5 Quantitative Approaches to Phylogenetics

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1 Quantitative Approaches to Phylogenetics Kaila E. Folinsbee. David C. Evans. Jörg Fröbisch. Linda A. Tsuji. Daniel R. Brooks Abstract We review Hennigian, maximum likelihood, and different Bayesian approaches to quantitative phylogenetic analysis and discuss their strengths and weaknesses. We also discuss various protocols for assessing the relative robustness of one s results. Hennigian approaches are justified by the Darwinian concepts of phylogenetic conservatism and the cohesion of homologies, embodied in Hennig s Auxiliary Principle, and applied using outgroup comparisons. They use parsimony as an epistemological tool. Maximum likelihood and Bayesian likelihood approaches are based on an ontological use of parsimony, choosing the simplest model possible to explain the data. All methods identify the same core of unambiguous data in any given data set, producing highly similar results. Disagreements most often stem from insufficient numbers of unambiguous characters in one or more of the data types. Appeals to Popperian philosophy cannot justify any kind of phylogenetic analysis, because they argue from effect to cause rather than cause to effect. Nor can any approach be justified by statistical consistency, because all may be consistent or inconsistent depending on the data being analyzed. If analyses based on different types of data or using different methods of phylogeny reconstruction, or some combination of both, do not produce the same results, more data are needed..1 Introduction Formalized by Willi Hennig in 190, phylogenetic systematics has emerged as a universal and transparent method for generating and evaluating phylogenetic hypotheses. Over the last 0 years, it has developed into a research program ostensibly embraced by a majority of evolutionary biologists who are interested in exploring the patterns and processes of evolution. However, this apparent unity is misleading. As researchers from diverse fields began putting the theory and methods into practice, they approached their studies from different perspectives # Springer-Verlag Berlin Heidelberg 2007

2 168 Quantitative approaches to phylogenetics and using different types of data. This has generated a multitude of quantitative methods and research strategies, the efficiency and validity of which are fiercely debated in the literature. These debates stem partly, perhaps largely, from the nature of phylogeny reconstruction. Unlike much of physics and chemistry (astronomy and astrophysics being notable exceptions), in which experiments in a hypotheticodeductive framework aim to be predictive with respect to spatiotemporally invariant laws, the reconstruction of phylogenies deals with a singular history and thus is descriptive and retrodictive. Attempts to infer causal processes for evolutionary patterns must be based not only on evolutionary patterns of relationships but also on corroboration from independent data (Brooks and McLennan 2002). Despite these obstacles, evolutionary biologists strive to provide as accurate an approximation of evolutionary history as possible. Paleoanthropologists, like all paleontologists, should be especially concerned with the accurate reconstruction of primate evolution, partially because fossils provide some of the most powerful evidence supporting evolution, the unifying theory of biology. More particularly, paleoanthropologists are the curators of information about the most fascinating evolutionary story of all, the story of Us. As a critical part of evolutionary biology, paleoanthropology thus stands to benefit enormously from participation in phylogenetic research programs. Collaborating with researchers in diverse fields and exploring new methods will allow paleoanthropologists to refine their hypotheses and methods while placing them within the larger context of biotic evolution on this planet..2 Fount of stability and confusion: A synopsis of parsimony in systematics The principle of parsimony (Latin parcere, to spare) is also known as the principle of simplicity. The principle is often connected to the English philosopher and Franciscan monk William of Ockham (ca ), who advocated the use of the principle so forcefully that it is also known as Ockham s razor : Pluralitas non est ponenda sine neccesitate (plurality should not be posited without necessity) and non sunt multiplicanda entia praeter necessitatem (entities should not be multiplied unnecessarily). In this sense, the principle represents an epistemological tool or rule of thumb, which obliges us to favor theories or hypotheses that make the fewest unwarranted, or ad hoc, assumptions about the data from which they are derived. This does not necessarily imply that nature itself is parsimonious. Aristotle (30 BCE) articulated an ontological basis for the principle of parsimony, the postulate that nature operates in the shortest way possible

