SELECTION FOR CHARACTER DISPLACEMENT IS CONSTRAINED BY THE GENETIC ARCHITECTURE OF FLORAL TRAITS IN THE IVYLEAF MORNING GLORY

Authors


Abstract

Evolutionary theory predicts that interactions between species such as resource competition or reproductive interference will generate selection for character displacement where similar species co-occur. However, the rate and direction of character displacement will depend not only on the strength of selection for trait divergence, but also on the amount of genetic variation for selected traits and the nature of genetic correlations between them. To assess the importance of genetic constraints for the evolution of character displacement, we examined the genetic architecture of a suite of floral traits previously shown to be under selection in the annual plant Ipomoea hederacea when this species co-occurs with Ipomoea purpurea. We found that the six floral traits we measured are all positively genetically correlated. We also demonstrate, using new statistical approaches, that the predicted response to selection for four of these six traits is substantially constrained by their genetic correlation structure. Most notably, the response to selection for reduced separation of the tallest and shortest anthers, which reduces the degree of detrimental heterospecific pollen flow, is substantially constrained. Our results suggest that the rate of evolution of reproductive character displacement in I. hederacea is limited by the genetic architecture of floral traits.

Ecologists and evolutionary biologists have long been interested in the role of species interactions (i.e., competition, mutualism, predation, parasitism, etc.) in driving evolutionary change. One major category of interaction-driven change is character displacement, which is expected to yield a characteristic pattern of variation: for a given character or set of characters, differences between two closely related species are more pronounced in sympatry than in allopatry (Brown and Wilson 1956). Two types of character displacement are recognized: ecological character displacement, which results from selection generated by competition for shared resources; and reproductive character displacement, which results from selection to avoid disadvantageous hybridization. In plants, both types of character displacement can arise when species share common pollinators, leading to evolutionary divergence in characters like flowering phenology, attractiveness to specific pollinators, rates of self-fertilization, and positioning of anthers and stamens (Rathcke 1983; Waser 1983; Levin1985).

A substantial literature documents patterns of selection for character displacement (i.e., Schluter 1994, 2003; Bolnick 2004; Pfennig et al. 2007), although studies involving plants are uncommon (but see Caruso 2000; Smith and Rausher 2008). It thus seems reasonable to conclude that interactions between species often generate selection that favors character divergence. However, interacting species do not always exhibit greater character divergence in sympatry than in allopatry (Walker 1974; Noor 1997); we suspect that such cases are under-reported because of bias against negative results. The absence of character displacement in these examples may of course be due to absence of selection for divergence, although such absence is seldom documented. Alternatively, as for any type of evolutionary change, absence of character displacement may also be due to constraints on the response to selection. Evolutionary biologists have long recognized that developmental and genetic constraints can prevent natural selection from producing optimally adapted phenotypes (Darwin 1859; Gould and Lewontin 1979; Cheverud 1984; Maynard Smith et al. 1985; Barker and Thomas 1987; Clark 1987). Genetic constraints can inhibit evolutionary change in two ways, either through lack of additive genetic variation or through genetic correlations between traits that run counter to the direction of selection on those traits (Lande and Arnold 1983). With respect to character displacement, the rate and direction of trait divergence will thus depend not only on the strength of selection, but also on the amount of genetic variation for selected traits and the nature of genetic correlations between them. Because few attempts have been made to assess the magnitude of genetic constraints on traits under selection for character displacement, the role of genetic constraints in limiting character displacement is poorly understood.

More generally, the role of genetic constraints in limiting the rate of response to selection in any character remains unclear. Evolutionary biologists have known for several decades that genetic correlations among traits may either constrain or facilitate the response to selection, depending on the orientation of the genetic correlation structure to the selection gradient (Lande 1979). Only recently, however, attempts have been undertaken to quantify the degree to which constraint or facilitation occur. These attempts have generally adopted one of two approaches. The first approach has been to compare the predicted univariate response to a given pattern of selection with the predicted multivariate response to selection. Although the univariate response predicts the response to direct selection alone, the multivariate response also takes into account indirect selection due to genetic covariances among traits (e.g., Campbell 1996; Mitchell et al. 1998; Etterson and Shaw 2001; Caruso 2004). The second approach has been to compare the orientation of the selection gradient to the orientation of the genetic variance–covariance structure of the traits (e.g., Blows and Higgie 2003; Blows et al. 2004; Blows and Hoffmann 2005). Although some of these investigations suggest that constraint may be substantial, none, except for Blows et al (2004), incorporate any statistical testing that takes into account uncertainty in the measurement of the variance–covariance structure. It is thus unclear how reliable inferences of either the existence or magnitude of constraint are for these studies.

Here we present and use a new statistical approach to evaluate the degree to which the genetic architecture of floral traits constrains the expected response to selection for character displacement in the common morning glory, Ipomoea hederacea, when this species co-occurs with its congener I. purpurea. These two species are commonly found together throughout the southeastern United States. We have demonstrated previously that the presence of I. purpurea alters patterns of selection on floral traits of I. hederacea (Smith and Rausher 2007, 2008). More specifically, we found that the presence of I. purpurea flowers generates selection to reduce the spatial separation between anthers and stigma in I. hederacea, an arrangement thought to be an adaptation for minimizing the amount of detrimental heterospecific pollen flow by maximizing the rate of self-fertilization and/or by acting as a barrier to incoming pollen (Ennos 1981; Stucky 1985). A response to this selection would thus constitute reproductive character displacement.

