Selective Sweeps in the Presence of Interference Among Partially Linked Loci
 ^{*} Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, New York 14853
 ^{†} Department of Biology II, Section of Evolutionary Biology, University of Munich, Munich, 80333 Germany
 1 Corresponding author: Department of Biological Statistics and Computational Biology, Warren Hall 440, Cornell University, Ithaca, NY 14853. Email: yk249{at}cornell.edu
Abstract
Recurrent directional selection on a partially recombining chromosome may cause a substantial reduction of standing genetic variation in natural populations. Previous studies of this effect, commonly called selective sweeps, assumed that at most one beneficial allele is on the way to fixation at a given time. However, for a high rate of selected substitutions and a low recombination rate, this assumption can easily be violated. We investigated this problem using fullforward simulations and analytical approximations. We found that interference between linked beneficial alleles causes a reduction of their fixation probabilities. The hitchhiking effect on linked neutral variation for a given substitution also slightly decreases due to interference. As a result, the strength of recurrent selective sweeps is weakened. However, this effect is significant only in chromosomal regions of relatively low recombination rates where the level of variation is greatly reduced. Therefore, previous results on recurrent selective sweeps although derived for a restricted parameter range are still valid. Analytical approximations are obtained for the case of complete linkage for which interference between competing beneficial alleles is maximal.
GENETIC linkage causes a correlation of ancestral histories among neighboring loci. The behavior of a neutral allele thus reflects that of a selected allele at a closely linked locus. Standing variation at neutral sites is suddenly wiped out when a rapid fixation of a strongly selected beneficial mutation occurs in this region. This “hitchhiking” effect of a beneficial mutation or “selective sweep” (Maynard Smith and Haigh 1974; Kaplanet al. 1989; Stephanet al. 1992; Barton 2000), along with “background selection” caused by recurrent purifying selection (Charlesworthet al. 1993), may be responsible for a substantial reduction of genetic variation in a genomic region of low recombination (Begun and Aquadro 1992). As the degree of the positive correlation between variation and recombination is determined by the strength and rate of directional and purifying selection, polymorphism data from various genomic regions can be used to estimate these parameters (Wiehe and Stephan 1993; Stephan 1995; Charlesworth 1996; Andolfatto 2001).
The action of natural selection is readily detectable as a reduction of variation in regions of low recombination rates. However, it is difficult to identify the form of natural selection responsible for this reduction because several selective forces (including those from distant chromosomal regions) may act simultaneously on variation at a given neutral site when the rate of recombination is reduced. As a consequence, interactions between selective forces such as between positive directional selection and recurrent purifying selection may occur. This problem was investigated in a previous study (Kim and Stephan 2000). The present article addresses the interaction among competing beneficial mutations.
Previous analyses of recurrent selective sweeps (Kaplanet al. 1989; Wiehe and Stephan 1993) are based on theories developed for rare selected substitutions. In these theories, one assumes that a neutral site is under the influence of at most one linked beneficial mutation at any given time. This assumption is satisfied when the rate of selected substitutions is low and the length of the selective phase is short. However, selected substitutions causing hitchhiking effects need to occur at least once in 2N generations to substantially reduce the level of variation. With such a rate of substitutions, the selective phases of different beneficial mutations will overlap with each other with high probability, if the length of the selective phase is not sufficiently short compared to 2N generations. Then the current theories of recurrent selective sweeps may not be applicable.
Only a few authors have analyzed the dynamics of competing beneficial alleles. Barton (1995) investigated the effect of one selected substitution on the fixation probability of a beneficial allele at a linked locus. The fixation probability is increased when the beneficial allele occurs in the genetic background of the previous beneficial allele that is on the way to fixation but decreased in the other background (repulsion phase of two beneficial alleles). The net effect, averaging over the two backgrounds, is a reduction of the fixation probability. Gerrish and Lenski (1998) showed that the trajectories of allele frequency changes are also affected by this interference between selected loci. We extend this work by analyzing the effect of competing beneficial alleles on neutral variation. We specifically ask whether the available theories of recurrent selective sweeps are still valid in the presence of interference.
