Joint Effects of Genetic Hitchhiking and Background Selection on Neutral Variation

STATIONARY LEVEL OF HETEROZYGOSITY CAUSED BY BACKGROUND SELECTION AND RECURRENT HITCHHIKING EVENTS (MODEL 1)

In this section, we investigate the stationary level of heterozygosity determined by recurrent substitutions of favorable alleles and continuous removal of deleterious alleles by background selection. We use a simple discrete-generation model of a diploid population of size N. (Note that a list of parameters is provided in Table 1.) Since we assume that the fitness effects of alleles within a locus combine multiplicatively, this model is equivalent to that of a haploid population of size 2N that undergoes random conjugation and recombination. We consider a three-locus model such that the three loci are located on a chromosome in the following order: The first locus (Del) experiences recurrent deleterious mutations. The mutation of the wild-type allele (A) to a deleterious allele (a) with selective disadvantage t occurs at a rate u (per gene per generation). That is, as in Stephan et al. (1999), background selection acts at a single locus. The second locus (Fav) is under positive selection such that a favorable allele (B) with selection coefficient s is introduced as described below. Mutation from an ancestral (m) to a derived neutral allele (M) occurs at the third locus (Neu). The genetic background (haplotype) on which the mutations to B and to M occur is chosen randomly in proportion to its frequency. The recombination fractions between Del and Fav and between Fav and Neu are r₁ and r₂, respectively. The two other possibilities of gene order, i.e., Del-Neu-Fav and Fav-Del-Neu, were also studied. For all three gene orders, the simulation and theoretical methods are similar. So we focus on the case of Del-Fav-Neu.

Simulation methods: For this model, there are eight possible haplotypes (ABM, ABm, AbM, Abm, aBM, aBm, abM, and abm) in the population. Therefore, the dynamics of the system can be completely described by the changes of eight haplotype frequencies. It is straightforward to derive a set of equations describing the deterministic change of haplotype frequencies by selection, recombination, and deleterious mutations (appendix a). When one copy of B or M is introduced in the population, haplotype frequencies are changed accordingly before they are subjected to selection. To incorporate the effects of finite population size, multinomial sampling of different haplotypes was simulated after their frequencies were changed according to the deterministic equations. We used the random binomial number generator of Press et al. (1992) with some modification.

Each simulation starts with a population of Abm and abm haplotypes. The initial frequency of a is given as u/t. Then, one copy of the favorable allele (B) is introduced in the population at rate u_f if the population is fixed for b (note that u_f is a mutation rate per population per generation, whereas u is per gene per generation). If B is fixed, all B's are converted to b. Therefore, with u, u_f > 0, a mutation-selection balance at the Del locus and occasional directional selection at the Fav locus occur simultaneously. To measure the standing level of genetic variation at Neu, we used the method suggested by Charlesworth et al. (1993). An allele M is introduced at the beginning of each simulation run and introduced again whenever it is lost in the population by drift. If M is fixed in the population, at the next generation all M's are converted to m and then another M is introduced. The frequency of M, y, is monitored until M is lost or fixed. During this period, heterozygosity, 2y(1 − y), is summed over generations. The expected value of this sum, H, in the neutral model is 2.0 (Kimura 1969, 1971). Heterozygosity per site per generation, π, for a given hypothetical process of mutation, with rate μ, is then obtained by “spreading out” trajectories of M over time according to the mutation process; i.e., π = 2NμH. To observe the change of the level of genetic variation, we only need to measure H without modeling a specific mutation process (Charlesworthet al. 1993).

This procedure is based on the principle of ergodicity, which says that averaging a random variable of a stationary stochastic process over time leads to the same result as averaging this quantity at any given time point over different realizations of the process. Assuming that the selective phase during hitchhiking is very short, we may consider each selective sweep instantaneous. Since these hitchhiking events are modeled as a time-homogeneous Poisson process, their effect is a shortening of the trajectories of the neutral allele M, independent of time. As a consequence, the expectation of H can be found by evaluating this random variable at arbitrary time points during a particular realization of the process and by averaging over these values, or by evaluating an ensemble of realizations of the process at a particular time. A total of 10⁸ introductions of M were made consecutively in each simulation run and the mean value of H was obtained. The mean numbers of generations until loss and until fixation were also recorded.

