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Keywords:

  • common principal component analysis;
  • conservation biology;
  • habitat fragmentation;
  • quantitative genetics

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Abstract The present study of Brassica cretica had two objectives. First, we compared estimates of population structure (Qst) for seven phenotypic characters with the corresponding measures for allozyme markers (Fst) to evaluate the supposition that genetic drift is a major determinant of the evolutionary history of this species. Secondly, we compared the genetic (co)variance (G) matrices of five populations to examine whether a long history of population isolation is associated with large, consistent differences in the genetic (co)variance structure. Differences between estimates of Fst and Qst were too small to be declared significant, indicating that stochastic processes have played a major role in the structuring of quantitative variation in this species. Comparison of populations using the common principal component (CPC) method rejected the hypothesis that the G matrices differed by a simple constant of proportionality: most of the variation involved principal component structure rather than the eigenvalues. However, there was strong evidence for proportionality in comparisons using the method of percentage reduction in mean-square error (MSE), at least when characters with unusually high (co)variance estimates were included in the analyses. Although the CPC and MSE methods provide different, but complementary, views of G matrix variation, we urge caution in the use of proportionality as an indicator of whether genetic drift is responsible for divergence in the G matrix.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Numerous species exist as small and isolated patches because of the increasing fragmentation of their natural habitats. Such populations are expected to lose a proportion (1/2Ne, where Ne is the effective population size) of their heterozygosity and additive genetic variance each generation owing to random genetic drift and inbreeding (Wright, 1951). As future evolutionary adaptation depends on the existence of additive variance within populations (Lynch & Walsh, 1998), it is possible that the genetic effects of landscape fragmentation represent a significant threat to the long-term maintenance of some species (Lande & Barrowclough, 1987; Schemske et al., 1994). Yet, despite evidence for a link between small population size and low diversity at selectively neutral marker loci (Ellstrand & Elam, 1993), it is still uncertain whether drift can reduce additive genetic variation – and adaptive potential – over timescales considered by most conservation biologists (Lande, 1988). Furthermore, only a few authors have used a multivariate approach to examine how the stochastic processes alter the genetic variance–covariance matrix (G) for suites of phenotypic characters (Roff, 2000; Phillips et al., 2001).

The expected reduction in the additive (co)variances as a result of genetic drift does not follow the same pattern as the loss of single-locus heterozygosity when there are high levels of dominance or epistatic variance in the character and when populations are forced through extreme bottlenecks, a process that might convert nonadditive (co)variance into additive (co)variance (Bryant & Meffert, 1988; Goodnight, 1988; Willis & Orr, 1993). Moreover, as drift is a stochastic process, one would expect wide variation around the theoretical expectations, so that some populations (or characters) might experience an increase in the (co)variance, even if on average the additive (co)variance decreases (Avery & Hill, 1977; Lynch, 1988; Zeng & Cockerham, 1991; Whitlock & Fowler, 1999; Phillips et al., 2001). For these reasons, it probably takes many generations before drift causes a decline in the entire G matrix.

The Mediterranean region was not glaciated during the Pleistocene and therefore provides ideal model systems for the testing of hypotheses about the long-term genetic effects of population isolation. Species with a large number of spatially isolated populations are particularly common in the flora of limestone cliffs such as those occurring in the central Aegean (Runemark, 1971). Many of these ‘chasmophytes’ are taxonomically isolated and show extensive differentiation in morphological characters (e.g. Snogerup, 1967). For example, available data for Brassica cretica Lam., a close relative of cabbage B. oleracea L. (Snogerup et al., 1990), indicate a mosaic of variation reflected in sharp random differentiation among populations and no obvious relationship with local habitat conditions (B. Widén, unpublished data). Although adaptive explanations cannot be ruled out, it seems reasonable to invoke genetic drift as a major evolutionary force in this and other chasmophytes (Snogerup, 1967).

The present study of B. cretica had two primary objectives. First, we compared estimates of population divergence for seven phenotypic characters with the corresponding data for eight putatively neutral allozyme markers, to test the supposition that genetic drift is a major factor in the differentiation of B. cretica, and that this species therefore serves as a suitable model for studying how a long period of population fragmentation influences the genetic (co)variance structure. Secondly, we used two approaches, the common-principal-components (CPC) technique (Flury, 1988; Phillips & Arnold, 1999) and the method of percent reduction in mean-square error (Roff, 2000), to compare the G matrices of five B. cretica populations. These approaches enabled us to distinguish between two hypotheses: (i) that the matrices differ by simple constants of proportionality (as predicted under additive, neutral models), and (ii) that most of the variation involves principal component structure (as predicted under scenarios involving stochastic, element-specific changes in the G matrix).

