Abstract
Pleiotropy allows for the deterministic fixation of bidirectional mutations: mutations with effects both in the direction of selection and opposite to selection for the same character. Mutations with deleterious effects on some characters can fix because of beneficial effects on other characters. This study analytically quantifies the expected frequency of mutations that fix with negative and positive effects on a character and the average size of a fixed effect on a character when a mutation pleiotropically affects from very few to many characters. The analysis allows for mutational distributions that vary in shape and provides a framework that would allow for varying the frequency at which mutations arise with deleterious and positive effects on characters. The results show that a large fraction of fixed mutations will have deleterious pleiotropic effects even when mutation affects as little as two characters and only directional selection is occurring, and, not surprisingly, as the degree of pleiotropy increases the frequency of fixed deleterious effects increases. As a point of comparison, we show how stabilizing selection and random genetic drift affect the bidirectional distribution of fixed mutational effects. The results are then applied to QTL studies that seek to find loci that contribute to phenotypic differences between populations or species. It is shown that QTL studies are biased against detecting chromosome regions that have deleterious pleiotropic effects on characters.
A major goal of modern biology is to identify the individual mutations that are the genetic basis of phenotypic differences among individuals, populations, and species. An important class of mutations that cause phenotypic differences between populations and species are fixed mutational differences, i.e., mutations that are fixed in one population and absent in the other.
Often quantitative trait locus (QTL) studies seek the fixed mutational differences that are the genetic basis of phenotypic differences between two interfertile species. Many of these studies observe bidirectional QTL effects; i.e., some QTL for a trait in a species have an effect in the direction of the difference between it and the other species and some QTL have effects in the opposite direction. For instance, consider the study by Bradshaw et al. (1998) that determined the QTL that distinguish floral characters between the sibling species Mimulus lewisii and M. cardinalis (Figure 1a). The effects are for those QTL present in M. cardinalis and are standardized such that negative effects are more M. lewisiilike and positive effects are more M. cardinalislike. Overall, out of the 11 characters studied, 8 had all of the QTL with positive effects and 3 had some positive and some negative effects. Likewise, bidirectional effects were found in the QTL that cause species differences in male sexual characters between Drosophila simulans and D. sechellia (Figure 1b; MacDonald and Goldstein 1999). Overall, out of 6 characters studied, 2 had only positive effects, 3 had bidirectional effects, and 1 had all negative effects.
The presence of bidirectional QTL effects extends from domesticated organisms undergoing artificial selection to phenotypically divergent populations or species (Doebley and Stec 1993; Tanksley 1993; Orr 1998b; Rieseberget al. 2003). For instance, in a review of 96 QTL studies, 22.1% of the 3252 QTL had negative effects (Rieseberget al. 2003). The interpretation of the origin and maintenance of bidirectional effects has not been clearly presented. The pleiotropic nature of mutation and evolutionary forces such as stabilizing selection, hitchhiking, and random genetic drift have been invoked to explain QTL with bidirectional effects. Stabilizing selection may yield bidirectional effects because an average individual in a population may sometimes be above and sometimes below an optimum for a character. Hitchhiking may lead to bidirectional effects because a mutation with a negative effect for a character may be linked to a mutation with a larger beneficial effect for the same or another character. With random genetic drift, a mutation fixes independently of the magnitude of its effect and its direction. Yet no study has quantified how these processes and forces individually or collectively shape the expected distribution of fixed mutational effects on phenotypic characters. In this article, we quantify analytically the contribution pleiotropy and directional selection make in generating a bidirectional distribution of fixed mutational effects. Through simulation, we quantify contributions made during stabilizing selection and random genetic drift.
Kimura (1983) derived the distribution of the magnitude of the overall effect of the next fixed mutation on a phenotype by employing Fisher's (1930) geometrical model and found that mutations of intermediate effect fix most frequently. Recently, Orr (1998a), who also used Fisher's model, derived the distribution of fixed mutational magnitudes during a bout of adaptation in which more than one mutation fixes and found the overall distribution to be approximately exponentially distributed. Later Orr (1999) showed that the distribution of the magnitude of fixed effects on specific dimensions is also approximately exponentially distributed.
With the exception of Orr (1999), none of the results are directly relevant to empirical studies that seek to understand the evolution of single characters because the effect of a mutation is the length of the vector it makes in an ndimensional hypersphere—not its effect on a single character. Orr's (1999) result is useful in that it predicts the magnitude of fixed effects on single characters.
