Rapid and unpredictable changes of the G-matrix in a natural bird population over 25 years

Authors


  • Data deposited at Dryad: do:10.5061/dryad.s55c4

Correspondence: Mats Björklund, Department of Animal Ecology, Evolutionary Biology Centre, Uppsala University, Norbyvägen 18D, SE-752 36 Uppsala, Sweden.

Tel.: +46 18 471 2666; fax: +46 18 471 6484;

e-mail: mats.bjorklund@ebc.uu.se

Abstract

Knowledge of the genetic variances and covariances of traits (the G-matrix) is fundamental for the understanding of evolutionary dynamics of populations. Despite its essential importance in evolutionary studies, empirical tests of the temporal stability of the G-matrix in natural populations are few. We used a 25-year-long individual-based field study on almost 7000 breeding attempts of the collared flycatcher (Ficedula albicollis) to estimate the stability of the G-matrix over time. Using animal models to estimate G for several time periods, we show that the structure of the time-specific G-matrices changed significantly over time. The temporal changes in the G-matrix were unpredictable, and the structure at one time period was not indicative of the structure at the next time period. Moreover, we show that the changes in the time-specific G-matrices were not related to changes in mean trait values or due to genetic drift. Selection, differences in acquisition/allocation patterns or environment-dependent allelic effects are therefore likely explanations for the patterns observed, probably in combination. Our result cautions against assuming constancy of the G-matrix and indicates that even short-term evolutionary predictions in natural populations can be very challenging.

Introduction

To understand the evolutionary response to selection in a given population, it is of fundamental importance to know the magnitude of the additive genetic variances and the structure of genetic covariances among traits, summarized by the G-matrix (e.g. Lande, 1979; Lande & Arnold, 1983). For long-term predictions of selection to be accurate, we need to know whether the G-matrix is stable over time (e.g. Steppan et al., 2002). If the G-matrix is highly integrated, then the response to selection is directed towards the direction of most genetic variation (‘lines of least resistance’, Schluter, 1996; Björklund, 1996, 2003), rather than in the direction of selection. Thus, long-term stability of the G-matrix can help to explain the observation of long-term stasis (e.g. Lande, 1979), and macro-evolutionary changes in terms of size rather than shape (e.g. Björklund, 1994). If the G-matrix is robust over time, so will the constraints be and we will see this as a lack of variation among taxa for certain trait combinations, which indeed has been observed (e.g. Björklund & Merilä, 1993; Björklund, 1994). On the other hand, if the G-matrix can change quickly over time, then the constraints imposed by the structure of the G-matrix will probably be important on a short ecological time scales, but is then unlikely to account for long-term stasis.

A number of theoretical models have tried to understand the evolutionary dynamics of the G-matrix (e.g. Bohren et al., 1966; Bulmer, 1980; Steppan et al., 2002; Jones et al., 2003, 2007; Björklund, 2004; Revell, 2007; see review by Arnold et al., 2008). The precise model of the genetic background is important for the detailed predictions of change over time (Barton & Turelli, 1987), and the time scale of interest matters for the mechanism of change. Hence, on longer evolutionary time scales, and in large populations, mutational input and selection are the main factors of change due to changing allele frequencies (e.g. Barton & Turelli, 1989; Arnold et al., 2008). On the much shorter ecological time scale, mutations and changes in allele frequencies are of less importance and too slow in relation to transient factors such as build-up of linkage disequilibrium. On this shorter time scale, stabilizing and directional selection will reduce variance, whereas disruptive selection will increase the variance, but because this is due to the build-up of linkage disequilibrium, this will disappear after a few generations of recombination, assuming the infinitesimal model (Bulmer, 1980). Hence, a short bout of directional selection will temporarily decrease genetic variance, and this is well known empirically (e.g. Gibson & Thoday, 1963; Sorensen & Hill, 1982; Hill & Caballero, 1992). Directional and quadratic selection can also create, or increase, genetic covariances (Bohren et al., 1966; for empirical data see e.g. Thoday, 1959; Scharloo et al., 1967), whereas correlated selection can drive the covariances towards zero (Villanueva & Kennedy, 1992) or strengthen them (Roff & Fairbairn, 2012), hence changing the shape of the G-matrix. Thus, given the ubiquity of selection in nature, there are many reasons to believe that the G-matrix will change over time (Barton & Turelli, 1989). In addition, the G-matrix can change even in the absence of selection and drift due to changes in the variance in allocation and acquisition as this can quickly change the sign of the genetic covariances (Houle, 1991; de Jong & van Noordwijk, 1992; Björklund, 2004). Thus, fluctuations in the G-matrix can be expected on these grounds, but it is unclear how large and fast these changes will be in any given natural population.