Plant material

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Brassica cretica is a diploid (2 n  = 18), largely self-incompatible, long-lived herbaceous plant with a native distribution in Greece (Crete, N. Peloponnisos), where it occurs in isolated ravines and cliff systems, largely inaccessible to humans and grazing animals. The plants are up to 150 cm high, with a variable leaf shape and branching pattern. The white or yellow, protandrous flowers are arranged in branched inflorescences (racemes) and develop into siliques, each consisting of 15–25 seeds with no particular adaptations for long-distance dispersal ( Snogerup et al., 1990 ).

This study involves seven Cretan populations (Table 1), sampled by collecting cuttings or seeds from 15 to 30 spatially separated (>10 m) plants in each locality, and grown in a greenhouse at the University of Lund, Sweden. The populations are interfertile (Rao et al., 2002) and can be assigned to one western [Topolia (To), Roka (Ro), Macheri (Ma)] and one eastern group [Miliardo (Mi), Gonies (Go), Monastriaki (Mo), Moni Kapsa (Mk)](Fig. 1).

Table 1.  Information on the study populations. Sample sizes refer to the number of individuals used in the allozyme study ( Nalloz ) and the number of paternal halfsib families used in the quantitative genetic analyses ( Nsires ).
Population/locationLongitudeLatitudeSample sizes
NallozNsires
  • *

    Populations excluded from the matrix comparisons because of small sample sizes for reproductive characters.

Topolia (To)23°40′35°27′4332
Roka (Ro)23°43′35°29′2525
Macheri (Ma)24°07′35°25′3731
Miliardo (Mi)*25°24′35°05′3730
Gonies (Go)25°27′35°14′4634
Monastriaki (Mo)*25°50′35°05′4529
Moni Kapsa (Mk)26°03′35°01′4131
image

Figure 1. Map showing Crete with the locations of the B. cretica populations used in this study. To = Topolia, Ro = Roka, Ma = Macheri, Mi = Miliardo, Go = Gonies, Mo = Monastriaki, Mk = Moni Kapsa.

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Allozyme analyses

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Allozyme polymorphism was assessed with standard horizontal starch gel electrophoresis using extracts from young flower buds. Four enzyme systems were investigated: Pgi, Pgm, Pgd, and Aco. Staining and buffer recipes are as described by Wendel & Weeden (1989). The level of population differentiation at these loci, assumed to be selectively neutral, was quantified as the standardized genetic variation between populations (Fst; Wright, 1951) according to the variance component approach of Weir & Cockerham (1984). Estimates of Fst (and their 95% confidence intervals, CI) were obtained with FSTAT (Goudet, 1995). This programme was also used to test for deviations from Hardy–Weinberg expectation and linkage disequilibrium. A total of 274 plants was used for the allozyme analyses (Table 1).

We cannot rule out the possibility that geographically varying selection has contributed to the structuring of allelic variation at some allozyme loci, but assume that these effects are too weak to influence the overall Fst estimate (averaged over loci).

Crosses and measurements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Quantitative genetic data were obtained with a nested mating design in which 25–34 pollen donors from each population (Table 1) were crossed to distinct pairs of (nonsib) seed parents from the same population. Hence, each progeny could be assigned to a population, a paternal halfsib family (sire) and a maternal fullsib family (dam). Four seedlings per dam, planted in separate pots and placed in random positions in a greenhouse, were scored for the following characters: (1) the mean length and (2) the mean level of dissection of the ninth and tenth leaf, (3) the distance between the 10th and 11th node (internode length), (4) the number of nodes to the first flower, (5) date of first flowering, (6) the number of lateral racemes in the central (terminal) inflorescence (inflorescence size), and (7) petal size. These characters were chosen to include several aspects of the life-cycle, thereby minimizing the risk of excluding critical parts of the G matrix. Leaf dissection was quantified as the length of the outer contour divided by the square root of leaf area, measured with a video camera and an image analysis system on a Macintosh computer (IMAGEGRABBERTM, Neotech Limited, UK; OPTILABTM, Graftek, France). Petal size refers to the mean length of the four petals in one of the first flowers on each plant. Phenotypic data were obtained for a maximum of 1535 individuals.

