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Genetics, Vol. 168, 2363-2372, December 2004, Copyright © 2004
doi:10.1534/genetics.104.029488
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,2,3
* Department of Statistics, University of Oxford, Oxford, OX1 3TG, United Kingdom
Program in Molecular and Computational Biology, University of Southern California, Los Angeles, California 90089-1340
3 Corresponding author: Molecular and Computational Biology, University of Southern California, SHS 172, 835 W. 37th St., Los Angeles, CA 90089-1340.
E-mail: magnus{at}usc.edu
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ABSTRACT |
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 TOP ABSTRACT BASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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A few clear cases of trans-specific polymorphism have been found, in particular, in the MHC (e.g., FIGUEROA et al. 1988) and plant self-incompatibility loci (e.g., IOERGER et al. 1990). At the same time, studies of sequence variability in several genes that might a priori be considered good candidates for trans-specific polymorphism have failed to find strong evidence for this hypothesis. Examples include the primate ABO blood group system (SAITOU and YAMAMOTO 1997) and red/green color vision polymorphism in New World monkeys and lemurs (BOISSINOT et al. 1998; TAN and LI 1999). In these cases, are the functionally similar alleles in different species examples of trans-specific polymorphism, or are they due to convergent evolution? The purpose of this article is to develop a modeling framework that allows us to address these questions. We focus in particular on our ability to detect trans-specific polymorphism when it exists and how this is determined by the length of the chromosomal region that is affected by the presence of a trans-specific polymorphism.
Throughout, we discuss relatedness in the genealogical sense, i.e., with reference to "descent" rather than to allelic "state." Thus, when we say that two homologous copies of a site (or locus or nonrecombining sequence) are more closely related to each other than to a third copy, we mean that the most recent common ancestor (MRCA) of these two is more recent than the MRCA of either of them and the third copy. This does not necessarily mean that the two copies are more similar to each other than either is to the third copy (although if they are not, we would typically not be able to infer the true relationship).
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BASIC MODEL |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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units of time ago. For simplicity, both descendant populations are assumed to be of the same size as the original one. All populations evolve according to a standard neutral model, the coalescent approximation is employed, and time is measured in units of the effective number of homologous chromosomes in each of the current populations. We consider both selective neutrality and various forms of balancing selection. We consider the following questions for samples of homologous sequence taken from the two species:
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PROBABILITY OF TRANS-SPECIFICITY |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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, and then calculating the probability that these lineages coalesce in the ancestral species in a trans-specific manner. Assuming neutrality, the numbers of surviving lineages in each species at are independent, identically distributed random variables whose distribution is given by TAVARé (1984), and the conditional probability of trans-specificity can be found using the results of SAUNDERS et al. (1984). The expression for the total probability can easily be evaluated numerically: see NORDBORG (2001), for example. Numerous treatments of the probability of trans-specificity exist (e.g., PAMILO and NEI 1988; TAKAHATA 1989; HEY 1994; HUDSON and COYNE 2002; ROSENBERG 2002); our main purpose here is to introduce the basic concepts and to enable comparisons with later results.
Trans-specificity is impossible unless there are at least three ancestral lineages at the time of speciation. If there are two lineages in one species and one in the other, then the probability of trans-specificity is 2/3 (trans-specificity is avoided if and only if the two lineages from the same species coalesce with each other; this happens in one out of three equally likely topologies). The probability of trans-specificity is higher if there are more than three ancestral lineages. Thus the probability of trans-specificity is at least 2/3 given that there are at least three lineages at the time of speciation. The probability that there are at least three lineages at the time of speciation, on the other hand, decreases sharply with and can be vanishingly small. The probability of trans-specificity is thus mainly determined by this latter probability.
To put this into context, consider a sample of size two. The probability that the MRCA of the sample predates speciation is e–. Let T be the time until the MRCA for two genes, and note that under our model, E[T] = 1 for two genes sampled from the same species, whereas E[T] = + 1 for two genes sampled from different species. Since the average number of pairwise differences between sequences is proportional to pairwise coalescence times under neutrality, an estimate of can be obtained as the ratio of the average number of pairwise differences between and within species, –1. For example, if, on average, humans and chimps are at most 99% identical, and humans and humans are at least 99.9% identical, then 
10–2/10–3 – 1 = 9. Let us say = 8 to be on the safe side. Then the probability that the MRCA of a sample from humans predates speciation from chimps would be e–8 = 3.3 x 10–4 (and probably much smaller).