To assess the degree to which genetic architecture constrains the response to selection in I. hederacea, we first quantified patterns of genetic variance and covariance for a suite of floral traits shown previously to be under selection for character displacement in this species. We then compared statistically the expected responses to selection in the presence and absence of genetic covariation among the characters. Our results indicate that the expected response to selection for reduced separation of anthers and stigma, as well as for most other traits measured, is greatly constrained by positive genetic covariances among the characters.

Materials and Methods

STUDY SYSTEM

Ipomoea hederacea (L.) Jacquin and I. purpurea (L.) Roth (Convolvulaceae) are self-compatible annual vines that commonly co-occur as weeds of agricultural fields and along roadsides throughout the southeastern United States. Both species have trumpet-shaped flowers that are showy but short-lived, typically opening before dawn and wilting by the afternoon of the same day. Ipomoea hederacea flowers are light to dark blue in color with a white throat. Ipomoea purpurea flowers are slightly larger in size and range in color from blue to purple to bright pink to white. Flowers of both species have five stamens of varying heights and one style, but whereas in I. purpurea the stigma is generally exserted above the anthers, in I. hederacea the stigma is commonly found at the same level as, and tightly surrounded by, the anthers (Ennos 1981). These differences in proximity of anthers and stigma are thought to explain the differences in self-fertilization rate between these two species: I. purpurea has a mixed mating system, with a selfing rate typically ranging between 65% and 70% (Ennos 1981; Brown and Clegg 1984), whereas I. hederacea is highly selfing (selfing rate = 93%; Ennos 1981) and is capable of high levels of autonomous seed set in the absence of pollinators.

In North Carolina, seeds of both species germinate between May and August. Plants begin flowering about six weeks after germination and continue to produce flowers until they are killed by the first hard frost in the fall. Once fertilized, seeds take approximately four weeks to mature. Individual fruits typically contain one to six seeds, although fruits containing as many as eight seeds have been observed (R. A. Smith, pers. obs.).

Premating barriers between these two species are extremely weak. Although I. hederacea typically initiates blooming 1–2 weeks earlier than I. purpurea (R. A. Smith, pers. obs.), their flowering phenologies are largely overlapping, and they share the same bumblebee pollinators, which account for more than 75%–95% of visits to both species (Stucky 1984; Wolfe and Sowell 2006). In addition, pollen flow from I. purpurea to I. hederacea has potentially detrimental effects on I. hederacea seed production because it germinates on I. hederacea stigmas and grows through the stylar tissue, but results in hollow, inviable seeds (Guries 1978). By contrast, I. hederacea pollen does not fertilize I. purpurea ovules and has not been found to reduce I. purpurea seed set (Guries 1978; Stucky 1985). The asymmetry between I. hederacea and I. purpurea is not entirely surprising. In hybrid crosses in plants, pollen from the longer-styled taxon frequently outcompetes pollen from the shorter-styled taxon (Levin 1971; Kiang and Hamrick 1978; Gore et al. 1990; Williams and Rouse 1990), and I. purpurea styles are generally longer than those of I. hederacea.

EXPERIMENTAL DESIGN

We studied I. hederacea growing intermixed at roughly equal frequency with I. purpurea in an experimental field population in central North Carolina in the summer of 2004. Natural selection on floral traits was measured on the same plants at this site and has been reported previously (Smith and Rausher 2008). The I. hederacea seeds used for both our previous selection analysis and for the present study were drawn from a collection of self-sib lines. To generate these lines, single seeds from each of 40 different maternal plants were collected from a total of six natural populations in central North Carolina. Each seed was planted and grown under identical conditions in a greenhouse to equalize maternal effects, allowed to self-fertilize, and propagated through single-seed descent for eight generations. The resulting seeds from a given maternal plant constitute a family of self-siblings and will hereafter be referred to as a family.

Because our experimental seeds are highly inbred, variation between families reflects genetic variation in the broad sense, including both additive and nonadditive components as well as any maternal effects. Although we have no data on the nature of maternal effects in our experiment, we attempted to equalize their importance by raising all maternal plants under identical conditions in a climate-controlled greenhouse. Measures of broad-sense genetic variation are appropriate for our purposes for two reasons. First, although the efficiency of natural selection in outcrossing species depends specifically on additive genetic variation (Falconer and Mackay 1996), natural selection in highly selfing organisms such as I. hederacea acts on total genetic variation, including both additive and nonadditive components (Roughgarden 1979; Hartl and Clark 1997). In addition, the crossing design needed to partition I. hederacea broad-sense genetic variation into its additive and nonadditive components would generate lines with such unusually elevated levels of heterozygosity as to be of questionable relevance to natural populations of this species (Bright 1998; Mauricio 1998; Stinchcombe and Rausher 2001; Stinchcombe and Rausher 2002).

In June of 2004, 25 seeds from each family were planted in a randomized block design (with one seed from each family per block) in a ploughed agricultural field in Orange County, North Carolina. Seeds were planted in a grid with 75 cm separating the rows and 100 cm separating the columns. To maintain a uniform interplant distance and to prevent plants from getting tangled with each other, each plant that germinated was allowed to twine up a 1.5-m-tall wooden stake placed 5–7 cm from the base of the plant. The site was enclosed by a 3-m-tall chain link fence to exclude deer. All nonexperimental Ipomoea plants that germinated in our site during the course of the experiment were removed, but other vegetation was left undisturbed to mimic conditions in natural populations.