The lack of long-term studies of the stability of the G-matrix in natural populations is not surprising because the number of long-term studies of natural populations (i.e. more than 10 years; Clutton-Brock & Sheldon, 2010) is relatively few. Furthermore, the estimation of the G-matrix requires data on a large number of individuals with known relatedness, which is a challenging task in most natural populations. There is an increasing number of studies where the G-matrix, or parts of it such as the genetic variances, has been estimated in different environments in different populations and species (e.g. Cano et al., 2004 in Rana temporaria, Caruso et al. (2005) in Lobelia siphilitica and L. cardinalis, Estes & Phillips, 2006 in C. elegans). A frequent result is that the G-matrices differ to some extent (for a review, see Arnold et al., 2008), suggesting that at least some components of the G-matrix are rather labile and subject to environmental influence, in accordance with theory. Even if these studies are not strictly speaking a test of the temporal stability of the G-matrix in the same population, it can be argued that because most natural populations live in a temporally fluctuating environment, any environmentally induced changes in elements of the G-matrix are likely to have an impact on the temporal stability of G as well.

The results from the few studies that have examined temporal stability of G so far are inconclusive. For example, Pfrender & Lynch (2000) found significant changes between generations in Daphnia pulex. Divergence of the G-matrix was found over a 20-year period in the soil nematode Acrobeloides nanus (Doroszuk et al., 2008). In contrast, a study of a great tit (Parus major) population over a 40-year period showed no obvious changes in the G-matrix for three important life-history traits (Garant et al., 2008). On shorter time scale (3 years), a significant difference in G-matrices between years was found in the collared flycatcher (Merilä & Gustafsson, 1996).

To address this lack of empirical knowledge on temporal changes in, and hence stability of, the G-matrix, we used data from a long-term individual-based study of almost 7000 breeding events over the last 25 years in the collared flycatcher (Ficedula albicollis) on the island of Gotland, Southern Sweden (e.g. Merilä & Gustafsson, 1996; Qvarnström et al., 2006). We used an animal model approach to estimate the G-matrix (Henderson, 1950; Kruuk, 2004) using four different morphological traits for the whole 25-year period and separate G-matrices for five different 5-year periods. We then compare these matrices in terms of the size, shape and orientation and test whether the differences observed are due to drift or selection. The main finding was that the G-matrix changes in size and structure over the time periods in an unpredictable way.

Materials and methods

Study population

Pedigree information and phenotypic measures were collected using standardized methods as part of a 25-year (1980–2004) long-term study of a wild collared flycatcher population breeding on the island of Gotland, Sweden (57°30′N, 18°33′E (e.g. Merilä & Gustafsson, 1996; Qvarnström et al., 2006). Only adult individuals were included in this study, and sample sizes for each trait are given in Table A1.

Estimation of G-matrices

We used multivariate animal models (Henderson, 1950; Kruuk, 2004) to estimate the G-matrix as it is well accepted that this method returns the most accurate and unbiased estimates of additive genetic variances and covariances from natural populations (Kruuk, 2004). This is because information from many different types of relatives (occupying different habitats and time periods) can be simultaneously utilized so as to reduce confounding environmental factors that may otherwise inflate genetic parameter estimates (Kruuk & Hadfield, 2007). Information about the relatedness between individuals was collected for the entire study period based on field observations. The pedigree for the entire period spanned ten generations and consisted of 6727 individuals with 1911 maternities, 1828 paternities, 513 full sibs, 379 maternal half sibs, 398 paternal half sibs and an overall mean relatedness of 1.54 × 10−4. Extra-pair paternity (EPP) occurs in this population (~15%, e.g. Sheldon & Ellegren, 1999), and the pedigree will therefore contain some errors through the paternal line. Simulations have shown that EPP rates < 20% have a small impact (< 5%) on heritability estimates when sample sizes are large and heritability is as large as in the traits studied (Charmantier & Réale, 2005). Moreover, because the rate of EPP is not expected to change over years in a systematic way, this will not affect the comparisons of the G-matrices between time periods.