3 Quantitative approaches to phylogenetics and the more limited, if adequate, is always preferable (Charlesworth 196). This sense of the principle postulates that nature is itself parsimonious in some manner. Phylogeneticists have used the term parsimony in both senses, resulting in much confusion and, from our perspective, unnecessary conflict. The most important concept introduced by Hennig (190, 1966) was the stipulation that we should assume homology in the absence of contradictory evidence. Although now known as Hennig s Auxiliary Principle, this concept lies at the foundation of evolutionary theory, [p]erhaps the correct way of viewing the whole subject, would be, to look at the inheritance of every character whatever as the rule, and noninheritance as the anomaly (Darwin 189 p 13), and Mr. Waterhouse has remarked that, when a member belonging to one group of animals exhibits an affinity to a quite distinct group, this affinity in most cases is general and not special (Darwin 1872 p 409). Hennig s argumentation method is clearly intended to maximize hypotheses of homology and minimize hypothesis of homoplasy, which invokes the principle of parsimony by avoiding the assumption of unnecessary ad hoc hypotheses of parallelism. In the Hennigian system, if evolution were parsimonious, all traits would be logically consistent with the true phylogeny there would be no conflicting relationships suggested by any set of traits, that is, there would be no homoplasy. The Auxiliary Principle implies that there will be conflicts in the data, which should be resolved in favor of the hypothesis postulating the fewest number of assumptions of multiple origins (homoplasy) over single origins (homology). Contemporary Hennigians assert that both the Auxiliary Principle and the use of parsimony are logical requirements of any attempt to reconstruct phylogeny; if one were to assert that all similarities were due to homoplasy, there would be no evidence of common descent, and thus no evidence of evolution. Therefore, if one is going to study evolution, one must use a method that is capable of finding evidence of evolution. Likewise, if one is going to invoke the Auxiliary Principle, one must invoke it for all traits, thereby choosing the phylogenetic hypothesis that minimizes the total number of violations of the Auxiliary Principle for a given set of data. In this manner, the Auxiliary Principle is an epistemological tool practically synonymous with the principle of parsimony (Farris 1983; Wiley et al. 1991). Wiley (1981) suggested four main assumption of phylogenetics: (1) evolution occurs and has occurred, documented by the characters of different species; (2) each species is a historically unique mosaic of plesiomorphic, synapomorphic, and autapomorphic traits; (3) before the analysis we do not have knowledge about which characters are homologous and homoplasious; and (4) we do not know beforehand what the phylogenetic relationships are, nor do we know the relative or absolute rates of divergence. The presumption of homology embodied in Hennig s Auxiliary Principle is not an a priori assumption in the sense of a formal 169

4 170 Quantitative approaches to phylogenetics model, because the method is designed in part to recognize all mistakenly presumed homologies as homoplasies. Edwards and Cavalli Sforza (1963, 1964) reconstructed a tree of extant human populations based on frequencies of blood group alleles, using an approach they developed and called the Method of Minimum Evolution. Their studies originally aimed to present a maximum likelihood method for phylogeny reconstruction, but their algorithm for a likelihood approach did not work. Edwards (1996 p 83) later emphasized that [t]he idea of the method of minimum evolution arose solely from a desire to approximate the maximum likelihood solution, that is, from a maximum likelihood model based on the assumption that evolution has been parsimonious. Felsenstein (2004 p 127) characterized the method of minimum evolution as a parsimony method, while at the same time not seeing a direct connection between Hennig s Auxiliary Principle and the principle of parsimony, e.g., [i]t is not obvious how to get from this auxiliary principle to the parsimony criterion (Felsenstein 2004 p 138). This reveals that for Felsenstein and like minded phylogeneticists, parsimony is an ontological issue, whereas Hennigians see it as an epistemological issue. There are two critical distinctions between these positions. The ontological perspective on parsimony requires first that evolution be parsimonious in some manner, usually as defined by certain assumptions and parameters of a model; and second, that the resulting phylogenetic hypothesis be accepted as true so long as the model is accepted as true. Practitioners are thus preoccupied with the accuracy of their results, and believe it is possible to develop means by which their preferred hypotheses can be verified with respect to the true phylogeny. The Hennigian or epistemological use of parsimony does not imply that the evolutionary process itself is parsimonious. In fact, it suggests that evolution has been so complex that we should always expect to find conflicts in the data, which will require the use of a logical decision making principle to resolve. An important corollary of this perspective is that there need be no necessary connection between the most parsimonious hypothesis and truth. Practitioners are thus preoccupied with the empirical robustness of their results. The expectation is that if the most parsimonious hypothesis is not true, the accumulation of additional data will force phylogeneticists to abandon it in favor of a new most parsimonious hypothesis; they do not believe that their hypotheses can be verified, but do believe that they can use new data to falsify all or parts of previous hypotheses. Phylogeny reconstruction is thus an open ended process involving a potentially endless search for information. If, at some point in the future, the accumulation of data has led to a situation in which the phylogenetic hypothesis for a given group is no longer changing with the addition of new data, Hennigians

5 Quantitative approaches to phylogenetics may express the belief that the hypothesis has approached the truth as closely as possible, but in principle it is never appropriate for a Hennigian to claim to have the true phylogeny. Hennigians do feel justified in claiming that they have the most robust hypothesis possible for any set of data. Today numerous quantitative methods for reconstructing phylogenetic trees are applied to multiple kinds of characters. These methods can be divided into two main types, commonly called parsimony (invoking epistemological parsimony) and likelihood (invoking ontological parsimony) approaches Epistemological parsimony: The Wagner algorithm In September 196 two articles on phylogeny and parsimony appeared. Wilson (196) introduced a consistency test for phylogenies based on contemporaneous species. His null hypothesis was that all characters that are used for a phylogenetic analysis are unique and unreversed. In order to pass Wilson s consistency test, the taxa defined by these characters must be nested and these nested conditions must persist as new species are added to the tree. Colless (1966) was concerned that more than one cladogram might pass the consistency test, that a polyphyletic character state might mistakenly be regarded as unique and unreversed, and that the taxa are in the first place grouped on the basis of similarities. Wilson (1967 p 104) asserted that his consistency test was internally sound, but that he shared one of Colless main concerns, which is the lack of efficient methods for selecting the character states. That concern was discussed in the second article published in 196 in which Camin and Sokal (196) presented the first algorithm for applying the parsimony criterion to phylogenetics and first applied the term parsimony to a method of phylogenetic inference. They used a group of imaginary animals (Caminalcules) possessing a number of morphological characters that could change according to particular rules. Thus, the true phylogenetic tree was known and could be compared to trees that were achieved by different methodologies. Camin and Sokal (196) found that the trees that most closely resembled the true phylogeny required the least number of changes in the morphological characters, which seems to invoke an epistemological use of parsimony. However, they claimed that their technique examined the possibility of reconstructing cladistics by the principle of evolutionary parsimony (p 312), saying that the correctness of our approach depends on the assumption that nature is, indeed, parsimonious (pp ), an appeal to ontological parsimony. Significantly, Camin and Sokal produced a computer program implementing their method, demonstrating for the first time that quantitative phylogenetic analysis could be performed on as