SIMULATIONS
Examination of genetic variation by fullforward simulation: Genetic variation under complex models for which an analytical method is not readily available is usually investigated by computer simulations. Fullforward simulation (FFS), where processes of the entire population are simulated generation by generation forward in time, can accommodate any complex feature in the population. Usually a mutant allele at a neutral locus is introduced in the simulation and the frequency change of that allele is monitored in FFS. The number of generations is on the order of N_{e}, the effective population size, before a significant change in the allele frequency is obtained. Therefore, it requires a long simulation time for a population of a realistic size. However, it is possible to measure the level of genetic variation without introducing mutants in FFS. In coalescent simulations, the amount of variation is directly proportional to the size of the coalescent tree at the neutral locus under the infinitesite model. Similarly, identity by descent (IBD) can be measured and substituted as a measure of genetic variation in FFS. The relationship between IBD and coalescent time is well known (Slatkin 1991; Barton 1998). Assume that the number of generations is counted backward in time. Two genes randomly selected from a population find a common ancestor at generation T, where T is a random variable. Then, g(t) = P[T ≤ t] is the probability of IBD by generation t. According to the standard coalescent theory,
Using Equation 1, it follows that the effective population size, N_{e}, can be estimated by
Single selective sweeps: We first investigate the effect of a single selective sweep using a twolocus simulation. A diploid population of size N is simulated according to the WrightFisher model of reproduction. A beneficial mutation occurs at a locus linked to the neutral locus where genetic variation is measured by I_{2}. The recombination rate between the two loci is r. The fitness of an individual heterozygous for the beneficial allele is given by 1 + 2ηs (0 <η< 1), and that of homozygous individuals by 1 + 2s. To reduce the simulation time, 10 copies of beneficial alleles are introduced in the population at the beginning of the simulation. Chromosomes carrying these 10 copies share the same ancestral number at the neutral locus. Therefore, this has the same effect as a beneficial allele producing 10 descendants immediately, such that there is no opportunity for recombination to separate the association between the beneficial and neutral alleles. This procedure is justified since it is known that, conditional on its fixation, the initial copy number of a beneficial allele usually increases quickly by drift (Barton 1998). Furthermore, we examined the effect of initial copy number by introducing 1, 2, 5, and 10 copies of beneficial alleles (N = 10^{4}, s = 0.05, η = 0.5, and r = 0.005) and found no significant difference in the mean I_{2} measured after fixation (results not shown). Even though 10 copies are given initially, the beneficial mutation may still fail to be fixed. If all beneficial alleles are lost, simulation starts again from the beginning. The observed frequency of this loss is given by l_{10}. As the early branching processes of these 10 initial copies are largely independent of each other, the fixation probability of each beneficial allele, Φ, can be obtained from l_{10} = (1 –Φ)^{10}. I_{2}(t) is measured when the fixation of the beneficial allele occurs, where t is the number of generations until fixation. I_{2}(t) is a measure of the cumulative coalescent events during the time interval [0, t]. As coalescence between a pair of lineages may occur with probability 1 – exp(–t/2N) at this interval without hitchhiking, the net effect of hitchhiking is measured as
Table 1 shows the comparison between the prediction and the simulation results. For η ≠ 0.5, we still used the equation above but replaced s by 2ηs. This may be justified if the hitchhiking effect is determined mainly at the early stage of the selective phase when the beneficial mutation is in low frequency and thus found mainly in heterozygotes (Stephanet al. 1992). Agreement between the simulation result and prediction is good when η is close to 0.5. However, values of I_{2}_{h} are smaller (larger) than the prediction when η is greater (smaller) than 0.5. A particularly large discrepancy between the simulation results and the prediction for η ≪ 0.5 indicates that the hitchhiking effect caused by a recessive beneficial mutation is not determined mainly at the early stage of the selective phase but at a later time when the beneficial allele is in substantially higher frequency such that homozygotes start appearing in the population. Table 1 also shows that 1 – e^{–4η}^{s} approximates the fixation probability of the beneficial allele quite well if η does not deviate too much from 0.5. For the remainder of this article, we consider only genic selection (η = 0.5).
Two overlapping selective sweeps: The simulation scheme described above is extended to investigate the effect of two overlapping substitutions of strongly selected beneficial alleles on a linked neutral locus. Let the locus of the first beneficial mutation be S1 and that of the following beneficial mutation be S2. We consider all three chromosomal arrangements of the loci: NeuS1S2, NeuS2S1, and S1NeuS2, where Neu represents the neutral locus. The selection coefficients for both selected loci are identical. For this threelocus model (and also the other multilocus models in this study), effects of beneficial alleles on fitness combine multiplicatively. The recombination rate between adjacent loci is r. Ten beneficial alleles are introduced in the population at each locus just as in the simulation of single selective sweeps. However, the beneficial mutation at S2 does not occur until the allele frequency at S1 exceeds a certain value, Q. These beneficial alleles at S2 are initially in complete linkage with either the beneficial (background of 1) or the ancestral (background of 0) alleles of S1. This process is repeated until the fixations at both loci are completed. Then I_{2}_{h} is observed as explained above. The fixation probability, Φ_{2},at S2 conditional on the fixation of the preceding beneficial allele at S1, is measured using the same method as in the case of single selective sweeps. The lengths of the selective phases at S1 and S2, t_{S1} and t_{S2}, respectively, are also recorded.