Theoretical predictions: We combined the previous theoretical results of hitchhiking and background selection using the following assumptions. Deleterious mutations occur very frequently, leading to the establishment of a mutation-selection balance. We assume that this mutation-selection balance at Del affects the fixation probability and the effective population size at the Fav locus. On the other hand, we assume that directional selection at Fav has no influence on the mutation-selection balance at Del. Therefore, the equilibrium frequency of the deleterious allele (a) is maintained during the selective phase. The combined effects of background selection and hitchhiking on heterozygosity may therefore be approximated by well-known formulas (Kaplanet al. 1989; Stephanet al. 1992; Wiehe and Stephan 1993) that describe the effects of hitchhiking on neutral variation as a function of linkage, effective population size, and strength of selection, except that these latter quantities have to be corrected to allow for the occurrence of background selection.

The effect of background selection at a single locus on a linked locus is given by fB(r)=1−ut{1+r(1−t)∕t}2 (1) (Hudson and Kaplan 1995; Nordborget al. 1996), where f_B(r) is the relative reduction of heterozygosity or effective population size as a function of the recombination fraction, r, between the two loci. Then, without hitchhiking, heterozygosity at Neu is predicted to be π₀f_B(r₁ + r₂), where π₀ is the average nucleotide heterozygosity in the neutral equilibrium model. To incorporate the effect of hitchhiking, it follows from the above considerations that the strength of directional selection at Fav is characterized by α₁ = 2N₁s, where N₁ = Nf_B(r₁), and that the fixation probability of B is Φ=(1−e−2s)Π, (2) where Π = Π (u, t, s, r₁) is the reduction of the fixation probability due to background selection (may be obtained by solving Equations 15a and 15b of Barton 1995 numerically). To obtain the effect of repeated substitutions at Fav on average heterozygosity at Neu, we use a theoretical argument by Wiehe and Stephan (1993). The reduction of expected heterozygosity by a single hitchhiking event is given by h=h(r1,r2)=2r2sα1−2r2∕sΓ(−2r2s,1α1), (3) (Stephanet al. 1992), where Γ(. , .) is the incomplete gamma function. Using coalescent arguments (Kaplanet al. 1989; Wiehe and Stephan 1993), h can be related to a sample quantity. For a sample of size 2, consider a coalescent process, with time running backward. Then it can be shown that h is equal to the probability of entering the selective phase with two ancestral genes and exiting it with two ancestral genes. In other words, h is equal to the probability that the neutral locus escapes hitchhiking by recombining away from the favored allele, while the latter is on its way to fixation. The rate of hitchhiking events per generation experienced by the Neu locus is therefore given by 1 − h times the probability of selective fixations at the Fav locus, which is u_fΦ. As the history of the sample is traced back, regular coalescent events may occur at the rate of 1/2N₂, parallel to hitchhiking events, where N₂ = Nf_B(r₁ + r₂). Without hitchhiking, the expected time to the most recent common ancestor of a sample of size 2 is 2N₂. Therefore, on the time scale of 2N₂ generations, the rate of occurrence of a coalescent or a hitchhiking event is 1 + 2N₂ u_fΦ(1 − h), and the expected time back until one of these events occurs is 1/[1 + 2N₂ u_fΦ(1 − h)]. Thus, due to hitchhiking the expected total time of the genealogy of the Neu locus is reduced by the factor fH(r1,r2)=11+2N2ufΦ(1−h). (4) Thus, combining the effects of background selection and hitchhiking, the value of H is predicted to be E[H]=2fB(r1+r2)fH(r1,r2). (5) In summary, we predicted the combined effect of hitchhiking and background selection by using a previously known analytic solution of the hitchhiking effect and by modifying the effective population sizes at the linked loci and the fixation probability of the favorable allele such that background selection is taken into account.

