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# Improving the accuracy of PSI-BLAST protein database searches with composition-based statistics and other refinements

^{a}To whom correspondence should be addressed. Tel: +1 301 435 5884; Fax: +1 301 480 2918; Email: vog.hin.xileh@reffahcs

## Abstract

PSI-BLAST is an iterative program to search a database for proteins with distant similarity to a query sequence. We investigated over a dozen modifications to the methods used in PSI-BLAST, with the goal of improving accuracy in finding true positive matches. To evaluate performance we used a set of 103 queries for which the true positives in yeast had been annotated by human experts, and a popular measure of retrieval accuracy (ROC) that can be normalized to take on values between 0 (worst) and 1 (best). The modifications we consider novel improve the ROC score from 0.758 ± 0.005 to 0.895 ± 0.003. This does not include the benefits from four modifications we included in the ‘baseline’ version, even though they were not implemented in PSI-BLAST version 2.0. The improvement in accuracy was confirmed on a small second test set. This test involved analyzing three protein families with curated lists of true positives from the non-redundant protein database. The modification that accounts for the majority of the improvement is the use, for each database sequence, of a position-specific scoring system tuned to that sequence’s amino acid composition. The use of composition-based statistics is particularly beneficial for large-scale automated applications of PSI-BLAST.

## INTRODUCTION

BLAST and PSI-BLAST are widely used programs for detecting sequence similarities, including subtle ones, in searches of protein sequence databases (1,2). PSI-BLAST’s basic strategy is to construct a multiple alignment from the output of a BLAST protein database similarity search, abstract a position-specific score matrix from this multiple alignment, and search the database anew using the score matrix as query. The process may be iterated many times, as new significant similarities are found.

The ways in which PSI-BLAST estimates the statistical significance
of the similarities it finds, and constructs position-specific score
matrices from multiple alignments, are amenable to a variety of
modifications. Some of these involve simple parameter adjustments,
but others are based on deeper theoretical considerations. We sought
to investigate the effect of a large number of proposed refinements
on PSI-BLAST’s sensitivity to distant sequence relationships,
with the aim of incorporating any that proved beneficial. To this
end, we constructed a test set of query sequences, and annotated
for each those yeast sequences that were related. We tested various versions
of PSI-BLAST with a protocol described in the next section. To evaluate
the performance of PSI-BLAST versions, we pooled the output from
all queries, ranked by *E*-value, and plotted false
positives versus true positives.

The outlined procedure was used to test over a dozen ideas for improving PSI-BLAST search accuracy. Among these, fewer than half proved of value. Nevertheless, PSI-BLAST version 2.2.1, which incorporates all useful changes found, is substantially more accurate than the original program. Among the modifications tested are four that we considered ‘known’ and could reasonably have been implemented in the original PSI-BLAST (version 2.0.1) or were published elsewhere (3). We included these four improvements first in what we call the ‘baseline’ version of PSI-BLAST, and in what follows we do not count the improvement that they yield. The remaining modifications were added, usually one at a time, in what seemed to us a plausible order. Among the modifications we included in the baseline version of PSI-BLAST are: (i) the ability to run the Smith–Waterman algorithm on all alignments reported, and (ii) the filtering of database sequences for the presense of ‘low-entropy’ segments of restricted amino acid composition. The modification that yields most of the improvement above the baseline version is the use of statistics tuned to the composition of the query and database sequences to evaluate the significance of local alignments.

## THE TEST SET AND SEARCH PROTOCOL

We used as a primary test set some queries originally constructed
for the purpose of detecting proteins with widespread regulatory
and signal-transduction domains in the yeast *Saccharomyces
cerevisiae* and the nematode worm *Caenorhabditis
elegans* (4). Lists of true
positives in yeast were delineated and curated by human experts
(L.A. and E.V.K.).

