ERROR and UNCERTAINTY in COMPUTATIONAL PHYLOGENETICS by VICTOR HANSON-SMITH a DISSERTATION Presented to the Department of Comput

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ERROR AND UNCERTAINTY IN COMPUTATIONAL PHYLOGENETICS

VICTOR HANSON-SMITH

A DISSERTATION

Presented to the Department of Computer and Information Science and the Graduate School of the University of Oregon in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

December 2011 DISSERTATION APPROVAL PAGE

Student: Victor Hanson-Smith

Title: Error and Uncertainty in Computational Phylogenetics

This dissertation has been accepted and approved in partial fulﬁllment of the requirements for the Doctor of Philosophy degree in the Department of Computer and Information Science by:

John Conery Chair Daniel Lowd Member Sara Douglas Member Joseph W. Thornton Outside Member and

Kimberly Andrews Epsy Vice President for Research and Innovation/ Dean of the Graduate School

Original approval signatures are on ﬁle with the University of Oregon Graduate School.

Degree awarded December 2011

ii c 2011 Victor Hanson-Smith

iii DISSERTATION ABSTRACT

Victor Hanson-Smith

Doctor of Philosophy

Computer and Information Science

December 2011

Title: Error and Uncertainty in Computational Phylogenetics

The evolutionary history of protein families can be diﬃcult to study because necessary ancestral molecules are often unavailable for direct observation.

As an alternative, the ﬁeld of computational phylogenetics has developed statistical methods to infer the evolutionary relationships among extant molecular sequences and their ancestral sequences. Typically, the methods of computational phylogenetic inference and ancestral sequence reconstruction are combined with other non-computational techniques in a larger analysis pipeline to study the inferred forms and functions of ancient molecules. Two big problems surrounding this analysis pipeline are computational error and statistical uncertainty. In this dissertation, I use simulations and analysis of empirical systems to show that phylogenetic error can be reduced by using an alternative search heuristic. I then use similar methods to reveal the relationship between phylogenetic uncertainty and the accuracy of ancestral sequence reconstruction. Finally, I provide a case-study of a molecular machine in yeast, to demonstrate all stages of the analysis pipeline.

This dissertation includes previously published co-authored material.

iv CURRICULUM VITAE

NAME OF AUTHOR: Victor Hanson-Smith

GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED: University of Oregon, Eugene, OR Seattle University, Seattle, WA

DEGREES AWARDED: Doctor of Philosophy in Computer and Information Science, 2011, University of Oregon Master of Science in Computer and Information Science, 2007, University of Oregon Bachelor of Science in Computer Science and Software Engineering, 2003, Seattle University

AREAS OF SPECIAL INTEREST: Evolution, genomic systems, high-performance computing, scientiﬁc software engineering

PROFESSIONAL EXPERIENCE:

Graduate Research Fellow, National Science Foundation IGERT program in evolution, development, and genomics, 2008-2011

Graduate Teaching Assistant, Department of Computer and Information Science, University of Oregon, 2007-2009

Graduate Research Fellow, University of Oregon Neuroinformatics Lab, 2006- 2007

Software Architect, Authora Inc., Seattle, WA, 2003-2005

Undergraduate Fellow, Microsoft Corp., Seattle, WA, 2002-2003

Undergraduate Fellow, Intel Corp., Folsom, CA, 2000

GRANTS, AWARDS AND HONORS:

National Science Foundation IGERT training grant, 2008 - 2011

Intel Foundation Science and Engineering Scholar, 1999 - 2003 v PUBLICATIONS:

G.C. Finnigan, V. Hanson-Smith, T.H. Stevens, and J.W. Thornton. Evolution of increased complexity in a molecular machine In press, Nature, 2011.

G.C. Finnigan, V. Hanson-Smith, B.D. Houser, H.J. Park, and T.H. Stevens. The reconstructed ancestral subunit a functions as both V-ATPase isoforms vph1p and stv1p in S. Cerevisiae. Molecular Biology of the Cell, 22(17), July 2011.

V. Hanson-Smith, B. Kolaczkowski, Bryan and J.W. Thornton. Robustness of ancestral sequence reconstruction to phylogenetic uncertainty. Molecular Biology and Evolution, 27(9), Apr 2010.

vi ACKNOWLEDGEMENTS

Special thanks to John Conery for facilitating interdisciplinary research at the University of Oregon between the departments of biology and computer and information Science. John’s graduate-level courses in bioinformatics and computational science sparked my initial interest in this ﬁeld of research. Thanks to Joe Thornton for allowing me to work in his lab, and for constantly challenging me to pursue excellence – it has made me a better thinker, writer, and scientist.

Thanks to members of the Thornton Lab for constructive comments on my work, and for fostering an overall supportive and productive work environment over the past four years. Thanks to members of the Cresko and Phillips labs for comments.

For the experiments discussed in chapters 2 and 3, I thank Geeta Eick,

Christopher Baker, Eric Gaucher, Belinda Chang, and Mikhail Matz for curating and sharing sequence data. Thanks to Katrina Ray for working with me to develop an early software prototype. For chapter 4, I thank Greg Finnigan for his collaboration on a challenging – but ultimately rewarding – project.

This work has been supported by the NSF IGERT training grant DGE-

9972830 to the University of Oregon.

vii TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION TO COMPUTATIONAL PHYLOGENETICS . . . 1

The Role of Computer Science in Evolutionary Studies ...... 1

ErrorandUncertainty...... 2

A: Sequence Sampling ...... 5

B: Sequence Alignment ...... 5

C: Phylogenetic Inference ...... 6

D: Ancestral Sequence Reconstruction ...... 7

E: Functional Characterization ...... 8

II. REDUCING ERROR IN PHYLOGENETIC INFERENCE ...... 10

Materials and Methods ...... 14

Results ...... 19

Discussion ...... 39

III. ANCESTRAL RECONSTRUCTION AND TREE UNCERTAINTY . . 41

Materials and Methods ...... 45

Results ...... 54

Discussion ...... 64

viii Chapter Page

IV. CASE STUDY: EVOLUTION OF INCREASED COMPLEXITY . . . 69

Materials and Methods ...... 74

Results ...... 79

Discussion ...... 85

V. CONCLUSION ...... 89

Error versus Uncertainty ...... 90

Implications for the Future of Systems Biology ...... 91

Software ...... 92

APPENDICES

A. MARKOV MODELS OF SEQUENCE EVOLUTION ...... 93

B. COMPUTING THE LIKELIHOOD OF A PHYLOGENY ...... 95

C. ALIGNMENT ERROR & ANCESTRAL RECONSTRUCTION . . . 97

D. V-ATPASE SUBUNIT PROTEIN SEQUENCES ...... 105

REFERENCESCITED ...... 109

ix LIST OF FIGURES

Figure Page

1. A multi-step analysis pipeline for evolutionary studies ...... 3

2. Flowchart of Unimax and Multimax optimization ...... 12

3. Discordance between Unimax and Multimax trees ...... 20

4. Phylogenies of Steroid-Hormone Receptor Family ...... 21

5. Phylogenies of Mcm1 protein family ...... 22

6. Phylogenies of Thioredoxin protein family ...... 23

7. Phylogenies of V-ATPase subunits c, c’, and c” ...... 24

8. Phylogenies of the TMO-4c4 gene family in Actinopterygii ...... 25

9. Tree error for Unimax and Multimax ML trees ...... 26

10. Maximum likelihood values of phylogenies for empirical alignments. . . 27

11. Schematic illustration of reciprocal restart analysis on ML trees . . . . 28

12. Schematic illustration of path restart analysis on UM trees ...... 29

13. Schematic illustration of reciprocal restart analysis on initial trees . . . 32

14. Treelengthaccuracy ...... 33

15. The length of ML trees over the duration of tree search ...... 34

16. Unimax suﬀers from an order-dependence problem ...... 35

17. Reconstructed ancestral androgen receptor sequences ...... 37

18. Four-taxon simulated conditions ...... 50

19. Empirical phylogenies used for simulations ...... 53

20. Integrating over phylogenetic uncertainty rarely changes ancestors . . . 56

21. ML and TEB diﬀerences versus phylogenetic support ...... 58

22. ASRerrorrates...... 60 x Figure Page

23. Relationship of ancestral PP to accuracy ...... 62

24. Phylogenetic uncertainty versus alternate ancestral reconstructions . . 63

25. Inferred ancestral states are the same across uncertain trees ...... 65

26. Structure and evolution of the V-ATPase complex ...... 72

27. The maximum likelihood phylogeny of V-ATPase subunits ...... 73

28. Reconstructed V-ATPase ancestors replace extant versions ...... 81

29. Increasing complexity by complementary loss of interactions ...... 84

30. Genetic basis for functional diﬀerentiation of Anc.3 and Anc.11 . . . . 86

32. Alignment length versus insertion-deletion rate ...... 103

33. Ancestral length error versus insertion-deletion rate ...... 104

xi CHAPTER I

INTRODUCTION TO COMPUTATIONAL PHYLOGENETICS

Over the last two centuries of human thought, the discovery of biological evolution profoundly changed – and continues to change – our perception of the living world (Darwin (1859); Huxley (1942); Lewontin (1972)). Combined with a modern understanding of molecular biology and genetic architecture, an evolutionary perspective allows us to investigate the genesis and function of diverse biological forms (Raﬀ (1996); Carroll et al. (2005)). Further, an evolutionary perspective is useful to learn how the overall ecology of our planet is interconnected, and how our own bodies interact with that ecology. The study of evolution can be challenging, however, because many interesting and relevant biological systems evolved over timescales that are vastly longer than the length of a human lifetime. The challenge is that necessary ancestral forms are often unavailable for direct study because they existed millions – or billions – of years ago.

The Role of Computer Science in Evolutionary Studies

We can time travel, in a sense, using computational models of molecular evolution. The ﬁeld of computational phylogenetics has developed Markov models to infer evolutionary history from contemporary molecular sequence data. These types of Markov models are used to reconstruct the phylogenetic history of gene families and to reconstruct ancestral gene sequences. These two in silico techniques

– phylogenetic inference and ancestral reconstruction – are often combined with in vivo or in vitro molecular techniques to generate and then test hypotheses about

1 the evolutionary trajectory of protein families. This combination of computational analysis with wet-lab experimentation is the cornerstone to an emerging paradigm for studying functional molecular evolution (Dean and Thornton (2007)).

A multi-algorithm analysis pipeline combines the methods of phylogenetic inference and ancestral reconstruction with non-computational molecular techniques

(Fig. 1.) (Thornton (2004)). This pipeline begins with a family of molecular sequences whose evolution we wish to investigate. Sequences are typically chosen whose functions vary across a family, and we wish to know how those functions shifted over evolutionary time. Here I brieﬂy describe the pipeline stages. We

ﬁrst align homologous sites in the sequences, infer the phylogeny that give rise to the sequences, and then reconstruct sequences for ancestral species. An ancestral gene sequence that has been computationally reconstructed can be physically

“resurrected” by synthesizing its coding sequence onto a plasmid, transfecting that plasmid into a living cell, and then allowing the cell’s native genetic machinery to transcribe and translate the reconstructed gene into real protein product.

Ancestral proteins can be functionally characterized using many diﬀerent assays; the appropriate assay depends on the type of protein under scrutiny. Overall, this analysis pipeline allows us to observe ancient protein functions before and after signiﬁcant milestones in evolutionary history.

Error and Uncertainty

Two big problems surrounding this analysis pipeline are computational error and statistical uncertainty. Error can be introduced at every stage in the pipeline; the methods of sequence alignment, phylogenetic inference, and ancestral reconstruction use heuristic algorithms that are known to produce errored results

2 A molecular sequences Scelortinia sclerotiorum Mus musculus from related lineages MSLTYGIGGATTALA... MTGLELLYLGIFVAFWACM...

Danio rerio Homo sapiens MNGHAILYTGVTLAFWSTM... MTGLALLYSGVFVAFWACA...

Meleagris gallopavo Hydra magnipapillata MTGLALLYSGLGLAFWTTL... LLDAGIFVCYMLLD...

Saccharomyces cerevisiae MNKESKDDDMSLGKFSFSH...

H. sapiens MTGLALLYSGVFVAFWACA. . . M. musculus MTGLELLYLGIFVAFWACM. . . M. gallopavo sequence MTGLALLYSGLGLAFWTTL. . . B D. rerio MNGHAILYTGVTLAFWSTM. . . alignment H. magnipapillata -----LLDAGIFVCYMLLD. . . S. cerevisiae MNKESKDDDMSLGKFSFSH. . . S. scelortium MSLTY----GIGGATTALA. . .

S. scelortium C phylogenetic S. cerevisiae inference 1.0, 18.4 H. sapiens 1.0, 7.2 M. musculus 0.98, 21.3 M. gallopavo 1.0, 27.4 D. rerio H. magnipapillata D ancestral L: 0.62 H: 0.34 sequence F: 0.04 reconstruction 0.1 subs/site

MNGLALLYSGVTLAFWACA...

DNA-binding assay E synthesis of ancestral protein, 2 growth assay in vivo expression, bits and functional characterization 0

structural analysis ligand-binding assay response ligand dose

FIGURE 1. A multi-step analysis pipeline for evolutionary studies. This pipeline is used to infer the evolution and function of ancestral proteins. Shown here is the analysis of seven eukaryotic amino acid sequences, arbitrarily chosen for this example. See text in chapter I for description of the pipeline steps.

3 in some conditions. Identifying and eliminating sources of error is critical because inaccurate inferences made at early stages in the pipeline will lead to inaccurate inferences at downstream stages. Related to computational error, statistical uncertainty measures the degree to which we think a particular inference is accurate. The presence of signiﬁcant uncertainty implies that alternate solutions should be considered in addition to the best solution. Statistical uncertainty, like error, can emerge at every stage in the pipeline; uncertainty comes in the form of mismatch costs (for alignments), likelihood scores (for trees), and posterior probabilities (for ancestral sequences). These types of uncertainty can be explicitly propagated down the pipeline by performing each downstream stage on the distribution of possible inputs from the upstream stage. Uncertainty propagation, however, incurs non-trivial computational costs and becomes intractable when taken to its philosophical extreme. It is therefore important to know when uncertainty matters, and when uncertainty can be ignored.

This dissertation addresses the role of error and uncertainty within this pipeline. I am broadly interested in two questions: (i) How do we make the results of the pipeline more accurate? (ii) When is it appropriate to propagate uncertainty from an early pipeline stage to downstream stages? In this introduction, I describe each stage of the analysis pipeline in more detail. In chapter II, I discuss improving the accuracy of phylogenetic inference, the role of heuristic search algorithms in introducing ML phylogenetic error, and I propose a more accurate heuristic.

In chapter III, I discuss the role of phylogenetic uncertainty on the accuracy of ancestral sequence reconstruction; I show that phylogenetic uncertainty can be ignored due to a seemingly paradoxical relationship between trees and ancestral sequences. The material in chapter III was previously published with co-authors

4 (Hanson-Smith et al. (2010)). In chapter IV, I provide a case-study demonstrating an analysis using all stages of the pipeline. The material in chapter IV was is co- authored with collaborators in Tom Stevens lab at University of Oregon, and

– at the time of this writing – is currently in-press at Nature (2011). Curious readers may also ﬁnd appendix C useful, in which I show there is a complex, but signiﬁcant, relationship between alignment accuracy and ancestral reconstruction accuracy.

A: Sequence Sampling

The ﬁrst step of the pipeline is to collect molecular sequences – typically nucleotides or amino acids – that are evolutionary related and whose encoded functions are of experimental interest (Fig. 1.A). The amino acid sequences shown in Fig. 1.A encode an arbitrary protein fragment in seven Eukaryotic species: S. sclerotiorum (pathogenic plant fungus), D. rerio (zebraﬁsh), M. gallopavo (turkey),

S. cerevisiae (budding yeast), M. musculus (house mouse), H. sapiens (humans), and H. magnipapillata (fresh water polyp). Collecting molecular sequences is labor intensive, and most evolutionary studies use sequences that have been previously uploaded to sequence repositories. The database GenBank is the dominant worldwide repository, storing millions of protein sequences from thousands of species across the tree of life (Burks et al. (1992); Benson et al. (1999)).

B: Sequence Alignment

The second step of the analysis pipeline is to infer the relatedness – or homology – between individual sites within the set of collected sequences (Fig. 1.

B). Families of sequences drift away from each other over evolutionary time, and

5 it may not be clear how two or more related sequences are, in fact, related. This problem is typically solved with dynamic string-matching algorithms, of which there exist many varieties (Batzoglou (2005); Notredame (2007)). The result of sequence alignment is a matrix of size M N, where M is the length of the longest × sequence in the collection, and N is the number of sequences. Each cell in this matrix contains a single evolutionary character, and all the characters in each column are assumed to be homologous. Over the course of evolution, insertion and deletion events alter the length of a molecular sequence, such that some members of a sequence family can have extra characters (in the event of insertions) or have missing characters (in the event of deletions). Alignment algorithms place “gap” characters to indicate the location of insertions and deletions.

C: Phylogenetic Inference

Given an alignment of sequences, the next step in the pipeline is to infer the phylogeny that gave rise to the alignment (Fig. 1.C). A primitive approach is to cluster sequences according to their pairwise distances, measured as percentage sequence dissimilarity (Cavalli-Sforza and Edwards (1967); Sokal and Sneath

(1963); Saitou and Nei (1987)). However, distance-based methods have limited utility because they compress the sequence alignment into a matrix of pairwise distances between sequences, and thus discard potentially useful information about evolutionary constraints at individual sequence sites. Rather, the dominant paradigm for phylogenetic inference is to begin with a distance-based tree and then optimize this tree using a Markov model in a likelihood framework. Unlike simpler distance-based approaches, Markov models explicitly consider the substitutional

6 process by which sequences evolved. The statistical foundations of molecular

Markov models are described in more detail in Appendix A.

Phylogenetic Markov models include parameters whose values are typically unknown. These parameters include the phylogenetic topology, branch lengths on that topology, the relative substitution rates between sequence states, and— depending on the particular model—other parameters to account for various evolutionary processes. A likelihood function is used to calculate the likelihood of a particular set of parameter values (Felsenstein (1981)); a search function is then used to ﬁnd the set of values with the maximum likelihood. Given a sequence alignment D, the likelihood L(t, θ D) of a topology t with model parameters θ, | is deﬁned as the probability P (D t, θ) of observing D given t and θ. Likelihoods | are calculated using an algorithm that recursively traverses the phylogenetic tree

(described in Appendix B). The goal of maximum likelihood (ML) phylogenetics is to ﬁnd values for t and θ that maximize the function L(t, θ D). The search | strategies used to ﬁnd ML phylogenies are the subject of Chapter II.

D: Ancestral Sequence Reconstruction

Once a phylogeny is found, the next step in the pipeline is to infer the sequences for ancestral nodes (Fig. 1.D). Although any ancestor on a phylogeny can be reconstructed, usually only a few ancestors are experimentally relevant, depending on the speciﬁc hypothesis under scrutiny. Most ancestral queries target historical shifts in protein function; these types of studies require at least two ancestors: one immediately before the shift and another immediately after the shift took place. Other hypotheses may require more than two ancestors.

For example, more than a dozen reconstructed opsin protein ancestors were

7 used to reveal that vertebrates historically evolved through a variety of aquatic environments (Yokoyama et al. (2008)). In another example, eleven reconstructed elongation factor protein ancestors were used to infer a historical shift in Earth’s paleotemperature (Gaucher et al. (2003, 2007)).

Ancestral sequences are reconstructed using the same types of Markov models that are used to infer ML phylogenies. However, rather than searching for the

ML tree, ancestral reconstruction uses a ﬁxed ML tree and searches for the ML ancestral states. The computational mechanics of ML ancestral reconstruction are discussed in Chapter III.

E: Functional Characterization

After ancestral sequences have been computationally reconstructed, the function of those ancestors can be experimentally observed using molecular techniques (Fig. 1.E) (Thornton (2004); Liberles (2007)). Speciﬁcally, ancestral sequences can be physically synthesized, subcloned onto plasmids, and transfected into living cells. The cells’ native genetic machinery will then transcribe and translate the plasmid sequence into actual proteins. In situations where the cell naturally contains a contemporary descendant of the ancestral sequence, the cell’s native copy can be disabled such that the only functional copy is the ancestral gene on the plasmid. Once an ancestral protein is expressed, its function can be studied using a variety of assays. The appropriate assay depends on the protein family.

Transcription factor proteins, for example, can be assayed to determine their binding preference for diﬀerent DNA motifs (Stormo and Zhao (2010)). Nuclear proteins can be assayed to determine their binding-response to variable ligand doses (Bridgham et al. (2006)). Some protein families can be studies at a coarse

8 level, where simple cell growth can be taken as a proxy for function (Finnigan and

Hanson-Smith 2011 ). Finally, some proteins are amenable to structural analysis

– using techniques like X-ray crystallography – and the biophysical determinants of their speciﬁc functions can be investigated (Ortlund et al. (2007); Harms and

Thornton (2010)). Taken together, this interdisciplinary analysis pipeline opens a window into the evolutionary past and allows for the direct observation of hypothesized protein functions that have not existed in millions – or billions – of years.

9 CHAPTER II

REDUCING ERROR IN PHYLOGENETIC INFERENCE

Phylogenetic tree structures are the de facto representation for evolutionary relationships among related molecular sequences. Knowing the correct tree is the necessary ﬁrst step for many useful downstream analyses, including ancestral sequence reconstruction (Liberles (2007)), phylogeography (Avise et al. (1987);

Avise (1998)), and estimation of species divergence times (Taylor and Berbee

(2006)). The correct phylogeny, however, is often unknowable because the ancestral species necessary to determine historical branching patterns are typically unavailable. Rather, phylogenies are usually inferred computationally from contemporary sequences. The dominant paradigm for phylogenetic inference is to use a parametric Markov model that describes the relative substitution rates between diﬀerent molecular states — typically, nucleotides, amino acids, or codons.

Given a molecular sequence alignment, the likelihood of a particular tree and model equals the probability of observing the alignment, given the tree topology, branch lengths on that topology, and speciﬁc values for substitution rates between states

(Felsenstein (1981); Bryant et al. (2005)). The true values for the tree, branches, and parameters are typically unknown, and search and optimization algorithms are necessary to ﬁnd the combination with the maximum likelihood (ML) value. To the extent that the likelihood function correlates with the accuracy of evolutionary history, the ML phylogeny is the most probable explanation for the evolution of a given sequence alignment.

The ML phylogeny, however, is not always easy to ﬁnd because the space of possible trees and parameter values is so immense that brute-force search strategies

10 are computationally intractable for all but trivial-sized problems. Instead, search heuristics are employed to constrain the ML exploration to high-likelihood regions of parameter space (Felsenstein (2004)). The search for an ML tree is typically decomposed into two nested problems: optimizing tree topologies, and optimizing continuous parameters (Figure 2.). The primary problem is to search the space of tree topologies and ﬁnd the topology with the highest likelihood. The search for the ML topology typically begins with an initial tree constructed using a neighbor- joining algorithm (Saitou and Nei (1987); Gascuel (1997)). From this initial tree, the space of possible tree topologies can be traversed by swapping tree branches in order to transform one topology into a diﬀerent topology. The secondary problem is to optimize the branch lengths and other model parameters on each explored tree.

Virtually everyone solves the secondary problem using an approach I refer to as

Unimax, in which free parameters are sequentially optimized individually. Unimax is typically implemented using the van Wijngaarden-Deker-Brent method, which combines three unique hill-climbing algorithms – inverse quadratic interpolation, root bracketing, and bisection – to ﬁnd the maximum of a function with one free parameter (Brent (1972)). Unimax is the default option in popular phylogenetic software packages PhyML (Guindon et al. (2010)), RaxML (Stamatakis (2006); Ott et al. (2007)), Garli (Zwickl (2006)), PAML (Yang (2007)), and PAUP (Swoﬀord

(2003)).

