ELECTRONIC SUPPLEMENTARY MATERIAL
Nonlinear selection model
The LandeArnold approximation (Lande and Arnold 1983; see also Phillips and Arnold 1989) of selection on two quantitative traits is:
(1) w = α + β_{d}x_{d} + β_{v}x_{v} + ½ γ_{d}x_{d}^{2} + ½ γ_{v}x_{v}^{2} + γ_{dv}x_{d}x_{v}
where β and γ are the linear and nonlinear selection coefficients, respectively, α is the intercept and w is fitness. The variable ‘x’, denotes the value of a trait in an individual and subscripts ‘d’, and ‘v’ are days to first flower and vegetative size at first flower, respectively. We consider only positive values of x_{d} and x_{v} since it is impossible to flower before day zero, or to have negative size. In Lythrum salicaria fitness effects for these two traits are additive, such that γ_{dv} = 0 in Eq. 1 (O’Neil 1997).
We approximated the fitness function for flowering time as a decelerating curve with a maximum fitness at x_{d} ≤ 0 (i.e. no delay to flowering) and vegetative size as an accelerating function with a minimum at x_{v }≤ 0. Therefore, γ_{d} < 0 and β_{d} ≤ 0, γ_{v} > 0 and β_{v} ≥ 0. These two traits tradeoff if there is a positive genetic correlation, in which case x_{d }and x_{v} can each be rescaled by their intercepts and variances to a common principal component (PC1); for simplicity we substitute PC1 into Eq. 1 and redefine α, β and γ for this new scale of measurement (i.e. standardized selection coefficients), simplifying Eq. 1 to:
(2) w = (β_{d} + β_{v}) PC1 + ½ (γ_{d} + γ_{v}) PC1^{2}
Assuming there is an optimum (θ_{pc}) for Eq. 2, fitness is maximized when:
(3) PC1 =  (β_{d }+ β_{v}) / (γ_{d} + γ_{v})
Southern populations of L. salicaria experience a longer growing season, so the marginal cost of flowering later will be smaller in the south, resulting in weaker stabilizing selection on flowering time, or γ_{d,s} < γ_{d,n}, where the subscripts ‘s’ and ‘n’ denote hypothetical southern and northern environments, respectively. Given that γ_{d,s} < γ_{d,n}, it is obvious that θ_{pc,n} < θ_{pc,s}, if we hold constant the selection coefficients for vegetative size (i.e. β_{v} and γ_{v}), and the linear selection coefficient for flowering time (β_{d}). Therefore, weaker stabilizing selection in the southern environment alone is sufficient to shift the optimum (θ_{pc}) towards smaller plants that flower earlier in northern populations.
In the absence of a mutation threshold preventing early flowering, the expected mean of a population is given by θ_{pc} (Eq. 3). This remains a good approximation in the presence of a threshold, provided θ_{pc} is larger than the threshold value (figure 1). The expected variance of PC1 is proportional to the curvature of the fitness function, effective population size, the distribution of mutational effects, and the rate of new mutations (Kimura 1965, Lande 1975, Turelli 1984). We assume that genetic variance for PC1 follows a Gaussian distribution and that the effects of population size and mutation do not correlate strongly with latitude. Variance of PC1 for northern vs. southern sites will be proportional to the xintercepts of Eq. 2 (where PC1 > 0), which are inversely proportional to the strength of stabilizing selection. However, a threshold prevents small plants from flowering early, resulting in a positive skew that is inversely proportional to the difference between the optimum and the threshold size. That is, a population with an optimum much larger than the threshold will have a skew closer to zero. This occurs because the probability of observing the phenotypic value PC1 is zero for all values of PC1 below the threshold (figure 1).