Furthermore, all of the previous results have focused on the magnitude of fixed effects, that is, their unsigned absolute value. Biologically, of course, whether a fixed effect increases or decreases the value of a trait is a crucial bit of information. In this article we distinguish between the size and magnitude of mutational effects. With “size” we refer to the signed value of the effects, while with “magnitude” we refer to its absolute value. To understand the evolution of bidirectional effects, we of course focus on size rather than magnitude.
Here we employ a model that is conceptually simpler than Fisher's, yet allows for the retention of information about the direction fixed mutations have on characters. The comparable results to Fisher (1930), Kimura (1983), and Orr (1998a, 1999) are duplicated despite the simpler approach. In our model, mutations are assumed to independently affect some number of characters. The results from this analysis are more relevant to empirical studies that measure characters. Analysis shows that information about both the direction and the magnitude of fixed mutations can potentially be used to estimate how characters are associated pleiotropically through mutation and to infer the role selection has taken in shaping the evolution of a character. At the end of the analysis we scale up from individual mutations to QTL.
In this study, three factors affecting the distribution of fixed mutational effects are evaluated: pleiotropy, drift, and stabilizing selection. By varying these factors separately and together we evaluate how they contribute to the existence of fixed bidirectional mutational effects and change the shape of the distribution of fixed mutational effects.
We assume that mutation acts independently on characters and that a mutation has an effect on a character that is a random draw from a bilaterally symmetric gamma distribution (in the appendix we allow for the possibility of asymmetric mutation). Empirical evidence supports that the effects of mutation may be bilaterally symmetric; Mackay et al. (1992) and Lyman et al. (1996) found that random Pelement inserts in Drosophila affected bristle number in a bilaterally symmetric manner with the frequency of effects with larger magnitude decreasing in an exponentiallike way or possibly with even higher kurtosis. Fitness is modeled as a linearly decreasing function of the phenotype's distance from the optimum in simulations and as a linear function of a mutation's individual effects on characters in the analytical analyses. Following previous work, we assume that beneficial mutations are rare, so that there is no selective interference among mutations (Hill and Robertson 1966).
Figure 2 illustrates how the processes of random genetic drift, stabilizing selection, and directional selection with pleiotropy by themselves or collectively lead to the fixation of bidirectional mutations. In Figure 2, a–d, the axes are characters, the optimum character state is at the origin for both characters, and the current phenotype, i.e., the combination of two character states, is represented by a point. Mutations that bring the phenotype to be within the circle are beneficial overall because the new phenotype would be closer to the optimum. Random genetic drift leads to the fixation of bidirectional effects because chance plays a role in which mutations fix, allowing for the fixation of mutations in all directions (Figure 2a). Stabilizing selection becomes important when the probability that a mutation overshoots the optimum becomes nonnegligible, and then the sequential overshooting of the optimum leads to bidirectional fixed effects (Figure 2b). Pleiotropy allows for the fixation of bidirectional effects, in the absence of drift and stabilizing selection, because even though one or a few characters may have effects in the opposite direction to their optima, other characters may have stronger effects in the direction of their optima (Figure 2c). Unidirectional effects are expected under pure directional selection and when mutations that fix have effects only in the direction of the optima for both characters (Figure 2d).
Analytical expressions are derived for varying levels of pleiotropy in the absence of drift and stabilizing selection for exponentially distributed mutations. Some useful approximations cannot be made when mutation is gamma distributed with shape parameters <1.0, and expressions giving the distribution of fixed mutational effects are presented in the appendix under these conditions that require numerical integration. Simulations are used to determine distributions under drift and stabilizing selection and mixtures of varying levels of drift, stabilizing selection, and pleiotropy. We characterize the effects of drift, stabilizing selection, and pleiotropy on fixed mutational effects with three measures: (1) the expected size of a fixed mutational effect on a character, (2) the distribution of fixed mutational effects, and (3) the relative frequencies of bidirectional fixed mutational effects, i.e., the frequency of positive vs. negative fixed effects.
MODEL
We assume that organisms are haploid. The current state of n characters pleiotropically affected by mutation is represented by a vector z = {z_{1}, z_{2},..., z_{n}}. The effect of a random mutation on a character (δ_{i}) is a random draw from a bilaterally symmetric gamma distribution, f(δ; α, β), with shape parameter α and scale parameter β. The result is a mutation vector δ that adds to the vector z. The selection coefficient of a new mutant phenotype is determined by the relationship w_{wild type}(1 + s) = w_{mutant}, where w_{wild type} is the fitness of the phenotype without the mutation and w_{mutant} is the fitness with the mutation.