Partitioning of quantitative genetic variation

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

The phenotypic data were subjected to restricted maximum likelihood (REML) analyses (using the REML procedure in GENSTAT 5, 1998) to partition the total variance in each character into variance among populations (Vpop), among sires within population (Vsire), among dams within sire, and among offspring within dams. The level of population divergence (referred to as Qst following Spitze, 1993) was quantified according to Wright (1951):

  • image

where Va is the additive variance within populations, calculated as 4Vsire (Lynch & Walsh, 1998). Qst is the quantitative genetic analogue of Fst and therefore allows direct comparison with levels of differentiation at marker loci. A close similarity between estimates of Qst and Fst suggests that the metric characters have been influenced by the same evolutionary force (genetic drift) as the marker genes (Spitze, 1993; Waldmann & Andersson, 1998; Whitlock, 1999). Approximate 95% CIs of the Qst estimates were obtained with the VFUNCTION option (GENSTAT 5, 1998), a procedure based on the delta method (Lynch & Walsh, 1998).

The relationships between Qst, Vpop, Vsire and Va used in the present investigation are valid only for randomly mating populations. This assumption should be at least partly satisfied, given the high levels of self-incompatibility in most populations (B. Widén, unpublished data) and the small number of significant deviations from Hardy–Weinberg equilibrium at the allozyme loci (see Results).

Comparison of (co)variance matrices

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Genetic (co)variance matrices were compared with CPC technique outlined by Flury (1988) and applied to G matrix analysis by Phillips & Arnold (1999). This maximum-likelihood method tests whether two or more (co)variance matrices have completely unrelated structures, whether the matrices share one or more principal components (eigenvectors), whether the matrices differ by a simple constant of proportionality (matrices share identical principal components but their eigenvalues differ by a proportional constant), or whether the matrices are identical. Using a log-likelihood statistic to quantify the fit of each model to the observed matrices, the testing procedure in a CPC analysis progresses upwards from unrelated structure through partial common principal components (PCPC), CPC, proportionality and ends at equality. Following the ‘jump-up’ approach (Flury, 1988; Phillips & Arnold, 1999), we determined the highest point in the Flury hierarchy at which accumulated differences in the matrices became significant and used the model immediately below as the best-fitting model for the observed matrices. Covariance component matrices, based on one-way analyses of (co)variance among fullsib families, were compared with the CPCrand program (Phillips, 1998a), which uses a randomization approach to evaluate each hypothesis in the Flury hierarchy and a ‘matrix bending’ procedure to eliminate negative eigenvalues (Phillips & Arnold, 1999). As a complementary approach, we estimated product-moment (co)variance matrices from the sire means (averaged across dams) and utilized the CPC program (Phillips, 1998b) to compare these matrices. The latter approach allows direct tests of significance using parametric methods, but depends on the assumption of multivariate normality (Flury, 1988). Sire mean (co)variances have a stronger additive component than estimates from fullsib analyses, but include fractions of the within-family (co)variance (Lynch & Walsh, 1998). Current versions of CPCrand and CPC do not allow nesting of dams within sires.

Matrix permutation procedures (2000 permutations) were applied to examine whether the degree of matrix similarity in pair-wise CPC analyses (quantified as the number of shared components) was significantly correlated with the geographical distance separating the parent populations (Fig. 1) or the distance between population mean phenotypes (as revealed by a canonical variate analysis using population as class variable, the seven phenotypic traits as independent variables, and sire means as replicates).

The CPC method often underestimates the degree of structure shared by different matrices, particularly at intermediate levels of similarity (Houle et al., 2002). For this reason, we also used a second approach, the method of percent reduction in mean-square error (Roff, 2000), to evaluate the type of differences between matrices. In this approach, the difference between two ‘vectorized’ matrices is partitioned into two components, one corresponding to a model that only assumes proportional changes vs. one in which differences between matrices are assumed to be nonproportional. This technique is based on the comparison of mean-square errors (MSE) for three regression models:

Model 1 (equal matrices):

  • image

Model 2 (proportional matrices):

  • image

Model 3 (different matrices):

  • image

where C is the number of variances and covariances, Xi and Yi are the values of a particular element (i) in the two matrices (X, Y), b0 and B0 are the slopes of the reduced major axis regression forced through the origin, and a, b, A and B are the parameters of the reduced major axis regression with the intercept included. On the assumption that selection causes nonproportional changes, the percent reduction in MSE from model 1 to model 2 can be used as an estimator of the effect of drift alone, while the reduction from model 1 to model 3 can be attributed to both drift and selection (Roff, 2000).

