Often we do not expect the whole of one sequence to align well with the other. For example, the proteins may have just one domain in common, in which case we
136
want to find this high-scoring zone, referred to as a local alignment (see Section 4.5). In a global alignment, those regions of the sequences that differ substantially will often obscure the good agreement over a limited stretch. The local alignment will identify these stretches while ignoring the weaker alignment scores elsewhere. It turns out that a very similar dynamic programming algorithm to that described above for global alignments can obtain a local alignment. Smith and Waterman first proposed this method. However, it should be noted that the method presented here requires a similarity-scoring scheme that has an expected negative value for random alignments and positive value for highly similar sequences. Most of the commonly used substitution matrices fulfill this condition. Note that the global alignment schemes have no such restriction, and can have all substitution matrix scores posi- tive. Under such a scheme, scores will grow steadily larger as the alignment gets larger, regardless of the degree of similarity, so that long random alignments will ulti- mately be indistinguishable by score alone from short significant ones.
The key difference in the local alignment algorithm from the global alignment algorithm set out above is that whenever the score of the optimal sub-sequence alignment is less than zero it is rejected, and that matrix element is set to zero. The scoring scheme must give a positive score for aligning (at least some) identical residues. We would expect to be able to find at least one such match in any align- ment worth considering, so that we can be sure that there should be some posi- tive alignment scores. Another algorithmic difference is that we now start traceback from the highest-scoring matrix element wherever it occurs.
Figure 5.15
The dynamic programming calculation for determining the optimal local alignment of the two sequences THISLINE and ISALIGNED. (A) The completed matrix using the BLOSUM-62 scoring matrix with a linear gap penalty, defined in Equation EQ5.13 with E set to -8. (B) The optimal alignment, determined by the highest-scoring element, which has a score of 12.
Figure 5.16
Optimal local alignment calculation identical to Figure 5.15, except with a linear gap penalty with E set to -4. (A) The completed matrix for determining the optimal local alignment of THISLINE and ISALIGNED using the BLOSUM-62 scoring matrix. (B) The optimal alignment, identified by the highest- scoring element in the entire matrix, which has a score of 19.
Figure 5.17
The dynamic programming calculation that follows on from the calculation shown in Figure 5.16 to find the best suboptimal local alignment. (A) The completed matrix for determining the best suboptimal local alignment of THISLINE and ISALIGNED using the BLOSUM-62 scoring matrix with a linear gap penalty with E set to -4. The matrix elements that were involved in the optimal local alignment have been set to zero and are shown with bold font. The matrix elements that have changed value from Figure 5.16(A) are also shown with bold font, and extend below and to the right of the optimal alignment. (B) The best suboptimal alignment, identified by the highest- scoring element in the entire matrix, has a score of 5.
The extra condition on the matrix elements means that the values of S, 0 and S0j are
set to zero, as was the case for global alignments without end gap penalties. The formula for the general matrix element Stj with a general gap penalty function
gtttgap) IS
which only differs from Equation EQ5.18 by the inclusion of the zero. The same modifications as above can be applied for the cases of linear gap penalty given in Equation EQ5.13 and affine gap penalty given in Equation EQ5.14.
Figures 5.15 and 5.16 show the optimal local alignments for our usual example in the two cases of linear gap penalties g(«gap) = -8ngap and -4rcgap, respectively. Both
result in removal of the differing ends of the sequences. In the first case, the higher gap penalty forces an alignment of serine (S) and alanine (A) in preference to adding a gap to reach the identical IS sub-sequence. Lowering the gap penalty in this instance improves the result to give the local alignment we would expect. Sometimes it is of interest to find other high-scoring local alignments. A common instance would be the presence of repeats in a sequence. There will usually be a number of alternative local alignments in the vicinity of the optimal one, with only slightly lower scores. These will have a high degree of overlap with the optimal alignment, however, and contain little, if any, extra information beyond that given by the optimal local alignment. Of more interest are those suboptimal local align- ments that are quite distinct from the optimal one. Usually their distinctness is defined as not sharing any aligned residue pairs.
An efficient method has been proposed for finding distinct suboptimal local align- ments. These are alignments in which no aligned pair of residues is also found aligned in the optimal or other suboptimal alignments. They can be very useful in a variety of situations such as aligning multidomain proteins. Sometimes a pair of proteins has two or more domains in common but other regions with no similarity. In such cases it is useful to obtain separate local alignments for each domain, but only one of these will give the optimal score, the others being suboptimal align- ments. The method starts as before by calculating the optimal local alignment. Then, to ensure that any new alignment found does not share any aligned residues
138