Unimax assumes free parameters are separable. In other words, Unimax assumes that the ML solution can be found by individually optimizing each parameter while holding all other parameters constant. For branch lengths, at least, this assumption seems to be incorrect because the likelihood of a phylogeny is computed via a postorder traversal of the tree – using Felsenstein’s so-called

11 A ? Build initial START: converged on NJ tree T Examine n no maximum input alignment d possible tree yes swaps T = best swap

optimize B DONE: outputs Unimax Multimax

Propose Propose ? converged on max ? no

yes ? converged on max ? Propose no

? converged on max ? yes no

yes etc.

Propose

? converged on max ? no no ? converged on max ? yes yes

FIGURE 2. Flowchart of Unimax and Multimax optimization. Unimax and Multimax use different approaches to optimize model parameters. Finding the ML phylogeny involves two nested problems. (A) The high-level problem is find the ML topology T . Starting with a multiple sequence alignment d, propose a topology T , and optimize all free model parameters and branch lengths Θ on T . This process stops when we converge on an ML T . Otherwise, we examine possible topological rearrangements to T (tree swaps), and repeat the overall process. (B) The low-level problem is optimize all continuous parameters on a fixed T . Unimax sequentially optimizes each free parameter θ , θ , ..., θ Θ, using fixed values for all other free 1 2 n ∈ parameters. Multimax simultaneously optimizes all parameters θi Θ,ateach iteration proposing a new set of parameter values. ∈

12 pruning algorithm – in which conditional probabilities of ancestral states at internal nodes are propagated up the tree (Felsenstein (1981)); adjusting one branch length aﬀects the conditional probabilities at nearby internal nodes, which ultimately aﬀects the likelihood of nearby branch lengths. The recursive relationship between branches means that branch lengths may be correlated parameters. Unimax ignores this recursive relationship, and instead optimizes each parameter in isolation.

Stated formally, Unimax assumes that the ML solution for a parameter θi can be found using ﬁxed values for the alignment d, the topology t, and all other free parameters θj in the set of parameters Θ (Eq. 2.1).

P (θˆ d, t, θ Θ,i= j) (2.1) i| j ∈

In contrast, a non-separable optimization method simultaneously seeks the ML solutions for all free parameters θˆ1, ..., θˆn, using a ﬁxed alignment d and topology t (Eq. 2.2).

P (θˆ , ..., θˆ Θ d, t) (2.2) 1 n ∈ |

Although previous scholarship has recognized faulty logic in Unimax’s assumption of separability, the degree to which this assumption impairs the accuracy of ML phylogenetic inference has not been systematically investigated (Yang (2000);

Bryant et al. (2005)).

In order to assess the eﬀect of separability on phylogenetic accuracy,

I implemented an alternative ML optimization algorithm that optimizes all parameters simultaneously rather than separably. This approach, referred to here as Multimax, is formally based on the conjugate-gradient approach of the Broyden-

13 Fletcher-Goldfarb-Shanno (BFGS) algorithm (Press et al. (1992); Nocedal and

Wright (1999)). BFGS operates by estimating the ﬁrst- and second-derivates of the likelihood function locally around the current parameter values. BFGS then uses these derivates to estimate an ML solution. BFGS jumps to this estimated optimum, and recomputes the local derivatives. If the functional gradient at the new point is zero or within an acceptable margin, BFGS has converged upon an

ML solution. Otherwise, BFGS uses the new gradient to estimate an updated ML solution. BFGS repeats this process until it converges on an optimum or reaches a user-speciﬁed maximum number of iterations.

I compared the performance of Multimax (MM) to Unimax (UM) under a range of conditions, both empirical and simulated. Across these conditions, I observed that UM’s assumption of separability signiﬁcantly impaired the accuracy of phylogenetic inference. UM was less accurate than MM because UM leads to poor ML branch lengths, which ultimately drives the tree search algorithm into suboptimal regions of tree space.

Materials and Methods

Unimax

I used PhyML’s implementation of Unimax using Brent’s method, as described in chapter 9.3 of Numerical Recipes in C (Press et al. (1992)).

Multimax

Multimax using BFGS works as follows. Given a sequence alignment d, a ﬁxed tree topology T, and a set of starting values for all free parameters Θ,

BFGS estimates the ﬁrst- and second-derivate gradients of the likelihood function 14 L(Θ T,d). My software implementation estimates these gradients as a Hessian | matrix, constructed with local secant approximation (Nocedal and Wright (1999)).

The likelihood gradients are used to ﬁnd a multidimensional uphill direction p in parameter space (Eq. 2.3).

Hp = L(Θ T,d) (2.3) −∇ | where H is the Hessian matrix of second-order partial derivates between all parameters θ Θ, p is the direction of the functional optima relative to our current ∈ values of Θ, and L is the ﬁrst-derivate of the likelihood function. After solving ∇ for p, BFGS performs a line search in the direction of p in order to propose a new optimum. BFGS then jumps to this proposed point. BFGS repeats these four steps

– calculating gradients, solving equation 2.3, proposing an optimum, and then jumping – until it arrives at a point whose gradient is zero or 200 iterations have been performed.

Tree Swapping

I used a combination of nearest neighbor interchange (NNI) and subtree pruning and regrafting (SPR) to swap branches and propose new topologies

(Swoﬀord and Olsen (1990)). I used the default implementations of NNI and SPR within PhyML version 3.0. PhyML’s swap algorithm calculates a swap score for every possible topology rearrangement; this score estimates the potential likelihood increase of accepting the swapped tree relative to the pre-swapped tree. Swap scores were calculated as the estimated likelihood of the swapped tree (Lt+1) minus the likelihood of the current tree (Lt). The values for Lt+1 were estimated by optimizing only the four branches directly aﬀected by the swap. 15 Simulated Sequence Alignments

In order to compare the accuracy of UM and MM on simpliﬁed conditions,

I simulated amino acid and nucleotide sequences evolving on randomly generated trees whose sizes varied from 8 to 1024 terminal branches. For each tree size, I generated twenty random trees and generated a random amino acid sequence at the root of the tree. I simulated the ancestor evolving across the branches, without insertion/deletion events, to produce an alignment of descendant sequences.

All branches were individually drawn from the uniform distribution [0.00,0.05] for pinnate trees and [0.00,0.50] for balanced trees. Nucleotide sequences were simulated using the JC69 model to create alignments with 1000 sites. Amino acid sequences were simulated using the JTT model to create alignments with 400 sites. Simulations were performed using INDELible with insertion/deletion events disabled (Fletcher and Yang (2009)).

Empirical Sequence Alignments

In order to determine if UM biased our evolutionary interpretation of real sequence data, I used MM and UM to infer ML trees for sequence alignments from six gene families: (i) steroid-hormone receptor ligand binding domains from across Metazoa (Bridgham et al. (2008)), (ii) Mcm1 transcription factors from twelve species of Fungi Ascomycete Saccharomycotina (Baker et al. (2011)), (iii) vacuolar ATP-ase subunits c, c’, and c” sampled broadly from Opisthokonts (cite

XX), (iv) thioredoxins from species across the tree of life (Perez-Jimenez et al.

(2011)), (v) ribosomal 16S nucleotide sequences sampled from across proteobacteria

(cite XX), and (vi) TMO-4c4 gene nucleotide sequences, sampled broadly from the mitochondria of ray-finned fish (Scorpaeniformes) (Smith and Wheeler (2004)). 16 I compared the MM and UM ML trees using four criterion: their congruity with our a priori expectations, their maximized likelihood scores, their complementary topological differences, and their unique paths taken through tree space during tree search.

Phylogenetic Inference

I inferred ML phylogenies for empirical alignments using our own in-house modifications to PhyML version 3.0. Tree search began with the neighbor-joined tree, as implemented in PhyML. The search was then driven by either UM or MM until convergence on an optimum. The best-fitting Markov model was found by repeating the search with different substitution matrices and levels of heterogeneity, and then using the Akaike Information Criterion to find the model with the highest likelihood without overparameterization (Akaike (1973)). Using the best-fitting model, ML phylogenetic inference was performed with full tracing enabled, in which

PhyML records path taken through tree space.

Reciprocal Restart Analysis

In order to determine if UM or MM could further optimize the other method’s ML tree, I restarted UM from the MM ML tree and restarted MM from the UM ML tree. In order to determine if UM or MM could optimize the other method’s ML branch lengths on the initial topology (before any swaps had been performed), I disabled topology search and restarted MM from the UM ML branch lengths on the initial topology and restarted UM from the MM ML branch lengths.

17 Ancestral Sequence Reconstruction

I reconstructed ancestral sequences using maximum likelihood as implemented in PAML version 4.2 and an in-house GUI – named Lazarus – that controls PAML

(Yang (2007); Hanson-Smith et al. (2010)).

Tree Error

For every simulated alignment, I measured relative tree error as the symmetric difference between the neighbor-joined topology and the true tree, divided by the number of branches in the tree. I computed symmetric differences using the function dendropy.Tree.symmetric difference as implemented in the

DendroPy library (Sukumaran and Holder, 2010). Relative tree error, informally, measures the proportion of spurious clades in a tree. In order to determine statistical significance between Unimax and Multimax tree error, I used a paired two-tailed T-test to measure if the mean tree errors were significantly different. T-values and derived P-values were computed using the function cogent.stats.math.t paired in the PyCogent library (Knight et al., 2007).

I measured the error in overall tree length by dividing the sum of branch lengths on each ML tree by corresponding sum on its true tree. This product is reported in this paper as length error. In order to determine statistical significance between Unimax and Multimax tree length error, I used a paired two-tailed T-test to measure if the mean tree errors were significantly different. T-values and derived

P-values were computed using the function cogent.stats.math.t paired within the

PyCogent library (Knight et al., 2007).

18 Results

Separable ML optimization impairs evolutionary interpretation

Unimax (UM) and Multimax (MM) drove the tree search to find different ML trees, so the assumption of separability does matter. For sequences simulated under controlled conditions, the difference between UM and MM ML trees was greater for large trees and for pinnate-shaped trees (Fig. 3.). For sequences of gene families evolved under real conditions, UM and MM drove the tree search algorithm to

ﬁnd disagreeing topologies for ﬁve out of six families (Figs. 4. - 8.). In most cases, there was strong support for the disagreeing branches; in a typical analysis, the placement of these discordant branches would not be interpreted as uncertain, nor would their placement warrant further investigation regarding sequence choice.

MM was superior in ﬁnding trees with high likelihood scores and with less error. This means the assumption of separability not only matters, but actually impairs phylogenetic inference. For empirical sequences, MM-driven search led to trees with higher likelihoods, indicating a more eﬀective search of space (Fig. 10.).

For simulated sequences, in which the true is known, I found MM trees had fewer erroneously placed clades than UM trees (Fig. 9.).

Separable ML optimization leads to suboptimal trees

I next sought why UM led to poor ML solutions. Was it that UM foiled the search by getting stuck on ridges, saddles, or other non-peak features in parameter space? Or, did UM irreversibly drive the search into poor regions of tree space whose highest summits were non-global optima? I disqualified the first theory by performing reciprocal restart analysis on ML trees (Fig. 11.). For the five empirical

19 nucleotides amino acids 1.0

proportion discordance

0 8 16 32 64 96 128 160 200 256 384 512 8 16 32 64 72 96 128 192 256 512 N taxa

amino acids nucleotides 0.1

0 8 16 32 64 128 192 256 512 712 1024 8 16 32 64 384 512 576 640 720 1024

FIGURE 3. Discordance between Unimax and Multimax trees. Unimax and Multimax found discordant ML topologies in many cases. N taxa is the number of sequences in the simulated alignment. Proportion discordance, the mean proportion of clades that diﬀered between the UM and MM ML trees.

20 Unimax Multimax

8.2 34.8 34.7 69 Glucocorticoid Receptors (23) 8.2 69.1 Glucocorticoid Receptors (23) 1.8 Mineralocorticoid Receptors (21) 1.8 Mineralocorticoid Receptors (21) 101.9 PetMarPR 102.2 PetMarPR 30.3 29.7 A MyxGluSR MyxGluSR 0.5 subs 37.6 37.8 1.9 PetMarCR 2.4 PetMarCR 14.9 14.5 MyxGluCR MyxGluCR 165.9 Progesterone Receptors (22) 165.2 Progesterone Receptors (22) 16.4 9.7 16.6 9.7 53.2 Androgen Receptors (39) 53.1 Androgen Receptors (39) 126.6 BraFloSR 127.4 BraFloSR 8.8 BraBelSR 8.8 BraBelSR 1.5 Estrogen Receptors (92) 1.6 60.7 76 60.6 Estrogen-related Receptors (13) 76.2 23 23.2 Estrogen-related Receptors (13) TriAdhERR TriAdhERR 147 147.5 170.2 Steroidogenic Factors (5) 170.2 Steroidogenic Factors (5) 54.5 54.1 Retinoid X Receptors (10) Retinoid X Receptors (10) 147 Chicken Ovalbumin Upstream Promoters (COUP) (6) 147.5 Chicken Ovalbumin Upstream Promoters (COUP) (6)

18.5 PagMajAR 18.5 PagMajAR 0.7 AcaSchAR 0.8 AcaSchAR 5.2 MicUndAR 5.2 Androgen Receptors (39) MicUndAR 2 DicLabAR 2.2 9.7 B DicLabAR GasAcuARb 0.1 subs 10.4 GasAcuARb 75.8 HalTriAR 64.4 HalTriAR OreNilARb OreNilARb OryLatAR 7.1 3.2 OryLatAR KryMarAR 9.2 KryMarAR 3.7 4 0.4 185.8 GamAffAR 0.5 0.4 185.7 GamAffAR HapBurARb 2.3 39.2 HapBurARb 9.4 11.9 PimProAR AngJapARb 96.2 DanRerAR OncMykARb2 14.9 Salmonidae OncMykARa2 CarAurAR 21.6 PimProAR Salmonidae OncMykARb2 2.1 12.3 24.1 DanRerAR Node 254 33.9 OncMykARa2 AngJapARb Node 254 1.3 79.9 CarAurAR GarAR Actinopterygii AngJapARa GamAffARa Actinopterygii 272.2 GamAffARa 47.4 224.7 53.6 HapBurAr 0.7 HapBurAr 6.8 TakrubAR 7.1 79.7 TakrubAR 83 OreNilARa OreNilARa Node 253 AngJapARa GarAR 3.8 ManVitAR 6.6 ManVitAR 2.5 82.5 SerCanAR Node 253 6.8 78.8 SerCanAR TaeGutAR TaeGutAR 27 CotJapAR 29.3 CotJapAR Node 252 SusScrAR SusScrAR RatNovAR Node 252 RatNovAR 0.01 OryCunAR 0.09 23.3 121.8 125.3 OryCunAR Node 251 HomSapAR 22.3 HomSapAR CrocroAR (AncAR) CrocroAR CanFamAR CanFamAR 9.7 17.4 EngPusAR Node 251 RanCatAR 9.7 EngPusAR 4.2 XenLaeAR (AncAR) 15.2 RanCatAR NeoFosAR 4.3 XenLaeAR 83 SquAcaAr NeoFosAR LeuEriAR 83.9 SquAcaAr LeuEriAR

FIGURE 4. Phylogenies of steroid-hormone receptor family. Shown also are related outgroup genes. (A) The ML tree inferred using Unimax is shown on the left, and the ML tree inferred using Multimax is on the right. The clade in red is expanded in part (B). In the clades of androgen receptors, colors indicate branches that were placed diﬀerently on the two trees. All support values on internal branches are approximate likelihood ratio test values. Relevant ancestral nodes (see main text) are labeled as Node 251, 252, 253, and 254.

21 Unimax Y_lipolytica_Mcm1 C_albicans_Mcm1 K_lactis 0.5 10.6 K_thermotolerans K_waltii_Mcm1 Pre-duplication 14.5 E_gossyppi 2.0 subs/site 35.4 S_kluyveri_Mcm1 5.9 S_mikatae_paralog1 34.8 S_bayanus_paralog1 3.1 S_cerevisiae_paralog1 4.1 S_paradoxus_paralog1 Z_rouxii_paralog1 Paralog 1 C_glabrata_paralog1 8.8 K_polysporus_paralog1a 1.9 K_polysporus_paralog1b 14.7 4.5 11.1 S_castellii_paralog1 K_polysporus_paralog2 25.4 2.4 C_glabrata_paralog2 9.1 Z_rouxii_paralog2 1.5 S_castellii_paralog2 11.9 S_mikatae_paralog2 S_bayanus_paralog2 Multimax 0.7 S_cerevisiae_arg80_paralog2 S_paradoxus_paralog2 Paralog 2 Y_lipolytica_Mcm1 23.0 C_albicans_Mcm1 K_lactis 0.6 K_thermotolerans 9.8 K_waltii_Mcm1 Pre-duplication 14.9 E_gossyppi 35.7 S_kluyveri_Mcm1 6.3 S_mikatae_paralog1 34.7 3.0 S_bayanus_paralog1 5.1 S_cerevisiae_paralog1 S_paradoxus_paralog1 Z_rouxii_paralog1 Paralog 1 8.9 C_glabrata_paralog1 2.1 K_polysporus_paralog1a 4.5 K_polysporus_paralog1b 13.5 11.2 S_castellii_paralog1 5.2 C_glabrata_paralog2 0.6 K_polysporus_paralog2 3.7 Z_rouxii_paralog2 90.0 S_castellii_paralog2 S_bayanus_paralog2 10.1 S_mikatae_paralog2 2.6 S_cerevisiae_arg80_paralog2 22.8 S_paradoxus_paralog2 Paralog 2

FIGURE 5. Phylogenies of Mcm1 protein family. Terminal amino acid sequences come from twelve species of Fungi Ascomycete Saccharomycotina. A gene duplication occurred sometime after the clade containing S. kluyveri, S. gossyppi, K. thermotolerans, K. walti, and K. lactis branched from the other species. The ML tree optimized using Unimax is shown on top, and the Multimax ML tree is shown on bottom. Red branches are discordant between the two trees. Support values on internal branches are approximate likelihood ratio test values.