An earlier version of this query set with 105 sequences was used
to test IMPALA in Schäffer *et al.* (5) and called the aravind105 set. For the
tests reported here we used 103 queries of which 91 are common to
the aravind105 set. The changes in the new query test set are due
to: 12 useful queries added after the IMPALA project started; nine
queries dropped due to having no apparent true positives in yeast;
five queries dropped due to overlaps with other queries and/or
difficulties in identifying all true positives because of the PSI-BLAST
corruption problem discussed below. Even though the methods presented here
largely solve the PSI-BLAST problems that made some annotations
difficult, we did not reintroduce the problematic query sequences.
Starting with the copy of the yeast proteome used by Schäffer *et al.* (5) and
containing 6406 proteins, we dropped 65 proteins that were identical
or nearly identical to substrings of proteins remaining in the yeast
database to avoid double counting what were really single true positives.

At the start of this project, the 103 lists of true positives included 918 total matches. As a result of testing with hundreds of PSI-BLAST versions we identified 87 more true positives, giving a total of 1005 for a mean of 9.8/query (range: 2–123). Improved software is expected to detect new true positives, and the ability to add these to the original true positive set is important. Therefore, we chose to use match lists developed and curated in house in lieu of fixed external ones. The 103 queries, copy of the yeast database, and lists of true positives are available from ftp.ncbi.nlm.nih.gov under directory pub/impala/blasttest.

For our 103 queries, we had comprehensive lists of true positives only for the yeast database. However, PSI-BLAST achieves greater sensitivity to distant relationships when it constructs its score matrices from larger and more diverse sets of related sequences (6). If one is ultimately interested in relationships between a query and the members of a restricted database, such as the proteins in PDB (7), it pays to search a comprehensive sequence database with PSI-BLAST through several iterations, save the resulting position-specific score matrix (PSSM) as a ‘checkpoint’, and then restart using the checkpoint matrix to search the database of interest. We have used this general protocol in evaluating PSI-BLAST search accuracy. To evaluate each version of PSI-BLAST, we first compare every query sequence to the NCBI nr protein sequence database (8) (frozen on 2 February 2000 for consistency across tests) and save the matrix computed after the fifth PSI-BLAST iteration, or when the program stops because no further significant similarities are found. After this, each saved PSSM is compared to the yeast database, and the program’s accuracy in detecting distant relationships is evaluated through an analysis of the pooled search outputs.

## EVALUATING PSI-BLAST SEARCH ACCURACY

Given a classification of (query sequence, database sequence) pairs
as related or unrelated, it is useful to have an objective criterion
for comparing the performance of database search programs. Eliding
for the moment the precision of reported *E*-values,
one may treat them simply as a means of ordering database search
results, and then plot the number of false versus the number of
true relationships found as *E*-values increase.
Such a procedure yields a ‘sensitivity curve’ such
as those shown in Figures Figures11–4. In general, one prefers to find as many true
positives as possible before a given number of false positives,
so the farther to the right the sensitivity curve lies the better.
A problem in assessing the relative merit of different search procedures
is that their sensitivity curves are high-dimensional objects subject
to a variety of reasonable linear orderings (9–13). A measure of search ‘accuracy’ derived
from sensitivity curves that has gained fairly wide use is the truncated
receiver operating characteristic (ROC) (13).
Let *T* be total number of true positives available
to be found. For a fixed number of false positives *n*,
the quantity ROC* _{n}* is the proportion
of the area in the rectangle [0,

*T*][0,

*n*] that lies to the left of the sensitivity curve. ROC

*is a proportion, so its values lie between 0 (worst) and 1 (best). See Appendix A for more information.*

_{n}^{–3}, FWh10

^{–6}and FWh10

^{–9}. The sensitivity

**...**

The upper bound on false positives in calculating ROC values
has a practical justification. Programs such as BLAST (1,2)
report similarities only above a fairly high threshold score, so
only a limited number of false positives appear in the output. Moreover,
a researcher performing a database search generally is unwilling
to analyze many weak matches, most of which are false positives,
in the hope of finding a few more true positives. For comprehensive
sequence database searches with a single query, Gribskov and Robinson
(13) have recommended ROC_{50} as
a figure of merit. Since we use pooled results from approximately
100 searches here, this would appear to translate into a recommendation
of ROC_{5000}. Because the database we ultimately search
consists of only 6341 proteins and contains a mean of <10
true relationships per query, diminishing returns have a much earlier
onset. After a total of about 100 pooled false positives have been
observed, the number of additional false positives found per additional true
positive escalates beyond a level most researchers would consider
fruitful for detailed analysis. Thus, the figure of merit we will
report for our pooled results is ROC_{100}.