To test for scaling effects arising from differences in the mean phenotype, we repeated some of the matrix analyses with log-transformed data. Low flowering percentage reduced the number of individuals that could be scored for all characters, making it necessary to exclude two populations (Mi, Mo; n < 110) from the matrix comparisons.

Analysis of population structure

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Eight putative loci were resolved from the four enzyme systems: Pgi-2 : 3, Pgm-2 : 5, Pgm-3 : 3, Pgd-1 : 3, Pgd-2 : 5, Aco-1 : 3, Aco-2 : 5, and Aco-4 : 4 (the first number denotes the locus and the second the number of alleles found at this locus). The different loci appeared to provide more or less independent information on single-locus diversity: none of the pair-wise combinations of loci were found to be in significant linkage disequilibrium after Bonferroni correction. Genotypic frequencies at polymorphic loci (data not shown) generally conformed to Hardy–Weinberg predictions, the only exceptions being a statistically significant deficiency of heterozygotes at the Pgm-2 locus in the Ro, Go and Mi populations (inline image = 5.3–7.2, P < 0.01–0.05) and at the Pgi-2 locus in the Go population (inline image = 6.0, P < 0.05). The Go population was polymorphic at all eight loci, whereas the remaining populations showed variable levels of fixation, the most extreme being Ro (six monomorphic loci). Hierarchical analyses of allozyme diversity revealed a high between-population component of diversity, with locus-specific estimates of Fst ranging from 0.08 (Aco-1) to 0.93 (Pgi-2), and a mean Fst (0.63) significantly greater than 0 (95% CI 0.46–0.76). Exclusion of the two loci showing an excess of homozygous had little effect on the estimate of population structure (mean Fst 0.59, 95% CI 0.42–0.69).

The phenotypic characters showed high levels of population differentiation (Table 2), with estimates of Qst ranging from 0.35 (inflorescence size) to 0.79 (leaf dissection). All values were sufficiently large to be declared significant (95% CI excluding 0 in all cases), but there were no significant differences between the Qst values and the overall Fst value (overlapping CI in all cases). Patterns of differentiation among mean phenotypes (Fig. 2) showed no relationship to the place of origin (Fig. 1). Hence, there was no tendency for the degree of divergence to increase with the geographical distance separating the parent populations (r = 0.12, P = 0.31). Inspection of character loadings (Table 3) and character means (data not shown) indicated relatively long internodes, early flowering and small inflorescences for Ma plants, relatively weakly dissected leaves and small petals for Mk plants, and relatively long leaves and a large number of nodes for plants representing other populations (Ro, Go, To).

Table 2.  The level of population structuring for vegetative and floral characters in B. cretica , as determined by Qst and its 95% confidence intervals (CI).
CharacterQstCI
Leaf length0.5620.222, 0.902
Leaf dissection0.7940.528, 1.059
Internode length0.5550.231, 0.879
Node number0.7500.505, 0.994
Flowering date0.7720.522, 1.022
Inflorescence size0.3520.048, 0.656
Petal size0.5570.244, 0.869
image

Figure 2. Canonical variates plot of the five B. cretica populations used in the analyses of G matrices. The populations are represented by their centroids: Ma = Macheri, Mk = Moni Kapsa, Ro = Roka, Go = Gonies, To = Topolia.

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Table 3.  Character loadings for the seven vegetative and floral characters on the first canonical variates (CV) in a canonical variate analysis using population as class variable.
CharacterCV1CV2
Leaf length0.4150.261
Leaf dissection0.5010.458
Internode length −0.2510.517
Node number0.654 −0.492
Flowering date0.268 −0.553
Inflorescence size0.080 −0.325
Petal size0.4540.493

Variances and covariances

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Collapsing the data into sire means did not substantially influence the comparison of matrices, so only the covariance component matrices are presented (Table 4). Estimates of the genetic variance, obtained with the program H2boot (Phillips, 1998c), differed between characters and were significantly greater than zero for most characters (P < 0.001–0.05, Table 4), according to a bootstrapping procedure involving 5000 resamplings from the original data sets (Phillips, 1998c). Apart from a few negative covariances, most of which involved flowering date, the majority of the covariances were positive. However, all but one covariance (leaf dissection vs. node number in the Go population) failed to reached significance. Different populations showed high (co)variances for different characters, as demonstrated by the Mk population, which had high (co)variances for reproductive characters, and the Go population, which had high (co) variances for vegetative characters.

