Quantitative genetics of growth and cryptic evolution of body size in an island population
ABSTRACT While evolution occurs when selection acts on a heritable trait, empirical studies of natural systems have frequently reported
phenotypic stasis under these conditions. We performed quantitative genetic analyses of weight and hindleg length in a freeliving
population of Soay sheep (Ovis aries) to test whether genetic constraints can explain previously reported stasis in body size despite evidence for strong positive
directional selection. Genetic, maternal and environmental covariance structures were estimated across ontogeny using random
regression animal models. Heritability increased with age for weight and hindleg length, though both measures of size were
highly heritable across ontogeny. Genetic correlations among ages were generally strong and uniformly positive, and the covariance
structures were also highly integrated across ontogeny. Consequently, we found no constraint to the evolution of larger size
itself. Rather we expect size at all ages to increase in response to positive selection acting at any age. Consistent with
expectation, predicted breeding values for agespecific size traits have increased over a twentyyear period, while maternal
performance for offspring size has declined. Reexamination of the phenotypic data confirmed that sheep are not getting larger,
but also showed that there are significant negative trends in size at all ages. The genetic evolution is therefore cryptic,
with the response to selection presumably being masked at the phenotypic level by a plastic response to changing environmental
conditions. Densitydependence, coupled with systematically increasing population size, may contribute to declining body size
but is insufficient to completely explain it. Our results demonstrate that an increased understanding of the genetic basis
of quantitative traits, and of how plasticity and microevolution can occur simultaneously, is necessary for developing predictive
models of phenotypic change in nature.
 [Show abstract] [Hide abstract]
ABSTRACT: Exaggerated sexual displays are often supposed to indicate the indirect benefits females may receive from sexual reproduction with displaying males, but empirical evidence for positive relationships between the genetic quality and sexual trait quality is scant. The explanation for this might lie in the fact that mixing of reproductive individuals whose development has been influenced by genotypebyenvironment interactions (GEIs) can blur the relationship between the individual male genetic quality and phenotype as perceived by females. Strong GEIs can generate an ecological crossover, where different genotypes are superior in environments that are separated either in space or time. Here, we use a stochastic simulation model to show that even a weak GEI, which does not generate an obvious ecological crossover, can neutralize or even reverse the relationship between genetic quality and sexual trait size in the presence of environmental heterogeneity during development. Our model highlights the importance of developmental selection in evolution of traits and allows us to predict the situations in which sexual displays might not be reliable indicators of genetic quality.Proceedings of the Royal Society B: Biological Sciences 01/2009; 276(1659):11539. · 5.68 Impact Factor  SourceAvailable from: Denis J.F. RéaleAlastair J Wilson, Denis Réale, Michelle N Clements, Michael M Morrissey, Erik Postma, Craig A Walling, Loeske E B Kruuk, Daniel H Nussey[Show abstract] [Hide abstract]
ABSTRACT: 1. Efforts to understand the links between evolutionary and ecological dynamics hinge on our ability to measure and understand how genes influence phenotypes, fitness and population dynamics. Quantitative genetics provides a range of theoretical and empirical tools with which to achieve this when the relatedness between individuals within a population is known. 2. A number of recent studies have used a type of mixedeffects model, known as the animal model, to estimate the genetic component of phenotypic variation using data collected in the field. Here, we provide a practical guide for ecologists interested in exploring the potential to apply this quantitative genetic method in their research. 3. We begin by outlining, in simple terms, key concepts in quantitative genetics and how an animal model estimates relevant quantitative genetic parameters, such as heritabilities or genetic correlations. 4. We then provide three detailed example tutorials, for implementation in a variety of software packages, for some basic applications of the animal model. We discuss several important statistical issues relating to best practice when fitting different kinds of mixed models. 5. We conclude by briefly summarizing more complex applications of the animal model, and by highlighting key pitfalls and dangers for the researcher wanting to begin using quantitative genetic tools to address ecological and evolutionary questions.Journal of Animal Ecology 01/2010; 79(1):1326. · 4.84 Impact Factor  SourceAvailable from: Alastair J Wilson
Article: New answers for old questions: the evolutionary quantitative genetics of wild animal populations
[Show abstract] [Hide abstract]
ABSTRACT: Recent years have seen a rapid expansion in the scope of quantitative genetic analyses undertaken in wild populations. We illustrate here the potential for such studies to address fundamental evolutionary questions about the maintenance of genetic diversity and to reveal hidden genetic conflicts or constraints not apparent at the phenotypic level. Tradeoffs between different components of fitness, sexuallyantagonistic genetic effects, maternal effects, genotypebyenvironment interactions, genotypebyage interactions, and variation between different regions of the genome in localized genetic correlations may all prevent the erosion of genetic variance. We consider ways in which complex interactions between ecological conditions and the expression of genetic variation can be elucidated, and emphasize the benefits of conducting selection analyses within a quantitative genetic framework. We also review potential developments associated with rapid advances in genomic technology, in particular the increased availability of extensive marker information. Our conclusions highlight the complexity of processes contributing to the maintenance of genetic diversity in wild populations, and underline the value of a quantitative genetic approach in parameterizing models of lifehistory evolution.Annu. Rev. Ecol. Evol. Syst. 01/2008; 39:52548.
Page 1
Abstract
studies of natural systems have frequently reported phenotypic stasis under these
conditions. We performed quantitative genetic analyses of weight and hindleg length
in a freeliving population of Soay sheep (Ovis aries) to test whether genetic con
straints can explain previously reported stasis in body size despite evidence for
strong positive directional selection. Genetic, maternal and environmental covari
ance structures were estimated across ontogeny using random regression animal
models. Heritability increased with age for weight and hindleg length, though both
measures of size were highly heritable across ontogeny. Genetic correlations among
ages were generally strong and uniformly positive, and the covariance structures
were also highly integrated across ontogeny. Consequently, we found no constraint
to the evolution of larger size itself. Rather we expect size at all ages to increase in
response to positive selection acting at any age. Consistent with expectation, pre
dicted breeding values for agespecific size traits have increased over a twentyyear
period, while maternal performance for offspring size has declined. Reexamination
of the phenotypic data confirmed that sheep are not getting larger, but also showed
that there are significant negative trends in size at all ages. The genetic evolution is
therefore cryptic, with the response to selection presumably being masked at the
phenotypic level by a plastic response to changing environmental conditions.
