Distinguishing the Hitchhiking and Background Selection Models

Corresponding author: Hideki Innan, School of Public Health, University of Texas Health Science Center, 1200 Hermann Pressler Dr., Houston, TX 77030., hideki.innan{at}uth.tmc.edu (E-mail)

Communicating editor: M. AGUADÉ

	ABSTRACT

TOP ABSTRACT LITERATURE CITED

A simple method to distinguish hitchhiking and background selection is proposed. It is based on the observation that these models make different predictions about the average level of nucleotide diversity in regions of low recombination. The method is applied to data from Drosophila melanogaster and two highly selfing tomato species.

ONE of the signatures of genome-wide selection is the positive correlation between the amount of polymorphism and recombination rate, which was found in Drosophila (BEGUN and AQUADRO 1992 ), humans (NACHMAN 2001 ), the partial selfer Caenorhabditis elegans (CUTTER and PAYSEUR 2003 ), and several outcrossing and partially selfing plant species (reviewed in NORDBORG and INNAN 2002 ), including wild tomatoes (STEPHAN and LANGLEY 1998 ). Hitchhiking (HH) and background selection (BS) are considered as the most important forces causing this positive correlation. Under the HH model, adaptive fixations of strongly favored mutations reduce the level of variation, because such fixations sweep out neutral polymorphisms in the surrounding region while some of them "hitchhike" with the favored mutations (MAYNARD SMITH and HAIGH 1974 ; KAPLAN et al. 1989 ). The BS model considers negative (purifying) selection against deleterious mutations as the cause of the reduction of the amount of variation (CHARLESWORTH et al. 1993 ). Negative selection works to eliminate deleterious mutations together with linked neutral variants. Recombination is a very important factor in determining the degree of reduction in the amount of neutral variation in both models. The lower the recombination rate is, the more variation is swept out by a single adaptive fixation. The probability that a neutral polymorphism is eliminated by a single deleterious mutation also increases as the recombination rate decreases. Thus, the two modes of selection may explain the positive correlation between the level of variation and recombination rate. There is no doubt that both selection processes occur. The joint action of the two modes of selection also creates a positive correlation between recombination rate and levels of variation (KIM and STEPHAN 2000 ). However, the relative importance of these two models is not well understood and still vigorously debated (reviewed in ANDOLFATTO 2001 ). In this note, we present an approach to distinguish the two selection models on the basis of data of levels of DNA polymorphism and recombination rates.

Theoretical studies have shown that, in a diploid population (with constant effective size N_e) undergoing recurrent hitchhiking events or background selection, the expected degree of reduction in neutral polymorphism (f) is a function of , the recombination rate per site per generation. That is, the expectation of the amount of variation in a region with a local recombination rate is given by

where _neu = 4N_eµ and µ is the neutral mutation rate per generation. On the basis of the work of KAPLAN et al. 1989 and STEPHAN et al. 1992 , WIEHE and STEPHAN 1993 obtained a simple formula for f in the HH model,

(1)

where a is a parameter that depends on the product of the population selection parameter (population size times selection coefficient) and the rate of sweeps per generation. It should be noted that this equation has several assumptions. First, _neu and a are constant over the genome. Second, the equation considers only selective sweeps but other types of selection (e.g., negative selection, balancing selection) are neglected. Third, the local recombination rate has a uniform distribution (i.e., recombination hot and cold spots are ignored). Last, multiple concurrent selective sweeps are not allowed, so that (1) does not hold when recombination rate is very small due to interference among them. KIM and STEPHAN 2003 showed that if the asymptotic value of f, f_|=0, is 0.1, then (1) holds for all values such that f 0.1. Furthermore, if f_|=0 0.1, (1) holds approximately for all values for which f f_|=0.

Under the BS model and the assumption that recombination rate is not extremely low, f is approximately given by

(2)

where u is the deleterious mutation rate per site per generation (HUDSON and KAPLAN 1995 ). This equation considers the effect of negative selection alone and also requires assumptions of constant _neu and u values and a uniform distribution of . It is known that these two Equation 1 and Equation 2 produce similar functional relationships for a wide range of . In Fig 1A, the solid curve represents the function of f for the HH model obtained from Equation 1 with a = 5 x 10^-9, which is an estimate for Drosophila melanogaster (STEPHAN 1995 ). Fig 1A also shows that we can generate a very similar curve using Equation 2 with u = 4 (or 5) x 10^-9 (broken line). In the case of HH, f converges to 1 - a/ when gets large, while for BS f converges to 1 - u/ for large . That means for a = u the two functions are asymptotically identical. Therefore, given a data set of levels of DNA polymorphism and recombination rates, we can fit both Equation 1 and Equation 2 to the data. For this reason, it is impossible to distinguish the two selection models on the basis of these types of data for large values.