Seven floral traits were measured as described in our previous analysis of selection (Smith and Rausher 2008): corolla diameter, corolla tube length, corolla tube width, stigma height, two measures of anther height (based on the distance between the base of the flower and the tops of both the shortest and tallest anthers), and flower number. We measured one flower per plant, passing down each row of our experimental population in turn over the course of several days until all I. hederacea plants had been measured. We swept through the population in this manner two times over the course of the flowering season such that two flowers per plant were sampled. A pair of hand-held digital calipers was used to measure all traits except flower number, which was measured by counting the number of open flowers on a plant on the day of census. The trait means for each plant in a family were then averaged to calculate a mean for that family.

DATA ANALYSIS

Genetic correlations

In a previous study, we reported that all seven characters exhibited genetic variation in our experimental population, as reflected by highly significant differences in the trait means among families (Smith and Rausher 2008). Here we estimate genetic correlations between the characters as the correlations among family means of the traits (e.g., Stinchcombe and Rausher 2001).

Estimating genetic variances, covariances, and constraint

Genetic constraints can be caused by either or both of two factors: (1) limited genetic variation for individual characters, and (2) genetic correlations among characters. Such constraints can have either or both of two effects on the response to selection: (1) they may alter the direction of the response, or (2) they may alter the magnitude of the response (Arnold 1992). To quantify the effect of constraints, the predicted response to selection with constraint present must be compared to some reference reflecting the expected response in the absence of constraint. Two types of reference seem appropriate. The first is the selection gradient vector, the direction of which represents the direction of the expected response to selection if the genetic variances are equal for all the characters and if all genetic covariances are zero. With this reference, one may ask two relevant questions: (1) do unequal genetic variances cause the direction of the expected response to selection to differ from the reference direction? and (2) do nonzero genetic covariances cause the direction of response to differ from that which would be expected in the absence of covariances?

A second possible reference is the expected response based on the observed genetic variances but with all covariances equal to zero. This reference represents the direction of the expected response in the absence of genetic correlations. Using this reference, one may ask whether the response in the presence of correlations differs from the response in the absence of correlations by comparing the actual response vectors. This comparison allows direct determination of whether genetic correlations affect the magnitude of the response, and indirect determination of whether genetic correlations affect the direction of the response. This type of comparison cannot be used, however, to determine the effect of unequal genetic variances on the response to selection. This is not possible because the reference vector would have to be compared to the expected response to selection if all genetic variances are equal (and covariances are all zero), and there is no nonarbitrary common variance that can be assumed. In this section, we describe methods to implement both of these approaches. In the Discussion, we evaluate the relative usefulness of the two approaches under different circumstances.

Method 1

This method compares the predicted response to selection in the presence versus absence of genetic covariances, and thus asks specifically how genetic correlations constrain evolutionary change. Although genetic correlations cannot normally be manipulated experimentally, it is possible, using the framework of quantitative genetics, to ask what effect trait correlations have on the expected response to selection. We adopt this approach and quantify this type of constraint as the decrease in predicted response that occurs when genetic covariances are taken into account.

Specifically, we compare two vectors, Vc (expected response with genetic covariances) and Vnc (expected response without covariances). The vector Vc is calculated by the standard equation for multivariate response to selection (Lande 1979),

image(1)

where Gc is the genetic variance–covariance matrix of the traits and β is the vector of unstandardized selection gradients taken from our previous analysis of selection (Smith and Rausher 2008). The vector Vnc is calculated similarly as

image(2)

where Gnc is a matrix with the genetic variances of the traits on the diagonal, and the off-diagonal elements (the genetic covariances between traits) set to zero. The null hypothesis of no constraint corresponds to Vdiff=VncVc=0.

Because there is error associated with estimating Gc and Gnc, an assessment of the validity of this null hypothesis must take this error into account. To accomplish this, we constructed 5000 bootstrap samples of these matrices using a bootstrap resampling procedure in MATLAB (MathWorks, Inc., Natick, MA). For each bootstrap sample, 40 families were drawn randomly with replacement from the original data. Each randomly chosen family was then reconstructed by sampling randomly with replacement from among the original members of that family, until the family had the same number of individuals as it did in the original dataset. For each bootstrap sample, among-family components of variance and covariance, which in our case estimate the genetic variances and covariances of the G-matrix, were computed using the method of Ahrens (1976), as implemented by an APL program written by MDR. For each sample, we then calculated Vc and Vnc from equations (1) and (2). Using these bootstrap samples, we tested the null hypothesis in two ways.

First, a necessary condition for the null hypothesis to be true is that the lengths of Vc and Vnc are equal. The lengths are given by |Vc | = (VTcVc)1/2 and |Vnc | = (VTncVnc)1/2, where the superscript T indicates the transpose of the vector. To test statistically whether this is the case, we calculated the proportion of bootstrap samples for which |Vc | > |Vnc |. If this proportion was less than 0.05, we rejected the hypothesis that the two vectors do not differ in length in favor of the hypothesis that Vc is shorter than Vnc, that is, that there is constraint. Next, we evaluated the null hypothesis VncVc=0 directly by assessing whether 0 lies outside the 95% confidence surface for Vdiff (See Appendix for details). Finally, we evaluated whether each character was significantly constrained by assessing the proportion of bootstrap samples for which the predicted response to selection with genetic covariances was greater than the predicted response to selection without covariances. If this proportion was less than 0.05, we rejected the null hypothesis of no constraint.