Simulation results for NeuS1S2 are shown in Table 2. The interference between substitutions causes modification of the fixation probability and the length of the selective phases. Table 2 shows that Φ_{2} increases (decreases) in the beneficial (ancestral) background of S1. Interaction of two beneficial alleles either speeds up or slows down the course of substitution, as revealed by changes in t_{S1} and t_{S2} as a function of the genetic background. The net probability of fixation at S2 is given by QΦ_{2,1} + (1 – Q)Φ_{2,0}, where Φ_{2,1} and Φ_{2,0} are the fixation probabilities in the beneficial and the ancestral backgrounds, respectively. Figure 1 shows the comparison of the simulation results with the theoretical prediction obtained by numerically solving Equations 6a and 6b of Barton (1995). Interference produces the greatest effect on net fixation probability when Q ≈ 0.2.
The effect of two overlapping selective sweeps on neutral variation can be quantified using I_{2}_{h}. I_{2}_{h} averaged over the two genetic backgrounds at S1,
Multilocus simulation of selective sweeps: To further investigate the effect of interference on genetic variation under recurrent selective sweeps, FFS described in the previous section is extended to a multilocus model. The neutral locus under investigation is located in the middle of a chromosome. Thirty loci where beneficial mutations can occur are on each side of the neutral locus. Mutation occurs at a rate u per gene per generation if the beneficial allele is not already segregating at the same locus in the population. The recombination rate between adjacent loci is uniformly r. The first phase of the simulation, which is t_{1} generations long, brings the population into an equilibrium state in which there is a constant flux of beneficial alleles reaching fixation. Then, at the beginning of the second phase, which takes t_{2} generations, ancestral numbers are assigned to genes at the neutral locus. Fixation probability, Φ, of the beneficial allele is measured by counting the number of introduced and fixed alleles during the second phase of the simulation. At the end of the second phase, I_{2} = I_{2}(t_{2}) is recorded. The effective population size is estimated using Equation 3, where instead of t and I_{2}(t) we use t_{2} and the observed mean of I_{2}(t_{2}) over replicates, respectively. Preliminary study showed that N̂_{e} obtained from Equation 3 is a decreasing function of t_{2} if t_{2} is small compared to the length of the single selective phase (t_{s}), but converges to the expected value when t_{2} > 2t_{s} (results not shown).
Using previous results on the coalescent effective population size under the model of selective sweeps (Wiehe and Stephan 1993; Gillespie 2000b; Kim and Stephan 2000), we predict the expectation of N̂_{e} as
First, we consider a uniform selection coefficient for all loci (Table 3A, s = 0.1, N = 5000). The beneficial mutation rates are high enough to cause interference among closely linked sites (see discussion). As expected, the fixation probability averaged over all sites decreased as r decreased. The average length of the selective phase, t_{s}, increased as r decreased. Therefore, interference slows down the course of the selected substitutions. N̂_{e} is consistently >E_{1}[N̂_{e],} confirming that the standing level of genetic variation is not as much decreased as expected without interference. E_{2}[N̂_{e}] is generally closer to N̂_{e} than is E_{1}[N̂_{e].} Therefore, a large part of the difference between N̂_{e} and E_{1}[N̂_{e],} presumably due to interference, can be explained by the reduction in the rate of substitution. The remaining discrepancy between N̂_{e} and E_{2}[N̂_{e]} might be explained by the reduced hitchhiking effect of a given substitution if N̂_{e} > E_{2}[N̂_{e]} is observed. However, N̂_{e} < E_{2}[N̂_{e]} in many cases when r = 0 and u is large. One possible explanation is the presence of a “leapfrog” effect, which is described below.