Simulation results: Table 2 shows the results of simulations for the gene arrangement Del-Fav-Neu in model 1. In most cases, we used t = 0.02, which is close to the mean heterozygote effect of deleterious mutations estimated from D. melanogaster (Crow and Simmons 1983). The choices of s, r₁, r₂, u, and u_f are rather arbitrary. Since simulation time increases with population size, we used 2N = 10⁵, which is smaller than typical Drosophila population sizes. We obtained accurate results for H, the fixation probability of B, and the mean time to loss (T₀) and fixation (T₁) of M expected under standard theories (simulations 1-1, 1-4, and 1-8). The fixation probability of M was close to 1/2N in all the simulation runs for model 1 (data not shown), which agrees with the fact that selection at linked loci does not affect the fixation probability of neutral mutants (Birky and Walsh 1988). The H values obtained for all parameter values agree well with our simple theoretical predictions (Equation 5). This result indicates that the effects of background selection and hitchhiking can be combined in a predictable way. However, the combined effects are not simply multiplicative. For example, comparing simulations 1-3, 1-4, and 1-6, the H values were reduced by the factors of 0.67 and 0.84 by background selection and hitchhiking, respectively, when each process took place without the other. However, when both processes occurred at the same time, the reduction factor of H was 0.61, larger than the product 0.67 × 0.84 = 0.57. This discrepancy can be fully explained by the expected change of f_H from 0.836 (for 1-4) to 0.894 (for 1-6). This suggests that the effect of hitchhiking diminishes with background selection. This nonmultiplicative combination of hitchhiking and background selection produces even more striking results when the effect of hitchhiking is very strong: In simulations 1-9 and 1-10, it is shown that the reduction of heterozygosity is smaller when hitchhiking is combined with background selection than when hitchhiking occurs in the absence of background selection. Simulations for the other arrangements of genes, i.e., Del-Neu-Fav and Fav-Del-Neu, gave qualitatively the same results (data not shown).

In simulations 1-13, 1-14, and 1-15, where s > t, the equilibrium frequency of the deleterious allele at Del is likely to be perturbed by the substitution of B, which violates the assumptions of our theoretical predictions. However, for these parameter values theoretical predictions given by (5) still agree well with simulation results (Table 2). We further address this problem for model 2 below.

Generalizations and implications: The results obtained from our three-locus analysis suggest the following generalizations. Consider a chromosome of a finite physical length throughout which the recombination rate per nucleotide per generation, ρ, is constant. A position on the chromosome is described by the number of nucleotides, l, away from the reference locus Neu, where a positive (negative) value of l defines a locus to the right (left) of Neu. Neu is located l_L and l_R nucleotides away from the left and right ends of the chromosome, respectively. Deleterious mutations can occur at any position along the chromosome at a rate u (per nucleotide per generation). Single hitchhiking events may occur according to a time-homogeneous Poisson process caused by advantageous substitutions at randomly chosen loci. The question we ask is, What is the joint effect of these forces on neutral polymorphism at Neu? Equation 5 suggests that nucleotide diversity, π, can be approximated as π≈π0fB(ρ)ρρ+kαfB(ρ)ν(ρ), (6) where π₀ is the neutral equilibrium value of nucleotide diversity, α = 2Ns, k is a constant (see below), ν(ρ) is the expected number of selected substitutions per nucleotide per generation at Neu, and f_B(ρ) is the reduction factor of effective population size at Neu due to background selection.

