Topological properties of binary trees grown with order-dependent branching probabilities

This paper describes a growth model for binary topological trees. The model defines the branching probability of all segments in the tree. The branching probability of a segment is formulated as a function of two variables, one indicating its type (intermediate or terminal), the other representing its order, i.e. the topological distance to the root segment. The function is determined by two parameters, namely the ratio of branching probabilities of intermediate and terminal segments and the strength of the order dependency, implemented in an exponential form. Expressions are derived for the calculation of symmetry properties of the partitions and it is indicated which part of the parameter domain results in predominantly symmetrical trees.     

Refining the standardized growth change method for pointer year detection: Accounting for statistical bias and estimating the deflection period

Detecting pointer years in tree-ring data is a central aspect of dendroecology. Pointer years are usually represented by extraordinary secondary tree growth, which is often interpreted as a response to abnormal environmental conditions such as late-frosts or droughts. Objectively identifying pointer years in larger tree-ring networks and relating those to specific climatic conditions will allow for refining our understanding of how trees perform under extreme climate and consequently, under anticipated climate change. Recently, Buras et al. (2020) demonstrated that frequently used pointer-year detection methods were either too sensitive or insensitive for such large scale analyses. In their study, Buras et al. (2020) proposed a novel approach for detecting pointer years – the standardized growth change (SGC) method which outperformed other pointer-year detection methods in pseudopopulation trials. Yet, the authors concluded that SGC could be improved further to account for the inability to detect pointer years following successive growth decline. Under this framework, we here present a refined version of the SGC-method – the bias-adjusted standardized growth change method (BSGC). The methodological adjustment to the SGC approach comprises conflated probabilities derived from standardized growth changes with probabilities derived from a time-step specific global standardization of growth changes. In addition, BSGC allows for estimating the length of the deflection period, i.e. the period before extraordinary growth values have reached normal levels. Application of BSGC to simulated and measured tree-ring data indicated an improved performance in comparison to SGC which allows for the identification of pointer years following years of successive growth decline. Also, deflection period lengths were estimated well and revealed plausible results for an existing tree-ring data set. Based on these validations, BSGC can be considered a further refinement of pointer-year detection, allowing for a more accurate identification and consequently better understanding of the radial growth response of trees to extreme events.     

Uncovering and Testing the Fuzzy Clusters Based on Lumped Markov Chain in Complex Network

Identifying clusters, namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. By means of a lumped Markov chain model of a random walker, we propose two novel ways of inferring the lumped markov transition matrix. Furthermore, some useful results are proposed based on the analysis of the properties of the lumped Markov process. To find the best partition of complex networks, a novel framework including two algorithms for network partition based on the optimal lumped Markovian dynamics is derived to solve this problem. The algorithms are constructed to minimize the objective function under this framework. It is demonstrated by the simulation experiments that our algorithms can efficiently determine the probabilities with which a node belongs to different clusters during the learning process and naturally supports the fuzzy partition. Moreover, they are successfully applied to real-world network, including the social interactions between members of a karate club.     

Predicting tree death for Fagus sylvatica and Abies alba using permanent plot data

Question: How well can mortality probabilities of deciduous trees(Fagus sylvatica) and conifers (Abies alba) be predicted using permanent plot data that describe growth patterns, tree species, tree size and site conditions? Location: Fagus forests in the montane belt of the Jura folds (Switzerland). Method: Permanent plot data were used to develop and validate logistic regression models predicting survival probabilities of individual trees. Backward model selection led to a reduced model containing the growth‐related variable ‘relative basal area increment’ (growth‐dependent mortality) and variables not directly reflecting growth such as species, size and site (growth‐independent mortality). Results: The growth‐mortality relationship was the same for both species (growth‐dependent mortality). However, species, site and tree size also influenced mortality probabilities (growth‐independent mortality). The predicted survival probabilities of the final model were well calibrated, and the model showed an excellent discriminatory power (area under the receiver operating characteristic curve = 0.896). Conclusion: Mortality probabilities of Fagus sylvatica and Abies alba can be predicted with high discriminatory power using a well calibrated logistic regression model. Extending this case study to a larger number of tree species and sites could provide species‐ and site‐specific tree mortality models that allow for more realistic projections of forest succession.     

