Sackin Index Research Articles

Detecting branching rate heterogeneity in multifurcating trees with applications in lineage tracing data.

Understanding cellular birth rate differences is crucial for predicting cancer progression and interpreting tumor-derived genetic data. Lineage tracing experiments enable detailed reconstruction of cellular genealogies, offering new opportunities to measure branching rate heterogeneity. However, the lineage tracing process can introduce complex tree features that complicate this effort. Here, we examine tree characteristics in lineage tracing-derived genealogies and find that editing window placement leads to multifurcations at a tree's root or tips. We propose several ways in which existing tree topology-based metrics can be extended to test for rate heterogeneity on trees even in the presence of lineage-tracing associated distortions. Although these methods vary in power and robustness, a test based on the statistic effectively detects branching rate heterogeneity in simulated lineage tracing data. Tests based on other common statistics ( and the Sackin index) show interior performance to . We apply our validated methods to xenograft experimental data and find widespread rate heterogeneity across multiple study systems. Our results demonstrate the potential of tree topology statistics in analyzing lineage tracing data, and highlight the challenges associated with adapting phylogenetic methods to these systems.

The Sackin index and depth of leaves in generalized Schröder trees

. Schröder trees are biological models of evolution, with internal nodes having two or three children. We generalize the model to grow from an arbitrary stochastic process of independent nonnegative integers (not necessarily identically distributed). We call such a process the building sequence. We study the depth of leaves and the Sackin index for some specific building sequences, such as constant additions, Bernoulli, and Poisson-like models. We include an example that shows that the methods can be extended to exchangeable sequences.

Two results about the Sackin and Colless indices for phylogenetic trees and their shapes.

The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of tree shapes (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short direct proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted trees with leaves being labelled with taxa) under the uniform model.

Limit laws for two distance-based indices in random recursive tree models

Abstract In this paper, we derive several results related to total path length and Sackin index in two classes of random recursive trees. A limiting distribution of the normalized version of the Sackin index is given by the contraction method in random recursive trees. Also, we show the normalized total path length converges in L2 and almost surely to a limiting random variable in plane-oriented recursive trees via martingales.

Robust, Universal Tree Balance Indices.

Balance indices that quantify the symmetry of branching events and the compactness of trees are widely used to compare evolutionary processes or tree-generating algorithms. Yet, existing indices are not defined for all rooted trees, are unreliable for comparing trees with different numbers of leaves, and are sensitive to the presence or absence of rare types. The contributions of this article are twofold. First, we define a new class of robust, universal tree balance indices. These indices take a form similar to Colless' index but can account for population sizes, are defined for trees with any degree distribution, and enable meaningful comparison of trees with different numbers of leaves. Second, we show that for bifurcating and all other full m-ary cladograms (in which every internal node has the same out-degree), one such Colless-like index is equivalent to the normalized reciprocal of Sackin's index. Hence, we both unify and generalize the two most popular existing tree balance indices. Our indices are intrinsically normalized and can be computed in linear time. We conclude that these more widely applicable indices have the potential to supersede those in current use. [Cancer; clone tree; Colless index; Sackin index; species tree; tree balance.].

Combinatorial perspectives on Dollo-k characters in phylogenetics

Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-k characters, i.e. traits that are gained at most once and lost precisely k times throughout evolution. We first introduce an algorithm based on the notion of spanning subtrees for finding a Dollo-k labelling for a given character and a given tree in linear time. We then compare persistent characters (consisting of the union of Dollo-0 and Dollo-1 characters) and general Dollo-k characters. While it is known that there is a strong connection between Fitch parsimony and persistent characters, we show that Dollo parsimony and Fitch parsimony are in general very different. Moreover, while it is known that there is a direct relationship between the number of persistent characters and the Sackin index of a tree, a popular index of tree balance, we show that this relationship does not generalize to Dollo-k characters. In fact, determining the number of Dollo-k characters for a given tree is much more involved than counting persistent characters, and we end this manuscript by introducing a recursive approach for the former. This approach leads to a polynomial time algorithm for counting the number of Dollo-k characters, and both this algorithm as well as the algorithm for computing Dollo-k labellings are publicly available in the Babel package for BEAST 2.