Densitydependence, coupled with systematically increasing population size, may
contribute to declining body size but is insufficient to completely explain it. Our
results demonstrate that an increased understanding of the genetic basis of
While evolution occurs when selection acts on a heritable trait, empirical
A. J. Wilson (&) Æ J. M. Pemberton Æ J. G. Pilkington Æ L. E. B. Kruuk
Institute of Evolutionary Biology, University of Edinburgh, West Mains Road, Edinburgh EH9
3JT, UK
email: Alastair.Wilson@ed.ac.uk
T. H. CluttonBrock
Department of Zoology, University of Cambridge, Downing Street, Cambridge, UK
D. W. Coltman
Department of Biological Sciences, University of Alberta, Edmonton, AB, Canada T6G2E9
123
Evol Ecol
DOI 10.1007/s106820069106z
ORIGINAL PAPER
Quantitative genetics of growth and cryptic evolution
of body size in an island population
A. J. Wilson Æ Æ J. M. Pemberton Æ Æ J. G. Pilkington
T. H. CluttonBrock Æ Æ D. W. Coltman Æ Æ
L. E. B. Kruuk
Received: 25 April 2006/Accepted: 2 August 2006
? Springer Science+Business Media B.V. 2006
Page 2
quantitative traits, and of how plasticity and microevolution can occur simulta
neously, is necessary for developing predictive models of phenotypic change in
nature.
Keywords
growth
heritability Æ Ovis aries Æ ontogeny Æ cryptic evolytion Æ
Introduction
Large size frequently confers fitness advantages in the form of increased survival or
fecundity in animals. Consequently, while not ubiquitous (Gaillard et al. 2000),
positive directional selection on size traits has been commonly reported (e.g.,
Guinness et al. 1978; Sogard 1997; Merila ¨ et al. 2001a) giving rise to interest in the
question of what keeps organisms small (Blanckenhorn 2000). In this study we
examine size in a freeliving ungulate population in which animals are not getting
bigger despite such directional selection. We test two alternate hypotheses that might
explain this observation; firstly, that genetic constraints prevent the evolution of body
size; and, secondly, that genetic evolution of larger size is occurring but that these
changes are masked at the phenotypic level by changes in environmental conditions.
We focus on body size in a freeliving population of Soay sheep, Ovis aries,
resident on the Scottish island of Hirta in the St. Kilda archipelago. While Soay
sheep are small relative to other breeds of domestic sheep, size is known to be under
selection in this population. Although it is difficult to absolutely preclude the pos
sibility that large size could impose a fitness cost that is yet to be measured, prior
studies of this system have consistently provided evidence for positive directional
selection on body size. For example, increased birth weight is associated with higher
juvenile viability and lifetime fitness (CluttonBrock et al. 1992; Wilson et al. 2005c),
and adult size traits are positively correlated with both survival and reproductive
success (Coltman et al. 1999; Milner et al. 1999). Evolution is expected to occur
when selection acts on a trait that has a heritable component of phenotypic variation
(Falconer and Mackay 1996), and significant heritabilities for size traits (e.g. weight,
leg length) have also been estimated in this population (Milner et al. 2000). While
theory therefore predicts the evolution of larger size, previous analysis has found no
evidence that sheep are getting bigger (Milner et al. 2000).
Phenotypic stasis in traits under selection has been frequently reported in natural
populations (Merila ¨ et al. 2001b; Kruuk et al. 2002). One possible explanation for
this phenomenon is that stasis reflects genetic constraints arising either from a lack
of heritable variation, or from genetic correlations between traits under selection
(Arnold 1992). The latter will occur if genetic correlations are negative between
traits under similar directions of selection, or if positively correlated traits are sub
ject to antagonistic selection regimes (e.g., Wilson et al. 2003). To date, analyses of
genetic correlations between body size and other phenotypic traits have provided
little evidence for this type of constraint in natural systems (e.g., Coltman et al.
2005). However, while important traits may have been overlooked in studies to date,
it has also been argued that genetic constraints will not always be apparent from
consideration of h2and rG alone (Pease and Bull 1988), and that multivariate
statistical methods (e.g., eigenvector analysis of genetic variance–covariance matri
ces) should additionally be used for their detection (Blows and Hoffman 2005).
123
Evol Ecol
Page 3
Significant heritabilities (h2) for body size have been shown in a number of wild
vertebrate systems (e.g., Kruuk et al. 2001; Pakkasmaa et al. 2003), though most
studies to date have been performed in a univariate framework. Specifically, they
have either focused on size as defined at a single point in ontogeny, or have used
analytical methods that implicitly assume constancy of h2with age. However, if size
is viewed as a series of agespecific traits, then genetic and environmental compo
nents of phenotypic variance (and hence h2) are frequently found to vary over
ontogeny (Cheverud et al. 1983a; Re ´ale et al. 1999). Failing to account for this may
result in misleading expectations of evolutionary change, particularly if selection acts
differently at different ages. Furthermore, genetic covariance structures among age
specific size traits might impose nowlater tradeoffs (Wilson et al. 2003), or more
generally limit the potential for growth trajectories to evolve in certain directions
(Kirkpatrick and Lofsvold 1992).
It is therefore expected that multivariate approaches should provide more flexible
(and biologically realistic) models of genetic architecture over ontogeny. However,
application to natural systems has been limited by statistical considerations since
individuals are frequently not sampled at all measurement ages (due to both mor
tality and fieldsampling limitations). This is particularly problematic for determin
ing genetic covariances among ages since accurate and precise estimation generally
requires large sample sizes (Lynch and Walsh 1998). Consequently, comparatively
few studies have estimated genetic covariance structures for trait ontogenies outside
the laboratory, except over limited ontogenetic periods where repeated sampling is
relatively feasible (e.g. analyses of nestling growth in passerines; Bjo ¨rklund 1997;
Badyaev and Martin 2000).