22 0.97 Chloroplasts (10) Chloroplasts (10) Bacteria.Cyanobacteria.17227548.Nostoc 0.2 Bacteria.Cyanobacteria.17227548.Nostoc Bacteria.Cyanobacteria.16331825.Synechocystis Bacteria.Cyanobacteria.22298354.Thermosynechococcus 2.31 Bacteria.Cyanobacteria.22298354.Thermosynechococcus Bacteria.Cyanobacteria.17229358.Nostoc Bacteria.Cyanobacteria.17229358.Nostoc 3.51 Bacteria.Cyanobacteria.126696505.Prochlorococcus Bacteria.Cyanobacteria.126696505.Prochlorococcus Bacteria.Cyanobacteria.16331825.Synechocystis 2.19 Gammaproteobacteria (10) Gammaproteobacteria (10) Bacteria.Cyanobacteria.17229833.Nostoc Unimax Bacteria.Proteobacteria.Deltaproteobacteria.120602 Bacteria.Cyanobacteria.16329883.Synechocystis Bacteria.Acidobacteria.94970094.Acidobacteria Bacteria.Acidobacteria.116619449.Solibacter Multimax Bacteria.Proteobacteria.Deltaproteobacteria.399983 1.0 Bacteria.Proteobacteria.Deltaproteobacteria.399983 3.03 Bacteria.Cyanobacteria.17229833.Nostoc Bacteria.Acidobacteria.94970094.Acidobacteria Bacteria.Cyanobacteria.16329883.Synechocystis Bacteria.Proteobacteria.Deltaproteobacteria.120602 2.1 Bacteria.Acidobacteria.116619449.Solibacter Bacteria.Chloroflexi.147669275.Dehalococcoides Bacteria.Chloroflexi.147669275.Dehalococcoides 0.06 Bacteria.Chlorobi.119357012.Chlorobium Bacteria.Chlorobi.119357012.Chlorobium Bacteria.Firmicutes.20807685.Thermoanaerobacter Bacteria.Planctomycetes.32477354.Rhodopirellula Bacteria.Actinobacteria.62391938.Corynebacterium 6.47 Bacteria.Aquificae.15606934.Aquifex Bacteria.Firmicutes.15896334.Clostridium Bacteria.Chloroflexi.118048687.Chloroflexus Bacteria.Firmicutes.110798962.Clostridium Bacteria.Chloroflexi.118047160.Chloroflexus Bacteria.Proteobacteria.Alphaproteobacteria.156038 5.2 Bacteria.Chloroflexi.118046691.Chloroflexus 1.44 Bacteria.Firmicutes.110800418.Clostridium Bacteria.Acidobacteria.116619824.Solibacter Bacteria.Firmicutes.15894825.Clostridium 7.34 0.44 Bacteria.Proteobacteria.Alphaproteobacteria.179883 Bacteria.Firmicutes.153179313.Listeria Bacteria.Proteobacteria.Alphaproteobacteria.159184 Bacteria.Firmicutes.28379765.Lactobacillus Bacteria.Proteobacteria.Alphaproteobacteria.150398 Bacteria.Firmicutes.28377165.Lactobacillus 1.23 Bacteria.Proteobacteria.Deltaproteobacteria.425239 11.24 Bacteria.Firmicutes.28378864.Lactobacillus Bacteria.Deinococcus.Thermus.46199419.Thermus Bacteria.Firmicutes.29375972.Enterococcus Actinobacteria (4) Bacteria.Firmicutes.15901605.Streptococcus 0.01 Bacteria.Deinococcus.Thermus.15807833.Deinococcus 2.08 Bacteria.Firmicutes.30264587.Bacillus Bacteria.Firmicutes.15896334.Clostridium Bacteria.Firmicutes.16079902.Bacillus Bacteria.Firmicutes.110798962.Clostridium Bacteria.Firmicutes.138896249.Geobacillus 2.37 Bacteria.Proteobacteria.Alphaproteobacteria.156038 Bacteria.Firmicutes.150393692.Staphylococcus Bacteria.Firmicutes.110800418.Clostridium 3.79 Bacteria, Deinococcus (2) Bacteria.Firmicutes.15894825.Clostridium Bacteria.Cyanobacteria.17229859.Nostoc Bacteria.Firmicutes.153179313.Listeria 2.96 Bacteria.Aquificae.15606934.Aquifex 14.79 Bacteria.Firmicutes.28379765.Lactobacillus Bacteria.Planctomycetes.32477354.Rhodopirellula Bacteria.Firmicutes.28377165.Lactobacillus Bacteria.Proteobacteria.Alphaproteobacteria.179883 Bacteria.Firmicutes.28378864.Lactobacillus Bacteria.Proteobacteria.Alphaproteobacteria.159184 Bacteria.Firmicutes.29375972.Enterococcus 13.00 Bacteria.Proteobacteria.Alphaproteobacteria.150398 Bacteria.Firmicutes.15901605.Streptococcus Bacteria.Chloroflexi.118048687.Chloroflexus Bacteria.Firmicutes.30264587.Bacillus 2.14 Bacteria.Chloroflexi.118047160.Chloroflexus Bacteria.Firmicutes.16079902.Bacillus Bacteria.Chloroflexi.118046691.Chloroflexus Bacteria.Firmicutes.138896249.Geobacillus 1.76 Bacteria.Acidobacteria.116619824.Solibacter Bacteria.Firmicutes.150393692.Staphylococcus Bacteria.Proteobacteria.Deltaproteobacteria.425239 Bacteria, Deinococcus (2) Actinobacteria (4) Bacteria.Deinococcus.Thermus.46199419.Thermus Gammaproteobacteria (6) Bacteria.Deinococcus.Thermus.15807833.Deinococcus Bacteria.Actinobacteria.72163169.Thermobifida Gammaproteobacteria (6) Bacteria.Proteobacteria.Deltaproteobacteria.425218 Bacteria.Actinobacteria.72163169.Thermobifida Bacteria.Actinobacteria.72160576.Thermobifida Bacteria.Proteobacteria.Deltaproteobacteria.425218 Bacteria.Actinobacteria.21219405.Streptomyces Bacteria.Actinobacteria.72160576.Thermobifida Bacteria.Actinobacteria.62391823.Corynebacterium Bacteria.Actinobacteria.21219405.Streptomyces Bacteria.Proteobacteria.Betaproteobacteria.7431624 Bacteria.Actinobacteria.62391823.Corynebacterium Bacteria.Proteobacteria.Betaproteobacteria.3360200 Bacteria.Proteobacteria.Betaproteobacteria.7431624 Bacteria.Actinobacteria.57116870.Mycobacterium Bacteria.Proteobacteria.Betaproteobacteria.3360200 Bacteria.Actinobacteria.15608608.Mycobacterium Bacteria.Actinobacteria.57116870.Mycobacterium Bacteria.Proteobacteria.Deltaproteobacteria.399985 Bacteria.Actinobacteria.15608608.Mycobacterium Bacteria.Chlorobi.119357517.Chlorobium Bacteria.Proteobacteria.Deltaproteobacteria.399985 0.49 Bacteria.Bacteroidetes.34540117.Porphyromonas Bacteria.Chlorobi.119357517.Chlorobium Bacteria.Bacteroidetes.29346866.Bacteroides Bacteria.Bacteroidetes.34540117.Porphyromonas Bacteria.Bacteroidetes.29345628.Bacteroides 0.12 Bacteria.Bacteroidetes.29346087.Bacteroides 1.51 Bacteria.Bacteroidetes.29346087.Bacteroides Bacteria.Bacteroidetes.29346866.Bacteroides Archaea.Euryarchaeota.48477193.Picrophilus Bacteria.Bacteroidetes.29345628.Bacteroides 1.94 Bacteria.Firmicutes.20808289.Thermoanaerobacter Epsilonproteobacteria (3) Chrenarchaeota (5) Archaea.Euryarchaeota.48477193.Picrophilus 16.18 Archaea.Crenarchaeota.159040636.Caldivirga 7.98 4.09 Archaea.Crenarchaeota.159040636.Caldivirga Archaea.Crenarchaeota.15899007.Sulfolobus Bacteria.Firmicutes.20808289.Thermoanaerobacter 13.37 Archaea.Crenarchaeota.15922449.Sulfolobus 0.01 Archaea.Crenarchaeota.15899007.Sulfolobus Archaea.Crenarchaeota.70607552.Sulfolobus 1.72 Archaea.Crenarchaeota.15922449.Sulfolobus 2.87 Archaea.Euryarchaeota.11498883.Archaeoglobus 5.91 Archaea.Crenarchaeota.70607552.Sulfolobus Archaea.Euryarchaeota.11499727.Archaeoglobus Archaea.Euryarchaeota.11498883.Archaeoglobus 3.15 Archaea.Euryarchaeota.154149646.Candidatus 0.33 Archaea.Euryarchaeota.11499727.Archaeoglobus Archaea.Euryarchaeota.88603734.Methanospirillum Archaea.Euryarchaeota.116754023.Methanosaeta 6.19 Archaea.Euryarchaeota.116754023.Methanosaeta Archaea.Euryarchaeota.91773622.Methanococcoides Archaea.Euryarchaeota.91773622.Methanococcoides 0.04 Epsilonproteobacteria (3) Archaea.Euryarchaeota.154149646.Candidatus Bacteria.Proteobacteria.Epsilonproteobacteria.1562 Archaea.Euryarchaeota.88603734.Methanospirillum 0.07 Bacteria.Firmicutes.20807685.Thermoanaerobacter Chrenarchaeota (5) Bacteria.Actinobacteria.62391938.Corynebacterium Bacteria.Proteobacteria.Epsilonproteobacteria.1562 Bacteria.Cyanobacteria.17229859.Nostoc Bacteria.Bacteroidetes.34539910.Porphyromonas 7.45 Bacteria.Bacteroidetes.34539910.Porphyromonas Bacteria.Bacteroidetes.150024368.Flavobacterium Bacteria.Bacteroidetes.150024368.Flavobacterium Bacteria.Bacteroidetes.29347639.Bacteroides Bacteria.Bacteroidetes.29347639.Bacteroides Bacteria.Cyanobacteria.22299630.Thermosynechococcu 4.77 Bacteria.Cyanobacteria.22299630.Thermosynechococcu 7.76 Bacteria.Cyanobacteria.16329237.Synechocystis Bacteria.Cyanobacteria.16329237.Synechocystis Bacteria.Proteobacteria.Betaproteobacteria.7431862 Bacteria.Proteobacteria.Betaproteobacteria.7431862 Bacteria.Firmicutes.153181008.Listeria 3.03 Bacteria.Firmicutes.153181008.Listeria Bacteria.Planctomycetes.32476401.Rhodopirellula 7.65 Bacteria.Cyanobacteria.22297898.Thermosynechococcu 1.48 Archaea.Euryarchaeota.150401020.Methanococcus 6.11 Archaea.Euryarchaeota.13541608.Thermoplasma Archaea.Euryarchaeota.13541608.Thermoplasma Archaea.Crenarchaeota.119720035.Thermofilum Archaea.Crenarchaeota.119720035.Thermofilum Bacteria.Firmicutes.29377495.Enterococcus 2.63 Bacteria.Cyanobacteria.22297898.Thermosynechococcu 0.8 Bacteria.Firmicutes.15901736.Streptococcus 0.5 0.08 Bacteria.Cyanobacteria.17229697.Nostoc 3.03 Bacteria.Firmicutes.16077522.Bacillus Bacteria.Firmicutes.29377495.Enterococcus Archaea.Crenarchaeota.118431901.Aeropyrum Bacteria.Firmicutes.15901736.Streptococcus 2.17 Archaea.Crenarchaeota.126465005.Staphylothermus 7.34 Bacteria.Firmicutes.16077522.Bacillus Bacteria.Firmicutes.15894111.Clostridium Bacteria.Firmicutes.16079205.Bacillus 2.32 Bacteria.Firmicutes.16079205.Bacillus Archaea.Crenarchaeota.126465005.Staphylothermus Bacteria.Actinobacteria.21219599.Streptomyces 1.69 Bacteria.Firmicutes.15894111.Clostridium Bacteria.Actinobacteria.15607956.Mycobacterium Archaea.Crenarchaeota.118431901.Aeropyrum Bacteria.Proteobacteria.Epsilonproteobacteria.3453 6.43 Bacteria.Actinobacteria.21219599.Streptomyces Bacteria.Proteobacteria.Gammaproteobacteria.145629 Bacteria.Actinobacteria.15607956.Mycobacterium Bacteria.Proteobacteria.Epsilonproteobacteria.3452 Bacteria.Proteobacteria.Epsilonproteobacteria.3453 Archaea.Euryarchaeota.124485138.Methanocorpusculum Bacteria.Proteobacteria.Gammaproteobacteria.145629 Archaea.Euryarchaeota.116754438.Methanosaeta Bacteria.Proteobacteria.Epsilonproteobacteria.3452 Cyanobacteria (4) Archaea.Euryarchaeota.124485138.Methanocorpusculum Mitochondrion (6) Archaea.Euryarchaeota.116754438.Methanosaeta Bacteria.Chlamydiae.76789276.Chlamydia Cyanobacteria (4) 2.57 Bacteria.Chlamydiae.15836191.Chlamydophila Mitochondrion (6) Archaea.Euryarchaeota.76802694.Natronomonas Bacteria.Chlamydiae.76789276.Chlamydia Archaea.Euryarchaeota.76802488.Natronomonas Bacteria.Chlamydiae.15836191.Chlamydophila Archaea.Euryarchaeota.16120325.Halobacterium Archaea.Euryarchaeota.76802694.Natronomonas 0.3 Archaea.Euryarchaeota.110667588.Haloquadratum Archaea.Euryarchaeota.76802488.Natronomonas Archaea.Euryarchaeota.55380304.Haloarcula Archaea.Euryarchaeota.16120325.Halobacterium Bacteria.Cyanobacteria.17229697.Nostoc Archaea.Euryarchaeota.110667588.Haloquadratum Bacteria.Planctomycetes.32476401.Rhodopirellula Archaea.Euryarchaeota.55380304.Haloarcula Archaea.Euryarchaeota.150401020.Methanococcus 1.16 Eukaryota.Fungi.46441186.Candida 0.19 Eukaryota.Fungi.46441186.Candida Eukaryota.Fungi.126213085.Pichia Eukaryota.Fungi.126213085.Pichia Eukaryota.Fungi.50309357.Kluyveromyces Eukaryota.Fungi.50309357.Kluyveromyces Eukaryota.Fungi.50291653.Candida Eukaryota.Fungi.a.Saccharomyces Eukaryota.Fungi.151941211.Saccharomyces Eukaryota.Fungi.151941211.Saccharomyces Eukaryota.Fungi.151943486.Saccharomyces 1.0 Eukaryota.Fungi.50291653.Candida Eukaryota.Alveolata.46226985.Cryptosporidium Fungi, Aspergillus & related (5) Fungi, Aspergillus & related (5) Eukaryota.Alveolata.46226985.Cryptosporidium Eukaryota.Viridiplantae.1620905.Fagopyrum Eukaryota.Viridiplantae.1620905.Fagopyrum Chordates (19) Chordates (19) Eukaryota.Amoebozoa.67479051.Entamoeba Eukaryota.Amoebozoa.67479051.Entamoeba Eukaryota.Viridiplantae.145351136.Ostreococcus Eukaryota.Viridiplantae.145351136.Ostreococcus Eukaryota.Viridiplantae.53801490.Helicosporidium Eukaryota.Viridiplantae.53801490.Helicosporidium 2.22 Eukaryota.Mycetozoa.165988451.Dictyostelium Eukaryota.Mycetozoa.165988451.Dictyostelium Arthropods (8) Arthropods (8) Eukaryota.Metazoa.Porifera.30580603.GEOCY Eukaryota.Metazoa.Porifera.30580603.GEOCY Eukaryota.Alveolata.148804689.Plasmodium 1.0 Eukaryota.Alveolata.148804689.Plasmodium Eukaryota.Alveolata.68350806.Theileria Eukaryota.Alveolata.68350806.Theileria Eukaryota.Fungi.150951554.Pichia Eukaryota.Fungi.150951554.Pichia Eukaryota.Choanoflagellida.167537844.Monosiga Eukaryota.Choanoflagellida.167537844.Monosiga

FIGURE 6. Phylogenies of Thioredoxin protein family. Terminal sequences are sampled from across the tree of life. The ML tree on the left was optimized using Unimax, and the tree on the right was optimized using Multimax. Red branches disagree between the two trees. Related Chrenarchaeota species are highlighted in purple, Fungi are highlighted in Green, and Cryptosporidium is highlighted in Orange. Support values on branches are approximate likelihood ratio test values.

23 Unimax 0.8 Multimax Amoebozoa, 246.5 Apicomplexa 246.6 sub. c” 0.9 9.3 9.4 1.1 8.5 Animals, 8.6 Choanoflagellates 17.5 sub. c” 17.3 c”/c gene 2.1 1.9 c”/c gene duplication 5.7 Fungi 5.4 duplication 122.9 sub. c” 122.9

21.7 Amoebozoa, 21.7 Apicomplexa sub. c Animals, 17.2 Choanoflagellates 18.9 13.4 sub. c 13.5 0.8 0.9 Fungi 38.1 sub. c 36.7 6.6 6.6 c/c! gene c/c! gene duplication Fungi duplication 71.9 sub. c! 75.3

FIGURE 7. Phylogenies of V-ATPase subunits c, c’, and c”. Sequences are samples from across Opisthokonts. The Unimax ML tree is on the left, the Multimax ML tree is on the right. The two trees have identical topologies, but diﬀerent branch lengths and approximate likelihood ratio test values.

24 Unimax Multimax 0.1 Hoplostethus_mediterraneus Hoplostethus_mediterraneus Polymixia_lowei Polymixia_lowei 0.22 Pegasus_volitans 41.52 Chilara_taylori 24.76 Chilara_taylori 0.17 Labrisomus_multiporosus 0.4 Labrisomus_multiporosus 63.81 Menidia_menidia Pegasus_volitans 17.76 Dactylopterus_volitans Menidia_menidia 8.6 Dactylopterus_volitans 73 Aulostomus_maculatus Aulostomus_maculatus 10.25 Epinephelus_adscensionis 1.86 Morone_saxatilis Grammistes_sexlineatus 4.7 Cheilodactylus_variegatus 23.1 Pseudogramma_thaumasium 11.73 0 Hoplolatilus_purpureus Cheilodactylus_variegatus 8.03 Kuhlia_rupestris 16.5 11.03 Hoplolatilus_purpureus Centrogenys_vaigiensis Centrogenys_vaigiensis 12.6 10.6 Haemulon_plumieri 9.8 Haemulon_plumieri 8.6 Epinephelus_adscensionis Perciformes: Morone_saxatilis 27.1 Grammistes_sexlineatus 5.8 Kuhlia_rupestris 6.68 Pseudogramma_thaumasium 29.7 Hemanthias_leptus 16 37.9 Hemanthias_leptus 15.03 2.84 Zalanthias_kelloggi Zalanthias_kelloggi 11.39 Apeltes_quadracus 0.93 Normanichthys_crockeri Percoidei 35.67 Aulorhynchus_flavidus 50.6 62.2 Elates_ransonnetii Cirrhitus_rivulatus Platycephalus_bassensis Parapercis_ramsayi 22.8 Trachinus_draco 11.7 160.1 Gymnodraco_acuticeps 0 Iracundus_signifer Harpagifer_kerguelensis Scorpaenopsis_macrochir Blenniodei 1.13 3.62 45.3 Gymnapistes_marmoratus Caracanthus_maculatus Ablabys_taenionotus 2.3 Taenianotus_triacanthus 6.35 12.03 Aploactisoma_milesii 6 Scorpaena_plumieri 8.96 13.43 16.97 Aetapcus_maculatus 0 Scorpaena_brasiliensis Notothenioidei 6.05 Synanceia_verrucosa 11.8 Scorpaena_guttata 9.74 106.1 Zanclorhynchus_spinifer 1.22 Dendrochirus_brachypterus 7.86 Gramma_loreto Champsodon_atridorsalis Pterois_volitans 25.12 18.95 11.6 Pontinus_longispinis 7.16 Setarches_guentheri Trachinoidei 82.21 Elates_ransonnetii Salarias_fasciatus 5 Sebastolobus_macrochir Platycephalus_bassensis 11.05 Trachyscorpia_cristulata Normanichthys_crockeri 112.5 Helicolenus_dactylopterus 0 Trachinus_draco 9.55 Thysanichthys_sp._WLS-2004 208.7 Peristedion_gracile 0 Sebastes_elongatus Zoarcoidei Peristedion_miniatum 31 Sebastes_matsubarae 8.57 Scorpaena_brasiliensis 1.14 Maxillicosta_whitleyi 20.9 Scorpaena_guttata 0 Diplectrum_formosum 0.4 Scorpaena_plumieri 24.3 Etheostoma_blennioides 0 6.9 7.7 Taenianotus_triacanthus 8.24 Perca_flavescens 51.8 Caracanthus_maculatus Plectrogenium_nanum Iracundus_signifer 4.55 11.9 Apeltes_quadracus 2.09 Scorpaenopsis_macrochir 30.7 Aulorhynchus_flavidus 9.05 104.25 Dendrochirus_brachypterus Cirrhitus_rivulatus Pterois_volitans 12.69 166.2 Peristedion_gracile 112.29 Helicolenus_dactylopterus 5.22 Peristedion_miniatum 3.49 31 Sebastes_elongatus Parapercis_ramsayi Sebastes_matsubarae 5.82 Gymnodraco_acuticeps 10.05 9.37 0 Thysanichthys_sp._WLS-2004 164.0 Harpagifer_kerguelensis 5.53Sebastolobus_macrochir 1.03 9.3 Synanceia_verrucosa 5.21 Trachyscorpia_cristulata 16.4 Aetapcus_maculatus 14.0 9.47 Setarches_guentheri 45.9 Aploactisoma_milesii Pontinus_longispinis 8.98 Ablabys_taenionotus Maxillicosta_whitleyi Gymnapistes_marmoratus 1.14 0 Diplectrum_formosum Zanclorhynchus_spinifer 24.35 8.9 Etheostoma_blennioides 9.6 Gramma_loreto Champsodon_atridorsalis Perca_flavescens 23.4 19.3 9.27 Plectrogenium_nanum Bembras_japonica Bembras_japonica Salarias_fasciatus 5.33 Anoplopoma_fimbria Anoplopoma_fimbria 7.26 36 Bathymaster_signatus 9.3 36 Bathymaster_signatus 66.18 Lycodes_diapterus 69.4 Lycodes_diapterus Zaniolepis_frenatus Zaniolepis_frenatus 2.5 70.7 Hexagrammos_decagrammus 2.5 70.7 Hexagrammos_decagrammus Pleurogrammus_azonus Pleurogrammus_azonus 6.9 Trichodon_trichodon 6.9 Trichodon_trichodon 7.0 Aptocyclus_ventricosus 7.0 Aptocyclus_ventricosus 5.6 27.9 Cyclopterus_lumpus 5.6 27.9 Cyclopterus_lumpus 29.5 Careproctus_melanurus Scorpaeniformes 29.5 Careproctus_melanurus Liparis_mucosus Liparis_mucosus 17.25 Artedius_fenestralis 17.3 Artedius_fenestralis 0.03 Radulinus_asprellus 0.03 Radulinus_asprellus 14.18 Icelinus_filamentosus 0 Icelinus_filamentosus Myoxocephalus_polyacanthocephalus 22.24 14.18 Myoxocephalus_polyacanthocephalus 7.08 Taurulus_bubalis 7.1 Taurulus_bubalis Triglops_scepticus 22.24 Triglops_scepticus 7.12 Cottunculus_thomsonii Cottunculus_thomsonii Dasycottus_setiger 7.12 Dasycottus_setiger 96.5 Cottus_carolinae Cottus_carolinae 6.72 Cottus_bairdii 6.72 Cottus_bairdii Cottus_poecilopus 96.5 25.4 25.4 Cottus_poecilopus 0.0 Comephorus_baicalensis 0 Comephorus_baicalensis Batrachocottus_baicalensis Batrachocottus_baicalensis Asprocottus_pulcher Asprocottus_pulcher Leptocottus_armatus Leptocottus_armatus Jordania_zonope Jordania_zonope Hemitripterus_americanus Hemitripterus_americanus Hemilepidotus_jordani Hemilepidotus_jordani 11.15 Nautichthys_pribilovius 11.15 Nautichthys_pribilovius Rhamphocottus_richardsoni Rhamphocottus_richardsoni Hypsagonus_quadricornis Hypsagonus_quadricornis 7.09 Sarritor_frenatus 7.09 Sarritor_frenatus 6.2 Xeneretmus_latifrons 6.2 Xeneretmus_latifrons

FIGURE 8. Phylogenies of the TMO-4c4 gene family in Actinopterygii (ray- ﬁnned ﬁsh). The Unimax ML tree is on the left; the Multimax ML tree is on the right. Perciformes and suborders are highlighted in blue, purple, and green. Scorpaeniformes are highlighted in orange. Support values on internal branches are approximate likelihood ratio test values.

25 0XOWLPD[PRUHDFFXUDWH 0.0

8QLPD[PRUHDFFXUDWH

0.1 nucleotides 0.2 DPLQRDFLGV ¨ WUHH error 0.0 0.0

-0.025 8 16 32 64 96 128 160 200 256 384 512 -0.12 8 16 32 64 72 96 128 192 256 512 1WD[D

0.01 0.005 DPLQRDFLGV nucleotides

0.0

-0.001 8 16 32 64 128 192 256 512 712 1024 -0.005 8 16 32 64 384 512 576 640 720 1024

FIGURE 9. Tree error for Unimax and Multimax ML trees. N taxa, the number of sequences in the simulated alignment. ∆ tree error, the mean diﬀerence of UM relative tree error and MM relative tree error (see Methods). Values above 0.0 indicate that MM was more accurate; values below 0.0 indicate that UM was more accurate. Error bars are standard error of the mean.

26 Steroid hormone receptors Thioredoxins -33215 -39600 Unimax -33215 -39590 ML ¨ ¨

-33230 -39630 50 82 -34036 Multimax 45 68 -40553 0 trees 82 0 68 Mcm1 V-ATPase -7510 -16477 -16475 -7510 ¨

¨ -16483 -7515 35 55 65 80 -7977 -16973 0 80 0 55

TMO-4c4 Proteo 16S -9400 -55176

-9400 -54175

¨ ¨ -9500 -54200 20 28 15 30 -10028 -55461 0 28 0 30

FIGURE 10. Maximum likelihood values of phylogenies for empirical alignments. ML, the maximum log-likelihood of the best tree during the tree search. trees, the unique topologies accepted by UM and MM during their search of tree space. ∆, the ﬁnal diﬀerence in log(L) scores between the Unimax and Multimax solutions.

27 MM on t ˆUM

ˆtUM

UM on t ˆMM Unimax

t ˆtMM start Multimax

FIGURE 11. Schematic illustration of reciprocal restart analysis on ML trees. ML search begins from an initial tree (tstart), typically found using a neighbor-joining algorithm. UM and MM drive the search into different regions of tree space. UM’s ML tree (tÛM) is different from MM’s ML tree (tˆMM). In this reciprocal restart analysis, MM-driven search was restarted from tÛM and UM-driven search was restarted from tMM. In no examined cases, could UM or MM drive tree search to improve the other algorithm’s ML tree.

28 ˆtUM k N optima tUM .

Unimax 1 tUM t ˆtMM start Multimax

alignment k N optima N>tUM steroid-hormone receptors 31 12 4 Mcm1 20 7 4 thioredoxins 139 107 33 16S 46 33 5 TMO-4c4 30 29 7 V-ATpase subunits c, c’, c” 31 0 0

FIGURE 12. Schematic illustration of path restart analysis on UM trees. MM was 1 k restarted from all trees along UM’s path through tree space (tUM through tUM). The table shows path restart data from six sequence families evolved under real conditions (see Methods). k, the number of trees unique to the UM path, excluding tÛM. N optima, the number of new optima found by MM. The fourth column, N>tÛM, is the number of new optima found by MM that also had log-likelihood scores better than the UM ML tree (tÛM).

29 sequence alignments in which UM and MM ML trees disagreed, I restarted UM from the MM-driven ML tree and restarted MM from the UM-driven ML tree. For all ﬁve alignments, UM and MM were unable to drive the tree search algorithm to

ﬁnd a better topology, indicating that UM-driven and MM-driven search indeed found topological optima (but the MM optimum was better). Further restart analysis revealed the second theory to be true: Unimax led to suboptimal trees.

Speciﬁcally, I restarted MM from every tree along the UM search path; MM-driven search then found several additional optima with higher ln(L)s than the original

UM optimum (Fig. 12.). This reveals that UM-driven search repeatedly chose poor topologies.

Separable optimization impairs branch length accuracy

In order to determine why UM-driven search chose suboptimal trees, I examined the choices UM and MM made immediately before they diverged in tree space (Fig. 13.). Tree search began from an initial tree (tstart) with initial branch lengths (blNJ), constructed using a neighbor-joining algorithm. UM and

MM optimized the values of blNJ to arrive at different ML branch lengths blˆ UM and blˆ MM, respectively. From the ML branch lengths on tstart, the tree search algorithm then evaluated a list of candidate tree swaps for the next tree (t1). Each potential swap was scored according to its estimated likelihood improvement. PhyML calculated a swap score for each proposal by first making the swap, optimizing only the branch lengths directly affected by the swap, recording the likelihood of this partially-optimized tree, and then restoring the tree to its pre-swap condition.

Because proposed trees are partially optimized – rather than fully optimized – swap scores are highly dependent on the ML branch lengths of the pre-swap tree. So if

30 one uses an optimization algorithm that systematically ﬁnds erred branches, then one might expect to make poor topology swaps.

In order to determine if UM branch lengths were erred, I measured the accuracy of the sum of all branch lengths on the ML trees. I found UM ML branches to be less accurate than MM ML branches (Fig. 14.). Speciﬁcally, the sum of all branch lengths on the UM ML trees were too long, especially on pinnate- shaped trees. On the six empirical alignments, branch length error cannot be decisively tested because true branch lengths are unknown. Instead, I recorded the sum of all branch lengths on the UM and MM trees over the duration of tree search. Consistent with my observations from simulated alignments, the UM ML trees were systematically longer than the MM ML trees (Fig. 15.).

