© 2005 Society of Systematic Biologists
A Tale of Two Processes
Edited by Jack Sullivan
1 Institute for Molecular BioSciences, Allan Wilson Centre for Molecular Ecology and Evolution, Massey University Palmerston North, New Zealand; E-mail: P.J.Lockhart{at}massey.ac.nz
2 Biomathematics Research Centre, Allan Wilson Centre for Molecular Ecology and Evolution, University of Canterbury New Zealand; E-mail: M.Steel{at}math.canterbury.ac.nz
Received March 13, 2005; Revised April 21, 2005; Accepted May 3, 2005 The "long-branch attraction" (LBA) phenomenon in phylogeny reconstruction is well cited but its causes have been poorly characterized. In this article, we point out that different biological processes can lead to similar forms of long-branch attraction. That is, although sequences generated by different processes look similar "through the eyes of parsimony," the ensemble of sequence site patterns (not just the parsimony sites) can distinguish between these processes.
In 1978, Felsenstein described an evolutionary scenario under which unequal amounts of change in nonadjacent lineages would mislead tree-building methods based on parsimony (or on uncorrected distances). Other authors have since shown that when substitution models are misspecified, maximum likelihood and distance-based methods can be similarly misled (e.g., Hillis et al., 1994; Lockhart et al., 1996; Bruno and Halpern, 1999; Swofford et al., 2001; Sullivan and Swofford, 2001; Ho and Jermiin, 2004). LBA problems may also arise because of sparse and/or unbalanced taxon sampling (Hendy and Penny, 1989; Holland et al. 2003; Lockhart and Penny, 2005) and/or because of lineage-specific differences in rates or processes of evolution (e.g., Hasagawa and Hashimoto, 1993; Steel et al., 1993, 2000).
Felsenstein (1978) assumed that whilst evolutionary rates varied across a tree, individual sites in sequences could be ascribed a rate of change that was the same at other sites in the same sequence. That is, if a lineage was fast (or slow) evolving, then the evolution of sites in a sequence belonging to that lineage was also fast (or slow). A site position that has evolved under this scenario can be seen as a special case of "heterotachy." This is a property of individual sequence positions, which literally means different speeds. It is the concept of sequence evolution at a given site undergoing substitution at different rates in different parts of the tree (Lopez et al., 2002). Interestingly, Simon et al. (1996) have also described this phenomenon and referred to it as "mosaic evolution." It is important to note that variation in the substitution rate of a site throughout the tree is distinct from rate variation across sites (as modeled, for example, by a gamma distribution). In the latter case there is a site-specific substitution rate that varies randomly across the sites, but at any site it applies equally to all the branch lengths of the tree (the branch lengths at the site are all multiplied by the site-specific rate). Consequently, the ratio of substitution rates on two different branches is constant across sites in such models, even when one allows both rate variation in the tree (as in Felsenstein's scenario) as well as an independent process of rate variation (e.g., gamma distribution) across sites. In contrast, with more general forms of heterotachy, the ratio of substitution rates on different branches of the tree may vary across sites.
The special case of heterotachy assumed by Felsenstein (1978) is different from another very special type of heterotachy recently explored in simulations by Kolaczkowski and Thornton (2004). These authors envisaged a four-taxon tree, for which the external lineages evolved in such a way that sites in the sequences accumulated substitutions at one of two rates, either slow or fast. As Spencer et al. (2005) point out, the frequencies of patterns expected under the simulation model studied by Kolaczkowski and Thornton (2004) are a small snapshot of the full range of possibilities when all possible combinations of short and long branches are considered, and most of these do not cause LBA (further concerns regarding the findings of Kolaczkowski and Thornton [2004] have been discussed by Steel, 2005).