Nevertheless, the statistical problems associated with incomplete sampling can be
reduced through recently developed random regression animal models. If growth is
viewed as an infinitedimensional trait, then each individual’s ontogenetic trajectory
is defined by a potentially infinite series of size traits along a temporal axis corre
sponding to age (Kirkpatrick et al. 1990). Within this framework it is possible to
model an individual’s genetic merit, or ‘‘breeding value’’ as a covariance function of
age (Kirkpatrick et al. 1990; Meyer 1998; Schaeffer 2004), a technique that allows
estimation of agespecific heritabilities and genetic correlations between ages.
Importantly, in comparison to more traditional multipletrait analyses, random
regression allows more efficient use of the incomplete data sets typically obtained in
natural populations (Wilson et al. 2005b).
Here we apply the random regression approach to quantitative genetic analyses of
size and growth in Soay sheep. We estimate the additive genetic variance–covariance
matrix for agespecific size traits in order to test for genetic constraints in a multi
variate framework. Specifically we test whether genetic constraints can explain the
lack of a systematic increase in Soay sheep body size. Furthermore, since our
hypothesis of genetic constraint is based on previously reported phenotypic stasis
(Milner et al. 2000), we reexamine phenotypic trends (or lack thereof) using an
extended data set, and explicitly test for genetic responses to selection using pre
dicted breeding values (PBV). The predicted breeding value is a measure of the
additive genetic merit of an individual for the trait in question (Lynch and Walsh
1998), and can be thought of as an estimate of the individual’s phenotype corrected,
albeit imperfectly (Postma 2006), for environmental effects. In the absence of any
genetic constraint it is possible that a genetic response to selection has occurred
but is masked at the phenotypic level by environmental effects on body size.
Evol Ecol
123
Page 4
Consideration of predicted breeding values allows us to explicitly test this alternate
hypothesis for phenotypic stasis.
Materials and Methods
Data and pedigree structure
The Village Bay population of Soay sheep (Ovis aries) is resident on the Scottish
island of Hirta, in the St. Kilda archipelago in the North Atlantic (57(49(N,
08(34(W). This population has been subject to longterm study, with morphological,
life history and genetic data collected on individually tagged sheep throughout their
lives since 1985. We used two different measures of body size, weight and hindleg
length, that are phenotypically and genetically correlated (Milner et al. 2000; Colt
man et al. 2001). Body weight is measured in lambs (within a few days of birth), and
both weight and hindleg length are recorded during annual round ups of older
animals conducted each August, during which over 50% of the population is cap
tured. More extensive details of the project and field methodology are presented
elsewhere (see CluttonBrock and Pemberton 2004).
Soay sheep are sexually dimorphic, and early size traits are also influenced by
natal litter size (twins being smaller than singletons; CluttonBrock et al. 1992). We
removed these known effects prior to estimation of genetic parameters, by fitting
linear models of size at each measurement age in which sex and natal litter size were
fitted as explanatory variables. Residuals from these models were then used as the
measures of ‘‘corrected weight’’ (WTAGE) and ‘‘corrected hindleg length’’ (HLAGE)
at each age. For weight at birth (i.e. AGE = 0) the linear model included the
additional explanatory term of capture age (in days) as many individuals are not
captured for several days after birth and lambs grow at an average of 0.135 kg/day.
Note that there is no available data on leg length at birth.
In order to avoid sample size issues that arise from a lack of records on older
animals, we restricted our attention to size traits expressed from birth (AGE = 0 
months) until August in the fifth year of life (AGE = 64 months). Although Soay
sheep can sometimes live for 10 years or more, 80% of weight records and 85% of
hindleg records relate to this period of ontogeny. Thus we defined corrected size
traits at ages 0 months (i.e., at birth which is typically in April) and at 4, 16, 28, 40, 52
and 64 months (i.e. successive August weights up to 5 years old). Analyses of weight
are based on 6,871 measurements made on 3,749 distinct individuals between 1985
and 2005. The number of phenotypically informative individuals is 3,054, 1,670, 760,
515, 327, 297 and 248 for WT0, WT4, WT16, WT28, WT40, WT52and WT64respec
tively. Analyses of hindleg length are based on 3,453 measurements made on 1,985
distinct individuals, with the number of phenotypically informative individuals being
1,536, 671, 464, 291, 272 and 219 for HL4, HL16, HL28, HL40, HL52 and HL64
respectively.
Quantitative genetic analyses also require pedigree information which has been
determined through field observations of maternity and molecular paternity
assignment. Paternity assignment was performed using microsatellite data and the
maximumlikelihood method CERVUS (Marshall et al. 1998). Paternities were
assigned for individuals born before and after 1997 using overlapping panels of 14
and 18 microsatellite loci respectively, and accepted if assigned at a pedigreewide
123
Evol Ecol
Page 5
confidence level of 80% with a maximum of one locus showing an allelic incom
patibility between offspring and putative sire. Complete details of microsatellite
protocols and paternity assignment methods are presented elsewhere (Overall et al.
2005). The pedigree structure determined in this way contains 6342 individual
records with 3,541 maternal links, and 1,615 paternal links (from 807 distinct dams
and 495 distinct sires respectively), and has a maximum depth of 10 generations.
Quantitative genetic analysis
Phenotypic variance for weight and hindleg length was partitioned into genetic and
environmental components using animal models (Kruuk 2004). Two modelling ap
proaches were employed for each measure of size: firstly, we estimated variance
components under a conventional repeated measures animal model in which all
variance components are assumed to be constant across ontogeny; and secondly, we
used random regression to fit a series of less constrained animal models in which
variance components can vary across ontogeny. Fixed effects were included in all
animal models to account for increasing average size with age (i.e., growth) and
phenotypic differences caused by variation in environmental conditions among dif
ferent years of the study. Thus age (in months) and birth year were fitted as factors,
and the interaction between these terms was also included. Birth year was fitted as 30
level factor and its inclusion as a fixed effect also accounts for any temporal trend in
mean phenotype, reducing the possibility of spurious trends in breeding values
resulting from environmentallyinduced phenotypic trends (Postma 2006). In all
cases parameter estimates were solved for using restricted maximum likelihood
(REML) implemented in the program ASReml (VSN International Ltd 2002). The
two modelling approaches are presented more fully below. Parallel analyses were
performed for corrected weight and hindleg length but for simplicity we show the
former only.