Simulations: w_{wild type} is scaled to equal one and w_{mutant} is equal to 1 + (z – z + δ)σ, where σ is the magnitude the slope of the fitness function and x denotes the length of a vector x. The selection coefficient of a mutation is then s = (z – z + δ)σ. A mutation that causes the phenotype to be greater than z + 1/σ is assumed to have zero fitness. A mutation with selection coefficient s fixes with probability (1 – e^{–2Nes/N})/(1 – e^{–2Nes}) (Kimura 1957), where N_{e} and N are the effective and census population sizes. When a mutation fixes, the phenotype of all individuals in the population takes on the value z + δ.
Each character begins adaptation the same distance from its optimum. The beginning distances and stopping points vary among simulations to model the effects of the role of stabilizing selection in shaping the distribution of fixed mutational effects. Effective and census population sizes are varied to model the effects of random genetic drift.
Mathematical analysis: Here we assume that there is no random genetic drift or stabilizing selection and investigate how pleiotropy alone causes the fixation of bidirectional effects. Because each character experiences directional selection only, we employ the convention that mutations with positive effects are in the direction of the optimum and mutations with negative effects are in the opposite direction of the optimum. For both the onecharacter case and the multicharacter cases w_{wild type} is scaled to equal one and w_{mutant} is equal to
No pleiotropy: Prior work, under the assumption of directional selection and that the fitness effects of random mutation are gamma distributed with shape parameter α and scale parameter β, showed that the distribution of fixed mutational effects is gamma distributed with shape parameter α+ 1 and scale parameter β (Otto and Jones 2000). It can be shown that under our assumptions, the same distribution is obtained. No bidirectional fixed mutations are expected.
Effects of stabilizing selection: The previous analysis (Otto and Jones 2000) assumes only directional selection is happening. Another way of viewing this is that the population is sufficiently far from the optimum that overshooting of the optimum and, therefore, stabilizing selection is not happening. It is of interest to determine how far a population needs to be to safely assume stabilizing selection is unimportant. The probability that stabilizing selection is initiated, i.e., the probability of overshooting the optimum with the next fixed mutation, is
With pleiotropy: Twocharacter case: In the absence of random genetic drift, mutations with nonzero probabilities of fixation satisfy the condition δ_{1} +δ_{2} > 0. Thus, if a mutation fixes and its fitness effect on one character is negative, its effect on the other character must be positive with a greater magnitude.
Given that a mutation arises and is bilateral and exponentially distributed, the probability that it has an effect of x on a character and fixes is
More than two characters: For n characters, the selection coefficient of a new mutation is
When each effect, δ_{i}, of the other n – 1 characters is a bilaterally symmetric exponentially distributed random variable, then by the central limit theorem, the sum of their effects is approximately normally distributed with mean zero and variance θ= 2β^{2}(n – 1). Using this distribution of effects for the other component characters, analysis proceeds in a similar manner as in the twocharacter case. For a gammadistributed mutation with shape parameters <1 the normal approximation is poor, and the distribution of fixed effects is given in the appendix. Let h(y; θ) be the probability that the sum of the effects on the n – 1 other characters is y. h(y; θ) will be approximately normally distributed with mean zero and variance θ. Given a mutation, the probability that it has effect x on a component character and fixes is
RESULTS
No pleiotropy: During directional selection and in the absence of drift and pleiotropy, no bidirectional effects fix (Figure 4a). For exponentially distributed mutation, the analytical expectation of Otto and Jones (2000) closely follows the simulated distribution, and the theoretical expectation of the scaled mean fixed effect (2.0) is close to the simulated average (1.98). When adaptation begins closer to the optimum (Figure 4, b and c) and proceeds until the population travels 90% of its original distance to the optimum, bidirectional mutations fix because of stabilizing selection: lowering the scaled average fixedeffect size to 1.43 for Figure 4b and 0.78 for Figure 4c. The distribution of the magnitudes of fixed mutations becomes approximately exponentially distributed when adaptation begins the average magnitude of two or less random mutations from the optimum. When mutation is more leptokurtic, such that the shape parameter of gammadistributed random mutation is 0.5, the scaled average fixed mutation in simulations (2.98) agrees closely with the prediction of 3.0 by Otto and Jones (2000). Note that Otto and Jones (2000) predict that the average fixed effect is (1 +α)β, and when this is scaled by the average magnitude of a random mutation (αβ), the expected scaled effect is (1 + α)β/αβ = 1/α+ 1, where α is the shape parameter and β is the scale parameter of random mutation that is bilaterally symmetric and gamma distributed.