Thus, the patterns that Kolaczkowski and Thornton (2004) studied are different from those expected under the standard stationary covarion (or "covarion drift") models, which have been the subject of much recent study (e.g., Tuffley and Steel, 1998; Penny et al., 2001; Gaucher et al., 2001; Huelsenbeck, 2002; Galtier, 2001; Misof et al., 2002; Inagaki et al., 2004; Ané et al., 2005; Guindon et al., 2004). These standard covarion models have reversible stationary substitution rates among character states that are switched "on" (variable), and a reversible stationary process between the state of "off" (invariable) and "on." The latter condition will maintain the same proportion of variable sites in different lineages, though some models also allow variation in the substitution rate of sites that are "on." Under these standard covarion models, LBA might occur if faster rates of change at variable sites occur in some lineages. In this situation, tree building will be problematic in a manner similar to that previously described (Felsentein, 1978; Lockhart et al., 1996; Bruno and Halpern, 1999; Swofford et al., 2001). Evidence for covarion-like evolution in sequences has been reported by a number of authors (e.g., Fitch and Markowitch, 1970; Miyamoto and Fitch, 1995; Simon et al., 1996; Lockhart et al., 2000; Gaucher et al., 2001; Penny et al., 2001; Huelsenbeck, 2002; Misof et al., 2002; Inagaki et al., 2004; Ané et al., 2005; Brown, 2005).
The property of heterotachy might also arise in data if there is a change in the proportion of sites free to vary in different lineages. Such a property of data is presumably the basis of "covarion shifts" (Inagaki et al., 2004) and also an explanation for why, in the sequences of some taxa, there are relatively more positions with "shared-ancestral" character states than in others (Stiller et al., 2001). Concern that the proportions of variable sites can differ in biological orthologues was an initial motivation for the word "heterotachy" (Philippe and Germot 2000; Lopez et al., 2002). The phenomenon is a concern for phylogeny estimation, because if proportions of variable sites do change in different evolutionary lineages, then misleading parsimony patterns will arise similar to those that arise under the scenario described by Felsenstein (1978). Although we pointed this out previously (Lockhart et al.1998), we have only recently recognized that considering the ensemble of pattern frequencies will distinguish between the evolutionary model studied by Felsenstein (1978) and an evolutionary model in which proportions of variable sites differ between lineages. We contrast these two models as follows.
Model 1
Under this model, sequence sites evolve under a standard Markov process, with all sites evolving according to the same process. Two situations that can lead to long branch attraction are shown in Figure 1 where two nonadjacent branches undergo substitution at a markedly higher rate (1a) or where there is a long edge leading to an outgroup, and one of the two sister taxa has an elevated substitution rate (1b). However, these are just particular cases, and in Model 1 no constraints are imposed a priori on branch lengths.
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Model 2
Under our second model we assume that there are two classes of sites. Proportion a evolve on the tree as in Model 1, but with branch lengths that satisfy a molecular clock. The remaining proportion (1–a) of the sites evolve on the same tree and with the same branch lengths, except that on certain branches the sites are invariable (which can be modeled by setting those particular branch lengths equal to zero). For a four-taxon tree 12 vert 34, we assume that the branches on which the sites are invariable include at least one branch that connects taxa 1 and 2, and at least one branch that connects taxa 3 and 4. We also assume that a and 1–a are both strictly positive (since if a = 0 or 1, then Model 2 is just a special case of Model 1).
A special case of Model 2 is the analogue of (1a), which is shown as the mixed process of (2a) in Figure 1, where dotted edges show branches on which the second class of sites are invariable—this also leads to long-branch attraction and causes parsimony and uncorrected distance methods to infer the same incorrect tree as (1a). Indeed, if one just considers parsimony informative patterns or pairwise distances, the models appear to be essentially similar, yet we will see shortly that consideration of the full site pattern distribution allows them to be distinguished. The Model 2 analog of (1b) is (2b) shown in Figure 1—however, in contrast to what happens in the Model 1 setting, (2b) does not lead to long-branch attraction. That is, maximum parsimony (and uncorrected distances) will converge on the correct tree for (2b) (for parsimony this follows immediately from the linearity of the parsimony-score function and the observation that the site patterns produced by the tree on the right of (2b) are not parsimony patterns; a similar simple argument applies for uncorrected distances). As with Model 1, (2a) and (2b) are just two particular instances of Model 2.