Our main aim was to estimate the additive genetic variance–covariance matrix (G) and examine its temporal stability over the study period. To do so, we firstly used all the available data (Table A1) and estimated G for the full 25-year period (hereafter GT). Secondly, to estimate time-specific G-matrices, we divided our 25-year longitudinal time series data into five 5-year periods (1980–1984, 1985–1989, 1990–1994, 1995–1999, 2000–2004, Table A1). We chose to use 5-year periods because this offered the best trade-off between having large numbers of relatives within each time period to allow us to estimate G with precision while at the same time examine relatively fine scale temporal change in G. We then used the data for each 5-year period in a multivariate animal model using the available pedigree for each time period. In total therefore, we estimated six G-matrices, one for the total period and one for each of the periods given above. Because we only use a subset of the data to estimate G for each time period, the pedigree structure for the different time periods differs because only related individuals with phenotypic information (within the specific time period) will contribute to the estimates of the genetic variances and covariances in G (‘pruned’ pedigree statistics are reported in Appendix S1).

There was no relationship between the pedigree structure (measured as either mean relatedness, number of individuals, or number of sibs) and the estimated evolvability, respondability or conditional evolvability for the different time periods (all P > 0.2). Consequently, the observed changes in G were not simply a result of changes in pedigree structure in the different time periods. The fitted multivariate animal model was the following:

display math

where X is the design matrix for the fixed effects vector (β), Z1 is the design matrix for the additive genetic effects u, Z2 is the design matrix for the repeated measures effects (‘permanent environment effects’) c and e is the error vector term. We included sex as a two-level fixed effect factor to correct for the slight sexual dimorphism in this species (Merilä & Gustafsson, 1996; Sheldon et al., 1998). The permanent environment effect accounts for fixed differences between individuals that are due to environmental (or nonadditive genetic) effects (Kruuk, 2004). Thus, for each time period, we estimated a 4 × 4 G-matrix, a 4 × 4 permanent environment matrix and a 4 × 4 error variance–covariance matrix. The estimated G-matrices are given in Table A3–4. The animal models were fitted in ASREML v3.0 (Gilmour et al., 2009), and standard errors of the (co)variance components were estimated based on the average information matrix (Gilmour et al., 2009).

Statistics

Univariate statistics

Because our measurements are on different scales, we need to transform the matrices to make the results interpretable. Here we use transformation by the mean as this transformation has a natural extension to evolutionary theory (Hansen & Houle, 2008). To test differences between successive time periods in phenotypic and genetic variance, we resampled the genetic variances by sampling from a normal distribution with the same mean as in the estimated matrix and the estimated error variance. We calculated the difference between years for the original data (largest minus the smallest) and counted the number of times we got a larger difference in the randomized data set. This was repeated 10 000 times, and the P-value is then simply the proportion of times we found a larger difference. To test for differences in mean trait values, we used two-tailed t-tests.

Multivariate descriptive statistics

We used a number of multivariate statistics to describe the matrices taken from Hansen & Houle (2008, 2009). These were mean evolvability, which is the mean amount of evolutionary change that can be expected for a random selection vector. Evolvability was estimated as the arithmetic mean of the eigenvalues of the G-matrix. The next statistic is mean respondability, which estimates the mean response to a random selection vector taking the genetic covariance among the traits into account. Mean respondability was estimated as

display math

where λ is the eigenvalues of the G-matrix, and I[x] is var(x)/E(x)2, and E[] is the expectation (mean). We also used the mean conditional evolvability, which is the mean evolvability of each trait holding correlations with other traits constant. Mean conditional evolvability was estimated as

display math

where H[λ] is the harmonic mean of the eigenvalues. To get an estimate of the uncertainty of the estimates, we permuted the matrices element-wise as above 10 000 times. We recorded the mean value and the 95% interval. We used modified intervals using the BCa –algorithm (Efron & Tibshirani, 1993), which gives a better 95% coverage of the distribution than by simply taking the upper and lower 2.5 percentiles. To test differences between successive time periods in these summary statistics, we used the same approach as above using element-wise permutation of the estimated G-matrices. The procedure of permuting the matrices element-wise is justifiable if the sampling covariance is low. In our case, these were indeed low (mean absolute correlation of residuals = 0.10, 95% CI = 0.075, 0.13; mean raw correlation = 0.031, 95% CI = −0.0081, 0.07). In addition, we estimated the effective number of dimensions, nD, (Kirkpatrick 2009) defined as the sum of the eigenvalues divided by the largest eigenvalue. This will range from 1.0, if the matrix only consists of a single dimension, to the number of traits measured if there are no covariances. A low number therefore indicates that covariances among traits are important.