When analyzing database search programs, it is useful not only to compare their ROC values, but also to know when these values are significantly different. We approximate the ‘random’ distribution of ROC values for a given program through bootstrap resampling of the false positive database search results it returns. As described in Appendix A, this distribution should be approximately normal, so that we need estimate only its standard deviation, which can be calculated analytically.

As we investigate various methods for improving PSI-BLAST, we in essence are exploring the high-dimensional space of different possible combinations of methods for constructing PSSMs. Due to the combinatorial explosion, it would be impractical to consider all points in this space and find a version of PSI-BLAST that is on average optimal. Accordingly, we generally content ourselves with examining one proposed change at a time, and accepting it if it produces a program with improved search accuracy. Therefore we claim only that the version of PSI-BLAST we finally produce is a significant improvement over the baseline version of the program.

## THE ‘BASELINE’ PSI-BLAST PROGRAM

To keep track of the various versions of PSI-BLAST we will be studying, we will use a shorthand to describe modifications to the original program. Single upper case letters will refer to modifications that we programmed so that they may be easily turned on or off in our test versions. Lower case letters will refer to changes in a numerical parameter, and will be followed by a numerical argument. In the released code, we do not enable the user to turn on/off each option individually because most combinations are of interest only in testing. As modifications to PSI-BLAST are described, letters signifying these changes will be introduced, but for reference purposes the meanings of all these codes are collected in Table Table11.

The ‘baseline’ version of PSI-BLAST with which
we will begin our analysis is modified in four ways from that described by
Altschul *et al.* (2).
Because the first two changes are incorporated in all versions of
PSI-BLAST we will study, neither will be assigned a letter for program
designation. The other two changes will be assigned a letter because
we will consider the effect of suppressing them.

### Estimation of statistical parameters

BLAST and PSI-BLAST *E*-values are calculated
using statistical and edge-effect parameters (3).
These parameters cannot be calculated analytically, but must be
estimated by prior random simulation for any amino acid substitution
matrices and gap costs to be supported. BLAST release 2.2.1 employs parameter
values estimated by a more accurate procedure (3,14) than that used for earlier releases
(15).

### Numerical precision and amino acid pair frequency ratios

The scores *s** _{ij}* in
the PAM (16,17)
and BLOSUM (18) amino acid substitution
matrices are constructed using the formula

* s** _{ij}* = log

*r*

_{ij}**1**

where *r*_{ij } is
the ratio of the estimated frequency with which the amino acids *i* and *j* are aligned due to evolutionary
descent, to the frequency with which they would be aligned by chance.
All local alignment substitution matrices are implicitly of this form,
with the base of the logarithm simply providing a scale factor (19,20).

For convenience, scores constructed using equation **1** are generally
rounded to the nearest integer. However, as described below, we
will have occasion to change, sometimes by small degrees, the scale
of the matrices employed by BLAST and PSI-BLAST. Integral substitution
scores of the usual scale discard too much precision for our purposes,
so our new baseline version of BLAST and PSI-BLAST represents substitution
matrices internally by the floating point *r*_{ij} rather than by the integral *s*_{ij}. This
provides the programs with two opportunities for greater precision. [Note
that the *r*_{ij} used
in constructing substitution matrices generally do not appear explicitly
in the literature. For the PAM matrices (16,17) we inferred these values from other
published data, while for the BLOSUM matrices (18)
we obtained these values from the authors, S. Henikoff and J. G.
Henikoff.]

