POPULATION DIVERGENCE ALONG LINES OF GENETIC VARIANCE AND COVARIANCE IN THE INVASIVE PLANT LYTHRUM SALICARIA IN EASTERN NORTH AMERICA

ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2011.01313.x

POPULATION DIVERGENCE ALONG LINESOF GENETIC VARIANCE AND COVARIANCEIN THE INVASIVE PLANT LYTHRUM SALICARIAIN EASTERN NORTH AMERICARobert I. Colautti1,2,3 and Spencer C. H. Barrett1

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks St. Toronto, Ontario M5S 3B2, Canada2E-mail: [email protected]

Received September 28, 2010

Accepted March 28, 2011

Evolution during biological invasion may occur over contemporary timescales, but the rate of evolutionary change may be inhibited

by a lack of standing genetic variation for ecologically relevant traits and by fitness trade-offs among them. The extent to which

these genetic constraints limit the evolution of local adaptation during biological invasion has rarely been examined. To investigate

genetic constraints on life-history traits, we measured standing genetic variance and covariance in 20 populations of the invasive

plant purple loosestrife (Lythrum salicaria) sampled along a latitudinal climatic gradient in eastern North America and grown

under uniform conditions in a glasshouse. Genetic variances within and among populations were significant for all traits; however,

strong intercorrelations among measurements of seedling growth rate, time to reproductive maturity and adult size suggested

that fitness trade-offs have constrained population divergence. Evidence to support this hypothesis was obtained from the

genetic variance–covariance matrix (G) and the matrix of (co)variance among population means (D), which were 79.8% (95% C.I.

77.7–82.9%) similar. These results suggest that population divergence during invasive spread of L. salicaria in eastern North America

has been constrained by strong genetic correlations among life-history traits, despite large amounts of standing genetic variation

for individual traits.

KEY WORDS: Fitness trade-off, G matrix, purple loosestrife, quantitative genetics.

The invasive spread of introduced species usually occurs across

heterogeneous landscapes and often along large-scale environ-

mental gradients that correlate with latitude. Introduced species

may be aided in their spread in heterogeneous environments by

phenotypic plasticity, including general purpose genoypes (Baker

1965; Williams et al. 1995; Parker et al. 2003; Ross et al. 2009).

However, if plasticity is costly or poorly tracks environmental con-

ditions then local adaptation may evolve (Via and Lande 1985;

Scheiner 1993; Tufto 2000), potentially increasing invasive spread

3Current Address: Department of Biology, Duke University, PO Box

90338, Durham, North Carolina 27708.

(Garcıa-Ramos and Rodriguez 2002). The extent to which local

adaptation increases fitness in invasive species confronting novel

environments will depend on the availability of standing genetic

variation within populations for ecologically relevant traits (Fisher

1930; Lande 1979; Lee 2002; Lee et al. 2007).

Evidence from neutral genetic markers indicates that bot-

tlenecks during introduction are common in invasive species

(e.g., Barrett and Shore 1989; Barrett and Husband 1990;

Malacrida et al. 1998; Tsutsui et al. 2000; Novak and Mack

2005; Zhang et al. 2010). Nevertheless, genetic variation within

introduced populations can sometimes be greater than that occurs

in native source regions as a result of multiple introductions of

2 5 1 4C© 2011 The Author(s). Evolution C© 2011 The Society for the Study of Evolution.Evolution 65-9: 2514–2529

GENETIC CONSTRAINTS ON POPULATION DIVERGENCE

differentiated populations followed by admixture among them

(reviewed in Lee et al. 2004; Dlugosch and Parker 2008a; see also

Keller and Taylor 2010; Verhoeven et al. 2011). However, genetic

variation inferred from neutral markers poorly predicts genetic

variation for quantitative traits (Lande 1988; Merila and Crnokrak

2001; McKay and Latta 2002) and, consequently, determining the

potential for adaptive evolution during the geographical spread of

invasive species requires measurements of the standing genetic

variation of ecologically relevant traits sampled from across the

range.

The occurrence of genetic variation for quantitative traits

within populations may not guarantee contemporary evolution in

introduced species because fitness trade-offs have the potential

to impede local adaptation. Trade-offs among two or more life-

history traits are often manifested as positive or negative genetic

correlations that prevent natural selection from simultaneously

improving correlated traits (Dickerson 1955; Lande 1982; Blows

et al. 2004; Roff and Fairbairn 2007). For example, the classic

life-history trade-off between age and size at maturity constrains

individuals to reproduce early at a small size, or later at a large

size. In such cases, population divergence will be constrained

along the axis of covariance between age and size (Mitchell-Olds

1996; Schluter 1996). Fitness trade-offs can therefore serve to

constrain the direction of contemporary adaptive evolution and

may influence survival and reproduction at range limits (Etter-

son and Shaw 2001; Etterson 2004) with consequences for the

geographical spread of invasive species.

Understanding genetic constraints on population divergence

requires the simultaneous comparison of correlations among mul-

tiple traits. The G matrix is a convenient mathematical repre-

sentation of the genetic variances (diagonal cells) for a set of

quantitative traits and the covariances (off-diagonal cells) among

them (Lande 1979). However, the use of the G matrix to di-

rectly infer constraints imposed by genetic trade-offs presents

two experimental challenges. First, variance and covariance com-

ponents of G (hereafter “(co)variances”) are estimated from ge-

netic lines or families, making empirical estimates imprecise

without large sample sizes (Shaw 1991). Second, in addition to

fitness trade-offs, the covariance structure of G is also shaped

by natural selection, genetic drift, and migration (e.g., Turelli

1988; Phillips et al. 2001; Jones et al. 2003; Guillaume and

Whitlock 2007). These processes are likely to be important during

rapid range expansion by invasive species, which often includes

founder events and changes in the selective landscape. As a re-

sult, the G matrix of a randomly chosen population is likely to be

a poor predictor of constraints on adaptive evolution across the

range of a species. Therefore, both methodological (e.g., sample

size) and biological (e.g., stochastic forces) factors complicate the

identification of genetic constraints on the divergence of natural

populations.

An alternative approach for identifying genetic correlations

that may constrain population divergence is to investigate trade-

offs that are expected a priori to be of likely ecological impor-

tance (e.g., age vs. size at reproduction) and then estimate Gfor these traits averaged across a sample of populations. Here,

the idiosyncratic effects of selection, migration, and drift in any

one population should be “averaged out” (see Chenoweth and

Blows 2008). For example, bottlenecks significantly changed

the (co)variance structure of G in experimental populations of

Drosophila melanogaster, yet the average G across populations

remained similar to the ancestral outbred population (Phillips

et al. 2001). Additional insight can also be obtained by investi-

gating the primary eigenvectors (i.e., principal components) of

G because these are likely to remain more stable over time than

are particular (co)variance estimates (reviewed in Arnold et al.

2008). If the principal component eigenvectors of the “average”

G are relatively stable over time, then populations should diverge

in a predictable fashion, resulting in a correlation between the

eigenvectors of G and those of the matrix of (co)variance among

population means (D). The D matrix characterizes population di-

vergence and is similar to G, but instead represents variances

and covariances among population means, rather than genetic

families within populations. If divergence among populations is

completely constrained by genetic variance within and covariance

among life-history traits, then G and D should be identical. Con-

versely, evolution that is not constrained by genetic (co)variances

within populations will result in less similarity between G and D.