The prior probabilities of phylogenetic trees

Bayesian methods have become among the most popular methods in phylogenetics, but theoretical opposition to this methodology remains. After providing an introduction to Bayesian theory in this context, I attempt to tackle the problem mentioned most often in the literature: the “problem of the priors”—how to assign prior probabilities to tree hypotheses. I first argue that a recent objection—that an appropriate assignment of priors is impossible—is based on a misunderstanding of what ignorance and bias are. I then consider different methods of assigning prior probabilities to trees. I argue that priors need to be derived from an understanding of how distinct taxa have evolved and that the appropriate evolutionary model is captured by the Yule birth–death process. This process leads to a well-known statistical distribution over trees. Though further modifications may be necessary to model more complex aspects of the branching process, they must be modifications to parameters in an underlying Yule model. Ignoring these Yule priors commits a fallacy leading to mistaken inferences both about the trees themselves and about macroevolutionary processes more generally.     

Summarizing a posterior distribution of trees using agreement subtrees

Bayesian inference of phylogeny is unique among phylogenetic reconstruction methods in that it produces a posterior distribution of trees rather than a point estimate of the best tree. The most common way to summarize this distribution is to report the majority-rule consensus tree annotated with the marginal posterior probabilities of each partition. Reporting a single tree discards information contained in the full underlying distribution and reduces the Bayesian analysis to simply another method for finding a point estimate of the tree. Even when a point estimate of the phylogeny is desired, the majority-rule consensus tree is only one possible method, and there may be others that are more appropriate for the given data set and application. We present a method for summarizing the distribution of trees that is based on identifying agreement subtrees that are frequently present in the posterior distribution. This method provides fully resolved binary trees for subsets of taxa with high marginal posterior probability on the entire tree and includes additional information about the spread of the distribution.     

Branch-length prior influences Bayesian posterior probability of phylogeny

The Bayesian method for estimating species phylogenies from molecular sequence data provides an attractive alternative to maximum likelihood with nonparametric bootstrap due to the easy interpretation of posterior probabilities for trees and to availability of efficient computational algorithms. However, for many data sets it produces extremely high posterior probabilities, sometimes for apparently incorrect clades. Here we use both computer simulation and empirical data analysis to examine the effect of the prior model for internal branch lengths. We found that posterior probabilities for trees and clades are sensitive to the prior for internal branch lengths, and priors assuming long internal branches cause high posterior probabilities for trees. In particular, uniform priors with high upper bounds bias Bayesian clade probabilities in favor of extreme values. We discuss possible remedies to the problem, including empirical and full Bayesian methods and subjective procedures suggested in Bayesian hypothesis testing. Our results also suggest that the bootstrap proportion and Bayesian posterior probability are different measures of accuracy, and that the bootstrap proportion, if interpreted as the probability that the clade is true, can be either too liberal or too conservative.     

Is there a star tree paradox?

Concerns have been raised that posterior probabilities on phylogenetic trees can be unreliable when the true tree is unresolved or has very short internal branches, because existing methods for Bayesian phylogenetic analysis do not explicitly evaluate unresolved trees. Two recent papers have proposed that evaluating only resolved trees results in a "star tree paradox": when the true tree is unresolved or close to it, posterior probabilities were predicted to become increasingly unpredictable as sequence length grows, resulting in inflated confidence in one resolved tree or another and an increasing risk of false-positive inferences. Here we show that this is not the case; existing Bayesian methods do not lead to an inflation of statistical confidence, provided the evolutionary model is correct and uninformative priors are assumed. Posterior probabilities do not become increasingly unpredictable with increasing sequence length, and they exhibit conservative type I error rates, leading to a low rate of false-positive inferences. With infinite data, posterior probabilities give equal support for all resolved trees, and the rate of false inferences falls to zero. We conclude that there is no star tree paradox caused by not sampling unresolved trees.     

Topological analysis of binary tree structures when occasional multifurcations occur

The occurrence of multifurcations in essentially binary trees is investigated with respect to two methods for testing growth models,viz. subtree partition analysis and vertex analysis. It is shown that under certain conditions multifurcations may be incorporated in the analysis. Although the conditions are more restrictive for subtree partition analysis only minor loss of information occurs if forbidden multifurcations are simply ignored.     