To describe variation among G-matrices between time periods, we used the method of genetic covariance tensors (Hine et al., 2009), and the reader is advised to consult this article for a full description of the method. Briefly, the tensor, ΣG, is fourth-order dimensional and describes the variances and covariances of the variances and covariances in the different G-matrices. In practice, a matrix S of order n × n × n × n is created where the variances of the variances are on the diagonal and the variances of the covariances, and covariances of variances and covariances constitute the off-diagonal elements. From the S-matrix, eigenvalues and eigenvectors, or eigentensors EG, can be extracted. The eigentensors can in turn be transformed into n × n symmetrical matrices for further decomposition into eigenvalues and eigenvectors. The eigentensors describe the way the G-matrices have changed over time, and in particular which traits or trait combinations have changed. As with regular G-matrices, if one or two eigenvalues are substantially larger than the rest, it implies that the pattern of change has not been in all directions, but in some directions more frequently than other in other directions. This in turn can give important information of which traits or trait combinations are more flexible and amendable for change over time, and which are not. Finally, it is of interest to know how much each individual G-matrix contributes to the variance captured by each eigentensor, and this can be estimated by noting that the jth G-matrix is a linear combination of the eigentensors of ΣG

display math

where math formula is the kth eigentensor. The scaling factor C is calculated as the Frobenius inner product of math formula and Gj and can be used to calculate the proportion of total genetic variance each individual G-matrix contributes to a specific eigentensor. Thus, the proportion can be expressed as

display math

where the denominator is the Frobenius norm of the G-matrix. To get a picture of the change in genetic variance for a given trait combination by using the projecting

display math

This gives the genetic variance associated with the kth eigentensor for population j. Likewise, we can get the mean, phenotypic and genetic variance associated with each eigentensor by projecting the eigenvectors through the mean values, and the phenotypic and genetic covariance matrices, respectively. Because we are interested in the change over time, all values over the time periods are scaled by the values from the first time period. This means that the first time periods have all values equal to 1.0 and subsequent values represent differences in relation to the first time period.

Test of matrix differences

There are many different tests of matrix differences available, and they differ in terms of statistical properties as well as what they actually are testing (e.g. Roff et al., 2012). We used an approach where we started with the null hypothesis that the estimated G-matrices are all the same, but differ only as a result of sampling. If so, the mean matrix over time can be used as a summary (Hohenlohe and Arnold 208). We calculated the variance among the different time periods for each summary statistic (genetic variances, evolvability, etc.). We compared this variance to the variance of five matrices randomly drawn from the mean G-matrix over time with each element sampled from a normal distribution with the same mean as in the estimated matrix, and the mean estimated error variance. Because the time-specific matrices have an error variance as well, we used the same approach for these. Thus, we take into account the uncertainty of both the time-specific matrices and the mean matrix. We repeated this process 10 000 times and thus got a distribution of observed variances from the different time periods, and a distribution of variances from the five matrices randomly drawn from the mean G-matrix. The latter distribution was then used as the null distribution. To test the null hypothesis, we calculated the likelihood that the observed variance comes from the null distribution and compared this to the likelihood that it comes from a different distribution with the mean and variance estimated from the observed data. We compared the likelihoods by means of differences in the Akaike Information Criterion (AIC). The calculation of likelihoods for each distribution followed Lynch & Walsh (1998).

In addition to the summary statistic described above, we also used a version of the random skewer approach (Cheverud & Marroig, 2007). The idea of this test is that two matrices with a similar covariance structure will also give a similar response to a given selection vector. Likewise, if they differ, there will be a difference in the directions of the two response vectors. We simulated random selection vectors (see below) and estimated the angle between the response vector and the selection vector. It is known that vectors composed of random numbers scaled to unit length are not truly random, and therefore, we used the hypersphere point picking algorithm (Marsaglia, 1972; Weisstein, 2011). These vectors were then used as selection gradient vectors, β. We calculated response vectors using the multivariate version of the Breeder's equation, dz = Gβ. If the matrices from the same time periods have the same covariance structure, then all response vectors, dz, would be parallel. Due to sampling, they will differ and hence there will be a variance among the five different response vectors. This variance was compared with the variance of five matrices randomly drawn from the matrix estimate over the whole time period using a likelihood approach (see above for details).

To test for genetic drift, we used the results from Lande (1979, see also Hohenlohe & Arnold, 2008) who showed that after t generations of drift on a single trait, the expected change is mean zero but with a variance D(t) = Gt/Ne, where G is the additive genetic variance for the trait, and Ne is the effective population size. This can easily be generalized to a multivariate case replacing G with the genetic variance–covariance matrix G, and if G fluctuates over time, we can use the mean G. Now D is a matrix of variances and covariances. This can be extended to test whether an observed difference in a vector of m trait means math formula over time is due to drift,

display math

(Hohenlohe & Arnold, 2008). Unfortunately, we do not know Ne but used a very conservative estimate of 100. The true value is probably much higher (as yearly about 4000–5000 pairs are breeding), but this gives a test that is biased in favour of drift. Thus, if the test is still significant, we can clearly reject drift as a cause of the divergence. We used a generation time of 1.3 years, thus between the time periods the number of generations equals three.

