Unpredictability of correlated response to selection: pleiotropy and sampling interact.
An informative example of the variability of correlated responses involves the red and yellow eye pigments (the drosopterins) of D. melanogaster. Both family and individual selection were used to select for levels of red and yellow eye pigments (Chauhan and Robertson 1966). Later, Barthelmess and Robertson (1970) found that the levels of a variety of other fluorescent pteridines changed in selection lines; they verified that this was not due to inadvertent direct selection. As shown in their figure 2 (Barthelmess and Robertson 1970), correlated responses of these pteridines in the two low lines were dissimilar. Some of the pteridines (biopterin and AH) increased in abundance with respect to the control in one low line but did not change significantly in the other. For other pteridines (xanthopterin and xanthopterin-like), one line significantly increased and the other significantly decreased.
Another aspect of the variability of correlated response is evident in studies that compare genetic correlations in a base population (estimated from half-sib or parent-offspring designs) to correlated responses obtained from selection. For instance, selection on wing length in milkweed bugs produced the expected correlated responses in several characters (Palmer and Dingle 1986). However, the genetic correlation between wing length and fecundity was highly significant and negative in the base population, whereas the actual correlated response was highly significant and positive. My work on courtship and reproductive behaviors in D. melanogaster also produced several examples of this kind (Gromko 1987, 1989; Gromko et al. 1991), and has led me to pursue the causes of variability of correlated response, as will be reported in this paper.
The most common explanation for the high variability of correlated responses is sampling error. Genetic correlation has a high standard error and this comes into play repeatedly in a selection experiment because each generation of selected parents constitutes a sample of the genetic correlation. The values of the genetic correlation in successive generations of selection depend upon one another sequentially. Consequently, changes in the value of the genetic correlation will have some of the properties of a random walk. In this view, variation in correlated responses to selection is the consequence of the large standard error of the genetic correlation.
In contrast (or in addition) to this view, Bohren and coworkers argue that "some mechanism other than genetic sampling is affecting the correlated response" in their selection lines (Bohren et al. 1966). They found discrepancies in correlated responses between "up" and "down" lines and discrepancies resulting from changes in the focal character (selection on trait A produced correlated responses in B that were difficult to reconcile with the correlated responses in trait A produced by selection on trait B). However, that report did not extend to variation among replicates selected in the same direction for the same character.
Like Bohren et al. (1966), I suspect that the variation in correlated responses is more than a statistical phenomenon, although sampling issues are clearly involved. Relying on Barthelmess and Robertson's (1970) interpretation of the correlated responses of pteridines in the eyes of D. melanogaster, Gromko et al. (1991) proposed the variable pleiotropy hypothesis: that variability of pleiotropic effects among loci may account for the variability of correlated responses to selection (Gromko et al. 1991). Variability of pleiotropic effects is biochemically justified (Barthelmess and Robertson 1970; Kacser and Burns 1981; Noordwijk and Gebhardt 1987), and empirically supported (Caballero et al. 1991; Santiago et al. 1992). Loci with contrasting pleiotropic effects contribute differently to correlated responses in replicate selection lines as follows. Selection experiments usually restrict the size of the breeding population; consequently, some loci become fixed for the allele "unfavorable" to the response (Falconer 1989, pp. 220-221). The frequency of fixation for unfavorable alleles should be greater given smaller numbers of individuals selected. Furthermore, different alleles fix in different selection replicates, as demonstrated by crossing replicate selection lines that have reached a selection limit to produce further progress in selection (Mather 1983, p. 211). Because different loci fix in different selection replicates and these loci have different pleiotropic effects, correlated responses will vary among selection replicates, even though direct responses do not vary. Variable correlated responses are expected, on this model, at some combination of population size, selection intensity, number of loci, and variability of pleiotropic effects despite repeated, stable estimates of genetic correlation from parent-offspring or sib analysis in the base population (Gromko et al. 1991).
The purpose of the computer simulations reported here is to investigate the theoretical consequences of the variable pleiotropy hypothesis. Genetic correlation due to linkage may also contribute to variation of correlated response; however, this simulation has focused on pleiotropy. I found that the variability of correlated responses is affected by the details of the pleiotropic relationships. Given two sets of pleiotropic effects that produce the same genetic correlation, one set of pleiotropic effects could constrain correlated response and the other could allow a great deal of variation in the correlated response. Such differences in variability of correlated response are not predictable from knowledge of the genetic correlation. In the extremes, (1) constraint due to pleiotropy can occur in the absence of genetic correlation, and (2) correlated responses can be opposite in sign to the genetic correlation.
