Let D=(d1,d2,…,dn) and F=(f1,f2,…,fn) be two sequences of positive integers. We consider the following decision problems: is there a (i) multigraph, (ii) loopless multigraph, (iii) simple graph, (iv) cycle-free graph (forest or tree), (v) caterpillar G=(V,E) such that for all k, d(vk)=dk and ∑w∈N(vk)d(w)=fk (d(v) is the degree of v and N(v) is the set of neighbors of v). Here we show that all these decision problems can be solved in polynomial time if Δ≔maxkdk is bounded. The problems are converted into an integer programming feasibility problem in which both the number of variables and the number of inequalities depend only on Δ but not on n. The problem is motivated by NMR spectroscopy of hydrocarbons. The algorithm has been implemented in the ZIMPL language, and its applicability is demonstrated on trees up to n=1000 vertices. The average reconstruction time for trees with 1000 vertices is still less than 40 ms.

Edge disjoint realization problems have connections for example to discrete tomography. In this paper, we consider the edge disjoint caterpillar realizations of tree degree sequences. We give necessary and sufficient conditions when two tree degree sequences have edge disjoint caterpillar realizations. We conjecture that an arbitrary number of tree degree sequences have edge disjoint realizations if every vertex is a leaf in at most one tree. We prove that the conjecture is true if the number of tree degree sequences is at most four. We also prove that the conjecture is true if n≥max{22k−11,396}, where n is the number of vertices and k is the number of tree degree sequences.

BACKGROUND: A standard procedure in many areas of bioinformatics is to use a single multiple sequence alignment (MSA) as the basis for various types of analysis. However, downstream results may be highly sensitive to the alignment used, and neglecting the uncertainty in the alignment can lead to significant bias in the resulting inference. In recent years, a number of approaches have been developed for probabilistic sampling of alignments, rather than simply generating a single optimum. However, this type of probabilistic information is currently not widely used in the context of downstream inference, since most existing algorithms are set up to make use of a single alignment. RESULTS: In this work we present a framework for representing a set of sampled alignments as a directed acyclic graph (DAG) whose nodes are alignment columns; each path through this DAG then represents a valid alignment. Since the probabilities of individual columns can be estimated from empirical frequencies, this approach enables sample-based estimation of posterior alignment probabilities. Moreover, due to conditional independencies between columns, the graph structure encodes a much larger set of alignments than the original set of sampled MSAs, such that the effective sample size is greatly increased. CONCLUSIONS: The alignment DAG provides a natural way to represent a distribution in the space of MSAs, and allows for existing algorithms to be efficiently scaled up to operate on large sets of alignments. As an example, we show how this can be used to compute marginal probabilities for tree topologies, averaging over a very large number of MSAs. This framework can also be used to generate a statistically meaningful summary alignment; example applications show that this summary alignment is consistently more accurate than the majority of the alignment samples, leading to improvements in downstream tree inference. Implementations of the methods described in this article are available at http://statalign.github.io/WeaveAlign .

BACKGROUND: Even for moderate size inputs, there are a tremendous number of optimal rearrangement scenarios, regardless what the model is and which specific question is to be answered. Therefore giving one optimal solution might be misleading and cannot be used for statistical inferring. Statistically well funded methods are necessary to sample uniformly from the solution space and then a small number of samples are sufficient for statistical inferring. CONTRIBUTION: In this paper, we give a mini-review about the state-of-the-art of sampling and counting rearrangement scenarios, focusing on the reversal, DCJ and SCJ models. Above that, we also give a Gibbs sampler for sampling most parsimonious labeling of evolutionary trees under the SCJ model. The method has been implemented and tested on real life data. The software package together with example data can be downloaded from http://www.renyi.hu/~miklosi/SCJ-Gibbs/.