Table 4.  Genetic (co)variances from one-way analyses of (co)variance among fullsib families (raw data). The two largest absolute values for each element are in italics. Estimates were obtained with the program H2boot ( Phillips, 1998c ).
Character(s)Population
MaToMkRoGo
  • *

    P  < 0.05,

  • * *

    P  < 0.01,

  • * * *

    P  < 0.001.

Leaf length (LL)2.309 * 1.909*0.8140.7192.116 * *
LL – Leaf dissection0.2571.4270.3080.0441.313
LL – Node number −1.744 −1.5721.20010.0324.228
LL – Internode length2.704 −0.075 −0.3930.3960.112
LL – Flowering date22.1352.01957.68213.167 −3.047
LL – Inflorescence size0.414 −0.1300.1330.8640.591
LL – Petal size0.8410.364 −0.012 −0.0700.195
Leaf dissection (LD)0.1863.316 * * 0.437* −0.2882.074 * *
LD – Node number0.688 −5.1130.0108.8225.891 *
LD – Internode length −0.6250.939 −0.5540.5250.658
LD – Flowering date −12.72820.95047.82512.307 −13.036
LD – Inflorescence size0.2110.654 −0.1040.485 −0.227
LD – Petal size0.1090.897 −0.071 −0.1140.762
Node number (NN)19.041**43.916 * * * 29.156***64.375 * * * 12.895***
NN – Internode length3.6770.765 −2.9758.778 −0.445
NN – Flowering date42.83151.444 −0.11781.840 −0.707
NN – Inflorescence size1.1415.8754.7534.0991.012
NN – Petal size −1.1550.0510.9753.4991.191
Internode length (IL)10.978 * 1.102**1.583**2.4250.111
IL – Flowering date −42.00093.49288.979 −12.864 −2.011
IL – Inflorescence size0.1411.399 −0.5391.603 −0.067
IL – Petal size1.275 −0.151 −0.4460.972 −0.017
Flowering date (FD)518.140***619.860 * * * 1798.600 * * * 280.010***312.320***
FD – Inflorescence size16.276100.330121.070  8.874 −2.723
FD – Petal size −1.575132.54025.06220.6893.608
Inflorescence size (IS)1.424**3.742 * * * 2.565 * * * 1.1040.527**
IS – Petal size0.2800.3400.657 −0.1840.069
Petal size1.750 * * * 1.424***0.885***2.170 * * * 0.427*
Mean absolute value25.23739.13678.13919.33313.299

After removing the sign of each covariance estimate and averaging over characters, the (co)variances were found to be highest for the Mk population (mean 78.1) and lowest for the Ro and Go populations (mean < 20), largely because of (co)variances related to flowering time (Table 4). These patterns were also apparent in the sire mean analyses (data not shown), the major exception being the Ma population, which had the highest mean (co)variance of all populations after log-transformation (cf. Table 4).

CPC analyses

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

The CPC analyses involving all (co)variance matrices provided little support for the equality or proportionality hypotheses, regardless of whether the comparisons were based on randomization or parametric tests (P < 0.001 in all cases). These analyses also rejected the hypothesis of shared structure, as shown by the significant P-value for a model involving a single common component (PCPC1 < 0.05).

Pair-wise analyses on raw data generally revealed low levels of matrix similarity (Table 5), the most important exceptions being the Ma matrix, which had the same CPC structure as the To, Mk and Ro matrices in the randomization tests (PCPC > 0.30), and the To matrix, which was found to share its CPC structure with the Ro and Go matrices in the parametric analyses (PCPC > 0.24). When each character was transformed to logarithms, the Ma and Mk matrices turned out to be statistically indistinguishable (Pequality = 0.19) and to share all of their CPC structure with the To matrix (PCPC > 0.75). For three comparisons (Ma–Ro, Ma–Go, Mk–Ro), the best-fitting model for the parametric analyses changed from no shared structure (natural scale: PCPC1 < 0.02) to a model involving a single common component (log scale: PCPC1 > 0.25). None of the pair-wise CPC analyses supported the proportionality hypothesis (Pproportionality < 0.05).

