Introduction

There are three types of projects in this collection: short projects, long projects, and projects that involve the development of novel methods. Each project requires data analysis, either on real or simulated data, and also writing. Therefore, even the short projects will require about a week for completion.

The main purpose of the short projects is to familiarize the student with the process of computing and interpreting alignments and trees on datasets. Because the data analysis part of these projects should be fast to complete, they are focused on relatively small nucleotide datasets. If the student has access to sufficient computational resources, then analyses of larger datasets or amino acid datasets are possible. Each short project also asks the student to explore the impact of method choice (i.e., alignment method or tree estimation method) or dataset on the resultant tree, typically using visualization tools.

The long projects build on the short projects, but do more exploration of the impact of method choice (for alignment estimation or tree estimation) or dataset on phylogeny estimation. Some of these projects examine scalability of methods to large datasets, and so will require substantial computational resources. As the student will learn, the degree to which the method selection impacts the final phylogeny can depend on the properties of the data, such as number of sequences, number of sites (i.e., sequence length), rate of evolution, percentage of missing data, etc. The use of both biological and simulated data will help the students evaluate the impact of the different factors on the final outcomes.

The projects aimed at novel method development are likely to be the most difficult, and success in these projects will probably require substantial effort beyond the period of the course. However, a student who wishes to do a novel method development project is usually best served by starting with a long project to identify the competing methods and select datasets that are best able to differentiate between methods.

Final projects for the course are typically long projects rather than novel method development projects, and are focused on comparisons of leading computational methods on simulated or biological datasets, with an eye toward assessing the relative performance of these methods, and gaining insight into the conditions that impact each method.

Introduction

The construction of large phylogenies is of increasing interest, and the size of these datasets can be enormous. For example, the insect portion of the Tree of Life already contains “nearly a million described species,” and the evolutionary relationships between these species is far from resolved (Maddison, 2016). Yet methods for constructing large trees are typically based on approaches that were designed for much smaller datasets, and very few available software packages have adequate accuracy on large datasets. Thus, there is a gap in terms of basic algorithmic strategies and in terms of software development for large-scale phylogeny estimation.

In this chapter, we investigate techniques for constructing phylogenetic trees for large datasets.We explore algorithm design, including standard heuristics used in many software packages, and also divide-and-conquer techniques used to scale computationally intensive methods to large datasets. Some of the initial parts of this chapter appear in earlier chapters, and are repeated here for the sake of completeness.

Standard Approaches

Many phylogeny estimation methods fall into the following categories: distance-based methods, subtree assembly-based methods, heuristics for NP-hard optimizationmethods, or Bayesian methods. Understanding each of these types of methods is helpful in developing methods for large datasets.

Distance-based methods. Generally the fastest phylogeny estimation methods operate by computing a matrix of pairwise “distances” between every pair of taxa, and then using that distance matrix to compute a tree. Distance-based methods (described in Chapter 5) typically run in O(n3) time, but some even run in O(n2) time, and many are fast enough to be used on even very large datasets.

Subtree assembly-based methods. Some methods operate by computing trees on a collection of small subsets of the taxon set, and then combine the subset trees together into a tree. We call these “subtree assembly-based methods” since they operate by estimating subtrees and then assembling them into a larger tree. Many subtree assembly-based methods compute quartet trees using maximum likelihood, and then combine the quartet trees using some quartet amalgamation method, such as Quartet Puzzling (Strimmer and von Haeseler, 1996), Weight Optimization (Ranwez and Gascuel, 2001), Quartet Joining (Xin et al., 2007), Quartets MaxCut (Snir and Rao, 2010), and Quartet FM (Reaz et al., 2014).

Overview

The evolutionary history of a set of genes, species, or individuals provides a context in which biological questions can be addressed. For this reason, phylogeny estimation is a fundamental step in many biological studies, with many applications throughout biology, such as protein structure and function prediction, analyses of microbiomes, inference of human migrations, etc. In fact, there is a famous saying by Dobzhansky that “Nothing in biology makes sense except in the light of evolution” (Dobzhansky, 1973).

Because phylogenies represent what has happened in the past, they cannot be directly observed but rather must be estimated. Consequently, increasingly sophisticated statistical models of sequence evolution have been developed, and are now used to estimate phylogenetic trees. Indeed, over the last few decades, hundreds of software packages and novel algorithms have been developed for phylogeny estimation, and this influx of computational approaches into phylogenetic estimation has transformed systematics. The availability of sophisticated computational methods, fast computers and high-performance computing (HPC) platforms, and large sequence datasets enabled through DNA sequencing technologies, has led to the expectation that highly accurate large-scale phylogeny estimation, potentially answering open questions about how life evolved on earth, should be achievable.