First, as described below, we re-evaluate all database sequences
that participate in an alignment with good initial score. This is
done using an integral score matrix but, since the *r*_{ij} are available, we employ a
scaled-up version of the standard matrix. For greater precision,
the log-odds scores to the usual scale derived from equation **1** are
multiplied by 32 before rounding, as are any gap costs. (The usual
scale varies among matrices.) To avoid confusion, ‘raw’ alignment
scores are returned to the usual scale before printing.

Secondly, the availability of the *r*_{ij} gives
PSI-BLAST an opportunity for greater precision in constructing amino
acid frequency ratios for individual columns. In contrast to the frequency
ratio *R*_{i} = *Q*_{i}/*P*_{i} for amino
acid *i* implied by equations 3–5 of Altschul *et al.* (2), where *Q*_{i} and *P*_{i} are the
predicted and background frequencies, respectively, for amino acid *i*, the baseline PSI-BLAST now derives this ratio
using the equation:

** ** **2**

Here, as defined and discussed further by Altschul *et
al.* (2), *f** _{i} *is the weighted observed frequency
of amino acid

*i*for the column in question, α reflects the effective number of independent observations for that column, and β is the pseudocount parameter, which balances the relative importance given to data from the multiple alignment, and prior information on amino acid mutation propensities implicit in the reference substitution matrix. The ratio

*r*

_{ij}here replaces the earlier

*e*

^{λuSij}. The earlier formulation was inferior both because the

*s*

_{ij}in general have low precision, and because the scale parameter λ

_{u}was derived using standard amino acid background frequencies

*P*

_{i}(21) as opposed to the frequencies implicit in the original construction of the substitution matrix.

### Sequence filtering

Alignments involving ‘low entropy’ protein
segments with highly restricted or biased amino acid composition
generally are not of interest, and BLAST has traditionally filtered
such segments out of query sequences using the SEG program (22,23).
However, because PSI-BLAST PSSMs are constructed from the output
of database searches, it has been possible for low-entropy segments
from database sequences to strongly influence the scores in these
PSSMs, which are then aligned with unfiltered segments of other
database sequences in further rounds of PSI-BLAST searching. Accordingly,
we apply SEG filtering in our baseline program to database sequences
as opposed to the query sequence in the original PSI-BLAST program;
this change is designated with the letter ‘F’.
For SEG filtering of the database we use a window-size of 10, a
trigger complexity of 1.8 and an extension complexity of 2.1; this
compares with the default values of 12, 2.2 and 2.5, respectively.
The SEG parameters were tuned by testing them on all proteins in *Mycoplasma genitalium* plus some problematic queries
that arose in studies by Y.I.W. We considered SEG parameter values
to be safe if we could run all queries for five iterations without
obvious signs of PSSM corruption, such as a program crash or massive
increase in the size of the output file. We tried to find the lowest
safe setting of the window and trigger-complexity, while keeping
the difference between extension complexity and trigger complexity
at 0.3 as suggested by the developer (J. Wootton, personal communication).
It is not surprising that one can use more permissive SEG parameters
for filtering the database because the composition-based statistics
also help correct for composition biases. Applying SEG filtering
to both query and database sequences appears to be overkill, and
reduces search accuracy (data not shown).

### Smith–Waterman alignments

Database search heuristics such as FASTA (24) and BLAST (1,2) are used in place of the Smith–Waterman local alignment algorithm (25) primarily for the purpose of speed. These heuristics may completely miss some significant alignments, and may produce non-optimal alignments for some sequence similarities they find. At a small price in speed it is possible to avoid the second of these problems in BLAST. One simply runs the full Smith–Waterman algorithm on the small fraction of database sequences the heuristic algorithm chooses to report. We have implemented this option, and represent it with the letter ‘W’; thus the baseline version of PSI-BLAST is designated FW.

