Hidden Markov chains and the analysis of genome structure

A powerful approach to identify replicable variants in genome-wide association studies

Replicability is the cornerstone of modern scientific research. Reliable identifications of genotype-phenotype associations that are significant in multiple genome-wide association studies (GWASs) provide stronger evidence for the findings. Current replicability analysis relies on the independence assumption among single-nucleotide polymorphisms (SNPs) and ignores the linkage disequilibrium (LD) structure. We show that such a strategy may produce either overly liberal or overly conservative results in practice. We develop an efficient method, ReAD, to detect replicable SNPs associated with the phenotype from two GWASs accounting for the LD structure. The local dependence structure of SNPs across two heterogeneous studies is captured by a four-state hidden Markov model (HMM) built on two sequences of p values. By incorporating information from adjacent locations via the HMM, our approach provides more accurate SNP significance rankings. ReAD is scalable, platform independent, and more powerful than existing replicability analysis methods with effective false discovery rate control. Through analysis of datasets from two asthma GWASs and two ulcerative colitis GWASs, we show that ReAD can identify replicable genetic loci that existing methods might otherwise miss.

Inference of genomic landscapes using ordered Hidden Markov Models with emission densities (oHMMed)

BACKGROUND

Genomes are inherently inhomogeneous, with features such as base composition, recombination, gene density, and gene expression varying along chromosomes. Evolutionary, biological, and biomedical analyses aim to quantify this variation, account for it during inference procedures, and ultimately determine the causal processes behind it. Since sequential observations along chromosomes are not independent, it is unsurprising that autocorrelation patterns have been observed e.g., in human base composition. In this article, we develop a class of Hidden Markov Models (HMMs) called oHMMed (ordered HMM with emission densities, the corresponding R package of the same name is available on CRAN): They identify the number of comparably homogeneous regions within autocorrelated observed sequences. These are modelled as discrete hidden states; the observed data points are realisations of continuous probability distributions with state-specific means that enable ordering of these distributions. The observed sequence is labelled according to the hidden states, permitting only neighbouring states that are also neighbours within the ordering of their associated distributions. The parameters that characterise these state-specific distributions are inferred.

RESULTS

We apply our oHMMed algorithms to the proportion of G and C bases (modelled as a mixture of normal distributions) and the number of genes (modelled as a mixture of poisson-gamma distributions) in windows along the human, mouse, and fruit fly genomes. This results in a partitioning of the genomes into regions by statistically distinguishable averages of these features, and in a characterisation of their continuous patterns of variation. In regard to the genomic G and C proportion, this latter result distinguishes oHMMed from segmentation algorithms based in isochore or compositional domain theory. We further use oHMMed to conduct a detailed analysis of variation of chromatin accessibility (ATAC-seq) and epigenetic markers H3K27ac and H3K27me3 (modelled as a mixture of poisson-gamma distributions) along the human chromosome 1 and their correlations.

CONCLUSIONS

Our algorithms provide a biologically assumption free approach to characterising genomic landscapes shaped by continuous, autocorrelated patterns of variation. Despite this, the resulting genome segmentation enables extraction of compositionally distinct regions for further downstream analyses.

Whole genome/proteome based phylogeny reconstruction for prokaryotes using higher order Markov model and chaos game representation

Traditional methods for sequence comparison and phylogeny reconstruction rely on pair wise and multiple sequence alignments. But alignment could not be directly applied to whole genome/proteome comparison and phylogenomic studies due to their high computational complexity. Hence alignment-free methods became popular in recent years. Here we propose a fast alignment-free method for whole genome/proteome comparison and phylogeny reconstruction using higher order Markov model and chaos game representation. In the present method, we use the transition matrices of higher order Markov models to characterize amino acid or DNA sequences for their comparison. The order of the Markov model is uniquely identified by maximizing the average Shannon entropy of conditional probability distributions. Using one-dimensional chaos game representation and linked list, this method can reduce large memory and time consumption which is due to the large-scale conditional probability distributions. To illustrate the effectiveness of our method, we employ it for fast phylogeny reconstruction based on genome/proteome sequences of two species data sets used in previous published papers. Our results demonstrate that the present method is useful and efficient.