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes $n$. To prove fast mixing, we first prove a general factorization theorem similar to the Martin--Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

HD amino acid duplex has been found in the active center of many different enzymes. The dyad plays remarkably different roles in their catalytic processes that usually involve metal coordination. An HD motif is positioned directly on the amyloid beta fragment (Aβ) and on the carboxy-terminal region of the extracellular domain (CAED) of the human amyloid precursor protein (APP) and a taxonomically well defined group of APP orthologues (APPOs). In human Aβ HD is part of a presumed, RGD-like integrin-binding motif RHD; however, neither RHD nor RXD demonstrates reasonable conservation in APPOs. The sequences of CAEDs and the position of the HD are not particularly conserved either, yet we show with a novel statistical method using evolutionary modeling that the presence of HD on CAEDs cannot be the result of neutral evolutionary forces (p<0.0001). The motif is positively selected along the evolutionary process in the majority of APPOs, despite the fact that HD motif is underrepresented in the proteomes of all species of the animal kingdom. Position migration can be explained by high probability occurrence of multiple copies of HD on intermediate sequences, from which only one is kept by selective evolutionary forces, in a similar way as in the case of the "transcription binding site turnover." CAED of all APP orthologues and homologues are predicted to bind metal ions including Amyloid-like protein 1 (APLP1) and Amyloid-like protein 2 (APLP2). Our results suggest that HDs on the CAEDs are most probably key components of metal-binding domains, which facilitate and/or regulate inter- or intra-molecular interactions in a metal ion-dependent or metal ion concentration-dependent manner. The involvement of naturally occurring mutations of HD (Tottori (D7N) and English (H6R) mutations) in early onset Alzheimer's disease gives additional support to our finding that HD has an evolutionary preserved function on APPOs.

MOTIVATION: When comparing the organization of two genomes, it is important not to draw conclusions on their modes of evolution from a single most parsimonious scenario explaining their differences. Better estimations can be obtained by sampling many different genomic rearrangement scenarios. For this problem, the Double Cut and Join (DCJ) model, while less relevant, is computationally easier than the Hannenhalli-Pevzner (HP) model. Indeed, in some special cases, the total number of DCJ sorting scenarios can be analytically calculated, and uniformly distributed random DCJ scenarios can be drawn in polynomial running time, while the complexity of counting the number of HP scenarios and sampling from the uniform distribution of their space is unknown, and conjectured to be #P-complete. Statistical methods, like Markov chain Monte Carlo (MCMC) for sampling from the uniform distribution of the most parsimonious or the Bayesian distribution of all possible HP scenarios are required. RESULTS: We use the computational facilities of the DCJ model to draw a sampling of HP scenarios. It is based on a parallel MCMC method that cools down DCJ scenarios to HP scenarios. We introduce two theorems underlying the theoretical mixing properties of this parallel MCMC method. The method was tested on yeast and mammalian genomic data, and allowed us to provide estimates of the different modes of evolution in diverse lineages. AVAILABILITY: The program implemented in Java 1.5 programming language is available from http://www.renyi.hu/~miklosi/DCJ2HP/.

BACKGROUND: In this paper, we introduce a progressive corner cutting method called Reticular Alignment for multiple sequence alignment. Unlike previous corner-cutting methods, our approach does not define a compact part of the dynamic programming table. Instead, it defines a set of optimal and suboptimal alignments at each step during the progressive alignment. The set of alignments are represented with a network to store them and use them during the progressive alignment in an efficient way. The program contains a threshold parameter on which the size of the network depends. The larger the threshold parameter and thus the network, the deeper the search in the alignment space for better scored alignments. RESULTS: We implemented the program in the Java programming language, and tested it on the BAliBASE database. Reticular Alignment can outperform ClustalW even if a very simple scoring scheme (BLOSUM62 and affine gap penalty) is implemented and merely the threshold value is increased. However, this set-up is not sufficient for outperforming other cutting-edge alignment methods. On the other hand, the reticular alignment search strategy together with sophisticated scoring schemes (for example, differentiating gap penalties for hydrophobic and hydrophylic amino acids) overcome FSA and in some accuracy measurement, even MAFFT. The program is available from http://phylogeny-cafe.elte.hu/RetAlign/ CONCLUSIONS: Reticular alignment is an efficient search strategy for finding accurate multiple alignments. The highest accuracy achieved when this searching strategy is combined with sophisticated scoring schemes.

