation. Changes in trait values resulting from
directional selection that are comparable with di€erences
observed among species are readily obtainable in short
geological time spans (<5000 generations) with selection
intensities so small that they would have an insigni®cant
e€ect on the G matrix (of course it is possible that such
changes came about by strong selection over a few genera-
tions, followed by long periods of stasis; there is insucient
evidence to presently distinguish these two possibilities). On
the other hand, observed e€ective population sizes are
suciently small that considerable changes in G can be
expected from drift alone. The action of drift can be
distinguished from selection because the former produces a
proportional change in G whereas the latter, in general, will
not. A survey of studies examining variation in G suggests
that the null hypothesis that most of the variation can be
attributed to drift rather than selection cannot be rejected.
However, more research on the predicted statistical distribution of G as a result of selection and/or drift is required and
further development of statistical tests to distinguish these
two forces needs to be made.
What is the G matrix?
Of what use is G?
Quantitative genetics has long made signi®cant contributions
to the development of domesticated plant and animal species.
In the last two decades there has been an increasing interest
in the use of quantitative genetics to understand the evolution
of quantitative traits in natural populations. For domestic
organisms the central equation of quantitative genetics is
the relationship, R ˆ h2S (response ˆ heritability ´ selection
di€erential), or less frequently the two-trait formulation
PY
CRY ˆ rA hX hY rrPX
SX , where CRY is the correlated response of
trait Y to direct selection on trait X, rA is the genetic
correlation between X and Y, and rPi is the phenotypic
standard deviation of trait i. However, natural selection does
not typically act upon single or even pairs of traits, but rather
the whole organism and hence it is more appropriate to
consider the multivariate extension of the above equations
(Lande, 1979a), Dz ˆ GPÿ1 S where Dz is the vector of mean
responses (conventionally symbolized by z), G is the matrix of
genetic variances and covariances, P is the matrix of phenotypic variances and covariances, and S is the vector of selection
di€erentials. The matrix combination GP±1 can thus be viewed
as the multivariate `equivalent' of h2, with G replacing the
additive genetic variance in the numerator and P±1 replacing
the phenotypic variance in the denominator.
The multivariate response equation has been used theoretically
to explore evolutionary trajectories with particular reference to
constraints on the direction of evolutionary change (e.g. Via &
Lande, 1985; Arnold, 1992; Bjorklund, 1996; Schluter, 1996).
With respect to the analysis of natural populations, the response
equation has been used to retrospectively reconstruct the
selection regime that has given rise to a change over time or
among populations (Arnold, 1992), or to predict responses to
selection (Morris, 1971; Grant & Grant, 1993; Ro€ & Fairbairn,
1999). Retrospective selection analysis assumes that both G and
P remain constant and that there are no dramatic reversals in
the direction of selection (Shaw et al., 1995). Depending upon
the nature of the analysis, prediction of response requires
either that both G and P remain constant or change in a
proportional manner relative to the original matrices.
There are two defences for the assumption of constancy. The
®rst is the same as applied to arti®cial selection experiments,
namely that although selection will change allelic frequencies
and hence genetic variance and covariances (and thus also
their phenotypic counterparts), the change will be so small per
generation that the equations will remain correct at least for a
dozen or so generations (see Ro€, 1997 for a review providing
support for this assumption). This argument clearly cannot be
applied to those cases in which the equations are used to
predict changes taking place over hundreds or thousands of
generations. For this the assumption is made that directional
Keywords: G matrix, genetic covariance, genetic drift,
heritability, mutation, selection
*Correspondence: E-mail: dro€@bio1.lan.mcgill.ca
Ó 2000 The Genetical Society of Great Britain.
135
136
D. ROFF
or stabilizing selection is suciently weak that any erosion of
variance is countered by new variation added by mutation
(Lande, 1976a).
How weak is weak selection?
