Identification of phylogenetic trees of minimal length.

The problem of determining an optimal phylogenetic tree from a set of data is an example of the Steiner problem in graphs. There is no efficient algorithm for solving this problem with reasonably large data sets. In the present paper an approach is described that proves in some cases that a given tree is optimal without testing all possible trees. The method first uses a previously described heuristic algorithm to find a tree of relatively small total length. The second part of the method independently analyses subsets of sites to determine a lower bound on the length of any tree. We simultaneously attempt to reduce the total length of the tree and increase the lower bound. When these are equal it is not possible to make a shorter tree with a given data set and given criterion. An example is given where the only two possible minimal trees are found for twelve different mammalian cytochrome c sequences. The criterion of finding the smallest number of minimum base changes was used. However, there is no general method of guaranteeing that a solution will be found in all cases and in particular better methods of improving the estimate of the lower bound need to be developed.     

A general approach to proving the minimality of phylogenetic trees illustrated by an example with a set of 23 vertebrates

Summary We have recently described a method of building phylogenetic trees and have outlined an approach for proving whether a particular tree is optimal for the data used. In this paper we describe in detail the method of establishing lower bounds on the length of a minimal tree by partitioning the data set into subsets. All characters that could be involved in duplications in the data are paired with all other such characters. A matching algorithm is then used to obtain the pairing of characters that reveals the most duplications in the data. This matching may still not account for all nucleotide substitutions on the tree. The structure of the tree is then used to help select subsets of three or more. characters until the lower bound found by partitioning is equal to the length of the tree. The tree must then be a minimal tree since no tree can exist with a length less than that of the lower bound.The method is demonstrated using a set of 23 vertebrate cytochrome c sequences with the criterion of minimizing the total number of nucleotide substitutions. There are 131130 7045768798 9603440625 topologically distinct trees that can be constructed from this data set. The method described in this paper does identify 144 minimal tree variants. The method is general in the sense that it can be used for other data and other criteria of length. It need not however always be possible to prove a tree minimal but the method will give an upper and lower bound on the length of minimal trees.     

A neural network for the Steiner minimal tree problem

The problem of finding the shortest tree connecting a set of points is called the Steiner minimal tree problem and is nearly three centuries old. It has applications in transportation, computer networks, agriculture, telephony, building layout and very large scale integrated circuit (VLSI) design, among others, and is known to be NP-complete. We propose a neural network which self-organizes to find a minimal tree. Solutions found by the network compare favourably with the best known or optimal results on test problems from the literature. To the best of our knowledge, the proposed network is the first neural-based solution to the problem. We show that the neural network has a built-in mechanism to escape local minima.     

Probability Steiner trees and maximum parsimony in phylogenetic analysis

The phylogenetic tree (PT) problem has been studied by a number of researchers as an application of the Steiner tree problem, a well-known network optimisation problem. Of all the methods developed for phylogenies the maximum parsimony (MP) method is a simple and commonly used method because it relies on directly observable changes in the input nucleotide or amino acid sequences. In this paper we show that the non-uniqueness of the evolutionary pathways in the MP method leads us to consider a new model of PTs. In this so-called probability representation model, for each site a node in a PT is modelled by a probability distribution of nucleotide or amino acid states, and hence the PT at a given site is a probability Steiner tree, i.e. a Steiner tree in a high-dimensional vector space. In spite of the generality of the probability representation model, in this paper we restrict our study to constructing probability phylogenetic trees (PPT) using the parsimony criterion, as well as discussing and comparing our approach with the classical MP method. We show that for a given input set although the optimal topology as well as the total tree length of the PPT is the same as the PT constructed by the classical MP method, the inferred ancestral states and branch lengths are different and the results given by our method provide a plausible alternative to the classical ones.     

Steiner minimal trees, twist angles, and the protein folding problem

The Steiner Minimal Tree (SMT) problem determines the minimal length network for connecting a given set of vertices in three-dimensional space. SMTs have been shown to be useful in the geometric modeling and characterization of proteins. Even though the SMT problem is an NP-Hard Optimization problem, one can define planes within the amino acids that have a surprising regularity property for the twist angles of the planes. This angular property is quantified for all amino acids through the Steiner tree topology structure. The twist angle properties and other associated geometric properties unique for the remaining amino acids are documented in this paper. We also examine the relationship between the Steiner ratio rho and the torsion energy in amino acids with respect to the side chain torsion angle chi(1). The rho value is shown to be inversely proportional to the torsion energy. Hence, it should be a useful approximation to the potential energy function. Finally, the Steiner ratio is used to evaluate folded and misfolded protein structures. We examine all the native proteins and their decoys at http://dd.stanford.edu. and compare their Steiner ratio values. Because these decoy structures have been delicately misfolded, they look even more favorable than the native proteins from the potential energy viewpoint. However, the rho value of a decoy folded protein is shown to be much closer to the average value of an empirical Steiner ratio for each residue involved than that of the corresponding native one, so that we recognize the native folded structure more easily. The inverse relationship between the Steiner ratio and the energy level in the protein is shown to be a significant measure to distinguish native and decoy structures. These properties should be ultimately useful in the ab initio protein folding prediction.     