To test for the possible effects of selection on the changes in the G-matrices over time, we used the approach by Hine et al. (2009; see also Lande & Arnold, 1983; Phillips & Arnold, 1989) where the expected change due to selection is given by

display math

where β is the selection gradient vector and γ is the matrix of second-order selection on means and on covariances. We compared the observed change in mean evolvability, mean respondability and mean conditional evolvability. We estimated γ using data on females from 1987 (n = 382) to get an indication of the magnitude of selection. The full details are given in Appendix S2.

Results

Univariate results

Most changes in population means between successive years are significant even though the magnitude of the differences is small, and never larger than one standard deviation between time periods (Fig. 1). Overall, the changes in genetic variance were larger than the changes in phenotypic variance, even though there were more often significant changes in phenotypic variance, probably as a result of the smaller standard errors associated with the phenotypic variance estimates. The observed variance among time periods in genetic variance was between 2.6 (mass) and 13 times (tarsus length) larger than the variance of genetic variances obtained from the mean G-matrix (Fig. 2), and the difference in AIC between the two models was at the lowest 2.46 (mass), 12.2 for wing, 12.25 for tarsus length and 12.48 for tail length. Thus, the null hypothesis of constant genetic variances differing only as a result of sampling could be clearly rejected, except for mass where the difference between the two models is small.

Figure 1.

Change in mean, genetic and phenotypic values over time periods expressed in relation to the first time period for each trait. Mean values (filled squares, solid line) are standardized in relation to the mean and standard deviation of the first time period hence all differences are in units of standard deviations. To increase visibility, variances were standardized by dividing by the variance of the first time period. Open circles: phenotypic variance, filled circles, dotted line: genetic variance. (*)P < 0.1, * P < 0.05, **P < 0.01, ***P < 0.001.

Figure 2.

The observed (dotted line) and estimated variance in genetic variance from the mean G-matrix (solid line) based on 10 000 parametric bootstrap samples. See text for details.

Multivariate results

The time-specific summary statistics are given in Table 1 in addition to the values for the whole period. The mean evolvability over time periods was on average 40% of maximum evolvability (Figs 3 and 4a). The observed variance among matrices was twice as high as expected assuming a single constant matrix over time, but the observed variance is very large and hence the two distributions did not differ. Respondability varied between time periods accordingly (Fig. 4b) and was on average 51% of the maximum variance (Fig. 3). The observed variance in respondability was twice as high as the expected variance, but again the variance was large. Conditional evolvability was on average 11.5% of maximum evolvability (Fig. 3), with the observed variance among periods being 8% larger than the expected variance (Fig. 4c). The effective number of dimensions ranged between 1.39 (1980–1984) to 1.72 (1990–1994; Table 1).

Table 1. Mean and 95% intervals as estimated from 10 000 parametric bootstrap estimates for mean evolvability (e), mean respondability (r) and mean conditional evolvability (c) (all × 104), the percentage of total variance accounted for by the first (gmax), the second eigenvalue (g2) and the effective number of dimensions from mean standardized G-matrices from each time period and from the total matrix
Time period r 95% r 95% c 95%gmax95%g295%nD
1980–19844.883.55, 7.436.714.53, 10.680.540.12, 3.1575.163.9, 89.719.213.1, 36.01.38
1985–19897.565.64, 9.5210.067.15, 13.413.440.24, 4.0269.558.6, 75.918.013.8, 24.81.44
1990–19944.883.44, 7.446.084.10, 10.002.110.17, 4.0862.547.2, 78.129.319.1, 47.81.77
1995–19997.095.24, 9.078.986.93, 12.052.570.13, 4.0160.448.4, 69.229.122.7, 40.21.73
2000–20044.853.64, 7.556.124.58, 9.971.680.15, 3.9565.049.9, 90.728.418.6, 43.51.67
Total7.556.53, 8.5610.048.51, 11.683.241.68, 3.7569.264.6, 73.018.315.7, 21.51.46
Figure 3.

Evolvability, respondability and conditional evolvability expressed in per cent of maximum evolvability over the time periods.

Figure 4.

The observed (dotted line) and estimated variance (solid line) among time periods in (a) evolvability, (b) respondability, (c) conditional evolvability and (d) mean response to random selection. Note the difference in scale on the x-axis in (d), which is made to improve visibility.