Model 1: Repeated measures animal model
Under the repeated measures animal model the corrected weight of individual i with
mother j is given as:
WTiAGE?ðAGEþBIRTHYEARþAGE?BIRTHYEARÞiAGEþaiþmjþpeiþei
where aiis the breeding value of individual i which has a population mean of zero
and variance of r2A (the additive genetic variance). Estimating r2A is possible
because the variance–covariance matrix of additive genetic effects is expected to
equal to Ar2Awhere A is the additive numerator relationship matrix containing the
individual elements Aij= 2Qij, and Qij is the coefficient of coancestry between
individuals i and j obtained from the pedigree structure. Permanent environment
effect pei, maternal performance mj, and residual error eiterms also contribute to the
phenotype. These terms were assumed to be normally distributed with means of zero
and variance–covariance matrices of Ir2PE, Ir2Mand Ir2R, where r2PE, r2Mand r2E
are the permanent environment, maternal, and residual (temporary environmental)
variance components respectively, and I is an identity matrix with order equal to the
number of records as appropriate. The phenotypic variance of weight (r2P) was
Evol Ecol
123
Page 6
determined as the sum of its estimated components, and the direct heritability (h2),
permanent environment effect (pe2), maternal effect (m2) and ratio of residual
variance (r2) were then calculated as the ratio of the relevant variance component to
r2P.
Model 2: Random regression animal model
To test for ontogenetic variation in quantitative genetic parameters and estimate
genetic correlations between agespecific weights, additive and maternal effects on
the phenotype of individual i were then modelled by regressing on orthogonal
(Legendre) polynomials of standardized age (AGESD), defined as age in months
standardized to the interval –1 £ AGESD£ 1 (Kirkpatrick et al. 1990; Meyer 1998).
Consequently, at the individual level, the corrected weight (WTAGE) of individual i
with mother j at any age is given as:
WTiAGE? ðAGE þ BIRTHYEAR þ AGE ? BIRTHYEARÞiAGE
þ f1ðai;n1;AGESDÞ þ f2ðmj;n2;AGESDÞ þ eiAGE
where f1(ai,n1, AGESD) is the random regression function, on orthogonal polyno
mials of AGESDwith order n1, of additive genetic merit values of individuals; and
f2(mj,n2, AGESD) is a random regression function with order n2of maternal per
formance values of individuals on AGESD; and eiAGEis the agespecific residual
error for individual i. The latter term was modelled using a 7 · 7 unstructured matrix
to permit a multivariate error structure, with eiAGEseparately estimated at values
corresponding to 0, 4, 16, 28, 40, 52 and 64 months. The unstructured matrix allows
residual errors to be correlated across ages within individuals (removing the
requirement for including a permanent environment effect).
We first fitted a model without additive or maternal effects such that all pheno
typic variance is allocated to the residual structure (Model 2.0 in Table 1). The
resultant residual matrix is therefore a description of the phenotypic variance–
covariance surface for WT where –1 < AGESD< 1 (i.e., ages 0–64 months).
Subsequently, a forward selection procedure was used to compare a series of
successively more complex random regression models that differed in the order of
polynomial function used to fit additive and maternal effects (see Table 1). Values
of n1from 0 (aias constant) to 3 (aias a cubic function of ASD), and n2from 0 (mjas
a constant) to 1 (mjas a linear function of AGESD) were used. Increasing values of
n1 and n2 result in an increase in the number of (co)variance components estimated.
Consequently, model selection was performed using likelihoodratio tests to com
pare pairs of models (Meyer 1992), with the number of degrees of freedom deter
mined as the difference in the number of (co)variance parameters estimated
between models.
Following selection of appropriate values of n1and n2, the variance–covariance
matrices of random regression parameters obtained for the additive genetic effect
(matrix Q with dimensions (n1+ 1) · ( (n1+ 1) was used to derive agespecific
genetic parameters. Specifically the additive genetic variance–covariance matrix, G,
for WTAGE(for ages 0–64 months) was obtained as G = Z Q Z’, where Z is the
vector of orthogonal polynomials evaluated at values of standardized age that cor
respond to 0, 4, 16, 28, 40, 52 and 64 months (and Z’ is the transpose of Z). An
Evol Ecol
123
Page 7
analogous procedure was used to obtain the maternal genetic variance–covariance
matrix M, and approximate standard errors for the elements of G and M matrices
were determined (according to Fischer et al. 2004). The multivariate residual error
structure obtained by solving the random regression model represents the envi
ronmental variance–covariance matrix R.
At each age of measurement, the phenotypic variance was then determined as the
sum of agespecific variance components, and the ratios h2, m2and r2were calcu
lated. We also calculated phenotypic (CVP), additive (CVA), maternal genetic
(CVM) and residual (CVR) coefficients of variation [where the coefficient of
variation is found as 100 · (variance0.5)/sample mean]. By comparison to the raw
variances, coefficients of variation are expected to be less sensitive to scale effects
arising from an increasing mean weight with age (Houle 1992). Finally, between each
pair of ages, covariance components were rescaled to give genetic (rG) and maternal
(rM) correlations, and the variance–covariance matrices G and M were also sub
jected to eigenvector decomposition in order to summarise the major patterns of
variation for individual growth trajectories (following e.g., Cheverud et al. 1983b).
Determination of phenotypic and genetic temporal trends
To test for temporal trends at the phenotypic level, linear regressions on birth year
were performed for annual means of each agespecific corrected weight (WTAGE)
and hindleg length (HLAGE) with the degrees of freedom determined from the
number of years represented in the phenotypic data. Trends at the genetic level were
similarly tested for using annual means of the predicted breeding values at each age
(PBVAGE) as the dependent variable, and the number of degrees of freedom
determined from the number of birth years represented in the full pedigree struc
ture, which includes animals some born prior to 1985 (N = 30 years).
