Bayesian restoration of a hidden Markov chain with applications to DNA sequencing.

Hidden Markov models (HMMs) are a class of stochastic models that have proven to be powerful tools for the analysis of molecular sequence data. A hidden Markov model can be viewed as a black box that generates sequences of observations. The unobservable internal state of the box is stochastic and is determined by a finite state Markov chain. The observable output is stochastic with distribution determined by the state of the hidden Markov chain. We present a Bayesian solution to the problem of restoring the sequence of states visited by the hidden Markov chain from a given sequence of observed outputs. Our approach is based on a Monte Carlo Markov chain algorithm that allows us to draw samples from the full posterior distribution of the hidden Markov chain paths. The problem of estimating the probability of individual paths and the associated Monte Carlo error of these estimates is addressed. The method is illustrated by considering a problem of DNA sequence multiple alignment. The special structure for the hidden Markov model used in the sequence alignment problem is considered in detail. In conclusion, we discuss certain interesting aspects of biological sequence alignments that become accessible through the Bayesian approach to HMM restoration.     

Detecting recombination with MCMC

MOTIVATION: We present a statistical method for detecting recombination, whose objective is to accurately locate the recombinant breakpoints in DNA sequence alignments of small numbers of taxa (4 or 5). Our approach explicitly models the sequence of phylogenetic tree topologies along a multiple sequence alignment. Inference under this model is done in a Bayesian way, using Markov chain Monte Carlo (MCMC). The algorithm returns the site-dependent posterior probability of each tree topology, which is used for detecting recombinant regions and locating their breakpoints. RESULTS: The method was tested on a synthetic and three real DNA sequence alignments, where it was found to outperform the established detection methods PLATO, RECPARS, and TOPAL.     

A Bayesian approach to DNA sequence segmentation

Many deoxyribonucleic acid (DNA) sequences display compositional heterogeneity in the form of segments of similar structure. This article describes a Bayesian method that identifies such segments by using a Markov chain governed by a hidden Markov model. Markov chain Monte Carlo (MCMC) techniques are employed to compute all posterior quantities of interest and, in particular, allow inferences to be made regarding the number of segment types and the order of Markov dependence in the DNA sequence. The method is applied to the segmentation of the bacteriophage lambda genome, a common benchmark sequence used for the comparison of statistical segmentation algorithms.     

On the representability of complete genomes by multiple competing finite-context (Markov) models

A finite-context (Markov) model of order k yields the probability distribution of the next symbol in a sequence of symbols, given the recent past up to depth k. Markov modeling has long been applied to DNA sequences, for example to find gene-coding regions. With the first studies came the discovery that DNA sequences are non-stationary: distinct regions require distinct model orders. Since then, Markov and hidden Markov models have been extensively used to describe the gene structure of prokaryotes and eukaryotes. However, to our knowledge, a comprehensive study about the potential of Markov models to describe complete genomes is still lacking. We address this gap in this paper. Our approach relies on (i) multiple competing Markov models of different orders (ii) careful programming techniques that allow orders as large as sixteen (iii) adequate inverted repeat handling (iv) probability estimates suited to the wide range of context depths used. To measure how well a model fits the data at a particular position in the sequence we use the negative logarithm of the probability estimate at that position. The measure yields information profiles of the sequence, which are of independent interest. The average over the entire sequence, which amounts to the average number of bits per base needed to describe the sequence, is used as a global performance measure. Our main conclusion is that, from the probabilistic or information theoretic point of view and according to this performance measure, multiple competing Markov models explain entire genomes almost as well or even better than state-of-the-art DNA compression methods, such as XM, which rely on very different statistical models. This is surprising, because Markov models are local (short-range), contrasting with the statistical models underlying other methods, where the extensive data repetitions in DNA sequences is explored, and therefore have a non-local character.     

A compression-based approach for coding sequences identification. I. Application to prokaryotic genomes.