The difference in the orientation of the matrices can be seen by the eigenvectors of the time-specific matrices (Table 2). In all years, the leading eigenvector (gmax) was dominated by mass and tarsus length to a varying degree over time periods. The second largest eigenvector was generally dominated by wing and tail length, but the loadings differed between time periods.

Table 2. Loadings at the first two eigenvectors from each time period and the total matrix
Trait1980–19841985–19891990–19941995–19992000–2004Total
(a) gmax
Tail length0.140.065−0.032−0.052−0.0700.015
Tarsus length0.420.120.350.260.650.25
Wing length0.140.0770.150.024−0.0090.046
Mass0.880.990.920.960.750.97
(b) g2
Tail length0.810.820.770.860.650.88
Tarsus length−0.200.240.390.13−0.510.037
Wing length0.530.510.460.490.200.046
Mass0.12−0.120.2000.520.97

We have summarized the differences between the G-matrices in Fig. 5, where the two dimensions are the first and second eigenvectors of the G-matrix. The orientation is in relation to the first time period. The amount of genetic variance is given as the size of the ellipse, and it is clear that this varies substantially among time periods, with a large amount of variance in 1985–1989 compared with 2000–2004. The strength of the genetic covariances can be seen in the shape of the ellipses (ratio between the first and second eigenvalue), where more elongated ellipses means stronger covariances among traits. Again, there are clear differences, for example between 1980 and 1984 with strong covariances and 2000–2004 with weaker covariances. If the covariances were constant over time periods, all ellipses would align with the first time period. This is not the case as the orientation changes between time periods.

Figure 5.

Graphical description of the different G-matrices. The orientation has the first time period as the reference.

The variance in the response to random selection vectors was larger using the time-specific matrices (mean 4.51 compared to 3.24 for the mean matrix, Fig. 4d), but the difference is not significant. This is exemplified in Fig. 6a where we have to use the first eigenvector of the G-matrix (gmax) from the first time period as the selection gradient vector. The response vectors from the other time periods differed both in direction and in length, up to about 45 degrees in two time periods. We did the same using the eigenvector with the smallest eigenvalue (gmin). Because this was close to zero in the first time period, no response is expected in that time period. The predicted response of the other matrices differed substantially both in length and direction (Fig. 6b).

Figure 6.

Graphical examples of the differences in G-matrices. (a) the selection vector, β, is the eigenvector connected to the largest eigenvalue in the time period 1980–1984. (b) the selection vector, β, is the eigenvector connected to the smallest eigenvalue in the time period 1980–1984.

Eigentensors

To summarize variation among G-matrices over time, we used the eigentensor approach. First, we can note that the different time periods contribute about an equal amount to the overall variance among time periods, with the possible exception of the last time period, which seems to contribute less (Fig. 7). We found that the first eigenvalue of the S-matrix accounted for 69.1% and the second eigenvalue for 17.3% (total 86.4%) of the total variance. The first eigenvalue of math formula accounted for 89.6% of the variance and second eigenvalue 5.5% (in total 95.0%). Again, the first eigenvector captured basically all variation among years, and we will only consider this vector. The first eigenvector of math formula was [-0.014, -0.12, 0.047, 0.99]. The main change in variance was in terms of mass. The first eigenvalue of math formula accounted for 57.9% of the variance, and second eigenvalue 39.5% (in total 97.4%). The first eigenvector was [0.14, 0.61, 0.12, 0.77]. Thus, the vector describes changes in variance in tarsus and mass over time periods. The second eigenvector was [-0.31, -0.72, 0.17, 0.60], which describes changes in tail and tarsus in relation to mass and to minor degree wing length. The changes over time are visualized in Fig. 8 and show that, first, the change in genetic and phenotypic variance is not related to the changes in mean values (Fig. 8a,b) and second, the leading eigenvector of the first eigentensor very closely follows the genetic variance.

Figure 7.

The percentage of the total variance that can be accounted for by each time period in the analysis of eigentensors (see text for details).

Figure 8.

The time-period-specific overall means (filled squares), genetic variance (solid circles) and phenotypic variance (open circles) estimated by a projection of the first eigenvector of the first (a) and second (b) eigentensor matrix.