Table 1 Random regression model selection for weight and hindleg length showing: order of
polynomials used for additive and maternal effects, number of (co)variance parameters estimated,
and associated loglikelihoods scores (LnLK) for each model
ModelPolynomial orderWEIGHTHINDLEG
Additive n1
Maternal n2
Parameters LnLK ParametersLnLK
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
NF
0
0
0
1
1
1
2
2
2
3
3
3
NF
NF
0
1
NF
0
1
NF
0
1
NF
0
1
28
29
30
32
31
32
34
34
35
37
38
39
41
–4644.54
–4608.14
–4556.34
–4555.03
–4590.24
–4538.09
–4536.72
–4580.87
–4528.42
–4527.26
–4575.3
–4522.55*
–4521.17
21
22
23
25
24
25
27
27
28
30
31
32
34
–8258.19
–8221.23
–8215.58*
–8215.17
–8221.12
–8215.28
–8213.46
–8218.57
–8212.84
–8209.44
–8216.92
–8211.11

NF indicates that an effect was not fitted. Note that convergence of Model 2.12 was not achieved for
hindleg length
*Statistically best model
Evol Ecol
123
Page 8
For each individual, PBV was determined from the animal model as the best
linear unbiased predictor (BLUPs) of the breeding value (ai). The repeated mea
sures animal model yields a single PBV per individual (since aiis necessarily con
stant over ontogeny). In contrast, under the random regression model aiAGE, the
breeding value of individual i at a given age, is specified by a polynomial function of
AGESDwith order n1. Thus BLUPs of n1+ 1 coefficients were obtained for each
individual, and by evaluating the polynomial we obtained PBVs for each animal at
ages 0, 4, 16, 28, 40, 52 and 64 months. Equivalent analyses were then performed to
test for temporal trends in maternal performance for offspring size. Thus, mean
predicted maternal performance ( PMV) was regressed on birth year with individual
PMVs determined from the BLUPs of the n2+ 1 coefficients in the polynomial
function for mjA, the maternal performance at age A.
Results
Genetic and environmental covariance structures for weight
Animal model analyses supported the presence of additive and maternal effects on
corrected body weight. Under the repeated measures approach (Model 1), variance
components (± standard errors) were estimated as r2A= 0.363 ± 0.077, r2M=
0.080 ± 0.038, r2PE= 0.764 ± 0.083 and r2R= 2.552 ± 0.059. The ratios of additive
and maternal varianceto
r2P
were
m2= 0.021 ± 0.010 respectively, while most of the phenotypic variance was attrib
uted to permanent and temporary environmental effects (pe2= 0.203 ± 0.022,
r2= 0.679 ± 0.015). Note that while the maternal effect accounted for only 2% of the
phenotypic variance, comparison to a reduced model showed this to be statistically
significant (v21= 3.37, P = 0.009).
The random regression models also demonstrated additive and maternal effects
on weight. Furthermore, using this approach provided important evidence that
variance components and associated quantitative genetic parameters change across
ontogeny, a phenomenon an evolutionary importance that cannot be detected using
the conventional repeated measures model (Model 1). Based on loglikelihood tests,
Model 2.11 was selected as the best model (Table 1), performing significantly better
than models 2.0–2.10, while Model 2.12 was not a significantly better fit (v22= 1.38,
P = 0.251). Thus agespecific genetic parameters were estimated using Model 2.11 in
which the breeding value was modelled with a third order polynomial regression on
standardized age (n1= 3), and the maternal performance was modelled using a zero
order polynomial (n2= 0).
Under Model 2.11, the additive genetic variance r2Aincreases with age (Table 2,
Fig. 1), while the residual (environmental) variance also rises from 0 to 4 months
before showing relative stability. Maternal variance is constrained to be constant by
the choice of n2= 0. However, since variance is expected to increase over ontogeny
as a scaledependent consequence of increasing phenotypic mean, the coefficients of
variation are more appropriate for comparing among measurement ages. The
coefficient of phenotypic variation (CVP) shows a general pattern of decline from
age 0 to 64 months, caused by reduction in maternal and residual environmental
variation (measured by CVMand CVRrespectively; Table 2). It is these changes,
rather than increases in CVA, that cause heritability of weight to increase from
123
calculated ash2= 0.096 ± 0.020 and
Evol Ecol
Page 9
Table 2 Estimated phenotypic means, variance components, coefficients of variation and ratios to phenotypic variance for agespecific size traits. Estimates are
based on Model 2.11 for weight and Model 2.2 for hindleg, and approximate standard errors are shown in parentheses where available
Age
Mean
r2A
r2M
r2R
CVP
CVA
CVM
CVR
h2
m2
r2
Weight (kg)
0
2.13
0.025 (1.263)
0.101 (0.013)
0.231 (0.011)
28.0
7.36
14.94
22.6
0.069
0.284
0.647
4
13.3
0.284 (0.566)
0.101 (0.013)
5.59 (0.211)
18.4
4.00
2.39
17.8
0.047
0.017
0.936
16
19.5
2.07 (0.113)
0.101 (0.013)
6.43 (0.531)
15.1
7.38
1.63
13.0
0.240
0.012
0.748
28
23.9
2.99 (0.321)
0.101 (0.013)
5.66 (0.592)
12.4
7.24
1.33
9.96
0.342
0.012
0.647
40
24.8
3.02 (0.534)
0.101 (0.013)
4.34 (0.595)
11.0
7.00
1.28
8.39
0.405
0.014
0.581
52
25.5
3.14 (0.719)
0.101 (0.013)
5.51 (0.707)
11.6
6.94
1.25
9.20
0.359
0.012
0.630
64
25.7
4.30 (1.202)
0.101 (0.013)
4.39 (0.916)
11.6
8.08
1.24
8.16
0.489
0.012
0.499
Hindleg (mm)
4
160
28.4 (5.14)
6.83 (2.44)
65.3 (3.86)
6.28
3.34
1.64
5.06
0.283 (0.045)
0.068 (0.024)
0.649 (0.045)
16
177
28.4 (5.14)
6.83 (2.44)
41.1 (3.69)
4.93
3.01
1.48
3.62
0.372 (0.057)
0.090 (0.030)
0.538 (0.056)
28
182
28.4 (5.14)
6.83 (2.44)
66.3 (5.77)
5.54
2.93
1.44
4.47
0.280 (0.046)
0.067 (0.024)
0.653 (0.046)
40
181
28.4 (5.14)
6.83 (2.44)
30.3 (3.96)
4.46
2.94
1.44
3.03
0.434 (0.065)
0.104 (0.036)
0.462 (0.062)
52
182
28.4 (5.14)
6.83 (2.44)
29.7 (4.08)
4.43
2.93
1.44
2.99
0.437 (0.066)
0.105 (0.036)
0.457 (0.063)
64
181
28.4 (5.14)
6.83 (2.44)
28.7 (4.11)
4.42
2.94
1.44
2.96
0.444 (0.066)
0.107 (0.037)
0.449 (0.069)
Evol Ecol
123
Page 10
h2= 0.069 at birth to h2= 0.489 for WT64(Fig. 1). As a proportion of phenotypic
variance, the maternal effect m2declines from 0.284 to 0.012 over the same period
(Table 2).