Under conditions allowing random genetic drift, but in the absence of stabilizing selection, bidirectional effects also can fix (Figure 5a; Table 1). When N_{e} = 200, 7.0% of the mutations have negative effects and the average effect size is 1.74, or 13% less than that in the absence of drift (Table 1). Note that the average magnitude of a fixed effect decreases with a decrease in N_{e} (Table 1). Under conditions of genetic drift and stabilizing selection in Figure 5b, the frequency of negative effects increased to 13.7% and the average effect size decreased to 1.15.
Mutations pleiotropically affecting two characters: When mutation is bilateral and exponentially distributed, bidirectional mutational effects fix in the presence of pleiotropy and in the absence of drift and stabilizing selection (Figure 6a). The average scaled fixed effect in the simulations (1.32) agrees with our prediction (1.34). The frequency of negative effects in the simulations (16.1%) is accurately predicted by the theory presented here (16.67%). The average scaled fixed effect given that it is positive in the simulations (1.69) agrees with our prediction (1.70). When mutation is more leptokurtic, the average scaled fixed mutational effect increases. For instance, when the shape parameter of the mutational distribution is 0.5, the average scaled fixed mutational effect is 1.80 in simulations, which is in agreement with the value (1.83) predicted by the equation presented in the appendix. Although the average fixed effect increases, the frequency of negative effects also increases ∼3% to 19.2% in simulations (19.4% by the equations presented in the appendix). The increase in average fixed effect, despite an increase in the frequency of negative effects, is due to the fact that larger mutations arise at higher probabilities when mutation is more leptokurtically distributed, conditioned on the same average random effect, and these larger mutations, if beneficial, have a high probability of fixation.
When N_{e} is small, but there is no stabilizing selection, an increasing fraction of negative effects occur, the average size of a fixed effect decreases, and the average magnitude of a fixed effect decreases (Table 1). When stabilizing selection occurs because adaptation begins closer to the optimum, the average fixed effect size decreases 62% for the conditions presented in Table 2. Correspondingly, the distribution becomes more balanced because of the sequential overshooting of the optimum such that ∼26% of the mutations are in one direction and 74% in the other. The magnitude of fixed effects also decreased by 53%.
Mutations pleiotropically affecting multiple characters: The relative frequency of negative effects increases as the degree of pleiotropy increases (Figure 6, b and c). Both simulation and mathematical analyses show that, for exponentially distributed mutation, the average scaled fixed mutational effect decreases as pleiotropy increases: from 2.0 with no pleiotropy, to 0.8 when mutations affect 5 characters, to 0.35 when mutations affect 25 characters. When mutations affect 5 characters, simulations show that 30.1% of fixed mutations have negative effects, agreeing with the analytical expectation of 30.0%. For 25 characters, 40.0% were negative, in agreement with the analytical expectation of 40.1%. When random mutation is more leptokurtic such that the shape parameter is 0.5, the average scaled fixed effect is 1.0 in simulations (1.0 by the equations in the appendix) when mutations affect 5 characters and 0.44 (0.44 by the equations in the appendix) when mutations affect 25 characters. Again, despite the increase in the average fixed effect, the frequency of negative effects also increases such that when mutations affect 5 characters it is 32.2% (32.5% by the equations in the appendix) and when they affect 25 characters it is 42.6% (42.6% by the equations in the appendix).
When random genetic drift occurs in the absence of stabilizing selection, the average fixed effect size decreases, the average magnitude of a fixed effect decreases, and the frequency of negative effects increases (Table 1). Stabilizing selection drops the average size of a fixed effect 64% under the conditions presented in Table 2 for mutations affecting five characters, and the distribution of fixed effects becomes more balanced with 37% of the effects being negative. The overall magnitude of fixed effects also drops, this time by 50%.
CONSEQUENCES FOR INFERENCE
The following analyses are applicable to QTL studies that cross individuals from isolated populations or species that differ phenotypically. Under these circumstances, fixed mutational differences are one source of the phenotypic difference between populations or species.