6 172 Quantitative approaches to phylogenetics objective a basis as phenetics, thereby undermining one of the strongest arguments in favor of phenetics over evolutionary approaches to systematics (Sokal and Sneath 1963). Their algorithm was unwieldy and inefficient for larger data sets, and was never fully adopted nor effectively programmed and made available for widespread use. Soon afterward, Kluge and Farris (1969; also Farris 1970) presented a new algorithm for reconstructing phylogenetic trees as well as searching among several trees for the most parsimonious tree for a given data set. They named their method Wagner parsimony in honor of W.H. Wagner, who formalized an older approach (Mitchell 1901, 190; Tillyard 1921; Sporne 1949, 193; Danser 190) called the groundplan divergence method (Wagner 192, 1961, 1969, 1980), which formed the basis for Kluge and Farris algorithm. Kluge and Farris (1969) also discussed explicitly their perspective that the use of the parsimony criterion did not assume that evolution itself is parsimonious, clearly invoking an epistemological use of the principle..3.1 The Wagner algorithm Kluge and Farris (1969) method minimizes the Manhattan distance between members of a set of taxa via the creation of hypothetical taxonomic units. The first Wagner algorithm (in later papers termed the simple Wagner algorithm ) is constructed as follows: Definition: X(A, i); state ¼ X, of character i, for Taxon A D(A, B) is the difference between taxon A and taxon B; DðA; BÞ ¼ X i bxða; iþ XðB; iþc ð1þ D(A, INT(B)) is the difference between A and interval B; DðA; INTðBÞÞ ¼ DðA; BÞþDðA; ANCðBÞÞ DðB; ANCðBÞÞ 2 ð2þ Step 1. Choose an ancestral taxon. Step 2. Find the operational taxonomic unit (OTU) that is the least distance from the ancestor, using Eq. 1. Connect the taxon to the ancestor, forming an interval. Step 3. Find the next taxon that is next least distant from the ancestor. Step 4. Find the interval from which the taxon found in step 3 differs least, using Eq. 2. Step. Attach the taxon found in step 3 to the interval found in step 4 by constructing an intermediate (hypothetical ancestor), Y, and insert it into the

7 Quantitative approaches to phylogenetics tree at this interval. The character states of Y will be the median of the three nodes surrounding this newly created intermediate. Step 6. If any taxa remain unplaced, go to step 3, otherwise stop. Example: Using the same four taxon, five character statement as was used in the groundplan divergence example (> Table.1) 173. Table.1 Exemplar of a four taxon, five character statement K L M N Step 1. As in the example for groundplan divergence, K is determined to be the ancestor. Step 2. Requires us to find the difference between the ancestor (or outgroup) and each of the other taxa using equation 1. DðL; KÞ ¼ P bxðl; iþ XðK; iþc i ¼ b1 0cþ b0 0cþ b0 þ 0cþ b0 þ 0cþ b0 þ 0c ¼ 1 So the difference between L and ancestor K is 1. We must then determine the difference between the ancestor (K), and each of the other taxa in the matrix, using the same equation. DðM; KÞ ¼3 DðN; KÞ ¼4 We find that the difference between L and K is the least, so we connect L to K via an internode, as shown below. This interval will be known as interval L (INT (L)) (> Figure.1): Step 3. The next step is to determine which taxon to add to the cladogram next. It is the one that is the next closest (least different) from ancestor K. In this case, it happens to be taxon M (with a difference of 3 from above). Step 4. To figure out the interval to which we can add M, we must calculate the distance between M and each of the intervals on the diagram. At this point, there

8 174 Quantitative approaches to phylogenetics. Figure.1 Interval between taxon L and taxon K. Figure.2 Three taxon cladogram with hypothetical ancestor Y is only one interval so it is self evident to which interval the taxon will be added, but the calculation is: DðM; LÞþDðM; ANCðLÞÞ DðL; ANCðLÞÞ DðM; INTðLÞÞ ¼ 2 ¼ 2 þ ¼ 2 Step. Taxon M is then added to INT(L) via a hypothetical intermediate, Y (> Figure.2). Hypothetical intermediate Y is given the character states that are the medians of those taxa to which it is connected (L, M, and K). It is then added to the character matrix as in > Table.2: Step 6. The next taxon to be added to the diagram is the last remaining one, N. We now must determine the interval to which we must add N. To do this, we find the difference between N and each of the intervals. INT(M), INT(L), and INT(Y) using equation 2.