Our predicted latitudinal trends in mean, variance and skew (hereafter ‘qualitative predictions’) are robust to a wide range of parameters and require only that:
(4)  (β_{d,n }+ β_{v,n}) / (γ_{d,n} + γ_{v,n}) <  (β_{d,s }+ β_{v,s}) / (γ_{d,s} + γ_{v,s})
If β_{v} and γ_{v} do not differ predictably with latitude then it is clear from Eq. 4 that our qualitative predictions hold wherever β_{d,n }/ γ_{d,n} > β_{d,s }/ γ_{d,s}. Recall that both γ_{d} and β_{d} are less than zero; thus, in addition to stabilizing selection, stronger directional selection for early flowering will result in similar fitness functions and qualitative predictions (i.e. latitudinal clines). We have assumed no latitudinal change in selection on vegetative size. However, if there is a latitudinal trend it is likely towards a decrease in β_{v} and γ_{v} as latitude increases because plants from southern populations experience a longer growing season, and therefore have more opportunity for vegetative growth and competitive interactions, increasing the marginal benefit of being large when they initiate flowering. A correlated decrease in β_{v} and/or γ_{v} with increasing latitude would strengthen our predicted latitudinal gradients. Our results are also robust to the assumption of the form of selection on vegetative size. An optimum for Eq. 2 will only occur when γ_{v} <  γ_{d}. Since γ_{d} is negative in our model, Eq. 2 will have an optimum for all γ_{v} ≤ 0. This suggests that our assumption of an accelerating fitness function for vegetative size is conservative, since an optimum will also exist for any linear or decelerating fitness function. Therefore, from our interpretation of Eq. 4 it appears that our qualitative predictions are robust to a wide range of model assumptions.
Genetic estimates of variance and skew
We used BLUPs of population means and estimated the coefficient of skew from BLUP estimates of family means using the model:
PC_{ijk} = Bl + P_{i}+ F_{j}(P_{i}) + ε_{ijk}
where Bl is the fixed effect of experimental block, population (P) i and seed family (F) j (nested within population i) are random factors, and ε is the error term of individual k. Variances among populations and among families were estimated directly by ReML; coefficient of skew was calculated from the best linear unbiased predictor (BLUP) of family means.
We also estimated variances among families for each population separately by ReML using the statistical model:
PC_{jk} = Bl + F_{j} + ε_{jk}
and a separate analysis for each population. Latitudinal trends using genetic estimates support our phenotypic analysis (figure 3), with similar clines in the mean, variance and skew (see figure S1).
Scaling effects
Positive correlations between the mean and central moments (i.e. variance, skew and kurtosis) of morphological and lifehistory traits is primarily a problem for comparing statistics based on central moments (e.g. heritability) among different traits, or different species, where the underlying genetic architecture may differ significantly. The issue of scaling is less clear for comparisons of days to flower and vegetative size among populations of the same species, where differences are more likely due to variation in allele frequencies than genetic architecture. Scaling can occur for traits like biomass or flower number that are lognormal or Poissondistributed if data are assumed to be Gaussian. However, there was no significant correlation between the mean and variance (n = 20, r = 0.065, p = 0.785), or between the mean and skew (n = 20, r = +0.047, p = 0.843) of PC2 allowing us to reject a general scaling relation in our data. Furthermore, a BoxCox (Box & Cox 1964) transformation of traits (λ: days = 0.75; vegetative size = 0.25; inflorescence length = 0.5) stabilized the variance and skew of PC1 and still supported the results in table 1 and figure 2, with significant latitudinal gradients in the mean (n = 20, r = 0.764, p < 0.001), variance (n = 20, r = 0.484, p < 0.05) and skew (n = 20, r = +0.782, p < 0.001) of PC1. Therefore, statistical artefacts are unlikely to explain latitudinal gradients in the mean, variance and skew of PC1.
Controlled pollinations
All of our field experiments used openpollinated seed families and therefore maternal effects could have contributed to the latitudinal clines in days to flower and vegetative size, as well as the genetic correlation between them. For example, seeds formed under cooler conditions, or under rapid fruit maturation at higher latitudes, could conceivably be of lower quality, resulting in earlier maturation at smaller size. Maternal effects should be most evident in seed size and at early life stages (Roach & Wulff 1987). However, seed size did not correlate with latitude (n = 22, r = +0.150, p = 0.500; Montague et al. 2008), and in the glasshouse latitude did not correlate with time to germination (n = 20, r = +0.275, p = 0.241) or seedling height at transplant (n = 20, r = 0.349, p = 0.132). The concordance between glasshouse and field studies (table 2) also suggests that plasticity in growth and flowering time across environments is minor relative to genetic differences among populations.
