Clades, clans, and reciprocal monophyly under neutral evolutionary models

Clade size distribution under neutral evolutionary models

Given a labeled tree topology t, consider a population P of k leaves chosen among those of t. The clade of P is the minimal subtree of t containing P and its size is given by the number of leaves in the clade. When t is selected under the Yule or uniform distribution among the labeled topologies of size n, we study the "clade size" random variable determining closed formulas for its probability mass function, its mean, and its variance. Our calculations show that for large n the clade size tends to be smaller under the uniform model than under the Yule model, with a larger variability in the first scenario for values of k≥5. We apply our probability formulas to investigate set-theoretic relationships between the clades of two populations in a random tree, determining how likely one clade is contained in or it is equal to the other. Our study relates to earlier calculations for the probability that under the Yule model the clade size of P equals the size of P - that is, the population P forms a monophyletic group - and extends known results for the probability that the minimal (non-trivial) clade containing a random taxon has a given size.

The Probability of Joint Monophyly of Samples of Gene Lineages for All Species in an Arbitrary Species Tree

Monophyly is a feature of a set of genetic lineages in which every lineage in the set is more closely related to all other members of the set than it is to any lineage outside the set. Multiple sets of lineages that are separately monophyletic are said to be reciprocally monophyletic, or jointly monophyletic. The prevalence of reciprocal monophyly, or joint monophyly (JM), has been used to evaluate phylogenetic and phylogeographic hypotheses, as well as to delimit species. These applications often make use of a probability of JM under models of gene lineage evolution. Studies in coalescent theory have computed this JM probability for small numbers of separate groups in arbitrary species trees and for arbitrary numbers of separate groups in trivial species trees. In this study, generalizing existing results on monophyly probabilities under the multispecies coalescent, we derive the probability of JM for arbitrary numbers of separate groups in arbitrary species trees. We illustrate how our result collapses to previously examined cases. We also study the effect of tree height, sample size, and number of species on the probability of JM. We obtain relatively simple lower and upper bounds on the JM probability. Our results expand the scope of JM calculations beyond small numbers of species, subsuming past formulas that have been used in simpler cases.

Measuring the external branches of a Kingman tree: A discrete approach

The Kingman coalescent process is a classical model of gene genealogies in population genetics. It generates Yule-distributed, binary ranked tree topologies - also called histories - with a finite number of n leaves, together with n-1 exponentially distributed time lengths: one for each layer of the history. Using a discrete approach, we study the lengths of the external branches of Yule distributed histories, where the length of an external branch is defined as the rank of its parent node. We study the multiplicity of external branches of given length in a random history of n leaves. A correspondence between the external branches of the ordered histories of size n and the non-peak entries of the permutations of size n-1 provides easy access to the length distributions of the first and second longest external branches in a random Yule history and coalescent tree of size n. The length of the longest external branch is also studied in dependence of root balance of a random tree. As a practical application, we compare the observed and expected number of mutations on the longest external branches in samples from natural populations.

The probability of reciprocal monophyly of gene lineages in three and four species

Reciprocal monophyly, a feature of a genealogy in which multiple groups of descendant lineages each consist of all of the descendants of their respective most recent common ancestors, has been an important concept in studies of species delimitation, phylogeography, population history reconstruction, systematics, and conservation. Computations involving the probability that reciprocal monophyly is observed in a genealogy have played a key role in criteria for defining taxonomic groups and inferring divergence times. The probability of reciprocal monophyly under a coalescent model of population divergence has been studied in detail for groups of gene lineages for pairs of species. Here, we extend this computation to generate corresponding probabilities for sets of gene lineages from three and four species. We study the effects of model parameters on the probability of reciprocal monophyly, finding that it is driven primarily by species tree height, with lesser but still substantial influences of internal branch lengths and sample sizes. We also provide an example application of our results to data from maize and teosinte.

The genomic landscape and evolution of endometrial carcinoma progression and abdominopelvic metastasis

Recent studies have detailed the genomic landscape of primary endometrial cancers, but the evolution of these cancers into metastases has not been characterized. We performed whole-exome sequencing of 98 tumor biopsies including complex atypical hyperplasias, primary tumors and paired abdominopelvic metastases to survey the evolutionary landscape of endometrial cancer. We expanded and reanalyzed The Cancer Genome Atlas (TCGA) data, identifying new recurrent alterations in primary tumors, including mutations in the estrogen receptor cofactor gene NRIP1 in 12% of patients. We found that likely driver events were present in both primary and metastatic tissue samples, with notable exceptions such as ARID1A mutations. Phylogenetic analyses indicated that the sampled metastases typically arose from a common ancestral subclone that was not detected in the primary tumor biopsy. These data demonstrate extensive genetic heterogeneity in endometrial cancers and relative homogeneity across metastatic sites.