9 Quantitative approaches to phylogenetics 17. Table.2 New character matrix including hypothetical ancestor Y K Y L M N DðN; INTðMÞÞ ¼ 2 DðN; INTðLÞÞ ¼ 2: DðN; INTðYÞÞ ¼ 3 N is found to differ least from INT(M), so it is added to that interval via a hypothetical intermediate, X, whose character states are the median of M, N, and Y. The network is now complete (> Figure.3). Farris (1970) concluded that it was unnecessary to have an ancestor from which to begin the construction of the tree. He observed that the choice of. Figure.3 Cladogram of relationships between taxa in > Table.2 ancestor of a given group of taxa changed the topology of the tree. Since the simple algorithm did not impose directionality to the evolution of the group, he reasoned that the choice of ancestor is not crucial. Since parsimony assumes the least about the way evolution works, then choosing one taxon as an ancestor would be assumption about the status of that taxon. He thus argued that a rootless network would reduce the dependency of the form of the tree on the

10 176 Quantitative approaches to phylogenetics ancestor. For the creation of networks, he used a method for creating networks that minimized the length of the intervals between taxa (symbolized by nodes), using the shortest network connections method of Prim (197; Sokal and Sneath 1963). Farris differentiated his use of this method from previous phenetic applications by its use of shared, derived characters, and also by the evolutionary implications of the method. This new Wagner algorithm differed from that of the simple Wagner algorithm as follows: 1. Find the pair of OTU s that differs the most (using equation 1 from above). 2. Compute the advancement index of each taxon from the interval formed between the two initial taxa. 3. Take the taxa with the largest advancement index and add it to the interval via an HTU. 4. Find the next unplaced OTU with the largest advancement index, find the interval from which it differs least. This produces a network, rather than a tree, and does not assume that any of the taxa are ancestral. Farris suggested that the network could be converted into a phylogenetic tree by rooting it at one of the taxa within the tree, or an interval within the network. Completing the process of constructing phylogenies using this method requires that the characters be optimized onto the tree. The earliest programs implementing the Wagner algorithm did not necessarily find the most parsimonious tree for large data sets. The program needed to run multiple times and have a method of comparison in order to determine whether it has indeed found the shortest tree, or if there were multiple equally parsimonious trees. In a large matrix, examining every possible tree could require an enormous amount of computer time, and thus it became necessary to develop heuristic methods to try to find the shortest tree. Today s parsimony programs, such as those in PAUP, Hennig 86, and NONA, use a variety of heuristic algorithms to rerun the data to attempt to ensure that the most parsimonious tree or trees are found. For small numbers of taxa and characters, the Branch and Bound algorithm (Hendy and Penny 1982), which guarantees finding the shortest tree, or the Exhaustive Search option, which enumerates all possible trees, can be employed. As phylogeneticists began to analyze increasingly larger and more complicated data sets, shortcomings in the original computer programs became evident. In the decade following Farris (1970) contribution, a number of algorithms were developed, such as Fitch parsimony (Fitch 1971) and Dollo parsimony (Farris 1977), which were incorporated into the existing programs as alternatives to Wagner parsimony. These differed primarily in their assumptions and restrictions regarding character evolution and are discussed in more detail by Wiley et al. (1991).

11 Quantitative approaches to phylogenetics The first iteration of the Wagner algorithm did not take into account multistate characters, and therefore technically it was not possible to have unordered states, since polarized binary characters are automatically ordered. Initially, before more variations were developed for the algorithm, it was suggested that all multiple character states be divided into multiple binary characters [e.g., a single multistate transformation series of an imaginary character (absent (0), short (1), long (2)) would be divided into two separate characters (absent (0), present (1)) and (short (0), long (1)]. Current algorithms allow for multistate transformation series and allow characters to be run either polarized or unpolarized, and either ordered or unordered, at the discretion of the user. Again, the advantage of phylogenetic methodology is that these decisions are transparent (if they are reported) and repeatable; with the same data set, anyone can rerun an analysis using the same settings to check the reliability of the analysis, or change the settings to see if the results are different. Whatever algorithm you use to build a tree, in most cases some characters will not be decisive at every node (Farris 1970). It is therefore important for the purpose of studying character evolution to be able to optimize characters on a tree. There are two types of optimization, ACCTRAN (Farris 1970; Wiley et al. 1991) and DELTRAN (Swofford and Maddison 1987; Wiley et al. 1991). The ACCTRAN setting accelerates the transformation of a character on a tree, pushing the evolution toward the root. This is equivalent to preferring parallelisms to reversals, if the choice does not affect the tree length. DELTRAN delays the transformation of a character on a tree, essentially choosing reversals over parallelisms when they are equally parsimonious (Wiley et al. 1991). When there are no equally parsimonious alternatives, both ACCTRAN and DELTRAN w ill prov ide the same result ( > Figures.4 and.).. Figure.4 DELTRAN tree (length ¼ 9 steps). Redrawn and modified from Wiley et al. (1991) 177