Method 2

This method compares the direction of the predicted response to selection with genetic constraints to the direction of one of the reference vectors. In particular, three comparisons of direction are performed: (1) between the selection gradient vector and the predicted response with genetic covariances (Vc), which quantifies the combined effect of unequal genetic variances and nonzero genetic covariances on the direction of the response; (2) between the predicted response with genetic covariances (Vc) and the predicted response without genetic covariances (Vnc), which quantifies the effect of genetic covariances on the direction of response; and (3) between the selection gradient vector and the predicted response without genetic covariances, which quantifies the effect of unequal genetic variances on the direction of response.

For each bootstrap sample, Vc and Vnc vectors were converted to hyperspherical angular coordinates by projecting the vectors onto each (v2, vi) plane (i= 1, 3, 4, 5, 6) and calculating the angle of the projected vector with the v2 axis. This yielded a five-vector of angles, θc or θnc, that specify the direction of the vector in the original coordinate system. Three vectors of angular differences, θdiff, corresponding to the three comparisons above, were then calculated by subtracting the angle vectors pairwise: (1) 1θdiff=θβθc, representing comparison 1, where θβ is the angle five-vector for the selection gradient vector; (2) 2θdiff=θncθc, representing comparison 2; and (3) 3θdiff=θβθnc, representing comparison 3. The null hypotheses 1θdiff=0, 2θdiff=0, and 3θdiff=0, corresponding to no angular differences between the pairs of vectors compared, were then tested using the same approach for testing Vdiff=0 as described for Method 1 above and in the Appendix. Finally, the overall vector angle between pairs of vectors was calculated as angle = arcos(θi·θj/ | θi | | θj |), where | θ | is the length of vector θ, equal to (θ·θ)1/2, and i, j=β, c, or nc.

Results

GENETIC ARCHITECTURE OF FLORAL TRAITS

Genetic correlations were positive for all pairs of traits except those including flower number, which was weakly negatively correlated with corolla tube length, corolla tube width, and both measures of anther height (Table 1). Because all genetic correlations with flower number were nonsignificant, we omit this trait from our subsequent analysis of selection response and constraint.

Table 1.  Genetic correlations between floral traits of Ipomoea hederacea grown in the field. Values shown are Pearson product moment correlation coefficients between family means (N=40 families).
 Corolla tube lengthCorolla tube widthStigma heightAnther heighttallestAnther heightshortest Flower  number
  1. 1P<0.01; 2P<0.001; 3P <0.05.

Corolla diameter0.47110.57220.32530.30.152 0.069
Corolla tube length 0.42510.59820.68520.5592−0.061
Corolla tube width  0.2520.59120.6322−0.116
Stigma height   0.37530.273 0.116
Anther heighttallest    0.9282−0.095
Anther heightshortest     −0.025

ANALYSIS OF CONSTRAINT

Method 1

Because all correlations among the analyzed characters are positive, although the selection gradients for some of the characters are negative, the evolutionary response of at least some of the characters is expected to be constrained. The magnitude of this constraint is revealed by comparing the expected response to selection with genetic covariances included (Vc) with the expected response in the absence of covariances (Vnc), using the G matrix presented in Table 2A. For all traits the expected response with covariances is substantially less than that expected with no covariances (Table 2C). Moreover, for the height of the shortest anther, the expected response with covariances is in the opposite direction as the expected response without covariances.

Table 2.  Genetic variance-covariance matrix (G-matrix), selection gradients, and predicted evolutionary responses to selection across one generation (V) for six floral traits in I. hederacea. In the genetic variance-covariance matrix, the diagonal entries are the variances, and the off-diagonal entries give the covariances between traits. Selection gradients were obtained from Smith and Rausher (2008). Vc[i] and Vnc[i] are, respectively, the predicted change in character i with and without constraint due to genetic covariances between traits. All predicted responses are in units of measurement for the trait (mm). Part D of the table lists, for tall and short anthers, the indirect responses to selection due to selection on each of the traits listed at the top of the table. (Values in bold are direct responses.)
 Corolla diameterCorolla tube lengthCorolla tube widthStigma heightAnther heighttallestAnther heightshortest
  1. *Proportion of bootstrap samples with−Vc[i]>−Vnc[i]

A. G-matrix
  Corolla diameter2.30 0.340.17 0.30 0.25−0.16
  Corolla tube length  0.400.06 0.25 0.40 0.24
  Corolla tube width  0.05 0.04 0.13 0.11
  Stigma height    0.50 0.23 0.12
  Anther heighttallest     0.94 0.70
  Anther heightshortest      0.60
B. Unstandardized selection gradients (β)+0.034−0.147+0.102+0.058−0.140+0.141
C. Predicted responses
  Vc (with covariances) .025−.048−.003−.009−.056−.032
  Vnc (no covariances) .078−.058 .005 .029−.131 .084
  Vdiff=Vnc−Vc .053−.001 .003 .038−.075 .116
  Vc/Vnc .325 .814 .492−.303 .426−.374
  Prop. bootstrap samples with Vc[i] > Vnc[i]0.062 0.079*0.295 0.009 0.00* 0.00
D. Indirect responses to selection (Vc broken down by trait)
  Anther heighttallest .009−.059 .013 .013−.131 .099
  Anther heightshortest .005 .035 .011 .007−.098.084