Most of the gene prediction algorithms for prokaryotes are based on Hidden Markov Models or similar machine-learning approaches, which imply the optimization of a high number of parameters. The present paper presents a novel method for the classification of coding and non-coding regions in prokaryotic genomes, based on a suitably defined compression index of a DNA sequence. The main features of this new method are the non-parametric logic and the costruction of a dictionary of words extracted from the sequences. These dictionaries can be very useful to perform further analyses on the genomic sequences themselves. The proposed approach has been applied on some prokaryotic complete genomes, obtaining optimal scores of correctly recognized coding and non-coding regions. Several false-positive and false-negative cases have been investigated in detail, which have revealed that this approach can fail in the presence of highly structured coding regions (e.g., genes coding for modular proteins) or quasi-random non-coding regions (e.g., regions hosting non-functional fragments of copies of functional genes; regions hosting promoters or other protein-binding sequences). We perform an overall comparison with other gene-finder software, since at this step we are not interested in building another gene-finder system, but only in exploring the possibility of the suggested approach.     

A segmental maximum a posteriori approach to genome-wide copy number profiling

MOTIVATION: Copy number profiling methods aim at assigning DNA copy numbers to chromosomal regions using measurements from microarray-based comparative genomic hybridizations. Among the proposed methods to this end, Hidden Markov Model (HMM)-based approaches seem promising since DNA copy number transitions are naturally captured in the model. Current discrete-index HMM-based approaches do not, however, take into account heterogeneous information regarding the genomic overlap between clones. Moreover, the majority of existing methods are restricted to chromosome-wise analysis. RESULTS: We introduce a novel Segmental Maximum A Posteriori approach, SMAP, for DNA copy number profiling. Our method is based on discrete-index Hidden Markov Modeling and incorporates genomic distance and overlap between clones. We exploit a priori information through user-controllable parameterization that enables the identification of copy number deviations of various lengths and amplitudes. The model parameters may be inferred at a genome-wide scale to avoid overfitting of model parameters often resulting from chromosome-wise model inference. We report superior performances of SMAP on synthetic data when compared with two recent methods. When applied on our new experimental data, SMAP readily recognizes already known genetic aberrations including both large-scale regions with aberrant DNA copy number and changes affecting only single features on the array. We highlight the differences between the prediction of SMAP and the compared methods and show that SMAP accurately determines copy number changes and benefits from overlap consideration.     

A Hidden Markov Model approach to variation among sites in rate of evolution

The method of Hidden Markov Models is used to allow for unequal and unknown evolutionary rates at different sites in molecular sequences. Rates of evolution at different sites are assumed to be drawn from a set of possible rates, with a finite number of possibilities. The overall likelihood of phylogeny is calculated as a sum of terms, each term being the probability of the data given a particular assignment of rates to sites, times the prior probability of that particular combination of rates. The probabilities of different rate combinations are specified by a stationary Markov chain that assigns rate categories to sites. While there will be a very large number of possible ways of assigning rates to sites, a simple recursive algorithm allows the contributions to the likelihood from all possible combinations of rates to be summed, in a time proportional to the number of different rates at a single site. Thus with three rates, the effort involved is no greater than three times that for a single rate. This "Hidden Markov Model" method allows for rates to differ between sites and for correlations between the rates of neighboring sites. By summing over all possibilities it does not require us to know the rates at individual sites. However, it does not allow for correlation of rates at nonadjacent sites, nor does it allow for a continuous distribution of rates over sites. It is shown how to use the Newton-Raphson method to estimate branch lengths of a phylogeny and to infer from a phylogeny what assignment of rates to sites has the largest posterior probability. An example is given using beta-hemoglobin DNA sequences in eight mammal species; the regions of high and low evolutionary rates are inferred and also the average length of patches of similar rates.      