Table 5.  Results of hierarchical comparison of G matrices (jump-up approach). Entries give the best-fitting model for the observed data. CPC indicates a full model with six common components, while CPC1, CPC2, etc. indicate partial models involving one, two or more common components.
ComparisonRandomization testsParametric tests
Raw dataRaw dataLog data
  1. P -values are given in parentheses.

All populationsUnequalUnequalUnequal
Ma–ToCPC (0.307)UnequalCPC (0.975)
Ma–MkCPC (0.476)UnequalEqual (0.189)
Ma–RoCPC (0.344)UnequalCPC1 (0.465)
Ma–GoCPC1 (0.161)UnequalCPC1 (0.535)
To–MkUnequalUnequalCPC (0.790)
To–RoUnequalCPC (0.298)CPC (0.198)
To–GoUnequalCPC (0.241)CPC (0.157)
Mk–RoUnequalUnequalCPC1 (0.260)
Mk–GoCPC (0.566)UnequalUnequal
Ro–GoUnequalUnequalUnequal

Judging from the randomization tests, there was a tendency for the difference between two G matrices to increase with the geographical distance between the parent populations: comparisons involving one western and one eastern population (Ma–Mk, Ma–Go, To–Mk, To–Go, Mk–Ro, Ro–Go) generally supported models involving no shared structure or a single common component, whereas most of the within-region comparisons (Ma–To, Ma–Ro, To–Ro, Mk–Go) supported the CPC model (Table 5). When quantified as the number of shared components, the degree of matrix similarity showed a marginally significant negative correlation with geographical distance (r = −0.44, P = 0.06). No such pattern was observed in the parametric analyses (P > 0.19). Results of the CPC analyses showed no significant relationship with patterns of variation among mean phenotypes (P > 0.10). For example, the Ro and Go populations had similar mean phenotypes (Fig. 2), but widely different G matrices, regardless of how the matrices were compared (Table 5).

Percent reduction in MSE

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

The percent reduction in mean-square error (MSE) from model 2 (proportional matrices) to 3 (different matrices) was small (<16%), compared with the reduction in MSE from model 1 (equal matrices) to model 2 (range = 35.5–99.5%; Table 6), the major exception being the Ro–Go comparison, which showed the opposite pattern (reduction in MSE = 15.2% for model 2–3 vs. 1.1% for model 1–2). Hence, proportionality was the best-fitting model in almost all comparisons, a finding that also applies to analyses of sire mean matrices (data not shown). Most of these patterns can be attributed to unusually high (co)variances involving node number and flowering time (Table 4). Exclusion of these variables from the covariance component matrix reduced the average reduction in MSE attributable to proportional changes (model 1 vs. 2) from 73.2 to 17.2%, whereas the reduction attributable to nonproportional changes (model 2 vs. 3) increased from 3.4 to 27.5% (Table 6).

Table 6.  Percent reduction in mean square error for pair-wise comparison of covariance component matrices (Table 4). Results are given for reduction from model 1 (equal matrices) to model 2 (proportional matrices) or model 3 (different matrices). Reduced matrices include all characters except node number and flowering time.
ComparisonOriginal matricesReduced matrices
2323
Ma–To35.540.48.430.9
Ma–Mk98.098.139.158.4
Ma–Ro73.478.150.664.1
Ma–Go93.393.39.531.3
To–Mk90.290.524.268.0
To–Ro69.876.57.339.1
To–Go81.482.331.057.0
Mk–Ro89.791.42.037.0
Mk–Go99.599.5032.4
Ro–Go1.116.30.328.7
Mean73.276.617.244.7

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

Understanding the genetic consequences of habitat fragmentation is fundamental to conservation biology, especially in regions where many populations have become fragmented through human activities (Lande & Barrowclough, 1987). Although the role of genetic diversity has become a major topic in studies of small, isolated populations (Schemske et al., 1994), only a few authors have considered variation in potentially adaptive phenotypic characters and there is still a paucity of studies relating estimates of genetic (co)variances to population structure (Widén & Andersson, 1993; Waldmann & Andersson, 1998; Podolsky, 2001). The major goal of the present study was to decide whether the null hypothesis of equal G matrices could be rejected for spatially isolated populations of B. cretica and whether genetic drift has caused large, consistent changes in the genetic (co)variances. However, before addressing these questions, it is necessary to ask whether genetic drift has played a major role in the past evolution of B. cretica.