The probability of monophyly of a sample of gene lineages on a species tree

Monophyletic groups-groups that consist of all of the descendants of a most recent common ancestor-arise naturally as a consequence of descent processes that result in meaningful distinctions between organisms. Aspects of monophyly are therefore central to fields that examine and use genealogical descent. In particular, studies in conservation genetics, phylogeography, population genetics, species delimitation, and systematics can all make use of mathematical predictions under evolutionary models about features of monophyly. One important calculation, the probability that a set of gene lineages is monophyletic under a two-species neutral coalescent model, has been used in many studies. Here, we extend this calculation for a species tree model that contains arbitrarily many species. We study the effects of species tree topology and branch lengths on the monophyly probability. These analyses reveal new behavior, including the maintenance of nontrivial monophyly probabilities for gene lineage samples that span multiple species and even for lineages that do not derive from a monophyletic species group. We illustrate the mathematical results using an example application to data from maize and teosinte.

Clades and clans: a comparison study of two evolutionary models

The Yule-Harding-Kingman (YHK) model and the proportional to distinguishable arrangements (PDA) model are two binary tree generating models that are widely used in evolutionary biology. Understanding the distributions of clade sizes under these two models provides valuable insights into macro-evolutionary processes, and is important in hypothesis testing and Bayesian analyses in phylogenetics. Here we show that these distributions are log-convex, which implies that very large clades or very small clades are more likely to occur under these two models. Moreover, we prove that there exists a critical value κ(n) for each n ≥ 4 such that for a given clade with size k, the probability that this clade is contained in a random tree with n leaves generated under the YHK model is higher than that under the PDA model if 1 < k < κ(n),and lower if κ(n) < k < n. Finally, we extend our results to binary unrooted trees, and obtain similar results for the distributions of clan sizes.

Yule-generated trees constrained by node imbalance

The Yule process generates a class of binary trees which is fundamental to population genetic models and other applications in evolutionary biology. In this paper, we introduce a family of sub-classes of ranked trees, called Ω-trees, which are characterized by imbalance of internal nodes. The degree of imbalance is defined by an integer 0 ≤ ω. For caterpillars, the extreme case of unbalanced trees, ω = 0. Under models of neutral evolution, for instance the Yule model, trees with small ω are unlikely to occur by chance. Indeed, imbalance can be a signature of permanent selection pressure, such as observable in the genealogies of certain pathogens. From a mathematical point of view it is interesting to observe that the space of Ω-trees maintains several statistical invariants although it is drastically reduced in size compared to the space of unconstrained Yule trees. Using generating functions, we study here some basic combinatorial properties of Ω-trees. We focus on the distribution of the number of subtrees with two leaves. We show that expectation and variance of this distribution match those for unconstrained trees already for very small values of ω.

Root location in random trees: a polarity property of all sampling consistent phylogenetic models except one

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we show that among models that treat all taxa equally, and are sampling consistent (i.e. the distribution on trees is not affected by taxa yet to be included), all such models, except one (the so-called PDA model), convey some information as to the location of the ancestral root in an unrooted tree.

Multiple merger gene genealogies in two species: Monophyly, paraphyly, and polyphyly for two examples of Lambda coalescents

Probabilities of monophyly, paraphyly, and polyphyly of two-species gene genealogies are computed for modest sample sizes and compared for two different Λ coalescent processes. Coalescent processes belonging to the Λ coalescent family admit asynchronous multiple mergers of active ancestral lineages. Assigning a timescale to the time of divergence becomes a central issue when different populations have different coalescent processes running on different timescales. Clade probabilities in single populations are also computed, which can be useful for testing for taxonomic distinctiveness of an observed set of monophyletic lineages. The coalescence rates of multiple merger coalescent processes are functions of coalescent parameters. The effect of coalescent parameters on the probabilities studied depends on the coalescent process, and if the population is ancestral or derived. The probability of reciprocal monophyly tends to be somewhat lower, when associated with a Λ coalescent, under the null hypothesis that two groups come from the same population. However, even for fairly recent divergence times, the probability of monophyly tends to be higher as a function of the number of generations for coalescent processes that admit multiple mergers, and is sensitive to the parameter of one of the example processes.