I next determined if UM branch lengths were less accurate simply because

UM failed to climb peaks in the space of continuous parameters. I reciprocally restarted UM from blˆ MM and MM from blˆ UM, disabling the tree search algorithm (Fig. 13.). For all six empirical alignments, UM and MM were unable to ﬁnd better reciprocal ML branch lengths on tstart, indicating that blˆ UM and blˆ MM were indeed optima in the space of continuous parameters.

Taken together, these observations suggest that UM chose suboptimal trees because UM’s ML branches were erred.

Separable optimization ﬁnds ML values that are order dependent

Unimax’s ML branch lengths depend on the sequential order in which parameters are optimized. To illustrate this point, I implemented alternative versions of UM with diﬀerent sequential orderings for branch lengths. I measured how these alternative optimization orders aﬀected ML values for a branch labeled

31 1 tUM

MM on bl bl ˆ UM ˆUM UM on bl ˆMM Unimax

Multimax 1 blNJ bl t ˆ MM MM tstart

FIGURE 13. Schematic illustration of reciprocal restart analysis on initial trees. Tree search began on an initial tree tstart, with initial branch lengths blNJ. UM and MM found ML branch lengths on tstart, labeled blˆ UM and blˆ MM, respectively. When MM was restarted from blˆ UM, MM was unable to improve the values of blˆ UM. Similarly, UM restarted from blˆ MM was unable to improve the values of blˆ MM. UM- driven search and MM-driven search next chose diﬀerent topology swaps, labeled 1 1 tUM and tMM.

32 Unimax ML Multimax ML

3.5 nucleotides 3.5 amino acids length error = estimated BL true BL

1.0 1.0

0.0 0.0 8 16 32 64 96 128 160 200 256 384 512 8 16 32 64 72 96 128 192 256 512 N taxa

nucleotides amino acids 1.25 1.25 1.0 1.0

0.0 8 16 32 64 128 192 256 512 712 1024 0.0 8 16 32 64 384 512 576 640 720 1024

FIGURE 14. Tree length accuracy. Length error is ratio of the estimated (ML) branch lenghs to the true branch lengths. N taxa is the number of sequences in the simulated alignment. Error bars are standard error of the mean.

33 Multimax Unimax

Steroid hormone receptors Thioredoxins tree ¨ length ¨ tree search 0FP V-ATPase subunits

¨ ¨

TMO-4C4 3URWHREDFWHULD6 7

¨ ¨ 3

FIGURE 15. The length of ML trees over the duration of tree search. Tree search on the horizontal axis corresponds to the set of unique topologies explored by Unimax and Multimax. Tree length on the vertical axis is the sum of all branch lengths.

34 Unimax, Unimax Multimax bjarÀUVW Unimax, random order initial length ML bjar subs/site 0.1030.104 0. 0. 0.121 117 118

FIGURE 16. Unimax suﬀers from an order-dependence problem. Multimax and three versions of Unimax arrived at diﬀerent ML lengths for the branch bjar, attached to the sequence AngJapARb in Fig. 4.. The initial length of bjar was determined by the neighbor-joining algorithm.

35 bjar, attached to the protein sequence for the Anguilla japonica androgen receptor beta (AngJapARb) in the tree of steroid hormone receptor proteins (Fig. 4.). The branch bjar began with the length 0.104 substitutions per site (subs/site) in the initial neighbor-joined tree (Fig. 16.). MM reduced bjar to 0.13 subs/site. UM, in contrast, ﬁrst optimized branches 1 through 444, then increased bjar (numbered 445) to length 0.121 subs/site, and then optimized branches numbered 446 through

473. I modified UM to optimize bjar first, and then optimize the other branches afterwards. This version of UM arrived at an ML length for bjar of 0.117 subs/site. I next modified UM to optimize branch lengths in a random order. This version of

UM optimized bjar to 0.118 subs/site. The fact that three diﬀerent UM orderings arrived at three diﬀerent ML lengths for bjar indicates that UM’s ML solutions depend on the order in which parameters are sequentially optimized.

Eﬀect of Multidimensional Optimization on Ancestral Sequence Reconstruction

In order to determine if UM’s assumption of parameter separability has downstream consequences for ancestral sequence reconstruction, I inferred ancestral sequences on the UM and MM ML trees for the steroid hormone receptor alignment. For several ancestors of interest, I compared their ML states and their statistical support on the UM and MM ML trees.

I observed that UM and MM ML trees yielded signiﬁcantly diﬀerent ancestral sequences. For the last-shared ancestor of all Androgen receptors (AncAR, node251), the ancestral sequences disagreed at 5% (17 of 359) of the sites (Fig.

17.). Twelve of these sites form a motif at the beginning of the AR sequence; this motif is missing in sharks, skates, gar, and other early-branching ﬁsh. The inferred presence or absence of this motif in ancestral sequences is primarily contingent on

36 hr h Lsae ge,adoesaevco u o h te introduces – other the not but – state. vector ML state the one about and uncertainty infer agree, signiﬁcant sequences states both ML disagree. but the tree trees, where UM between ( and disagree support tree states low MM ML complementary the the deletions. on posterior where or states its sites insertions ML expresses indicates indicate the character sites each where Empty of sites site. height indicates indicated The the 4.. names at Fig. Node probability in sequences. labels receptor to androgen correspond ancestral Reconstructed 17. FIGURE Multimax Multimax Multimax Multimax Unimax Unimax Unimax Unimax P P P P P P P P 1.0 0.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 0.0 0.0 0.0 Node 251(AncAR) G G R R K K F F F Node 252 F F Node 254( Node 253 K K Q Q G G K G K G G G K 201 201 K H 101 H 101 L L P L L H P P H P P H L H 301 301 201 201 H 1 1 H 201 101 101 L 201 1 301 301 1 201 Y 1 1 1 L 1 201

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204 204 T S S G G A A

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313 313 C C A 13

13 H C A A 113 A C 13 A A 13 13 313 313 13 13 213 213 113 13 H 213 213 113

113 113 H 313 H 313 313 313 113

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316 T

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238 238 I I R R I 138 138 I K K R R K K K R K R R 338 338 38 38 338 338 238 238 138 138 38 38 338 338 238 238 138 138 38 38 338 338 238 238 138 R 138 38 38 I I I I A A 239 239 I I N N I 139 139 I H S S N H N Q Q A A N S S Q N 339 339 239 239 N 39

39 N A A Q Q Q Q Q A 139 39

239 S 239 339 339 39 39 239 139 39 A 39 39 139 139 S 339 239 139 139 S 339 S 339 339 L 240 L 240 L 140 L 140 L L L K L K L L L K L L L L K K K K K L V 340 V 340 240 240 240 140 40 240 40 40 V 40 40 340 340 V 40 40 240 140 40 240 140 140 140 A 140 340 A 340 37

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342 V 142

342 V

342 V F 243 F 243 D 143 D 143 F F P P F F F P P D F C D C C D 343 343 243 243 P 43 D D C C 43 C D C C 343 343 43 43 243 243 143 143 43 P 243 43 43 43 P 243 143 143 143 343 P 343 343 343 143 S 244 S 244 D 144 D 144 K K S S D D K K S S A D A 344 344 244 244 S D D

S 44 44 A K 144 D A A A 44 A A 44 44 344 344 44 44 244 244 144 44 K 244 244 144

144 144 K 245 245 344 K 344 344 344 144 Q 145 Q 145 S S S S S Q 345 345 245 245 S S S 45 Q 145 45 45 345 345 245 145 245 Q Q Q 45 45 245 45 45 45 145 145 345 345 345 345 245 145 Q 145 I I I I

246 246 I I I I M 146 M 146 L L R R R L L R R R R R M M 346 346 246 246 46 L 46 M 146 L M M 246 146 246 46 346 346 246 M 46 46 46 46 L 46 246 146 146 L 146 247 247 346 346 346 346 146 T 147 T 147 N N N N N N A A N 347 347 N 247 247 47 47 A A 247 147 T 247 47 47 47 347 347 47 47 247 147 T 47 247 147 T A T 147 147 T

147 T 347 A 347 A 248 248 347 A 347 V 148 V 148 L D L D D L G G D D D L 348 348 D D 248 248 48 48 G G 148 V 348 348 48 48 248 248 148 V 48 248 48 48 48 V G 248 148 V 148 148 V V 348 G 348 148 G 348 G 348 249 249 149 149 M M C C C M 349 349 M 249 249 49 49 C C C C C 349 349 249 149 149 249 49 49 49 249 49 M I I 49 49 249 149 149 149 M 349 149

M 349 349 M 349 I I I I I 250 250 I

150 Q 150 Q V V Q

350 350 250 250 T Q 50 T 50

150 T

Q 50 V V 250 150 250 Q Q T T 350 350 50 50 250 50 T T T 50 50 250 Q 150 150 A 150 V A 350 150 350 V V 251 251 350 V 350 H 151 H 151 K K K 351 351 K 251 251 51

51 K Y

Y 151 51 351 351 251 251 151 K 251 51 51 51 51 51 251 151 Y Y Y K

151 151 Y 351 K 351 351 351 151 I I I I I I I 252 252 I S 152 S 152 P P D D P P D S D D D S D D 352 352 252 252 P 52 52 S P S S 152 52 352 352 52 52 252 252 152 52 252 S 52 52 P 252 152 152 152 P 253 253 352 352 352 352 152 W 153 W 153 K K K K K K K K 353 353 253 253 L 53 W 53 L 253 153 W 153 253 53 53 53 353 353 53 53 253 53 W W W 253 153 153 153 353 353 353 353 153 W I I I I I

254 254 T I I I M 154 M 154 L L F F L L F F L L I I F L L M F L M V T

254 254 V 354 354 S 54 F

54 L M 154 54 V M M T 254 154 254 T V F M

354 354 L 54 54 254 T 54 F

54 54 L 154 154 354 354 354 354 254 T 154 154 I I I I I I I I I I I I 255 255 I G 155 155 G F F R R F F I I R R R R R R G F

355 355 255 255 G 55

55 F

155 G 55

G 255 255 G 155

355 355 F 55 55 255 55 G 55 55 155 155 355 F 355 355 355 255 155 155 I I P P I I I 256 256 I V 156 156 V P P R H H R L L R P P P R L R R H H R R P L L 356 356 256 256 L 56

56 H 156 V 56 V 356 356 56 56 256 256 156 56 H 256 56 56 256 156 V H V 156 156 V V 356 H 356 356 356 156 V 257 V 257

157 M 157 M K K K K K K K K K K M K K M V 357 357 257 257 V 57

57 K 157 M V V M M 57

257 K 257

157 M 357 357 57 57 257 V 57 57 57 257 V 157 157 157 K 357 K 357 357 357 157 258 E 258 E V 158 V 158 E E N N E E E N E N N N N 358 358 N 258 258 58 Q Q 58 158 V 58 358 358 58 58 258 258 158 V 58 258 V 58 58 Q 258 158 V 158 158 V V Q 358 158

Q 358 358 Q 358 G 259 G 259 F 159 F 159 F F F G G F F C C F C G 359 359 259 259 59 G 59 G C C C C C 59 59 359 359 159 159 59 59 359 359 259 259 159 159 59 359 359 259 G 259 159 159 59 260 L 260 L A 160 A 160 L L P P L L P L L P P P P P 260 260 A 60 60 260 160 A 160 260 60 60 60 260 60 A A 60 60 A 260 160 160 160 A 160 K 261 K 261 L 161 L 161 K K S S K K K S M K M S S S S S 261 261 61 61 161 M M M 261 161 261 61 61 61 161 161 61 61 261 61 261 161 M 161 N N S 262 S 262 G 162 G 162 N N N N N C C N C 262 262 G G 62 62 C C C C C 162 262 162 262 G 62 G G 62 62 262 62 62 62 162 162 262 162 G 162 Q 263 Q 263 W 163 W 163 R R R R R R R R Q 263 263 Q Q Q 63

W 63 Q

W 163 263 163 Q 263 63 63 63 63 63 263 63 263 W W W 163 163 163 163 W K 264 K 264 R 164 R 164 L L K K L R L L L K K R L L K K R R 264 264 R 64 64 164 164 64 64 264 264 164 164 64 64 264 264 164 R 164 64 64 F Y 265 Y 265 S 165 S 165 F R R F F R R R R R R S K S Y 265 265 Y 65 65 S Y Y S S 165 65 S Y 265 165 265 65 65 165 165 65 65 265 65 265 Y 165 165 F 266 F 266 Y 166 Y 166 F F K F F K F K F K K K K K 266 266 66 66 Y 266 266 166 Y 166 266 66 66 66 66 66 66 266 166 Y Y Y 166 166 166 Y D 267 D 267 K 167 K 167 D D K K C D D C R D K C D K K

267 267 R

K 67 C C 67 C C 67 C 67 167 167 67 67 267 267 167 167 67 267 267 167 167 67 E 268 E 268 N 168 N 168 E E F F F E E N E F F F F F N E H N N 268 268 N H 68 68 168 168 68 68 268 268 168 168 68 68 268 268 168 N 168 68 68 L 269 L 269 V 169 V 169 L L E E E L L E E E L E E L Q A 269 269 A 69 A 69 V A 169 A

A 69 69 69 269 269 169 V 69 269 69 69 269 169 V V 169 169 V 169 V R 270 R 270 N 170 N 170 R R N R R N R R N A A N 270 270 N N 70 A 70 A A 170 A 70 A 70 A 70 170 170 70 70 270 270 170 70 270 270 170 170 M M T 271 271 T S S A 171 A 171 M M G S G S G M M

271 271 A

M 71 71 S G G A G M G G S S 171 71 271 S 71 71 271 171 71 271 A A 71 71 A 271 171 171 171 A 171 N N S S

272 272 T 172 R 172 R N N R R N N M M N R N R R M 272 272 R 72 M M 72 M M 72 M 72 172 172 72 72 272 272 172 172 72 272 272 172 172 72 Y 273 Y 273

173 M 173 M M Y Y 273 273 M 73 T 73 T M Y Y 173 73 T M M

Y 273 173 273 M T T T 73 73 273 73 Y T 73 T 73 173 173 273 173 173 274 274 174 174 L L L I I L L L L 74 L 74 274 274 174 174 74 74 274 274 174 174 74 74 274 274 174 174 74 74 I I H N N I I I I 275 Q 275 L 175 L 175 K K H N N L L K K L K Q Q G G L L L G Q 275 275 Q 75

Q 75 G G G G G 75 75 175 175 75 75 275 275 175 175 75 275 Q 275 175 175 75 E 276 E 276 Y 176 Y 176 E E E E E E G A A G

276 276 A 76 Y 76 Y A A 176 A A A 276 176 276 76 76 76 276 76 Y Y 76 76 Y 276 176 176 176 176 Y L 277 L 277 F 177 F 177 L L R F R L L F R L L F R R R R R F F 77 77 277 277 177 177 77 77 277 277 177 177 77 77 277 277 177 F 177 77 77 D 278 D 278 A 178 A 178 K K D D K K K K D D K K D D

278 278 A 78 A 178 78 78 278 278 178 278 A A 78 78 A 78 78 78 178 178 278 178 A 178 R 279 R 279 P 179 P 179 L R R L L P R R P L L L L L R R P P P 79 79 279 279 179 179 79 79 279 279 179 179 79 79 279 279 179 P 179 79 79 L 280 L 280 180 D 180 D K K K D K K K D K K D V 280 280 V D D 80 80 180 D V 280 V 180 280 80 80 80 280 V 80 80 80 180 180 280 V 180 180 I I V 281 V 281 L 181 L 181 K L K L K I I L K K K L L K K L V 281 281 V 81 81 V 281 181 281 V 81 81 81 281 V 181 81 81 181 181 81 281 V 181 181 I I I I I S S 282 282 I V 182 182 V L L L L L L S S L M M S S

282 282 S 82 82 S 182 V 82 82 282 282 182 V 82 282 82 82 82 282 182 V V 182 182 V 182 V F F I I I I I I I I I I I C I Y 283 Y 283 F 183 F 183 F F P F P F F F P F F F L P P P C F F P P F Y 283 Y 283 S 183 83 83 83 83 183 183 83 83 283 Y 283 183 83 283 283 183 183 83 R R K K G 284 284 G 184 184 L F L R R K K F L F L L L R R K K M R K R K

284 284 V V M M

Q 84 Q

M 84 Q 184 84 V 284 184 Q 284 V V V

V 284 84 84 V 184 184 84 84 84 284 184 184 I I R R I I I I I 285 285 I N 185 N 185 F F R R F E N R F F R F F F N E R R N N N

285 285 G 85 N 85 85 85 185 185 85 85 285 285 185 185 85 285 G 285 185 185 85 K 286 K 286 186 186 L L K K L L L L L L K K K N 86 86 286 286 186 186 86 86 286 286 186 186 86 86 286 K 286 186 186 86 86 N N S T 287 T 287 187 187 N N N N S N N N D D N N N D N S N A N N D 287 T 287 87 87 87 187 187 87 87 287 287 187 187 87 87 287 287 187 187 87 A P P S S A A T 288 T 288 188 188 P P P P S S P P A A S S Green V S T 288 288 T V 88

88 S A A V

A 188 88

288 T 288 T

188 A V V V 88 88 88 88 288 T 88 188 188 288 T 188 188 I I I I I I S S I I T 289 289 T 189 189 L L L S S L L L L L A A S S T 289 T 289 89 A 89 A

A 189 89 T T 289 189 A 289 89 89 289 T 89 89 89 189 189 289 T 189 189 N N S 290 S 290 190 190 E H N N S E E E S H H N N E E H N Q S S Q H N S Q

290 290 Q 90 S 90 Q Q Q 90 90 190 190 90 90 290 290 190 190 90 290 Q 290 190 190 90 C C 291 Y 291 191 191 N N S S S S N N C C D N S A D A S S N S S S S C S S 291 291 A

S 91

C 91 A A 191 A A 91 91 A 191 191 91 91 291 291 191 91 291 291 191 191 91 S 292 S 292 E E D 192 D 192 E E P S S E P E E S S E 292 292 S 92 92 192 192 92 92 292 292 192 192 92 92 292 S 292 192 192 92 92 I I I I I I I I Q 293 Q 293 H 193 193 H E E E E E E H E E H Q Q H H 293 293 H Q Q 93 H Y 93 Q Y 193 Q 93 93 293 293 193 93 Q 293 93 93 93 Q Y 193 193 Y 293 193 193 R 294 R 294 R 194 R 194 R R P P P R R R R R P P P P P R R R R 94 94 294 294 194 194 94 94 294 294 194 194 94 94 294 294 194 R 194 94 94 F 295 F 295 M 195 M 195 F F E E E F F D D F E E E E E D F D D D M 295 295 M 95 95 195 M

M 95 M 295 195 295 95 95 195 195 95 95 295 95 295 195 M 195 F Y 296 Y 296 H H Q 196 196 Q H M H M Y Y M Q

296 296 V H H V M M 96 M Q

96 H Y Y

196 V

Y 96 296 196 296 Q Y V V V Q Q 96 96 296 96 V V 96 96 196 196 296 196 Q 196 Q 297 Q 297 197 197 Q Q V

297 297 V Q 97 Q 97 Q 197 V 297 197 Q 297 97 V V V 97 V 97 V 197 197 97 97 297 97 297 197 197 L 298 L 298 198 198 L L L L L L N N N N Y 298 298 98 Y 98 C C 198 98 Y Y Y 98 98 198 198 98 98 298 298 198 98 Y 298 298 198 198 T 299 T 299 199 199 A A T 299 299 T 99 A 99 A A 199 A A A 99 T 299 199 299 T 99 99 99 99 299 T 99 199 199 299 T 199 300 300 199 200 200 100 100 300 300 200 200 100 100 300 300 200 200 100 100 300 300 200 200 100 100 niae sites indicates Orange Red the phylogenetic placement of gar ﬁsh. The UM tree placed gar deep inside the

Neopterygii clade, and AncAR was therefore reconstructed to contain the motif.

On the MM tree, however, gar was placed in a more basal position and AncAR was reconstructed to not contain this motif. The UM and MM trees also disagreed about the presence of this motif at two nodes descendant from AncAR (nodes 252 and 253). Aside from this motif, the last-shared ancestor of Neopterygii (node 254) was reconstructed with nineteen disagreeing sites between the UM and MM trees.

At seven of these sites, the dissenting state was not signiﬁcantly supported in the reconstruction on the other tree.

The role of line search, versus quadratic interpolation

My particular implementation of MM using BFGS diﬀers from UM using

Brent’s method not only in their treatment of parameter separability. Speciﬁcally,

BFGS and Brent’s method use different mechanisms to move forward in parameter space. BFGS uses line search to move in a direction informed by the functional gradient, whereas Brent’s method uses quadratic interpolation to move in single dimensions. I determined that this difference is not the primary reason that UM and MM drove the tree search to different ML trees. I implemented an alternative version of UM that uses line search, such that UM (with line search) and MM would be identical except in their assumption of separability. I used this alternative version of UM to infer ML phylogenies for the six empirical alignments in this study. I observed that UM using line search and UM using quadratic interpolation led to the same ML topologies for all six empirical alignments. Further, the final

ML values between the two methods diﬀered by less than 0.0001 log likelihood units. This means that the use of line search, rather than quadratic interpolation, is

38 not the reason that UM and MM led to diﬀerent ML trees. This result indicates that UM’s assumption of parameter separability is the primary cause of UM’s phylogenetic impairment.

Discussion

My results demonstrate that UM’s assumption of parameter separability impairs its ability to find accurate ML phylogenies. By using an alternative ML algorithm that does not assume separability, I was able to find more accurate and higher-likelihood trees. UM repeatedly made suboptimal topology choices during its search of tree space. UM ML branch lengths are less accurate than MM ML branch lengths, and this difference is likely to be a mechanism by which UM-driven tree search choose poor topologies. The assumption of parameter separability not only affects phylogenetic accuracy, it also affects the downstream inference of ancestral states.

Prior work has shown that ML phylogenetic inference is NP-hard (Chor and

Tuller (2005); Roch (2006)). Consequently, I know with certainty that no ML search heuristic running in polynomial time – including UM and MM – can be guaranteed to ﬁnd the true ML phylogeny for every alignment. Indeed, I observed that UM and MM were not accurate all the time, but MM was more accurate more often.

An open question in phylogenetics is whether likelihood landscapes are simple or complex (Fukami and Tateno (1989); Steel (1994); Rogers and Swoﬀord (1999);

Yang (2000); Chor et al. (2000); Billera et al. (2001)). I observed that UM and MM arrived at unique ML trees, and at unique ML branch lengths for ﬁxed trees. These results indicate that the phylogenetic likelihood landscape is complex on both the

39 space of topologies and also on the subspace of continuous model parameters, at least for the conditions studied here.

Although MM has been previously implemented in software for ML phylogenetics, its performance heretofore has not been rigorously investigated.

The software suite PAML implements MM using BFGS as an option called

“simultaneous update.” PAML provides this option as a sort of algorithmic curio, without any deeper analysis of the algorithm’s eﬃcacy and accuracy.

Because BFGS – and MM in general – has not been deeply studied as a tool for phylogenetics, there has not been any real motivation for evolutionary biologists to use PAML’s implementation of BFGS. Rather, the scientiﬁc community has continued to use the default UM implementations provided by PhyML, RaxML, and GARLI (Guindon et al. (2010); Stamatakis (2006); Zwickl (2006)); perhaps this is because PhyML, RaxML, and GARLI provide other useful features, including support for large sequence datasets and command-line facilities for high-throughput batch analysis. I implemented our own version of BFGS, rather than using PAML’s code, in order to ensure computational eﬃciency and in order to instrument the algorithm to report useful metrics, including likelihood gradients and neighbor- swap tree scores. Our MM implementation is built within the open source code for

PhyML. I encourage you to use our software. It is available at the following URL: http://markov.uoregon.edu/software/m3l.

40 CHAPTER III

ANCESTRAL RECONSTRUCTION AND TREE UNCERTAINTY

In this chapter, I show that ancestral sequence reconstruction is robust to phylogenenetic uncertainty. Speciﬁcally, I discuss the relationship between poorly- supported phylogenies and the downstream accuracy of ancestral reconstruction.

This work was previously published in the Oxford Journal Molecular Biology and

Evolution (Hanson-Smith et al. (2010)).