MATERIALS AND METHODS
This is a computer simulation of divergent selection on a polygenic trait. It specifies genetic values of alternative alleles with a completely additive basis within and across loci. The critical feature of the simulation is the specification of pleiotropic effects of alleles on a second trait and the consequent correlated response to selection in that second trait. I wrote the program in Pascal and ran the simulation on a VAX 8530. Copies of the program are available on request.
The four alternative types of alleles are: positive effects on both traits (1-1); negative effects on both traits (0-0); positive on trait one and negative on trait two (1-0); and negative on trait one and positive on trait two (0-1). Four types of loci are constructed from specific combinations of alleles [ILLUSTRATION FOR FIGURE 1 OMITTED]. "Positive" loci have alleles that affect both traits in the same way (1-1 and 0-0 alleles). "Negative" loci have alleles that affect the two traits in opposite ways (1-0 and 0-1 alleles). "One-trait-only" loci segregate for alleles that make a difference to one trait but not to the other (1-0 and 0-0 alleles for loci that affect trait one; 0-1 and 0-0 alleles for loci that affect trait two). The number of loci affecting trait one is made equal to the number affecting trait two. Consequently, for every locus that has 1-0 and 0-0 alleles (affecting trait one only) there is a locus that has 0-1 and 0-0 alleles (affecting trait two only), as shown in figure 1. Categorizing loci on the basis of different fixed pleiotropic effects can be justified functionally (Houle 1991).
The genetic system is described by the number of positive, negative, and one-trait-only loci. For instance, the genetic system shown in figure 1 consists of two positive loci, two negative loci, and four trait-one-only loci (two affecting trait one only, and two affecting trait two only) and is written 2,2,4 in tables to follow. It has eight loci in all, four of which affect both traits. Similarly, a genetic system comprised of two positive loci, zero negative loci, and 12 trait-one-only loci (six affecting trait one only and six affecting trait two only) is written 2,0,12, has eight loci per trait, and fourteen loci in all.
Inheritance is Mendelian (50:50 segregation ratios at each locus) with each locus segregating independently. Genetic effects are additive within and across loci. Each locus is segregating for two alleles with initial frequencies of p = q = 0.5 in the base population. There is no input of new variation by mutation; I simulated only the first ten generations of selection and in this time frame mutation does not contribute to selection response (Hill and Keightley 1988).
For each trait and individual separately, the simulation samples an environmental deviation from a normal distribution with a mean of zero and a variance scaled to produce the desired initial heritability. The environmental deviations for the two traits are independent of one another and can be scaled to different heritabilities. Additive genetic variance decreases over the course of selection; environmental variance remains at its initial value. Each individual's phenotypic value is the sum of its genotypic value and its environmental deviation.
Values of the genetic correlation are not input parameters in the simulation. Genetic correlations are consequences of the number of positive, negative and one-trait-only loci. Whereas it might seem that the genetic correlation could be calculated directed from the allelic effects (as given, for instance in fig. 1), the correlation of allelic effects is not equal to the genetic correlation in all cases. Only where there are no loci that affect one trait only does the correlation of allelic effects equal the genetic correlation. Consequently, a calibration step is necessary in which the genetic correlation corresponding to different genetic systems is calculated using parent-offspring covariances. Each of these genetic correlations is based on 1000 midparent-offspring pairs and is calculated using standard midparent-offspring formulas (Falconer 1989, p. 317).
I was also interested in the generation-to-generation variation in the value of the genetic correlation over the course of selection. For these runs, each genetic correlation is based on 10,000 simulated midparent-offspring pairs.
Selection is by truncation, for trait one only. Population size of the progeny generation is held constant. Sex ratio is not allowed to vary; equal numbers of sons and daughters are always produced. Selection is divergent; selection lines are run in pairs, with one line selected for increased values of trait one and one line selected for decreased values of trait one. For most analyses, divergence scores (difference between means of the paired up and down lines) are calculated. The variability of selection response was evaluated in response to systematic variation in several input parameters: population size; the proportion selected; the number of loci per trait; heritability; and pleiotropic effects.