To mimic a more realistic situation, we also performed simulations where beneficial mutations occur with two different selection coefficients. The beneficial mutation under relatively strong directional selection occurs with selection coefficient s_{s} at rate u_{s} and the mutation under weaker directional selection with s_{w} at rate u_{w} (>u_{s}). These two different mutations occur at 30 “strong” and 30 “weak” loci, respectively, which alternate with each other along the chromosome. It should be understood that weak means only the relative strength of a mutation when compared to a strong mutation. We still consider 2Ns_{w} ≫ 1. Table 3B shows the result for s_{s} = 2s_{w} and u_{w} = 2u_{s}. As expected (Barton 1995), the decline of the relative fixation probability at the weak loci is much greater than that in the case of equally strong beneficial alleles (Table 3A). However, differences among N̂_{e,} E_{1}[N̂_{e],} and E_{2}[N̂_{e]} are not much greater than those among the uniform selection coefficients. For r > 0, the reduced rate of substitutions at weak loci might be unimportant in modifying the effective population size because the latter is determined mainly by the hitchhiking effects from strongly selected loci, the fixation rate of which does not change as much as that of weakly selected loci. However, this cannot be an explanation for the case of r = 0 because both strong and weak selections wipe out standing variation completely. This issue is further investigated below.
Next, the joint effect of selective sweeps and background selection was investigated by a similar simulation scheme in which half of the selected loci are now under recurrent purifying selection. A total of 48 selected loci, where loci under directional selection are alternating with those under purifying selection, were used and the neutral locus was inserted in the middle of the arrangement. Deleterious mutations with selective disadvantage s_{d} were introduced at a rate u_{d} per locus per generation. Table 4 shows the results. Φ decreased with background selection as expected (Barton 1995). N̂_{e} obtained from the simulation was in good agreement with the theoretical prediction (E[N̂_{e]} in Table 4), which was obtained by modifying Equation 6 (see Kim and Stephan 2000, Equation 6). Selective sweep and background selection act nonadditively on N_{e}: With low recombination, the joint effect of the two forces is almost the same as that of hitchhiking alone (compare the third, sixth, and ninth rows of Table 4). This result can be explained by the fact that the increasing effect of background selection is offset by a decreasing effect of recurrent selective sweeps, as background selection causes a reduction of the fixation probability with lower recombination rates. This multilocus simulation confirms the results obtained by a threelocus model of the joint effects of background selection and hitchhiking investigated in a previous study (Kim and Stephan 2000).
THEORY OF RECURRENT SELECTIVE SWEEPS FOR ZERO RECOMBINATION
The multilocus model described above is further investigated in the case of zero recombination for which the effect of interference is expected to be maximal. Equation 6 with r = 0 gives the expected level of variation under the model of recurrent selective sweeps with complete linkage. Since the substitution of any selected allele with complete linkage has the same hitchhiking effect, i.e., the complete removal of genetic variation, only the rate of fixation of beneficial mutations is expected to determine the level of standing variation. The fixation probability of beneficial mutations for a nonrecombining chromosome was studied in Barton (1995) and Gerrish and Lenski (1998). However, their approximate solutions are either inaccurate or not applicable to our multilocus model. Here we present an alternative derivation of the fixation probability. The derivation assumes a haploid population of 2N individuals.
There are two possibilities by which the fixation probability of a new beneficial mutation, B_{1}, is affected by another beneficial mutation, B_{2}, at a linked site. First, if B_{2} is already segregating in the population when B_{1} arises, the initial selective advantage of the chromosome carrying B_{1} relative to others is modified depending on the frequency of B_{2} and depending on which chromosome B_{1} occurs. Conditional on fixation, the frequency of B_{1} drifts quickly to a certain threshold above which the chance of B_{1} being lost by drift is negligible. Therefore the fate of B_{1}, loss or fixation, is decided in a short initial period. Second, B_{1} while on the way to fixation (after this short initial phase) may be displaced by another beneficial mutation that arises after B_{1} (Gerrish and Lenski 1998). Therefore we approximate the fixation probability under interference as Φ = f_{1} f_{2}, where f_{1} is the fixation probability that takes into account only the initial competition with preexisting alleles and f_{2} is the probability that the allele that survived the initial drift is not lost in the competition with latearising alleles.