Bayesian phylogenetic inference using DNA sequences: a Markov Chain Monte Carlo Method

An improved Bayesian method is presented for estimating phylogenetic trees using DNA sequence data. The birth-death process with species sampling is used to specify the prior distribution of phylogenies and ancestral speciation times, and the posterior probabilities of phylogenies are used to estimate the maximum posterior probability (MAP) tree. Monte Carlo integration is used to integrate over the ancestral speciation times for particular trees. A Markov Chain Monte Carlo method is used to generate the set of trees with the highest posterior probabilities. Methods are described for an empirical Bayesian analysis, in which estimates of the speciation and extinction rates are used in calculating the posterior probabilities, and a hierarchical Bayesian analysis, in which these parameters are removed from the model by an additional integration. The Markov Chain Monte Carlo method avoids the requirement of our earlier method for calculating MAP trees to sum over all possible topologies (which limited the number of taxa in an analysis to about five). The methods are applied to analyze DNA sequences for nine species of primates, and the MAP tree, which is identical to a maximum-likelihood estimate of topology, has a probability of approximately 95%.      

Parameter estimation in topological analysis of binary tree structures

Different types of random binary topological trees (like neuronal processes and rivers) occur with relative frequencies that can be explained in terms of growth models. It will be shown how the model parameter determining the mode of growth can be estimated with the maximum likelihood procedure from observed data. Monte Carlo simulations were used to study the distributional properties of this estimator which appeared to have a negligible bias. It is shown that the minimum chi-square procedure yields an estimate that is very close to the maximum likelihood estimate. Moreover, the goodness-of-fit of the growth model can be inferred directly from the chi-square statistic. To illustrate the procedures we examined axonal trees from the goldfish tectum. A notion of complete partition randomness is presented as an alternative to our growth hypotheses.     

The role of forest tent caterpillar defoliations and partial harvest in the decline and death of sugar maple

Background and Aims

Natural and anthropogenic disturbances can act as stresses on tree vigour. According to Manion''s conceptual model of tree disease, the initial vigour of trees decreases as a result of predisposing factors that render these trees more vulnerable to severe inciting stresses, stresses that can then cause final vigour decline and subsequent tree death. This tree disease model was tested in sugar maple (Acer saccharum) by assessing the roles of natural and anthropogenic disturbances in tree decline and death.

Methods

Radial growth data from 377 sugar maple trees that had undergone both defoliations by insects and partial harvest were used to estimate longitudinal survival probabilities as a proxy for tree vigour. Radial growth rates and survival probabilities were compared among trees subjected to different levels of above- and below-ground disturbances, between periods of defoliation and harvest, and between live and dead trees.

Key Results

Manion''s tree disease model correctly accounts for vigour decline and tree death in sugar maple; tree growth and vigour were negatively affected by a first defoliation, predisposing these trees to death later during the study period due to a second insect outbreak that initiated a final vigour decline. This decline was accelerated by the partial harvest disturbance in 1993. Even the most severe anthropogenic disturbances from partial harvest did not cause, unlike insect defoliation, any growth or vigour declines in live sugar maple.

Conclusions

Natural disturbances acted as predisposing and inciting stresses in tree sugar maple decline and death. Anthropogenic disturbances from a partial harvest at worst accelerated a decline in trees that were already weakened by predisposing and inciting stresses (i.e. repeated insect defoliations). Favourable climatic conditions just before and after the partial harvest may have alleviated possible negative effects on growth resulting from harvesting.Key words: Manion, tree disease model, disturbance, Acer saccharum, tree mortality, tree vigour     