Ancestral sequence reconstruction (ASR) is widely used to formulate and test hypotheses about the sequences, functions, and structures of ancient genes.

Ancestral sequences are usually inferred from an alignment of extant sequences using a maximum likelihood (ML) phylogenetic algorithm, which calculates the most likely ancestral sequence assuming a probabilistic model of sequence evolution and a speciﬁc phylogenytypically the tree with the ML. The true phylogeny is seldom known with certainty, however. ML methods ignore this uncertainty, whereas Bayesian methods incorporate it by integrating the likelihood of each ancestral state over a distribution of possible trees. It is not known whether

Bayesian approaches to phylogenetic uncertainty improve the accuracy of inferred ancestral sequences. Here, I use simulation-based experiments under both simpliﬁed and empirically derived conditions to compare the accuracy of ASR carried out using ML and Bayesian approaches. I show that incorporating phylogenetic uncertainty by integrating over topologies very rarely changes the inferred ancestral state and does not improve the accuracy of the reconstructed ancestral sequence.

Ancestral state reconstructions are robust to uncertainty about the underlying tree because the conditions that produce phylogenetic uncertainty also make the

41 ancestral state identical across plausible trees; conversely, the conditions under which different phylogenies yield different inferred ancestral states produce little or no ambiguity about the true phylogeny. These results suggest that ML can produce accurate ASRs, even in the face of phylogenetic uncertainty. Using Bayesian integration to incorporate this uncertainty is neither necessary nor beneficial.

The properties and evolution of ancient genes and proteins can seldom be directly studied, because such molecules are rarely preserved intact over very long periods of time. In 1963, Pauling and Zuckerkandl proposed that ancestral molecules could one day be “resurrected” by inferring their sequences and then synthesizing them (Pauling and Zuckerkandl (1963)). Decades later, the methods of ancestral sequence reconstruction (ASR) have emerged as important tools for examining the trajectory of molecular sequence evolution and testing hypotheses about the functional evolution of ancient genes (Thornton (2004); Liberles (2007);

Dean and Thornton (2007)). Among numerous examples, ASR has been used in the last decade to investigate the evolution of elongation-factor proteins (Gaucher et al.

(2003), Gaucher et al. (2007)), steroid hormone receptors (Thornton et al. (2003),

Bridgham et al. (2006), Ortlund et al. (2007)), visual pigments (Shi and Yokoyama

(2004); Chang et al. (2002)), ﬂuorescent proteins (Ugalde et al. (2004)), and alcohol dehydrogenases (Thomson et al. (2005)).

Although the ﬁrst ASR practitioners used parsimony methods (e.g., Jermann et al. (1995)), most modern studies use maximum likelihood (ML) (Yang et al.

(1995); Koshi and Goldstein (1996); Pupko et al. (2000)). ML begins with an alignment of extant gene sequences, a phylogeny relating those sequences, and a statistical model of evolution. For each internal node in the phylogeny and each site in the sequence, the likelihood of each possible ancestral state—deﬁned as the

42 probability of observing all the extant states given that ancestral state, the tree, and the model—is calculated. The ML ancestral state is the state with the highest likelihood. Conﬁdence in any ancestral state inference is typically expressed as its posterior probability, deﬁned as the likelihood of the state (weighted by its prior probability) divided by the sum of the prior-weighted likelihoods for all states.

The ML approach to ancestral reconstruction assumes that the alignment, tree, model, and model parameters are known a priori to be correct. In practice, this assumption is often not valid; for many real-world datasets, alternatives to the ML tree and parameter values cannot be ruled out. To accommodate these sources of uncertainty, Bayesian methods have been proposed. Whereas ML assumes the most likely estimate of the tree and model parameters, Bayesian approaches incorporate uncertainty by summing likelihoods over a distribution of possible trees or parameter values, each weighted by its posterior probability.

Pagel et al. proposed a Bayesian method for integrating topological uncertainty into inference of ancestral states for binary and other discrete characters (Pagel et al. (2004)). Schultz and Churchill proposed a Bayesian method to integrate uncertainty about the parameters of the evolutionary model into discrete character reconstructions (Schultz and Churchill (1999)). For inference of ancestral DNA and protein sequences, Huelsenbeck and Bollback developed a Bayesian method to integrate uncertainty about the tree topology, branch lengths, and model parameters (Huelsenbeck and Bollback (2001)).

It is not known how Bayesian approaches affect the accuracy of reconstructed ancestral sequences. Here I focus on the specific effects of one source of uncertainty—the phylogeny. There have been a few attempts to characterize the robustness of reconstructed ancestral sequences with respect to phylogenetic

43 uncertainty in speciﬁc cases: Gaucher et al. reconstructed ancestral elongation factor proteins on two plausible phylogenies (Gaucher et al. (2003)), and Bridgham et al. reconstructed the ancestral corticosteroid receptor on all trees within the 95% conﬁdence interval from a Bayesian phylogenetic analysis (Bridgham et al. (2006)).

In both cases, the maximum a posteriori ancestral sequences changed very little when different phylogenies were assumed, and the functions of the reconstructed proteins in experimental assays were also unchanged. Huelsenbeck and Bollback used simulations to show that integrating uncertainty about the phylogeny, branch lengths, and model parameters can affect the posterior probabilities of ancestral states (Huelsenbeck and Bollback (2001)), but they did not study the effect of integration on the inferred maximum a posteriori state or the accuracy of those inferences.

To determine the causal eﬀects of integrating over phylogenetic uncertainty on

ASR accuracy, I implement a topological empirical Bayesian method for ancestral reconstruction that is identical to the ML algorithm, except that it integrates over topologies. This approach allows us to directly infer the eﬀects of incorporating phylogenetic uncertainty on ASR accuracy. I simulate and record the evolution of sequences under a variety of simpliﬁed and empirically derived conditions and infer ancestral states from the evolved alignments, allowing us to characterize the accuracy of each approach to ASR by comparing inferred ancestral sequences to the

”true” ancestors recorded during the simulation.

44 Materials and Methods

Ancestral State Reconstruction Algorithms

The ML method for ancestral sequence reconstruction, also called the

empirical Bayes method (Yang et al. (1995)), calculates the posterior probability

that some ancestral node contained state a at a sequence site of interest, given

the observed sequence data d, an evolutionary model m, a topology tˆ, and a set

of branch lengths and other model parameters θˆ; the topology and parameters

are those that maximize the likelihood over all data columns in the alignment.

The conditional likelihood of a equals the probability of observing d given a, m,

tˆ, and θˆ. The prior-weighted conditional likelihood of a is the conditional likelihood

of a multiplied by the prior probability of observing a, which is given by πa, the equilibrium state frequency of a. The posterior probability of a equals the prior-

weighted conditional likelihood of a divided by the sum of the prior-weighted

conditional likelihoods for all possible ancestral state assignments (4 for nucleotides

or 20 for amino acids) (Equation 3.1).

P (d a, m, t,ˆ θˆ)π P (a d, m, t,ˆ θˆ)= | a (3.1) | P (d a, m, t,ˆ θˆ)πa a | The ML state assignment is the state with the highest prior-weighted likelihood

(and necessarily the highest posterior probability, as well). The ML sequence is the

string of ML states. To reconstruct ML ancestral sequences, I used PAML v.4.1

(Yang (1997, 2007)).

The Topological Empirical Bayes (TEB) approach to ASR diﬀers from ML

only by integrating ancestral reconstructions over a distribution of trees (Equation

2). The TEB posterior probability of ancestral state a is the weighted average of 45 the posterior probability of a over all possible trees, where the weights are given

by the empirical Bayes posterior probability of each tree t. The empirical Bayes

posterior probability PEB of a tree assumes the maximum likelihood estimate of

branch lengths and other model parameters θˆt on each tree (Kolaczkowski and Thornton (2008), Kolaczkowski and Thornton (2009)):

PTEB(a d, m)= P (a d, t, m, θˆt) PEB(t d, m, θˆt) (3.2) | t | × | Equation 3.2 takes a diﬀerent form from but is equivalent to (see Supplemental

Note 1) the expression used by others (Pagel et al. (2004), Huelsenbeck and

Bollback (2001)) for ancestral state reconstructions integrated over topologies:

P (d a, t, m, θˆt)πaP (t) t | PTEB(a d, m)= (3.3) | P (d a, t, m, θˆt)πaP (t) t a | The ML method also has an empirical Bayesian interpretation, because Equation

3.1 calculates a posterior probability and uses priors on ancestral states. For

simplicity, I will refer to the approach which uses only the ML tree as the “ML

method” and the approach which integrates over trees as the “TEB method.”

One issue with estimating ancestral states from a distribution of trees is that

every topology contains diﬀerent ancestral nodes. I accommodate this problem by

deﬁning an ancestral node to be reconstructed as the most recent common ancestor

(MRCA) of a speciﬁed set of descendants (Pagel et al. (2004)). On any rooted tree,

the clade descending from the speciﬁed ancestor will contain all members of this

set; additional sequences may also be included in that clade, depending on the

topology. A similar approach can be used to describe internal nodes on unrooted

46 trees in relation to the split that places a speciﬁed set of terminal sequences into the smallest possible partition of the tree.

I implemented both the TEB and ML method in a new software package called Lazarus. This package spawns, manages, and then parses large batches of parallelized PAML jobs, one for each of a set of user-speciﬁed topologies. For each topology, branch lengths and model parameters are optimized by ML, the maximum likelihood of the tree is calculated, and the posterior probability of each ancestral state is calculated on that topology. Lazarus then parses these results to calculate the posterior probability of each ancestral state integrated over topologies.

Lazarus includes a modular Python API with object classes for quickly abstracting ancestral reconstruction data and is available at

Simulations

I compared the ancestral states reconstructed by the ML and TEB methods on data simulated under both controlled and empirically-derived conditions. The correct evolutionary model was assumed for all ancestral reconstructions.

Four-Taxon Phylogenetic Uncertainty

I simulated sequence evolution on four-taxon ultrametric trees of variable height and internal branch length (Fig. 18.A) and on four-taxon trees with randomly generated branch lengths. I examined ultrametric trees because they can be described by specifying only the total height of the tree and the lengths of the internal branches; the limited number of free parameters allows a detailed investigation of ancestral reconstruction methods as phylogenetic signal varies.

Further, ultrametric trees represent the most diﬃcult conditions for ancestral

47 sequence reconstruction. For a pair of terminal branches with any given sum of lengths descending from an internal node, the ultrametric case represents the greatest total loss of character information about the ancestor; conversely, as some branches descending from an ancestral node become longer and others shorter, the information in the short branch has a more determinative eﬀect on the inferred ancestral state. In the limit as one descendant branch length approaches zero, the ancestral state is inferred without ambiguity or error as the state in the sequence at the end of that branch.

48 On ultrametric trees, the internal branch length (labeled ’r’ in Fig. 18.A) was varied from (0.01, 0.02, 0.03, 0.05, 0.1, 0.2), and the overall height of the descendant clade (labeled ’h’ in Fig. 18.A) varied from 0.25 to 0.75 substitutions per site in intervals of 0.125. For each combination of ’r’ and ’h,’ I used Seq-Gen

(Rambaut and Grassly (1997)) to generate 100 sets of replicate descendant amino acid sequences of length 400 sites, using the JTT evolutionary model (Jones et al.

(1991)). For the non-ultrametric simulations, 1000 four-taxon trees were generated by randomly drawing an internal branch length from the uniform distribution

U[0.01, 0.1] and drawing four terminal branches from the uniform distribution

U[0.25, 0.75]. Seq-Gen was then used to simulate the evolution of sequences 400 amino acids long on each tree (Fig. 18.C).

For each replicate, I used ML and TEB ASR to infer the posterior probability of reconstructed ancestral states in the most-recent-common ancestor of taxa

A, B, C , of A, B , of A, C , and of B,C . Depending on the tree, some { } { } { } { } of these ancestors are the same. For example, on the tree (((A, B),C,D), the ancestor of A, C is the same node as the ancestor of A, B, C . However, on tree { } { } (((A, C),B,D), the ancestors for A, B, C and A, C are unique. I compared the { } { } maximum a posteriori ancestral state from TEB and ML to each other and to the true state, which was recorded at all nodes during the simulation. I analyzed the concordance and accuracy of TEB and ML ancestral states across all replicates and in relation to the values of ’r’ and ’h,’ the state pattern in descendant taxa, and whether the set of taxa in the clade descending from the ancestral node of interest in the ML tree is identical to that set in the true tree. With respect to the last criterion, the membership may be correct, a spurious taxon may be included as a

49 A h-r taxon A r h-r 0.2 taxon B

h randomly generated taxon C ancestral sequence 0.75 taxon D (outgroup) B tree #1 tree #2 tree #3

A A B B C C C B A D D D C [0.25, 0.75] taxon A [0.01, 0.1] [0.25, 0.75] [0.01, 0.1] taxon B

[0.25, 0.75] randomly generated taxon C ancestral sequence [0.25, 0.75] taxon D (outgroup)

FIGURE 18. Four-taxon simulation conditions. (A) I seeded randomly-generated amino acid sequences at the root of an ultrametric tree with four terminal branches. I simulated the ancestral sequences evolving across the branches to produce four descendant sequences (including one outgroup descendant). Simulations were performed under a variety of conditions by adjusting the internal branch length r and the overall height of the descendant clade h. (B) For each set of replicate sequences, I estimated the ML branch lengths and calculated the posterior probability of all three possible topologies. (C) Sequences were also simulated using non-ultrametric four-taxon trees with terminal branch lengths drawn from the uniform interval [0.25, 0.75] and internal branch lengths from the interval [0.01, 0.1]. 50 descendant (mem+), or a taxon may be incorrectly excluded from the clade (mem-

Empirically Derived Phylogenetic Uncertainty

I also compared the accuracy of ML and TEB reconstructions inferred from sequences simulated on empirically-derived trees. I used phylogenies inferred from the extant sequences of alcohol dehydrogenase (ADH) proteins (Thomson et al.

(2005)), steroid hormone-receptors (Bridgham et al. (2006)), green ﬂuorescent-like proteins (GFP) (Kelmanson and Matz (2003); Ugalde et al. (2004)), and Tu family elongation factor (EF-Tu) proteins (Gaucher et al. (2003)). For each gene family, the phylogeny and branch lengths were calculated by ML using Phyml version 2.4.4

(Guindon and Gascuel (2003)). The posterior probabilities of phylogenies in the

95% credible set (1,195 trees for ADH, 3,335 for steroid hormone receptors, 655 for GFP, and 544 for EF-Tu) were inferred using empirical Bayes Markov Chain

Monte Carlo (BMCMC), which integrates over topologies, each of which is assigned its maximum likelihood branch lengths (Kolaczkowski and Thornton (2007)). The

ML phylogenies for AFH, GFP, and EF-Tu (Fig. 19.) diﬀer only slightly from the original ML phylogenies shown in those datasets’ corresponding publications. On each ML phylogeny, 100 replicates of protein sequences 400 amino acids long were then evolved by simulation, using the JTT model of evolution, to yield terminal descendant sequences. For each replicate, ancestral sequences at all internal nodes were then reconstructed using ML and TEB. I examined only the uncertain nodes

(with Bayesian posterior probability less than 1.0) and their immediate neighboring nodes; nodes with PP =1.0 have no uncertainty over which to integrate, and therefore the TEB and ML reconstructions are identical.

51 State Pattern Analysis

To illustrate how integrating over topologies affects ancestral reconstruction for different data patterns under specific conditions, I performed ASR using ML and TEB and calculated the probability of each ancestral state for each of the possible state patterns of four nucleotides. I simulated DNA sequences 50,000 nucleotides long using the JC69 model on four taxon ultrametric trees with high phylogenetic uncertainty (h=0.3, r=0.01) or virtually no phylogenetic uncertainty

(h=0.3, r=0.2). I then examined the posterior probability of each ancestral state inferred using ML and TEB for each of the possible state patterns for four- state data. Character state patterns are indicated using variables representing nucleotides of the same type: for example, pattern xyxy for the four-taxon case stands for the realizations ACAC, AGAG, ATAT, CACA, ...TGTG at that site in the four leaves, respectively.

Statistical Analysis

The correspondence between posterior probabilities (PPs) and the frequency of correct inferences for TEB and for ML were analyzed by binning inferences according to their PPs and calculating the mean PP (x) and the fraction of correct reconstructions (y) in each bin. The fit of the resulting points to the function y=x was evaluated using a chi-square distribution with degrees of freedom equal to the number of bins. The significance of the difference between ML and TEB in fit to the function y=x was assessed by evaluating the ratio of the chi-square statistics for the two methods using an F-distribution with degrees of freedom equal to the number of bins. To compare the differences in mean accuracy of the ML and TEB

52 A B

0.65 0.97

0.99 0.86 0.98 0.83

0.68 0.61 0.74

0.94 0.72 0.45 0.99 0.42 0.87 0.83

0.99 0.75 0.48 0.71 0.76

0.95 0.90 0.98 0.59 0.56

0.71

0.58 1

C D 1

1 1 --

0.2 -- 1

1.0 1

1 1 0.36

0.31 1 1 1

1 1 0.37

0.02 1 0.29 -- 0.99 -- 0.98 -- 0.99 1 -- 0.9

FIGURE 19. Empirical phylogenies used for simulations. Internal nodes are labeled with their empirical Bayes posterior probability; circles indicate nodes at which ancestral sequences were reconstructed. A) Steroid hormone receptors (Bridgham et al. (2006)). The tree and branch lengths were inferred from empirical protein sequences using the JTT+G model. B) Alcohol dehydrogenases (Thomson et al. (2005)). The tree and branch lengths were inferred from empirical DNA sequences using GTR+G. C) Green ﬂuorescent-like proteins Kelmanson and Matz (2003). The tree and branch lengths were inferred from empirical DNA sequences using GTR+G. D) EF-Tu family elongation factors (Gaucher et al. (2003)). The tree and branch lengths were inferred from empirical protein sequences using JTT+G.

53 reconstructions, I conducted a paired two-sample t-test against the null hypothesis of no signiﬁcant diﬀerence in accuracy between the two methods.

Results

Eﬀect of Incorporating Phylogenetic Uncertainty

To determine how incorporating topological uncertainty affects ancestral sequence reconstruction, I first examined the extent to which ancestors inferred using ML and TEB differ from each other under a range of conditions. I found that integrating over trees only rarely affected the inferred state at ancestral nodes

(Fig. 20.A). In simulations on ultrametric four-taxon trees with varying levels of phylogenetic noise, the ancestral states inferred by ML and TEB diﬀered at only

0.4% of sites. On non-ultrametric trees, they diﬀered at 0.7% of sites. On larger trees derived from empirical datasets of four gene families previously analyzed using

ASR—steroid hormone receptors, alcohol dehydrogenases, green ﬂuorescent-like proteins, and Tu family elongation factors—ML and TEB reconstructions diﬀered by one percent or less (Fig. 20.A).

54 To determine whether certain phylogenetic conditions cause integrating over topological uncertainty to have a stronger eﬀect on inferred ancestral states, I decomposed the results of the ultrametric four-taxon simulations according to the state patterns in the terminal sequences that descend from the reconstructed ancestor, the length of the branches on the tree, and the ways (if any) that the

ML tree diﬀers from the true tree (Supplemental Table 2). There were no state patterns that resulted in diﬀerences between ML and TEB ancestors greater than

0.5 percent. The eﬀect of integrating over uncertainty was slightly greater for divergent state patterns in which all ingroup descendants have diﬀerent states

(pattern xyz) than for patterns that contain phylogenetic signal (xxx or xxy, Fig.

20.B). Similarly, no branch length conditions examined caused ML and TEB to differ by more than 0.5 percent; ML and TEB ancestors differed least when the total root-to-tip branch length was short, and they differed to a slightly greater extent as the terminal branches became very long (Fig. 20.C). When the ML tree was correct (as it was in the majority of cases), integrating over uncertainty had a particularly weak effect on the inferred ancestor; however, even when the

ML phylogeny erroneously inferred a spurious sequence as a descendant of the ancestor of interest or excluded a true descendant, the two methods still produced identical inferences at > 99% of sites (Fig. 20.B). Together, these data indicate that integrating over topological uncertainty per se does not strongly aﬀect ancestral reconstructions; the eﬀects are weak under conditions that cause the traces of the ancestral state to be lost in descendant sequences and virtually non-existent under those that preserve phylogenetic signal about the ancestral state.

I next analyzed whether integrating over topological uncertainty tends to aﬀect sites that are strongly or weakly supported by ML. Most ASR practitioners

55 ( * 1.000 different state 99.6% 99.3% of sites same state ultrametric non-ultrametric four taxon four taxon proportion of sites 0.990 0.0 0.2 0.4 0.6 0.8 1.0

99.1% 99.0% 99.3% 99.5% height of descendant clade

alcohol steroid- green Tu family + dehydrogenase hormone flourescent elongation receptor protein factors 1.000 ) proportion of sites 0.990 0.00 0.05 0.10 0.15 0.20 0.25 length of internal branch

FIGURE 20. Integrating over phylogenetic uncertainty rarely changes ancestors. (A) Proportion of sites simulated under a variety of conditions at which ML and TEB methods inferred the same or diﬀerent states. (B, C, D) Details of similarity between ML and TEB reconstructions for the ultrametric four-taxon simulations. (B) Proportion of sites at which ML and TEB infer identical states is shown in terms of descendant state patterns and types of phylogenetic error. Each row presents results for sites in which the descendant taxa A, B, and C have the speciﬁed state pattern (where pattern xxx corresponds to AAA, CCC, GGG, or TTT; xxy corresponds to AAC, AAG, AAT, ... or TTG). Columns indicate whether the set of taxa descending from the reconstructed node in the ML tree corresponds to those in the true tree: clade ok means the descendant membership is correct, mem. + means the ML descendant set spuriously includes an extra taxon, and mem. – means the ML descendant set incorrectly excludes a taxon. (C) Similarity between ML and TEB reconstructions is plotted against the height of the descendant clade (“h” in Fig. 18.). (D) Similarity between ML and TEB reconstructions is shown versus the length of the internal branch (“r” in Figure 1).

56 examine the support for ancestral state inferences and experimentally characterize the robustness of their inferences to alternate reconstructions that have posterior probability above some deﬁned plausibility cutoﬀ (Bridgham et al. (2006); Ortlund et al. (2007); Thomson et al. (2005); Chang et al. (2002); Ugalde et al. (2004)). I found that ML and TEB reconstructions disagreed only at sites that were already ambiguous in the ML reconstruction (Fig. 21.). In both ultrametric and non- ultrametric four-taxon simulations, the ML and TEB reconstructions agreed at all sites at which the ML reconstruction had posterior probability (PP) greater than 0.70. In the ADH, GFP, and EF-Tu simulations, the two methods agreed at all sites with PP greater than 0.76, 0.63, and 0.71, respectively. In the steroid hormone simulation, the methods agreed at all sites with PP greater than 0.87, and they disagreed at only 0.003% of all sites reconstructed with posterior probability

> 0.80. Over all four-taxon reconstructions, the maximum a posteriori ancestral state from TEB was different from the first- or second-best state using the ML method at only 0.001625% of sites. These data indicate that integrating over topological uncertainty never causes inferred ancestral states that are strongly supported by ML to be revised. Rather, TEB inferred a state different from the

ML state only when that state was ambiguously reconstructed anyway, switching the favored state from one weakly supported possibility to another.

57 KPMMLYLU[Z[H[L 43 ( ZHTLZ[H[L ( ;,) + + = 43Z[H[L;,)Z[H[L WYVWVY[PVUVMZP[LZ^OLYL 7HPU[LNYH[PUN[YLLZ

7H\ZPUN43[YLL 7H ) ) , ,

0 0 4 0 0 3 0 0 2 0 0 1

0.0 0.2 0.4 0.6 0.8 1.0

* * - - 5 3 5 0 3 4 5 2 3 0 2 5 1 2 0 1 1 5 0 0

0.0 0.2 0.4 0.6 0.8 1.0

FIGURE 21. ML and TEB differences versus phylogenetic support. ML and TEB infer different ancestral states only when posterior probabilities are low. In each pair of plots, the left plot (A1, B1, etc.) compares the posterior probability of the maximum a posteriori state inferred by ML to that inferred by TEB. Black points show sites at which ML and TEB methods inferred the same state; green diamonds indicate that the two methods inferred different states. The right plots (A2, B2, etc.) are histograms of the green points in the left plot: I grouped all ASR inferences into 5%-sized bins based on their posterior probability and counted the proportion of sites at which ML and TEB inferred different states. Results are shown for simulations on ultrametric four-taxon trees (A1,A2), non-ultrametric four-taxon trees (B1,B2), and the steroid-hormone receptor (C1,C2), ADH (D1,D2), GFP (E1,E2), and EF-Tu phylogenies (F1,F2).