Extremal Values of the Sackin Tree Balance Index

Tree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Also, concerning ordered search trees, more balanced ones allow for more efficient data structuring than imbalanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have only been provided for some of them, and only in the context of ordered binary (search) trees, not for general rooted trees. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves has been completely unknown. In this manuscript, we extend the findings on trees with minimal and maximal Sackin indices from the literature on ordered trees and subsequently use our results to provide formulas to explicitly calculate the numbers of such trees. We also extend previous studies by analyzing the case when the underlying trees need not be binary. Finally, we use our results to contribute both to the phylogenetic as well as the computer scientific literature using the new findings on Sackin minimal and maximal trees to derive formulas to calculate the number of both minimal and maximal phylogenetic trees as well as minimal and maximal ordered trees both in the binary and non-binary settings. All our results have been implemented in the Mathematica package SackinMinimizer, which has been made publicly available.

Inferring transmission heterogeneity using virus genealogies: Estimation and targeted prevention.

Spread of HIV typically involves uneven transmission patterns where some individuals spread to a large number of individuals while others to only a few or none. Such transmission heterogeneity can impact how fast and how much an epidemic spreads. Further, more efficient interventions may be achieved by taking such transmission heterogeneity into account. To address these issues, we developed two phylogenetic methods based on virus sequence data: 1) to generally detect if significant transmission heterogeneity is present, and 2) to pinpoint where in a phylogeny high-level spread is occurring. We derive inference procedures to estimate model parameters, including the amount of transmission heterogeneity, in a sampled epidemic. We show that it is possible to detect transmission heterogeneity under a wide range of simulated situations, including incomplete sampling, varying levels of heterogeneity, and including within-host genetic diversity. When evaluating real HIV-1 data from different epidemic scenarios, we found a lower level of transmission heterogeneity in slowly spreading situations and a higher level of heterogeneity in data that included a rapid outbreak, while R0 and Sackin's index (overall tree shape statistic) were similar in the two scenarios, suggesting that our new method is able to detect transmission heterogeneity in real data. We then show by simulations that targeted prevention, where we pinpoint high-level spread using a coalescence measurement, is efficient when sequence data are collected in an ongoing surveillance system. Such phylogeny-guided prevention is efficient under both single-step contact tracing as well as iterative contact tracing as compared to random intervention.

Allopatry increases the balance of phylogenetic trees during radiation under neutral speciation

The shape of a phylogenetic tree is defined by the sequence of speciation events, represented by its branching points, and extinctions, represented by branch interruptions. In a neutral scenario of parapatry and isolation by distance, species tend to branch off the original population one after the other, leading to highly unbalanced trees. In this case the degree of imbalance, measured by the normalized Sackin index, grows linearly with species richness. Here we claim that moderate values of imbalance for trees with large number of species can occur if the geographic distribution involves more than one deme (allopatry) and speciation is parapatric within demes. The combined values of balance (normalized Sackin index) and species richness provide an estimate of how many demes were involved in the process if it happened in such neutral scenario. We also show that the spatial division in demes moderately slows down the diversification process, portraying a neutral mechanism for structuring the branch length distribution of phylogenetic trees.

Combinatorial views on persistent characters in phylogenetics

The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well-known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one 0→1 transition followed by at most one 1→0 transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize persistent characters in terms of the first phase of the famous Fitch algorithm. Afterwards we focus on the number of persistent characters for a given phylogenetic tree. We show that this number solely depends on the balance of the tree. To be precise, we develop a formula for counting the number of persistent characters for a given phylogenetic tree based on an index of tree balance, namely the Sackin index. Lastly, we consider the question of how many (carefully chosen) binary characters together with their persistence status are needed to uniquely determine a phylogenetic tree and provide an upper bound for the number of characters needed.