View larger version (24K): In this window In a new window Download PPT slide   Figure 1. (A) Genetic variation f as a function of the recombination rate for the HH and BS models. The solid curve is for the HH model obtained from (1) with a = 5 x 10-9, and the broken curve is for the BS model obtained from (2) with u = 4 x 10-9. It should be noted that (1) and (2) are not very good approximations when is very small (those parts of f are shown by thin lines). (B) Genetic variation f as a function of the recombination rate for the HH and BS models in regions of low recombination ( 3 x 10-9). a = 5 x 10-9 and u = 2.4 x 10-9 are assumed. (C–F) Distributions of the correlation of coefficient r under the HH (solid bars) and BS (open bars) models. See text for details.

However, a close examination reveals a difference between the two functions in regions of low (but not too low) recombination rates such that both equations are still valid (see above). Fig 1B shows f under the two models with a = 5 x 10^-9 for the HH model and u = 2.4 x 10^-9 for the BS model. Although the two parameters are chosen to give the same average level of f for 1 x 10^-9 3 x 10^-9, it is evident that the shapes of the two curves are different. The curve of the HH model is convex in this parameter range while that of the BS model is concave, suggesting that polymorphism data from regions of low recombination might be useful to distinguish the two selection models. Focusing on this difference between the two functions, we propose a simple method with which to distinguish the two selection models. The idea is from STEPHAN 1995 , who suggested applying Equation 1 to data by transforming it into the following linear regression formula:

(3)

Suppose now that polymorphism data from n independent loci (DNA regions) are available from a single species. Let _i be the estimated amount of DNA polymorphism () at the ith locus (i = 1, 2, 3, ... , n). Let _i be the recombination rate () at the ith locus. We assume that recombination rates are known. Equation 3 indicates that has a negative linear correlation with /. That is, the correlation coefficient between and / is -1 if _i = _HHi at all the loci, where _HHi is the expectation of _i under the HH model given _i (obtained from Equation 1). On the other hand, we expect that r is relatively close to +1 under the BS model because of the concave behavior of f in regions of low recombination.

In practice, is never exactly the same as the theoretical expectation due to genetic drift. Therefore, we investigate the distribution of the correlation coefficient (r) between and /, taking the variance of into account. First, the distribution of r is investigated under the HH model assuming _i has a normal distribution with mean _HHi and standard deviation (SD) k_HHi, where k is a constant value. A computer simulation is carried out in the following way:

1. Determine _neu, a, and k.

2. Simulate for the n loci. _i is assumed to be a random variable on the basis of a normal distribution with mean _HHi and SD k_HHi. If _i < 0, _i = 0 is set.

3. Calculate the correlation coefficient, r, between and / using the simulated for the n loci.

Steps 2 and 3 are repeated 10,000 times and the distribution of r is obtained. The procedure to obtain the null distribution of r under the BS model is almost identical to this. That is, _i is simulated as a random variable on the basis of a normal distribution with mean _BSi and SD k_BSi, where _BSi is the expectation of _i under the BS model given _i according to Equation 2.

Fig 1C and Fig D, shows the results of the distributions of r. Fig 1C and Fig E, investigates the distributions when regions of high recombination are studied, and Fig 1D and Fig F, is for regions of low recombination. a and u in Fig 1C and Fig E, are the same as in Fig 1A, while in Fig 1D and Fig F, we use the same a and u as those in Fig 1B. We consider n = 20 loci, whose recombination rates are assumed to be _i = (i + 5) x 10^-9 so that the range of is 6–25 x 10^-9 in Fig 1C and Fig E. In Fig 1D and Fig F, _i is assumed to be (i + 10) x 10^-10 so that _i ranges from 1.1 x 10^-9 to 3 x 10^-9. Fig 1C and Fig D, studies a case of small k (k = 0.1), while k = 1 is assumed in Fig 1E and Fig F.

First, we consider Fig 1D, which shows the distributions of r when regions of low recombination are studied and k is small. The distribution of r under the HH model is nearly symmetrical and the average is -0.33, while r under the BS model has a relatively narrow distribution close to +1. This result indicates that r might be a useful summary statistic to distinguish the HH and BS models since the distributions of r are completely different in the two models. However, it should be noted that this method does not work when applied to regions of high recombination. As shown in Fig 1C, the two distributions of r are very similar as expected (discussed above).