Methods for comparing G and D among natural populations is

an area of active research (Shaw 1991; Phillips and Arnold 1999;

Roff et al. 1999; McGuigan et al. 2005; Chenoweth and Blows

2008; Calsbeek and Goodnight 2009; Simonsen and Stinchcombe

2010), but to our knowledge none of these approaches has yet been

used to study adaptive evolution during biological invasion. Inva-

sive species may be well-suited for studying genetic constraints

on the evolution of natural populations because multiple intro-

ductions and admixture are likely to weaken phylogeographical

relationships and the extent of population structure that can com-

plicate comparisons of G and D.

We investigated genetic constraints associated with life-

history trade-offs within and among invasive populations of the

wetland plant Lythrum salicaria L. (Lythraceae) in eastern North

America. Previous work on L. salicaria identified genetically

based latitudinal clines for days to first flower and several mea-

surements of plant size in both native European (Olsson and Agren

2002; Olsson 2004) and introduced North American populations

(Montague et al. 2008; Colautti et al. 2010). Moreover, measure-

ments of phenotypic selection indicate that introduced populations

of L. salicaria are under strong selection for earlier flowering

time and larger size (O’Neil 1997; Colautti and Barrett 2010). A

study of populations from eastern North America indicates that

EVOLUTION SEPTEMBER 2011 2 5 1 5

R. I . COLAUTTI AND S. C. H. BARRETT

evolution of earlier flowering in northern populations is associ-

ated with a correlated change in vegetative size (Colautti et al.

2010). However, the extent to which multitrait genetic constraints

may limit population divergence was not previously investigated

and this is one of the main goals of this study.

Here, we evaluate constraints on population divergence dur-

ing biological invasion in L. salicaria by measuring the standing

genetic variance and covariance of 12 ecologically relevant life-

history traits. This was undertaken by sampling 20 populations

along a latitudinal gradient of 10 degrees latitude from south-

ern Maryland (U.S.A.) to Timmins, Ontario (Canada) and con-

ducting a quantitative genetic experiment under uniform growth

conditions in a glasshouse. The specific questions we addressed

in our study were: (1) Is there evidence for quantitative genetic

variation within and among populations for life-history traits as-

sociated with growth and reproduction? The rapid reestablishment

of latitudinal clines in North America (Montague et al. 2008) im-

plicates the occurrence of significant standing genetic variation

within populations; however, genetic drift and founder events are

also known to play an important role in eroding diversity in in-

troduced populations of this species (Eckert and Barrett 1992;

Eckert et al 1996). (2) Are life-history traits correlated among

populations, and among families within populations? Within pop-

ulations, fitness trade-offs among traits such as seedling growth,

time to maturity, and plant size should result in significant genetic

correlations. Correlations among population means could arise

from constraints on population divergence and/or selection for

particular trait combinations. We predicted that selection for early

flowering and larger size at all latitudes should result in weaker

correlations between these traits relative to within-population ge-

netic correlations. (3) To what extent is any correlated divergence

among population means (i.e., the D matrix) constrained by the

structure of genetic correlations among families within popula-

tions (i.e., the G matrix)? Constraints on adaptive evolution re-

sulting from trade-offs among life-history traits should result in a

strong correlation between D and G, whereas divergence resulting

from selection for larger plants that flower sooner should not.

MethodsSTUDY SYSTEM AND SAMPLING

Lythrum salicaria (purple loosestrife) is an insect-pollinated, out-

crossing, autotetraploid, perennial herb native to Eurasia. Dur-

ing the last century it has become a successful invader of North

American wetland habitats, roadside ditches and other moist dis-

turbed sites, particularly in eastern North America (Thompson

et al. 1987; Mal et al. 1992; Blossey et al. 2001; USDA 2009).

Herbarium records indicate initial introduction to ports along the

eastern seaboard at the end of the 18th century, with more recent

(< 70 years) spread into central and northern Ontario, Canada

(Thompson et al. 1987). Both herbarium records and studies using

molecular markers suggest multiple introductions to North Amer-

ica with admixture among introduced populations (Thompson et

al. 1987; Houghton-Thompson et al. 2005; Chun et al. 2009).

Lythrum salicaria is tristylous and self-incompatible with suc-

cessful mating largely between plants of different style morphs.

All shoot growth is localized within a few centimeters of the pri-

mary stem and therefore population growth occurs exclusively

through seed production (Yakimowski et al. 2005).

We used populations previously studied by Montague et al.

(2008) in which latitudinal clines for several life-history traits

were detected; details of population sampling and seed collec-

tions used in this experiment are given therein. Briefly, in autumn

2003, open-pollinated infructescences were collected from 25

populations in eastern North America. Populations were chosen

to represent a latitudinal cline from Timmins, Ontario (48◦48′N,

81o30′W) to Easton, Maryland (38o75′N, 75o99′W). The latitude

of these populations is a strong predictor of season length (see

Montague et al. 2008). From the initial 25 populations, we chose

a subsample of 20 representing the entire latitudinal gradient.

EXPERIMENTAL DESIGN AND MEASUREMENTS

In June 2004, we planted 20 seeds from 20 families from each

of 20 populations (8000 total) into 2 cm × 2 cm plug trays filled

with Pro-MixTM “BX” peat containing vermiculite and perlite.

The position of each seed family was randomized across trays

and trays were placed on a glasshouse bench at the University

of Toronto and rotated thrice weekly to reduce position effects.

We monitored seeds daily for germination and after two weeks

we randomly selected eight seedlings from each of 17 families,

which we transplanted into 10 cm pots filled with Pro-MixTM.

The experiment involved a randomized block design with two

individuals per family per population per block (N = 4 blocks).

The final dataset contained 2623 (of 2720) individuals from 339

(of 340) seed families due to mortality. All blocks received 1.5

g/L of fertilizer (ratio 20N:20P:20K) every three weeks and were

kept partially flooded so that the lower one-fourth of each pot

was in water. To prevent aphid outbreaks, we treated plants with

Dursban 2E (chlorpyrifos) pesticide at a rate of 4 mL/L, once in

July and again in August.

Following transplant, we measured seedling height and esti-

mated total seedling leaf area as the diameter of the largest leaf

multiplied by the largest total width of two-leaved seedlings. Two

and four weeks after transplant we measured seedling height. On-

set of flowering of plants occurred from July to September 2004

and during this period we monitored plants daily. On the first

day of anthesis for a given plant, we measured stem width of the

primary stem and the height of the primary stem, divided into

two segments. The first, “vegetative size” was measured from

2 5 1 6 EVOLUTION SEPTEMBER 2011


the soil surface to the base of the inflorescence. The second,

“inflorescence length” was measured from the base of the in-

florescence to its tip. Division of the primary stem height into

these two measures is a convenient division between resource-

accumulation structures (i.e., leaves) and resource sinks (i.e.,

flowers and fruits). Field surveys of natural populations, and a

common garden experiment, confirm that the height of the veg-

etative portion of the primary stem is a strong predictor of to-

tal vegetative growth (see Colautti and Barrett 2010; Colautti

et al. 2010). In October, we again measured vegetative height and

inflorescence length; we then harvested and oven-dried plants to

a constant weight to measure biomass. Similar to measurements

of stem length, we divided shoot biomass into vegetative and

inflorescence structures.