AVAILABILITY AND IMPLEMENTATION

The source codes for our algorithm to get the distance matrix and genome/proteome sequences can be downloaded from ftp://121.199.20.25/. The software Phylip and EvolView we used to construct phylogenetic trees can be referred from their websites.

Phasic Triplet Markov Chains

Hidden Markov chains have been shown to be inadequate for data modeling under some complex conditions. In this work, we address the problem of statistical modeling of phenomena involving two heterogeneous system states. Such phenomena may arise in biology or communications, among other fields. Namely, we consider that a sequence of meaningful words is to be searched within a whole observation that also contains arbitrary one-by-one symbols. Moreover, a word may be interrupted at some site to be carried on later. Applying plain hidden Markov chains to such data, while ignoring their specificity, yields unsatisfactory results. The Phasic triplet Markov chain, proposed in this paper, overcomes this difficulty by means of an auxiliary underlying process in accordance with the triplet Markov chains theory. Related Bayesian restoration techniques and parameters estimation procedures according to the new model are then described. Finally, to assess the performance of the proposed model against the conventional hidden Markov chain model, experiments are conducted on synthetic and real data.

Investigating genomic structure using changept: A Bayesian segmentation model

Genomes are composed of a wide variety of elements with distinct roles and characteristics. Some of these elements are well-characterised functional components such as protein-coding exons. Other elements play regulatory or structural roles, encode functional non-protein-coding RNAs, or perform some other function yet to be characterised. Still others may have no functional importance, though they may nevertheless be of interest to biologists. One technique for investigating the composition of genomes is to segment sequences into compositionally homogenous blocks. This technique, known as 'sequence segmentation' or 'change-point analysis', is used to identify patterns of variation across genomes such as GC-rich and GC-poor regions, coding and non-coding regions, slowly evolving and rapidly evolving regions and many other types of variation. In this mini-review we outline many of the genome segmentation methods currently available and then focus on a Bayesian DNA segmentation algorithm, with examples of its various applications.

Multiscale DNA partitioning: statistical evidence for segments

MOTIVATION

DNA segmentation, i.e. the partitioning of DNA in compositionally homogeneous segments, is a basic task in bioinformatics. Different algorithms have been proposed for various partitioning criteria such as Guanine/Cytosine (GC) content, local ancestry in population genetics or copy number variation. A critical component of any such method is the choice of an appropriate number of segments. Some methods use model selection criteria and do not provide a suitable error control. Other methods that are based on simulating a statistic under a null model provide suitable error control only if the correct null model is chosen.

RESULTS

Here, we focus on partitioning with respect to GC content and propose a new approach that provides statistical error control: as in statistical hypothesis testing, it guarantees with a user-specified probability [Formula: see text] that the number of identified segments does not exceed the number of actually present segments. The method is based on a statistical multiscale criterion, rendering this as a segmentation method that searches segments of any length (on all scales) simultaneously. It is also accurate in localizing segments: under benchmark scenarios, our approach leads to a segmentation that is more accurate than the approaches discussed in the comparative review of Elhaik et al. In our real data examples, we find segments that often correspond well to features taken from standard University of California at Santa Cruz (UCSC) genome annotation tracks.

AVAILABILITY AND IMPLEMENTATION

Our method is implemented in function smuceR of the R-package stepR available at http://www.stochastik.math.uni-goettingen.de/smuce.