On Sackin\u2019s original proposal: the variance of the leaves\u2019 depths as a phylogenetic balance index

BackgroundThe Sackin indexS of a rooted phylogenetic tree, defined as the sum of its leaves’ depths, is one of the most popular balance indices in phylogenetics, and Sackin’s paper (Syst Zool 21:225–6, 1972) is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves’ depths, but their “variation”. This proposal was later implemented as the variance of the leaves’ depths by Kirkpatrick and Slatkin in (Evolution 47:1171–81, 1993), where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin’s original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal.ResultsIn this paper we study the properties of the variance of the leaves’ depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space mathcal {BT}_{n} of bifurcating rooted phylogenetic trees with n≤183 leaves at the so-called “maximally balanced trees” with n leaves, this property fails for almost every n≥184. We provide then an algorithm that finds the trees in mathcal {BT}_{n} with minimum V value in time O(n log(n)). Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index (Mir et al., Math Biosci 241:125–36, 2013) of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature.ConclusionThe phylogenetics community has been wise in preferring the sum S(T) of the leaves’ depths of a phylogenetic tree T over their variance V(T) as a balance index, because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But it is still a valid and useful shape index.

On the minimum value of the Colless index and the bifurcating trees that achieve it.

Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (Syst Zool 31:100-104, 1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with n leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the tree shapes that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

Phylogenies from dynamic networks.

The relationship between the underlying contact network over which a pathogen spreads and the pathogen phylogenetic trees that are obtained presents an opportunity to use sequence data to learn about contact networks that are difficult to study empirically. However, this relationship is not explicitly known and is usually studied in simulations, often with the simplifying assumption that the contact network is static in time, though human contact networks are dynamic. We simulate pathogen phylogenetic trees on dynamic Erdős-Renyi random networks and on two dynamic networks with skewed degree distribution, of which one is additionally clustered. We use tree shape features to explore how adding dynamics changes the relationships between the overall network structure and phylogenies. Our tree features include the number of small substructures (cherries, pitchforks) in the trees, measures of tree imbalance (Sackin index, Colless index), features derived from network science (diameter, closeness), as well as features using the internal branch lengths from the tip to the root. Using principal component analysis we find that the network dynamics influence the shapes of phylogenies, as does the network type. We also compare dynamic and time-integrated static networks. We find, in particular, that static network models like the widely used Barabasi-Albert model can be poor approximations for dynamic networks. We explore the effects of mis-specifying the network on the performance of classifiers trained identify the transmission rate (using supervised learning methods). We find that both mis-specification of the underlying network and its parameters (mean degree, turnover rate) have a strong adverse effect on the ability to estimate the transmission parameter. We illustrate these results by classifying HIV trees with a classifier that we trained on simulated trees from different networks, infection rates and turnover rates. Our results point to the importance of correctly estimating and modelling contact networks with dynamics when using phylodynamic tools to estimate epidemiological parameters.

Modelling tree shape and structure in viral phylodynamics

Epidemiological models have highlighted the importance of population structure in the transmission dynamics of infectious diseases. Using HIV-1 as an example of a model evolutionary system, we consider how population structure affects the shape and the structure of a viral phylogeny in the absence of strong selection at the population level. For structured populations, the number of lineages as a function of time is insufficient to describe the shape of the phylogeny. We develop deterministic approximations for the dynamics of tips of the phylogeny over evolutionary time, the number of ‘cherries’, tips that share a direct common ancestor, and Sackin's index, a commonly used measure of phylogenetic imbalance or asymmetry. We employ cherries both as a measure of asymmetry of the tree as well as a measure of the association between sequences from different groups. We consider heterogeneity in infectiousness associated with different stages of HIV infection, and in contact rates between groups of individuals. In the absence of selection, we find that population structure may have relatively little impact on the overall asymmetry of a tree, especially when only a small fraction of infected individuals is sampled, but may have marked effects on how sequences from different subpopulations cluster and co-cluster.