The power of this method to distinguish the two models depends on k. Fig 1E and Fig F, shows the results of the same analysis as those in Fig 1C and Fig D, respectively, but k = 1 is assumed instead of k = 0.1. The two distributions of r for the case of low recombination are quite similar, although not identical (Fig 1F). Fig 1E shows the distributions of r when regions of high recombination are investigated. The two distributions are very similar again, except that the means have moved to 0.7. These results suggest that the two selection models can be best distinguished under the following two conditions: (1) Polymorphism data from regions of low (but not too low) recombination are available; (2) the variances of the estimates of variation are sufficiently small.

Next we discuss how this method may be applied to data. Suppose that we have a data set of estimates of from n independent loci and that we know the local recombination rates for the n loci. First, the correlation coefficient between and /, r_obs, is calculated, and then r_obs is compared with the null distributions of r under the HH and BS models. The procedure described above is modified because we need to estimate a, u, and k from the data. That is, step 1 in the procedure should be replaced by the following two steps:

1. 1a. Determine _neu.

2. 1b. For the HH model, find a, which gives the best fit of Equation 1 to the data by a least-squares method, which also gives an estimate of k. In a similar way, for the BS model, u is estimated using Equation 2 together with k.

Then, we can follow steps 2 and 3.

We apply this method to the data of D. melanogaster from ANDOLFATTO and PRZEWORSKI 2001 . We use the 10 X-linked loci with < 7 x 10^-9 [yellow and su(s) are excluded because of too low recombination rates; see Table 2 in ANDOLFATTO and PRZEWORSKI 2001 for details]. Fig 2A shows the observed levels of polymorphism in the 10 loci scaled by _neu, which is assumed to be 0.03 (e.g., ANDOLFATTO 2001 ; ANDOLFATTO and PRZEWORSKI 2001 ). The correlation coefficient of this data set is r_obs = 0.40. Following the procedure described above, we found the best-fit functions of Equation 1 and Equation 2 when a = 2 x 10^-8 and u = 0.8 x 10^-8, respectively (solid and dashed lines in Fig 2A). Fig 2B shows the null distributions of r under the HH and BS models with these estimated a and u. The null distribution of the HH model has a relatively wide range and r_obs is almost in the middle of the distribution. On the other hand, the BS model predicts a narrow distribution of r around +1 and r_obs is too small (P < 0.0001). Similar results are obtained when _neu = 0.02 and 0.04. These results indicate that the observed distribution of the amount of variation in the 10 loci is explained better by a convex function than by a concave one, suggesting that it is very difficult to explain the observation by background selection alone. Hitchhiking might be the dominant force creating the pattern of standing polymorphism on the X chromosome of D. melanogaster. This is consistent with the conclusion of ANDOLFATTO and PRZEWORSKI 2001 , who used a different way to distinguish the two selection models.

View larger version (15K): In this window In a new window Download PPT slide   Figure 2. Application of the test to Drosophila.

Other interesting species to study are partially selfing plants, in which recombination is "effectively" reduced in the whole genome. We use two tomato species, Lycopersicon pimpinellifolium and L. chmielewskii, whose selfing rate (S) is 0.9 (RICK 1966 , RICK 1983 ). The polymorphism data are from MILLER and TANKSLEY 1990 (also summarized in Table 1 of STEPHAN and LANGLEY 1998 ). They studied restriction fragment length polymorphism in 40 loci of nine tomato species including the two highly selfing species mentioned above.

NORDBORG 1997 suggested that population genetics theory for outcrossing species can be easily applied to partially selfing species by rescaling parameters using the inbreeding coefficient, F = S/(2 - S). That is, the recombination rate is decreased to , the effective population size decreases to , and selection intensity increases by a factor 1 + F when the effect of selection is additive. Then, is defined as the reduction of the amount of polymorphism in comparison with the rescaled neutral expectation, . Note that the rescaled parameters are represented by a bar. Thus, plugging these rescaling coefficients into (1) and (2), we can study the joint effects of the two mechanisms, selfing and selection, both of which decrease the level of polymorphism.