Eﬀect of Incorporating Phylogenetic Uncertainty on ASR Accuracy

Although the ML and TEB methods inferred the same state at most sites, it is possible that TEB might produce more accurate reconstructions at the rare sites where the two methods diﬀer. I measured accuracy as the proportion of sites at which the reconstructed state was identical to that of the true ancestor, which I recorded during each simulation. In the four-taxon and GFP simulations, ML was 58 slightly, but not signiﬁcantly, more accurate than TEB (Fig. 22.A, Supplemental

Table 7). In the ADH, steroid hormone receptor, and EF-Tu simulations, there was no diﬀerence in accuracy between the methods. The accuracy of both ML and TEB declined as terminal branch lengths grew longer, causing multiple substitutions to occur (Fig. 22.B). ML’s superiority to TEB was greatest when the membership of the descendant clade was correct (Fig. 22.C), presumably because when the

ML topology is the true tree, integrating phylogenetic uncertainty serves only to introduce error. Even when the ML tree was incorrect, however, TEB generally decreased accuracy; integrating over uncertainty improved accuracy only under the rare condition that the descendant state pattern was textitxyz and a spurious taxon had been included as a descendant of the node of interest. Under these conditions, both methods performed poorly, because little or no phylogenetic signal of the ancestral state was retained in the descendants. For all other state patterns and forms of phylogenetic error, ML had accuracy equal to or slightly greater than that of TEB.

Eﬀect of Incorporating Phylogenetic Uncertainty on ASR Posterior Probabilities

I next examined whether TEB or ML yielded more accurate estimates of statistical conﬁdence in inferred ancestral states. For all simulations, I binned reconstructed ancestral sites by their posterior probability and counted the proportion of accurate inferences in each bin (Fig. 23.). If posterior probability is an accurate predictor of the probability that an inferred state is correct, the mean

PP in that bin should equal the proportion of correct ancestral state inferences. I observed that the ML and TEB methods generally produced similar PP values, and both types of PP were good predictors of mean accuracy. The major exception

59 43 ;,) ( ) * WYVWVY[PVUJVYYLJ[

7H + , -

FIGURE 22. ASR error rates, measured as the proportion of sites at which the maximum a posteriori reconstructions differ from the true ancestral state. (A) Results from the four-taxon and empirically-derived conditions are averaged over all replicates. None of the differences between ML and TEB are statistically significant. (B) Results from the ultrametric four-taxon simulation are shown versus the height of the descendant clade (where height equals “h” in Fig. 18.. Error bars for ML and TEB are nearly identical. (C) Detailed results from the ultrametric four-taxon simulation. Each cell reports two values: the proportion of sites incorrectly reconstructed by ML (top) and TEB (bottom). Bold values indicate the method with higher accuracy. Data are sorted according to the same criteria in Fig. 20.B.

60 to this pattern was the four-taxon simulation on ultrametric trees, in which integrating over trees slightly inﬂated support for reconstructions with PP > 0.5

(Fig. 23.A); a chi-square test indicates that ML’s posterior probabilities fit the ideal better than TEB’s PPs do, but the difference is small and does not reach statistical significance (P =0.16, Supplemental Table 1). When the ML tree was correct, ML’s PPs were more accurate than TEB, but TEB was more accurate when the ML tree was wrong; because the former conditions are more frequent than the latter, however, ML’s accuracy was higher overall. For the empirically derived conditions, ML’s PPs were slightly more accurate, but the difference was again small and not statistically significant (Supplemental Tables 3, 4, 5, 6).

An Intrinsic Tradeoﬀ Explains Why Incorporating Uncertainty Does Not Aﬀect

ASR

In order to understand why integrating over phylogenies has such a weak eﬀect on ancestral reconstruction, I examined the relationship between the plausibility of alternate phylogenies and the dependence of the reconstructed state on the assumed phylogeny. I conjectured that as phylogenetic uncertainty increases, the same state will be reconstructed on the plausible trees. To test this hypothesis,

I grouped all the replicates from the ultrametric four-taxon simulations according to the posterior probability of their ML tree. For each replicate, I counted the proportion of sites at which the inferred ancestral state diﬀers between the ML tree and the tree with the next-highest PP (Fig. 24.A). I observed that when the ML tree was uncertain (PP <1.0), the ancestral states among trees rarely disagreed. In contrast, when the ML tree was absolutely certain (PP = 1.0), the ancestral states on the ML tree and the second-best tree disagreed at up to 25%

61 ( 0.20 43 ;,)

0.15

0.10

0.05 proportion wrong

0.00 \S[YHTL[YPJ UVU\S[YHTL[YPJ HSJVOVS Z[LYVPK NYLLU ;\MHTPS` MV\Y[H_VU[YLLZ MV\Y[H_VU[YLLZ KLO`KYVNLUHZL OVYTVUL MSV\YLZJLU[ LSVUNH[PVU YLJLW[VY WYV[LPU MHJ[VY ) * 0.30 43 ;,) 0.25

0.20

0.15

0.10 proportion wrong

0.05

0.00

height of clade

FIGURE 23. Relationship of ancestral PP to accuracy. The posterior probability (PP) of inferred ancestral states is plotted against the probability that those states are correct. For both ML and TEB, I grouped all ancestral state inferences by their PP into 5%-sized bins. Within each bin, I calculated the proportion of inferred states that match the true state. Bins with fewer than 50 members were excluded. Data are shown for simulations on (A) ultrametric four-taxon, (B) non-ultrametric four-taxon, (C) ADH, (D) steroid hormone receptors, (E) GFP, and (F) EF-Tu phylogenies.

62 of sites; however, because the posterior probability of the second tree was so low, it contributed virtually zero weight to the TEB reconstruction. Support measures showed a similar trade-off: only when there was little or no uncertainty about the tree did the PP of an ancestral reconstruction differ among phylogenies. These results indicate that there is a trade-off between phylogenetic uncertainty and the extent to which ancestral state reconstruction depends on the phylogeny assumed.

( ) proportion of sites

tree uncertainty (log scale) tree uncertainty (log scale)

FIGURE 24. Phylogenetic uncertainty versus alternate ancestral reconstructions. Each point corresponds to one set of replicate descendants in the ultrametric four- taxon simulation. Tree uncertainty for each replicate is measured as 1.0 minus the posterior probability of the ML tree. (A) Tree uncertainty is plotted versus the proportion of sites at which the most likely ancestral state on the ML tree disagrees with the most likely ancestral state on the second-best tree. (B) Tree uncertainty is plotted versus the average absolute diﬀerence between the posterior probability of the most likely state on the ML tree minus the posterior probability of this same state on the second-best tree.

To understand this trade-oﬀ in detail, I examined ancestral reconstructions under two contrasting four-taxon conditions with diﬀerent degrees of phylogenetic uncertainty (Fig. 25.). In one condition, the true phylogeny had a long internal branch, so the ML tree was inferred with no uncertainty (PP = 1.0); in the other,

63 the true phylogeny had a very short internal branch, so the ML tree was inferred with considerable uncertainty (PP = 0.384). For each state pattern, I reconstructed the ancestral state on all three possible topologies. I found that when there was no phylogenetic uncertainty, the probability of an ancestral state can differ radically given different trees; for three of the state patterns, the maximum a posteriori ancestral state inferred on the ML tree differed from that inferred on alternate trees. Because the internal branch was long, however, these alternate trees had zero posterior probability, so incorporating them into TEB reconstruction produces ancestral state inferences and posterior probabilities identical to the ML inference.

In contrast, when the internal branch was short and the phylogeny was uncertain, all three topologies were close to being star trees. In this case, the probability of the ancestral state inferred on the ML tree was almost identical to the probability of that state given any other tree. Because the inferred ancestral state did not diﬀer among phylogenies, TEB and ML again yielded the same reconstruction.

Discussion

My results demonstrate that a Bayesian approach to incorporating uncertainty about the underlying phylogeny is not necessary for ancestral state reconstruction. By comparing two methods of ancestral sequence reconstruction that differ only in that one assumes the ML phylogeny while the other integrates over phylogenies, I were able to determine the specific effect of incorporating phylogenetic uncertainty on ancestral state inferences, their statistical support, and their accuracy. I found that using TEB virtually never changes the inferred ancestral state; when it does, the reconstruction was already ambiguous using ML.

64 ( ) * ()*+ (*)+ +$JHZLZ^OLYL[OLILZ[Z[H[L *)(+ VU43[YLLKVLZUV[TH[JO[OL [Y\L[YLL 43[YLL HS[LYUH[L[YLLZ PU[LNYH[PUN[YLLZ ILZ[Z[H[LVUHS[LYUH[L[YLLZ

+ + + ( * + + ) ) ) * ( ( +

+ ) ( * * ____ _`__ __`_ ___` __`` _`_` _``_ _`à __à _à_ _àa ^_à MYLX\LUJ`$ UVWO`SVNLUL[PJ\UJLY[HPU[`

( + + + * ( ) )

+ ) ( * ) ( * ____ _`__ __`_ ___` __`` _`_` _``_ _`à __à _à_ _àa ^_à MYLX\LUJ`$ OPNOWO`SVNLUL[PJ\UJLY[HPU[`

FIGURE 25. Inferred ancestral states are the same across uncertain trees. The conditions that produce phylogenetic uncertainty cause ancestral state inferences to be identical across trees. (A) I simulated sequences on trees with long (top) and short (bottom) internal branches. On each, I randomly generated an ancestral sequence 50,000 nucleotides long and simulated sequence evolution. (B) From the descendant sequences, I inferred the empirical Bayes posterior distribution of the three trees, each with its maximum likelihood branch lengths. (C) On each tree, I used the true model to reconstruct the common ancestor of descendants A, B, and C for all possible descendant state patterns (xxxx, xyxx, wxyz, etc.). Each bar corresponds to the posterior probability of the best ancestral state on the ML tree (blue), on the alternate trees (yellow and red), and integrated over all trees (green). Stars indicate state patterns for which the maximum a posteriori ancestral state on one of the alternate trees is diﬀerent from that on the ML tree.

65 ML has slightly higher accuracy, and its posterior probabilities provide a slightly better predictor of the probability that an ancestral state inference is correct.

These analyses show that incorporating phylogenetic uncertainty only weakly aﬀects ASR because the conditions that cause phylogenetic uncertainty also make the ancestral state the same across trees. This phenomenon occurs because when internal branches are short, the distance in tree-space is small between the ancestor on the ML tree and the ancestor on the second-best tree (Felsenstein

(2004)). At the limit, the true tree is a star tree with a zero-length internal branch, and all resolved topologies have equal posterior probability, leading to maximal phylogenetic uncertainty; however, the ancestral nodes on the diﬀerent topologies are identically located in tree space. In contrast, under the conditions that cause inferred ancestral states to diﬀer among trees, there is typically no phylogenetic uncertainty to integrate over.

Prior work has shown that ASR is generally most accurate on star-like trees, because the descendant sequences contain maximum mutual information about the ancestral state when those descendants are completely independent phylogenetically (Blanchette et al. (2004); Lucena and Haussler (2005)). Those studies, however, assumed that the true phylogeny was known a priori, which is particularly unlikely for star-like trees with short internal branches. My work shows that phylogenetic uncertainty, which is inevitable under these conditions, is not expected to undermine the accuracy of ancestral state reconstruction on star-like trees. These results underscore the potential to accurately reconstruct ancestral sequences at the base of rapid phylogenetic radiations despite phylogenetic uncertainty, such as the ancestors of all mammals (Blanchette et al. (2004)) or all metazoans (Rokas et al. (2005)).

66 Previous work by Huelsenbeck and Bollback, like ours, showed a close relationship between ancestral posterior probabilities estimated using the ML tree and integrating over trees (Huelsenbeck and Bollback (2001)). Those authors did suggest, however, that uncertainty in the phylogeny might lead to significantly different interpretations of the ancestral state. This suggestion was illustrated using trees with arbitrarily assigned branch lengths and posterior probabilities; for all topologies in the illustration, the internal branch lengths were of significant length and the posterior probabilities were substantial. In reality, it is unlikely that any data set would support such a distribution of posterior probabilities over this set of tree/branch length combinations, because non-trivial posterior probabilities on

”next-best” trees typically arise only when internal branches are short. My results show that when the posterior probabilities on trees are derived from sequence data rather than arbitrarily assigned, integrating over uncertainty has a negligible eﬀect on ancestral sequence inference and a negative impact, if any, on accuracy.

My results should not be interpreted as an endorsement for sloppy analysis.

Although incorporating phylogenetic uncertainty does not improve the accuracy of ancestral reconstruction, this does not mean the phylogeny is unimportant.

Because ancestral reconstructions can vary across trees under some conditions, arbitrarily choosing an incorrect and implausible phylogeny could yield inaccurate reconstructions.

These ﬁndings should not be taken as evidence that ancestral reconstruction never errs. There are numerous potential sources of error that I did not evaluate, including use of incorrectly parameterized evolutionary models, which could yield incorrect (and strongly supported) inferences of phylogeny (Kolaczkowski and Thornton (2004)) or incorrect ancestral state reconstructions even when

67 the true tree is assumed. ASR practitioners should continue to use rigorous statistical practices, such as formal evaluation of a wide range of models that incorporate evolutionary heterogeneity (Posada (2001); Lartillot and Philippe

(2004); Kolaczkowski and Thornton (2008)) and dense, targeted taxon sampling

(Hillis (1998); Pollock et al. (2002); Heath et al. (2008)). My analyses were speciﬁc to Bayesian integration over uncertainty about the underlying phylogeny: I did not address the eﬀect on ancestral reconstructions of integrating over uncertainty about branch lengths, the substitution model, or its parameters. Whether a Bayesian approach to these sources of uncertainty would improve or degrade ASR accuracy warrants further research.

In summary, incorporating phylogenetic uncertainty by integrating over topologies does not improve the accuracy of ancestral sequence reconstruction, because the conditions that cause phylogenetic uncertainty make the ancestral state the same across trees. Using the ML tree will typically yield the best ancestral reconstruction, even when the ML tree is uncertain. A Bayesian approach to phylogenetic uncertainty is intuitively appealing but computationally demanding and, in this case, unnecessary.

68 CHAPTER IV

CASE STUDY: EVOLUTION OF INCREASED COMPLEXITY

In this chapter, I provide a case study demonstrating the analysis pipeline previously described (Fig. 1.). I applied the methods in this pipeline to investigate the evolution of increased complexity in a molecular machine with multiple interacting parts. This project is groundbreaking because this is the ﬁrst time the evolution of an entire molecular complex has been traced through history, and this is the ﬁrst application of phylogenetic ancestral resurrection to all the members of a complex. Collaborators in the the Stevens Lab performed the in vitro experimentation. This work has been accepted for publication in the journal Nature.

Many cellular processes are carried out by molecular machines – assemblies of diﬀerentiated proteins that physically interact to execute biological functions

(Pallen and Matzke (2006); Mulkidjanian et al. (2007); Forgac (2007); Dolezal et al.

(2006); Clements et al. (2009); Archibald et al. (2000)). Despite much speculation, evidence is lacking concerning the mechanisms by which their complexity evolved.

Comparative genomic approaches suggest that the the components of many molecular machines appeared sequentially during evolution, implying gradual increases of complexity by incorporating new parts into simpler machines (Pallen and Matzke (2006); Mulkidjanian et al. (2007); Dolezal et al. (2006); Clements et al. (2009); Gabald´on et al. (2005); Liu and Ochman (2007)). These horizontal analyses, however, are unable to decisively test these hypotheses or reveal the mechanisms by which additional parts became obligate components of existing systems. Here we perform the ﬁrst vertical evolutionary analysis of a molecular machine by using ancestral gene resurrection (Thornton (2004); Frattini et al. 69 (2000)) and manipulative genetic experiments to reconstruct all components of a complex – the hexameric transmembrane ring of the vacuolar H+-ATPase (V-

ATPase) proton pump – and identify the specific genetic and functional changes that caused an increase in complexity hundreds of millions of years ago. We show that the transmembrane ring of Fungi which is composed of three paralogous proteins evolved from a two-paralog ancestral complex because of a very small number of degenerative mutations, without the evolution of apparent new functions by the parts. After a gene duplication, both descendant proteins lost some of the specific inter-subunit interfaces required for their interactions with other ring proteins; these losses were complementary, so both copies became obligate components with restricted spatial roles in the complex. Reintroducing a single historical mutation from each paralog lineage into the resurrected ancestral proteins is sufficient to recapitulate this asymmetric degeneration and trigger the requirement for the more complex three-component ring. Our experiments show that increased complexity in an essential molecular machine evolved by simple, high-probability evolutionary processes and suggest a plausible mechanism for the evolution of complexity in other multi-paralog machines whose parts function in specific spatial orientations.

The V-ATPase proton pump is a multi-subunit protein complex that pumps hydrogen ions across membranes to acidify subcellular compartments; this function is required for intracellular protein trafficking, coupled transport of small molecules, and receptor-mediated endocytosis (Forgac (2007)). V-ATPase dysfunction has been implicated in human osteopetrosis, acquired drug resistance in human tumors, and pathogen virulence (Frattini et al. (2000); Pérez-Sayáns et al. (2009); Xu et al.

(2010)). A key component of the V-ATPase is the V0 subcomplex, a hexameric

70 protein ring that utilizes a rotary mechanism to move protons across organelle membranes (Fig. 26.A) (Hirata et al. (2003); Imamura et al. (2005)). Although the V-ATPase is found throughout Eukaryotes, the V0 ring varies in subunit composition among lineages. In animals and most other eukaryotes, the ring consists of one subunit of Vma16 protein and ﬁve copies of its paralog, Vma3

(Fig. 26.B) (Forgac (2007)). In Fungi, the ring consists of one Vma16 subunit, four copies of Vma3, and one Vma11 subunit, arranged in a speciﬁc orientation relative to each other (Powell et al. (2000)). All three proteins are required for V-ATPase to function in Fungi (Umemoto et al. (1990, 1991)), but the mechanisms by which both Vma3 and Vma11 became obligate components with speciﬁc positions in the complex are unknown.

To address this issue, we reconstructed the ancestral ring proteins from periods just before and after the increase in complexity (Harms and Thornton

(2010)), synthesized and functionally characterized them (Thornton (2004); Liberles

(2007)), and used manipulative methods to identify the genetic and molecular mechanisms by which their functions changed (20). We ﬁrst inferred the phylogeny and best-ﬁt evolutionary model of the Vma3/11/16 protein family from the sequences of all 139 extant family members available in Genbank. The maximum likelihood phylogeny (Fig. 27.) indicates that Vma11 and Vma3 are sister proteins produced by a duplication of an ancestral gene (Anc.3-11) deep in the Fungal lineage, before the last common ancestor of all Fungi ( 800 million years ago) but after the divergence of Fungi from other eukaryotes ( 1 billion years ago) (Taylor and Berbee (2006)). The Vma11/Vma3 and Vma16 lineages, in turn, descend from an even older gene duplication deep in the Eukaryotic lineage (Fig. 26.B). We then used a maximum likelihood algorithm (Yang et al. (1995)) to infer the ancestral

71 A B 9.5 223.6 Amoebozoa, Apicomplexa 3.0 4.0 26.0 Animals, Choanoflagellates subunit 13.8 V1 Anc.16 1.6 16 1.8 Fungi 152.1

14.1 Amoebozoa, Apicomplexa 223.6 14.5 Animals, Choanoflagellates 7.5 subunit 11 0.8 46.3 Fungi 3 16 3 membrane Anc.3-11 8.8 Anc.3 V0 subunit 0.8 subs./site 54.44 Fungi 11 H+ Anc.11

FIGURE 26. Structure and evolution of the V-ATPase complex. (A) In S. cerevisiae, the V-ATPase is assembled from fourteen subunits; the V1 subdomain is on the cytosolic side of the organelle membrane, and V0 is membrane-bound. (B) The maximum likelihood phylogeny of V-ATPase subunits Vma3, Vma11, and Vma16. Amoebozoa, Apicomplexa, Animals, and Choanoﬂagellates contain subunits 3 and 16, whereas Fungi contain 3, 11, and 16. Black circles show ancestral proteins reconstructed in this study. Colors correspond to those of the subunits in panel A; green lineages are the unduplicated orthologs of Vma3 and Vma11. Decimal values at internal branches express the approximate-likelihood ratio support for the monophyly of the descendant clade. A more detailed phylogeny is found in Fig. 27..

72 1.0 Amoebozoa & 0.76 1.0 Apicomplexa subunit 16 (c”) Opisthokont Anc.16 1.0 73.2% of sites with P 1.0 0.94 1.0 Animals & 1.0 Choanoflagellates subunit 16 (c”)

proportion of sites 1.0 0.0 0.0 1.0 ASR posterior probability (in 5% bins)

1.0 0.75 Fungi subunit 16 (c”) Fungi Anc.3 90.2% of sites with PP 1.0 0.38

Amoebozoa & 1.0 Apicomplexa

proportion of sites subunit 3 (c) 0.0 0.0 1.0 ASR posterior probability (in 5% bins) 1.0 Opisthokont Anc.3-11 Animals & 93.5% of sites with PP 1.0 Choanoflagellates 1.0 subunit 3 (c)

0.99 proportion of sites 0.0 0.54 0.0 1.0 ASR posterior probability (in 5% bins) Fungi subunit 3 (c) 1.0 Fungi Anc.11 RIVLWHVZLWK33 1.0 1.0 0.99 1.0

proportion of sites 0.78 Fungi subunit 11 0.0 0.0 1.0 (c!) ASR posterior probability (in 5% bins)

1.0

0.58 VXEVVLWH

FIGURE 27. The maximum likelihood phylogeny of V-ATPase subunits Vma3, Vma11, and Vma16.

73 amino acid sequences with the highest probability of producing all the extant sequence data, given the best-ﬁt phylogeny and model. We reconstructed the two paralogs that composed the ancient eukaryotic ring – Anc.3-11 (the last common ancestral protein from which the Vma3 and Vma11 lineages descend) and Anc.16

(the ancestral proteins from which all Vma16s descend); we also reconstructed the duplicated paralogs comprised by the three-member fungal ring in the ancestor of all Fungi – Anc.3 and Anc.11, the progenitors of the diﬀerentiated subunits

Vma3 and Vma11, respectively, after Anc.3-11 was duplicated (Taylor and Berbee

(2006)). We synthesized DNAs that code for these reconstructed proteins and assayed their functions in vivo using extant yeast S. cerevisiae.

Materials and Methods

In Silico Reconstruction of Ancestral Protein Sequences

V0 complex subunits Vma3, Vma11, and Vma16 are sometimes referred to as subunits c, c, and c in the specialty literature. We queried GenBank for all

Eukaryote V-ATPase V0 ring sequences. Our query returned subunit 3, 11, and 16 protein sequences for twenty-six species in Fungi, and subunit 3 and 11 sequences for thirty-ﬁve species in Metazoa, Amoebozoa, and Apicomplexa. We aligned the sequences using PRANK v0.081202 (Loytynoja and Goldman (2005, 2008)).

This alignment is best-ﬁt by the Whelan-Goldman matrix (WAG) with gamma- distributed rate variation (+G) and proportion of invariant sites (+I), according to the Akaike Information Criterion as implemented in PROTTEST (Abascal et al. (2005)). Using WAG+G+I, we used PhyML v3.0 to infer the maximum likelihood (ML) topology, branch lengths, and model parameters (Guindon and

Gascuel (2003)). We optimized the topology using the best result from Nearest- 74 Neighbor-Interchange and Subtree Pruning and Regrafting; we optimized all other free parameters using the default hill-climbing algorithm in PhyML. Phylogenetic support was calculated as the approximate likelihood ratio (36). Our ML analysis inferred the Nematoda 3 and 11 sequences to be connected by a very long branch basal to the Chromalveolata lineages; this result is inconsistent with our expectation that Nematoda are animals (Aguinaldo et al. (1997)) and we therefore excluded Nematoda data from further downstream analysis.