A new balance index for phylogenetic trees

Several indices that measure the degree of balance of a rooted phylogenetic tree have been proposed so far in the literature. In this work we define and study a new index of this kind, which we call the total cophenetic index: the sum, over all pairs of different leaves, of the depth of their lowest common ancestor. This index makes sense for arbitrary trees, can be computed in linear time and it has a larger range of values and a greater resolution power than other indices like Colless’ or Sackin’s. We compute its maximum and minimum values for arbitrary and binary trees, as well as exact formulas for its expected value for binary trees under the Yule and the uniform models of evolution. As a byproduct of this study, we obtain an exact formula for the expected value of the Sackin index under the uniform model, a result that seems to be new in the literature.

A MODEL OF MACROEVOLUTION AS A BRANCHING PROCESS BASED ON INNOVATIONS

We introduce a model for the evolution of species triggered by generation of novel features and exhaustive combination with other available traits. Under the assumption that innovations are rare, we obtain a bursty branching process of speciations. Analysis of the trees representing the branching history reveals structures qualitatively different from those of random processes. For a tree with n leaves generated by the introduced model, the average distance of leaves from root scales as ( log n)2to be compared to log n for random branching. The mean values and standard deviations for the tree shape indices depth (Sackin index) and imbalance (Colless index) of the model are compatible with those of real phylogenetic trees from databases. Earlier models, such as the Aldous' branching (AB) model, show a larger deviation from data with respect to the shape indices.

Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees

The stationary birth-only, or Yule–Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or “ancestors”, and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of “birth” of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred. The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations. For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves. In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.

Central Moments and Probability Distributions of Three Measures of Phylogenetic Tree Imbalance

Several recent studies have included attempts to use the balance of phylogenetic trees, i.e., the extent to which sister groups within a tree tend to be the same size, to test hypotheses about the macroevolutionary processes that produced them. Such tests require measures of balance or imbalance and the moments or probability distributions of these measures under some null models. In earlier work, I developed recursion equations for the mean, variance, skewness, and complete probability distribution of Colless's coefficient of imbalance (I) (Rogers, 1994, Evolution 48:2026–2036). In this paper, I report the extension of these techniques to two additional imbalance measures, the number of unbalanced nodes on a tree (J) and Sackin's index (K), under both the equal-rates Markov (ERM) model and the equal probability (EP) model. I also show how to find the correlations and joint probability distributions of all pairs of these three coefficients. I and K are so highly correlated for trees of all sizes that K contains little additional information about tree balance that is not conveyed by I. The correlation of I and J, however, decreases rapidly with increasing tree size, indicating that the testing of macroevolutionary hypotheses may be refined by employing the joint distribution of these two coefficients. The results of two simulation studies of non-ERM speciation processes are used to illustrate how the joint distribution of I and J may be used

Central Moments and Probability Distributions of Three Measures of Phylogenetic Tree Imbalance

Several recent studies have included attempts to use the balance of phylogenetic trees, i.e., the extent to which sister groups within a tree tend to be the same size, to test hypotheses about the macroevolutionary processes that produced them. Such tests require measures of balance or imbalance and the moments or probability distributions of these measures under some null models. In earlier work, I developed recursion equations for the mean, variance, skewness, and complete probability distribution of Colless's coefficient of imbalance (I) (Rogers, 1994, Evolution 48:2026–2036). In this paper, I report the extension of these techniques to two additional imbalance measures, the number of unbalanced nodes on a tree (J) and Sackin's index (K), under both the equal-rates Markov (ERM) model and the equal probability (EP) model. I also show how to find the correlations and joint probability distributions of all pairs of these three coefficients. I and K are so highly correlated for trees of all sizes that K contains little additional information about tree balance that is not conveyed by I. The correlation of I and J, however, decreases rapidly with increasing tree size, indicating that the testing of macroevolutionary hypotheses may be refined by employing the joint distribution of these two coefficients. The results of two simulation studies of non-ERM speciation processes are used to illustrate how the joint distribution of I and J may be used