The simulation evaluates the variability of selection response at the tenth generation in three ways. First, the mean and variance of divergence scores for traits one and two are calculated for 100 pairs of simulated "up" and "down" selection lines. A two-tailed F-test is used to compare the variance of divergence scores between two combinations of positive, negative and one-trait-only loci that produce the same genetic correlation. Second, the number of significant divergences between up and down lines is recorded. Significance is determined by a two-tailed t-test (alpha = 0.05). The test is two-tailed because the correlated response is sometimes in the direction opposite to the sign of the genetic correlation. Because simulation runs are based on 100 pairs of up and down lines the maximum possible number of divergences is 100. Third, the number of "wrong-way" responses is recorded. These are determined by calculating the difference between the mean value of the trait immediately before selection in the base population and the value of the trait after ten generations of selection. If that difference is zero or has a sign opposite to the genetic correlation, it is scored as being "wrong way." Because each individual selection line is evaluated for the direction of response, the maximum possible number of wrong-way responses equals the number of selection lines, or 200 for all runs reported here.
TABLE 1. Different genetic systems produce the same genetic correlation.
Number of loci Loci per Genetic trait Positive Negative One-trait-only correlation
4 3 1 0 0.5 2 0 4 0.5 8 6 2 0 0.5 4 0 8 0.5 16 12 4 0 0.5 8 0 16 0.5 4 2 1 2 0.25 1 0 6 0.25 8 4 2 4 0.25 2 0 12 0.25 16 8 4 8 0.25 4 0 24 0.25 4 2 2 0 0.0 0 0 8 0.0
Genetic correlations varied depending upon the number of positive, negative and one-trait-only loci. Results for four, eight, and sixteen loci are shown in table 1. The same genetic correlation was produced by different genetic systems. Genetic systems that produced the same genetic correlation (table 1) were compared in the selection part of the model.
The genetic correlation varied over the course of selection [ILLUSTRATION FOR FIGURE 2 OMITTED]. Two sources of sampling error contributed to variation in the genetic correlation: the "statistical" error of estimate in the parent-offspring covariance and the "biological" sampling error that is due to the use of a finite population size. The parent-offspring covariances were based on 10,000 families, making the statistical error quite small; repeated estimates of the genetic correlation in the same population were within a range of 0.04 of one another. Consequently, most of the generation to generation variation evident in figure 2 was due to finite population size.
The simulated selection runs produced results that were similar to actual selection experiments in a number of ways [ILLUSTRATION FOR FIGURE 3A OMITTED]. The regression of cumulative response on cumulative selection differential (i.e., the realized heritability) calculated over the first five to ten generations of selection was approximately equal to the input heritability. The additive genetic variance decreased as selection progressed, yielding a diminishing response as selection continued. Variation among replicates was evident both in initial and maximum response; as expected, this variation was due to sampling error and so was more evident at small population size or high selection intensity. Generation to generation variation in selection response was greater for smaller values of heritability. One aspect of selection response that was not obtained in the simulation was the appearance of a plateau in response followed by additional selection gain (Mather 1983). The addition of either linkage or mutation would be necessary to reproduce that aspect of selection response.
Correlated responses to selection were also variable [ILLUSTRATION FOR FIGURE 3B OMITTED]. One of the observations that motivated this simulation study was that correlated responses are sometimes in a direction [TABULAR DATA FOR TABLE 2 OMITTED] opposite the sign of the genetic correlation, as discussed in the introduction. Such wrong-way correlated responses were sometimes obtained in the simulation; an example is evident in figure 3B.
Two combinations of positive, negative, and one-trait-only loci that produced the same genetic correlation resulted in different degrees of variation in the correlated response; variance ratios for all simulations run at heritability of 0.5 and genetic correlation of 0.5 were highly significant (table 2). In contrast, the variance of the divergence scores of the trait directly selected was unaffected by differences in genetic systems. For the correlated response, the genetic system producing greater variability was always the one with more loci affecting one trait only. The 4,0,8 condition (4 positive, 0 negative, and 8 one-trait-only loci) produced more variable correlated responses than the genetic system 6,2,0 (6 positive, 2 negative and 0 one-trait-only). The magnitude of the effect was relatively independent of population size and selection [TABULAR DATA FOR TABLE 3 OMITTED] intensity at the values of heritability and genetic correlation represented in table 2.
Despite the variability in correlated response for the genetic system 4,0,8, the relatively high values of heritability and genetic correlation represented in table 2 produced significant divergences for both traits in almost all runs. The few wrong-way responses that were recorded were all for trait two and genetic system 4,0,8. The mean divergence for trait one is expected to be twice the number of loci per trait (16 in these runs) and for trait two twice the number of loci per trait times the genetic correlation (8 in these runs). These values were approached in all runs and no apparent difference was found in mean divergence between the contrasted sets of genetic systems.
