Growth models (including terminal and segmental branching) for topological binary trees

A growth model for topological trees is formulated as a generalization of the terminal and segmental growth model. For this parameterized growth model, expressions are derived for the partition probabilities (probabilities of subtree pairs of certain degrees). The probabilities of complete trees are easily derived from these partition probabilities.     

Constraining prior probabilities of phylogenetic trees

Although Bayesian methods are widely used in phylogenetic systematics today, the foundations of this methodology are still debated among both biologists and philosophers. The Bayesian approach to phylogenetic inference requires the assignment of prior probabilities to phylogenetic trees. As in other applications of Bayesian epistemology, the question of whether there is an objective way to assign these prior probabilities is a contested issue. This paper discusses the strategy of constraining the prior probabilities of phylogenetic trees by means of the Principal Principle. In particular, I discuss a proposal due to Velasco (Biol Philos 23:455–473, 2008) of assigning prior probabilities to tree topologies based on the Yule process. By invoking the Principal Principle I argue that prior probabilities of tree topologies should rather be assigned a weighted mixture of probability distributions based on Pinelis’ (P Roy Soc Lond B Bio 270:1425–1431, 2003) multi-rate branching process including both the Yule distribution and the uniform distribution. However, I argue that this solves the problem of the priors of phylogenetic trees only in a weak form.     

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.     

Numbering binary trees with labeled terminal vertices

For each rooted binary tree witht labeled terminal vertices (leaves) a natural number can be assigned uniquely. Unrooted trees witht labeled terminal vertices andt-2 unlabeled internal vertices of degree 3 can also be numbered uniquely using the same convention. Rooted trees in which the hights of the internal vertices are rank ordered are also considered. Applications to problems in taxonomy are discussed.     

Frequentist properties of Bayesian posterior probabilities of phylogenetic trees under simple and complex substitution models

What does the posterior probability of a phylogenetic tree mean?This simulation study shows that Bayesian posterior probabilities have the meaning that is typically ascribed to them; the posterior probability of a tree is the probability that the tree is correct, assuming that the model is correct. At the same time, the Bayesian method can be sensitive to model misspecification, and the sensitivity of the Bayesian method appears to be greater than the sensitivity of the nonparametric bootstrap method (using maximum likelihood to estimate trees). Although the estimates of phylogeny obtained by use of the method of maximum likelihood or the Bayesian method are likely to be similar, the assessment of the uncertainty of inferred trees via either bootstrapping (for maximum likelihood estimates) or posterior probabilities (for Bayesian estimates) is not likely to be the same. We suggest that the Bayesian method be implemented with the most complex models of those currently available, as this should reduce the chance that the method will concentrate too much probability on too few trees.     

The exact probabilities of branching patterns under terminal and segmental growth hypotheses

Information about the way of branching of dendritic arborizations may be obtained by comparing the frequency distributions of observed branching patterns with theoretical distributions based on well-defined growth models. Two models usually get much attention in geomorphological and (neuro)biological studies, viz. terminal growth and segmental growth. Formulae to construct the exact probability distributions for both growth models are presented. It is shown that ranking and lumping of the individual branching patterns enable the analysis of very large arborizations with relatively few data. The application of the Kolmogorov goodness-of-fit test for discrete distributions to the analysis is discussed.     

Confidence limits on the branching order of phylogenetic trees.

We describe a confidence test for branching order that can aid protein phylogeny reconstruction as well as the evaluation of the optimal tree. It is proposed that the process resulting in the observed amino acid residue differences, which is the basis for the identification of the order and relative times of divergence events, is appropriately described by a modification of the negative binomial distribution. The relative total numbers of mutations (accepted and nonaccepted), which result in a given number of amino acid differences, may be obtained as the expectation of this distribution. The associated variances enable significant differences in tree branching order to be established. If the total rates of mutation of the genes encoding the compared proteins are equal, the expected total mutations and their associated variances map identically to their relative times of divergence. In addition, significantly different rates of change (due to differences in total mutation rate and/or acceptance rate) may be identified without the requirement of outlying reference group. The method is equally applicable to phylogenies derived from DNA or RNA sequence information.     

The branching angles in computer-generated optimized models of arterial trees

The structure of a complex arterial tree model is generated on the computer using the newly developed method of "constrained constructive optimization." The model tree is grown step by step, at each stage of development fulfilling invariant boundary conditions for pressures and flows. The development of structure is governed by adopting minimum volume inside the vessels as target function. The resulting model tree is analyzed regarding the relations between branching angles and segment radii. Results show good agreement with morphometric measurements on corrosion casts of human coronary arteries reported in the literature.     