Additivegeneticcovariancesandcorrelationswerepositivebetweenallagespecific
traits (Table 3). Residual (environmental) and total phenotypic covariances between
ageswerealsouniformlypositive(resultsnotshown),whilethematernalcorrelationis
constrained to rM= +1 between any pair of ages under Model 2.11. Among traits
expressed from age 4 months, genetic correlations were effectively equal to +1
(Table 3). Note that the model 2.11 was unconstrained such that some estimates of rG
lie just outside the strictly permissible parameter space of –1 £ rG£ +1). In contrast,
416 28 405264
Age (months)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
4 164840 52 64
2
0
2
4
6
4
16284052 64
Age (months)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
41628 40 5264
6
12
18
24
30
36
42
48
WEIGHT HINDLEG
h2
Model 1
Model 1
Model 2.11
Model 2.2
Model 2.11
Model 1
Model 2.2
Model 1
σ2
A
Fig. 1 Estimated r2A(upper panels) and h2(lower panels) under Model 1 (dashed line), and best
fitting models for weight (Models 2.11) and hindleg (Model 2.2) (solid lines). 95% confidence
intervals as thinner lines where available
Table 3 Estimated additive genetic covariances (below diagonal) and correlations (above diagonal)
between agespecific weights estimated under Model 2.11. Approximate standard errors are shown in
parentheses where available
Age (months)
04 1628 40 5264
0
4

0.027 (0.375)
0.045 (0.387)
0.07 (0.629)
0.091 (0.706)
0.095 (0.744)
0.074 (0.913)
0.327

0.759 (0.264)
0.919 (0.505)
0.921 (0.582)
0.926 (0.551)
1.094 (0.66)
0.202
0.992

2.48 (0.389)
2.465 (0.478)
2.485 (0.465)
3.006 (0.513)
0.259
0.998
0.998

2.992 (0.494)
3.037 (0.497)
3.673 (0.528)
0.332
0.994
0.986
0.995

3.087 (0.611)
3.703 (0.622)
0.344
0.981
0.975
0.991
1.001

3.709 (0.786)
0.227
0.99
1.009
1.025
1.027
1.009

16
28
40
52
64
Evol Ecol
123
Page 11
genetic correlations between birth weight (WT0) and traits expressed later were sub
stantially lower (ranging from 0.202 to 0.343; Table 3).
Decomposition of the estimated genetic variance–covariance G for agespecific
weights showed that over 99.9 % of the variation in the data was explained by the
first eigenvector (Table 4). For agespecific weight traits, the loading coefficients
associated with this vector were uniform in sign with a general trend of increasing
magnitude with age consistent with the agerelated trend in r2A(Table 4). The
variation described by this eigenvector therefore corresponds to additive effects that
are highly integrated across ontogeny, with alleles having a uniformly positive or
negative influence on weight at all ages. Note since all estimated elements of matrix
M are equal, no eigenvector decomposition was performed.
Genetic and environmental covariance structures for hindleg length
Additive and maternal effects were also found to be significant for hindleg length.
Under Model 1, variance components (± standard errors) were estimated as
r2A= 19.8 ± 4.30, r2M= 7.41 ± 2.18, r2PE= 33.7 ± 3.67 and r2R= 20.9 ± 0.806,
withcorrespondingratios
r2P
of;
pe2= 0.412 ± 0.046, and r2= 0.256 ± 0.013). In contrast to weight, random regres
sion models did not provide evidence for changing levels of additive variance over
ontogeny, and the best model was found to be Model 2.2 in which both r2Aand r2M
are constrained to be constant (Table 2). However, Model 2.2 uses a multivariate
error structure, allowing r2Rto differ between measurement ages which provided a
significantly better fit to the data than the univariate error structure of Model 1
(Likelihood ratio test, v219= 528, P < 0.001). Consequently, while r2Ais constant,
we found a general (though imperfect) trend of increasing heritability of hindleg
length from h2= 0.283 for HL4to h2= 0.444 for HL64(Table 2, Figure 1). This
trend is driven by a decline in residual variation as indicated by CVR(Table 2). Note
that under Model 2.2, genetic correlations (rG) are by definition constrained to +1
between all pairs of measurement ages (as are the maternal correlations). Eigen
vector decompositions of G and M matrix were not performed since estimated
matrices contain no variation. Residual and total phenotypic covariances (and cor
relations) between ages were again positive (results not shown).