The small areas of most cliff systems and the pronounced local differentiation found in B. cretica (Snogerup et al., 1990) and other cliff elements (e.g. Snogerup, 1967) suggest a long history of population isolation and a great potential for stochastic factors to shape evolutionary change in these species. Genetic data from B. cretica are consistent with this hypothesis. First, analysis of variation at eight allozyme loci demonstrated high levels of between-population variance, with estimates of Fst (mean 0.63) approaching values normally found in selfing species (Hamrick & Godt, 1996). Second, comparison of mean phenotypes indicates a mosaic of variation reflected in random differentiation among populations, with estimates of Qst (0.35–0.79) not significantly different from the mean Fst value. Thirdly, crosses between populations of B. cretica have been found to induce significant heterosis effects in early performance characters (Rao et al., 2002), a pattern that can be attributed to the random fixation of deleterious recessive alleles in different populations and the masking of such genes in interpopulation hybridizations (Lynch & Walsh, 1998). On the basis of these observations, we believe that the populations of B. cretica have been genetically isolated for substantial periods of time and that this species therefore provides a useful system in which to test hypotheses about the long-term effects of habitat fragmentation.

Under a strictly additive model of genetic variation, the genetic variances and covariances for a set of phenotypic characters are expected to decrease in proportion to the inbreeding coefficient of the population (Wright, 1951; Lande, 1979; Roff, 2000). Genetic drift is also expected to cause wide fluctuations around the additive expectations, not only from one population (or generation) to another (Lynch, 1988; Zeng & Cockerham, 1991; Whitlock & Fowler, 1999), but also between genetically independent characters, a pattern that would lead to wide variance in the orientation and magnitude of genetic variance and covariance (Phillips et al., 2001). Based on these considerations and the wide variety of characters measured in this study (leaf shape, node density, flowering date, inflorescence size, petal length), one would expect the G matrices of B. cretica populations to differ in their principal component structure.

Judging from the results of the CPC analyses, the hypothesis of equal matrices could be rejected in almost all comparisons, regardless of whether or not the data were collapsed into sire means or transformed to logarithms. The best-fitting model was specific to each comparison, ranging from no common component to identical CPC structure, and there was no support for the proportionality hypothesis, despite a more than fivefold difference in the mean level of (co)variance. These results can be attributed to two major patterns in the G matrices: (i) different populations showed high (or low) (co)variance for different characters, and (ii) some characters (especially flowering time) had a disproportionately large influence on differences in the mean (co)variance. Some of the differences can also be interpreted as scaling effects arising from the wide separation of mean phenotypes: a log-transformation of all traits not only shifted the rank order of populations with respect to the mean (co)variance, but also influenced the results of the CPC analyses, especially those involving the Ma, To and Mk populations. When combined with the data on population structure, these observations lead to two suggestions: (i) that drift has caused large, idiosyncratic changes in the principal component structure, and (ii) that some of the changes can be attributed to shifts in the mean phenotype.

Comparison of matrices using the method of percent reduction in MSE (Roff, 2000) gave strong support for a model in which all changes are assumed to be proportional, contrasting with lack of proportionality in the CPC tests. Such incongruence has also been documented by Roff (2000) and can be attributed to a variety of factors (Bégin & Roff, 2001). As found in a recent simulation study (Houle et al., 2002), differences in one or a few principal components often prevent detection of shared structure lower in the Flury hierarchy. Furthermore, element-by-element approaches such as the MSE method are likely to be more strongly influenced by one or a few divergent elements than methods comparing the overall structure of G matrices. In the case of B. cretica, we found weaker support for the proportionality model when the high (co)variances of node number and flowering time were removed from the analyses. Collectively, the data from the CPC and MSE analyses indicate that the two methods perceive different aspects of matrix evolution (see also Bégin & Roff, 2001) and that both proportional and nonproportional changes contribute to the pattern of variation between matrices.