Yet large-scale phylogeny estimation turns out to be much more difficult than expected, for multiple reasons. First, all the best methods are computationally intensive, and standard techniques do not scale well to large datasets; for example, maximum likelihood phylogeny estimation is NP-hard, so exact solutions cannot be found efficiently (unless P = NP), and Bayesian MCMC methods can take a long time to reach stationarity. While massive parallelism can ameliorate these challenges to some extent, it doesn't really address the basic challenge inherent in searching an exponential search space. However, another issue is that the statistical models of sequence evolution that properly address genomic data are substantially more complex than the ones that model individual loci, and methods to estimate genome-scale phylogenies are (relatively speaking) in their infancy compared to methods for single gene phylogenies.

Introduction to Statistical Estimation in Phylogenetics

Phylogeny estimation from biomolecular sequences is often posed as a statistical inference problem where the sequences evolve down a tree via a stochastic process. Statistical estimation methods take advantage of what is known (or hypothesized) about that stochastic process in order to produce an estimate of the evolutionary history. When we consider phylogeny reconstruction methods as statistical estimation methods, many statistical performance issues arise. For example: Is the method guaranteed to construct the true tree (with high probability) if there are enough data (i.e., is the method statistically consistent under the model)? How much data does the method need to obtain the true tree with high probability (i.e., what is the sample complexity of the method)? Is the method still relatively accurate if the assumptions of the model do not apply to the data that are used to estimate the tree (i.e., is the method robust to model misspecification)?

Markov models of evolution form the basis of most computational methods of analysis used in phylogenetics. The simplest of these models are for characters with two states, reflecting the presence or absence of a trait. However, the most common models are for nucleotide (four-state characters) or amino acid (20-state characters) data. They can also be used (although less commonly) for codon data, in which case they have 64 states (Goldman and Yang, 1994; Yang et al., 2000; Kosiol et al., 2007; Anisimova and Kosiol, 2009; De Maio et al., 2013).

In Chapter 1, we described the Cavender–Farris–Neyman (CFN) model of binary sequence evolution, and a simple method to estimate the CFN tree from binary sequences. In this chapter we focus on sequence evolution models that are applicable to molecular sequence evolution. As we will see, the mathematical theorems and algorithmic approaches are very similar to those developed for phylogeny estimation under the CFN model.

Statistical identifiability is an important concept related toMarkov models.We say that a parameter (such as the tree topology) of the Markov model is identifiable if the probability distribution of the patterns of states at the leaves of the tree are always different for two Markov models that are different for that parameter. Thus, some parameters of a model may be identifiable while others may not be. For example, the unrooted tree topology is identifiable under the CFN model, but the location of the root is not.

Introduction

In essence, there is really one primary type of data used to construct trees – characters. An example of a character is the nucleotide (A, C, T, or G) that appears in a particular location within a gene, the number of legs (any positive integer), or whether the organism has hair (a Boolean variable). In each of these cases, a character divides the input set into disjoint subsets so that the elements of each subset (which could be molecular sequences or biological species) are equivalent with respect to that character. We will refer to the different sets defined by the character as the character states. In other words, a character is an equivalence relation on the set S of taxa, and the different equivalence classes are the different character states. However, characters can also be described by functions from the taxon set S to the set of character states.

The number of states the character can take is an important aspect for modeling purposes. Many characters are based on sites within a molecular sequence alignment, and thus have a maximum number of possible states (four for DNA or RNA, and 20 for amino acids). Morphological features can be multi-state (and perhaps even have an unbounded number of possible states), but many morphological features are just based on the presence or absence of a given characteristic, and hence are explicitly binary.

Most stochastic models assume that the taxa are related by a tree, and that a character evolves down the tree under a process that involves substitutions. However, not all evolution is treelike; for example, horizontal gene transfer (HGT) is common in some organisms (e.g., bacteria) and hybridization (whereby two species come together to make a new species) is also frequent for some organisms (e.g., plants).

Character-based methods are the basis of nearly all phylogenetic estimation methods, since characters form the way the input is nearly always described. If the evolutionary process operating on the characters can be modeled adequately, then phylogeny estimation – whether of a tree or of a phylogenetic network – can be performed using statistical methods, such as maximum likelihood or Bayesian MCMC. However, when no statistical model is available (or the statistical models are unreliable), then simple methods that are not based on explicit models of evolution can be used to estimate trees.

Introduction

The basic objective of most supertree studies is the assembly of a large species tree from a set of species trees that have been estimated on potentially smaller sets of taxa. Indeed, it is generally believed that construction of the Tree of Life, which will encompass millions of species, will require supertree methods, because no software will be able to deal with the computational challenges involved in such a difficult task. More recently, however, new uses of supertree methods have been discovered, especially in the context of divide-andconquer strategies. This chapter examines both applications of supertree methods.