We recorded seed germination day but found it had a negligi-

ble effect on seedling growth and time to maturity because germi-

nation date was highly skewed with no seed germinating before

day 3 and 90% of seeds germinating between days 3 to 6. In

total, our analysis included 12 traits: four involving seedling

growth and development (seedling leaf area, seedling height,

Height-2wk and Height-4wk), hereafter referred to as “seedling

traits”; as well as four “adult traits” (days to first flower, stem

width, vegetative size at flowering, and inflorescence length at

flowering); and four “harvest traits” (vegetative size at harvest,

inflorescence length at harvest, vegetative biomass, and reproduc-

tive biomass).

STATISTICAL ANALYSIS OF QUANTITATIVE TRAITS

Prior to analysis we log-transformed vegetative and inflorescence

biomass to meet assumptions of multivariate normality and then

standardized the phenotypic distributions of each trait to a mean

of zero and a standard deviation of one. We performed a statistical

linear mixed model using the MIXED procedure in SAS 9.1 (SAS

Institute Inc., Cary, NC) with population as a fixed effect, seed

family nested within population as a random effect, and repeated

measurements on each individual nested within a seed family.

This model estimates the “average” (co)variance matrix among

seed family means within populations (i.e., the G matrix), and the

residual (co)variance among individuals within a seed family (Rmatrix) by restricted maximum likelihood (ReML). Satterthwaite

(1946) correction was used for the degrees of freedom of the fixed

effects. We used best linear unbiased estimators (BLUEs) of popu-

lation means from this model to calculate the matrix of divergence

among population means (D matrix) because (co)variance compo-

nents are not estimated directly for fixed effects in a mixed model.

We used a Linux-based processor on the University of North

Carolina’s research computing cluster to run the model (20 GB

RAM; runtime ∼72 h). This was necessary due to the memory

and processing power required to simultaneously estimate two

separate 12 × 12 matrices using ReML: one from 339 seed fami-

lies (G) and one from 2623 individuals (R).

To investigate genetic variation within and among popula-

tions, we compared the fraction of total phenotypic variance man-

ifest as (1) variance among population means (Vpop), (2) variance

among seed families within populations (Vfam), and (3) variance

among individuals within seed families (Vres). These variance

components represent: (1) genetic differentiation among popula-

tions (Vpop), (2) ∼ 12 to 1

4 of the average standing genetic vari-

ation within populations (Vfam), and (3) the residual phenotypic

variance (Vres) among individuals within a seed family. Variance

among seed families represent 12 to 1

4 of standing genetic varia-

tion within populations because full- and half-siblings share on

average 12 or 1

4 of their genes, respectively.

Note that Vpop and 4 × Vfam are maximum estimates of the

additive genetic variance because they also include di-, tri-, and

tetragenic interactions, which are analogous to dominance vari-

ance in diploids but are much smaller in magnitude (Kempthorne

1955). Under random mating, with no epistasis or linkage dise-

quilibrium, genetic covariance (σG) between tetraploid half sibs x

and y is:

σG(x, y) = σ2A/4 + σ2

D/216

and for tetraploid full siblings is:

σG(x, y) = σ2A/2 + 2σ2

D/9 + σ2T /12 + σ2

Q/36

which are functions of the additive genetic variance (A) as well as

interactions among two (D), three (T), or four (Q) alleles at each

locus (Kempthorne 1955; Lynch and Walsh 1998).

Differences in maternally inherited genes or in maternal pro-

visioning to seeds can also inflate Vfam and Vpop. However, mater-

nal provisioning is minimal as seeds of L. salicaria are very small

(200 × 400 μm) and lack endosperm (Thompson et al. 1987).

Moreover, we detected no significant maternal effects on adult

size and flowering time in experimental crosses (see supplemen-

tary material in Colautti et al. 2010).

To investigate whether there was significant genetic variation

among and within populations, we ran a separate mixed model

for each trait with populations and seed families, both as random

effects. We tested whether the variance component for Vpop or Vfam

were significantly different from zero, using a likelihood ratio test

(LRT) with one degree of freedom. We treated population as a

random effect in this analysis because it is a more conservative

test of the null hypothesis that there is no significant variation

among population means.

We tested for latitudinal trends in each phenotypic trait using

generalized linear models in R (version 2.8.1), with the stan-

dardized (BLUE) estimates of population means of each trait as



dependent variables and latitude as the independent variable. We

fitted quadratic regression terms to allow for nonlinear relation-

ships with latitude and used an LRT to hierarchically test the

significance of the quadratic and linear regression coefficients.

The analysis comparing the similarity between G and D (see

below) uses the first six principal component (PC) eigenvectors of

these two matrices. To determine if all six PCs should be included

in the analysis, we determined whether there was significant ge-

netic variation in the sixth eigenvector of G using a LRT and a

factor analytical structure for G. This was particularly important

because several of the life-history traits we measured are likely

to be correlated (e.g., height measured at different stages), which

could result in nonsignificant genetic variation for higher PCs of

G. This procedure allowed us to test the significance of a statis-

tical model with six versus five principal components (see also

McGuigan and Blows 2010).

COMPARING G AND D

The classic test for evolution along lines of genetic variance

compares gmax and dmax, which are the first principal compo-

nent eigenvectors of G and D, respectively (Mitchell-Olds 1996;

Schluter 1996). The angle between gmax and dmax is calculated as

θ = cos−1[gmax]′[dmax] where ′ indicates a transpose of the gmax

vector so that the two vectors can be multiplied (Schluter 1996).

The angle θ ranges from 0 (complete similarity) to 90 (orthogonal,

no similarity) and therefore provides a useful and intuitive mea-

sure of similarity (see Schluter 1996 for details). Although gmax

by definition contains the most variation of any single eigenvector,

the interpretation of θ as a measure of similarity between G and

D ignores variance in other eigenvectors of G, which may be con-

siderable and together may account for more phenotypic variance

than gmax alone. In such cases, a more biologically informative

approach is to compare multiple eigenvectors of G and D. There-

fore, we used the Krzanowski (1979) method described in Blows

et al. (2004; see also McGoey and Stinchcombe 2009), with the

exception that we were interested in potential constraints on D (the

matrix of variance–covariance among population means) instead

of γ (the matrix of nonlinear selection gradients). If population

divergence is constrained by G, then population divergence will

occur along the major eigenvectors of G, resulting in an overall

similarity between G and D. In contrast, population divergence

that is not constrained by G will not constrain eigenvectors of D,

resulting in little similarity between G and D.

The Krzanowski method for comparing the similarity be-

tween two matrices begins with a separate principal components

analysis (PCA) for each matrix. The variance–covariance matri-

ces of G and D were standardized to the same scale (μ = 0, σ =1) prior to the mixed model analysis and thus traits had the same

total phenotypic variance but could differ in the relative amount

of variation among populations, among seed families, or within

seed families. We then used the principal components of G and

D to generate a “matrix of similarity” (S). S is calculated as

A′BB′A, where A and B are matrices containing principal com-

ponent eigenvectors of G and D, respectively, with each column

representing a separate eigenvector. Here again the symbol ′ indi-

cates matrix transposition. Technically S describes the similarity

of “subspaces” of G and D because only six of 12 eigenvectors

(i.e., principal components) from each of G and D were included.