We can examine this question by asking what selection is
necessary to produce a change in mean trait value that is
characteristic of some given taxonomic level. A survey of tests
of variation in G matrices at di€erent taxonomic levels
suggests that statistically signi®cant di€erences are found at
least at the level of species (Ro€ & Mousseau, 1999). Most
studies have focused upon morphological traits and thus
we must ask how much variation is characteristically found
among species within any given genus. An analysis of variation
among species within 32 di€erent genera covering a wide range
of animals suggests that a typical range is a two- to four-fold
di€erence in linear dimensions, though di€erences larger than
tenfold also occur (Fig. 1). For illustration, suppose that
h2 ˆ 0.5 (a typical heritability for morphological traits),
VA ˆ 25 and the initial trait value is 50. When the top 99.9%
of the population is selected each generation then after 5000
generations the trait value will be 50 + (5000)(0.00337)
(Ö[0.5][25]) ˆ 110, a two-fold increase, while if the top 99% is
selected the trait value will be 50 + (5000)(0.02692)(Ö[0.5][25])
ˆ 526, a 10-fold increase. Both of these changes are within the
range of interspeci®c variation in morphological traits and the
former is probably most typical (see above). On a geological
time-scale 5000 generations is short and changes of the above
magnitude would be considered rapid. A cull of 0.1% of the
population per generation is extremely weak (certainly below
present levels of detection for most studies). Lande (1976b)
estimated the required cull rate per generation to account for
change in the tooth dimensions during the evolution of the
horse: to obtain the observed changes requires selective deaths
of the order of 2 per million individuals, which is considerably
weaker than the above rates (99.9998% survive), but the timespan is in the millions rather than thousands of generations. Of
course, it is unlikely that selection acts at a constant, minute
rate each generation but the important point is that even very
low rates of selection can produce very large phenotypic
changes in what is no more than a geological `instant'. Thus
the hypothesis of weak selection is at least tenable, but requires
empirical support (Turelli, 1988).
Endler (1986; p. 210) surveyed a large number of studies in
which estimates of selection intensities in natural populations
were made and found that `natural selection is often as strong
as arti®cial selection'. This result does not, however, demonstrate that the assumption of weak selection is invalid. First, it
remains to be shown if these are typical or simply those that
Fig. 1 Distribution of the ratio R2
among 32 di€erent genera (lower panel)
with two examples of the distribution of
sizes within genera (upper panels). Data
sources: birds (Dunning, 1984: note that
weight was the trait given, and to make
it comparable to a length measurement I
used the cube root); mammals (Barbour
& Davis, 1969; Schober & Grimmberger,
1987; Kingdon, 1997); reptiles (Conant,
1975); amphibia (Barker et al., 1995);
®sh (Scott & Crossman, 1973); beetles
(Lindroth, 1966); leafhoppers (Beirne,
1956); grasshoppers (Otte, 1984); spiders
(Locket & Millidge, 1953); cone shells
(Wagner & Abbott, 1977).
Ó The Genetical Society of Great Britain, Heredity, 84, 135±142.
THE EVOLUTION OF THE G MATRIX
137
have been selected a priori because the researchers suspected
that the traits in question would be under strong selection.
There is little value in selecting traits for which selection
intensities cannot be measured although these may represent
the majority of traits. Secondly, single or short-term episodes
of strong selection will not greatly a€ect the additive genetic
variance (Ro€, 1997). What we need to know is the frequency
with which episodes of strong selection occur.
additive variance, r2A , after one generation to (Keightley &
Hill, 1987)
How does selection and/or drift alter G?
where h2t is the heritability at time t, which changes because r2A
changes, k ˆ i(i ) x), where i is the intensity of selection and
x is the standardized deviation of the truncation point of
selection and r2g;t is the genic variance in the population, which
is the genetic variance that would be present under linkage and
Hardy±Weinberg equilibria. Assuming this to be true in the
initial, unselected population we have r2A;0 ˆ r2g;0 . The genetic
variance introduced by mutation is r2M . At equilibrium the
additive genetic variance maintained by selection, drift and
mutation, r2A , is obtained from the solution to the quadratic
When genetic correlations arise principally from pleiotropy,
populations are large and mating is random, the G matrix is
determined by the joint e€ects of pleiotropic mutation and
multivariate selection (Lande, 1980). Changes in the selective
regime will produce changes in G but there is no a priori
general prediction that can be made (or has so far been
published). If selection is strong enough to signi®cantly alter
the gene frequencies of the traits subject to selection there will
be a change in that portion of the G matrix in which the genes
have in¯uence, i.e. the additive genetic variance of the target
trait plus the additive genetic covariances between the target
trait and other nontarget traits. Thus, in general, the change in
the G matrix will show no overall general pattern, but will
depend upon the selection regime and the traits included in the
matrix. In contrast, genetic drift will cause G matrices to
diverge but remain proportional to each other (Lande, 1979a).
Population bottlenecks can convert nonadditive genetic
(co)variance into additive genetic (co)variance (Ro€, 1997;
chapter 8) but the general e€ect of genetic drift is to erode
additive genetic variance. This should be particularly true for
morphological traits in which non additive e€ects are typically
small (Mousseau & Ro€, 1987; Crnokrak & Ro€, 1995;
Derose & Ro€, 1999). Non-additive e€ects are present in life
history traits (Crnokrak & Ro€, 1995; DeRose & Ro€, 1999)
and hence changes in the G matrix due to short episodes of
genetic drift could be unpredictable.