Equation 6 can be derived in a similar way as the corresponding equation without background selection (Wiehe and Stephan 1993). Let ν(l, ρ) and α(l, ρ) be the expected number of advantageous substitutions and the strength of selection at a position l, respectively. The rate at which a neutral polymorphism at Neu undergoes hitchhiking caused by selected substitutions between l and l + dl nucleotides away is then given by ν(l, ρ) [1 − h(l, ρ)]dl, where h(l,ρ)=2ρ∣l∣sα(1,ρ)−2ρ∣l∣∕sΓ(−2ρ∣l∣s,1α(1,ρ)). (7) The equation above assumes that the recombination fraction scales linearly with physical distance, which can be justified as the effect of hitchhiking is limited to a small genetic distance. The expected number of selected substitutions causing a hitchhiking effect at Neu is then obtained as kh=∫−M∗∕ρM∗∕ρν(1,ρ)[1−h(1,ρ)]dl (8a) ≈ν(ρ)∫−M∗∕ρM∗∕ρ[1−h(1,ρ)]dl, (8b) where M* is the maximal recombination distance allowed (see Wiehe and Stephan 1993). The approximation leading to (8b) works because the hitchhiking term [1 − h(l, ρ)] has its maximum at l = 0 and declines rapidly to zero for l ≠ 0 relative to the function ν(l, ρ), which varies slowly with l. The integral in (8b) can be evaluated for constant α(. , .), namely α(. , .) = α₀, because the integrand is insensitive to α(. , .). Therefore, we find approximately kh≈ksν(ρ)ρ, (9a) where k denotes the integral k=∫−2M∕α00[1+uα0uΓ(u,1α0)]du, (9b) and M = 2NM*. As shown in Wiehe and Stephan (1993), k depends only weakly on α₀.

Assuming deleterious mutations have a uniform selective disadvantage, t, and effects of background selection at many loci combine multiplicatively, it can be shown that fB(ρ)≈exp[−ut∫−lLlR1{1+ρ∣x∣(1−t)∕t}2dx] (10) (Nordborget al. 1996) and ν(ρ)≈ν0exp[−ut∫−lLlR{2t(t+s+ρ∣x∣)2−4st+t+s+ρ∣x∣}2dx] (11) (Barton 1995, Equation 17a), where ν₀ is the expected number of selected substitutions per nucleotide on a free-recombining chromosome. Incorporating (10) and (11) into (6), Figure 1 shows the relationship of nucleotide diversity π/π₀ and recombination rate ρ for background selection, hitchhiking, and for the joint action of both processes. It is noteworthy that for low recombination rates the function describing the joint effects of background selection and hitchhiking approaches that of hitchhiking alone, whereas for higher recombination rates background selection is the dominant force determining the level of diversity. This is because in regions of low recombination the background selection terms f_B(ρ) cancel, unless background selection is extremely strong and/or the rate of favorable substitutions is extremely low, such that f_B(ρ)ν(ρ) converges to zero. The expression resulting from (6) for small ρ suggests that background selection affects only the fixation probability of advantageous mutations, not their strength. As a consequence, background selection may reduce the effect of hitchhiking; that is, lower levels of nucleotide diversity are expected in regions of very low recombination when hitchhiking operates alone than in the presence of background selection. A similar result was found for our three-locus model (see above).

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Figure 1.

Relative nucleotide diversity, π/π₀, against pernucleotide recombination rate, ρ. The model described in the text is used with t = s = 0.02, u = 1.5 × 10⁻⁹, 1_L = 1_R = 10⁷, kαν₀ = 10⁻⁹. The continuous line (—), produced by Equations 6, 10, and 11, represents the joint effect of hitchhiking and background selection. The graphs of relative diversity determined by background selection alone (- ċ -) and hitchhiking alone (---) are produced by π = π₀f_B(ρ) (Equation 10) and π = π₀ρ/(ρ + kαν₀) (Wiehe and Stephan 1993), respectively.