However, this is a necessary limitation of the Krzanowski method

because including more than half of the eigenvectors of G and Din A and B will constrain the comparison to recover angles of 0o

(see Blows et al. 2004 for details). Therefore, we included only

the first six eigenvectors of each matrix.

Interpreting the matrix of similarity (S) in biological terms is

difficult, but the principal component eigenvectors of S, denoted

ai, have two useful properties. First, the sum of their eigenvalues

represents the overall similarity between G and D, in this case

ranging from 0 to 6. For ease of discussion, we refer to this

sum as �λi, or the “index of similarity.” Second, the eigenvalues

(λi) of each ai range from 0 (orthogonal, no similarity) to 1

(complete similarity), which can be translated into angles using

the equation cos−1√λi (Blows et al. 2004). These angles can

be compared directly with the angle between gmax and dmax (θ),

as described above. Furthermore, Blows et al. (2004) show how

each ai can be “projected” back into the subspace of G or D,

to identify the phenotypic traits responsible for the similarity (or

dissimilarity) between G and D. Here again “subspace” refers

to the fact that this analysis uses the first six eigenvectors of Gand D represented by each ai. As in Blows et al. (2004), we

chose to project ai into the subspace G because we wanted to

identify trait variances and covariances within populations that

might constrain divergence among populations. The projection

of ai into the subspace of G results in vectors (bi), which we

refer to as “vectors of similarity,” and are calculated as bi = Aai.

Factor coefficients of the vectors of similarity (bi) are biologically

meaningful as they can be interpreted as coefficients describing

the loadings of each phenotypic trait on bi. This is analogous to the

coefficients of a PCA, which describe the relative “loadings” of

each phenotypic trait on each principal component eigenvector. In

this way, we could identify individual traits and trait combinations

in G with variance–covariance structure most similar to D.

We used a bootstrap method to generate 95% confidence in-

tervals for the angle of similarity (θ) and the index of similarity

(�λi). For each iteration of the bootstrap model (10,000 itera-

tions total), 17 standardized BLUPs of seed family means were

resampled (with replacement) within each of the 20 populations.

These data were used to generate new G and D matrices and to

recalculate both �λi and θ. A second bootstrap model was used



Table 1 . The proportion of phenotypic variation of 12 traits asso-

ciated with growth and phenology within and among 20 pop-

ulations of Lythrum salicaria grown under uniform glasshouse

conditions.

Trait Vpop Vfam Vres H2

Leaf area at transplant 0.178 0.088 0.734 0.428Height at transplant 0.242 0.109 0.649 0.575Height at week 2 0.257 0.069 0.674 0.371Height at week 4 0.184 0.068 0.747 0.334Days to first flower 0.383 0.129 0.488 0.836Stem width at maturity 0.427 0.100 0.473 0.698Vegetative size at maturity 0.542 0.111 0.347 0.969Inflorescence length at maturity 0.103 0.044 0.853 0.196Vegetative size at harvest 0.513 0.104 0.383 0.854Inflorescence length at harvest 0.253 0.060 0.686 0.322Final vegetative biomass (ln) 0.377 0.070 0.553 0.449Final inflorescence biomass (ln) 0.197 0.057 0.745 0.284

Variance components are standardized to sum to one and were calculated

from a mixed model; they describe divergence among populations (Vpop),

variation among seed families within populations (Vfam), and residual vari-

ation (Vres). Average within-population broad-sense heritabilities (H2) are

estimated as 4×Vfam/(Vfam+Vres).

following Blows et al. (2004, electronic supplement), which is

more appropriate for testing the null hypothesis because it uses

an orthogonal factor rotation to force G and D into coincident

subspaces (for details see Cohn 1999 as described in Blows et al.

2004). Both bootstrap models were written and implemented in

R (version 2.8.1).

In summary, the key measurements of the similarity between

G and D are as follows. The first is θ, the angle of similarity

between gmax and dmax (Schluter 1996), which ranges from 0

(identical vectors) to 1 (orthogonal, no similarity). Second, angles

calculated from the eigenvalues of ai are analogous to θ, but allow

for multiple dimensions of G and D to be compared. Third, the

coefficients of bi show the “loadings” of each phenotypic trait

for the corresponding ai. Finally, the sum of eigenvalues of ai

represents an overall metric of similarity for the first six principal

components of G and D and ranges from 0 (no similarity) to 6

(completely identical).

ResultsQUANTITATIVE GENETIC VARIATION AND

LATITUDINAL CLINES

Variance–covariance matrices among populations (D), among

seed families within populations (G), and among individuals

within seed families (R) were estimated from a single mixed

model and standardized to sum to one (Table 1). The LRT of a

factor analytic model comparing six versus five principal compo-

nents was highly significant (χ2 = 47.4, df = 7, P < 0.001), indi-

cating significant standing genetic variation in all six eigenvectors

of G. Separate LRTs for each trait confirmed highly significant

effects of population (across all 12 traits: χ2 > 114.9, df = 1,

P < 0.001) and seed family within population (across all 12

traits: χ2 > 15.6, df = 1, P < 0.001). Adult vegetative size

and time to first flower exhibited the greatest level of divergence

among populations (i.e., Vpop), with variance among populations

explaining 38.3% (days to first flower) to 54.2% (vegetative size

at maturity) of the total phenotypic variance (Table 1). In con-

trast, “populations” explained only 17.8% (leaf area at transplant)

to 25.7% (height-2wk) of the phenotypic variance in seedling

traits, and 10.3% to 25.3% of the variance in inflorescence length

and biomass (Table 1), indicating less divergence among popu-

lations for these traits relative to vegetative size and time to first

flower.

Broad-sense heritability (H2), an estimate of standing ge-

netic variation within populations (4 × Vfam/[Vfam+Vres]), was

highest for days to flower (83.6%) and vegetative size measured

at harvest (85.4%) and at maturity (96.9%). In contrast, the lowest

H2 estimates were for inflorescence length measured at flowering

(19.6%) and at final harvest (32.2%), as well as inflorescence

biomass (28.4%). Seedling traits had intermediate heritabilities

ranging from height-4wk (33.4%) to height at transplant (57.5%).

Nine of the 12 traits we investigated were significantly corre-

lated with latitude (Fig. 1). Consistent with patterns of population

divergence and the relative amounts of standing genetic varia-

tion within populations, the four strongest clines involved days to

flower, stem width, vegetative size, and vegetative biomass (R2 =0.506 to 0.709). In contrast, the weakest clines (R2 = 0.031 to

0.279) were among seedling traits, inflorescence length, and in-

florescence biomass, which generally showed quadratic or non-

significant correlations with latitude. In general, these patterns

indicate that northern plants flowered earlier at a smaller size and

also remained small in stature until the end of the experiment.

