Methods for Calculating the Parameters of the Extreme Value Distribution

In the analysis by Altschul and Gish (1996), 10,000 random amino acid sequences of variable lengths were aligned using the Smith-Waterman method and a combination of the scoring matrix and a reasonable set of gap penalties for the matrix. The scores found by this method followed the same extreme value distribution predicted by the underlying statistical theory. Values of K and _ were then estimated for each combination by fitting the data to the predicted extreme value distribution. Some representative results are shown in .10. Readers should consult Tables V–VII in Altschul and Gish (1996) for a more detailed list of the gap penalties tested. Altschul and Gish (1996) have cautioned users of these statistical parameters. First, the parameters were generated by alignment of random sequences that were produced assuming a particular amino acid distribution, which may be a poor model for some proteins. Second, the accuracy of _ and K cannot be estimated easily. Finally, for gap costs that give values of H _ 0.15, the optimal alignment length is a significant fraction of the sequence lengths and produces a source of error called the edge effect. The effect occurs when the expected length of an alignment is a significant fraction of the sequence length, and, as discussed earlier, alignments between sequences that overlap at their ends cannot be completed. The expected length is then subtracted from the sequence length before _ is estimated.

If no such correction is done, _ may be overestimated. These values for gap penalties should also not be construed to represent the best choice for a given pair of sequences or the only choices, simply because the statistical parameters are available. The process of choosing a gap penalty remains a matter of reasoned choice. In trying the effects of varying the gap penalty, it is important to recognize that as the gap penalty is lowered, the alignments produced will have more gaps and will eventually change from a local to a global type of alignment, even though a local alignment program is being used. In contrast, higher H values are generated by a very large gap penalty and produce alignments with no gaps ( .10), thus suggesting an increased ability to discriminate between related and unrelated sequences. In this respect, Altschul and Gish (1996) note that beyond a certain point increasing the gap extension penalty does not change the parameters, indicating that most gaps in their simulations are probably of length 1. However, reducing the gap penalty can also allow an alignment to be extended and create a higher scoring alignment. Eventually, however, the optimal local alignment score between unrelated sequences will lose the log length relationship with sequence length and become a linear function. At this point, gap penalties are no longer useful for obtaining local alignments and the above statistical relationships are no longer valid. The higher the H value, the better the matrix can distinguish related from unrelated sequences. The lower the value of H, the longer the expected alignment.

These conditions may be better if a longer alignment region is required, such as testing a structural or functional model of a sequence by producing an alignment. Conversely, scoring parameters giving higher values of H should produce shorter, more compact alignments. If H _ 0.15, the alignments may be very long. In this case, the sequences have a shorter effective length since alignments starting near the ends of the sequences may not be completed. This edge effect can lead to an overestimation but was corrected (Altschul and Gish 1996). Unfortunately, the above method for calculating the significance of an alignment score may not be used to test the significance of a global alignment score. The theory does not apply when these same substitution matrices are used for global alignments. Transformation of these matrices by adding a fixed constant value to each entry or by multiplying each value by a constant has no effect on the relative scores of a series of global alignments. Hence, there is no theoretical basis for a statistical analysis of such scores as there is for local alignments (Altschul 1991). As discussed,

two programs are commonly used for database similarity searches: FASTA and BLAST. These programs both calculate the statistical significance of the higher scores found with similar sequences, but the types of analyses used to determine the statistical significance of these scores are somewhat different. BLAST uses the value of K and _ found by aligning random sequences, where n and m are shortened to compensate for inability of ends to align. FASTA calculates the statistical significance using the distribution of scores with unrelated sequences found during the database search. In effect, the mean and standard deviation of the low scores found in a given length range are calculated. These scores represent the expected range of scores of unrelated sequences for that sequence length (recall that the local alignment scores increase as the logarithm of the sequence length). The number of standard deviations to the high scores of related sequences in the same length range (z score) is then determined. The significance of this z score is then calculated according to the extreme value distribution expected of the z scores, given in it. This method is discussed in greater detail in . Pearson (1996) showed that these two methods are equally useful in database similarity searches for detecting sequences more distantly related to the input query sequence. Pearson (1996) has also determined the influence of scoring matrices and gap penalties on alignment scores of moderately related and distantly related protein sequences in the same family.

For two examples of moderately related sequences, the choice of scoring matrix and gap penalties (gap opening penalty followed by penalty for each additional gap position) did not matter, i.e., BLOSUM50 _12/_2, BLOSUM62 _8/_2, Gonnet93 _10/_2, and PAM250 _12, _2 all produced statistically significant scores. The scores of distantly related proteins in the same family depended more on the choice of scoring matrix and gap penalty, and some scores were significant and others were not. Pearson recommends using caution in evaluating alignment scores using only one particular combination of scoring matrix and gap penalties. He also suggests that using a larger gap penalty, e.g., _14, _2 with BLOSUM50, can increase the selectivity of a database search for similarity (fewer sequences known to be unrelated will receive a significant alignment score). A difficulty encountered by FASTA in calculating statistical parameters during a database search is that of distinguishing unrelated from related sequences, because only scores of unrelated sequences must be used. As score and sequence length information is accumulated during the search, the scores will include high, intermediate,

and sometimes low scores of sequences that are related to the query sequence, as well as low scores and sometimes intermediate and even high scores of unrelated sequences. As an example, a high score with an unrelated database sequence can occur because the database sequence has a region of low complexity, such as a high proportion of one amino acid. Regardless of the reason, these high scores must be pruned from the search if accurate statistical estimates are to be made. Pearson (1998) has devised several such pruning schemes, and then determined the influence of the scheme on the success of a database search at demonstrating statistically significant alignment scores among members of the same protein family or superfamily. However, no particular scheme proved to be better than another.

The above method does not necessarily ensure that the choice of scoring matrix and gap penalties provides a realistic set of local alignment scores. In the comparable situation of matching a test sequence to a database of sequences, the scores also follow the extreme value distribution. For this situation, Mott (1992) has explained that for local alignments the end point of the alignment should on the average be half- way along the query sequence, and for global alignments, the end point should be beyond that half- way point. Pearson (1996) has pointed out that the presence of known, unrelated sequences in the upper part of the curve where E _ 1 can be an indication of an inappropriate scoring system.

10.1.6 The Statistical Significance of Individual Alignment Scores between Sequences and