This is a small number, but the genome is large. If there are G sites in the genome, then we expect Ge– to have MRCAs that predate speciation. If we consider the whole population rather than just two copies of the genome, the expected number of sites with MRCAs that predate speciation increases about threefold: the time until there are two ancestral lineages is 
1, so the expected number of sites is Ge–(–1) 
3Ge–. For the ranges of we are interested in, qualitative conclusions are unaffected by sample size. For simplicity, we therefore discuss mainly samples of size two throughout this article.
The probability of trans-specificity for a site depends on whether the site is polymorphic or not. The calculations above assumed no knowledge of allelic state. What is the probability that T > for a polymorphic locus A, i.e., for a sample of two different alleles? We consider the process that keeps track of the number of ancestral lineages in each of the two allelic classes. Denote the state of the process at time t by Xt = (i, j), where i is the number of lineages in the first allelic class, and j is the number of lineages in the second allelic class. The probability we seek is
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= Q/Q0, with
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We consider two cases: unidirectional mutation and bidirectional mutation. For the case of unidirectional mutation, assume that allele A1 mutates into A2 at rate 
/2 and that further mutation in A2 does not change the allelic state (we think of A2 as a loss-of-function allele: this case is motivated by the observation that many examples of balancing selection appear to involve such mutations). Using standard population genetics theory (HUDSON 1990), we find
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For the case of bidirectional mutation, assume that alleles A1 and A2 mutate back and forth at rate /2. Here we find
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These results make intuitive sense: for small , P (1 + )e– in both cases. The probability of trans-specificity conditional on polymorphism is higher than the unconditional probability because the fact that a (rare) mutation must have occurred automatically pushes the time to the MRCA further back in time. For large , P 0 with unidirectional mutation and P e– with bidirectional mutation. In the former case, the MRCA must be recent or all A1 would have mutated to A2, whereas in the latter case, mutations occur so frequently that the allelic states tell us nothing about the age of the MRCA.
The main conclusion from the above discussion, however, is that no matter which model is used, the probability of trans-specificity under neutrality is always very low for large (for recent attempts to estimate it directly, see CHEN and LI 2001; O'HUIGIN et al. 2002). In contrast, if some form of balancing selection is acting, trans-specificity becomes highly probable. Selected alleles will of course also be lost through genetic drift, but this occurs over entirely different timescales (TAKAHATA 1990; VEKEMANS and SLATKIN 1994), and it is easy to imagine strengths of selection that make loss of polymorphism during speciation unlikely even if one believes that speciation is accompanied by genetic bottlenecks (VINCEK et al. 1997). Trans-specificity may therefore, in and of itself, be viewed as evidence for a history of balancing selection. But how do we detect trans-specificity? To consider the traces of trans-specificity in sequence data, we need to know something about its chromosomal extent. This is the topic of the following sections.
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THE EXTENT OF TRANS-SPECIFICITY |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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/2, where = 4Nr, N is the effective population size, and r is the recombination fraction, and consider a sample of size two. The site is trans-specific if no recombination occurs before coalescence at the other site. The probability of this is
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If, as suggested above, 8 for humans and chimps, and per site is 5 x 10–4 (PRZEWORSKI et al. 2000; PRITCHARD and PRZEWORSKI 2001; INNAN et al. 2003), then the probability is >60% for sites separated by 100 bp, but it decreases rapidly to 1% for 1 kb. Linkage to a trans-specific site increases the probability of trans-specificity for tighly linked sites dramatically, but we should not expect large chromosomal regions to be trans-specific (at least not due to linkage).
The probability just derived is an underestimate: the focal site can of course be trans-specific without being identical by descent (with respect to recombination) to the conditional one. Most importantly, whereas two lineages linked to a trans-specific site cannot coalesce before without at least one recombination, a single recombination does not allow them to coalesce unless it occurs between descendants of different trans-specific lineages ("moving" the two lineages into the same trans-specific lineage). The probability of this depends on the frequency of descendants of each trans-specific lineage in every generation back to .
To take this into account, we consider the model of balancing selection first described by HUDSON and KAPLAN (1988) and extended by NORDBORG and INNAN (2003). Imagine that some form of strong balancing selection maintains two alleles, A1 and A2, at a locus. Selection is strong enough to maintain the alleles at frequencies x and 1 – x, respectively. The recombination rate between the locus under selection and the locus of interest is /2, as before. Depending on the allelic state at the former locus, each haplotype belongs to one of two allelic classes. The state of a sample of size two from the focal locus can be described by (z1, z2), where zi denotes the number of lineages belonging to the Ai allelic class. The ancestry of the sample can be described by the Markov process z = (z1, z2) with states (1, 1), (2, 0), (0, 2), (1, 0), (0, 1). Let i = 1, 2, 3, 4, 5 refer to these states in the order given. The rate matrix Q = {qij}i,j of z is
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jqij. The states (1, 0) and (0, 1) are absorbing, and the process starts in (1, 1).