Then, we apply our method to L. pimpinellifolium and L. chmielewskii. To avoid the problem that Equation 1 and Equation 2 are invalid when recombination rate is very small, we use only 29 loci of MILLER and TANKSLEY's (1990) data set, excluding loci of very low recombination. and for the 29 loci are according to Table 1 in STEPHAN and LANGLEY 1998 . ranges from 0 to 0.0275 in L. pimpinellifolium and from 0 to 0.0143 in L. chmielewskii. The recombination rates are rescaled to per site per generation values by multiplying them by a factor of 12.1 x 10^-8. This factor results from the fact that the tomato genome size is 950 Mb (SHERMAN and STACK 1995 ; PILLEN et al. 1996 ). The recombination rates for the investigated 29 loci are in the range 1.1 x 10^-8 2.7 x 10^-8 (or 2 x 10^-9 4.9 x 10^-9). Then, we obtain the correlation coefficient r_obs = 0.945 and 0.949 for L. pimpinellifolium and L. chmielewskii, respectively. These very high values of r seem to favor the BS model.

To test this possibility, we investigate null distributions of r under each selection model. We use a relatively wide range of _neu since it is very difficult to estimate _neu for highly selfing species in which the level of polymorphism is reduced in the whole genome. The probabilities that r exceeds the observation (r_obs) are shown in Table 1. These probabilities are relatively low under the HH model for the two species, suggesting that r_obs may be too big to be expected under the HH model, especially for L. pimpinellifolium. P(r > r_obs) seems to be quite robust to _neu. Under the BS model, P(r > r_obs) is relatively sensitive to _neu. This may be because f under the BS model can be either concave or convex depending on _neu. The two models can be distinguished well for _neu values that generate a concave shape of f. r_obs for the two species may be in the acceptable range under the BS model unless a very small _neu is assumed. For example, r_obs = 0.945 of L. pimpinellifolium could be too big even under the BS model if _neu = 0.02 is assumed, but this small value of _neu seems to be quite unrealistic because can be as large as 1.38 x _neu in this highly selfing species. Thus, the results of Table 1 seem to suggest that background selection has played a larger role than hitchhiking in shaping genome-wide patterns of variation in the history of these two tomato species.

In this note, we proposed a method to distinguish the HH and BS models. Since the test looks at whether the level of polymorphism is a convex or concave function of the local recombination rate, we should have data from multiple regions in which the recombination rate is low (but not too low). The test is very powerful when the variance of is low, indicating that should be estimated from sufficiently long regions (such that the variances of are reduced due to intragenic recombination). INNAN et al. 2003 showed that the distribution of from 500-kb fragments on human chromosome 21 is very similar to a normal distribution with a quite small SD. The application of our method to Drosophila and tomatoes led to different results. That is, the HH model is preferred in Drosophila while the BS model could better explain the observation in tomatoes. This might be due to the difference in life style and mating system between animals and plants (e.g., BAUDRY et al. 2001 ). Plant populations (especially highly selfing species) are generally more structured and selective sweeps in such a structured population might not occur as quickly as in a random-mating population (e.g., CHERRY and WAKELEY 2003 ; WHITLOCK 2003 ).

However, there are some potential problems in the application of our method to the currently available data sets:

1. It was not possible to obtain correct estimates of _neu, especially in the highly selfing tomato species.

2. The variance of is relatively large, which decreases the power of the test. Also, our assumption of a normal distribution of may not be adequate. These problems could be fixed for Drosophila and other outcrossing species if data from very long regions of low recombination rates are available, together with data from regions of high recombination to estimate _neu. For highly selfing species, even with such data, it is very difficult to estimate _neu because the level of polymorphism is reduced in the whole genome.

3. The theory assumes an unstructured population of constant size. The relationship between levels of variation and recombination rate should also be studied in other population models (see also ANDOLFATTO and PRZEWORSKI 2001 ).

4. The theory assumes constant values of a and u in Equation 1 and Equation 2 across the genome. Variation in these parameters could increase the variance of the observed amounts of polymorphism, reducing the power of the test. In such a case, problem 2 becomes more serious.

	ACKNOWLEDGMENTS

We thank Yuseob Kim for his stimulating study of the hitchhiking process with interference among adaptive fixations. We also thank M. Aguadé and two anonymous reviewers for comments and suggestions. H.I. is supported by a fund from University of Texas. W.S. is grateful to the Erwin Schroedinger International Institute for Mathematical Physics in Vienna for support during his stay in winter 2002/2003 and to the Deutsche Forschungsgemeinschaft for funding (STE 325/5-1; Schwerpunktprogramm 1127).

Manuscript received April 29, 2003; Accepted for publication August 1, 2003.

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