Pattern recognition and probabilistic measures in alignment-free sequence analysis

With the massive production of genomic and proteomic data, the number of available biological sequences in databases has reached a level that is not feasible anymore for exact alignments even when just a fraction of all sequences is used. To overcome this inevitable time complexity, ultrafast alignment-free methods are studied. Within the past two decades, a broad variety of nonalignment methods have been proposed including dissimilarity measures on classical representations of sequences like k-words or Markov models. Furthermore, articles were published that describe distance measures on alternative representations such as compression complexity, spectral time series or chaos game representation. However, alignments are still the standard method for real world applications in biological sequence analysis, and the time efficient alignment-free approaches are usually applied in cases when the accustomed algorithms turn out to fail or be too inconvenient.

Interpreting genomic data via entropic dissection

Since the emergence of high-throughput genome sequencing platforms and more recently the next-generation platforms, the genome databases are growing at an astronomical rate. Tremendous efforts have been invested in recent years in understanding intriguing complexities beneath the vast ocean of genomic data. This is apparent in the spurt of computational methods for interpreting these data in the past few years. Genomic data interpretation is notoriously difficult, partly owing to the inherent heterogeneities appearing at different scales. Methods developed to interpret these data often suffer from their inability to adequately measure the underlying heterogeneities and thus lead to confounding results. Here, we present an information entropy-based approach that unravels the distinctive patterns underlying genomic data efficiently and thus is applicable in addressing a variety of biological problems. We show the robustness and consistency of the proposed methodology in addressing three different biological problems of significance—identification of alien DNAs in bacterial genomes, detection of structural variants in cancer cell lines and alignment-free genome comparison.

Mining genomic patterns in Mycobacterium tuberculosis H37Rv using a web server Tuber-Gene

Mycobacterium tuberculosis (MTB), causative agent of tuberculosis, is one of the most dreaded diseases of the century. It has long been studied by researchers throughout the world using various wet-lab and dry-lab techniques. In this study, we focus on mining useful patterns at genomic level that can be applied for in silico functional characterization of genes from the MTB complex. The model developed on the basis of the patterns found in this study can correctly identify 99.77% of the input genes from the genome of MTB strain H37Rv. The model was tested against four other MTB strains and the homologue M. bovis to further evaluate its generalization capability. The mean prediction accuracy was 85.76%. It was also observed that the GC content remained fairly constant throughout the genome, implicating the absence of any pathogenicity island transferred from other organisms. This study reveals that dinucleotide composition is an efficient functional class discriminator for MTB complex. To facilitate the application of this model, a web server Tuber-Gene has been developed, which can be freely accessed at http://www.bifmanit.org/tb2/.

Hidden Markov models in biology

Markov and Hidden Markov models (HMMs) are introduced using examples from linkage mapping and sequence analysis. In the course, the forward-backward, the Viterbi, the Baum-Welch (EM) algorithm, and a Metropolis sampling scheme are presented.

Comparative testing of DNA segmentation algorithms using benchmark simulations

Numerous segmentation methods for the detection of compositionally homogeneous domains within genomic sequences have been proposed. Unfortunately, these methods yield inconsistent results. Here, we present a benchmark consisting of two sets of simulated genomic sequences for testing the performances of segmentation algorithms. Sequences in the first set are composed of fixed-sized homogeneous domains, distinct in their between-domain guanine and cytosine (GC) content variability. The sequences in the second set are composed of a mosaic of many short domains and a few long ones, distinguished by sharp GC content boundaries between neighboring domains. We use these sets to test the performance of seven segmentation algorithms in the literature. Our results show that recursive segmentation algorithms based on the Jensen-Shannon divergence outperform all other algorithms. However, even these algorithms perform poorly in certain instances because of the arbitrary choice of a segmentation-stopping criterion.

