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The value in the last cell, the lower right —5 in our example , represents the score of the best alignment given the score function.

Methods, models, concepts, and strategies

Following a diagonal arrow indicates that the sites represented by that row and column of the matrix should be aligned. Following a vertical arrow indicates that the character in the sequence along the vertical axis the character represented by the row of the matrix should be aligned with a gap in the sequence represented by the horizontal axis. Following a horizontal arrow indicates that the character in the sequence along the horizontal axis the character represented by the column of the matrix should be aligned with a gap in the sequence represented by the vertical axis.

If there is more than one possible path back to the top of the matrix, this indicates that multiple pairwise alignments lead to the identical score and are equally optimal.

Smith–Waterman algorithm

Changing the cost function may change the result see below. Very few alignment programs produce more than one alignment, even if there are multiple equally optimal alignments; the sim algorithms of Huang et al. How the single resultant alignment produced by most programs is chosen from the universe of possible optimal alignments is usually not clear. This is the simplest approach to pairwise sequence alignment.

Obvious enhancements include the use of more complicated scoring functions. Not all mismatches are necessarily equal, and different types of mismatches could be given different scores depending on the properties of the characters. For DNA sequences, these differential scores might be based on standard models of sequence evolution. For example, it is well known that transitional substitutions occur more often than transversional substitutions, therefore a transversional mismatch might be given a higher cost than a transitional mismatch.

For protein sequences, empirically derived substitution matrices are usually used to determine relative costs of various mismatches; these matrices include the PAM Dayhoff et al. These matrices usually include estimated biological factors such as the conservation, frequency, and evolutionary patterns of individual amino acids. Another enhancement to the scoring function has to do with the gap costs. As described above, all gaps are treated as identical single Alignment Concepts and History 11 position events.

Biologically, we recognize that single insertion-deletion events may and often do cover multiple sites. We therefore may not want the cost of a gap that covers three sites to be triple the cost of a gap that covers only one site a linear gap cost.

Sequence alignment Methods - I

Some algorithms will also vary in how they treat terminal gaps that is, gaps that occur at the very beginning or ends of a sequence ; some algorithms will give these reduced cost even zero since they are not inferred to occur between observed characters this is sometimes known as semi-global alignment. In such a case, only subsections of the sequences may be homologous, or the homologous sections may be in a different order.

Illustration of global alignment problem. In each case, the other two sections cannot be aligned properly.

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An alternative approach to global alignment is local alignment. In a local alignment, subsections of the sequences are aligned without reference to global patterns. This allows the algorithm to align regions separately regardless of overall order within the sequence and to align similar regions while allowing highly divergent regions to remain unaligned. Early approaches for local alignment were developed by Sankoff and Sellers , , but the basic local alignment procedure most widely used was proposed by Smith and Waterman b.

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  • It is a simple adaptation to the standard Needleman—Wunsch algorithm. In addition to the three possible values described by the Needleman—Wunsch algorithm, the local alignment algorithm allows for a fourth possible value: zero. This prevents the alignment score from ever becoming negative; if this rule is invoked, no trace-back arrow is stored for the cell. The addition of the fourth rule substantially changes the structure of the scores and the trace-back arrows.

    In our example, this is the cell directly above the lower right cell, with a score of 3. Completed score and trace-back matrix for local alignment using the Smith and Waterman b algorithm. For this example, the local alignment is simply that shown in Figure 1.

    Only the aligned parts of the sequences are reported. Of course, additional local alignments of the sequences could be found if there are multiple cells with the same maximal score or by choosing submaximal starting points. As with global alignment, there have been major advances in approaches for local alignment; major local alignment programs and algorithms in use today include DIALIGN Morgenstern ; Morgenstern, Frech, et al. With the recent sequencing revolution, one bioinformatic challenge has been the comparison of full genome sequences.

    Many specialized programs for producing local alignments of entire genomes have been produced recently, include BlastZ Schwartz et al. Local Alignment vs. Database Searching Much of the work on local alignment has focused on database searching rather than simple sequence comparison. It was recognized very early on that algorithms would be necessary to retrieve sequences from a database with a pattern similar to that of a query sequence e.

    Comparing sequences for similar patterns requires, in some form, local alignment, and local alignment methods form the basis of all database searching algorithms. Although not discussed in any detail within this book, must of the major work on local alignment derives from interests in database searching, particularly in the development of both the BLAST and FASTx Lipman and Pearson ; Pearson and Lipman families of algorithms. As one would expect, changing the cost function may change which alignment is considered to be most optimal. A Four equally optimal global alignments.

    B The single optimal local alignment. Different sets of scoring values may lead to different optimal alignments; the best alignment is not only dependent on the algorithm global vs. How does one determine what values should be used for the cost function? Most users tend to use program defaults, in which case the problem of determining the proper values is just left to the program authors rather than the end user.

    It should be noted that the absolute magnitudes of the values are unimportant; it is the relative values that matter that is, multiplying all of the scores by a constant will not affect the resultant best alignment. As mentioned previously, relative match and mismatch values are usually determined from empirical substitution matrices for proteins or models of sequence evolution for DNA.

    Contrast with the local alignment in Figure 1. The values in the above example indicate that a gap is twice as costly as a mismatch. There is remarkably little data available on the observed ratio of indel events to point mutations. Additionally, there appears to be a bit of a discrepancy between the biological ratio of point mutations and indels and the actual cost structure used in the alignment. Methods for optimizing gap costs as well as other aspects of the cost function and their effects are an understudied aspect of sequence alignment.

    In general, we are often interested in aligning more than two sequences in general, called multiple alignment.

    Multiple sequence alignment: In pursuit of homologous DNA positions

    In principle, one could use a Needleman—Wunsch approach for more than two sequences for example, constructing a three-dimensional cubic matrix for three sequences Jue et al. Alignment Concepts and History 17 Early alternate approaches for multiple alignment required a known phylogenetic tree. Sankoff and colleagues Sankoff ; Sankoff et al.

    Waterman and Perlwitz suggested an alternate approach that used weighted averaging. Instead of using a known tree, Hogeweg and Hesper suggested an iterative method where one starts with a putative tree, aligns the data, uses the alignment to estimate a new tree, and then uses the new tree to realign the data, and so forth. The approach for multiple sequence alignment that eventually really caught on is known as progressive alignment Feng and Doolittle , These pairwise alignments are used to estimate a phylogenetic tree using a distance-based algorithm such as the unweighted pair group method with arithmetic mean UPGMA or neighbor joining.

    Using the tree as a guide, the most similar sequences are aligned to each other using a pairwise algorithm. One then progressively adds sequences to the alignment, one sequence at a time, based on the structure of the phylogenetic tree. Numerous multiple alignment programs have been based on a progressive alignment adaptation of the Needleman—Wunsch algorithm, including ClustalW Thompson et al.

    For example, a general disadvantage of the progressive alignment approach is that it is what is known as a greedy algorithm; any mistakes that are made in early steps of the procedure cannot be corrected by later steps. For example, take the case adapted from Duret and Abdeddaim , with three short sequences whose optimal alignment is as shown in Figure 1. Assuming the guide tree indicates we should start by aligning sequences 1 and 2, there are three possible alignments with the same score one transversional mismatch and one gap , shown in Figure 1.

    When adding sequence 3, the position of the gap cannot be changed. Thus, adding sequence 3 could lead to three possible multiple alignments, shown in Figure 1. Illustration of the progressive alignment problem. A The optimal multiple alignment of three sequences.