Traditionally, supertree methods were used to combine trees computed by different researchers that had already been estimated for different taxon sets. In this case, the person constructing the supertree has no control over the inputs, neither how the different subset trees were constructed nor how the taxon sets of the different subset trees overlap. Furthermore, the person constructing the supertree may not have easy access to the data (e.g., sequence alignments) on which the subset trees were constructed.

A modern and more interesting use of supertree methods is in the context of a divideand- conquer strategy to construct a very large tree, or to enable a statistically powerful but computationally intensive method to be applied to a larger dataset. In such a strategy, a large set of taxa is divided into overlapping subsets, trees are estimated (using the desired method) on the subsets, and the estimated subset trees are combined into a tree on the full set of taxa using a supertree method.

Divide-and-conquer techniques for constructing large trees have many desirable features: (1) the subsets can be made small enough that expensive methods can be used to construct trees on them; (2) different methods can be used on each subset, thus making it possible to better address heterogeneity within the full dataset; and (3) the subsets can be created so as to have desirable overlap patterns. The first two of these features tend to increase the accuracy of the estimated subset trees, while the third feature can make it easier to construct an accurate supertree from the subset trees (Wilkinson and Cotton, 2006). We will return to the topic of divide-and-conquer strategies and how to use them to construct large trees under a variety of scenarios in Chapter 11.

Introduction

One of the fascinating challenges in estimating the evolutionary history of a set of species is that different regions within the genomes can evolve differently due to various biological phenomena (Maddison, 1997; Mallo and Posada, 2016; Posada, 2016). One of the most obvious causes for this difference is horizontal gene transfer (Syvanen, 1985), whereby DNA is transferred from the genome of one species into that of another. Horizontal gene transfer (HGT) is especially frequent among prokaryotes (Gogarten et al., 2002), but occurs in other organisms as well. While it can be argued that a species tree is useful as a model (Mindell, 2013) and can make sense in the presence of HGT (even for prokaryotes) if it is based on genes that are resistant to HGT, a full depiction of evolutionary history when HGT has occurred requires a more general graphical model called a “phylogenetic network” (Morrison, 2014a).

Hybrid speciation, where two different species have viable offspring, is another biological process that requires a phylogenetic network. Well-known examples of hybridization include mules and hinnies (which are hybrids of horses and donkeys) and ligers and tigons (which are the hybrid offspring of lions and tigers). Hybrid speciation has long been known to be common in plants (Rieseberg, 1997), and its extent in other organisms is increasingly apparent (Pennisi, 2016). Like HGT, a proper representation of the evolutionary history of a dataset in which hybridization appears requires a phylogenetic network, rather than a tree.

There are also biological processes that cause different parts of the genomes to evolve differently, but where the species history is still correctly modeled as a tree. For example, under a gene duplication and loss model (Ohno, 1970), a gene evolves within the branches of a species tree with duplication events (that increase the number of copies of the gene within a lineage) and loss events (that reduce the number of copies of a gene), so that a given species can have multiple copies of the same gene within its genome.

Another process that can create discordance with the species tree is the multi-speciescoalescent (MSC) model (Maddison, 1997; Pamilo and Nei, 1998) (i.e., the multi-species version of the coalescent process), which models how alleles segregate into populations, and so reflects population-level processes.

In this first chapter, we give an introduction to random graphs and complex networks. We discuss examples of real-world networks and their empirical properties, and give a brief introduction to the kinds of models that we investigate in the book. Further, we introduce the key elements of the notation used throughout the book.

Motivation

The advent of the computer age has incited an increasing interest in the fundamental properties of real-world networks. Due to the increased computational power, large data sets can now easily be stored and investigated, and this has had a profound impact on the empirical studies of large networks. As we explain in detail in this chapter, many real-world networks are small worlds and have large fluctuations in their degrees. These realizations have had fundamental implications for scientific research on networks. Network research is aimed to both to understand why many networks share these fascinating features, and also to investigate what the properties of these networks are in terms of the spread of diseases, routing information and ranking of the vertices present.

The study of complex networks plays an increasingly important role in science. Examples of such networks are electrical power grids and telecommunication networks, social relations, the World-Wide Web and Internet, collaboration and citation networks of scientists, etc. The structure of such networks affects their performance. For instance, the topology of social networks affects the spread of information or disease (see, e.g., Strogatz (2001)). The rapid evolution and success of the Internet have spurred fundamental research on the topology of networks. See Barabási (2002) and Watts (2003) for expository accounts of the discovery of network properties by Barabási, Watts and co-authors. In Newman et al. (2006), you can find some of the original papers detailing the empirical findings of real-world networks and the network models invented for them. The introductory book by Newman (2010) lists many of the empirical properties of, and scientific methods for, networks.

One common feature of complex networks is that they are large. As a result, a global description is utterly impossible, and researchers, both in the applications and in mathematics, have turned to their local description: how many vertices they have, by which local rules vertices connect to one another, etc. These local rules are probabilistic, reflecting the fact that there is a large amount of variability in how connections can be formed