QTL effects: Provided that a QTL consists of one mutational effect, our results would predict the distribution of QTL effects as well as mutational effects. QTL may consist of more than one mutation in which the overall effect of a QTL is a function of the total mutational effects within that region of DNA (Nooret al. 2001). If there is more than one effect in a QTL region, the QTL will have a total effect that could mask some of the individual effects. For example, a negative effect on a character could be hidden by a larger positive effect in the same region. The probability that the overall effect of a QTL is negative decreases as the number of mutations per QTL increases (Figure 7). Although the overall effect of the QTL may be positive, by summing over the effects of two or more mutations, negative effects would be masked. Because mutations with negative effects are on average smaller than those with positive effects, they have less of a masking effect than those with positive effects. When the mutational distribution is more leptokurtic, there is a higher probability that a QTL will have a negative effect because as was shown earlier, there is a higher frequency of fixed mutations with negative effects on a character.
Of the detected QTL that affect a character, there is a bias for them to contain proportionally fewer mutations with negative pleiotropic effects than the true proportion if all of the regions of DNA that contain mutations that affect the character were detected (Figure 8). As the detection threshold of a QTL study becomes worse, i.e., the average magnitude of an effect that can be detected increases, the bias against detecting mutations with negative pleiotropic effects is magnified. The bias is magnified because positive fixed effects are on average larger than negative effects (on an absolute scale) and given that only mutations of larger absolute effect are detected, they are more likely to be positive.
Inferring directional selection on QTL loci: Orr (1998b) proposed that a sufficient number of unidirectional QTL effects would be evidence of directional selection acting on a character, while bidirectional effects would neither refute nor provide evidence for directional selection. When this method is tested against our results, in which only directional selection occurred and it is assumed that one fixed mutation is present per QTL, the ability to infer directional selection is weak (Table 3). The power of the sign test is <50% except when the random mutational distribution is more leptokurtic, mutations affect very few characters, and detection thresholds are poor. Power increases, despite poorer detection thresholds, because as the magnitude of an effect that is able to be detected increases, the probability the effect will be positive as opposed to negative increases. Unfortunately, it is not possible to correctly evaluate Orr's method when several mutations are fixed per QTL because the sign test assumes the distribution of random QTL effects is gamma distributed. When random mutational effects are gamma distributed and more than one mutation is present per QTL, the expected random QTL effect cannot be closely approximated by a gamma distribution. Qualitatively, though, power improves as the number of mutations fixed per QTL increases because as the number of fixed mutations increases per QTL, the probability that the overall effect of the QTL is positive increases.
DISCUSSION
This study quantified the degree to which pleiotropy contributes to the fixation of mutational effects opposite to the direction of selection for a character and compared it to the contributions of stabilizing selection and random genetic drift. As may be expected, the results have shown that pleiotropy can be a major cause of the fixation of effects opposite to the direction of selection even when the degree of pleiotropy is minimal; i.e., a random mutation affects merely two phenotypic characters. The results also show that despite an increase in the frequency of negative effects when the random mutational distribution is more leptokurtic, the average scaled size of a fixed effect does not decrease—it actually increases.
That the fraction of negative effects increases with the degree of pleiotropy is perhaps unsurprising given that pleiotropy clearly allows mutations with weakly deleterious effects to fix because there is the possibility that they also have stronger beneficial effects. Given a set of mutations that have a certain probability of having a positive effect on a character, as the degree of pleiotropy increases, the fraction of negative fixed effects will increase because there is a better chance that a negative effect will be counterbalanced by a positive effect. What is perhaps surprising is that mutations with deleterious effects, which are fixed by pleiotropic selection, can be so frequent.
Implications of model assumptions: Our model implies that the distribution of fixed effects is independent of the strength of selection (σ). This will not be strictly true if we relax the assumption that the mutation rate to beneficial mutations is slow relative to the rate at which they fix. When mutations cosegregate, there is the potential for selective interference (Hill and Robertson 1966). A consequence may be that beneficial mutations of larger effect may outcompete beneficial mutations of smaller effect.
The model assumes that random mutation is equally likely to be in one direction as another for a character. Results from Mackay et al. (1992) and Lyman et al. (1996) suggest this assumption may be reasonable, but more empirical study is necessary. Note that modeling mutation this way does not assume that it is equally likely to have a mutation arise that is beneficial overall vs. one that is deleterious. Deleterious mutations still occur at higher probability. Furthermore, in the appendix an alternate approach that allows for the possibility that negative effects occur at a high frequency is presented.
The distributions of fixed effects derived here are for a set of mutations that pleiotropically affect the same number of characters. Some characters may be affected by mutations that pleiotropically affect different numbers of characters. The resulting effects for such characters would be a mixed distribution over different degrees of pleiotropy.