CORRELATIONS AMONG TRAITS—G AND D

To visualize potential trade-offs among life-history traits and their

influence on population divergence, we present for each pair of

traits the bivariate plots of BLUPs for family means (Fig. 2, above

diagonal) and BLUEs for population means (Fig. 2, below diag-

onal), which were estimated from the large mixed model. Corre-

lations estimated from G may indicate genetic constraints among

life-history traits, whereas intercorrelations in D may arise as a

correlated response to selection or through stochastic processes

acting on genetically correlated traits. The G matrix estimated

directly from the mixed model is presented as Supporting infor-

mation (Table S1, above diagonal), as well as genetic correlation

coefficients calculated from G (Table S2, above diagonal). The

variance and covariance components of the D matrix are also



Figure 1 . Bivariate plots illustrating latitudinal clines in 12 phenotypic traits related to growth and phenology measured in 20 populations

of Lythrum salicaria grown under uniform glasshouse conditions. Measurements were made on developing seedlings (panels 1–4), on

mature individuals on the day of first flowering (panels 4–8), and on all individuals at the end of the experiment (panels 9–12). Linear

and quadratic approximations and R2 values are estimated by least-squared means regression. Phenotypic traits showing significant

quadratic relationships with latitude are indicated by curved lines; traits with significant linear but nonsignificant quadratic relationships

are indicated with straight lines. Significance of linear and quadratic terms was estimated by likelihood ratio tests using general linear

models. P-values correspond to the significance of the best-fit model based on likelihood ratio tests of a single intercept (i.e., mean) and

zero slope.

provided (Table S1, below diagonal) along with correlation coef-

ficients (Table S2, below diagonal).

The strongest correlations in G were between time to first

flower and stem width and between vegetative size and biomass

(+0.549 < R < +0.994) (Fig. 2 and Table S2, above diago-

nal). These traits all loaded positively on gmax, the first principal

component of the G matrix, which accounted for 45.7% of the to-

tal phenotypic variation among seed families within populations

(Table 2). Genetic correlations among seedling traits were gener-

ally weaker (+0.173 < R < +0.827) (Fig. 2 and Table S2, above



Figure 2 . Visualization of the genetic variance–covariance matrix (G) and the variance–covariance matrix of divergence (D) among 20

populations of Lythrum salicaria grown under uniform glasshouse conditions. Above Diagonal: G estimated from standardized means

of 339 seed families, estimated as best linear unbiased predictors (BLUPs) by restricted maximum likelihood (ReML), with seed family,

nested within population as a random effect. Below Diagonal: D estimated from standardized means of 20 populations, estimated best

linear unbiased estimators (BLUEs) by ReML, with population as a fixed effect.

diagonal). The second principal component of G (g2), which ac-

counted for 23.0% of the total variance among seed families, was

positively correlated with seedling traits (Table 2). Correlations

between size at transplant (i.e., transplant height and leaf area)

and adult traits were considerably weaker (−0.306 < R < +0.279) (Fig. 2 and Table S2, above diagonal). Combinations of

these traits define other eigenvectors of G (g3 - g6), each of which

accounts for less than 13% of the variation among seed families

(Table 2). Thus, genetic correlations were strongest among days

to flower and adult vegetative traits, intermediate among seedling

traits, and weakest between pairs of adult and seedling traits.

Population means were highly intercorrelated for most traits

and correlations were generally stronger among population means

(i.e., D matrix) than among seed families within populations

(Fig. 2 and Table S2, below diagonal). Similar to the correla-

tions evident in the G matrix: (1) flowering time and vegetative

size measurements exhibited the highest trait correlations among

populations (+0.698 < R < +0.999) (Fig. 2 and Table S2, below

diagonal), (2) these traits loaded primarily on dmax, which ex-

plained 67.8% of the total phenotypic variation among population

means (Table 3), and (3) seedling traits were not as strongly corre-

lated (+0.532 < R < +0.893; below diagonal). However, unlike

G, pairwise combinations of seedling and adult traits in D varied

markedly (−0.201 < R < +0.847) (Fig. 2 and Table S2, below

diagonal) and days to first flower loaded more heavily on d2 than

dmax (Table 3). Thus, in common with genetic correlations within

populations, population divergence was highly correlated among

adult traits and less correlated for seedling traits. However, unlike

genetic correlations, population divergence was strongly corre-

lated for a number of pairs of adult and juvenile traits, and days

to first flower was not as highly correlated with traits associated

with adult vegetative size.



Table 2 . Factor loadings, eigenvalues, and percent variation explained by the first six eigenvectors of the average matrix of broad-sense

genetic variance–covariance (G) within 20 populations of Lythrum salicaria grown under uniform glasshouse conditions.

Trait gmax g2 g3 g4 g5 g6

Leaf area at transplant −0.082 0.422 0.118 0.560 0.339 0.446Height at transplant −0.189 0.494 0.366 0.302 −0.294 −0.403Height at week 2 −0.147 0.420 0.214 −0.319 −0.024 0.002Height at week 4 −0.192 0.335 −0.087 −0.490 0.042 −0.004Days to first flower 0.515 −0.049 0.151 0.190 −0.165 0.053Stem width at maturity 0.374 0.174 −0.272 0.079 0.584 −0.302Vegetative size at maturity 0.426 0.237 0.110 −0.259 −0.122 0.310Inflorescence length at maturity 0.115 0.019 −0.299 0.267 −0.616 0.055Vegetative size at harvest 0.423 0.231 0.083 −0.254 −0.123 0.154Inflorescence length at harvest −0.210 0.153 −0.445 0.018 −0.054 0.541Final vegetative biomass (ln) 0.252 0.274 −0.328 0.093 −0.026 −0.274Final inflorescence biomass (ln) −0.110 0.220 −0.537 −0.018 −0.124 −0.231Eigenvalue 0.474 0.238 0.133 0.100 0.035 0.026Percentage of variation 45.7% 23.0% 12.8% 9.6% 3.4% 2.5%Cumulative percentage of variation 45.7% 68.7% 81.6% 91.2% 94.5% 97.0%

Table 3 . Factor loadings, eigenvalues, and percent variation explained by the first six eigenvectors of the matrix of population divergence

(D) calculated among 20 population means of traits associated with growth and phenology in Lythrum salicaria grown under uniform

glasshouse conditions.

Trait dmax d2 d3 d4 d5 d6

Leaf area at transplant 0.202 0.191 0.083 0.642 −0.245 −0.310Height at transplant 0.255 0.224 0.266 0.487 0.054 0.365Height at week 2 0.246 0.281 0.453 −0.191 0.018 −0.085Height at week 4 0.109 0.372 0.394 −0.393 −0.128 −0.320Days to first flower 0.311 −0.420 −0.152 0.051 0.048 −0.030Stem width at maturity 0.386 −0.170 −0.280 −0.006 −0.238 −0.607Vegetative size at maturity 0.453 −0.125 0.029 −0.253 −0.147 0.265Inflorescence length at maturity 0.092 0.239 −0.234 −0.132 0.665 −0.213Vegetative size at harvest 0.440 −0.132 0.019 −0.220 −0.133 0.289Inflorescence length at harvest −0.024 0.497 −0.538 −0.127 −0.439 0.263Final vegetative biomass (ln) 0.376 0.089 −0.118 0.100 0.418 0.147Final inflorescence biomass (ln) 0.164 0.380 −0.318 0.036 0.112 −0.046Eigenvalue 2.624 0.845 0.176 0.101 0.051 0.029Percentage of variation 67.8% 21.8% 4.6% 2.6% 1.3% 0.8%Cumulative percentage of variation 67.8% 89.6% 94.2% 96.8% 98.1% 98.9%