First we consider the case where all beneficial alleles are equally advantageous with selective coefficient s (such that Ns ≫ 1). Under no interference, the fixation probability of a beneficial allele starting at frequency x is given approximately by
Now suppose that B_{1} appears in the population when the frequency of B_{2} is x and that no additional beneficial mutation occurs after B_{1}.If B_{1} occurs on a chromosome carrying B_{2}, the relative fitness of this chromosome is ∼1 + 2s while the mean relative fitness of the population is given by 1 + sx, ignoring the contribution of the chromosome (with both B_{1} and B_{2}) to mean fitness. Assuming that the population size remains constant each generation, the absolute fitness (i.e., the mean number of its copies at the next generation) of this chromosome is thus (1 + 2s)/(1 + sx) ≈ 1 + 2s – sx. Then, the theory of branching processes (Barton 1995) predicts that the fixation probability of B_{1}, f, satisfies the equation 1 – f = exp[–(1 + 2s – sx)f]. f = f_{0}(1/2N, 2s – sx) = 1 – exp(–4s + 2sx) is a good approximate solution of this equation.
On the other hand, if B_{1} arises in repulsion phase with B_{2}, the fixation of B_{1} depends on two conditions: First, since one chromosome carrying B_{1} and 2Nx chromosomes carrying B_{2} are selectively equivalent, they comprise a subpopulation of effectively identical chromosomes. This subpopulation of chromosomes increases in frequency and eventually goes to fixation with probability f_{0}(x + 1/2N, s) = 1 – exp(–4Nsx – 2s). Second, B_{1} displaces B_{2} within that subpopulation by drift with probability 1/(2Nx + 1). Therefore, B_{1} is fixed approximately with probability
In the next step, we consider this latter possibility. A new beneficial mutation that arises in repulsion with B_{1} can compete with B_{1} for fixation. If the frequency of B_{1} is y when the new mutation occurs, the fixation probability of the latter is approximately f_{0}(y + 1/2N, s)/(2Ny + 1), following the argument above. Therefore, the probability that B_{1} is not lost in the competition with a latearising mutation is approximately
The effective population size under the model of recurrent sweeps for a nonrecombining chromosome and equally advantageous mutations is predicted from Equation 6 to be
Theoretical predictions can also be derived for the multilocus model when two different selection coefficients are used (Table 3B, r = 0). Here we are interested in whether a great reduction of the fixation probability at the weak loci can lead to a discrepancy between the results in the presence and absence of interference that is larger than that in the case of uniformly strong beneficial mutations. As previously, we put s_{s} = 2s_{w} and u_{w} = 2u_{s}; furthermore, we consider L_{s} (= 30) strong loci and L_{w} (= 30) weak loci. Because of the asymmetry of the effect of interference between weakly and strongly selected mutations (Barton 1995), it is assumed that the fixation probability at the strong loci is affected only by interference from other strong loci. Therefore Equation 11, with s_{s} and L_{s} instead of s and L, is used to obtain the average fixation probability, Φ_{s}, at the strong loci. On the other hand, the fixation probability at the weak loci, Φ_{w}, is determined mainly by interference due to strong beneficial mutations. If the weak mutation occurs on a chromosome carrying the strong allele, the absolute fitness of this chromosome becomes ∼1 + s_{w} + s_{s}(1 – x), where x is the frequency of the strong allele. On the other hand, if the weak allele occurs in repulsion with the strong allele, the weak allele can be fixed only if (1) the strong allele is lost and (2) the weak allele survives genetic drift. Therefore, a weak allele goes to fixation approximately with probability
It is consistently found in Figure 4A and Table 3, A and B (r = 0), that values of N̂_{e} are between the predictions of N_{e} with and without interference. This is unexpected because the reduction of the hitchhiking effect for a given substitution (Figure 2), which may further elevate N_{e}, was not considered in the prediction with interference. Therefore, another process causing an additional reduction of variation that has so far been neglected may play a role. One possible explanation is that many beneficial mutations increase in frequency substantially but fail to reach fixation due to the interference among them. This transient increase of mutations, called a “leapfrog” event (Gerrish and Lenski 1998), may also contribute significantly to the reduction of the level of genetic variation. To confirm that leapfrog effects occur, simulations were run using the same parameter values as in Table 3. The number, k_{0.5}, of beneficial alleles that increase above frequency 0.5 but fail to be fixed were counted. The number, k_{1}, of beneficial alleles that went to fixation during the same period was also recorded. For uniform selection coefficients, k_{0.5}/k_{1} was 0.025 and 0.043 for u = 2 × 10^{–8} and 4 × 10^{–8}, respectively. On the other hand, for unequal selection coefficients, k_{0.5}/k_{1} was 0.028, 0.055, and 0.173 for u_{w} = 10^{–8}, 2 × 10^{–8}, and 4 × 10^{–8}, respectively, for the weak loci.