12 178 Quantitative approaches to phylogenetics. Figure. ACCTRAN tree (length ¼ 9 steps). Redrawn and modified from Wiley et al. (1991).3.2 Development of outgroup comparison As noted above, Wagner algorithm generates a minimum length network (sometimes called an unrooted tree ). In order to convert a Wagner network into a phylogenetic tree, the network must be rooted in some manner. Increasingly, published studies convert the network into a tree by rooting it with an arbitrarily chosen single taxon not included in the group being analyzed (called the ingroup). This protocol should not be mistaken for the method of outgroup comparisons that emerged in phylogenetics during the 1970s. The distinction is slight, but significant, and must be understood in light of Hennig s perspective on the issue of ancestors. Hennig objected strongly to the notion that phylogeny reconstruction could be achieved by reconstructing a series of archetypal ancestors, from which particular descendant species could be derived. His position was that each species was a unique mosaic of plesiomorphic and apomorphic traits. Archetypes, defined as ancestral species exhibiting only plesiomorphic traits, thus did not exist; therefore, no single taxon could be used to determine the plesiomorphic and apomorphic traits for any analysis. Or, using current jargon, rooting a network with a single outgroup taxon is sufficiently robust in the Hennigian system only if that taxon is the archetype ancestor of the ingroup, something the Hennigian system disavows. As can be seen from the discussion above, the early development of the Wagner algorithm was not informed directly by Hennigian reasoning. Rather, it relied on the groundplan divergence method, based on a priori recognition of an

13 Quantitative approaches to phylogenetics archetypal ancestor. When Farris (1970) abandoned the a priori reliance on an ancestor, the Wagner algorithm reverted to a method for producing an unrooted network. Lundberg (1972) made a significant contribution to linking the results of Wagner analyses with Henngian analyses by differentiating ancestors from outgroups. He developed a method to determine an ancestor from within a network from the data within that same network. He opined that the structure of a network makes certain character states more likely to be ancestral, helping to determine which interval should form the root of the tree of a parsimonyoptimized network. The transition of emphasis from searching for ancestors to identifying outgroups was critical in linking Wagner with Hennig. The idea that similarity in traits even among distantly related species was due to homology (i.e., plesiomorphy) rather than independent evolution (homoplasy) was established before the development of Hennigian systematics "...it would in most cases be extremely rash to attribute to convergence a close and general similarity of structure in the modified descendants of widely distinct forms. The shape of a crystal is determined solely by the molecular forces and it is not surprising that dissimilar substances should sometimes assume the same form; but with organic beings we should bear in mind that the form of each depends on an infinitude of complex relations, namely on the variations that have arisen, these being due to causes far too intricate to be followed out, on the nature of the variations that have been preserved or selected, and this depends on the surrounding physical conditions, and in a still higher degree on the surrounding organisms with which each being has come into competition, and lastly, on inheritance (in itself a fluctuating element) from innumerable progenitors, all of which had their forms determined through equally complex relations. It is incredible that the descendants of two organisms, which had originally differed in a marked manner, should ever afterwards converge so closely as to lead to a near approach to identity throughout their whole organisation. If this had occurred, we should meet with the same form, independent of genetic connection, recurring in widely separated geological formations; and the balance of evidence is opposed to any such admission. Darwin (1872 pp ) Despite the fact that the connection between outgroups and the Auxiliary Principle had been around for a long time, there was no codification until the late 1970s. Engelmann and Wiley (1977) were the first to provide a rationale for outgroup comparisons. They pointed out that the reference to species outside the ingroup permits a researcher to distinguish traits that truly conflict with phylogeny (homoplasies) from those that only appear to conflict (plesiomorphies). This in turn creates the possibility that phylogenetic analysis could become testable, at 179

14 180 Quantitative approaches to phylogenetics least with respect to Darwinian concepts. Watrous and Wheeler (1981) expanded on this idea, suggesting a number of rules to determine ancestral states for each independent character on the basis of comparisons with an outgroup taxon. The first algorithm to determine ingroup relationships with reference to multiple outgroups was presented by Maddison et al. (1984), who showed that the most robust outgroup comparisons relied on two or more paraphyletic outgroups. This algorithm is incorporated in the program PAUP to root networks when outgroups are specified. Closely related to the issue of using outgroups to reconstruct ancestral character states are the terms and meanings of global and local parsimony, which were first applied by Maddison et al. (1984). They proposed a two step procedure that measures parsimony locally among the outgroups to determine ancestral states and given that these ancestral states then measures locally within the ingroup. This results in one or multiple ingroup cladograms that are most parsimonious globally, i.e., most parsimonious in the context of related groups. We previously addressed the connection between the Auxiliary Principle and epistemological parsimony. Linking the Auxiliary Principle to outgroup comparisons thus provides a connection, through the Auxiliary Principle, between outgroup comparisons and parsimony. It is the use of outgroups to root the shortest network that makes the Wagner algorithm Hennigian, accounting for high degrees of consistency between Wagner algorithm, groundplan divergence method, and Hennig argumentation of the same data (Churchill et al. 1984)..4 Evaluating the robustness of a parsimony analysis.4.1 Character evaluation As noted in the introduction, Hennigian phylogeneticists are preoccupied with assessing the empirical robustness of their results. There are various methods available to accomplish this goal. These goodness of fit measures are useful indicators of the degree of internal conflict among the data (characters) used. Measuring the robustness of the characters and knowing how they behave over the tree topology is one useful approach. The simplest summary statistic is the tree length; it is merely the number of steps required to produce a particular topology, and it is calculated by adding the number of character changes over the tree (Wiley et al. 1991). Parsimony analysis chooses the tree, or trees with the shortest overall length, given a set of characters. Consistency indices (Kluge and Farris 1969) attempt to quantify the amount of homoplasy on a particular tree. The original form of the consistency index (CI)