POPULATION DIVERGENCE ALONG GENETIC LINES

OF LEAST RESISTANCE

Quantitative estimates of G and D indicate significant simi-

larity between the two matrices (Fig. 3). Schluter’s (1996) θ,

which measures the angle of similarity between gmax and dmax

and ranges from 0o to 90o, suggested a moderate level of sim-

ilarity of ∼46o (mean = 45.78; bootstrap mean = 45.72o;

95% CI: 39.50–51.93o). In contrast, the index of similarity

(�λi) of the S-matrix, which compares the first six eigenvec-

tors of G and D, was 4.8 of 6.0 (mean = 4.79; bootstrap

mean = 4.80; 95% CI: 4.66–4.97), indicating a higher level

of similarity. Thus, while the first principal component eigen-

vectors of G and D were similar (θ = 45.8o/90o = 50.9%),

the first six eigenvectors of S were more so (�λi = 4.8/6.0 =80.0%), and �λi was less than any of the 10,000 iterations of the

orthogonal rotation bootstrap, indicating highly significant simi-

larity (P < 0.0001). The 10,000 bootstrap iterations of θ and �λi

indicated different levels of similarity when each was scaled from

0% (orthogonal) to 100% (identical) (Fig. 3).

To investigate the trait correlations responsible for the simi-

larity between G and D, we projected the eigenvectors of S into

the subspace defined by the first six eigenvectors of G (Table 4).



Table 4 . Summary of the eigenvectors of the matrix of similarity (S) projected onto the eigenvectors of the genetic variance–covariance

matrix (G) of 12 life-history traits in 20 populations of Lythrum salicaria grown under uniform glasshouse conditions.

Trait b1 b2 b3 b4 b5 b6

Leaf area at transplant 0.292 −0.307 0.195 0.594 0.282 0.420Height at transplant 0.412 −0.147 0.075 0.358 −0.640 −0.132Height at week 2 0.491 0.089 −0.300 0.019 −0.067 −0.051Height at week 4 0.440 0.011 −0.308 −0.233 0.137 −0.192Days to first flower −0.094 0.369 0.353 0.064 −0.147 0.248Stem width at maturity 0.070 0.182 0.425 0.230 0.448 −0.459Vegetative size at maturity 0.305 0.445 0.059 −0.156 0.090 0.318Inflorescence length at maturity −0.007 −0.202 0.419 −0.376 −0.358 0.266Vegetative size at harvest 0.287 0.440 0.097 −0.156 0.030 0.177Inflorescence length at harvest 0.186 −0.418 0.034 −0.303 0.357 0.362Final vegetative biomass (ln) 0.211 0.011 0.457 −0.106 0.002 −0.258Final inflorescence biomass (ln) 0.204 −0.316 0.261 −0.339 0.019 −0.307Eigenvalue 1.00 1.00 0.989 0.961 0.755 0.141Angle 0.04 1.25 5.99 11.40 29.65 67.93

Angle quantifies the orientation of a principal component axis from each of D and G and ranges from 0 (aligned) to 90 (orthogonal). Factor loadings show

the contribution of each trait to the similarity between G and D. For example, the closest principal components of G and D define b1 and are largely due to

the intercorrelation of stem width, biomass, and vegetative size measured at maturity and at harvest.

The first five of six eigenvectors of this projection (b1 to b5) were

close to the maximum of one (eigenvalues b1 = 1.00, b2 = 1.00,

b3 = 0.989, b4 = 0.969, b4 = 0.755) with corresponding angles of

0.04o to 29.65o (Table 4). Seedling height measurements loaded

most heavily on b1. Vegetative size at maturity and at harvest

loaded most heavily on b2, along with inflorescence length at

harvest (Table 4). Stem width, vegetative biomass, and inflores-

cence length at maturity loaded primarily on b3. Factor loadings

for days to first flowering were highest for b2 and b3 along with

measurements of adult size. The remaining eigenvectors of S (b4-

b6) were defined by a combination of seedling and adult traits.

DiscussionThe goals of this study on invasive populations of L. salicaria

from eastern North America were: (1) to estimate standing ge-

netic variation for ecologically relevant life-history traits using

seed families grown under uniform conditions, (2) to identify ge-

netic correlations for these traits within and among populations,

and (3) to determine to what extent divergence among popula-

tions may be constrained by the structure of genetic (co)variance

within populations. We found that despite evidence indicating that

founder events and genetic drift play an important role during the

invasion process in these populations (Eckert and Barrett 1992;

Eckert et al 1996), they maintained high levels of genetic variation

for all 12 of the quantitative traits that we examined (Table 1).

We also detected strong population differentiation for these traits,

most of which was manifested as geographical clines distributed

along latitudinal gradients in growing season length (Fig. 1).

Despite the availability of significant standing genetic variation

within and among populations, genetic correlations among traits

(Table 2, Fig. 2) appear to limit the “phenotypic space” avail-

able for populations to respond to natural selection, at least over

the short term. Evolutionary change during invasion has occurred

primarily along lines of greatest genetic (co)variance within pop-

ulations resulting in a high similarity between the first five eigen-

vectors of G and D (Table 4; Fig. 3). Below we discuss the im-

plications of these results and consider alternative hypotheses for

the similarity observed between G and D, particularly the possible

roles of migration and correlational selection.

GENETIC VARIATION IN LIFE-HISTORY TRAITS

There is considerable evidence for abundant genetic variation in

life-history traits within and among wild populations of plants

and animals (reviewed by Mousseau and Roff 1987; Houle 1992;

Mazer and LeBuhn 1999; Geber and Griffen 2003) although rel-

atively few studies have measured quantitative genetic variation

in populations of invasive species (but see Rice and Mack 1991;

Chen et al. 2006; Lavergne and Molofsky 2007; Dlugosch and

Parker 2008b; Facon et al. 2008; Chun et al. 2009). Previous work

on L. salicaria reported significant amounts of additive genetic

variation for age and size at flowering in two native populations

from Sweden (Olsson 2004) and in a single introduced popula-

tion from eastern North America (O’Neil 1997). Consistent with

a scenario of multiple introductions and admixture fostered by

the outcrossed mating system of L. salicaria (Thompson et al.

1987; Barrett 2000; Houghton-Thompson et al. 2005; Chun et al.

2009), we found high levels of genetic variation for vegetative and



Figure 3 . Bootstrap estimates of two measures of similarity be-

tween the matrix of genetic variance-covariance (G) and the ma-

trix of covariance among population means (D) estimated from 20

populations of Lythrum salicaria grown under uniform glasshouse

conditions. (A) The angle of orientation between gmax and dmax

(θ) examines only the first principal components of G and D (see

Schluter 1996) and ranges from 0◦ (complete similarity) to 90◦ (no

similarity). (B) The index of similarity compares the first six princi-

pal components of G and D (see Blows et al. 2004) and ranges from

0 (no similarity) to 6 (complete similarity). The bootstrap models

consist of 10,000 iterations of resampling, with replacement, from

339 standardized family means to generate a bootstrap sample of

10,000 G and D matrices.

reproductive traits both within and among the 20 populations that

we investigated (Table 1). Although hybridization with native L.

alatum has been proposed as a genetic mechanism contributing

towards invasion success in L. salicaria (Ellstrand and Schieren-

beck 2000; Houghton-Thompson et al. 2005), AFLP data do not

provide strong support for this hypothesis (see Fig. 2 in Houghton-

Thompson et al. 2005). Instead, it seems more likely that gene

flow among introduced genotypes and possibly ornamental vari-

eties may have contributed to the high genetic diversity of invasive

populations (Houghton-Thompson et al. 2005; Chun et al. 2009).