The following result shows that we can distinguish between these two models (1 or 2), provided we have enough data.
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Proposition |
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Suppose sequence sites evolve from either Model 1 or Model 2 and where variable sites evolve according to a simple (group-based) substitution models (e.g., Jukes-Cantor, Kimura 3ST). Then the model type (1 or 2) can be identified from sufficiently long sequences.
Proof
Under either model, consider the event E12 of a change of state between taxon 1 and 2 and the event E34 of change of state between taxa 3 and 4. Under Model 1 (and an underlying group-based process), it is well known (see e.g., Semple and Steel, 2003) that E12, E34 are statistically independent, so P[E12& E34] = P[E12]x P[E34].
In contrast, for Model 2, we claim that P[E12 & E34] > P[E12]x P[E34]. To see this, note that in Model 2 the sites are in two classes—the class A1 in which the site is variable in all branches, and the class A2 for which the site is variable in only some of the branches. For i = 1, 2, we have (for group-based models)
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Because we can estimate the probabilities P[E12], P[E34], and P[E12 & E34] arbitrarily accurately (with sufficiently long sequences) by the proportions of sites displaying these respective three patterns [p12, p34, and p12| 34], we can distinguish between the two models with sufficiently long sequences. This completes the proof.
Note that the ratio p12p34/p12| 34 is also the (Peterson) capture-recapture estimator for the proportion of variable sites under a mixed model in which a proportion of sites are variable (and evolving at a constant rate), whereas the remainder of sites are invariable (Steel et al., 2000). Curiously, although neither Model 1 nor 2 is of this type, the estimator still turns out to be useful since it is capable of distinguishing between them.
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Conclusions |
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TopPropositionConclusionsReferences |
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We have shown that it is possible in principle to distinguish between two possible causes of LBA. The simple and special case that we study is sufficient to demonstrate this principle. However, we note that our finding can also be generalized—for example, with less elegant analysis and formulae to allow an unknown proportion of invariable sites in both models. Also, it is possible to remove the restriction in Model 2 of a molecular clock; we imposed it here to keep Model 2 as simple as possible, and so that it involves no more parameters than Model 1.
Both models (1 and 2) can induce topological biases that will mislead tree building. The form identified by Felsenstein (1978) can be modeled with substitution models that are homogeneous across sites and are currently implemented in maximum likelihood and Bayesian inference software packages. However, a process whereby the proportion of variable sites change across the underlying phylogeny is not currently modeled (Inagaki et al., 2004). Such a model would involve either a mixture of different processes or a nonstationary process (such nonstationary processes have previously been applied to model the evolution of sequences exhibiting base composition variation).
In practice, distinguishing empirical examples where the proportion of variable sites changes from cases where the process of sequence evolution is described by the standard covarion model is complicated by the fact that the latter model can appear to indicate an increase in the proportion of variable sites in groups with larger evolutionary divergence. Nevertheless, we have recently attempted to make an objective test of whether or not proportions of variable sites change among orthologues, and we have reported observations suggesting lineage-specific differences among eubacterial and plastid sequences (Lockhart et al., submitted). In support of this inference we have also noted lineage-specific differences in the occurrence of indels and nonconservative substitutions, consistent with the relaxation of functional/structural constraints in lineages that have also increased their proportion of variable sites. These observations provide an indication of the importance of the relationship between sites that are free to vary in sequences and evolving evolutionary constraints. However, more detailed empirical studies are needed to investigate the degree to which heterotachy is nonhomogenous in nature. Further, the extent to which (a) homogenous maximum likelihood models can accommodate heterotachy in real data and (b) the extent to which nonreversible biological processes can be effectively modeled also requires further study. In respect to (a), the empirical findings that we report in Lockhart et al. (2005) are encouraging and are in contrast to the findings from simulations, reported by Kolaczkowski and Thornton (2004).
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