Probability of trans-specificity:
We are interested in P(, x), the probability that two lineages that start in (1, 1) are still distinct at the time of speciation. An exact solution can be found using standard methods. For x = 1/2 we find
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and 
. The solution for general x is highly intractable, but it can be shown that lim
P(, x) = e–, in agreement with Equation 2 and with our intuition for unlinked loci.
Several approximations are possible for 0. We consider two: the first is the best one we found; the second, the simplest.
Approximation 1:
The first approximation is obtained by modifying the rate matrix Q so that the recombination rate is set to zero once the process has left (1, 1). This prevents the process from reentering (1, 1), which simplifies calculations considerably. It is readily verified that this modified process corresponds to the original one in the limit x 0 or x 1, so the approximation is exact for these cases. Using the modified matrix, we find
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Approximation 2:
Assume that the lineages stay distinct if and only if no recombination occurs. This yields
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Equations 2–4 give the probability that two lineages linked to different alleles in a balanced polymorphism stay distinct until speciation. If this happens, trans-specificity is highly probable for the range of parameters in which we are interested ( 0, >> 1): one of the two lineages is likely to coalesce with lineages within the same allelic class from the other species long before a recombination event occurs. Equations 2–4 can thus be seen as approximations of the same probability as Equation 1.
Returning to our human-chimp example, and assuming x = 1/2, we find that Equations 2–4 give probabilities of trans-specificity of 69, 69, and 67%, respectively, for sites separated by 100 bp; and 6, 2, and 2%, respectively, for sites separated by 1 kb. As predicted, Equation 1 underestimates the probability of trans-specificity; however, the results are qualitatively similar. It can be shown that the probability is increased further when x 
1/2: intuitively, this is because the probability of recombination between the allelic classes is maximized when allele frequencies are even. With x = 0.01, Equation 3 gives 70 and 4%, respectively, in the above two cases.
Length of trans-specificity:
Let L(, x) be the length of the region on one side of the site under selection where two haplotypes from different allelic classes still have distinct lineages at time . L(, x) is possibly the total length of a number of disjoint intervals. We have 
. From this, and by considering the properties of P, it follows that for arbitrary 0 < x < 1,
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These equations are useful for evaluating approximations to 
[L(, x)], the exact value of which is not known for any x. The two approximations introduced above can be applied, however.
Approximation 1:
The density of L(, x) can be approximated by
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(u, x) is given by Equation 3, but the expectation cannot be obtained analytically. We refer to this expectation as 1[L(, x)]. Equations 6–8 hold for 1[L(, x)], but Equation 5 does not. Instead we have
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Note that Equation 5 holds if x 0 or x 1.
Approximation 2:
L(, x) is approximately exponential with intensity , Exp(), truncated at , and the expectation is
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(u, x) is given by Equation 4. Equations 6– 8 hold for 2[L(, x)], but instead of Equation 5 we have 
.
Approximation 3:
A third approximation comes from the expected coalescence time for a linked locus. As is discussed further below, this is 1 + 1/ if is large. Solving = 1 + 1/[L(, x)] gives the estimate 3[L(, x)] = 1/( – 1), which should be truncated at if greater than .
Table 1 shows how these approximations perform for a range of parameters. It can be seen that [L(, x)] > 1[L(, x)] > 2[L(, x)] (this can be proved for all r and x). 3[L(, x)] works surprisingly well as long as > 1/( – 1). The approximations can be extended to handle both sides of the balanced polymorphism simply by assuming independence of recombination on each side and multiplying by two.