Phylogenetic networks from multi-labelled trees

It is now quite well accepted that the evolutionary past of certain species is better represented by phylogenetic networks as opposed to trees. For example, polyploids are typically thought to have resulted through hybridization and duplication, processes that are probably not best represented as bifurcating speciation events. Based on the knowledge of a multi-labelled tree relating collection of polyploids, we present a canonical construction of a phylogenetic network that exhibits the tree. In addition, we prove that the resulting network is in some well-defined sense a minimal network having this property.     

Sequence length bounds for resolving a deep phylogenetic divergence

In evolutionary biology, genetic sequences carry with them a trace of the underlying tree that describes their evolution from a common ancestral sequence. The question of how many sequence sites are required to recover this evolutionary relationship accurately depends on the model of sequence evolution, the substitution rate, divergence times and the method used to infer phylogenetic history. A particularly challenging problem for phylogenetic methods arises when a rapid divergence event occurred in the distant past. We analyse an idealised form of this problem in which the terminal edges of a symmetric four-taxon tree are some factor (λ) times the length of the interior edge. We determine an order λ² lower bound on the growth rate for the sequence length required to resolve the tree (independent of any particular branch length). We also show that this rate of sequence length growth can be achieved by existing methods (including the simple ‘maximum parsimony’ method), and compare these order λ² bounds with an order λ growth rate for a model that describes low-homoplasy evolution. In the final section, we provide a generic bound on the sequence length requirement for a more general class of Markov processes.     

Accuracy of estimated phylogenetic trees from molecular data

Summary The accuracies and efficiencies of four different methods for constructing phylogenetic trees from molecular data were examined by using computer simulation. The methods examined are UPGMA, Fitch and Margoliash's (1967) (F/M) method, Farris' (1972) method, and the modified Farris method (Tateno, Nei, and Tajima, this paper). In the computer simulation, eight OTUs (32 OTUs in one case) were assumed to evolve according to a given model tree, and the evolutionary change of a sequence of 300 nucleotides was followed. The nucleotide substitution in this sequence was assumed to occur following the Poisson distribution, negative binomial distribution or a model of temporally varying rate. Estimates of nucleotide substitutions (genetic distances) were then computed for all pairs of the nucleotide sequences that were generated at the end of the evolution considered, and from these estimates a phylogenetic tree was reconstructed and compared with the true model tree. The results of this comparison indicate that when the coefficient of variation of branch length is large the Farris and modified Farris methods tend to be better than UPGMA and the F/M method for obtaining a good topology. For estimating the number of nucleotide substitutions for each branch of the tree, however, the modified Farris method shows a better performance than the Farris method. When the coefficient of variation of branch length is small, however, UPGMA shows the best performance among the four methods examined. Nevertheless, any tree-making method is likely to make errors in obtaining the correct topology with a high probability, unless all branch lengths of the true tree are sufficiently long. It is also shown that the agreement between patristic and observed genetic distances is not a good indicator of the goodness of the tree obtained.     

Techniques for the verification of minimal phylogenetic trees illustrated with ten mammalian haemoglobin sequences.

We have recently reported a method to identify the shortest possible phylogenetic tree for a set of protein sequences [Foulds Hendy & Penny (1979) J. Mol. Evol. 13. 127--150; Foulds, Penny & Hendy (1979) J. Mol. Evol. 13, 151--166]. The present paper discusses issues that arise during the construction of minimal phylogenetic trees from protein-sequence data. The conversion of the data from amino acid sequences into nucleotide sequences is shown to be advantageous. A new variation of a method for constructing a minimal tree is presented. Our previous methods have involved first constructing a tree and then either proving that it is minimal or transforming it into a minimal tree. The approach presented in the present paper progressively builds up a tree, taxon by taxon. We illustrate this approach by using it to construct a minimal tree for ten mammalian haemoglobin alpha-chain sequences. Finally we define a measure of the complexity of the data and illustrate a method to derive a directed phylogenetic tree from the minimal tree.     