To illustrate the potential importance of genetic drift when
there is only additive genetic variance, consider the change in
the additive genetic variance of a single trait as a result of drift
alone. The proportional decline in the additive genetic variance
after t generations is approximately e)t/2Ne (Robertson, 1960).
After 5000 generations the proportional decline for an e€ective
population size of 5000 will be 0.61, which is a substantial
reduction in a very short geological period. Thus even very
large populations can be subject to signi®cant erosion in
additive genetic variance due to genetic drift. A review of
estimated e€ective population sizes produced less than 20%
greater than 1000 and over 70% in the range 10±500 (see
table 8.3 and ®gs 8.2 in Ro€, 1997). The populations studied
do not represent a random sample (there are few invertebrates)
but do suggest that random genetic drift could play a very
important role in evolutionary changes in many populations
(see also Lande, 1979b).
The erosion of additive genetic (co)variance resulting from
the combined e€ects of selection (directional or stabilizing)
and drift will be compensated in part or whole by new
variation added by mutation. For the in®nitesimal model,
truncation selection, drift and mutation will change the
Ó The Genetical Society of Great Britain, Heredity, 84, 135±142.
r2A;t r2g;t
1
‡
‡ r2M
1ÿ
…1 ÿ h2t k†
2
2
Ne
1
ˆ r2g;tÿ1 1 ÿ
;
2Ne
r2A;t‡1 ˆ
r2g;t
r4A …1 ‡ k† ‡ r2A …r2E ÿ 2Ne r2M † ÿ 2Ne r2M r2E ˆ 0
…1†
…2†
which is
r2A ˆ
2Ne r2M ÿ r2E ‡
q
…2Ne r2M ÿ r2E †2 ‡ 8Ne r2M r2E …1 ‡ k†
2…1 ‡ k†
…3†
From the above it is evident that the equilibrium additive
genetic variance is approximately proportional to the e€ective
population size. For example, suppose the proportion culled
per generation is 0.1%, and the mutational variance is 0.002,
which is within the observed range (see tables 9.7 in Ro€,
1997), then for e€ective population sizes of 10, 100, 1000 the
equilibrium additive genetic variance is 0.04, 0.40 and 4.0,
respectively. Even if we make selection very strong by culling
50% of the population per generation, the respective additive
genetic variances are still approximately proportional to Ne
(0.04, 0.34, 2.73, respectively). BuÈrger (1993), assuming weak
selection, obtained the simple result that at equilibrium under
directional selection the additive genetic variance would be
equal to 2Ne r2M , which is the equilibrium variance under the
neutral mutation±drift balance as derived by Lynch & Hill
(1986).
What statistical methods are presently available
to test for variation among G matrices?
Given the G matrix of two or more populations consisting of
either the same or di€erent species we are in the ®rst instance
interested in two questions: ®rst, `can we reject the hypothesis
that the matrices came from the same statistical population?'.
This null hypothesis supposes that the populations do not
di€er in population size, selection history or mutational
variances. We might be inclined to entertain such a hypothesis
when comparing several populations of the same species but it
:
138
D. ROFF
seems an unlikely scenario when considering di€erent species.
The second question is, `can we reject the hypothesis that
di€erences among the G matrices can be accounted for on the
basis of drift alone?' This hypothesis does not preclude the
possibility that there are di€erences in selection history or
mutational variances among the populations, but postulates
that genetic drift will be relatively so large that it overwhelms
such e€ects. It is important to remember that proportionality
of the G matrices does not imply that the mean phenotypes
have not been drastically altered by selection. If equality and
proportionality can be rejected we accept the hypothesis that
di€erences are, at least in part, a consequence of variation in
selection history and/or mutational e€ects.
The statistical methods proposed to address the above
hypotheses can be classi®ed into four groups: element-byelement comparisons, correlation between matrices, maximum
likelihood, and the Flury hierarchical comparison. Most
analyses have involved only two populations and for simplicity
I shall assume that only two matrices are being compared and
that the elements of the matrices are `lined up' in two vectors.
Element-by-element comparison entails doing a separate test
on each element of the matrices (Brodie, 1993; Paulsen, 1996;
Ro€ & Mousseau, 1999; Ro€ et al., 1999). The simplest such
test is to check whether the di€erence between the elements is
greater than expected by chance, which may be done using a
bootstrap approach (Paulsen, 1996), randomization (Ro€
et al., 1999), or a t-test where the standard errors are derived
by using the jackknife (Brodie, 1993; Ro€, 1997). The problem
with the method is that the elements within a matrix are not
independent of each other and there are likely to be so many
tests that after correction for multiple tests there is too little
statistical power to be useful. However, while the method may
not serve as a means of testing for overall di€erences it is useful
as a `data-exploration' technique, serving to indicate where
large di€erences might lie.
