Description of the Algorithm

Alignment of two sequences without allowing gaps requires an algorithm that performs a number of comparisons roughly proportional to the square of the average sequence length, as in a dot matrix comparison. If the alignment is to include gaps of any length at any position in either sequence, the number of comparisons that must be made becomes astronomical and is not achievable by direct comparison methods. Dynamic programming is a method of sequence alignment that can take gaps into account but that requires a manageable number of comparisons. The method of sequence alignment by dynamic programming and the proof that the method provides an optimal (highest scoring) alignment. To understand how the method works, we must first recall what is meant by an alignment, using the two protein sequences as an example. The two sequences will be written across the page, one under the other, the object being to bring as many amino acids as possible into register. In some regions, amino acids in one sequence will be placed directly below identical amino acids in the second. In other regions, this process may not be possible and nonidentical amino acids may have to be placed next to each other, or else gaps must be introduced into one of the sequences. Gaps are added to the alignment in a manner that increases the matching of identical or similar amino acids at subsequent portions in the alignment. Ideally, when two similar protein sequences are aligned, the alignment should have long regions of identical or related amino acid pairs and very few gaps. As the sequences become more distant, more mismatched amino acid pairs and gaps should appear. The quality of the alignment between two sequences is calculated using a scoring system that favors the matching of related or identical amino acids and penalizes for poorly matched amino acids and gaps. To decide how to score these regions, information on the types of changes found in related protein sequences is needed. These changes may be expressed by the following probabilities: (1) that a particular amino acid pair is found in alignments of related proteins; (2) that the same amino acid pair is aligned by chance in the sequences, given that some amino acids are abundant in proteins and others rare; and (3) that the insertion of a gap of one or more residues in one of the

sequences (the same as an insertion of the same length in the other sequence), thus forcing the alignment of each partner of the amino acid pair with another amino acid, would be a better choice. The ratio of the first two probabilities is usually provided in an amino acid substitution matrix. Each table entry gives the ratio of the observed frequency of substitution between each possible amino acid pair in related proteins to that expected by chance, given the frequencies of the amino acids in proteins. These ratios are called odds scores. The ratios are transformed to logarithms of odds scores, called log odds scores, so that scores of sequential pairs may be added to reflect the overall odds of a real to chance alignment of an alignment. Examples are the Dayhoff PAM250 and BLOSUM62 substitution matrices described (p. 76). These matrices contain positive and negative values, reflecting the likelihood of each amino acid substitution in related proteins. Using these tables, an alignment of a sequential set of amino acid pairs with no gaps receives an overall score that is the sum of the positive and negative log odds scores for each individual amino acid pair in the alignment. The higher this score, the more significant is the alignment, or the more it resembles alignments in related proteins. The score given for gaps in aligned sequences is negative, because such misaligned regions should be uncommon in sequences of related proteins. Such a score will reduce the score obtained from an adjacent, matching region upstream in the sequences. The score of the alignment, using values from the BLOSUM62 amino acid substitution matrix and a gap penalty score of _11 for a gap of length 1, is 26 (the sum of amino acid pair scores) _11 _15. The value of _11 as a penalty for a gap of length 1 is used because this value is already known from experience to favor the alignment of similar regions when the BLOSUM62 comparison matrix is used. Choice of the gap penalty is discussed further below where a table giving suitable choices is presented. As shown in the example, the presence of the gap decreases significantly the overall score of the alignment. Although one may be able to align the two short sequences by eye and to place the gap where shown, the dynamic programming algorithm will automatically place gaps in much longer sequence alignments so as to achieve the best possible alignment. The derivation of the dynamic programming algorithm, using the above alignment as an example. Consider building this alignment in steps, starting with an initial matching aligned pair of characters from the sequences (V/V) and then sequentially adding a new pair until the alignment is complete, at each stage choosing a pair from all the possible matches that provides the highest score for the alignment up to that point. If the full alignment finally reached on the left side (I) has the highest possible or

optimal score, then the old alignment from which it was derived (A) by addition of the aligned Y/Y pair must also have been optimal up to that point in the alignment. If this were incorrect, and a different preceding alignment other than A was the highest scoring one, then the alignment on the left would also not be the highest scoring alignment, and we started with that as a known condition. Similarly, (II), alignment A must also have been derived from an optimal alignment (B) by addition of a C/C pair. In this manner, the alignment can be traced back sequentially to the first aligned pair that was also an optimal alignment. One concludes that the building of an optimal alignment in this stepwise fashion can provide an optimal alignment of the entire sequences. The example also illustrates two of the three choices that can be made in adding to an alignment between two sequences: Match the next two characters in the next positions in each sequence, or match the next character to a gap in the upper sequence. The last possibility, not illustrated, is to add a gap to the lower sequence. This situation is analogous to performing a dot matrix analysis of the sequences, and of either continuing a diagonal or of shifting the diagonal sideway or downward to produce a gap in one of the sequences.