Our previous studies revealed striking variation among pop-

ulations in their overall levels of quantitative genetic variation

(Colautti et al. 2010). In particular, we reported that genetic varia-

tion for days to first flower and vegetative size declined toward the

northern range limit. Levels of standing genetic variation and the

strength of genetic correlations may also differ among populations

for the traits examined here. For example, stronger stabilizing se-

lection on flowering time in northern populations may weaken

the strength of the genetic correlation between days to first flower

and vegetative size. Quantifying such changes in the variance and

covariance for genetically correlated traits is analytically difficult

and has only recently been attempted (see Hine et al. 2009). Such

an analysis was beyond the scope of the present study and instead

we examined the “average” genetic (co)variance matrix for the 20

populations we investigated.

TRAIT CORRELATIONS AS GENETIC CONSTRAINTS

Genetic correlations lower the “dimensionality” of available phe-

notypic space and can limit opportunities for local adaptation,

at least over short timescales (Dickerson 1955; Lande 1982; Orr

2000). The high intercorrelation among adult traits reported here

suggests that selection on earlier flowering in northern popula-

tions may be constrained by a trade-off with size at reproduction.

Specifically, we detected strong positive genetic intercorrelations

among days to first flower, stem width, vegetative size at flower-

ing, vegetative size at harvest, and vegetative biomass (Fig. 2 and

Table S2). Consequently, plants that flowered earlier are likely

to suffer a cost associated with their smaller size. The smaller

vegetative size of early-flowering genotypes was fixed at the time

of first flowering and did not increase throughout the growing

season, given the strong broad-sense genetic correlation between

vegetative height at first flowering and at harvest (r = 0.994). In

native populations of L. salicaria, Olsson (2004) also found sig-

nificant additive genetic correlations between time to first flower

and the number of vegetative nodes at maturity, a proxy for veg-

etative size. Thus, it appears that: (1) introduced populations are

subject to some of the same genetic constraints that are evident

in native populations, and (2) these trade-offs may be relatively

stable over longer timescales. Indeed, this may be a more general

phenomenon, as the evolution of earlier reproduction at higher lat-

itudes appears to be constrained in two other species—Xanthium

strumarium (Etterson and Shaw 2001) and Chamaecrista fasci-

culata (Griffith and Watson 2006). Constraints on the evolution

of early flowering because of shorter growing seasons may be an

important determinant of range limits generally.

A longer term genetic constraint involving early flowering

and size in L. salicaria may be associated with the architecture

of inflorescence development. In common with several other di-

cotyledonous species, for example, Antirrhinum majus and Ara-

bidopsis thaliana (reviewed in Yanofsky 1995; Ma 1998; Simpson

et al. 1999), organogenesis in L. salicaria proceeds in an acropetal

direction at the shoot apical meristem. In A. majus and A. thaliana,

maturation of the primary stem occurs in response to external

cues, particularly temperature and photoperiod, which trigger a



developmental phase change from resource-gathering structures

(i.e., shoots and leaves) to resource sinks (i.e., flowers, nectar,

seeds). This may result in a fitness trade-off because vegetative

size correlates strongly with flower and fruit production (O’Neil

1997; Colautti et al. 2010). In the plants we investigated, genetic

variation for vegetative growth after the onset of flowering must

be negligible compared to genetic variation for size at maturity,

otherwise there would be a weaker correlation between vegetative

size measured at first flower and at the end of the experiment. This

developmental constraint may help to explain, in part, the inter-

correlations among days to first flower and the measurements of

plant size (see Fig. 2).

Time to first flower and adult traits associated with vegeta-

tive size fit the model of plant development noted above, but the

results for inflorescence length and seedling traits (i.e., height at

transplant, leaf area at transplant, height-2wk, and height-4wk)

are harder to interpret. Variance–covariance components involv-

ing inflorescence length and biomass should be interpreted care-

fully because plants in our experiment had no opportunity to pro-

duce seeds, which require insect pollinators. Therefore, resources

available for seed production, unlike natural populations, would

not limit the length and biomass of inflorescences measured in

the glasshouse. Indeed, inflorescence measurements correlated

poorly between the glasshouse and the field, despite strong corre-

lations between glasshouse and field measurements for flowering

time and vegetative traits (Montague et al. 2008; Colautti et al.

2010).

Overall, seedling traits were less genetically intercorrelated

within populations than adult traits, but there were a few adult

traits that were strongly correlated with seedling traits (Fig. 2 and

Table S2, above diagonal). For example, seedling height mea-

surements were negatively correlated with days to first flower

(i.e., early-flowering plants grew faster), yet seedling height mea-

surements were only weakly correlated with adult height mea-

surements, despite a strong positive correlation between days to

flower and vegetative size at flowering. Thus, early-flowering

plants grow faster as seedlings but are still not able to grow as

large as later-flowering plants. This result is consistent with a

model of developmental constraint proposed by Colautti et al.

(2010) in which plants must reach a threshold size before they

begin to flower. Seedlings that grow faster are able to initiate

flowering sooner, whereas slower-growing seedlings delay flow-

ering until they reach a threshold size.

GENETIC VARIANCE–COVARIANCE (G) AND

POPULATION DIVERGENCE (D)

If the genetic correlations illustrated in Figure 2 represent con-

straints on population divergence then the genetic (co)variance

matrix (G) should be a reasonable predictor of trait (co)variance

among populations (D). Indeed, this is the case as the index of

similarity (�λi) between G and D was 4.8, which is close to the

theoretical maximum (6 = identical matrices) and highly signif-

icant based on a bootstrap model with orthogonal rotation (P <

0.001). The high value of �λi derives from the fact that five of the

six eigenvectors of S were close to the theoretical maximum of

one, with only the final eigenvector (b6) showing little similarity

between G and D (Table 4). The similarity in five of the six dimen-

sions of comparison between G and D is consistent with strong,

multitrait genetic constraint on population divergence during the

invasion of L. salicaria in eastern North America.

Interpreting the similarity between G and D as a constraint on

population divergence assumes that our estimate of G is a reason-

able approximation of the ancestral (co)variance structure of each

population (i.e., a 20-population polytomy). The genetic relation-

ships among the 20 populations used in this study have not been

characterized using neutral genetic markers. However, introduced

populations of invasive plants often show little geographical struc-

turing relative to native populations (Barrett and Husband 1990;

Dlugosch and Parker 2008a) and this is also true for other popula-

tions of L. salicaria in North America (Houghton-Thompson et al.

2005; Chun et al. 2009). Human-influenced gene flow in invasive

species tends to homogenize the phylogeographic relationships

among populations, which would otherwise confound interpre-

tation of similarities between G and D as genetic constraints on

evolution. Future work combining neutral markers with measure-

ments of natural selection at different points along the latitudinal

gradient would clarify the relative influence of genetic constraints,

natural selection, and stochastic processes on divergence of our

study populations (see methods in Chenoweth and Blows 2008;

Hohenlohe and Arnold 2008; Chenoweth et al. 2010).