The most commonly used test is to compare the correlation
between the matrices. To circumvent the problem of nonindependence, Lofsvold (1986) suggested the randomization test
known as Mantel's test. This test has been quite widely used
although its statistical validity has been questioned (Shaw,
1992). My objection to the method, apart from those enunciated by Shaw (1992) is that it cannot distinguish the
hypotheses presented above. A highly signi®cant correlation
between matrices could indicate equality (i.e. yi ˆ xi, where yi
and xi are the corresponding elements in the two `vectorized'
matrices), proportionality (i.e. yi ˆ bxi, where b ¹ 1), or the
joint e€ects of drift and selection/mutation (i.e. yi ˆ a + bxi,
where a ¹ 0). The three hypotheses can be distinguished by a
regression approach using the reduced major axis method and
randomization to estimate the required probabilities (Ro€
et al., 1999).
Maximum likelihood methods have become particularly
popular in quantitative genetic analyses in general; they have
been used to address the question of similarity between
matrices (e.g. Shaw, 1991; Holloway et al., 1993), but have
not been used to distinguish equality from proportionality vs.
other types of di€erences. A method that shows considerable
promise and is itself based on maximum likelihood is the Flury
approach which considers a hierarchy of possible di€erences.
This method looks at the matrices from the perspective of their
common principal components: starting from the `top' of the
hierarchy the matrices can be (a) identical, (b) proportional, in
which case the matrices share identical principal components
but their eigenvalues di€er by a proportional constant, (c) they
can have one or more, but not all, principal components in
common, or (d) they can have completely unrelated structures
(Phillips & Arnold, 1999; Fig. 2). The argument for analysing
the principle components is that evolutionary change will be
in¯uenced by the size of the eigenvalues corresponding to
the principal components (Bjorklund, 1996; Schluter, 1996).
Eigenvalues that are zero indicate directions in which there
is no genetic variation and hence in which there cannot be
evolutionary change. Very large eigenvalues indicate highly
correlated traits, and thus selection will be strongly in¯uenced
by the associated principal component.
Determining that two matrices di€er does not itself tell us
which process (drift vs. selection) is the more signi®cant. For
this purpose we require some method of partitioning the e€ect
that can be attributed to drift vs. that which can be attributed
to selection. The following approach based on mean-square
deviations is one possible method. First, we compute the
mean-square error for the three models
Model 1, Equal matrices MSE1 ˆ
1X
…xi ÿ yi †2
c
Model 2, Proportional matrices
1X
1X
MSE2 ˆ
…xi ÿ b0 yi †2 ‡
… yi ÿ B0 xi †2
…4†
2c
2c
Model 3, Different matrices
1X
1X
MSE3 ˆ
…xi ÿ ‰a ‡ byi Š†2 ‡
… yi ÿ ‰ A ‡ Bxi Š†2 ;
2c
2c
where c is the number of variances and covariances, b0 and B0
( ˆ 1/b0) are the slopes of the reduced major axis regression
forced through the origin, and a, b, A and B are the parameters
of the reduced major axis regression with the intercept
included. The MSE is calculated using both `x on y' and `y
on x', because these do not each give the same value. We can
now ask how much the MSE is reduced by the assumption of a
model that only assumes drift (model 2) vs. one that assumes
both drift and selection (model 3). To illustrate this I compare
the G matrices for two populations of garter snake studied by
Arnold & Phillips (1999). Using the Flury hierarchical method
the G matrices for females from a coastal population were
found to have common principal components but not to be
proportional (Arnold & Phillips, 1999). However, it is visually
evident that the elements of the matrices do not di€er much
from proportionality (Fig. 3). This is con®rmed by the
comparison of the percentage reduction: model 2 accounts
for a 17.1% reduction in the mean-square error whereas
model 3 accounts for a 18.3% reduction. Thus 17.1%
reduction can be attributed to drift while only 1.3% can be
attributed to selection. This does not mean that drift actually
caused the divergence, for it is possible that selection caused
a proportional change in the matrices. However, in general,
Ó The Genetical Society of Great Britain, Heredity, 84, 135±142.