Transfer properties of branching dendrites with tonically activated inputs

In the mathematical model of a neuron, properties of the active membrane during action of tonically activated synaptic inputs of an N-methyl-D-aspartate (NMDA) type were studied. In the dendrites with these properties, the impact of geometry on the somatopetal current transfer was investigated. The conditions were considered when the distal sites are more effective, as compared with the proximal ones, in their contribution to the somatopetal current. Contributions of “sister” branches to the net current transferred to the soma were compared at different levels of depolarization caused by action of homogeneously distributed NMDA synaptic inputs.     

An analytical method for investigating transient potentials in neurons with branching dendritic trees.

An analytical method is developed that allows one to explore the way in which the geometrical structure of a neuron's dendritic tree affects the time-course and amplitude of transient potentials generated at different locations on dendritic branches. The method requires that, for a given dendritic arborization, one associates a symmetric geometry for which exact mathematical expressions for time-varying dendritic potentials can be calculated. The value of the dendritic potential for the asymmetric geometry is evaluated by adding correction terms to the results for the symmetric geometry. Several model trees are examined, and in each case the analytical results are expressed in terms of two closely related families of functions. These functions provide a precise formulation for systematically analyzing the way in which the voltage transient at a given point depends upon the geometrical structure of the dentritic tree. Several numerical examples are presented. A discussion of how to generalize the method and of some potential applications are given.     

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.     

Protein classification based on propagation of unrooted binary trees

We present two efficient network propagation algorithms that operate on a binary tree, i.e., a sparse-edged substitute of an entire similarity network. TreeProp-N is based on passing increments between nodes while TreeProp-E employs propagation to the edges of the tree. Both algorithms improve protein classification efficiency.     

Estimating phylogenetic trees from pairwise likelihoods and posterior probabilities of substitution counts

The field of phylogenetic tree estimation has been dominated by three broad classes of methods: distance-based approaches, parsimony and likelihood-based methods (including maximum likelihood (ML) and Bayesian approaches). Here we introduce two new approaches to tree inference: pairwise likelihood estimation and a distance-based method that estimates the number of substitutions along the paths through the tree. Our results include the derivation of the formulae for the probability that two leaves will be identical at a site given a number of substitutions along the path connecting them. We also derive the posterior probability of the number of substitutions along a path between two sequences. The calculations for the posterior probabilities are exact for group-based, symmetric models of character evolution, but are only approximate for more general models.     

Topological properties of some PhSeX compounds

A theoretical study on the series of compounds “PhSeX”, where Ph = phenyl, Se = selenium and X = Cl, Br, I, CN or SCN, is reported and compared with previously reported experimental data. The molecular geometry for these PhSeX compounds was studied at the DFT/B3LYP level of calculation by means of the 6-311G(d,p) basis set. The equilibrium structures of the molecules were dependent on the method employed to compare the known solid structures. A topological study of the calculated PhSeX species, based on the AIM theory, was carried out to gain a deeper insight into the bonding nature and to find an explanation for the structural diversity exhibited by these PhSeX compounds. The results reported herein illustrate the subtle differences in the solid-state structures of PhSeX compounds.     

Mother rotor anchoring in branching tissue with heterogeneous membrane properties.

Abstract Current understanding of atrial fibrillation is based on the co-existence of multiple re-entrant waves propagating randomly throughout the tissue. However, recent experimental results indicate that in many cases one or a small number of periodic, high-frequency re-entrant sources (mother rotors) can drive the arrhythmia. Owing to the high activation rate, mother rotors seem to be located in regions of shortened action potential duration. In this study a computer model of cardiac propagation was applied to investigate mechanisms leading to the formation and maintenance of such mother rotors. For this purpose, a region of short action potential duration was generated by varying the acetylcholine concentration across the tissue. A mother rotor initiated in the center of this region drifts away, and the activation terminates. If an additional heterogeneity such as a bundle is included into the model, a further drift mechanism directed to the bundle is observed and the rotor can be stabilized. Therefore, bundle insertions may play an important role in the maintenance of mother rotors. The influence of the driving rotor on the activation pattern was studied in a three-dimensional model of rectangular shape and a monolayer model of anatomically correct atrial geometry.