When the same comparisons as described in the previous paragraph were repeated at a heritability of 0.25 and for a genetic system that produced a genetic correlation of 0.25, an effect of population size was apparent (table 3). Only at [TABULAR DATA FOR TABLE 4 OMITTED] higher population size was the variance of the divergence scores for the correlated trait influenced by the genetic system. The variance of the divergence scores for the trait directly selected remained unaffected by the details of the genetic system. The variance of the divergence scores was relatively large for trait two, at low population size, resulting in fewer significant divergences. Note especially that some of the significant divergences between up and down lines were in the wrong direction for the correlated trait (the "up" line was lower than the "down" line even though the genetic correlation was positive). So, for instance, in the first line of table 3, "55-4" means that 55 divergence scores were significant in the direction predicted and 4 were significant in the direction opposite to expectation. Similarly, for each response considered on its own, more wrong-way responses occurred at low population size than at high population size. As population size increased, a difference between the two genetic systems became apparent, just as it did for the variance ratio.
Variation in heritability and variation in number of loci (table 4) had little effect on the difference in variance produced by the two genetic systems with a genetic correlation of 0.25. Fewer wrong-way correlated responses were found at higher numbers of loci, and possibly at higher heritability. However, the difference between the two genetic systems was always evident. For each pair of contrasted genetic systems, one consistently produced greater variability in correlated response. The one with fewer loci that affect one trait only (e.g., 2,1,2 vs. 1,0,6) produced a less variable correlated response.
Why would two genetic systems that produce the same genetic correlation result in different amounts of variation in correlated response? To produce the same genetic correlation and to have the same number of loci per trait, two genetic systems must differ in the number of loci affecting one trait only. That difference was maximal for a genetic correlation of zero (table 5). One of the genetic systems that produced a genetic correlation of zero had only loci that affected one trait (0,0,8). The other genetic system had no loci of this kind (2,2,0). The variance in divergence scores for the correlated trait were one to two orders of magnitude greater for the case that had several loci affecting one trait only (table 5). In parallel fashion, many more significant divergences of trait two occurred for the 0,0,8 genetic system, a difference that was accentuated at large population size by the absence of significant divergences for the alternative genetic system, 2,2,0.
Divergences in trait two were due to drift in this case, because of the zero genetic correlation. However, given two positive loci and two negative loci (2,2,0), drift in trait two was constrained by selection on trait one [ILLUSTRATION FOR FIGURE 4A OMITTED]. If we consider a system with high heritability, in order to focus on the genetic effects, selection for increase or decrease in trait one will lead to fixation of 1s or 0s, respectively. Trait two will be fixed for 1s at two loci and for 0s at two loci, regardless of the direction of selection on trait one. The net correlated response is constrained to be zero. In contrast [ILLUSTRATION FOR FIGURE 4B OMITTED], a genetic system having only loci that affect one trait (0,0,8) was uncorrelated and unconstrained. Selection still produced fixation for trait one of 1s in the up line and 0s in the down line, but loci affecting trait two are not involved in this process and so are free to drift. Constraint without correlation (in the 2,2,0 system) is produced because pleiotropic effects are present on a locus by locus basis even though their net effect across loci (the genetic correlation) is zero. The determining cause of constraint is pleiotropy, to which the summary statistic of genetic correlation may be blind.
[TABULAR DATA FOR TABLE 5 OMITTED]
This simulation shows that (1) more than one combination of alleles results in a single value of genetic correlation, and (2) although these different genetic systems produce the same mean correlated response, they have substantially different effects on the variance of the correlated response. Given two genetic systems that produce the same genetic correlation, the condition with more loci affecting one trait only produces a more variable correlated response (tables 2-5). The loci that affect only the correlated character will not be subject to selection and so will be free to drift. Thus, when two traits share some loci with pleiotropic effects but have other loci affecting only one trait or the other, stochasticity of the correlated response (and not the direct response) will be apparent. In contrast, two traits that have many pleiotropic loci, even if these are a mix of positive and negative effects, will produce correlated responses constrained by selection.
In the extreme case, a genetic correlation of zero can be constructed entirely from equal numbers of positively and negatively pleiotropic loci, or it can be constructed entirely from loci that affect one trait only ([ILLUSTRATION FOR FIGURE 4 OMITTED], table 5). In the first case, where positive and negative pleiotropic effects balance, the correlated response is completely determined by selection on trait one, despite the net genetic correlation of zero. In the second case, selection on trait one is irrelevant to evolution of loci that affect the second trait, leaving the second trait free to drift. Hence, the limits of variation in the evolution of the second trait in this case fall between rigid constraint arising from balancing pleiotropic effects on the one hand and drift on the other.