h2= 0.242 ± 0.050,m2= 0.091 ± 0.026,
Phenotypic and genetic temporal trends for body size
Regressions of mean age specific size traits on birth year were all negative, and
significant (or marginally nonsignificant) in 11 of 13 cases (Table 5). Thus, as
measured by both weight and hindleg length, sheep are actually getting smaller at all
ages (Table 5, Fig. 2). Rates of phenotypic decline are substantial in some cases with
a maximum of –0.221 kg year–1for weight (at 16 months), and –0.635 mm year–1for
Table 4 Loading coefficients for the first two eigenvectors of the G matrix of agespecific weight
(WTAGE) as estimated under Model 2.11
Eigenvector% variation explained Loadings on agespecific weight Age
0416284052 64
1
2
99.95
0.04
–0.008
–0.102
–0.131
–0.077
–0.359
+0.126
–0.435
+0.362
–0.438
+0.423
–0.442
+0.104
–0.526
–0.804
Evol Ecol
123
Page 12
hindleg length (at 16 months). In contrast, at the genetic level, regressions of mean
predicted breeding values (as determined by the best fitting models) showed sig
nificant increases in genetic merit for size with time (Table 5, Fig. 2). Note that age
specific values of PBVAGEwere obtained for weight (Table 5) but not hindleg, since
using the zero order polynomial to model the additive covariance function (n1= 0 in
Model 2.2) constrains aito be constant over ontogeny. Similarly, since n2= 0 in both
Models 2.11 and 2.2 such agespecific analyses of maternal performance are not
applicable. In contrast to breeding values, the average maternal performance for size
was actually found to be declining over time. Significant declines in ( PMV) were
determinedfor weight (–0.005 kg year–1,
(–0.014 mm year–1, P = 0.002) with maternal performance determined under
Models 2.11 and 2.2 respectively.
Note that the above trends in breeding values and maternal performance for
weight and hindleg length were also found with analyses conducted under the more
conventional repeated measures model (Model 1). For comparison, under Model 1
the temporal trends were: for weight,
PBV trend = 0.003 kg year–1(P < 0.001),
PMV trend = –0.002 kg year–1(P < 0.001); for hindleg length,
0.009 mm year–1(P = 0.046), PMV trend = –0.020 mm year–1(P = 0.017).
P = 0.012) andhindleglength
PBV trend =
Discussion
Our results confirm that body size, as measured by both weight and hindleg length, is
heritable across ontogeny in Soay sheep. Furthermore, our estimation of the G
matrices revealed strong positive genetic correlations among ages, with highly
123
Table 5 Temporal trends in mean age–specific weight and hindleg traits estimated from linear
models with birth year fitted as an explanatory covariate. Also shown are equivalent models of mean
predicted breeding values ( PBV) and mean predicted maternal performance ( PMV) as determined
under Models 2.11 and 2.2 for weight and hindleg length respectively. Note agespecific predicted
breeding values were not obtained for hindleg because the best fit model suggested a constant
additive genetic variance with age. Significance of all temporal trends assessed from leastsquares
regression with residual degrees of freedom as indicated
AgeWeightHindleg
Trend (kg/year)DFP Trend (mm/year)DFP
Mean phenotype0
4
–0.016 (0.011)
–0.129 (0.036)
–0.221 (0.046)
–0.113 (0.054)
–0.092 (0.066)
–0.180 (0.050)
–0.146 (0.049)
0.001 (2 · 10–3)
0.003 (4 · 10–3)
0.006 (0.001)
0.008 (0.001)
0.008 (0.001)
0.009 (0.001)
0.010 (0.001)

–0.005 (0.002)
19
19
19
19
19
18
17
29
29
29
29
29
29
29
29
29
0.150
0.002
<0.001
0.051
0.179
0.002
0.008
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001

0.012

–0.431 (0.054)
–0.412 (0.059)
–0.281 (0.082)
–0.470 (0.097)
–0.182 (0.103)
–0.505 (0.106)







0.011 (0.005)
–0.014 (0.006)

0.002
<0.001
0.038
<0.001
0.066
0.002







0.004
0.002
16
16
16
16
15
15
16
28
40
52
64
0
4
16
28
40
52
64
PBV
Pooled
Pooled
29
29 PMV
Evol Ecol
Page 13
integrated ontogenies for both aspects of body size. Thus we found no support for
the hypothesis that a response to positive directional selection on size is limited by
genetic constraints between traits across ontogeny. Nevertheless, our analyses did
reveal important ontogenetic patterns in the quantitative genetic parameters that
will influence the evolution of body size. In contrast, our alternate hypothesis,
namely that selection is eliciting a genetic response but that this is masked at the
phenotypic level by opposing environmental effects, is supported for both weight
and hindleg length. In the following discussion we first consider the quantitative
genetic architecture of the body size ontogeny before focusing on the evidence for
cryptic evolution in this population.
Quantitative genetic architecture of body size
Genetic covariance structures for weight and hindleg length were found to be
broadly similar in a qualitative sense, with all analyses providing evidence of additive
1980198819962004
1.4
1.8
2.2
2.6
0.00
0.01
0.02
0.03
19801988 1996 2004
8
12
16
20
0.1
0.0
0.1
0.2
1980198819962004
12
16
20
24
0.1
0.0
0.1
0.2
1980 198819962004
18
22
26
30
0.1
0.0
0.1
0.2
1980198819962004
20
24
28
32
0.1
0.0
0.1
0.2
1980198819962004
20
24
28
32
0.1
0.0
0.1
0.2
19801988 19962004
20
24
28
32
0.1
0.0
0.1
0.2
0
WT
40
WT
4
WT
16
WT
28
WT
52
WT
WT64
0
PBV
4
PBV
16
PBV
28
PBV
40
PBV
52
PBV
64
PBV
Fig. 2 Temporal trends in phenotype and breeding value for weight. For each age, bars indicate
phenotypic mean by birth year (±SE). Lines show best fit from leastsquares regression of mean
phenotype (solid line), and mean predicted breeding value ( PBVAGE; dotted line) on birth year.
Note that Yaxis scaling at 0 months differs from other plots for clarity
Evol Ecol
123
Page 14
and maternal effects. Furthermore, for both measures of size the random regression
models showed a trend towards increasing heritability with age. By contrast, the
simpler models in which variance components are constrained to be constant pro
vided inferior fits to the data, and also yielded h2estimates heavily determined by
the earlymeasured phenotypes (i.e., at 0 and 4 months) that dominate the data set.