We directly tested for maternal effects and transgenerational plasticity using a diallel F_{1} crossing design. We also used these crosses to test the consistency of the genetic correlation estimated from openpollinated seed families. We created ten F_{1} families of L. salicaria by randomly choosing one of five plants from a northern population and crossing it with one of five plants from a southern population. Each of these 10 parental plants represents an openpollinated family used in the glasshouse and field experiments. Each plant was mated twice, once as a maternal plant and once as a pollen donor. We grew three offspring from each of the ten crosses. Our F_{1} crossing design is similar to a diallel in that each F_{1} family is crossed with itself and every other family. Because L. salicaria is tristylous successful crosses only occur between different floral morphs, limiting the number of total crosses to 88 of a possible 100 inter and intrafamily crosses.
We germinated 12 F_{2} seeds from each cross in 2cm x 2cm plug trays in early spring of 2007 and transplanted four offspring from each cross into a common garden at the Koffler Scientific Reserve at Jokers Hill in early June. We measured days to flower and vegetative size at flower in the first growing season using the same protocol as our other common garden studies (see methods). Our statistical method follows Fry (2004) and used restricted maximum likelihood (ReML) in SAS 9.1 (SAS Institute Inc., Cary, NC). For simplicity we assumed a diploid model. Since L. salicaria is an autotetraploid, this will underestimate the significance of digenic interactions (i.e. dominance variance) and additive variance, but should not underestimate maternal, paternal, or cytonuclear (i.e. maternal*paternal interactions) effects.
Results from our mixed model do not support maternal influences on days to flower (d.f. = 1, χ^{2} = 0.0, p = 0.963) or vegetative size at first flower (d.f. = 1, χ^{2} = 2.7, p = 0.102). ReML estimates of paternal effects and cytonuclear effects were near zero (χ^{2} < 0.01). Thus, our results do not support a strong effect of transgenerational plasticity and are more consistent with a genetic basis for latitudinal clines in mean, variance and skew of PC1 (and see further discussion in Montague et al 2008). In addition to examining the potential for transgenerational plasaticity, we tested for a genetic correlation between days to flower and vegetative size among F_{2} families. The Pearson productmoment correlation between these traits was highly significant across family means, where a family included reciprocal fullsibs (i.e. fullsibs with reversed ovule and pollen parents) (N = 34, R^{ }= 0.638, P < 0.001). Therefore, the genetic correlation between days to flower and vegetative size are the result of additive or dominant genes, with no detectable maternal effects.
Major genes and migration
Two other mechanisms could potentially explain the increase in skew and decrease in variance for PC1 in northern populations relative to southern populations. First, a change in allele frequencies for a few alleles of large effect would be qualitatively similar to a one locus, twoallele model, which can produce a correlated decline in the mean and variance with an increase in skew (Lynch and Walsh 1998). Second, a small proportion of longdistant migrants from southern populations could increase the skew of northern populations. However, the effects of migration on skew and variance should scale with each other, resulting in no change in the coefficient of skew, and the effect of major genes should be evident in frequency histograms of the phenotypic distributions within populations. The latitudinal clines observed for the coefficient of skew do not match the observed phenotype frequency histograms, which show continuous variation among PC1 phenotypes (figure S2). Variance among individuals from the five most northern population (V_{N} = 1.56) was lower than that of southern populations (V_{S} = 2.30), while the coefficient of skew was higher in the north (CS_{N} = 0.64) compared to the south (CS_{S} = 0.15). Removing rare individuals (i.e. bars with < 5 individuals in figure S2) had little effect on the relative variance (V_{N} = 1.29, V_{S} = 2.12) and skew (CS_{N} = 0.35, CS_{S} = 0.05). Our results suggest that a few alleles of large effect, or rare longdistance migration events, do not explain latitudinal changes in variance or skew of PC1 in our data.
References
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Figure S1. Latitudinal gradients in the mean, variance and skew of genetic estimates of PC1 and PC2 from statistical mixed models. (a) Observed standardized mean and central moments and leastsquares regression lines of twenty Lythrum salicaria populations for the observed first principal component (PC1) describing days to flower and vegetative size at flowering. (b) Observed mean and central moments and leastsquares quadratic regression lines of the second principal component (PC2).
Figure S2. Frequency histograms of PC1 score for individuals from five southern (red bars) and five northern (blue bars) populations of Lythrum salicaria from eastern North America.
Figure S1
Figure S2