The MinMax Squeeze: guaranteeing a minimal tree for population data

We report that for population data, where sequences are very similar to one another, it is often possible to use a two-pronged (MinMax Squeeze) approach to prove that a tree is the shortest possible under the parsimony criterion. Such population data can be in a range where parsimony is a maximum likelihood estimator. This is in sharp contrast to the case with species data, where sequences are much further apart and the problem of guaranteeing an optimal phylogenetic tree is known to be computationally prohibitive for realistic numbers of species, irrespective of whether likelihood or parsimony is the optimality criterion. The Squeeze uses both an upper bound (the length of the shortest tree known) and a lower bound derived from partitions of the columns (the length of the shortest tree possible). If the two bounds meet, the shortest known tree is thus proven to be a shortest possible tree. The implementation is first tested on simulated data sets and then applied to 53 complete human mitochondrial genomes. The shortest possible trees for those data have several significant improvements from the published tree. Namely, a pair of Australian lineages comes deeper in the tree (in agreement with archaeological data), and the non-African part of the tree shows greater agreement with the geographical distribution of lineages.     

Robustness of maximum likelihood tree estimation against different patterns of base substitutions

Summary In the maximum likelihood (ML) method for estimating a molecular phylogenetic tree, the pattern of nucleotide substitutions for computing likelihood values is assumed to be simpler than that of the actual evolutionary process, simply because the process, considered to be quite devious, is unknown. The problem, however, is that there has been no guarantee to endorse the simplification.To study this problem, we first evaluated the robustness of the ML method in the estimation of molecular trees against different nucleotide substitution patterns, including Jukes and Cantor's, the simplest ever proposed. Namely, we conducted computer simulations in which we could set up various evolutionary models of a hypothetical gene, and define a true tree to which an estimated tree by the ML method was to be compared. The results show that topology estimation by the ML method is considerably robust against different ratios of transitions to transversions and different GC contents, but branch length estimation is not so. The ML tree estimation based on Jukes and Cantor's model is also revealed to be resistant to GC content, but rather sensitive to the ratio of transitions to transversions.We then applied the ML method with different substitution patterns to nucleotide sequence data ontax gene from T-cell leukemia viruses whose evolutionary process must have been more complicated than that of the hypothetical gene. The results are in accordance with those from the simulation study, showing that Jukes and Cantor's model is as useful as a more complicated one for making inferences about molecular phylogeny of the viruses.     

NULL MODELS FOR THE NUMBER OF EVOLUTIONARY STEPS IN A CHARACTER ON A PHYLOGENETIC TREE

Random trees and random characters can be used in null models for testing phylogenetic hypothesis. We consider three interpretations of random trees: first, that trees are selected from the set of all possible trees with equal probability; second, that trees are formed by random speciation or coalescence (equivalent); and third, that trees are formed by a series of random partitions of the taxa. We consider two interpretations of random characters: first, that the number of taxa with each state is held constant, but the states are randomly reshuffled among the taxa; and second, that the probability each taxon is assigned a particular state is constant from one taxon to the next. Under null models representing various combinations of randomizations of trees and characters, exact recursion equations are given to calculate the probability distribution of the number of character state changes required by a phylogenetic tree. Possible applications of these probability distributions are discussed. They can be used, for example, to test for a panmictic population structure within a species or to test phylogenetic inertia in a character's evolution. Whether and how a null model incorporates tree randomness makes little difference to the probability distribution in many but not all circumstances. The null model's sense of character randomness appears more critical. The difficult issue of choosing a null model is discussed.     

拓扑树间的通经拓扑距离

给出了一种新的系统树间的拓扑距离,使用NJ,MP,UPGMA等3种方法对13种动物的线粒体中14个基因（含组合的）DNA序列数据进行系统树的构建,利用分割拓扑距离和本文给出的通经拓扑距离对这14种系统树这间及其与真树进行比较。结果显示,NJ法对获得已知树的有效率最高,MP法次之,UPGMA法最低。这14种DNA序列所构建的系统树与已知树的拓扑距离基本上是随其DNA序列长度增加而减小,但两者的相关系数并未达到显著水平,分割拓扑距离在总体上可反映树间的拓扑结构差异,但其测度精确度比通经拓扑距离要低。     

Implementation of a Markov model for phylogenetic trees

A recently developed mathematical model for the analysis of phylogenetic trees is applied to comparative data for 48 species. The model represents a return to fundamentals and makes no hypothesis with respect to the reversibility of the process. The species have been analysed in all subsets of three, and a measure of reliability of the results is provided. The numerical results of the computations on 17,296 triples of species are made available on the Internet. These results are discussed and the development of reliable tree structures for several species is illustrated. It is shown that, indeed, the Markov model is capable of considerably more interesting predictions than has been recognized to date.     