TRANSIENT PATTERNS OF HETEROZYGOSITY AFTER A SINGLE HITCHHIKING EVENT UNDER THE INFLUENCE OF BACKGROUND SELECTION (MODEL 2)

We use the same three-locus model as above, but assume different mutational processes for the Fav and Neu loci. We are interested in transient patterns of heterozygosity at Neu at given time points after a hitchhiking event, rather than the stationary level of heterozygosity measured in model 1. We define time, T, as the number of generations before present (T = 0). One selected substitution, from b to B, at Fav occurs at a fixed time in the past (T = τ). It is assumed that the previous substitution at Fav took place very long ago such that the level of genetic variation at Neu has recovered to its equilibrium level before the selected substitution occurs at T = τ; i.e., we are analyzing the effect of a single hitchhiking event. At the Neu locus, mutant alleles, M, are introduced in the past at T = λ (0 < λ < ∞), such that they may occur at any generation with an equal rate, μ (per gene). Each mutant has a certain probability of still segregating at T = 0, thus contributing to heterozygosity at T = 0. Therefore, heterozygosity can be determined by adding up all the contributions made by the mutations in the past (Kimura 1969, Equation 5). This suggests a simulation procedure in which heterozygosity at T = 0 is measured by summing over all trajectories of M that occurred in the past. That is, in each simulation run we follow the trajectory of allele M and record its frequency at T = 0. By repeating this procedure, we obtain the distribution of the frequency of M and thus the average heterozygosity at T = 0. As in model 1, recurrent deleterious mutations occur at the Del locus.

Simulation methods: The recurrence equations and multinomial sampling method that were described in model 1 are used to simulate the dynamics of allele frequency changes. In the simplest way, model 2 can be simulated by introducing a neutral mutant, M, at T = λ, where λ is uniformly distributed between 0 and L (⪢N), and by recording its frequency at T = 0. If M is lost or fixed before T = 0, its frequency is recorded as zero. Thus, the frequency distribution of M at T = 0 can be obtained by repeating the above procedure many times. However, this straightforward simulation scheme is too time-consuming because the length of the time window, L, has to be set to a large value. To circumvent this problem, we use the following procedures, which allow us to keep L reasonably small.

For simplicity, consider first the model without background selection. We assume that, at the beginning of the time window (T = L), the population is in mutation-drift equilibrium such that the number and the frequency distribution of a segregating allele at Neu can be described by the standard neutral theory (Kimura 1983). Then, we introduce mutants at T = L with initial frequency i/2N (i = 1, … , 2N − 1), where the probability of frequency i/2N is proportional to 1/i. The rest of the mutants are introduced at T = λ with initial frequency 1/2N, where λ is uniformly distributed between 0 and L − 1. The ratio of mutants appearing at T = L and T = λ is adjusted such that the expected number of segregating sites is at equilibrium. For a given mutation rate (per generation per locus), μ, the probability that allele M is segregating at the Neu locus at T = L is given by θa_2N, where θ = 4Nμ and an=∑i=1n−11∕i . The expected number of new mutants occurring between 0 and L is 2NμL. Therefore, the proportion of mutants, δ, introduced at T = L is 2a_2N/(2a_2N + L). Since one mutant occurs in each simulation run, the value of μ is given by (1 − δ)/(2NL). The expected value of heterozygosity at T = 0 at equilibrium (without hitchhiking) contributed by each M is θ (see appendix b). The favorable mutation at the Fav locus occurs at a fixed generation (τ < L) before present. The mutation to the favorable allele (B) occurs if a neutral mutant that appeared at T = λ (>τ) is still segregating in the population at T = τ. If B is lost, haplotype frequencies at T = τ (immediately before the favorable mutation occurred) are restored and the mutation to B occurs again. This procedure is repeated until B is fixed. While B is segregating, the frequency of M changes but the time counter T is arrested. It resumes decreasing after B is fixed. Therefore, this procedure simulates a situation where neutral allele frequencies change instantaneously in one generation by a hitchhiking effect. For example, if M is lost or fixed during the substitution of B, its frequency is recorded as 0 or 1, respectively, at T = τ − 1.