To examine the statistical significance of this constraint, we first compare the lengths of the multivariate response vectors, |Vc| and |Vnc|, for situations with and without genetic covariances, respectively. This length (in units of measurement for the floral traits in question) is 0.517 mm in the absence of covariances, but only 0.091 mm (< 20% as long) in the presence of covariances. For 4995 of 5000 bootstrap samples, |Vc| < |Vnc|, indicating that this constraint is highly significant (P= 0.001). Using the same bootstrap samples, we next determined that Vdiff=0 falls outside a critical hyper-elipsoid that contains all 5000 samples, also indicating highly significant constraint (P < 0.001). Analysis of each character separately indicates that for stigma height, height of the tallest anther, and height of the shortest anther, the proportion of bootstrap samples in which the predicted change with covariances was equal to or greater than the predicted change without covariances was less than 0.05, indicating significant constraint (P < 0.05) (Fig. 1).

Figure 1.

Frequency histograms for individual characters of (predicted change in character without constraints) − (predicted change in character with constraint), Vdiff, based on 5000 bootstrap samples. Shaded bars indicate samples for which the magnitude of change with constraints is greater than the magnitude of change without constraints. Units of Vdiff are millimeters.

Of particular interest in the context of reproductive character displacement in I. hederacea is the separation of anthers, because reduced separation reduces pollen flow from I. purpurea (Smith and Rausher 2008). This separation can be quantified as the distance between the tallest and shortest anthers. In the absence of genetic covariances between traits, this distance is predicted to decrease by 0.22 mm in one generation. In the presence of genetic covariances, however, the predicted decrease is only 0.024 mm, or 11% as much, indicating very severe constraint. In all 5000 bootstrap samples, the predicted response to selection with genetic covariances between traits is smaller than the predicted change without covariances (P < 0.001) (Fig. 2).

Figure 2.

Frequency histogram for (predicted change in anther separation without constraint) − (predicted change in anther separation with constraint), Vdiff, based on 5000 bootstrap samples. Note that the predicted change without constraints is negative. All sample values are < 0, indicating significant constraint on change in anther separation.

This constraint on the evolution of decreased anther separation arises largely from the positive correlation between the heights of the tallest and shortest anthers. For the tallest anther, the predicted response to direct selection alone (Vnc) is a reduction of 0.131 mm in height (Table 2C). By partitioning the indirect response to selection (Vc) into individual components by floral trait (Table 2D), it is seen that selection on the shortest anther favors a corresponding 0.099-mm increase in height of the tallest anther, greatly reducing the overall response to selection for this trait. By contrast, the indirect response due to selection on each of the other floral traits (Table 2D) is less than 25% as strong, and actually acts to decrease the height of the tallest anther by 0.024 mm. Similarly, for the height of the shortest anther, the direct response to selection (Vnc) is expected to be an increase of 0.084 mm (Table 2C). Upon examining the indirect response, however, we find that the predicted decrease of 0.11 mm due to selection on the height of the tallest anther completely offsets the expected direct response, whereas the combined effects of correlations with the other four characters contribute a net reduction in height of only 0.01 mm (Table 2D). It thus appears that genetic correlations between anther heights and the other characters (corolla diameter, corolla tube length and width, stigma height) contribute relatively little to the constraint on anther separation. This effect arises largely because the indirect responses associated with these other characters effectively cancel each other out due to differences in the direction of direct selection on those characters.

Method 2

Using this method, we made three comparisons. The first was between the direction of the selection gradient vector and the direction of the response vector under full constraint (e.g., unequal genetic variances, nonzero genetic covariances). The mean angle (across bootstrap samples) between these two vectors was 98.8°, a highly significant difference (Table 3). The combined effects of unequal genetic variances and nonzero covariances are thus to cause the response to selection to be at an angle of more than 90° away from the direction in which selection is acting to move this suite of traits. The angle involving the length of the shortest anthers (component 5 of 1θdiff; Table 3) makes the largest contribution to this effect, suggesting that this character is the most highly constrained by the combined effects of the genetic variances and covariances (but see below).

Table 3.  Comparisons of vector angles for reference and response vectors. A. Angles, in degrees, of five components of vector (θdiff) representing differences between pairs of vectors. Vector angle is the overall angle, in degrees, between the vectors. Note that the overall angle is not simply the sum of the five component angles. B. Statistical analysis of differences between vectors. Numbers give the probability, calculated as the proportion of bootstrap trials out of 5000, that the null hypothesis is true.
 A. Mean Angles between vectors
Comparison12θdiff component 345Vector angle
  1θdiffβ−θc)18.9−29.9−27.2−5.12−74.998.8
  2θdiffnc−θc)24.61.0630.6−15.3984.549.0
  3θdiffβ−θnc)43.6−28.83.4−20.59.573.0
B. Statistical analysis
   Prob. for θdiff components  Prob. for θdiff
Null hypothesis12345 
  1θdiff=0 0.2510.0020.0860.2410.000 0.0000
  2θdiff=0 0.1730.4130.0420.0060.000 0.0002
  3θdiff=0 0.0000.0000.2960.0010.101 0.0000

The second comparison was between the directions of the predicted response with character correlations and the predicted response without correlations. The mean angle between these two vectors was 49° and was significantly different from 0 (P= 0.0002; Table 3). Once again, the angle involving the length of the shortest anthers (component 5 of 2θdiff; Table 3) makes the largest contribution to this effect. These results are analogous to those obtained using method 1 for this comparison, and indicate that by themselves, genetic correlations alter the response to selection.