Drift and selection

The result from the test of genetic drift shows that the probability of finding a divergence in mean values due to drift alone ranged between 0.0096 and 10−32, clearly rejecting drift as a cause of the differences between time periods. We imposed selection (see Appendix S2) and analysed what levels of changes in G-matrices we can expect due to selection within a generation, bearing in mind that the time periods are longer than one generation. We used the first time period as a reference point, and the mean amount of change in genetic variance in a trait after one generation of selection was 14.4% (95% interval = 1.3–28.0%). This is smaller than the increase observed, for example, between 1990 and 1994 and 1995–1999 (see Fig. 1). The change after one generation of selection was 9.0% (95% interval = 0.9–16.9%) for evolvability and respondability. This is again considerably less than that observed in some periods (Table 1). Thus, given the selection observed in this population, one single bout of selection is not enough to change the G-matrices as much as we have observed.

Discussion

We found that the genetic variances in all four traits changed over time at an extent that could not be accounted for by sampling, but rather reflects a true change between time periods. The phenotypic variances and mean values changed as well, but there were no obvious patterns of covariation of changes across traits. The same was true for the multivariate descriptors; evolvability and respondability in particular varied among time periods and this was true also to a lesser extent for conditional evolvability, all of which suggests changes also in the covariance structure. The differences in the response to a given, random, selection vector between time periods support this conclusion.

The results show that genetic covariances will affect the outcome of selection in all time periods, but in different ways at each time period. This impact on evolutionary response can act in two ways; first, the response to selection in a trait is given by the selection on that trait but also by selection on genetically correlated traits, and hence, the response may differ from the particular selection. This is reflected in the metric respondability, which will be equal to the mean change due to selection, defined as evolvability, only if the genetic covariances are zero, and larger otherwise. In our study, respondability was always larger than evolvability (Fig. 3). Thus, the mean response is expected to be affected by the genetic covariances in a positive direction meaning that the response will, on average, be larger compared to a case where there are no genetic covariances. On the other hand, the genetic covariances can also have a negative impact constraining a response to selection. This is captured by the conditional evolvability, which in all years was considerably lower than evolvability and maximum evolvability. Thus, this shows that constraints in terms of genetic covariances are of a dual nature (Gould, 1989); promoting change in certain directions while constraining change in other directions. The relative importance can be seen when comparing respondability and conditional respondability to evolvability, and it is clear from Fig. 3 that the genetic covariances act more to constrain response to selection because the conditional evolvabilities are very low in all time periods. Taken together, the results show that the G-matrix is not stable over time, both the genetic variances and covariances change over time. In all time periods, the genetic covariances act as constraints to evolutionary change, even if the actual pattern of constraint changes over time.

What are the causes of changes in G over time? Additive genetic variance is defined as the sum of the product of allele frequencies and allelic effects (Falconer & Mackay, 1989). This means that differences can be due to changes in allele frequencies, allelic effects or both. Allele frequencies change due to drift and/or selection. There are good reasons to believe that we can exclude genetic drift because we found that the differences between matrices were highly significantly different from that expected if drift were causing the differences. This is not surprising because the population size is in the order of around 4–5000 individuals breeding every year, and even if the effective population size (Ne) most likely is lower than the number of individuals, drift will still be a weak force because the impact of drift is on the order of 1/Ne (e.g. Falconer & Mackay, 1989).

Short-term changes in G are expected due to selection (e.g. Bulmer, 1980) creating a build-up of linkage disequilibrium, and to a certain extent, the differences found can be obtained given the selection intensities. The changes in shape and orientation, which is a result of changing covariances, could be accounted for by selection to some extent, and this fits well with theory (Bohren et al., 1966). The studied population lives in a habitat that has remained relatively stable over the time period analysed; no major habitat shifts have occurred, the mean temperature has not changed significantly over time in the period when the young are growing [Temperature June, slope = 0.056 (SE = 0.042), P = 0.20, Fig. A3a; Precipitation June, slope = 0.39 (SE = 0.63), P = 0.54, Fig. A3b)]. The analysis of selection is complicated by the fact there seems to be significant indirect selection (Björklund and Gustafsson in prep.), and a full treatment is beyond the scope of this paper. On the basis of the preliminary data on selection, we see no reason to exclude selection as a viable explanation for some of the changes over time, even if selection seems to be rather moderate and is able to change the genetic parameters only to a minor extent. The larger changes that were observed might then be accounted for by bouts of even stronger selection or selection acting in the same direction for several years. However, correlated selection does not necessarily change mean trait values, but the genetic covariances, and there are models that show that the mean and the variance may evolve independently (De Vladar & Barton, 2011).