Markov chain Monte Carlo has been the standard technique for inferring the posterior distribution of genome rearrangement scenarios under a Bayesian approach. We present here a negative result on the rate of convergence of the generally used Markov chains. We prove that the relaxation time of the Markov chains walking on the optimal reversal sorting scenarios might grow exponentially with the size of the signed permutations, namely, with the number of syntheny blocks.

MOTIVATION: Bayesian analysis is one of the most popular methods in phylogenetic inference. The most commonly used methods fix a single multiple alignment and consider only substitutions as phylogenetically informative mutations, though alignments and phylogenies should be inferred jointly as insertions and deletions also carry informative signals. Methods addressing these issues have been developed only recently and there has not been so far a user-friendly program with a graphical interface that implements these methods. RESULTS: We have developed an extendable software package in the Java programming language that samples from the joint posterior distribution of phylogenies, alignments and evolutionary parameters by applying the Markov chain Monte Carlo method. The package also offers tools for efficient on-the-fly summarization of the results. It has a graphical interface to configure, start and supervise the analysis, to track the status of the Markov chain and to save the results. The background model for insertions and deletions can be combined with any substitution model. It is easy to add new substitution models to the software package as plugins. The samples from the Markov chain can be summarized in several ways, and new postprocessing plugins may also be installed.

BACKGROUND: Comparative methods have been the standard techniques for in silico protein structure prediction. The prediction is based on a multiple alignment that contains both reference sequences with known structures and the sequence whose unknown structure is predicted. Intensive research has been made to improve the quality of multiple alignments, since misaligned parts of the multiple alignment yield misleading predictions. However, sometimes all methods fail to predict the correct alignment, because the evolutionary signal is too weak to find the homologous parts due to the large number of mutations that separate the sequences. RESULTS: Stochastic sequence alignment methods define a posterior distribution of possible multiple alignments. They can highlight the most likely alignment, and above that, they can give posterior probabilities for each alignment column. We made a comprehensive study on the HOMSTRAD database of structural alignments, predicting secondary structures in four different ways. We showed that alignment posterior probabilities correlate with the reliability of secondary structure predictions, though the strength of the correlation is different for different protocols. The correspondence between the reliability of secondary structure predictions and alignment posterior probabilities is the closest to the identity function when the secondary structure posterior probabilities are calculated from the posterior distribution of multiple alignments. The largest deviation from the identity function has been obtained in the case of predicting secondary structures from a single optimal pairwise alignment. We also showed that alignment posterior probabilities correlate with the 3D distances between C alpha amino acids in superimposed tertiary structures. CONCLUSION: Alignment posterior probabilities can be used to a priori detect errors in comparative models on the sequence alignment level.

Computational methods for predicting evolutionarily conserved rather than thermodynamic RNA structures have recently attracted increased interest. These methods are indispensable not only for elucidating the regulatory roles of known RNA transcripts, but also for predicting RNA genes. It has been notoriously difficult to devise them to make the best use of the available data and to predict high-quality RNA structures that may also contain pseudoknots. We introduce a novel theoretical framework for co-estimating an RNA secondary structure including pseudoknots, a multiple sequence alignment, and an evolutionary tree, given several RNA input sequences. We also present an implementation of the framework in a new computer program, called SimulFold, which employs a Bayesian Markov chain Monte Carlo method to sample from the joint posterior distribution of RNA structures, alignments, and trees. We use the new framework to predict RNA structures, and comprehensively evaluate the quality of our predictions by comparing our results to those of several other programs. We also present preliminary data that show SimulFold's potential as an alignment and phylogeny prediction method. SimulFold overcomes many conceptual limitations that current RNA structure prediction methods face, introduces several new theoretical techniques, and generates high-quality predictions of conserved RNA structures that may include pseudoknots. It is thus likely to have a strong impact, both on the field of RNA structure prediction and on a wide range of data analyses.

SUMMARY: ParIS Genome Rearrangement is a web server for a Bayesian analysis of unichromosomal genome pairs. The underlying model allows inversions, transpositions and inverted transpositions. The server generates a Markov chain using a Partial Importance Sampler technique, and samples trajectories of mutations from this chain. The user can specify several marginalizations to the posterior: the posterior distribution of number of mutations needed to transform one genome into another, length distribution of mutations, number of mutations that have occurred at a given site. Both text and graphical outputs are available. We provide a limited server, a downloadable unlimited server that can be installed locally on any linux/Unix operating system, and a database of mitochondrial gene orders.