Finally, the third comparison was between the selection gradient vector and the direction of the response vector obtained when genetic covariances were set equal to zero. The mean angle between these two vectors was 73° (P= 0.0000; Table 3), indicating that unequal genetic variances alone cause the direction of the selection response to deviate substantially from the direction favored by selection. In this case, the angle involving corolla width (component 1 of 3θdiff;Table 3) contributes most to this effect, whereas the angle involving the height of the shortest anther (component 5) contributes minimally. Inequality of genetic variances thus seems to provide little constraint on the evolution of anther separation.

Discussion

CONSTRAINT ON THE EVOLUTION OF CHARACTER DISPLACEMENT

In a previous investigation (Smith and Rausher 2008), we found that there was selection for reproductive character displacement in I. hederacea when this species co-occurs with congener I. purpurea. Specifically, selection favored a decrease in anther separation by favoring a decrease in the height of the tallest anther, and an increase in the height of the shortest anther. Reduction of anther separation increases clustering of anthers around the stigma, which in turn reduces the susceptibility of plants to deleterious heterospecific pollen flow from I. purpurea (Smith and Rausher 2007, 2008). Reduced anther separation was not favored in the absence of I. purpurea.

In this study, we investigated the extent to which selection for character displacement in I. hederacea is constrained by the genetic architecture of floral traits in this species. The genetic variance–covariance structure among the six traits examined suggests that character displacement is constrained in this system. All traits are pairwise positively genetically correlated. However, for some characters, selection for character displacement favors an increase in size or height, whereas for other characters, selection favors a decrease in size or height. In particular, although the heights of the tallest and shortest anthers are strongly positively correlated (r= 0.93), selection favors a decrease in the height of the tallest anther, but an increase in the height of the shortest anther. In the face of the overall positive correlation structure for floral traits in this species, this antagonist selection is likely to result in constrained responses in many of the characters.

Analysis using method 2 indicates that genetic constraints in the form of both unequal genetic variances and nonzero genetic correlations are expected to affect the evolution of the floral traits we examined. Unequal genetic covariances cause the direction of the predicted selection response to deviate by more than 73° from the direction favored by selection. In addition, nonzero genetic covariances cause the expected response to deviate in direction from the response with all covariances equal to zero by 49°. These effects are not strictly additive, but they are also not completely compensatory: their combined effects cause the predicted response with both types of constraint to deviate by 99° from the direction of change favored by selection.

Analysis by method 1 confirms the conclusions that nonzero genetic covariances constrain the evolution of floral characters in I. hederacea; in addition it permits determination of the degree to which individual characters are constrained. By comparing the predicted response to selection with and without constraint due to genetic covariances between traits (Vc and Vnc), we found that for all but one character, the predicted response in the presence of constraint was less than half that expected in the absence of constraint; for two characters the predicted response is actually in the direction opposite to that favored by selection. For anther separation, the predicted response with constraint is only about 10% of the predicted response without constraint. Thus, the rate of evolution of increased reproductive character displacement in I. hederacea is expected to be markedly reduced from the rate one would expect based only on the magnitude of selection and the amount of genetic variation in floral traits alone. This result illustrates the principle that constraint is determined jointly by the pattern of genetic correlations and the direction of the selection gradient vector.

The pattern of constraint due to character correlations is similar to that found in other plant species. Contrary to the expectations of sex allocation theory, which predicts trade-offs in investment in male and female structures, positive genetic correlations between male and female floral traits are quite common in hermaphroditic plant species (Ashman and Majetic 2006). Genetic correlations among other floral traits (e.g., corolla and petal dimensions) are also typically quite high and positive (e.g., Connner and Via 1993; Campbell 1996; Elle 1998; Conner 2002), presumably because these traits share a common underlying developmental pathway (e.g., Hill and Lord 1989). By contrast, these floral characters are generally less strongly correlated with nonfloral traits such as leaf and stem dimensions (Meagher 1992; Conner and Via 1993; Mitchell et al. 1998; Ashman and Majetic 2006). The positive genetic correlations among floral morphological features has led several authors to suggest that these correlations are likely to constrain responses to selection that favor changes in relative shape, that is, selection for increasing some of these characters while decreasing others (Conner and Via 1993; Elle 1998). Our results provide some of the first evidence supporting this expectation. We know of only one other study to examine this question. Caruso (2004) reports that the expected response to selection in opposite directions on corolla and floral tube dimensions in Lobelia siphilitica is substantially reduced by the positive genetic correlations among these traits.

The highly constrained predicted response to selection for decreased anther separation in I. hederacea implies that despite the relatively high levels of genetic variation for short and tall anther heights individually, there is little genetic variation in the direction associated with decreasing separation. Although this may be an inherent property of these traits due to highly correlated development, it is also possible that genetic variation in this direction was higher in the past and has been reduced by selection for clustering. Anther clustering in I. hederacea appears to be a derived trait, with only I. hederacea exhibiting clustering among the six species in the clade to which it belongs (Smith and Rausher 2008). It is thus possible that in the past, before clustering evolved, that there was additional variation in this direction that was depleted as clustering evolved. If this hypothesis is true, it would predict that the genetic correlation between tall and short anther heights would be lower in the other species in this clade, a prediction that is open to experimental testing.