THE EVOLUTION OF THE G MATRIX
139
Fig. 2 Schematic view of the Flury
hierarchy for two covariance matrices
consisting of two traits. The ellipses
represent the covariance structure with
the axis orientation denoting the principal components and the spread of the
ellipses along each axis the eigenvalues.
The analysis proceeds from the top to
the bottom.
selection will cause nonproportional changes. Thus, in the
absence of evidence to the contrary, the hypothesis that
the major force causing divergence was drift cannot be
rejected.
What have the analyses of G matrices
told us to date?
Almost all studies have compared either di€erent populations
of the same species or di€erent species within the same genus.
Most have not distinguished between the e€ects of drift vs. the
e€ects of selection. More importantly many used a correlational
test, which as discussed above, cannot distinguish between
equality, proportionality and a general linear relationship.
The majority of studies comparing populations have found
Ó The Genetical Society of Great Britain, Heredity, 84, 135±142.
either no signi®cant di€erences or signi®cant correlations,
whereas comparisons between species have most frequently
found di€erences (Fig. 4). Of course this does not mean that
there are no di€erences between populations, simply that the
di€erence is smaller than can typically be detected given the
sample sizes typically used.
Another way of approaching the problem is to ask how
much of the di€erence at the various taxonomic levels is
attributable to drift vs. selection. Where signi®cant variation
between groups has been found it can almost all be accounted
for by drift (Fig. 5). Interestingly, the points showing the
greatest `e€ect' of drift and selection (as indicated by the
deviation from the line of equality) are all nonsigni®cant
indicating that analyses should consider both the results of
hypothesis testing and interval estimation. Also included in
140
D. ROFF
Fig. 3 Plot of genetic variances and
covariances of two populations of garter
snake (data from Arnold & Phillips
(1999).
Fig. 5 are the results from a selection experiment on body size
in D. melanogaster. Comparison of the G matrices of the base
and control populations showed signi®cant di€erences (Shaw
et al., 1995), which, as shown in the plot, can be attributed
to drift, as would be expected. Comparisons of the control
population with those selected for large or small size suggests
that drift has caused even greater deviation of the selected lines
from the control. Further, there appears to have been no e€ect
of selection on G in the population selected for small size but
an e€ect in the population selected for large size (Fig. 5).
However, these conclusions are suspect because in neither case
did the G matrix of the control population di€er from that of
the selected population (Shaw et al., 1995).
Where do we go from here?
There is considerable scope for further theoretical analysis of
the evolution of the G matrix. In particular, what is the
relationship between selection and the deviation from proportionality expected under drift alone? There is also need for
more research into both the statistical properties of the
proposed tests and further development of methods for
estimating the di€erence between matrices. I have presented
one measure, the reduction in mean-squared deviation, which
suggests that most of the di€erentiation observed between
populations and species is due to drift rather than selection.
Drift is a consequence of di€erences in e€ective population
Fig. 4 A survey of analyses comparing
genetic covariance matrices (data from
table 7 of Ro€ & Mousseau, 1999; plus
the garter snake data from Arnold &
Phillips, 1999). NSD, Not Signi®cantly
Di€erent; SC, Signi®cantly Correlated;
SD, SC, Signi®cantly Di€erent and
Signi®cantly Correlated; SD, Signi®cantly Di€erent; NSC, Not Signi®cantly
Correlated.
Ó The Genetical Society of Great Britain, Heredity, 84, 135±142.
THE EVOLUTION OF THE G MATRIX
141
Fig. 5 A comparison of percentage
reduction in mean-square error attributable to Drift vs. that attributable to
Drift and Selection. The data were taken
from those studies in table 7 of Ro€ &
Mousseau (1999) in which the necessary
variances and covariances were presented, Arnold & Phillips (1999) and
Shaw et al. (1995). Sig; matrices were
found to be signi®cantly di€erent from
each other. NS; the null hypothesis of
equality could not be rejected.
size, which are likely to be a consequence of ecological factors:
thus we might expect that variation in G should correlate with
both taxonomic and ecological variables. Support for the latter
is provided by the study of the genetic correlations between
morphological characters in the amphipod Gammarus minus
(Jernigan et al., 1994).
The G matrix is central to the quantitative genetics of
evolutionary processes. Recent developments have shown
great promise in improving our ability to analyse variation
among matrices. What is presently lacking is an examination
of a group incorporating several taxonomic levels (e.g. species
and genus) and ecological variables. While such a research
programme would be very labour-intensive, the rewards
certainly merit the required expenditure.
Acknowledgements
This work was supported by a grant from the Natural
Sciences and Engineering Council of Canada. I am grateful
to Mathieu Begin for helpful comments on a previous draft
of this paper.
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