重建系统演化树的一种新方法--试错法

重建系统演化树是进化研究的一个极为重要的方面。系统树的构建依赖于一定的方法和数据。在分子系统演化研究中,所使用的数据大多是ＤＮＡ序列、氨基酸序列和分子标记。而就构树方法来说,ＮＪ法、ＭＬ法和ＭＰ法是三种最为普遍使用的方法。本文给出了一种新的建树方法,即试错法。该方法不但具有与ＮＪ法一样好的建树效果,而且不存在难以解释的负枝长问题。     

Minimal model of a cell connecting amoebic motion and adaptive transport networks

A cell is a minimal self-sustaining system that can move and compute. Previous work has shown that a unicellular slime mold, Physarum, can be utilized as a biological computer based on cytoplasmic flow encapsulated by a membrane. Although the interplay between the modification of the boundary of a cell and the cytoplasmic flow surrounded by the boundary plays a key role in Physarum computing, no model of a cell has been developed to describe this interplay. Here we propose a toy model of a cell that shows amoebic motion and can solve a maze, Steiner minimum tree problem and a spanning tree problem. Only by assuming that cytoplasm is hardened after passing external matter (or softened part) through a cell, the shape of the cell and the cytoplasmic flow can be changed. Without cytoplasm hardening, a cell is easily destroyed. This suggests that cytoplasmic hardening and/or sol-gel transformation caused by external perturbation can keep a cell in a critical state leading to a wide variety of shapes and motion.     

Estimating errors and confidence intervals for branch lengths in phylogenetic trees by a bootstrap approach

A method, based on the bootstrap procedure, is proposed for the estimation of branch-length errors and confidence intervals in a phylogenetic tree for which equal rates of substitution among lineages do not necessarily hold. The method can be used to test whether an estimated internodal distance is significantly greater than zero. In the application of the method, any estimator of genetic distances, as well as any tree reconstruction procedure (based on distance matrices), can be used. Also the method is not limited by the number of species involved in the phylogenetic tree. An example of the application of the method in the reconstruction of the phylogenetic tree for the four hominoid species—human, chimpanzee, gorilla, and orangutan—is shown. Correspondence to: J. Dopazo     

Freeing phylogenies from artifacts of alignment.

Widely used methods for phylogenetic inference, both those that require and those that produce alignments, share certain weaknesses. These weaknesses are discussed, and a method that lacks them is introduced. For each pair of sequences in the data set, the method utilizes both insertion-deletion and amino acid replacement information to estimate a pairwise evolutionary distance. It is also possible to allow regional heterogeneity of replacement rates. Because a likelihood framework is adopted, the standard deviation of each pairwise distance can be estimated. The distance matrix and standard error estimates are used to infer a phylogenetic tree. As an example, this method is used on 10 widely diverged sequences of the second largest RNA polymerase subunit. A pseudo-bootstrap technique is devised to assess the validity of the inferred phylogenetic tree.     

Approximate Maximum Parsimony and Ancestral Maximum Likelihood

We explore the maximum parsimony (MP) and ancestral maximum likelihood (AML) criteria in phylogenetic tree reconstruction. Both problems are NP-hard, so we seek approximate solutions. We formulate the two problems as Steiner tree problems under appropriate distances. The gist of our approach is the succinct characterization of Steiner trees for a small number of leaves for the two distances. This enables the use of known Steiner tree approximation algorithms. The approach leads to a 16/9 approximation ratio for AML and asymptotically to a 1.55 approximation ratio for MP.     

Constructing and counting phylogenetic invariants.

The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of m taxa using nucleotide sequence data. Models for the respective probabilities of the 4m possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution occurs along the branches of a putative phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. If the invariant is typically non-zero for another phylogenetic tree, then estimates of the invariant can be used as evidence to support one phylogeny over another. Previous work of Evans and Speed showed that, for certain commonly used substitution models, the problem of finding a minimal generating set for the ideal of invariants can be reduced to the linear algebra problem of finding a basis for a certain lattice (that is, a free Z-module). They also conjectured that the cardinality of such a generating set can be computed using a simple "degrees of freedom" formula. We verify this conjecture. Along the way, we explain in detail how the observations of Evans and Speed lead to a simple, computationally feasible algorithm for constructing a minimal generating set.