We reconstructed ML ancestral states at each site for all ancestral nodes in our ML phylogeny using our own set of Python scripts, called Lazarus, which wraps

PAML version 4.1 (Yang (2007)). Lazarus parsimoniously places ancestral gap characters according to Fitchs algorithm (39). We characterized the overall support for Anc.3-11, Anc.16, Anc.3, and Anc.11 by binning the posterior probability of the

ML state at each site into 5%-sized bins and then counting the proportion of total sites within each bin (Supplement S2).

Robustness to Alignment Uncertainty

In order to assess if our ancestral reconstructions are robust to alignment uncertainty, we aligned our protein sequences using four diﬀerent alignment algorithms: CLUSTAL version 2.0.10 (Thompson et al. (1994)), MUSCLE v3.7

(Edgar (2004)), AMAP v2.2 (Do et al. (2005)), and PRANK v0.081202 (Loytynoja and Goldman (2005, 2008)). We then inferred the ML phylogeny and branch lengths for each alignment, using the methods described above. The resultant alignments varied in length from 347 sites (using CLUSTAL) to 683 sites (using

PRANK), but all four alignments yielded the same ML topology with nearly identical ML branch lengths.

75 In order to determine which alignment algorithm yields the most accurate ancestral inferences under V-ATPase phylogenetic conditions, we simulated sequences across the V-ATPase ML phylogeny using insertion and deletion rates ranging from 0.0 to 0.1 indels per site. For each indel rate, we generated ten random unique indel-free ancestral sequences 400 amino acids in length and then used INdelible (Fletcher and Yang (2009)) to simulate the ancestral sequence evolving along the branches of our ML phylogeny under the conditions of WAG+I+G model with indel events randomly injected according to the specified indel rate. The size of each indel event was drawn from a Zipfian distribution with coefficient equal to 1.1 and the maximum length limited to 10 amino acids. We aligned each replicates descendant sequences using AMAP, CLUSTAL, MUSCLEs, and PRANK; for each alignment, we inferred the ML topology, branch lengths, and model parameters using the methods described above. We used Lazarus to reconstruct all ancestral states, and queried Lazarus for the most-recent shared ancestor for Opisthokont subunit 3/11 and Opisthokont subunit 16 sequences. We measured the error of ancestral reconstructions as the proportion of ancestral sites that incorrectly contained an indel character (Fig. ??).

Plasmids and Yeast Strains

Bacterial and yeast manipulations were performed using standard laboratory protocols for molecular biology (Sambrook and Russel (2001)). Plasmids used can be found in Supplement S5. Ancestral sequences (pGF140, pGF139, pGF506, and pGF508) were synthesized by GenScript (Piscataway, NJ) with a yeast codon bias.

Triple hemagglutinin (HA) epitope tags were included prior to each stop codon.

The Anc.3-11, Anc.16, Anc.3, and Anc.11 genes were subcloned to single-copy,

76 CEN-based yeast vectors. The ADH terminator sequence (247 base pairs) and

NatR drug resistance marker (Goldstein and McCusker (1999)) were polymerase chain reaction (PCR) ampliﬁed with 40 bp tails homologous to the 3’ end of each coding region and vector sequence. Vectors were gapped, co-transformed into

SF838-1D yeast with PCR fragments, and cells were selected for NatR. A second round of in vivo ligation was used to place the ancestral genes under 500 bps of the VMA3 or VMA16 promoters to create pGF140 and pGF139, respectively. The following vectors all used a similar cloning strategy: pGF240 - pGF41, pGF252, pGF253, pGF503 - pGF508, pGF510, pGF512 - pGF515, pGF517 - pGF519, pGF521, pGF523, pGF528, pGF529, pGF531, pGF534 - pGF537, and pGF542.

Brieﬂy, the relevant locus (Anc.3-11, Anc.16, or Anc.3) was PCR ampliﬁed with

5’ and 3’ untranslated ﬂanking sequence and cloned into pCR4Blunt-TOPO

(Invitrogen, Carlsbad, CA). If necessary, a modiﬁed Quikchange protocol (Zheng et al. (2004)) was used to introduce point mutations before the gene was subcloned into a yeast vector (pRS316 or pRS415). To generate pGF502, codon 31 through the stop codon of Anc.16 were ampliﬁed with the ADH::NatR cassette from pGF139, cloned into TOPO, and in vivo ligated downstream of the VMA16 promoter (including a start codon) in pRS415.

A triple-fragment in vivo ligation was used to generate pGF646 - pGF651.

Gapped vector containing the VMA16 promoter was transformed into yeast with two PCR fragments of the ring genes to be fused. For pGF646, the coding region of (i) VMA16 (without codons 2-41) and (ii) the coding region of Anc.11

(without codons 2-5) were ampliﬁed by PCR. The proteolipid on the C-terminal portion of the gene fusion also contained the ADH terminator and NatR cassette; the ampliﬁed products contained PCR tails with homology to link the genes to

77 both the gapped vector and to each other. Gene fusions were modeled after the experimental design of Wang et al. (2007) where the lumenal protein sequence linking the two proteolipids was designed to be exactly 14 amino acids. To meet these criteria, additional amino acids were inserted into the following vectors linking the two subunits: pGF646 (TRVD), pGF648, pGF650 (TR), pGF649, pGF651 (GS).

Yeast strains used can be found in Supplement S5. Strains containing deletion cassettes other than KanR (Goldstein and McCusker (1999)) were constructed by

PCR amplifying the HygR or NatR cassette from pAG32 or pAG25, respectively, with primer tails with homology to ﬂanking sequences to the VMA11 or VMA16 loci. 11::KanR and 16::KanR strains (SF838-1D?) were transformed with the HygR and NatR PCR fragments, respectively, and selected for drug resistance. The

11::HygR locus was ampliﬁed and transformed into LGY113 (to create LGY125) and LGY115 (to create LGY124). This was repeated with the 16::NatR locus to create LGY139 and LGY143.

Yeast Growth Assays

Yeast were grown in liquid culture, diluted ﬁve-fold, and spotted onto YEPD media buﬀered to pH 5.0 or YEPD media containing 25 mM (Figs. 2, 3, 4) or 30 mM CaCl2 (Fig. 28.F).

Whole Cell Extract Preparation and Immunoblotting

Yeast extracts and Western blotting were performed as previously described

(47). Antibodies used in this study included monoclonal primary anti-HA (Sigma-

78 Aldrich), anti-Dpm1 (5C5; Invitrogen), and secondary horseradish-conjugated anti- mouse antibody (Jackson ImmunoResearch Laboratory, West Grove, PA).

Fluorescence Microscopy

Staining with quinacrine was performed as previously described (Ryan et al. (2008)). The cell wall (shown in red) was visualized using concanavalin A tetramethylrhodamine (Invitrogen). Microscopy images were obtained using an

Axioplan 2 ﬂuorescence microscope (Carl Zeiss, Thornwood, NY). A 100x objective,

AxioVision software (Carl Zeiss), and Adobe Photoshop CS (v. 8.0) were used.

Results

To functionally characterize the resurrected proteins, we transformed them into S. cerevisiae deﬁcient for various ring components and therefore incapable of growth in the presence of elevated CaCl2 (Kane (2006)). We found that the ancestral two-component ring can functionally replace the three-component ring of extant yeast; this result indicates that neither the complex nor its parts evolved new functions required for growth under the conditions in which the ring is known to be important. When the resurrected Anc.3-11 was transformed into yeast deﬁcient for Vma3 (3) or Vma11(11), growth in the presence of elevated CaCl2 was rescued, indicating that all the functions of the present-day Vma3 and Vma11 proteins were already present before their birth by gene duplication (Fig. 28.A).

Further, Anc.3-11 unlike either of its present-day descendants can partially rescue growth in yeast that are doubly deﬁcient for both Vma3 and Vma11 (311). In addition, the reconstructed Anc.16 rescued growth in Vma16-deﬁcient S. cerevisiae

(16) (Fig. 28.B), and co-expression of Anc.3-11 and Anc.16 together rescued

79 cell growth in 31116 yeast, which lack all three ring subunits (Fig. 28.C). The ancestral genes speciﬁcally restore proper V-ATPase function in acidiﬁcation of the vacuole lumen (Fig. 28.G). Further, mutation of the ancestral subunits to remove glutamic acids residues known to be essential for V-ATPase enzyme function

(16,24) abolished their ability to rescue growth on CaCl2 (Supplement S7). These inferences about the functions of Anc.3-11 and Anc.16 are robust to uncertainty about ancestral amino acid states. We reconstructed alternate versions of Anc.3-11 and Anc.16 by introducing amino acid states with posterior probability 0.2, but ≥ none of these abolished the ability of the ancestral genes to functionally substitute for the extant subunits (Supplement S8).

Similar experiments with the components of the ancestral three-component ring show that after Anc.3-11 duplicated, both Vma3 and Vma11 became necessary for a functional complex, because each lost speciﬁc ancestral functions that were maintained in the other. Unlike Anc.3-11, expression of Anc.3 can rescue growth and vacuole acidiﬁcation in 3 but not 11∆ yeast, and Anc.11 can rescue growth in 11∆ but not 3∆ yeast (Fig. 28.D,E,G). Further, both Anc.3 and Anc.11 are required to fully rescue growth in 311 yeast (Fig. 28.F). These data indicate that after their birth by gene duplication, Anc.11 lost the ancestral proteins ability to carry out at least some functions of Vma3, and Anc.3 lost the ancestral capacity to carry out those of Vma11.

We conjectured that the evolution of the specialized roles of Vma3 and

Vma11 reflected the loss of specific interaction interfaces required for ring assembly that were present in the ancestral protein. Previous experiments with fusions of extant yeast proteins have shown that the arrangement of subunits in the ring is constrained by the capacity of each subunit to partcipate in specific interfaces

80 A Plasmid Genotype YEPD Calcium D Plasmid Genotype YEPD Calcium G Quinacrine DIC none WT none WT WT none ¨ none ¨ none ¨ Anc.3-11 ¨ Anc.3-11 ¨ Anc.11 ¨ ¨ Anc.3-11 ¨ Anc.3-11 ¨ Anc.3-11 ¨¨ ¨ Anc.11 ¨ Anc.3-11 E Anc.3-11 ¨ B none WT ¨ Anc.3 ¨ none ¨ Anc.16 Anc.16 ¨ Anc.3-11 ¨ Anc.3 ¨ ¨ Anc.11 F Anc.3 ¨¨ C none WT Anc.11 ¨¨ ¨ none ¨¨¨ Anc.3 Anc.3-11 / Anc.16 ¨¨¨ Anc.3 / Anc.11 ¨¨

FIGURE 28. Reconstructed V-ATPase ancestors replace extant versions. The ancestral V0 subunits functionally replace the three-component ring in extant yeast. S. cerevisiae were plated on permissive media (YEPD) and media buffered with elevated CaCl2. (A) Anc.3-11 rescues growth in yeast deficient for endogenous subunit 3 (3?), subunit 11 (11?), or both (3?11?). Wild-type (WT) yeast growth is shown for comparison. (B) Anc.16 rescues growth in yeast deficient for subunit 16 (16?). (C) Expression of Anc.3-11 and Anc.16 together rescues growth in yeast deficient for subunits 3, 11, and 16. (D) Anc.11 rescues growth in 11? but not 3? yeast. (E) Anc.3 rescues growth in 3? but not 11? yeast. (F) Anc.3 and Anc.11 together rescue growth in 3 11 mutants. (G) Yeast expressing Anc.3-11, Anc.16, Anc.11, or Anc.3 properly acidified the vacuole lumen. Red signal labels yeast cell walls; green signal (quinacrine) labels acidified compartments. Yeast were visualized by differential interference contrast microscopy (DIC).

81 (which we labeled P, R, and Q) with the other subunits (Wang et al. (2007)).

Speciﬁcally, Vma11 is restricted to a single position between Vma16 and Vma3, because its clockwise interface can participate only in interface R with Vma16, and its counterclockswise interface can participate only in interface P with clockwise side of Vma3 (Fig. 29.). Copies of Vma3, in contrast, occupy several positions in the ring, because they form interface P with other copies of Vma3 or Vma11, as well as interface Q with Vma16; Vma3 cannot, however, form interface R with

Vma16. As a result, both Vma3 and Vma11 are required in extant yeast in order to form a complete ring with Vma16.

To test the hypothesis that specific interaction interfaces were lost during evolution, we engineered fusions using ancestral ring proteins to assay the capacity of each ancestral ring protein to form specific interfaces with the other subunits required for a functional complex. Because Anc.3-11 can complement the loss of both subunits 3 and 11, we hypothesized that Anc.3-11 subunit could participate in all three specific interaction interfaces, and that these capcities were then partitioned between Anc3 and Anc11 after the duplication of Anc.3-11 (Fig.

29.A,B). To test this hypothesis, we created six reciprocal gene fusions between yeast subunit 16 and subunits Anc.3-11, Anc.3, Anc.11 (Fig. 29.C). Each fusion constrains the structural position of subunits relative to extant subunit 16, allowing us to determine which arrangements yield a functional ring. As predicted, Anc.3-11 functioned on either side of subunit 16 (Fig. 29.D), indicating that it could form all three interfaces P, Q, and R. In contrast, Anc.3 functioned when constrained to participate in interface Q with Vma16 and interface P with subunit 3; however, ring function was lost when Anc.3 was constrained to form interface R with Vma16

(Fig. 29.E). Anc.11, in turn, functioned when constrained to participate in interface

82 R with Vma16 and interface P with Vma3, but ring function was lost when Anc.11 was constrained to participate in interface Q with Vma16 and interface P with

Vma3. This result indicates that Anc.11 lost the capacity to form one or both of these interfaces during its post-duplication divergence from Anc.3-11 (Fig. 29.F).

Taken together, these data indicate that the speciﬁcity of the ring arrangement and the obligate roles of Vma3 and Vma11 evolved by complementary loss of asymmetric interactions with other members of the ring (Fig. 29.G,H).

Before Anc.3-11 duplicated, the protein ring only contained an undiﬀerentiated subunit 3/11 and a subunit 16. Immediately after Anc.3-11 duplicated, the two descendant subunits must have been functionally identical, so the protein ring could have assembled with many possible combinations of the two descendants, including copies of only one of the descendant proteins. This ﬂexibility disappeared when Anc.3 lost the ancestral interface that allowed it to interact with the counterclockwise side of Vma16, and Anc.11 lost the ability to interact with

Vma16s clockwise side and/or Vma3s counterclockwise side. These complementary losses are suﬃcient to explain the speciﬁc arrangement of contemporary subunits in reconstructed and present-day fungal rings.

To determine the genetic basis for the partitioning of Anc.3-11s functions between Vma3 and Vma11, we introduced historical mutations into Anc3.11 by directed mutagenesis and determined whether they recapitulated the shifts in function that occurred during the evolution of Anc.3 and Anc.11. The two phylogenetic branches leading from Anc.3-11 to Anc.3 and to Anc.11 contain

25 and 31 amino acid substitutions, respectively, but only a subset of these were strongly conserved in subunits 3 or 11 from extant Fungi (Fig. 30.A). We introduced each of these diagnostic substitutions into Anc.3-11 and determined

83 IV II A Anc.3 cytosol P III I P C G II IV Anc I II .3 II III IV V I II III IV I II III IV II III IV V Anc.3-11 I/II Anc.16 II/III IV .3 I nc A II Anc.3-11III/IV P Q III I N C N C Anc.16 IV/V R --- Q P Vma16 Anc.X lumen II III Anc.X Q duplication I I Anc IV R Vma16 II .16 I .3 Anc IV V I III II Anc.3-11 I/II Anc.3-11 I/II Anc.16 II/III Anc.3-11 III/IV R Anc.11 P Plasmid Genotype YEPD Calcium P P Q Anc.3-11 III/IV P P Q II IV D Vma16, Vma11, Vma3 WT Anc.16 IV/V R R --- none 3∆ 11∆ 16∆ complementary losses Anc.3-11, [Vma16 ▃ Anc.3-11] 3∆ 11∆ 16∆ Anc.3-11 duplication and 3∆ 11∆ 16∆ Anc.3 I/II Anc.11 I/II Anc.16 II/III complementary losses Anc.3-11, [Anc.3-11 ▃ Vma16] Anc.3 III/IV P ! Q Anc.11 III/IV P ! ! Anc.16 IV/V ! R --- Vma16, Vma11, Vma3 WT IV II E Vma3, Vma11 16∆ Anc.3-11 B P P Vma3, Vma11, [Vma16 ▃ Anc.3] 16∆ I III II III I H IV Anc.3 Anc P, R P, Q .3-11I III 3∆ 16∆ ! IV .3-11II ▃ II IV Anc Anc.3, Vma11, [Anc.3 Vma16] III I Q P Vma16, Vma11, Vma3 WT I III II III A P, R Anc.3-11 P, Q I I nc IV F Vma3 II .16 I .3-11 11∆ 16∆ II IV Anc IV V II ▃ I III Vma3, [Vma16 Anc.11] 11∆ 16∆ I III R Anc.3-11 P Anc.11 ▃ 11∆ 16∆ P, R P, !Q II IV Vma3, [Anc.11 Vma16] ! II IV FIGURE 29. Increasing complexity by complementary loss of interactions in the fungal V0 ring. (A) Model of the protein ring composed of Anc.3, Anc.11, and Anc.16, arranged as in extant yeast (Wang et al. 2007). Unique intersubunit interfaces are labeled P, Q, and R. (B) Before duplication of Anc.3-11, the ring was assembled in similar fashion from two subunits. (C) To constrain the location of speciﬁc subunits, gene fusions were constructed by tethering an ancestral subunit to either the N or C-terminal side of yeast Vma16. Roman numerals indicate location of transmembrane helices (I through V) based on previous work (30). (D, E, F) Growth assays of yeast with fused V0 subunits to identify the interfaces in which ancestral subunits can participate. For each experiment, expressed V0 subunits are listed; tethered subunits are in brackets and connected by a thick line. Cartoons show the constrained location of the tethered subunit relative to Vma16. (D) Anc.3-11 can function on either side of Vma16. (E) Anc.3 can function only on the clockwise side of Vma16. (F) Anc.11 can function only on the counterclockwise side of Sc.16. (G) Interfaces formed by V0 subunits before and after duplication and complementary loss of interfaces, based on the data in panels D-F. Red Xs indicate lost interfaces. (H) Anc.3-11 participated in interfaces P and R on its clockwise side and interfaces P and Q on its counterclockwise side. Participation in R was lost along the lineage leading to Anc.3; participation in P and/or Q was lost along the lineage leading to Anc.11.

84 whether they recapitulated the loss by Anc.3 or Anc.11 of the capacity to complement Vma gene deletions. We found that a single amino acid replacement that occurred on the branch leading to Anc.11 (V15F) abolished the capacity of Anc.3-11 to function as subunit 3; it also enhanced the ability of Anc.3-11 as subunit 11 (Fig. 30.B). Moreover, a single historical replacement (M22I) from the branch leading to Anc.3 radically reduced the ancestral capacity to function as subunit 11 (Fig. 30.C); the Anc.3-11-M22I mutant retains some of the ancestral proteins capacity to rescue growth in the Vma11-deﬁcient background, suggesting that other mutations also contributed to the functional evolution of Vma3. One other historical mutation (N88T) on this branch also impaired Anc.3-11s capacity to function as Vma11, but it also reduced the proteins capacity to function as

Vma3, suggesting that epistatic interactions with other residues allow this mutation to be tolerated in Anc.3 and its descendants. Several of the replacements on the branch leading to Anc.11 display a similar pattern, reducing the proteins capacity to replace Vma3, suggesting that these historical replacements function better together than in isolation.

Discussion

How complexity and speciﬁc gene functions can evolve has long puzzled evolutionary biologists (Ohno (1970); Jacob (1977); Lynch (2007b)), because mutations that compromise function are far more frequent than those which generate functional novelty (Hietpas et al. (2011)). Our results indicate that increases in the architectural complexity of molecular assemblies can evolve due to a small number of simple, relatively high-probability mutations that degrade ancestral functions but leave other functions intact. The speciﬁc roles of subunits

85 A B Plasmid Genotype YEPD CalciumCalcium Yeast Growth Yeast Growth none WT Proteolipid in 3 in 11 none ¨ Anc.3-11 ++++ ++ Anc.11 none ++++++ Anc.3-11 ¨ Anc.3 ++++++ none Anc.3-11 V15F ¨ no proteolipid none none Anc.3-11 ¨ V15F none ++++ M16A ++ ++ Anc.3-11 V15F ¨ V38I ++++ ++ A42G +++ ++ V45T ++ ++ M46F ++ ++ I55L +++++ +++ Anc.3-11 A61S ++++ ++ Anc.11 Y87S ++ ++ F108Y + ++ Plasmid Genotype YEPD CalciumCalcium T121Y ++++ ++ C A122M + ++ none WT I132V +++++ ++ V15A + +++ none ¨ Anc.3 M22I +++++ + S25T +++ ++ Anc.3-11 ¨ M46L +++ ++ N88T ++ + Anc.3-11 M22I ¨ H92Q ++ ++ A120G ++ ++ Anc.3-11 ¨ N159D ++ ++ Anc.3-11 M22I ¨

FIGURE 30. Genetic basis for functional diﬀerentiation of Anc.3 and Anc.11. (A) Experimental analysis of historical mutations. Strongly conserved historical amino acid replacements from the branches on which Anc.3-11s functions were partitioned are listed in the table. Yellow, replacements on the branch leading to Anc. 11; blue, replacements on the branch leading to Anc.3. Each mutation was introduced singly into Anc.3-11; the variant genes were transformed into S. cerevisiae, and growth was assayed on elevated CaCl2. The table shows growth in semiquantitative terms from zero (none) to wild-type (++++++). Bold mutations recapitulate in whole or part the functional evolution of Anc.11 and Anc. 3. (B) Replacement V15F abolishes Anc.3-11s capacity to function as subunit 3 and enhances its capacity to function as subunit 11. (C) Replacement M22I impairs the capacity of Anc.3-11 to function as subunit 11 without aﬀecting its capacity to function as subunit 3.

86 Vma3 and Vma11 appear to have been acquired when duplicated genes lost some but not all of the ancestral proteins capacity to participate in interactions with copies of itself and another protein required for proper ring assembly. Because complementary losses occurred in both lineages, the two descendant subunits became obligate components, and the complexity of the ring increased. It is possible that specialization of the duplicated subunits allowed increases in ﬁtness, but genome-wide interaction screens and the phenotype of vma11∆ yeast provide no evidence that Vma11 evolved novel functions in addition to those it inherited from Anc.3-11 in the V0 ring (Tong et al. (2004)).

Our results indicate that increases in architectural complexity can evolve due to a small number of simple, relatively high-probability mutations that degrade ancestral functions but leave other functions intact. The speciﬁc roles of subunits

Vma3 and Vma11 appear to have been acquired when duplicated genes lost some but not all of the ancestral proteins capacity to participate in interactions with copies of itself and another protein required for proper ring assembly.

Because complementary losses occurred in both lineages, the two descendant subunits became obligate components, and the architectural complexity of the ring increased.

Because ours is the first mechanistic analysis of the evolutionary trajectory of a molecular machine, the generality of our observations is unknown. By definition, however, all molecular machines involve differentiated parts in specific spatial orientations, and many are composed in whole or part of paralogous proteins. In any such complex, additional paralogs could become obligate components due to gene duplication (Pereira-Leal et al. (2007)) and subsequent mutations that cause specific interaction interfaces among them to degenerate. For example, –

87 the V1 subcomplex, which represents the rest of the V-ATPase proton-translocating complex (Mulkidjanian et al. (2007)), as well as chaperonin complexes (Archibald et al. (2000)), the NADH:ubiquinone oxidoreductase (Gabald´on et al. (2005)), the mitochondrial import machinery (Dolezal et al. (2006)), and the bacterial

flagellums rod, hook, and filament (Pallen and Matzke (2006)) are all composed of paralogous subunits that can function only in specific spatial orientations; the mechanisms we observed that account for the increased complexity of the V0 ring could plausibly be involved in the evolution of these machines as well.