Probabilistic divergence measures for detecting interspecies recombination

This paper proposes a graphical method for detecting interspecies recombination in multiple alignments of DNA sequences. A fixed-size window is moved along a given DNA sequence alignment. For every position, the marginal posterior probability over tree topologies is determined by means of a Markov chain Monte Carlo simulation. Two probabilistic divergence measures are plotted along the alignment, and are used to identify recombinant regions. The method is compared with established detection methods on a set of synthetic benchmark sequences and two real-world DNA sequence alignments.     

Locating regions of differential variability in DNA and protein sequences.

In the comparison of DNA and protein sequences between species or between paralogues or among individuals within a species or population, there is often some indication that different regions of the sequence are divergent or polymorphic to different degrees, indicating differential constraint or diversifying selection operating in different regions of the sequence. The problem is to test statistically whether the observed regional differences in the density of variant sites represent real differences and then to estimate as accurately as possible the location of the differential regions. A method is given for testing and locating regions of differential variation. The method consists of calculating G(x(k)) = k/n - x(k)/N, where x(k) is the position of the kth variant site along the sequence, n is the total number of variant sites, and N is the total sequence length. The estimated region is the longest stretch of adjacent sequence for which G(x(k)) is monotonically increasing (a hot spot) or decreasing (a cold spot). Critical values of this length for tests of significance are given, a sequential method is developed for locating multiple differential regions, and the power of the method against various alternatives is explored. The method locates the endpoints of hot spots and cold spots of variation with high accuracy.     

Markov chains: computing limit existence and approximations with DNA

We present two algorithms to perform computations over Markov chains. The first one determines whether the sequence of powers of the transition matrix of a Markov chain converges or not to a limit matrix. If it does converge, the second algorithm enables us to estimate this limit. The combination of these algorithms allows the computation of a limit using DNA computing. In this sense, we have encoded the states and the transition probabilities using strands of DNA for generating paths of the Markov chain.     

DNA base-calling from a nanopore using a Viterbi algorithm

Nanopore-based DNA sequencing is the most promising third-generation sequencing method. It has superior read length, speed, and sample requirements compared with state-of-the-art second-generation methods. However, base-calling still presents substantial difficulty because the resolution of the technique is limited compared with the measured signal/noise ratio. Here we demonstrate a method to decode 3-bp-resolution nanopore electrical measurements into a DNA sequence using a Hidden Markov model. This method shows tremendous potential for accuracy (~98%), even with a poor signal/noise ratio.     

The isolation of the beta A-, beta C-, and gamma-globin genes and a presumptive embryonic globin gene from a goat DNA recombinant library

As an approach to understand how the expression of globin genes are regulated during development, clones containing globin DNA sequences were selected from a recombinant library of goat genomic DNA. The type of globin gene present in each of the recombinants was determined by cross-hybridization to the DNA of mouse alpha- and beta-globin cDNA-containing plasmids. Of 11 clones isolated, eight hybridized specifically to the DNA of the mouse beta-globin plasmid, while one clone hybridized only to the DNA of the alpha globin plasmid. The location of each globin sequence within its DNA insert was determined by a combination of restriction enzyme mapping and Southern transfer-hybridizations. Selected fragments were sequenced; comparisons of the amino acids coded for by these regions with those of the goat globins identified clones carrying beta A-, beta C-, and gamma-globin genes. Another recombinant coded for amino acid sequences resembling, but not identical with, the known goat globins, and was identified tentatively as containing an embryonic or epsilon-gene. Detailed analysis of the clone containing the beta C gene and an overlapping clone revealed that three other beta-like sequences are located 6, 12, and 21 kilobases on the 5'-side of the beta C gene. The globin sequence of the locus nearest to the beta C gene has an altered translation termination codon and, if transcribed and translated, would give a globin chain seven amino acids longer than the normal goat beta C-globin. In addition, the sequence following this termination codon is very AT-rich, unlike that of other globin genes. The recombinants described contain extensive regions of DNA surrounding the globin genes, making them useful for identifying regulatory sequences as well as determining the sequence organization of the goat globin genes.