The different levels of constraint suggested by θ and S are

difficult to reconcile with nonadaptive processes. The estimated

average angle (θ) between gmax and dmax was about half the the-

oretical minimum similarity (90o = no similarity) (Fig. 3) and

less similar than five of the six eigenvectors of S (0.04–29.65o;

Table 4). Stochastic processes such as bottlenecks, founder events,

and genetic drift should not affect the proportional relationship

between G and D and therefore the factor loadings and eigen-

values of gmax and dmax should be similar (Lande 1979; Jones

et al. 2004; Hohenlohe and Arnold 2008). Instead, dmax explains

67.8% of the total (co)variance among populations, whereas gmax

explains only 45.7% of total genetic (co)variance within popu-

lations, demonstrating that the first principal component eigen-

vectors of G and D are not proportional, despite the high degree

of similarity between G and D. Therefore, stochastic processes

alone would appear to be insufficient to explain the contrasting

estimates of similarity measured by θ and �λi.

In contrast to stochastic processes, strong natural selection

can result in divergence of populations away from the primary

eigenvectors of G (Lande 1979; Zeng 1988; Jones et al. 2004).



The difference in constraint suggested by θ and �λi could be

explained if: (1) selection on a few traits changed the orientation

of dmax relative to gmax, but (2) the evolutionary response to selec-

tion was still constrained by genetic correlations with other life-

history traits. This seems likely for two reasons. First, loadings

of traits differed between gmax (Table 2) and dmax (Tables 3),

suggesting a weak constraint in the first dimensions of G and D.

Indeed, the large factor loadings for days to first flower and other

adult size traits in gmax is consistent with a strong genetic con-

straint, whereas the weaker loading of days to flowering in dmax is

consistent with selection that breaks apart this constraint. Second,

the eigenvalues of the first five eigenvectors of S (i.e., b1, to b5 in

Table 4) were close to their theoretical maximum of one, suggest-

ing a strong constraint in five of the six principal eigenvectors Gand D. Therefore, selection appears to have played an important

role in the divergence of populations for a few traits, but overall

was highly constrained along genetic lines of (co)variance among

life-history traits.

ALTERNATIVE HYPOTHESES FOR G-D SIMILARITY

We have interpreted the similarity between G and D (Fig. 3B) as

a constraint on the evolution of D imposed by the genetic vari-

ance and covariance components of G. However, theory suggests

other factors that can also orient G toward D (reviewed in Arnold

et al. 2008). One hypothesis is that correlational selection favors

combinations of traits within populations that mirror the direc-

tion of divergence among populations. For example, the genetic

correlation between stem width and vegetative size at flowering

(Fig. 2) may be a result of correlational selection, as larger plants

may need larger stems to increase transport of more resources or

as structural support. However, time to flowering and vegetative

size are positively correlated in G and D (Fig. 2) yet selection

coefficients measured in L. salicaria confirm that selection favors

both early flowering and larger size (Colautti and Barrett 2010)

without strong correlational selection (O’Neil 1997). Therefore,

correlational selection alone is an unlikely explanation for the

strong similarity between G and D.

Effects of maternal environment on offspring growth and de-

velopment (i.e., maternal effects) could also result in correlations

among life-history traits resulting in similarity between G and

D. However, experimental evidence using the same populations

in this study does not support a significant influence of maternal

effects on adult traits (see Montague et al. 2008; Colautti et al.

2010). This conclusion was also supported by the weak corre-

lations between vegetative size at harvest and seedling traits in

G and D (Fig. 2 and Table S2). Maternal effects are known to

influence seedling traits in many plant species as higher quality

seeds germinate earlier and grow faster (Roach and Wulff 1987).

However, in our experiment the opposite was true, as germina-

tion date was positively correlated with relative growth rate from

transplant to week 2 (ρ = +0.383, df = 2767 P < 0.001) and

from week 2 to week 4 (ρ = +0.278, df = 2766, P < 0.001). Our

results are therefore not consistent with the hypothesis that the

similarity between G and D results from maternal influences on

correlations among life-history traits.

A third hypothesis that predicts similarity between G and Dis that gene flow along latitudinal gradients can create latitudinal

clines like those observed in our study (Fig. 1) and orient G in

the direction of D (Guillaume and Whitlock 2007). This scenario

would require a strong pattern of isolation-by-distance (IBD),

which is unlikely for a species that has spread over 1000 km in the

past 50–100 years and shows evidence of significant long-distance

dispersal (see Houghton-Thompson et al. 2005; Chun et al. 2009).

Moreover, strong IBD alone cannot explain the lower level of

divergence of θ relative to �λi (Fig. 3A) because the strength

of the covariance components of G should be proportional to

the same covariances of D, resulting in the greatest similarity

between gmax and dmax. The difference in factor loadings for days

to first flower in G (Table 2) versus D (Table 3) is contrary to

a scenario of IBD. Instead, our results are more consistent with

genetic constraints on an evolutionary response to selection on

population divergence.

CONCLUSIONS

Identifying genetic constraints on local adaptation is an important

step in understanding species’ range limits and for predicting the

rate and extent of spread in invasive species. However, identify-

ing constraints in natural populations is complicated by variation

among populations in gene flow, natural selection, and genetic

drift because these processes can lead to idiosyncratic differences

in the magnitude and direction of genetic correlations. We have

attempted to circumvent this problem by estimating an “average

G” in 20 populations of L. salicaria from eastern North Amer-

ica, a method that is particularly well-suited to introduced species

that are likely to have relatively weak phylogeographical struc-

turing as a result of human-mediated dispersal. Despite consider-

able standing genetic variation within populations of L. salicaria

for individual traits, genetic correlations among traits appear to

have limited the phenotypic space available for divergence among

populations, particularly for traits that display strong latitudinal

clines. The similarity in several of the genetic correlations be-

tween native and introduced populations of L. salicaria suggests

that genetic constraints on population divergence may play an

important role in the establishment of range limits in invasive

species.

ACKNOWLEDGMENTSWe thank M. Blows, J. Stinchcombe, K. Rice, A. Weis, and C. Eckert forcomments on the manuscript; L. Flagel for assistance running SAS onUniversity of North Carolina’s research computing cluster; the Ontario



Government and the University of Toronto for scholarship support to RIC;the Canada Research Chair program and an Ontario Premier’s DiscoveryAward for funding to SCHB; and the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for a graduate scholarship to RICand a Discovery Grant to SCHB.

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Associate Editor: M. Blows

Supporting InformationThe following supporting information is available for this article:

Table S1. Variances (diagonal) and covariances (off-diagonal) among 12 life-history traits for the G matrix (above diagonal)—the

(co)variance matrix among seed families within populations estimated by restricted maximum likelihood, and for the D matrix

(below diagonal)—the (co)variance matrix among standardized population means.

Table S2. Correlation coefficients for 12 life-history traits calculated from the G matrix (above diagonal)—the (co)variance matrix

among seed families within populations estimated by restricted maximum likelihood, and for the D matrix (below diagonal)—the

(co)variance matrix among standardized population means.

Supporting Information may be found in the online version of this article.

Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the

authors. Any queries (other than missing material) should be directed to the corresponding author for the article.


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POPULATION DIVERGENCE ALONG LINES OF GENETIC VARIANCE AND COVARIANCE IN THE INVASIVE PLANT LYTHRUM SALICARIA IN EASTERN NORTH AMERICA