Detecting species-site dependencies in large multiple sequence alignments

Multiple sequence alignments (MSAs) are one of the most important sources of information in sequence analysis. Many methods have been proposed to detect, extract and visualize their most significant properties. To the same extent that site-specific methods like sequence logos successfully visualize site conservations and sequence-based methods like clustering approaches detect relationships between sequences, both types of methods fail at revealing informational elements of MSAs at the level of sequence–site interactions, i.e. finding clusters of sequences and sites responsible for their clustering, which together account for a high fraction of the overall information of the MSA. To fill this gap, we present here a method that combines the Fisher score-based embedding of sequences from a profile hidden Markov model (pHMM) with correspondence analysis. This method is capable of detecting and visualizing group-specific or conflicting signals in an MSA and allows for a detailed explorative investigation of alignments of any size tractable by pHMMs. Applications of our methods are exemplified on an alignment of the Neisseria surface antigen LP2086, where it is used to detect sites of recombinatory horizontal gene transfer and on the vitamin K epoxide reductase family to distinguish between evolutionary and functional signals.

Relations between Shannon entropy and genome order index in segmenting DNA sequences

Shannon entropy H and genome order index S are used in segmenting DNA sequences. Zhang [Phys. Rev. E 72, 041917 (2005)] found that the two schemes are equivalent when a DNA sequence is converted to a binary sequence of S (strong H bond) and W (weak H bond). They left the mathematical proof to mathematicians who are interested in this issue. In this paper, a possible mathematical explanation is given. Moreover, we find that Chargaff parity rule 2 is the necessary condition of the equivalence, and the equivalence disappears when a DNA sequence is regarded as a four-symbol sequence. At last, we propose that S-2(-H) may be related to species evolution.

Markov models of genome segmentation

We introduce Markov models for segmentation of symbolic sequences, extending a segmentation procedure based on the Jensen-Shannon divergence that has been introduced earlier. Higher-order Markov models are more sensitive to the details of local patterns and in application to genome analysis, this makes it possible to segment a sequence at positions that are biologically meaningful. We show the advantage of higher-order Markov-model-based segmentation procedures in detecting compositional inhomogeneity in chimeric DNA sequences constructed from genomes of diverse species, and in application to the E. coli K12 genome, boundaries of genomic islands, cryptic prophages, and horizontally acquired regions are accurately identified.

Isochore structures in the chicken genome

The availability of the complete chicken genome sequence provides an unprecedented opportunity to study the global genome organization at the sequence level. Delineating compositionally homogeneous G + C domains in DNA sequences can provide much insight into the understanding of the organization and biological functions of the chicken genome. A new segmentation algorithm, which is simple and fast, has been proposed to partition a given genome or DNA sequence into compositionally distinct domains. By applying the new segmentation algorithm to the draft chicken genome sequence, the mosaic organization of the chicken genome can be confirmed at the sequence level. It is shown herein that the chicken genome is also characterized by a mosaic structure of isochores, long DNA segments that are fairly homogeneous in the G + C content. Consequently, 25 isochores longer than 2 Mb (megabases) have been identified in the chicken genome. These isochores have a fairly homogeneous G + C content and often correspond to meaningful biological units. With the aid of the technique of cumulative GC profile, we proposed an intuitive picture to display the distribution of segmentation points. The relationships between G + C content and the distributions of genes (CpG islands, and other genomic elements) were analyzed in a perceivable manner. The cumulative GC profile, equipped with the new segmentation algorithm, would be an appropriate starting point for analyzing the isochore structures of higher eukaryotic genomes.

Segmentation algorithm for DNA sequences

A new measure, to quantify the difference between two probability distributions, called the quadratic divergence, has been proposed. Based on the quadratic divergence, a new segmentation algorithm to partition a given genome or DNA sequence into compositionally distinct domains is put forward. The new algorithm has been applied to segment the 24 human chromosome sequences, and the boundaries of isochores for each chromosome were obtained. Compared with the results obtained by using the entropic segmentation algorithm based on the Jensen-Shannon divergence, both algorithms resulted in all identical coordinates of segmentation points. An explanation of the equivalence of the two segmentation algorithms is presented. The new algorithm has a number of advantages. Particularly, it is much simpler and faster than the entropy-based method. Therefore, the new algorithm is more suitable for analyzing long genome sequences, such as human and other newly sequenced eukaryotic genome sequences.