Increasing h2with age was particularly marked for weight where a rapidly dimin
ishing maternal influence on offspring phenotype was also found. Although the
declining maternal effect reflects a decrease in absolute levels of maternal variance,
the trend in heritability is actually driven by declines in nongenetic (i.e., maternal
and environmental) sources of variance rather than by increasing additive variation.
Similarly, the increase in h2of hindleg length is a consequence of diminishing
environmental variation rather than an increase in additive effects.
While qualitatively similar, some quantitative differences are apparent between
the genetic architecture of weight and hindleg length. For example, although a lack
of hindleg data at birth limits comparison, heritability is noticeably lower for weight
than hindleg length in lambs. This is likely because weight is more sensitive to short
term changes in the environment than skeletal traits (e.g. weight loss occurs in
response to temporary food shortage while bone resorption does not).
Overall phenotypic variances for both weight and hindleg length were found to
decline with increasing age, an observation consistent with the action of directional
viability selection which occurs in particular over the first part of ontogeny (i.e.
between measurement ages 0 and 16 months; (Overall et al. 2005; Wilson et al.
2005c). However, in addition to viability selection, compensatory growth processes
may cause a reduction in variance with age and have commonly been reported in
domestic sheep (Wilson and Re ´ale 2006). Although comparative studies are limited,
the ontogenetic patterns in quantitative genetic parameters found here are strikingly
similar to those reported for live weight in wild bighorn sheep, Ovis canadensis
(Wilson et al. 2005b), suggesting that they may be quite generalisable.
It should be noted that the viability selection discussed above might have the
potential to introduce bias into the estimates of covariance components. While
animal models are generally thought to be robust to selection through differential
reproductive success (Lynch and Walsh 1998), covariances between ages will
depend on agespecific phenotypic distributions that later in ontogeny might rep
resent postselection distributions. Intuitively, since a continuous covariance
function across ontogeny is estimated (i.e., using phenotypic data measured at all
ages), random regression models may be less susceptible to bias than a conven
tional two trait animal model. However, further investigation of these issues is
certainly warranted.
A general maternal effect was modelled in our analyses such that the maternal
variance may potentially include both environmental and genetic components.
Maternal genetic effects arise from allelic differences between individual mothers at
loci influencing offspring phenotype, and are themselves a heritable component of
phenotypic variance (Wolf et al. 1998). Here, the separation of the maternal vari
ance into genetic and environmental components was not statistically supported for
either size trait modelled across ontogeny (results not shown). However maternal
genetic effects on birth weight have previously been demonstrated (Wilson et al.
2005a), and failure to partition them here is likely due to statistical considerations
arising from the different definition of phenotype (i.e., weight traits across ontogeny
rather than at the single stage when maternal genetic effects are most important).
123
Evol Ecol
Page 15
This conclusion is supported by univariate analysis of WT0that confirm previous
findings using the slightly extended data herein (results not shown).
Since maternal genetic effects are heritable, the ‘‘total heritability’’ (incorpo
rating both additive genetic and maternal genetic variance; (Willham 1972) of
weight may be somewhat higher than the estimates of direct h2reported, partic
ularly for WT0 where the general maternal effect accounted for 28% of the
variance. Nevertheless, maternal genetic effects will not change the general pat
tern of increasing heritability with age, which would persist even under the
extreme (and highly unlikely) scenario of all maternal variance being attributable
to maternal genetic effects. All else being equal, selection on size later in life will
therefore elicit a more rapid evolutionary response due to higher heritabilities of
both weight and hindleg length. However, to offset this prediction, selection is
generally stronger on lamb and yearling size than on adult traits (Milner et al.
1999).
Strong positive genetic correlations between agespecific traits (fixed at +1 for
hindleg length) are such that that selection at any age should result in positively
correlated responses across all ages. A partial caveat to this is that genetic corre
lations were lower between birth weight and later traits, suggesting genetic influ
ences on WT0may differ somewhat from those determining later size and growth.
Although the magnitude of correlated responses by laterexpressed traits to selec
tion acting on birth weight will thus be reduced accordingly, positive selection should
increase the height of the population mean growth curve as a whole.
It is clear from our analyses that the ontogeny of Soay sheep body size is highly
genetically integrated (sensu Cheverud et al. 1983b). This conclusion is supported
by both the estimates of rGamong ages, and also the eigenvector decomposition of
the G matrix for agespecific traits. For weight, virtually all variation is explained by
a single eigenvector corresponding to genetic effects of uniform sign across
ontogeny (note that the first eigenvector necessarily explains all variation in G for
hindleg length and M for both size measures as a consequence of model selection).
Importantly, this lack of variation associated with additional eigenvectors can be
seen as a genetic constraint on the growth curve per se (Kirkpatrick and Lofsvold
1992; Blows and Hoffman 2005). Thus, while our results do not provide evidence of
a constraint to the evolution of larger size, they do indicate that evolutionary
change will be restricted to an increase (or decrease under negative selection) in the
height of the mean growth curve. In contrast, other laboratory and field studies have
found that additional eigenvectors can explain substantial genetic variance (e.g.,
Cheverud et al. 1983b; Ragland and Carter 2004; Wilson et al. 2005b) that may
reflect segregating alleles with antagonistic effects at different ages . Under such
conditions then the population growth curve might evolve in different directions
(e.g., increased size early in life and decreased size later) under an appropriate
selection regime.
It should be noted that we restricted our attention to the analysis of size traits
only. Though weight and hindleg length, the two aspects of size considered here, are
themselves positively genetically correlated (Milner et al. 2000; Coltman et al. 2001),
we cannot preclude the possibility of constraints arising from correlations between
size and other aspects of phenotype not considered. Furthermore, while accurately
estimating the G matrix for a wider set of (nonsize) phenotypic traits is not trivial in
this (or any other) natural system, eigenvector analyses of such a matrix may be
useful (Blows and Hoffman 2005).
Evol Ecol
123