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in Table 1. This is in agreement with Equation 7 and is perfectly intuitive: larger means more time for recombination to decrease the size of the trans-specific segment. However, we also see that [L(, x)] increases with , in particular for = 5. This may seem paradoxical: if we imagine that the recombination rate per base pair is constant, then increasing simply corresponds to looking at a larger section of the genome. The length of the trans-specific segment surrounding a balanced polymorphism should not depend on how large a section of the genome we look at (unless, of course, we are looking at a region that is too small to contain the entire trans-specific segment, but this hardly explains the difference between = 100 and = 1000). The reason for the increase is that L(, x) includes trans-specific regions that have nothing to do with the trans-specific polymorphism. As noted earlier, there is a small but positive probability that any site is trans-specific. The more of the genome we look at, the more of these we will encounter. The intuitive interpretation of Equation 5 is that, for sufficiently large regions, the fraction of the genome that has not coalesced by is simply e–, which is the probability that a particular site has not coalesced by . The case = 5, = 1000 is approaching this limit: 7.18 x 10–3 e–5 = 6.74 x 10–3. Thus, in this case, most of the fragments that have not coalesced by are not associated with the balanced polymorphism. These fragments may or may not be trans-specific (the probability for each fragment is 2/3), whereas the fragments that are linked to the balanced polymorphism are almost certain to be trans-specific. For = 5, the various approximations give a much better idea of the length of the trans-specific fragment that is associated with the balanced polymorphism than does the exact calculation. For approximations 1 and 2 this is not surprising, given that they are defined in terms of a region contiguous with the selected site. For larger values of , all expectations agree because the probability of noncoalescence that is not due to linkage to the balanced polymorphism is negligible. A slight increase is seen between = 0.1 and = 1: this is due to the former region being too small to contain the trans-specific region with sufficiently high probability.
Simulation results:
The process described here can be simulated, for example, using the algorithm described by NORDBORG and INNAN (2003). One simply simulates two independent realizations of balancing selection for time and then merges the states of the two processes and continues the simulation until all fragments have reached their MRCA.
We used simulations to investigate how well our analytical results concerning [L(, x)] predict the actual length of trans-specificity. Recall that L(, x) is the length of the region on one side of the site of selection where two haplotypes belonging to different allelic classes in a single species still have distinct lineages at the time of speciation, . To obtain the length that is trans-specific in samples, we have to consider L(, x) on both sides of the polymorphism, L(, x) in each species, the probability that lineages distinct at speciation actually coalesce in a trans-specific manner, and samples >2.
By assuming that the lengths of either side are independent, noting that the lengths in different species are independent, and ignoring the final two issues (i.e., we assume that lineages belonging to different allelic classes at speciation will almost always coalesce in a trans-specific manner and that samples >2 will have coalesced to 2 long before speciation for the parameters of interest here), we obtain the following approximation for the expected length of trans-specificity in a region of length surrounding a balanced polymorphism in a pair of species:
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Table 2 illustrates the performance of this approximation for various parameter values and sample sizes. In agreement with the argument just given, the expected length of trans-specificity increases only weakly with sample size. In general, the approximation is quite good, although it overestimates the length slightly. Whether this is due to nonindependence between the two sides or due to some distinct lineages not coalescing in a trans-specific manner is not clear.
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Two selected loci:
Our model can easily be extended to two or more selected sites, using the approach described in NORDBORG and INNAN (2003). This is of relevance because balancing selection may well act to maintain complex alleles that are distinguished by more than a single functionally important mutation (the MHC is a case in point). While it is perfectly possible under this model to obtain analytical results analogous to those presented for the single-locus model, they are too complicated to be useful except in very special cases. In particular, because coalescence between allelic classes in the two-locus model must often involve more than a single recombination event (e.g., for a site located between the selected loci sampled in A1B1 and A2B2), the simple approximations used above do not apply.
Because of this, and also due to space limitations, we content ourselves with showing simulation results that illustrate the main points. Figure 4 shows a straightforward extension of the other examples to two loci. As we would expect, there are now regions of trans-specificity around each selected site. In addition, the variance in time to MRCA in the general region has clearly increased due to the very complex history of recombination among the four haplotypic classes.
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DETECTING TRANS-SPECIFICITY |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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Table 3 shows the results of this study. It is clear that power decreases rapidly with recombination, as would be expected. It also decreases with increased mutation rate. This may seem counterintuitive given that more polymorphism should provide more information about the underlying genealogy. However, more mutation also means increased probability of repeat mutation, i.e., more noise from the point of view of phylogenetic reconstruction.
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DISCUSSION |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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Nonetheless, the basic conclusion that regions of trans-specificity are likely to be quite short seems hard to avoid. Suitable data for testing our predictions are available in primates, for the ABO system (SAITOU and YAMAMOTO 1997) and for color vision genes (SHYUE et al. 1995, 1998; BOISSINOT et al. 1998). Table 4 shows our rough estimates of the extent of trans-specificity surrounding putatively selected sites in these data. The extent of trans-specificity, if that is what it is, seems to be at most a few hundred base pairs.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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FOOTNOTES |
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2 Present address: Human Genetics Center, School of Public Health, University of Texas, Health Science Center, 1200 Hermann Pressler, Houston, TX 77030.
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LITERATURE CITED |
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TOPABSTRACTBASIC MODELPROBABILITY OF TRANS-SPECIFICITYTHE EXTENT OF TRANS-SPECIFICITYDETECTING TRANS-SPECIFICITYDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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