Sample sizes in the present investigation were reasonably large (c. 30 paternal halfsib families per population), yet the effect of sampling variability was substantial as indicated by the relatively few significant (co)variance estimates. Hence, the observed differences between G matrices may be a consequence of the large error associated with the estimation of (co)variance components. One way to address this problem is to compare the parametric and nonparametric results of the CPC analyses (Bégin & Roff, 2001). As shown in Table 5, the parametric analyses – which tend to be most powerful (Bégin & Roff, 2001) – indicate greater dissimilarity between G matrices than does the nonparametric version, the most notable exceptions being the To–Ro and To–Go comparisons, which showed the opposite pattern. Although these results agree with the low power available in analyses of (co)variance matrices, we also note that the simultaneous analysis of all five matrices and three of the pair-wise analyses (To–Mk, Mk–Ro, Ro–Go) always indicated low levels of matrix similarity, regardless of whether matrices were compared using parametric or nonparametric methods. In view of these observations, we consider the overall pattern of G matrix variation – as revealed in the combined analyses – to be more reliable than the results of the pair-wise comparisons.

We cannot distinguish between simple, stochastic fluctuations in allele frequencies and more complex mechanisms, e.g. bottleneck-induced release of additive (co)variance in characters with high levels of nonadditive (co)variance (Goodnight, 1988; Willis & Orr, 1993), as determinants of the divergence in the G matrices. Furthermore, it is too early to dismiss the possibility that selective forces are contributing to the wide separation of matrices, given the relatively high Qst estimates for some characters (Table 2) and the potentially large effect of selection on the structure of G matrices (e.g. Jernigan et al., 1994). However, based on the results of a large-scale experiment with 52 bottlenecked lines of Drosophila melanogaster (Phillips et al., 2001), there is a great potential for drift to cause rapid changes in the principal component structure. Whatever the mechanisms underlying the patterns seen in the present investigation, we urge caution in the use of proportionality (or lack of proportionality) as an indicator of whether drift is responsible for divergence in the G matrix (Roff, 2000).

Relatively few studies have related genetic variation in phenotypic characters to genetic drift or population structure. In a recent study of two Scabiosa species (Dipsacaceae) (Waldmann & Andersson, 1998), there was no apparent association between population size and the heritability for eight vegetative and floral characters, despite a more than 100-fold difference in population size. Moreover, there was no tendency for the geographically restricted S. canescens to possess lower levels of within-population variation than the more common and widespread S. columbaria. Other examples include a study of Widén & Andersson (1993), who found a small, subdivided population of Senecio integrifolius (Asteraceae) to retain higher levels of additive variance than a large, continuous population, and Podolsky (2001) who documented a weak, nonsignificant relationship between population size and quantitative genetic variation in the annual plant Clarkia dudleyana (Onagraceae). When combined with genetic evidence from B. cretica, there is little support for a consistent effect of habitat fragmentation on levels of variation in potentially adaptive characters, at least over the timescales considered by conservation biologists (Lande, 1988).

Quantitative genetic studies of other organisms have revealed a tendency for G matrices of natural populations to become more divergent as one progresses from lower to higher levels in the taxonomic hierarchy (Lofsvold, 1986; Arnold & Phillips, 1999; Roff, 2000; for an exception, see Bégin & Roff, 2001). However, only a few authors have utilized the CPC technique (Flury, 1988) to determine at which level differences between G matrices become statistically or biologically significant (Steppan, 1997; Arnold & Phillips, 1999; Waldmann & Andersson, 2000; Pigliucci & Hayden, 2001). The present study of B. cretica not only demonstrates extensive population divergence in the G matrix, but also indicates that the first changes involve principal component structure rather than the eigenvalues. Similar patterns have been recorded in Scabiosa columbaria (Waldmann & Andersson, 2000), while comparisons of garter snake (Thamnophis elegans) populations suggest conservation of principal component structure, despite large changes in eigenvalues (Arnold & Phillips, 1999). As more studies begin to compare (co)variance matrices in a standardized way, it should be possible to make broader generalizations regarding the rate of G matrix evolution and the evolutionary forces that influence genetic (co)variance structure.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References

We are indebted to Laura Bergvall, Alf Ceplitis, Nils Cronberg, Derek Roff, Gabrielle Rosquist, Linus Svensson, Rune Svensson and Patrik Waldmann for technical support in various ways. This study was supported by grants from the Swedish research councils NFR (to SA) and SJFR (to BW) and a scholarship from the scholar exchange program between Lund University and Peking University (to GYR).

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  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Plant material
  6. Allozyme analyses
  7. Crosses and measurements
  8. Partitioning of quantitative genetic variation
  9. Comparison of (co)variance matrices
  10. Results
  11. Analysis of population structure
  12. Variances and covariances
  13. CPC analyses
  14. Percent reduction in MSE
  15. Discussion
  16. Acknowledgments
  17. References
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