Fair-balance paradox, star-tree paradox, and Bayesian phylogenetics

The star-tree paradox refers to the conjecture that the posterior probabilities for the three unrooted trees for four species (or the three rooted trees for three species if the molecular clock is assumed) do not approach 1/3 when the data are generated using the star tree and when the amount of data approaches infinity. It reflects the more general phenomenon of high and presumably spurious posterior probabilities for trees or clades produced by the Bayesian method of phylogenetic reconstruction, and it is perceived to be a manifestation of the deeper problem of the extreme sensitivity of Bayesian model selection to the prior on parameters. Analysis of the star-tree paradox has been hampered by the intractability of the integrals involved. In this article, I use Laplacian expansion to approximate the posterior probabilities for the three rooted trees for three species using binary characters evolving at a constant rate. The approximation enables calculation of posterior tree probabilities for arbitrarily large data sets. Both theoretical analysis of the analogous fair-coin and fair-balance problems and computer simulation for the tree problem confirmed the existence of the star-tree paradox. When the data size n --> infinity, the posterior tree probabilities do not converge to 1/3 each, but they vary among data sets according to a statistical distribution. This distribution is characterized. Two strategies for resolving the star-tree paradox are explored: (1) a nonzero prior probability for the degenerate star tree and (2) an increasingly informative prior forcing the internal branch length toward zero. Both appear to be effective in resolving the paradox, but the latter is simpler to implement. The posterior tree probabilities are found to be very sensitive to the prior.     

Nuclear and mitochondrial genes contain similar phylogenetic signal for pigeons and doves (Aves: Columbiformes)

Molecular systematic studies generally assume that gene trees are reasonable estimates of species trees. We tested the validity of this assumption in the pigeons and doves (Aves: Columbiformes) by comparing phylogenies derived from nuclear (beta-fibrinogen intron 7) and mitochondrial (cytochrome b) genes. Trees derived from the two genes when analyzed separately contained many nodes in common. A partition homogeneity test revealed no significant incongruence between trees derived from the two genes; so, we combined nuclear and mitochondrial data in subsequent phylogenetic analyses. The resulting tree, which was highly resolved and generally well supported, contained a strong biogeographic component. The rate of nucleotide substitution for the nuclear intron was approximately six times slower than that of cytochrome b. This resulted in a much higher consistency index for trees derived from the intron because of the low level of multiple substitution. However, the degree of resolution and support for trees reconstructed from the two genes was similar. We also examined the transition and transversion substitution rates for the genes. Third position transversions for cytochrome b accumulated linearly with intron divergence, suggesting low levels of multiple substitution for third position transversions.     

Split Probabilities and Species Tree Inference Under the Multispecies Coalescent Model

Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we study the probabilities of splits on gene trees under the multispecies coalescent model, and how their features might inform species tree inference. After investigating the behavior of split consensus methods, we investigate split invariants—that is, polynomial relationships between split probabilities. These invariants are then used to show that, even though a split is an unrooted notion, split probabilities retain enough information to identify the rooted species tree topology for trees of 5 or more taxa, with one possible 6-taxon exception.     

Phylogeny of frogs of the Physalaemus pustulosus species group,with an examination of data incongruence

Characters derived from advertisement calls, morphology, allozymes, and the sequences of the small subunit of the mitochondrial ribosomal gene (12S) and the cytochrome oxidase I (COI) mitochondrial gene were used to estimate the phylogeny of frogs of the Physalaemus pustulosus group (Leptodactylidae). The combinability of these data partitions was assessed in several ways: measures of phylogenetic signal, character support for trees, congruence of tree topologies, compatibility of data partitions with suboptimal trees, and homogeneity of data partitions. Combined parsimony analysis of all data equally weighted yielded the same tree as the 12S partition analyzed under parsimony and maximum likelihood. The COI, allozyme, and morphology partitions were generally congruent and compatible with the tree derived from combined data. The call data were significantly different from all other partitions, whether considered in terms of tree topology alone, partition homogeneity, or compatibility of data with trees derived from other partitions. The lack of effect of the call data on the topology of the combined tree is probably due to the small number of call characters. The general incongruence of the call data with other data partitions is consistent with the idea that the advertisement calls of this group of frogs are under strong sexual selection.     