BACKGROUND: Two central problems in computational biology are the determination of the alignment and phylogeny of a set of biological sequences. The traditional approach to this problem is to first build a multiple alignment of these sequences, followed by a phylogenetic reconstruction step based on this multiple alignment. However, alignment and phylogenetic inference are fundamentally interdependent, and ignoring this fact leads to biased and overconfident estimations. Whether the main interest be in sequence alignment or phylogeny, a major goal of computational biology is the co-estimation of both. RESULTS: We developed a fully Bayesian Markov chain Monte Carlo method for coestimating phylogeny and sequence alignment, under the Thorne-Kishino-Felsenstein model of substitution and single nucleotide insertion-deletion (indel) events. In our earlier work, we introduced a novel and efficient algorithm, termed the "indel peeling algorithm", which includes indels as phylogenetically informative evolutionary events, and resembles Felsenstein's peeling algorithm for substitutions on a phylogenetic tree. For a fixed alignment, our extension analytically integrates out both substitution and indel events within a proper statistical model, without the need for data augmentation at internal tree nodes, allowing for efficient sampling of tree topologies and edge lengths. To additionally sample multiple alignments, we here introduce an efficient partial Metropolized independence sampler for alignments, and combine these two algorithms into a fully Bayesian co-estimation procedure for the alignment and phylogeny problem. Our approach results in estimates for the posterior distribution of evolutionary rate parameters, for the maximum a-posteriori (MAP) phylogenetic tree, and for the posterior decoding alignment. Estimates for the evolutionary tree and multiple alignment are augmented with confidence estimates for each node height and alignment column. Our results indicate that the patterns in reliability broadly correspond to structural features of the proteins, and thus provides biologically meaningful information which is not existent in the usual point-estimate of the alignment. Our methods can handle input data of moderate size (10-20 protein sequences, each 100-200 bp), which we analyzed overnight on a standard 2 GHz personal computer. CONCLUSION: Joint analysis of multiple sequence alignment, evolutionary trees and additional evolutionary parameters can be now done within a single coherent statistical framework.

BACKGROUND: Baum-Welch training is an expectation-maximisation algorithm for training the emission and transition probabilities of hidden Markov models in a fully automated way. It can be employed as long as a training set of annotated sequences is known, and provides a rigorous way to derive parameter values which are guaranteed to be at least locally optimal. For complex hidden Markov models such as pair hidden Markov models and very long training sequences, even the most efficient algorithms for Baum-Welch training are currently too memory-consuming. This has so far effectively prevented the automatic parameter training of hidden Markov models that are currently used for biological sequence analyses. RESULTS: We introduce the first linear space algorithm for Baum-Welch training. For a hidden Markov model with M states, T free transition and E free emission parameters, and an input sequence of length L, our new algorithm requires O(M) memory and O(LMTmax (T + E)) time for one Baum-Welch iteration, where Tmax is the maximum number of states that any state is connected to. The most memory efficient algorithm until now was the checkpointing algorithm with O(log(L)M) memory and O(log(L)LMTmax) time requirement. Our novel algorithm thus renders the memory requirement completely independent of the length of the training sequences. More generally, for an n-hidden Markov model and n input sequences of length L, the memory requirement of O(log(L)L(n-1) M) is reduced to O(L(n-1) M) memory while the running time is changed from O(log(L)Ln MTmax + Ln(T + E)) to O(Ln MTmax (T + E)). An added advantage of our new algorithm is that a reduced time requirement can be traded for an increased memory requirement and vice versa, such that for any c e {1, ..., (T + E)}, a time requirement of Ln MTmax c incurs a memory requirement of L(n-1) M(T + E - c). CONCLUSION: For the large class of hidden Markov models used for example in gene prediction, whose number of states does not scale with the length of the input sequence, our novel algorithm can thus be both faster and more memory-efficient than any of the existing algorithms.