15 Quantitative approaches to phylogenetics is the ratio of the total number of apomorphic states to the tree length. A high CI indicates there is little homoplasy (i.e., the tree length approaches the minimum number of steps required) and a low CI indicates there is a high degree of homoplasy. This measure is independent of a particular data set and thus can be used to compare trees produced by different data. However, the CI can be inflated by autapomorphies, which do not represent tests of relationships and thus are not informative with respect to the robustness of the tree. Farris (1989) therefore proposed the rescaled consistency index (RC). The RC is an adjusted version of the CI with the influence of characters that do not change the fit of the tree (e.g., autapomorphies) removed. It still gives a relative measure of the degree of homoplasy on a particular tree topology Tree evaluation A second type of evaluation assesses the robustness of the tree topology itself. Decay analysis (Bremer 1988) determines the number of steps required to collapse nodes. To perform a decay analysis, we increase the tree length by successive steps. This shows how many trees exist that are one or more steps longer than the most parsimonious tree (MPT); if there are a number of trees of similar length to the MPT, but with different topologies, we might place less confidence in the MPT. A decay analysis will also reveal how many added steps it takes to collapse individual nodes, and which specific characters influence those nodes. This in turn allows us to test whether a set of functionally correlated characters influences a particular node. TreeRot (Sorenson 1999) is a computer program that uses PAUP* to perform decay analysis, although it is possible (albeit time consuming) to do it manually by successively adding one to the tree length in the search parameters. Hennigian analysis of a data set may produce more than one tree with the same number of fewest possible steps, a phenomenon known as multiple most parsimonious trees. With a large number of taxa and characters, especially if they contain large amounts of homoplasy or missing data, parsimony frequently generates multiple MPTs. In these cases, it is not possible to designate a single preferred tree; however it is possible to generate a variety of consensus trees to delineate similarities in topologies of different MPTs (Adams 1972; Wilkinson 1994). There are different techniques to build consensus trees that combine the topological information from two or more trees to create a new summary tree. Strict consensus trees only include monophyletic groups that appear in all of the input trees, and thus usually result in a number of polytomies (Sokal and Rohlf 1981).

16 182 Quantitative approaches to phylogenetics Adams consensus trees are slightly more inclusive, purporting to give the most resolution between a set of trees (Adams 1972). These may, however, produce groupings not found in any of the input trees. Majority rule consensus trees are perhaps the most lenient and frequently used summary tree technique (Margush and McMorris 1981). They are created by building a new tree that contains all monophyletic groups that are supported by a majority of the set of input trees. This means that they may be logically inconsistent with the information produced by one of the MPTs. Consensus techniques are useful as visual summaries of points of agreement or logical consistency between MPTs, but they are not phylogenies, and they are not equivalent to what is produced by phylogenetic analysis of a data matrix. Occasionally, the consensus tree will be one of the MPTs in which case it is a summary tree as well as a phylogeny. Usually, the creation of consensus trees results in the creation of a number of polytomies, or nodes in which the relationships between taxa are unresolved. These are known as soft polytomies when they are created by lack of resolution due to insufficient data or methods. Hard polytomies are actual speciation events, in which a population divides simultaneously into three or more descendent species, or in which two sister species hybridize, forming a third species (Brooks and McLennan 2002). These are impossible to distinguish from soft polytomies with a phylogeny alone. Bootstrap and Jackknife analyses attempt to estimate the degree of sampling error in the original data set, by attempting to place confidence intervals on phylogenies by making inferences about the variability in the data set. Bootstrapping (Felsenstein 198a) samples the data set with replacement, that is, it allows for some characters to be sampled more than once, and some not to be included at all and constructs a new data set with the same number of characters. PAUP* (Swofford 1998) and other programs construct a series of these, and build a majority rules consensus tree that summarizes the results of the resampled data. The number of times a particular group is included in the set of trees that form the consensus is an estimate of the reality of that group, in that the process has measured the amount of variation between the newly sampled data sets. The bootstrap, then, is a measure of the confidence we can place in each node of the tree, like the decay index. Felsenstein (198a) suggested that a bootstrap value of 9% or greater offers statistically significant support for a clade. There are a number of caveats of which we must be aware before placing too much faith in the numbers generated by this analysis. Since the bootstrap measures the variation in one set of data, it does not allow us to choose between trees built from different data sets. Felsenstein (198a) stipulated that a bootstrap assumes characters are independent and equally distributed. He was explicit that the bootstrap indicates repeatability of an analysis given the data, and should not