Tree-based maximal likelihood substitution matrices and hidden Markov models

There has been considerable interest in the problem of making maximum likelihood (ML) evolutionary trees which allow insertions and deletions. This problem is partly one of formulation: how does one define a probabilistic model for such trees which treats insertion and deletion in a biologically plausible manner? A possible answer to this question is proposed here by extending the concept of a hidden Markov model (HMM) to evolutionary trees. The model, called a tree-HMM, allows what may be loosely regarded as learnable affine-type gap penalties for alignments. These penalties are expressed in HMMs as probabilities of transitions between states. In the tree-HMM, this idea is given an evolutionary embodiment by defining trees of transitions. Just as the probability of a tree composed of ungapped sequences is computed, by Felsenstein's method, using matrices representing the probabilities of substitutions of residues along the edges of the tree, so the probabilities in a tree-HMM are computed by substitution matrices for both residues and transitions. How to define these matrices by a ML procedure using an algorithm that learns from a database of protein sequences is shown here. Given these matrices, one can define a tree-HMM likelihood for a set of sequences, assuming a particular tree topology and an alignment of the sequences to the model. If one could efficiently find the alignment which maximizes (or comes close to maximizing) this likelihood, then one could search for the optimal tree topology for the sequences. An alignment algorithm is defined here which, given a particular tree topology, is guaranteed to increase the likelihood of the model. Unfortunately, it fails to find global optima for realistic sequence sets. Thus further research is needed to turn the tree-HMM into a practical phylogenetic tool.     

A biological and computational model of megakaryocyte development as a stochastic branching process

The purpose of this paper is to describe a model of megakaryocytopoiesis as a branching process with stochastic processes regulating critical control points of differentiation along the stem cell megakaryocyte platelet axis. Progress of cells through these critical control points are regulated by transitional probabilities, which in turn are regulated by influences such as growth factors. The critical control points include transition of resting megakaryocytic stem cells (CFU-meg) into proliferating stem cells, the cessation of cytokinesis, and the cessation of DNA synthesis. A computerized computational method has been developed for directly fitting the stochastic branching model to colony growth data. The computational model has allowed transitional probabilities to be derived from colony size data. The model provides a unifying explanation for much of the heterogeneity of stages of maturation within populations of megakaryocytes and is fully compatible with historical data supporting the stochastic nature of hematopoietic stem cell regulation and with modern molecular concepts about control of the cell cycle.     

Cell lineage tree models of neurogenesis

The production of neurons to form the mammalian cortex, known as embryonic cortical neurogenesis, is a complex developmental process. Insight into the process of cell division during neurogenesis is provided by murine cortical cell lineage trees, recorded through experimental observation. Recurring patterns within cell lineage trees may be indicative of predetermined cell behaviour. The application of mathematical modelling to this process requires careful consideration and identification of the key features to be incorporated into the model. A biologically plausible stochastic model of evolution of cell lineage trees is developed, based on the most important known features of neurogenesis. Tractable means of measuring lineage tree shape are discussed. Symmetry is identified as a significant feature of shape and is measured using Colless's Index of Imbalance. Distributions of tree size and imbalance for large tree sizes are computed and results compared to experimental data. Several refinements to the model are investigated, when the cell division probabilities are weighted according to cell generation. Two models involving generation-dependent cell division probabilities produce imbalance distributions which are the most consistent with the available experimental results. The results indicate that a stochastic cell division mechanism is a plausible basis of mammalian neurogenesis.     

The equilibrium partition function and base pair binding probabilities for RNA secondary structure.

A novel application of dynamic programming to the folding problem for RNA enables one to calculate the full equilibrium partition function for secondary structure and the probabilities of various substructures. In particular, both the partition function and the probabilities of all base pairs are computed by a recursive scheme of polynomial order N3 in the sequence length N. The temperature dependence of the partition function gives information about melting behavior for the secondary structure. The pair binding probabilities, the computation of which depends on the partition function, are visually summarized in a "box matrix" display and this provides a useful tool for examining the full ensemble of probable alternative equilibrium structures. The calculation of this ensemble representation allows a proper application and assessment of the predictive power of the secondary structure method, and yields important information on alternatives and intermediates in addition to local information about base pair opening and slippage. The results are illustrated for representative tRNA, 5S RNA, and self-replicating and self-splicing RNA molecules, and allow a direct comparison with enzymatic structure probes. The effect of changes in the thermodynamic parameters on the equilibrium ensemble provides a further sensitivity check to the predictions.