DISCUSSION
Kaplan et al. (1989) studied recurrent selective sweeps in a restricted parameter range for which an overlap of selective phases is minimal. They assumed the rate of selected substitutions to be under a certain limit such that the probability, P(Ω), of two gene lineages experiencing only nonoverlapping selective phases before they find a common ancestor (Equation 21 of Kaplanet al. 1989) remains close to 1.0. Wiehe and Stephan (1993) and Braverman et al. (1995) followed these guidelines. By applying the approach of Kaplan et al. (1989) to our multilocus model (uniform selection coefficients), we obtain
Little information is available to assess how widespread interference among beneficial mutations is in natural populations. However, there is some evidence that interference is common. Wiehe and Stephan (1993), Stephan (1995), and Andolfatto (2001) showed that for the recurrent selective sweep model to fully account for the positive correlation between variation and recombination in Drosophila melanogaster, the intensity of directional selection αν, where α = 2Ns and ν is the rate of selected substitution per nucleotide, should be somewhere between 10^{–8} and 10^{–7}. Assuming αν = 10^{–8}, N = 10^{6}, and s = 10^{–3}, P(Ω) calculated for a chromosomal region with a moderately low pernucleotide recombination rate ρ = 10^{–9} is 0.58 (Equation 21 of Kaplanet al. 1989). Therefore, as Przeworski (2002) pointed out, overlapping selective sweeps should occur with high probability if selective sweeps contribute significantly to the observed level of variation in D. melanogaster. Similarly, the patterns of variation in humans indicate the presence of overlapping selective sweeps (Przeworski 2002).
It is important to understand the relationship between the standing level of variation and local recombination rate in various models of selection. The level of variation determined by recurrent selective sweeps (without interference) may be much more sensitive to the change of local recombination rates over genomic regions than that determined by background selection (Kim and Stephan 2000). It was suggested that this difference may be used to distinguish selective sweeps and background selection as main contributors to the level of variation (H. Innan, personal communication). However, interference among beneficial mutations may slow down the reduction of variation in a region of very low recombination and thus make it difficult to distinguish sweeps from background selection. We examined where in the parameter space interference affects the relationship between recombination and variation. It appears that interference is important only in regions of low recombination where standing variation is highly reduced (Table 3 and Figure 4). This can be understood by examining Equation 6. A decrease of the fixation probability and the hitchhiking effect due to interference cannot significantly modify E[N̂_{e}] if the term 4N^{2}Σu_{i}Φ_{i}(1 – h_{i}) (in the absence of interference) is <1. However, interference becomes important when N_{e}/N has already been greatly reduced [4N^{2}Σu_{i}Φ_{i}(1 – h_{i}) ≫ 1]. This is consistent with the observation that the level of neutral variation decreases more rapidly with a decreasing recombination rate than does the fixation probability of beneficial mutation (compare the changes of Φ and N̂_{e} in Table 3). Therefore interference among beneficial mutations does not happen without having a great reduction in neutral variation.
Although we have shown that the reduction of effective population size or heterozygosity by selective sweeps is not much influenced by interference, it is not known at present whether other important aspects of variation, such as the frequency spectrum or linkage disequilibrium (Fay and Wu 2000; Kim and Stephan 2002; Przeworski 2002), are also insensitive to interference among beneficial mutations. With the forward simulation proposed in this study, one may detect a change in the frequency spectrum by observing higher moments of the frequency of ancestral numbers, namely I_{k}(t) (k ≥ 3). A preliminary study, using (I_{3}(t) – I_{4}(t))/(I_{2}(t)^{2} – I_{4}(t)) as a test statistic, showed that the frequency spectrum is less skewed if beneficial alleles arise on different chromosomes and thus compete with each other. More analysis of this statistic is in progress.
Acknowledgments
We thank Allen Orr for pointing out the leapfrog effect and Hideki Innan and Rasmus Nielsen for discussion. We also thank Deborah Charlesworth and two anonymous reviewers for comments that greatly improved the manuscript. This research was supported by funds from the Deutsche Forschungsgemeinschaft to W.S. and National Science Foundation grant DEB0089487 to Rasmus Nielsen.
Footnotes

Communicating editor: D. Charlesworth
 Received July 19, 2002.
 Accepted January 17, 2003.
 Copyright © 2003 by the Genetics Society of America