17 Quantitative approaches to phylogenetics imply the phylogenetic accuracy of a tree (Soltis and Soltis 2003). It will also be affected by biases such as long branch attraction (Swofford 1998). The jackknife is another mode of evaluation, similar to the bootstrap in that it estimates variability in the data set. It is a procedure to resample data by deleting a certain number of characters [either half (Felsenstein 198a) or another fraction (Farris et al. 1996)] and resampling the data without allowing characters to be duplicated. Characters are randomly and independently deleted from the original matrix to create a new resampled matrix, and like the bootstrap, many matrices are produced and the results are compiled into a consensus tree. For a review of the different kinds of jackknife resampling (delete half, parsimony, weighted) and the assumptions and problems with each see Efron (1979), Wu (1986), and Farris et al. (1996) Ontological Parsimony: Maximum likelihood and Bayesian likelihood..1 A precis of maximum likelihood in phylogenetics Microbiology made significant progress in the late 190s when the first proteins were sequenced. Molecular data were soon realized to be an important source of phylogenetic information useful in inferring evolutionary relationships (Sneath and Sokal 1973; Neyman 1974). Edwards and Cavalli Sforza (1963, 1964) first explored the idea that likelihood could be applied to phylogeny reconstruction. Edwards and Cavalli Sforza (1964; Cavalli Sforza and Edwards 1967) later described a likelihood method for phylogenetic inference using blood group allele frequency data in human populations. Neyman (1974) was the first to apply likelihood analysis to nucleotide sequences, and presciently suggested that this approach might become important in the future. Farris (1973) and Felsenstein (1973) published likelihood algorithms for phylogeny reconstruction, however problems of computational difficulties continued to limit a likelihood method for phylogenetic inference to the theoretical rather than practically operational. Felsenstein (1981b) introduced the first computationally efficient maximum likelihood algorithm for discrete character nucleotide sequence data. Just as the Wagner algorithm became the algorithm of choice for quantitative Hennigian analyses, nearly all phylogenetic applications of maximum likelihood are adapted from Felsenstein s early work. Since then, maximum likelihood methods have become increasingly popular in phylogenetic studies (Swofford et al. 1996; Huelsenbeck and Crandall 1997; Tuffley and Steel 1997; Felsenstein 2004). These approaches are most commonly used in molecular phylogenetics (Swofford

18 184 Quantitative approaches to phylogenetics et al. 1996; Huelsenbeck and Crandall 1997; Huelsenbeck et al. 2002; Ronquist 2004), but morphology based and combined likelihood and Bayesian methods have been proposed and are being refined (Lewis 2001; Nylander et al. 2004; Ronquist 2004)...2 Likelihood methods Several methods for inferring phylogenies from nucleotide sequence data are available, resulting in an often heated debate among evolutionary biologists over the best way to approach phylogeny reconstruction (Goldman 1990; Penny et al. 1992; Swofford et al. 1996; Huelsenbeck and Crandall 1997; Steel and Penny 2000). Maximum likelihood methods evaluate a hypothesis of evolutionary relationships using a presumed model of the evolutionary process and evaluate the probability that it would give rise to the observed data, which is typically DNA sequences of the terminal taxa (Felsenstein 1973, 1981b, 2004; Swofford et al. 1996; Huelsenbeck and Crandall 1997). It should be noted that there are several different types of the likelihood (Steel and Penny 2000; Goloboff 2003). Phylogenetic likelihood approaches use maximum average likelihood, a form of maximum relative likelihood (except Farris 1973, that adopted evolutionary pathway likelihood), and only this form applies to the discussion below (Steel and Penny 2000). The likelihood of a hypothesis (Fisher 1922) is the probability, P, of the data (D), given the hypothesis (H): L ¼ PðDjHÞ The likelihood of a parameter is proportional to the probability of the data and it gives a function that usually, but not always, has a single maximum value, which Fisher called the maximum likelihood. The likelihood does not estimate the probability of the hypothesis, which is assumed to be true in the likelihood formulation (Eq. 3). Likelihoods are calculated for possible tree topology, given the data and assuming a particular model of molecular evolution (Felsenstein 1973, 1981b, 2004; Swofford et al. 1996). In the likelihood equation, Eq. 3, the hypothesis, H, contains three distinct parts: (1) a mechanism or model of sequence evolution, (2) a tree or a hypothesis of relationships, and (3) branch lengths (Penny et al. 1992). For a given data set, likelihoods are calculated for each of the possible tree topologies, or a sample of them, and the tree topology with the highest overall likelihood is the preferred phylogenetic hypothesis. The number of possible tree topologies increases with the number of terminal taxa included in the analysis. ð3þ

19 Quantitative approaches to phylogenetics This can be computationally laborious if the data set is large, and especially if the maximum likelihood model uses rooted trees in its calculus. However, the most general and most commonly used models in molecular analyses are time reversible (Rodriguez et al. 1990; Swofford et al. 1996). With a time reversible model the probability of character state change from state i to state j is the same as the probability of state change from state j to state I (Felsenstein 1981b). Under this condition the likelihood of the tree does not depend on the position of the root, and the use of unrooted networks greatly reduce the total number of trees to be evaluated, and decrease computation time (Rodriguez et al. 1990; Swofford et al. 1996). In a four taxon statement, there are three possible unrooted networks (> Figure.6): 18. Figure.6 The three potential unrooted networks of the four taxon, four character statement For each network, likelihoods are evaluated on a site by site basis. The probabilities (P ef ) of all possible combinations of character states at the internal nodes are calculated for each site (i.e., each character) using a specified model of molecular evolution and estimated branch lengths (see below). The likelihood of the site is the sum total of all the probabilities for all the possible combinations of character states at the internal nodes of the network (Felsenstein 1981b; Swofford et al. 1996; Huelsenbeck and Crandall 1997). In the four taxon case, there are two internal nodes and therefore 16 possible combinations of character states and 16