There are alternative, but not mutually exclusive, explanations. First, the changes in the patterns of allocation and acquisition of resources are known to change the G-matrix (Houle, 1991; de Jong & van Noordwijk, 1992; Björklund, 2004). If the variance in acquisition is greater than the variance in allocation, then this will result in strong genetic correlations. If there is a variance in microhabitat quality, this can lead to a larger variance in acquisition in some years and less so in other years, which will result in a change in the strength of the genetic correlations over time (Björklund, 2004). Second, the expression of the genes underlying the traits analysed might be environmentally dependent, that is we have genotype–environment interactions of non-negligible magnitude (Falconer & Mackay, 1989; Lynch & Walsh, 1998). These interactions can work in many different ways, from a change in mean value to a change in the rank order of breeding values with the environment, to an increased variance of breeding values or combinations of these. This is supported by an increasing number of studies showing genotype–environment interactions and environment-dependent genetic variances and heritabilities in birds (e.g. Merilä & Fry, 1998; Husby et al., 2010, 2011).

Studies of the change in G-matrices over time for the same population in the same environment are very rare. Pfrender & Lynch (2000) found that the G-matrices changed significantly between clones of Daphina pulex within a season but not between generations of sexually reproducing Daphnia. This indicates that there is clonal selection building up genetic disequilibria that are broken up during the sexual phase. The study of the soil nematode Acrobeloides nanus (Doroszuk et al. 2008) showed a large change over a 20-year period, with the difference mainly due to depletion of genetic variance making the G-matrix singular (since if the variances become zero, covariances are undefined). In a long-term study of great tits (Parus major), Garant et al. (2008) found no significant difference in G for three important life-history traits over two time periods 40 years apart. This is surprising because life-history traits are generally more plastic and susceptible to differences in acquisition than are morphological traits.

The lack of any clear patterns across studies suggests that the actual expression of genes and the variance among individuals in their expression patterns is strongly time, population and environment specific (Falconer & Mackay, 1989) and can also be sex-specific (Hallsson & Björklund, 2012). If this is a general pattern in natural populations, then detailed predictions of the future responses to selection are difficult, if not impossible, to make. The findings in this study have bearings on the understanding of long-term stasis and the discussion of the importance of constraints to evolution. One argument for long-term stasis has been consistency of the G-matrix over time (e.g. Lande, 1979; Björklund, 1994), and hence, any selection in dimensions precluded by the structure of the G-matrix will not result in a measurable response. Furthermore, it has been suggested that the strong pattern of size-, but not shape-, related differences between various avian taxa (Björklund, 1994) is a result of stable genetic correlations over time. The fact that the patterns observed is consistent with a hypothesis of stable G-matrices over time does not mean we can exclude the possibility of flexible G-matrices. If the findings here are general, then the most likely explanation for stasis and size-related patterns is stabilizing selection that keeps the G-matrix stable over time, and selection on trait combinations because even fairly low amounts of quadratic selection can cause changes in the structure of the G-matrix. This fits well with the empirical studies aimed at understanding the causes of the patterns of variation found in nature (e.g. Björklund & Merilä, 1993; Björklund, 1994; Estes & Arnold, 2007).

In all the time-specific matrices were the smallest eigenvalue close to zero, which indicates that there are dimensions of trait combinations that will not respond to selection in that direction. As can be seen in Fig. 6b, this direction differs between time periods, but still represents a constraint to evolutionary change. This constraint is strong in each single case, but variable over time, which suggests that these constraints are to be taken seriously at short ecological time scales. However, what constitutes a constraint in one time is not necessarily a constraint at another time, and we have shown here that the covariances can change over time. This applies also to the dimension that host the most genetic variation (‘line of least resistance’ sensu Schluter, 1996), which also can change between time periods. Because the response to selection is often strongly determined by this dimension (Björklund, 1996), the main evolutionary trajectory will be affected by changes in the structure of the G-matrix. If the G-matrix changes quickly as a result of selection and/or changes in the variance of acquisition of resources, then our ability to predict long-term evolutionary change is limited and strictly speaking only valid for time period for which the G-matrix is stable. This has important consequences for theoretical work that often (for simplicity) assumes stable G-matrices (e.g. Duputie et al., 2012). Because we have shown that the G-matrix can change in all aspects even at very short (ecological) time scales, the impact of possible constraints and predictions of the directions of selection can only be seen in relation to the particular G-matrix at a given time. This is an empirical question that is likely to be unique for each study system and therefore has to be addressed in each particular case.

Acknowledgments

This research was funded by various grants from the Swedish Research Council to LG. AH was supported by a Marie Curie Re-Integration grant. We thank Ruth Shaw, Derek Roff and Bruce Walsh for stimulating discussion on this topic.

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