Others have made the point that genetic correlations are not necessarily constraints (Charlesworth 1990; Houle 1991; de Jong and van Noordwijk 1992). The simulation reported here is clearly in agreement with that result; it also establishes the converse, that genetic constraint of selection response can be present despite a genetic correlation of zero. The main point of the simulation, however, is to demonstrate that correlated responses to selection depend on the details of pleiotropic effects rather than on genetic correlation, and that because of this, correlated responses to selection can be more variable (and less predictable) than direct responses to selection.
Direct responses to selection are also variable. Cohan (1984) has shown that selection and sampling interact in an interesting way when the initial frequencies of the favored alleles are small in relation to population size and strength of selection. His model shows that uniform selection and drift together produce more divergence among populations than does drift alone. Empirical work bears out these predictions (Cohan and Hoffmann 1986): Correlated responses to selection were found to be variable, establishing that different genetic mechanisms were involved in different selection replicates to produce the same net direct response. However, Cohan's model limits the diversifying effect of selection to alleles at low frequency, whereas in the simulation reported here, initial allele frequencies were at 0.5 in all runs. Thus, heterogeneity of genetic response can produce variability of correlated response even in conditions that result in uniformity of direct response to selection.
Sampling error is clearly of fundamental importance in understanding variability of correlated responses to selection. However, the details of pleiotropy are of critical importance in determining the consequences of that sampling error. Two genetic systems subject to the same degree of sampling error produce different degrees of variability of the correlated response.
Others have recognized the importance of variation among pleiotropic effects. One of the premises of this simulation, that the genetic correlation is the net effect of many loci whose individual effects may differ and even cancel one another out, is well established (Falconer 1989; Aastveit and Aastveit 1993). Atchley and co-workers depend on variable pleiotropic effects, in a way similar to this simulation, to explain their correlated responses to selection in mice. In a recent example (Atchley et al. 1990), they showed that increase in body weight can be accomplished either by selection for high-fat content or for low-lean content. Alternatively, selection for low fat or high lean both produced reduction in body weight. Despite the similarity of endpoints reached for body weight, the selection lines exhibited different patterns of growth and morphological dimensions. Thus, they argue that different selection strategies involve different patterns of genetic correlation with skeletal, physiological and reproductive traits. "The consequences of this heterogeneity in genetic correlation structure is a heterogeneous correlated response to different selection strategies" (Atchley et al. 1990, p. 687). Where this example relies on purposeful modification of the selection strategy, I argue that stochasticity of the genetic response will cause heterogeneity of correlated response with a uniform selection strategy.
Several implications follow from this view of variable pleiotropic effects. First, it appears that actual correlated response could be an unreliable estimator of genetic correlation. Hence, contrary to the suggestions of some (e.g., Bell and Koufopanou 1986), half-sib and parent-offspring estimates of genetic correlation are more reliable estimators of the genetic structure of the base population than are correlated responses to selection. This observation could be of value in interpreting studies of life-history tradeoffs (Rose and Charlesworth 1981; Service et al. 1988; Roper et al. 1993). Second, and corollary to this, is the need for high degrees of replication in studies of correlated response to selection. Third, models of multivariate selection response (Lande 1979), in their reliance on genetic correlation, do not constitute causal models of selection response. They provide statistical predictions whose validity is dependent on population size, selection intensity, number of loci, and the details of pleiotropy. The next step is the development of a causal model of selection response that relies on pleiotropy and linkage directly, rather than on their statistical summary, genetic correlation. Houle (1991), for instance, has developed a model that distinguishes among loci affecting allocation versus acquisition of a limiting resource. And finally, genetic correlation is not a reliable indicator of constraint (Charlesworth 1990; Houle 1991; de Jong and van Noordwijk 1992). Genetic constraint can be present in the absence of genetic correlation. It is known that the presence of high genetic correlations can cause the persistence of neutral or even mal-adaptive traits (Price and Langen 1992); the same kinds of consequences might also occur in systems with very low values of genetic correlation but balancing influences of positively and negatively pleiotropic loci.
I am grateful to M. McPeek, G. Miller, and A. Porter for their critical reading of an earlier version of this manuscript, and to L. Leamy and two anonymous reviewers for comments and advice.
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|Author:||Gromko, Mark H.|
|Date:||Aug 1, 1995|
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