The chapter in BSA that introduces Markov chains and hidden Markov models plays a critical role in that book. The sequence comparison algorithms described in Chapter 2 could not be developed without the introduction of the theoretically justified similarity scores and statistical theory of similarity score distributions. These developments, in turn, are not feasible without rational choices of probabilistic models for DNA and protein sequences. Both Markov chains and hidden Markov models are often remarkably good candidates for the sequence models. Moreover, hidden Markov models (HMMs) are potentially a more flexible means for biological sequence analysis because they allow simultaneous modeling of observable and non-observable (hidden) states. The presence of the two types of states perfectly fits the need to model some important additional information existing beyond sequences per se, such as the functional meaning of the sequence elements, matches and mismatches of symbols in pairs of aligned sequences, evolutionary conserved regions in multiple sequences, phylogenetic relationships, etc.

Chapter 3 of BSA introduces the fundamental algorithms of HMM theory: the Viterbi algorithm, the forward and backward algorithms, as well as the Baum–Welch algorithm. All of these algorithms are amenable for a variety of applications in biological sequence analysis. Of course, some of these HMM constructions exist in parallel with their non-probabilistic counterparts; for example, consider the Viterbi algorithm for a pair HMM and the classic dynamic programming algorithm for pairwise alignment. Both HMM and non-HMM approaches are known for finding conserved domains, building phylogenetic trees, etc.

In this chapter, the BSA problems focus on deriving the formulas that support probabilistic modeling and the HMM algorithm construction.

The notion of sequence similarity is perhaps the most fundamental concept in biological sequence analysis. In the same way that the similarity of morphological traits served as evidence of genetic and functional relationships between species in classic genetics and biology, biological sequence similarity could frequently indicate structural and functional conservation among evolutionary related DNA and protein sequences. Introduction of the biologically relevant quantitative measure of sequence similarity, the similarity score, is not a trivial task. No simpler is the other task, developing algorithms that would find the alignment of two sequences with the best possible score given the scoring system. Finally, the third necessary component of the computational analysis of sequence similarity is the method of evaluation of statistical significance of an alignment. Such a method, establishing the cut-off values for the observed scores to be statistically significant, works properly as soon as the statistical distribution of similarity scores is determined analytically or computationally.

Chapter 2 of BSA includes twelve problems that require knowledge of the concepts and properties of the pairwise alignment algorithms. This topic is traditionally best known to biologists due to its utmost practical importance. Indeed, an initial characterization of any DNA or protein sequence starts with the BLAST analysis, utilization of a highly efficient heuristic pairwise alignment algorithm for searching for homologous sequences in a database.

Additional nine problems provide more information for understanding the protein evolution theory behind the log-odds scores of amino acid substitutions, as well as the models involved in the assessment of the statistical significance of the observed sequence similarity scores.

Establishing phylogenetic relationships between species is one of the central problems of biological science. While in Chapter 7 the reader was introduced to non-probabilistic methods of building phylogenetic trees for DNA and protein sequences, Chapter 8 continues the subject from the standpoint of consistent probabilistic methodology. The evolution of biological sequences has been largely viewed as a random process, and several probabilistic models with varying levels of complexity have been proposed. Therefore, the reconstruction of phylogenetic relationships can be formulated in probabilistic terms as well.

Several introductory BSA problems in Chapter 8 are concerned with the properties of the simplest probabilistic models of evolution, such as the Jukes–Cantor and the Kimura models.

Given a set of sequences (associated with the leaves of a tree) and a model of the process of substitutions in a DNA or protein sequence, it is important to know how to compute the likelihood of a tree with a given topology. The Felsenstein algorithm addresses this issue using the post-order traversal. Felsenstein also developed an EM-type algorithm for finding the optimal (maximum likelihood) lengths of the tree edges. However, as the number of leaves increases, the number of tree topologies grows too quickly to be processed in a reasonable time.

Therefore, finding the optimal tree among all possible trees for a rather large number of sequences (leaves) is one of the major challenges. The mainstream approach to managing such a problem is sampling from the posterior distribution on the space of trees.

The tree HMM concept described in BSA could be used for phylogenetic tree construction utilizing most general models of the sequence evolution.

The one-dimensional string is too simple a model to reflect fully the properties of a real biological molecule, which have, after all, been determined by its three dimensional structure selected in the course of evolution. Physical interactions of amino acids and nucleotides in the three-dimensional folds have to be described by the models that would go beyond the short range correlations which are the typical targets of the Markov chain models. The long range correlations are more important for proteins than for DNA, which has a rather uniform double helix structure. However, the structure of another nucleic acid, RNA, commonly has a significant number of long range interactions of special type, which could be a target for yet another class of probabilistic models.

Chapter 9 introduces the Chomsky hierarchy of deterministic transformational grammars, the models developed originally for natural languages and then applied to computer languages. These grammars could be readily used for the description of a protein (a regular grammar could generate amino acid sequences described as the PROSITE patterns) and RNA (a context-free grammar could generate RNA sequences with a given secondary structure).

Further generalization of these deterministic grammar classes to stochastic ones increases opportunities for sequence modeling. Stochastic regular grammars could be shown to be equivalent to hidden Markov models. Stochastic context-free grammars (SCFGs) are useful for modeling RNA sequences.

Classifying biological sequences into families is one of the major challenges of bioinformatics. In fact, to provide the exact definition of the family (for example, the protein family) is difficult. Even with the introduction of the notion of a conserved in evolution protein domain as a structural determinant of a protein family, to classify multidomain proteins consistently is not a simple task. For practical purposes, nevertheless, it is important to develop efficient computational tools able to assign a protein translated from a newly predicted gene to one of already established families, thus characterizing the protein based on its amino acid sequence alone.

Computational tools of protein characterization have to recognize the familyspecific features in a new protein sequence. Frequently, these detectable common features are manifested as statistically significant structural conservations. The computational tools that are required to solve the classification problem should be able to (i) make use of known structural patterns specific for a given family, (ii) detect the family patterns in the new protein sequence by alignment of the new protein to the family model, and (iii) assess the statistical significance of the detected similarity in order to help correctly identify the true family members.

These three properties of the protein characterization algorithm are similar to the properties of the pairwise sequence alignment algorithm, but there are significant differences. First, the availability of several sequences makes the differential scoring of amino acid matches feasible; with matches (mismatches) in conserved positions receiving higher (lower) scores than scores of similar events in nonconserved positions.