To incorporate background selection on Del, the same procedure is used but with the following changes. We assume that, when neutral mutants are introduced, the population is in a deleterious mutation-selection equilibrium. Therefore, the frequency of the deleterious allele (a), q, is set to u/t when each replicate of the simulation starts with a new introduction of M. One copy of M randomly associates with A or a at T = λ. The expected number of segregating sites at T = L is now ~4N₂μa_2K, where K is the closest integer to N₂ = Nf_B(r₁ + r₂) (see Equation 1); the latter is the effective population size at Neu when background selection has been taken into account. δ is now given by 2N₂a_2K/(2N₂a_2K + NL). The initial frequency of M at T = L is i/2K (i = 1, … , 2K − 1), where the probability of frequency i/2K is proportional to 1/i. We assume no linkage disequilibrium between Del and Neu at T = L; thus the frequency of the AM haplotype, for example, is given by (1 − u/t)(i/2K). However, we used L > τ + 10³, so that haplotype frequencies immediately before the hitchhiking event depend little on the initial frequencies.

Theoretical predictions: To predict average heterozygosity (π) at T = 0 for model 2, we used the same approach as for model 1 by modifying the effective population sizes at the linked loci due to background selection. A simple solution can be obtained as E[π]=∫012y(1−y)φi(y)he−τ∕2N2dy+2Nμ∫0τ212N(1−12N)e−t∕2N2dt, (12) where h = h(r₁, r₂) and N₂ = Nf_B(r₁ + r₂), as defined above, and ϕ(y) is the distribution of mutant allele frequencies immediately before the hitchhiking event. The first and second terms represent the heterozygosities contributed by neutral mutants that appeared before and after the hitchhiking event, respectively. Assuming that the equilibrium level of genetic variation has been attained before the hitchhiking event and background selection does not change the shape of the distribution of mutant frequencies from that of neutrality, ϕ(y) can be replaced by θ/y, where θ = 4N₂μ. Mean heterozygosity determined by ϕ(y) then becomes reduced by h and decays further by e^−τ/2N₂. The second term is derived similarly. Then, (12) reduces to E[π]=θ{1−(1−h)e−τ∕2N2}. (13) This equation describes the recovery of genetic variation as a function of time. It generalizes results obtained by previous studies of genetic hitchhiking (Wiehe and Stephan 1993; Perlitz and Stephan 1997) and population bottlenecks (Tajima 1989). In contrast to those studies that were derived under the assumption of neutrality, (13) takes background selection into account.

Simulation results: We introduced 2 × 10⁷ M alleles independently in a diploid population of a size 2N = 10⁵, as described above, and observed its frequency at T = 0 (Figure 2 and Table 3). The frequency distributions of allele M at T = 0, before and after the hitchhiking event, are shown in Figure 2. Comparing the observed and expected distributions before hitchhiking, it is clearly seen that background selection does not change the shape of the frequency distribution of a linked neutral locus. Significant excess of low-frequency alleles was not observed. Immediately after the hitchhiking event, however, the number of intermediate-frequency alleles was greatly reduced, and that of high-frequency alleles increased significantly. This effect was previously observed by J. Fay and C.-I Wu (personal communication). We further discuss this observation below.

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Figure 2.

Allele frequency distribution of M before and after a hitchhiking event. Shaded bars represent the frequency data obtained at T = 0 in simulation 3-2 of Table 3 (before hitchhiking). Small squares connected by lines show the expected number of M's segregating in each frequency interval. The expected number of M's segregating in the frequency interval (y, y + dy) is assumed to be θ/ydy, where θ = 4N₂μ. μ is determined by simulation, as explained in the text. Then, the expected number of M's in a frequency interval (i/20, (i + 1)/20], i = 0, … , 19, is given by . Solid bars represent the data obtained in simulation 3-3 (immediately after hitchhiking).

Table 3 summarizes the simulation results for heterozygosity (π), observed immediately after the hitchhiking event (except 2-2). Predicted values (using Equation 13) agree well with the simulation results. Figure 3A also shows that (13) accurately predicts heterozygosities at various time points after the hitchhiking event. For τ = 1, the average reduction of heterozygosity by a hitchhiking event, predicted by h (Equation 3), simply corresponds to π/θ. π/θ does not change significantly from 2-1 (0.48) to 2-3 (0.48) or from 2-4 (0.065) to 2-5 (0.067), which implies that the reduction of the effective population size at Fav by background selection does not weaken the effect of a single hitchhiking event, at least when s = t. This result differs from model 1, where the effect of hitchhiking decreased as the effect of background selection increased. This discrepancy is caused by the fact that N₂ and Φ are significantly reduced by background selection but h is relatively insensitive to changes of α₁ = 2N₁s (discussed above).