20 186 Quantitative approaches to phylogenetics probabilities for each site (when gaps are not considered) (Eq. 4). It is clear that some of these combinations, or scenarios, are more plausible than others given the data at the leaves. However, every combination of states at the internal nodes is theoretically possible and each is therefore given a probability (> Figure.7). L site ¼ P AA þ P AC þ P AG þ P AA þ P CA þ ð4þ. Figure.7 Illustration of the possible character states at the internal nodes for the four taxon, fourcharacter statement The overall likelihood of the network or tree is calculated as the product of the site likelihoods (L siten ) because each site is assumed to evolve independently of one another (Eq. ). L tr1 ¼ L site1 L site2 L site3 L siten Theoretically, this procedure is repeated for each possible unrooted network, although there are pruning algorithms that do not require every tree to be considered thereby improving computational efficiency in large data sets (Felsenstein 1981b; Swofford et al. 1996). The network with the highest overall likelihood is the maximum likelihood result, and the preferred phylogenetic hypothesis. It is the network topology that maximizes the likelihood function for the data given the specified model (Felsenstein 1973). It is possible that the network represents only a local maximum, or that it is one of a larger number of equally likely networks. It is important to remember that the likelihood of a network is ðþ

21 Quantitative approaches to phylogenetics 187 NOT L ¼ P(HjD), which is the probability of the hypothesis. The network is converted into a tree by rooting it with an outgroup or a molecular clock (Swofford et al. 1996; Felsenstein 2004)...3 Models of molecular evolution Likelihood analyses involve similar assumptions about the evolutionary process as other methods, including that evolution occurs in a branching pattern and is independent in different lineages (Swofford et al. 1996). The probability of a particular combination of character states at the internal nodes of the unrooted network (e.g., Eq. 4.), are calculated using a specified model of molecular evolution, which requires further assumptions about the nucleotide substitution process, including that sequence evolution can be modeled as a random, or stochastic, process (Rodriguez et al. 1990). Substitution models are typically based on a homogeneous Markov process (Rodriguez et al. 1990; Swofford et al. 1996) that assume the probability of a state change at one site does not depend on the history of that site and that probabilities of substitution do not change significantly in different parts of the tree (Felsenstein 1981b, 2004; Swofford et al. 1996). A DNA substitution model is expressed as a table of rates (substitutions per site per evolutionary distance unit) at which nucleotides are replaced by alternate nucleotides known as the Q matrix (Rodriguez et al. 1990; Swofford et al. 1996; Huelsenbeck and Crandall 1997). The term Q ij in the instantaneous rate matrix, Q, represents the rate of change from base i to base j over an infinitesimal evolutionary time period dt (Swofford et al. 1996). The Q matrix in Eq. 7 represents the general time reversible (GTR) model. Change probabilities are determined by the relative rate of change supported by the instantaneous rate matrix Q and estimated branch lengths. Q ¼ mðaf C þ bf G þ cf T Þ maf C mbf G mcf T maf A mðaf A þ bf G þ cf T Þ mdf G mef T mbf A mdf C mðbf A þ df C þ ff T Þ mff T mcf A mef C mff G mðcf A þ ef C þ ff G Þ ð6þ The rows and columns correspond to the bases A, C, G, and T. The mean instantaneous rate factor, m, is modified by the relative rate parameters a, b, c,..., f. The relative rate parameters correspond to each possible base substitution scenario. The rate parameter is the product of m and the relative rate parameters. The remaining terms (f A, f C, f G, f T ) represent the frequencies of the four bases.

22 188 Quantitative approaches to phylogenetics The base frequency parameters, calculated over all taxa, are assumed to be at equilibrium and that the rate of change from one base to another is proportional to the equilibrium frequency (Rodriguez et al. 1990; Swofford et al. 1996). The rates defined in the Q matrix are per instant of time dt, in order to calculate the likelihoods of each site, the probabilities (P ij ) of the possible state changes along a branch length of t (Swofford et al. 1996) must be determined. A substitution probability matrix (sensu Swofford et al. 1996) is calculated using Eq. 7: PðtÞ ¼e Qt Calculation of the exponential is complex, involving the decomposition of the Q matrix into is eigenvectors and eigenvalues (see Swofford et al for references). For the simple Jukes Cantor model, these values are relatively easily evaluated because there are only two probabilities, the probability of a state change and the probability of stasis, such that the transition probability matrix consists of two values, as in Eq. 8: ð7þ P ij ðtþ ¼ 0:2 þ 0:7e mt ði ¼ jþ 0:2 0:2e mt ði 6¼ jþ ð8þ The substitution probability matrix that corresponds to the GRT instantaneous rate matrix Q in Eq. 7 similarly has 12 values, 1 for each of the different substitution rates. The branch lengths are unknown prior to the analysis and must be estimated in the course of the likelihood calculation (Goloboff 2003). Estimation of branch lengths involves an iterative algorithm in which each branch is optimized separately (Felsenstein 1981b; Swofford et al. 1996). Unlike the rate and frequency parameters, branch lengths are specific to a particular tree topology. For each tree, multiple different branch lengths need to be evaluated, and branch lengths must be recalculated for each unrooted tree considered (Penny et al. 1992). Models employed in likelihood analyses make explicit assumptions regarding sequence evolution, the number of which depends on the particular model of sequence evolution used in the particular analysis (Swofford et al. 1996). The general time reversible model is the most general stochastic model of nucleotide substitution. It models base substitution as a random Markov process in which substitution rates are independent among sites, constant in time, equal in two lineages, and that the ancestral sequence base frequencies represent the equilibrium frequencies (Rodriguez et al. 1990). The GTR model has a maximum of 12 different substitution rates (estimated from the data and using the

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