The model assumes characters are independently affected by new mutations. Clearly this is not the case, in general. To overcome this problem, empirical studies need to employ statistical methods that orthogonalize their data if they wish to use the results presented here. Incorporating the effects of mutational covariance on the distribution of fixed effects represents a significant challenge for future work.
We have assumed that organisms are haploid. This assumption will likely not affect our overall conclusions under directional selection with no drift. In a diploid model, under directional selection with drift and recessivity, deleterious mutations have a higher probability of fixation, which may alter the distribution of fixed mutational effects. Likewise, in a diploid model under stabilizing selection and codominance, if mutations cosegregate and one mutation of large effect is paired with a mutation of small effect, they may have a combined effect that is beneficial; i.e., together they bring a phenotype to be closer to the optimum. But when the mutation with a large effect becomes homozygous, it may then overshoot the optimum to such an extreme that it is deleterious because the homozygous effect brings the phenotype to be farther from the optimum. This overdominance and its importance are left for future study.
Evolutionary consequences: Similar to the finding of Orr (2000), our results show that as the degree of pleiotropy increases the average scaled size of a fixed effect per character decreases. The scale is relative to the average magnitude of a random mutational effect per character, so the inference is not an artifact of the possibility that as pleiotropy increases the average magnitude of a random mutation may decrease.
The decrease in the average scaled fixed effect per character and the increase in frequency of negative effects present potential measures that can be used in comparative studies to determine whether characters are pleiotropically associated with more or less characters in one taxa vs. another, or whether, within a taxa, one character is pleiotropically associated with more characters than another. The implementation of such measures would assume that the characters of interest are experiencing the same evolutionary forces, that is, the same amount of drift, stabilizing selection, and directional selection.
Implications for inferences: Some studies rely on QTL analyses to narrow down regions of a genome that have mutations that affect a character and then perform positional cloning techniques to ultimately determine the effects of individual mutations on a character. Our results quantify a systematic bias for these studies to miss mutations with negative effects. Of the detected QTL, there is a bias for these to contain fewer negative effects than actually occur in the genome because the QTL that contain negative effects are less likely to be detected. Additionally, in missing regions with negative effects, the studies will also miss some positive effects because of masking by the negative ones.
The ability to tell whether directional selection is shaping the evolution of a character is made more complicated by pleiotropy. The frequency of negative pleiotropic effects is sufficient to cause high type II error rates with Orr's (1998b) method.
This study quantified how pleiotropy leads to the fixation of bidirectional mutational effects even in the absence of random genetic drift and stabilizing selection. Mutations pleiotropically affecting more characters fix more negative effects. The potential prevalence of bidirectional effects caused by pleiotropy leads to biases in QTL studies that seek to determine the genetic basis of phenotypic characters.
Acknowledgments
We thank B. Davis, C. Jones, A. Orr, S. Otto, A. Peters, and C. Spencer for very useful comments and discussions. A. Orr was especially useful in pointing out inconsistencies in an earlier manuscript. A reviewer's comments were also very useful in clarifying consequences of our model. This research was supported by a C. Sandercock Memorial Scholarship to C.K.G. and a grant from the National Science and Engineering Research Council (Canada) to M.C.W.
APPENDIX
Here we describe the distribution of fixed effects for mutations pleiotropically affecting two or more characters when the random mutational distribution is leptokurtic such that the shape parameter of the gamma distribution is less than one.
Mutations pleiotropically affecting two characters: The derivation of the distribution of fixed effects proceeds as in the exponentially distributed mutational case. The probability that a mutation has an effect of size x on a character and fixes is
Mutations pleiotropically affecting multiple characters: Here the normal approximation to the sum of the effects on the n – 1 other characters is poor. But we can make use of the property of gammadistributed random variables that given r effects are of the same sign and are drawn from a distribution with the same shape (α) and scale parameter (β); then the distribution of the absolute value of their sum is gamma distributed with shape parameter rα and scale parameter β. In our model up to now, the probability that a random mutational effect on a character is positive is 1/2 and correspondingly the probability that it is negative is also 1/2. We can further generalize and have the probability a random mutation is positive be p and the probability it is negative be q. Then given a mutation that pleiotropically affects n characters, we focus on the effect of that mutation on one character and ask what the probability is that, of the remaining n – 1 characters, t are positive. This probability is binomially distributed, or
Footnotes

Communicating editor: M. Noor
 Received April 4, 2003.
 Accepted August 7, 2003.
 Copyright © 2003 by the Genetics Society of America