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Figure 3.

Changes of heterozygosity, homozygosity, and fixation rates at Neu over time after a hitchhiking event. The parameters are 2N = 10⁵, L = 15,000, τ = 10,000, s = t = 0.02, u/t = 0.2, r₁ = 10⁻³, r₂ = 10⁻⁴, and number of M's introduced = 10⁸. (A) In addition to T = 0, the frequency of M was recorded every 500 generations. Mean heterozygosity and homozygosity were calculated at each generation. Solid squares represent observed heterozygosities. Lines were drawn for the expected values of heterozygosity using Equation 13. Shaded squares represent homozygosity. (B) Whenever the M allele is fixed, the time of this event was recorded. The histogram shows the number of fixation events at each time interval. The interval between 0 and 0.2N generations after the hitchhiking includes the fixation events occurring during the substitution of B.

We also investigated the perturbance of the mutation-selection balance at Del during the substitution of B at Fav and its effect on heterozygosity after the fixation of B. It was previously observed that the frequency of the deleterious allele, p_a, deviates most from its equilibrium value, u/t, when the increase of the frequency of B, p_B, is greatest. p_a returns toward u/t after p_B exceeds 0.5 (data not shown). We thus recorded p_a in the first generation after p_B became >0.5 during the substitution process of B. When s ≤ t, p_a remained very close to u/t. When s = 0.02 and t = 0.005 (2-7 and 2-8), the mean and standard deviation of p_a increased significantly, as expected. However, the observed and expected value of π still agreed very well. To further investigate the change of p_a, we conducted additional simulations in which B was introduced in initial linkage either with A or with a (2-7a and 2-7b and 2-8a and 2-8b). The initial linkage with A did not change p_a significantly. However, the linkage with a greatly elevated p_a. Surprisingly, mean heterozygosities after the hitchhiking event were relatively close to each other despite a large difference in p_a during the selective phase, although small increases in heterozygosity for the case of initial linkage with a were observed (2-7b and 2-8b). If the increase of the deleterious allele frequency caused the reduction of effective population size at linked loci, it would have resulted in a lower heterozygosity after the fixation of B. However, the result is in the opposite direction. One might argue that the reduction of effective population size will weaken the strength of directional selection and thus result in higher values of π. However, it was shown above that a decrease of α₁ does not change h significantly (2-4 and 2-5). Therefore, the agreement of the observed value of heterozygosity with its prediction needs to be further explored, as the underlying assumption—constant p_a during the selective phase—is violated (see discussion).

Finally, we investigated the increase of homozygosity of derived neutral alleles after a hitchhiking event. We observed the level of homozygosity of M at various time points before and after the hitchhiking event (Figure 3A). Homozygosity increased sharply immediately after hitchhiking. However, it dropped quickly and, after a short time (<0.5N generations), the homozygosity/heterozygosity ratio decreased below its standing level before the hitchhiking event. When we decreased the strength of the hitchhiking effect by increasing r₂ from 10⁻⁴ to 10⁻³, the immediate increase of homozygosity was smaller than shown in Figure 3A, but the decrease of homozygosity over time was slower than in Figure 3A (data not shown). The rapid change of homozygosity over time implies that the high-frequency alleles produced by hitchhiking are quickly fixed in the population. We confirmed this by recording fixation events of allele M over time (Figure 3B). There was a great increase of fixation events during and shortly after the substitution of B. As hitchhiking events cannot change the average substitution rate of neutral alleles (Birky and Walsh 1988), the transient increase of the fixation rate should be followed by a period of a low fixation rate. Indeed, we observed a reduction of fixation events following the period of a high fixation rate (Figure 3B). The same pattern was observed when we replaced the hitchhiking event with a population bottleneck in the simulation (data not shown).