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.

Use of Runs Statistics for Pattern Recognition in Genomic DNA Sequences

In this article, the use of the finite Markov chain imbedding (FMCI) technique to study patterns in DNA under a hidden Markov model (HMM) is introduced. With a vision of studying multiple runs-related statistics simultaneously under an HMM through the FMCI technique, this work establishes an investigation of a bivariate runs statistic under a binary HMM for DNA pattern recognition. An FMCI-based recursive algorithm is derived and implemented for the determination of the exact distribution of this bivariate runs statistic under an independent identically distributed (IID) framework, a Markov chain (MC) framework, and a binary HMM framework. With this algorithm, we have studied the distributions of the bivariate runs statistic under different binary HMM parameter sets; probabilistic profiles of runs are created and shown to be useful for trapping HMM maximum likelihood estimates (MLEs). This MLE-trapping scheme offers good initial estimates to jump-start the expectation-maximization (EM) algorithm in HMM parameter estimation and helps prevent the EM estimates from landing on a local maximum or a saddle point. Applications of the bivariate runs statistic and the probabilistic profiles in conjunction with binary HMMs for pattern recognition in genomic DNA sequences are illustrated via case studies on DNA bendability signals using human DNA data.

Applications of recursive segmentation to the analysis of DNA sequences

Recursive segmentation is a procedure that partitions a DNA sequence into domains with a homogeneous composition of the four nucleotides A, C, G and T. This procedure can also be applied to any sequence converted from a DNA sequence, such as to a binary strong(G + C)/weak(A + T) sequence, to a binary sequence indicating the presence or absence of the dinucleotide CpG, or to a sequence indicating both the base and the codon position information. We apply various conversion schemes in order to address the following five DNA sequence analysis problems: isochore mapping, CpG island detection, locating the origin and terminus of replication in bacterial genomes, finding complex repeats in telomere sequences, and delineating coding and noncoding regions. We find that the recursive segmentation procedure can successfully detect isochore borders, CpG islands, and the origin and terminus of replication, but it needs improvement for detecting complex repeats as well as borders between coding and noncoding regions.

Statistical measures of DNA sequence dissimilarity under Markov chain models of base composition

In molecular biology, the issue of quantifying the similarity between two biological sequences is very important. Past research has shown that word-based search tools are computationally efficient and can find some new functional similarities or dissimilarities invisible to other algorithms like FASTA. Recently, under the independent model of base composition, Wu, Burke, and Davison (1997, Biometrics 53, 1431 1439) characterized a family of word-based dissimilarity measures that defined distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters (i.e., n-words) in the two sequences. Specifically, they introduced the use of Mahalanobis distance and standardized Euclidean distance into the study of DNA sequence dissimilarity. They showed that both distances had better sensitivity and selectivity than the commonly used Euclidean distance. The purpose of this article is to extend Mahalanobis and standardized Euclidean distances to Markov chain models of base composition. In addition, a new dissimilarity measure based on Kullback-Leibler discrepancy between frequencies of all n-words in the two sequences is introduced. Applications to real data demonstrate that Kullback-Leibler discrepancy gives a better performance than Euclidean distance. Moreover, under a Markov chain model of order kQ for base composition, where kQ is the estimated order based on the query sequence, standardized Euclidean distance performs very well. Under such a model, it performs as well as Mahalanobis distance and better than Kullback-Leibler discrepancy and Euclidean distance. Since standardized Euclidean distance is drastically faster to compute than Mahalanobis distance, in a usual workstation/PC computing environment, the use of standardized Euclidean distance under the Markov chain model of order kQ of base composition is generally recommended. However, if the user is very concerned with computational efficiency, then the use of Kullback-Leibler discrepancy, which can be computed as fast as Euclidean distance, is recommended. This can significantly enhance the current technology in comparing large datasets of DNA sequences.