STATISTICAL ANALYSIS OF CONSTRAINT

We have presented two methods for assessing the effects of genetic constraints on the expected response to selection. These methods provide different information, and each has advantages and disadvantages. The primary advantage of method 2 is that it permits the partitioning of the overall genetic constraint into a component due just to nonequal genetic variances and a component due to nonzero genetic covariances, whereas method 1 only permits examining the effect of nonzero genetic covariances. This limitation occurs because although the selection gradient vector provides a direction for comparing the expected response vector in the absence of genetic covariances, it does not specify a magnitude for comparison. The magnitude of the unconstrained response (i.e., in the absence of unequal genetic variances and nonzero covariances) depends on the assumed common genetic variance of the characters, which cannot be specified nonarbitrarily.

The primary drawback of method 2 is that it does not permit a clear interpretation of how individual characters contribute to the observed directional deviations of the response vectors. For example, in the comparison of the selection gradient vector with the expected response under full constraint (i.e., the comparison represented by 1θdiff), the fifth component of 1θdiff is substantially larger than the other components (See Table 3A). This component represents the difference between two angles: (1) the angle between the axis corresponding to corolla tube length (which was used as the reference axis) and the projection of the selection gradient vector onto the plane represented by the axes corresponding to corolla tube length and height of the shortest anther; and (2) the corresponding angle for the response vector with full constraint. Because this difference could be caused by a constraint that affects either tube length or anther height or both, it is not clear which of these characters is constrained, and thus which character contributes more to the value of the fifth component of 1θdiff. Method 1 does not suffer from this limitation because it directly evaluates and compares the magnitudes of change in each character for different responses, although this comparison is limited to assessing the effect of genetic covariances. Moreover, method 1 allows comparisons of linear combinations of characters, as illustrated by the predicted responses of anther separation, which is the difference between the heights of the shortest and tallest anthers.

With method 1, deviation from equality of predicted response vectors with and without genetic covariances may be caused by facilitation as well as constraint. For example, in the two-character case in which the characters are positively correlated and selection acts to increase the value of each character, the multivariate response vector will be longer, and each character will increase to a greater extent, than if the characters were not correlated. In general, whether a deviation from the null hypothesis is due to constraint or facilitation can be determined by comparison of the response vectors. In the case reported here, none of the predicted responses with genetic covariances were greater than the predicted response without the covariances, so there was obviously no facilitation. In other cases, some characters may be constrained and others facilitated, depending on both the variance–covariance structure and the signs of the components of the selection gradient vector.

Our approaches to assessing genetic constraint differs in important ways from previous attempts. Notably, with method 1 we compare the predicted univariate response to a given pattern of selection with the predicted multivariate response using a statistical approach that takes into account uncertainty in the measurement of the variance–covariance structure. Etterson and Shaw (2001) also used this approach in examining constraints on response to selection caused by global warming in Chamaecrista fasciculata, although they do not compare the two responses statistically. Caruso (2004) also reported the relative magnitudes of both the univariate and multivariate responses to selection, although her statistical analysis compared the predicted multivariate response to the selection gradient (β) rather than to the predicted univariate response. Similarly, Mitchell et al. (1998) compared the predicted multivariate response to h2s, where h2 is the heritability of the character and s is the selection differential for that character. We believe that these latter two approaches are inappropriate for assessing the magnitude of constraint. Comparison of the multivariate response to β is equivalent to our approach only if all traits are standardized to have genetic variances equal to 1 (because then VAβ=β). However, it is not obvious how such standardization can be achieved. Standardizing the traits to have a phenotypic variance of 1 does not accomplish this objective, and failure to standardize VA to 1 means that β is not a prediction of response to selection. Similarly, the predicted univariate response h2s is not free from the influence of genetic correlations, and hence is not appropriate for comparison with the multivariate prediction to estimate constraint. In particular, the selection differential for a trait, s, is the phenotypic covariance of fitness and the trait (Lande 1979; Lande and Arnold 1983). This covariance is a composite of the effects of the covariances between each trait and fitness and between all traits and the focal trait, as reflected in the relationship s=Σ (covp)iβi, where (covp)i is the phenotypic covariance between the focal trait and trait i (for i= focal trait, covp= phenotypic variance). Because covp is partially determined by the additive genetic covariance, it is clear that s is not free from the influence of genetic covariances. Thus, h2s is not an appropriate predictor of the response to selection in the absence of genetic covariances.

Our method 2 shares features with the approach taken by Blows and colleagues (Blows and Higgie 2003; Blows et al. 2004; Blows and Hoffmann 2005) in that it focuses on the angular deviation between the selection gradient vector and the expected response vector. In their case, a constraint is inferred if the selection gradient vector is at a large angle relative to the principal axes of the G-matrix accounting for most of the character variation. In this situation, the selection gradient is essentially pointing in a direction in which there is little genetic variation to allow a response. In our analysis, we calculate the expected response vector with constraints present and determine the angle between the selection gradient and response vectors. If the angle is large, we infer constraint. One advantage of our approach is that it allows the effects of unequal genetic variances and nonzero genetic covariances to be distinguished.

Our results suggest that with reasonable sample sizes, G-matrix estimation error is small enough to allow detection of the effects of genetic constraints on the expected response to selection. One caveat to our analyses, however, is that because we are primarily interested in understanding how uncertainty associated with estimation of the G-matrix influences conclusions about constraints, we have not explicitly considered the effects of errors associated with estimating the selection gradient. One possibility for incorporating this uncertainty would be to bootstrap the selection gradient as well as the genetic variances and covariances. In our case, this does not seem to be a promising approach because the bootstrap error is much greater than the errors associated with parametric estimates of the selection gradient, and thus would lead to an overly conservative test. Other approaches will thus need to be developed. In spite of this caveat, our results do indicate that with our best estimate of the selection gradient, responses to selection on anther separation in I. hederacea are substantially constrained.