This view of the evolution of molecular machines is related to recent models that explain other biological phenomena such as the retention of large numbers of duplicate genes and mobile genetic elements within genomes as the product of degenerative processes acting upon biological systems with some degree of modularity (Lynch (2007a,b); Force et al. (1999)). Although mutations that enhanced the functions of individual ring components may also have occurred during evolution, our data indicate that simple degenerative mutations are suﬃcient to explain the historical increase in complexity of a crucial molecular machine. There is no need to invoke the acquisition of novel functions caused by low-probability mutational combinations.

88 CHAPTER V

CONCLUSION

In this dissertation I described a computational analysis pipeline for studying the evolutionary history of protein families. This pipeline begins with protein sequences that are evolutionarily related, and then proceeds to align the sequences, infer a phylogeny from the alignment, and then reconstruct ancestral sequences on the phylogeny. Reconstructed ancestral sequences can be physically synthesized and expressed in vivo in order to observe their ancient functions.

In chapter II, I showed that ML phylogenetic error can be partially ameliorated by using a multidimensional search heuristic. Virtually all implementations of ML phylogenetic inference use a simple heuristic that assumes parameters are separable. I implemented a multidimensional heuristic that does not assume parameter separability, and thus simultaneously optimizes all parameters. I observed that this heuristic found more accurate and higher-likelihood phylogenies more often than the simpler heuristic.

In chapter III, I showed that statistical uncertainty about ML phylogenies does not signiﬁcantly impact the downstream accuracy of ancestral reconstruction.

The conditions that cause phylogenetic uncertainty also create a situation in which ancestral sequences are the same across alternate phylogenies. Phylogenetic uncertainty is correlated with short tree branches, which eliminate opportunity for ancestral variance. This result is important because it allows experimentalists to avoid a very time-consuming computation that may require weeks or months to complete.

89 Finally, in chapter IV, I combined the pieces of this pipeline – in collaboration with molecular biologists in the Stevens Lab – to investigate the evolution of a molecular machine. Our work not only demonstrates the computational techniques I advocated in previous chapters, but also provides a novel biological result: molecular machines can evolve increased complexity through degenerative mechanisms.

Error versus Uncertainty

My results in chapter II may seem incongruous with my results in chapter

III. I want to dissuade readers from the following specious line of thinking:

Phylogenetic uncertainty does not signiﬁcantly aﬀect the accuracy of ancestral sequence reconstruction, so who cares about the accuracy of my ML phylogeny?

The key to this puzzle is that error and uncertainty are not the same thing.

Error is inaccuracy, whereas uncertainty is ambiguity. In an ideal world, our metrics for phylogenetic uncertainty would be perfect predictors of phylogenetic accuracy – but this is not the case. This means that a correct inference can sometimes be very uncertain, and an incorrect inference can be strongly supported.

Our metrics of phylogenetic uncertainty are imperfect because we do not have global complete knowledge of the phylogenetic likelihood landscape. Just as our optimization algorithms must be heuristic, our metrics of uncertainty must also be heuristic. Phylogenetic uncertainty typically arises when the ML phylogeny exists in a region of tree space with one or more strongly-supported nearby alternate trees. This region of space can be imagined as a broad hill, with the ML tree at the summit. In contrast, strongly-supported (i.e. certain) ML trees exists in a region of tree space that can be imagined as a sharp peak with no signiﬁcant support

90 for nearby trees. Phylogenetic error occurs when the ML tree exists on the wrong peak. An errored tree may be strongly supported (on a sharp peak) or ambiguously supported (on a broad hill) – but either way, it’s the wrong tree. My results in chapter II showed that Multimax does a better job ﬁnding the correct hill. My results in chapter III showed that trees on the same hill yield the same ancestral sequence.

Implications for the Future of Systems Biology

The analysis pipeline discussed in this dissertation was presented in the context of studying single gene families in isolation. However, nearly all biological phenotypes of scientiﬁc interest are produced by multiple genes working in concert.

A research frontier of biology is to study the evolution of complex phenotypes as a consequence of multi-gene systems responding to particular environmental cues. Future progress in this direction will be made by extending the methods of phylogenetic ancestral reconstruction from single-gene studies to multi-gene studies.

My results in chapter IV demonstrate this multi-gene paradigm for a simple system with three genes. I look forward to a not-so-distant future in which we can generate and test hypotheses about the evolution of complex ancestral systems, including ancestral regulatory networks, ancestral signal transduction networks, and the ancestral assembly of complex molecular machines.

Computational error and statistical uncertainty will play an important role in the future of systems-level evolutionary studies. When multiple genes are studied in tandem, error and uncertainty can become ampliﬁed. For example, in any reconstructed ancestral protein sequence, it is not uncommon for some proportion of sites to be ambiguously inferred (with low posterior probability support). A

91 rigorous experimentalist will explicitly test alternate molecular states at these ambiguous sites in order to observe their eﬀect on reconstructed protein function.

The total number of uncertain ancestral sites increases when reconstructing multiple ancestral proteins because there are simply more sequences being studied.

This means that reconstructing a complex multi-gene ancestral system increases the degrees of uncertainty that must be systematically explored. In extreme cases, the amount of labor required to test all ambiguous states could become overwhelming.

Therefore, as the ﬁeld of systems biology adopts the methods of phylogenetic ancestral reconstruction, it becomes increasingly important to minimize ancestral uncertainty and maximize ancestral accuracy.

Software

There are currently few – if any – available software tools that chain together all the algorithms necessary for multiple sequence alignment, phylogenetic inference, and ancestral reconstruction into a uniﬁed toolkit. My software tools

(URL = XX) provide a ﬁrst attempt at automating this pipeline. My tools automatically make “smart” decisions during each pipeline stage, including time- consuming tasks like ML model selection. I encourage you to use my software suite in your future ancestral reconstruction projects.

92 APPENDIX A

MARKOV MODELS OF SEQUENCE EVOLUTION

The core idea of molecular Markov models is that characters substitute to other characters over time with some probability. The frequency of substitution events is assumed to have a Poisson distribution, and the probability of k events occurring in time t is:

tke tµ P (k t)= − (A.1) | k! where µ is our assumed rate of evolution. The probability that any number of substitutions—from zero to inﬁnity—occur in time t can be calculated by summing over all values k:

k tµ ∞ µt e P (0 k t)= − =1.0 (A.2) ≤ ≤∞| k! k=0

Expression A.2 is used to calculate the likelihood of a single phylogenetic branch for a single sequence site as follows.

Suppose we observe an evolutionary character—a single nucleotide or an amino acid—currently in some state x, where x is one of the letters in the nucleotide or amino acid alphabet. Also suppose we have a matrix R expressing the relative substitution rates between states. R is an n-by-n matrix, where n is the size of the alphabet. Finally, we have a vector π expressing the expected frequencies of each state. Putting all these elements together, x will mutate to state y over time t with probability calculated as follows:

93 k t ∞ t e P (x y t)= (π π Rk ) (A.3) → | x y xy k! k=0 k . . . where Rxy is the relative rate of x transitioning to y, and (Rxy) the extrapolated rate of x y occurring over k steps. π and π are the frequencies → x y of states x and y, otherwise known as the stationary frequencies. Expression A.3 is

typically shown in a more compact form:

k ∞ Qµt P (t)= = eQµt (A.4) k! k=0 . . . where the matrix Q equals ΠR I. Π is the diagonal matrix, where Π[a, a] −

equals the equilibrium frequency πa for state a in our alphabet. I is the identity matrix. The value µ is chosen such that the total rate of possible mutation is one:

1=µ 1 πaRaa . Whereas Expression A.3 calculates a single ﬂoating- × − a point probability value, Expression A.4 calculates a matrix P of probability values

for any state x mutating to any other state y. A description of the matrix algebra

necessary to convert Expression A.3 into Expression A.4 is given in Bryant et al.

(2005).

94 APPENDIX B

COMPUTING THE LIKELIHOOD OF A PHYLOGENY

The likelihood of an entire phylogeny is calculated by recursively applying

Expression A.4 to all branches in the tree. The likelihood L(t, θ D) of the tree t | and model parameters θ, given sequence alignment D, is calculated as a product of

likelihoods L(t, θ D ) for each sequence site i (Expression B.1). | i i

L(t, θ D)= L(t, θ D ) (B.1) | | i i The likelihood L(t, θ D ) of the phylogeny at site i is the sum of partial likelihoods | i v Lx of the root node (call it node v) having state x at site i (Expression B.2). x

v L(t, θ Di)= Lx (B.2) | x v Each partial likelihood Lx is calculated recursively, by descending from v along its

branches t1 and t2 to nodes u1 and u2. Along each branch, we calculate the sum of probabilities of x mutating to some state y. (Expression B.3).

v u1 u2 Lx = P (x y t1)Ly P (x y t2)Ly (B.3) y → | y → |

u1 u2 Ly and Ly are the partial likelihoods of observing state y at nodes u1 and u2. These partial likelihoods are calculated by deeper recursion to the branches

descending from nodes u1 and u2. Eventually, the recurrence arrives at a leaf node

uT uT . The partial likelihood Ly of state y at node uT equals 1.0 if uT is state y in

uT the sequence data; otherwise Ly equals 0. Figure 31. illustrates the data structures involved in this recursion. 95 v v v v L =[Lx,Ly,...,Lz , ]

u1 u1 u1 u1 L =[Lx ,Ly ,...,Lz , ] u2 u2 u2 u2 v L =[Lx ,Ly ,...,Lz , ] t1 t2

u1 u2

FIGURE 31. The recursive data structure for the likelihood algorithm. We pick an arbitrary root node v, with descendant branches t1 and t2 leading to nodes u1 and u2. We recursively calculate a vector of partial likelihoods for each node on u1 u1 the tree. For example, the vector L contains the partial likelihood Lx of node u1 u1 existing as state x, the partial likelihood Ly of u1 existing as state y, etc.

Markov Models Make Simplifying Assumptions About Evolution

Markov models of molecular sequence evolution make three major several simplifying assumptions about the underlying evolutionary process. First, sites within an alignment are assumed to evolve independently from each. Second, the state-to-state substitution process is simpliﬁed to be time reversible. Finally, the substitution process is assumed to be ergodic, such that the expected frequency of each state is assumed to be static over evolutionary time. All three of these assumptions have been shown to be incorrect for speciﬁc empirical counterexamples, but for most protein families these assumptions seem to yield robust and accurate evolutionary inferences.

96 APPENDIX C

ALIGNMENT ERROR & ANCESTRAL RECONSTRUCTION

In this section I show there is a complex relationship between error in multiple sequence alignments and the downstream error in ancestral reconstruction. I observed that a collection of five alignment algorithms found significantly different alignments for simulated sequences evolved in a variety of controlled conditions.

Further, these ﬁve alignments yielded downstream ancestral sequences that varied in length and accuracy. My results show that the choice of alignment algorithm has signiﬁcant consequences for the accuracy of downstream evolutionary inference.

Further work is required to dissect the mechanisms of alignment algorithms in order to understand why their accuracy varies in diﬀerent evolutionary conditions.

Phylogenetic inference and all downstream analysis relies on the accuracy of the multiple sequence alignment (MSA). The goal of MSA is to identify the characters that are homologous – with shared evolutionary history – within a collection of sequences. MSA can be difficult because molecular sequences tend to acquire lineage-specific mutations over evolutionary time; the precise relationships between sequences may be unclear. There exist dozens of algorithms and software packages for MSA (Batzoglou (2005); Notredame (2007)). MSA algorithms primarily differ in two ways: their cost functions, and their strategies for hierarchically ordering the sequences.

Cost functions determine the relative penalties for inserting gap characters.

These functions are based on a dynamic string matching algorithms, using the

Needleman-Wunsch algorithm for global alignment (Needleman and Wunsch

(1970)) or the Smith-Waterman algorithm for local alignment (Smith and

97 Waterman (1981)). Cost functions can be made arbitrarily complex to reﬂect various biological similarities of molecular characters and the underlying insertion/deletion process. It is computationally intractable to apply the cost function to multiple sequences simultaneously, except in trivial-sized problems

(Gotoh (1990); Wang and Jiang (1994)). Rather, alignment software must employ an iterative approach, in which the most similar sequences are ﬁrst aligned and then less similar sequences are progressively added until all the sequences have been incorporated. The results of MSA are highly dependent on the order in which sequences are ordered (Berger and Munson (1991); Landan and Graur (2009)).

The relative accuracy of different MSA algorithms has not been comprehensively studied (Schwartz et al. (2005); Thompson et al. (2005)). Previous work showed that different MSA algorithms result in significantly different alignments, and these alignments ultimately yield different ML phylogenies (Wong et al. (2008)). However, the effects of different MSA algorithms on the downstream accuracy of ancestral reconstruction has not been investigated.

Here, I compared the performance of five different algorithms under a range of evolutionary conditions. I observed these algorithms created alignments that were significantly and consistently biased to be overaligned or underaligned, depending on the algorithm. I reconstructed ML phylogenies and ancestral sequences for these alignments. I observed that the accuracy of ancestral sequences varied among the

ﬁve alignment algorithms, but the accuracy of the alignment itself was a poor predictor of the ancestral accuracy. My results show that the choice of alignment algorithm has non-trivial ramiﬁcations for the accuracy of downstream evolutionary inference.

98 Methods and Materials

Simulated Sequence Alignment

In order to determine the extent to which a variety of alignment algorithms aﬀect the accuracy of ancestral sequence reconstruction, I simulated random amino acid sequences evolving on a four-taxon tree:

(((ta:0.2,tb:0.2):0.1,tc:0.25):0.1,td:0.3);

Simulations were initialized with random ancestral sequences 400 amino acids long.

The ancestors were then evolved along the branches of the tree using the JTT model of amino acid substitution, combined with one of six diﬀerent distributions of insertions/deletions (indels). I used (i) a small-mean and (ii) large-mean negative binomial distribution, (iii) a small-mean and (iv) large-mean Zipﬁan (power) distribution, and a (v) small-mean and (vi) large-mean uniform distribution. I varied the probability of indel events from 0.0 to 0.4, in increments of 0.04. I repeated the simulation twenty times for each combination of indel model and indel probability.

Alignment of Simulated Sequences

I aligned the simulated sequences using the default settings in the software

Amap (cite Do et al. (2005); Schwartz et al. (2005)), Clustal (Thompson et al.

(1994)) , Maﬀt (Katoh et al. (2002)), Muscle (Edgar (2004)), and Prank (Loytynoja and Goldman (2008)).

99 Phylogenetic Inference

I inferred ML phylogenies for each alignment, using my own in-house modiﬁcations to PhyML version 3.0. Trees were optimized using Multimax

(described in chapter II), using the default settings.

Ancestral Sequence Reconstruction

I reconstructed ancestral sequences for the most-recent-common ancestor of taxa ta and tb using maximum likelihood as implemented in PAML version 4.2 and an in-house GUI – named Lazarus – that controls PAML (Yang (2007); Hanson-

Smith et al. (2010)).

Ancestral Error

For every alignment, I measured the accuracy of the length of the ML ancestral sequence by counting the total number of sites in the ancestor, sans indel characters. I compared this value to the total number of sites in the true ancestral sequence, which had been recorded during its simulation.

Results

Variability of Alignment Length

The five alignment algorithms found significantly different alignments for sequences simulated under all six indel models (Fig. 32.). The algorithms Prank and Clustal systematically under-aligned sequences, whereas Amap, Muscle, and

Maﬀt systematically over-aligned sequences. Across all conditions, Maﬀtcreated

100 alignments with the most accurate lengths, whereas Prank created alignments with the least-accurate lengths.

Eﬀect on Ancestral Sequence Error

The accuracy of ancestral sequence lengths signiﬁcantly varied among the

ﬁve alignment algorithms (Fig. 33.). Across all methods, the absolute amount of ancestral error increased when the evolutionary conditions had large indel rates, and also when the true indel length distribution had a large mean value. Amap,

Clustal, Muscle, and Prank inferred alignments whose ML ancestral sequences were generally too long. Maﬀt, in contrast, inferred alingments whose ML ancestral sequences were nearly always too short. Overall, the most accurate ancestral sequences came from alignments inferred by Prank and Maﬀt.

Discussion

My results demonstrate that the choice of alignment algorithm has signiﬁcant consequences for the accuracy of the alignment, and for the downstream accuracy of the length of inferred ancestral sequences. Further, my results suggest that alignment error is a poor predictor of ancestral error. The least-accurate MSA algorithm – Prank – ultimately produced some of the most accurate alignments.

Prank’s superiority for ancestral reconstruction may be understood by comparing its underlying mechanisms to other algorithms. Whereas older MSA algorithms, such as Clustal, attempt to minimize the total number of indel characters in an alignment, the Prank algorithm attempts to minimize the total number of indel events – where a single event may include multiple contiguous indel characters.

101 Prank’s alignments seem to preserve a phylogenetic signal that is highly amenable to placing ancestral indel characters.

My results suggest that phylogenetic practitioners should embrace alignment uncertainty, and repeat their evolutionary analysis using several diﬀerent MSA algorithms in parallel.

102 True alignment AMAP CLUSTAL MUSCLE PRANK MAFFT

600 1.0 800 x = 1.1 1.0 ѫ x ѫ

0.0 0 30 0.0 0 30

0.0 indel rate 0.0

800 1.0 1.0 x x ѫ ѫ 0.0 0.0 0 0

0.0 0.0

0.2 x 0.2 ѫ x ѫ 0.0 0 30 0.0 0 30

0.0 0.0

FIGURE 32. Alignment length versus insertion-deletion rate. Sequences were simulated on the tree described in section C, using six diﬀerent models of insertion- deletion events (shown here in small insets). Vertical bars express the average length of the alignment for the given model, indel rate, and alignment algorithm. Error bars are standard error of the mean.

103 AMAP CLUSTAL MUSCLE PRANK MAFFT

1.0 1.0 x = 1.1 x 1.06x ѫ ѫ 0.0 1.1x 0.0 0 30 0 30

correct correct 0.97x 0.9x 0.0 indel rate 0.0

[ 1.3x 1.0 1.0 x x ѫ ѫ 0.0 0.0 0 0

correct correct [ 0.9x 0.0 0.0

1.6x 0.2 x 0.2 x ѫ [ ѫ 0.0 0.0 0 30 0 30

correct correct 0.9x [ 0.0 0.0

FIGURE 33. Ancestral length error versus insertion-deletion rate. Sequences were simulated on the tree described in section C, using six diﬀerent models of insertion- deletion events (shown here in small insets). Vertical bars express the scaling factor by which ancestral sequences are too long or too short. Error bars are standard error of the mean.

104 APPENDIX D

V-ATPASE SUBUNIT PROTEIN SEQUENCES

The following list contains the GenBank accession IDs for protein sequences used in chapter IV. The IDs are labeled with the ﬁrst letter of their genus, the full name of their species, and an integer number. 3, 11, and 16 indicate homology to yeast subunits c, c’, and c”, respectively.

M musculus 16 NP 291095 O mordax 3 ACO09611 D rerio 3 NP 001098606 M grisea 16 XP 369356 M grisea 11 XP 366989 A niger 3 XP 001399935 C glabrata 3 XP 447321 A terreus 11 XP 001214955 A terreus 16 XP 001211600 C parvum 16 XP 627363 P vivax 16 XP 001616329 T castaneum 3 XP 967959 X tropicalis 3 NP 988893 G zeae 3 XP 390178 L elongisporus 3 XP 001526092 A fumigatus 3 XP 001263225 M grisea 3 XP 365764 S pombe 3 NP 594799 C albicans 3 XP 721376 B bovis 16 XP 001612047 G zeae 11 XP 388749 C muris 3 XP 002141961 S purpuratus 3 XP 797801 C immitis 16 XP 001246494 M mulatta 16 XP 001097275 Y lipolytica 3 XP 505831 B fuckeliana 16 XP 001552198 C immitis 11 XP 001242880 A mellifera 16 XP 392599 L elongisporus 11 XP 001523616 T annulata 16 XP 953463 N vectensis 3 XP 001637733 S Salar 16 NP 001134021 L elongisporus 16 XP 001525467 C neoformans 16 XP 773114 105 C neoformans 11 XP 778255 P knowlesi 16 XP 002261350 A terreus 3 XP 001213329 A pisum 3 NP 001155531 S Salar 3 NP 001154112 P marneﬀei 3 XP 002152865 P stipitis 3 XP 001387092 A gossypii 3 NP 984787 C hominis 3 XP 667190 C muris 16 XP 002142524 X laevis 16 NP 001087741 M mulatta 3 XP 001088617 M brevicollis 16 XP 001742805 K lactis 3 XP 454966 T castaneum 16 XP 975026 C intestinalis 16 XP 002131348 E caballus 3 XP 001915231 G zeae 16 XP 385476 C globosum 3 XP 001229170 A clavatus 3 XP 001271234 N crassa 11 XP 965807 N crassa 16 XP 964449 X tropicalis 16 NP 001017064 P anserina 11 XP 001907168 P anserina 16 XP 001910317 H sapiens 16 AAP36886 T annulata 3 XP 952989 A gossypii 11 NP 985409 A gossypii 16 NP 983473 A pisum 16 NP 001155679 N crassa 3 XP 961418 V polyspora 3 XP 001642185 T adhaerens 3 XP 002112261 S cerevisiae 3 NP 010887 P marneﬀei 16 XP 002145395 D hansenii 3 XP 460869 S pombe 11 NP 593600 T guttata 3 ACH45347 S pombe 16 NP 594516 B fuckeliana 3 XP 001553113 T adhaerens 16 XP 002114348 S cerevisiae 16 NP 011891 R norvegicus 3 NP 033859 S cerevisiae 11 NP 015090 B taurus 3 NP 001017954

106 C familiaris 3 XP 537002 P anserina 3 XP 001911041 P marneﬀei 11 XP 002147471 P knowlesi 3 XP 002259621 T guttata 16 NP 001232246 B fuckeliana 11 CCD51873 C cinerea okayama 3 XP 001835649 P falciparum 16 XP 001350256 C intestinalis 3 XP 002132074 A niger 11 XP 001391591 A niger 16 XP 001397102 C parvum 3 XP 627909 S purpuratus 16 XP 790651 B ﬂoridae 3 XP 002598155 Y lipolytica 16 XP 505205 A mellifera 3 NP 001011570 A clavatus 16 XP 001275839 N vectensis 16 XP 001638230 A clavatus 11 XP 001274195 S sclerotiorum 3 XP 001588693 P falciparum 3 XP 001351750 D discoideum 16 XP 644318 C cinerea okayama 11 XP 001835902 D hansenii 16 XP 460013 D hansenii 11 XP 458901 C cinerea okayama 16 XP 001830694 P vivax 3 XP 001613765 C immitis 3 XP 001239974 A fumigatus 16 XP 755891 C neoformans 3 XP 772642 H sapiens 3 AAP36127 A fumigatus 11 XP 753781 Y lipolytica 11 XP 504637 M brevicollis 3 XP 001743042 D discoideum 3 XP 644319 P stipitis 16 XP 001386908 C albicans 16 XP 722165 C albicans 11 XP 721376 P stipitis 11 XP 001382501 X laevis 3 NP 001082675 C glabrata 11 XP 445959 E caballus 16 XP 001916016 C glabrata 16 XP 447739 V polyspora 11 XP 001645235 V polyspora 16 XP 001646358

107 C hominis 16 XP 665533 C globosum 11 XP 001222467 C globosum 16 XP 001223715 O mordax 16 ACO10130 D rerio 16 NP 955855 B bovis 3 XP 001609797 B ﬂoridae 16 XP 002610356 M musculus 3 NP 033859 C familiaris 16 XP 539645 K lactis 11 XP 452911 K lactis 16 XP 454470 B taurus 16 NP 001033127 S sclerotiorum 16 XP 001590765 S sclerotiorum 11 XP 001595091 R norvegicus 16 AAH09169

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