Inverse complementary homologues of short cysteine signatures

Inversions of short genomic sequences may play a central role in the generation of protein complexity. We report here the existence of an heterogeneous group of proteins (the trefoil precursors MUC-1 and MUA-1, six preproendothelins, and five classes of zinc finger knot proteins) having both cysteine signatures (Cs) and their inverse complementary sequences (Cs) in the same polypeptide chain. We have also found cases in which the (Cs) of a given signature is not present in the same protein, but elsewhere. TGEKPYK, a cysteine-free motif of the human transcription factor, Krab, coexists with its inverse complementary sequence in 31 proteins; the inverse complementary alone is present in a great number of proteins. Our findings suggest that short DNA inversions are a widespread feature of the genome.

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.

The study of correlation structures of DNA sequences: a critical review

The study of correlation structure in the primary sequences of DNA is reviewed. The issues reviewed include: symmetries among 16 base-base correlation functions; accurate estimation of correlation measures; the relationship between 1/f and Lorentzian spectra; heterogeneity in DNA sequences; different modeling strategies of the correlation structure of DNA sequences; the difference of correlation structure between coding and non-coding regions (besides the period-3 pattern); and source of broad distribution of domain sizes. Although some of the results remain controversial, a body of work on this topic constitutes a good starting point for future studies.

Finding genes in DNA with a Hidden Markov Model

This study describes a new Hidden Markov Model (HMM) system for segmenting uncharacterized genomic DNA sequences into exons, introns, and intergenic regions. Separate HMM modules were designed and trained for specific regions of DNA: exons, introns, intergenic regions, and splice sites. The models were then tied together to form a biologically feasible topology. The integrated HMM was trained further on a set of eukaryotic DNA sequences and tested by using it to segment a separate set of sequences. The resulting HMM system which is called VEIL (Viterbi Exon-Intron Locator), obtains an overall accuracy on test data of 92% of total bases correctly labelled, with a correlation coefficient of 0.73. Using the more stringent test of exact exon prediction, VEIL correctly located both ends of 53% of the coding exons, and 49% of the exons it predicts are exactly correct. These results compare favorably to the best previous results for gene structure prediction and demonstrate the benefits of using HMMs for this problem.

A statistical model for locating regulatory regions in genomic DNA

In addition to genes, chromosomal DNA contains sequences that serve as signals for turning on and off gene expression. These signals are thought to be distributed as clusters in the regulatory regions of genes. We develop a Bayesian model that views locating regulatory regions in genomic DNA as a change-point problem, with the beginning of regulatory and non-regulatory regions corresponding to the change points. The model is based on a hidden Markov chain. The data consist of nucleotide positions of protein-binding elements in a genomic DNA sequence. These positions are identified using a reference catalogue containing elements that interact with transcription factors implicated in controlling the expression of protein-encoding genes. Among the protein-binding elements in a genomic DNA sequence, the statistical model automatically selects those that tend to predict regulatory regions. We test the model using viral sequences that include known regulatory regions and provide the results obtained for human genomic DNA corresponding to the beta globin locus on chromosome 11.

Exceptional motifs in different Markov chain models for a statistical analysis of DNA sequences

Identifying exceptional motifs is often used for extracting information from long DNA sequences. The two difficulties of the method are the choice of the model that defines the expected frequencies of words and the approximation of the variance of the difference T(W) between the number of occurrences of a word W and its estimation. We consider here different Markov chain models, either with stationary or periodic transition probabilities. We estimate the variance of the difference T(W) by the conditional variance of the number of occurrences of W given the oligonucleotides counts that define the model. Two applications show how to use asymptotically standard normal statistics associated with the counts to describe a given sequence in terms of its outlying words. Sequences of Escherichia coli and of Bacillus subtilis are compared with respect to their exceptional tri- and tetranucleotides. For both bacteria, exceptional 3-words are mainly found in the coding frame. E. coli palindrome counts are analyzed in different models, showing that many overabundant words are one-letter mutations of avoided palindromes.