Conclusion

Whether particular traits are constrained by genetic correlations with other traits depends on the genetic variance–covariance structure (Lande 1979). Although a number of studies have examined the constancy of the genetic variance–covariance structure over time and/or environments by comparing G-matrices between divergent species or populations (Steppan et al. 2002; Bjorklund 2004), within-population variation in G is much less studied, despite the fact that estimates of G are subject to large measurement and sampling error (Lynch and Walsh 1998). The validity of any claim of constraint can be assessed only if this error is taken into account. The analyses described here account for this error and provide a statistical assessment of whether the hypothesis of no constraint can be rejected at a given, approximate level of significance. Moreover, they permit estimation of the degree to which characters are constrained.

Associate Editor: C. Goodnight

ACKNOWLEDGMENTS

We thank J. Bowsher, J. Rapp, and two anonymous reviewers for helpful comments on earlier drafts of this manuscript. We also thank T. Feldman for preliminary advice and guidance on data analysis in MATLAB. This work was made possible with financial support from National Science Foundation grant DEB 0308923 to RAS and MDR, and a National Science Foundation predoctoral fellowship to RAS (DGE 9818618).

Appendix

In this appendix, we describe our approach to testing the null hypothesis Vdiff=VncVc=0, that is, that the predicted change in the multivariate phenotype is the same whether genetic covariances are included in the G matrix when predicting the response to selection using equations (1 and 2). This is equivalent to testing the null hypothesis of no genetic constraint.

The general rationale for this approach is to use bootstrap resampling to calculate a critical (e.g., 1 −α= 95%, 99%, etc) confidence surface for Vdiff and determine whether the vector 0, corresponding to no difference between Vnc and Vc, lies within that surface. If it does not, then the null hypothesis can be rejected at significance level α. In our analysis, this confidence surface lies in six-dimensional space because there are six characters analyzed. For visual portrayal of the geometry of the analysis, however, we portray this surface as an ellipse in two dimensions in Figure A1.

Figure A1.

Schematic portrayal of constraint analysis for two characters. Arrows: representative values of Vdiff. Points: endpoints of Vdiff vectors. v1 and v2: axes for characters 1 and 2. 0: origin of original coordinate system. (A) Distribution of Vdiff vectors in original coordinate system. (B) New coordinate system resulting from translation of axes such that the new origin corresponds to the multivariate mean of the Vdiff vectors. Primes indicate coordinate values in new coordinate system. (C) Rotation of coordinate system to align axes with major axes of variation of Vdiff vectors. Double primes indicate coordinate values in resulting coordinate system. (D) Coordinate system as in (C). Ellipse is 95% critical surface, which encloses 95% of the Vdiff vectors. Note that the original coordinate origin, 0″, lies outside the 95% critical surface, indicating that Vdiff0 and hence that there is constraint.

The first step in the analysis is to resample the data to estimate the G matrix. For each sample, one then calculates the two predicted vectors of phenotypic change, Vnc and Vc using equations (1 and 2). The difference Vdiff=VncVc is then calculated. The tips of the resampled Vdiff vectors form a cloud of points, as portrayed in Figure A1A.

The next step is to transform the coordinate axes. First, from each vector, the mean vector, M, is subtracted, to yield the coordinates of Vdiff in the new coordinate system: V′diff=VdiffM. This transformation shifts the origin of the axes to the point corresponding to the original mean vector (Fig. A1B). Next, the variance–covariance matrix of the transformed vectors, Σ is calculated and the eigenvectors and eigenvalues corresponding to this matrix are obtained. The coordinate system is then rotated to make the axes correspond to the eigenvectors (Fig. A1C) using the transformation

image

where P is the matrix of eigenvectors. Applying this double transformation to the origin converts the original coordinates of the origin, 0, to the coordinates, 0″ in the new coordinate system (i.e., 0″= P (0 − M)).

The next step is to determine the equation of the critical hyper-ellipsoid that contains a fraction 1 −α of the points represented by the V″diff vectors (Fig. A1D). To do so, we assume that these points are multivariate-normally distributed. Projections of the cloud of points onto various pairs of untransformed axes (Fig. A2) suggest this assumption is a reasonable approximation. With this assumption, the equation for the family of concentric hyper-ellipsoid surfaces centered on the origin is given by

image

where ei and λi are the eigenvector and eigenvalue corresponding to axis i, and D is the distance from the origin to the surface. The value of D corresponding to the 1 −α confidence level is determined by trial and error by calculating the proportion of points that fall within the surfaces for different values of D.

Figure A2.

Representative bivariate scatter plots of 5000 bootstrap samples of Vdiff for pairs of traits. Note that for height of tallest anther versus height of shortest anther, the point (0,0) falls outside of the scatter of points, indicating that both of these characters are significantly constrained.

Once the critical surface has been determined, the last step is to ask whether the original origin, 0″ lies within this surface. If not, one infers that with a confidence level of approximately 1 −α, Vdiff0, and that there is significant constraint on the response to selection imposed by the genetic variance–covariance structure of the characters.

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