## THE CHOICE OF THRESHOLD PARAMETER AND THE PROBLEM OF CORRUPTION

A major potential problem for users of PSI-BLAST is that of PSSM ‘corruption’.
At each iteration, PSI-BLAST constructs a multiple alignment, from
which it then abstracts a PSSM. If a sequence *S* that
is unrelated to the original query sequence *Q* is included
in the multiple alignment, then the resulting PSSM will tend to
produce highly significant alignments to sequences related to *S* as well as those related to *Q*.
Such a PSSM is said to have been corrupted, and the search results
from further iterations are unreliable (26).
For the purpose of analysis, we shall say that the PSSM produced
by one of our query sequences after five rounds of comparison to
the nr database has become corrupted if it produces a false positive
alignment with *E*-value ≤10^{–4} when
compared to the yeast database.

With PSI-BLAST, a single corrupted query can yield many false
positives with very low *E*-values. Because we consider pooled
results, the sensitivity curves and corresponding ROC values may
be greatly affected. One might argue that this penalizes corruption
too heavily, and conclude that program evaluation should instead
be conducted on a query-by-query basis (12).
However, for researchers using PSI-BLAST on a large scale, even
a small percentage of corrupted queries can be a major problem,
and should therefore weigh heavily in any evaluation. First, the
results of a corrupted search can be almost completely meaningless,
and this casts considerable doubt on the reliability of results
from the large majority of searches that are uncorrupted. Secondly,
a corrupted search can consume a great quantity of computing time,
exhaust all virtual memory causing a crash, or produce a huge volume
of bogus output, limiting the applicability of PSI-BLAST to large-scale,
automatic annotation projects.

One may attempt to avoid PSSM corruption by setting to a sufficiently
low value the parameter *h*, which defines the maximum *E*-value a similarity may have for inclusion in
the multiple alignment. We use the letter ‘h’,
followed by its setting, in designating versions of PSI-BLAST. For
most queries, the threshold *h* = 10^{–3} is
sufficient to avoid corruption with the baseline program FW, but
a small percentage yield corrupted PSSMs at this and even much lower
values of *h*. Among our 103 query sequences, three
become corrupted with *h* = 10^{–3},
one with *h* = 10^{–6} and
none at *h* = 10^{–9}.
The corresponding sensitivity curves are shown in Figure Figure1.1. Although one may avoid most corruption by
lowering *h* sufficiently, one pays a price in search
accuracy for the majority of queries that do not get corrupted.
Among the thresholds considered, the highest ROC_{100} score,
0.758 ± 0.005, is for FWh10^{–6}.
Several of the refinements described below influence PSI-BLAST accuracy
to a large degree by suppressing the appearance of false positives
with low *E*-values in the iterated searches of the nr
database, thereby allowing the value of the *h* parameter
to be raised.

## COMPOSITION-BASED STATISTICS

Our most important refinement is to compute the statistical significance
of a match by taking into account the composition of the query and
database sequences. The statistical significance of a local alignment
produced by BLAST is assessed with an *E*-value,
calculated using the formula:

* E* = *Kmne*^{–}^{λ}^{S} **3**

where *m* and *n* are the effective
lengths of the query sequence and database, *S* is
the nominal score of the alignment, and λ and *K* are statistical parameters dependent upon the
scoring system used and the composition of the sequences being compared
(2). Because it enters equation **3** exponentially, the scale parameter λ is
the far more important.

For alignments that are not allowed to contain gaps, the parameters λ_{u} and *K*_{u} (where the subscript
indicates an ungapped scoring system) may be calculated analytically (19,27),
but for gapped alignments λ* _{g}* and

*K*

_{g}must be estimated (3,14,15,28–33). BLAST employs estimates of λ

_{g}and

*K*

_{g}precomputed from the comparison of a large number of ‘random protein sequences’ (3,15), generated using standard amino acid frequencies (21).

Occasionally, the amino acid compositions of a particular query
and matching database sequence pair imply a λ′_{g} substantially different from
the precomputed λ_{g},
rendering unreliable the estimates of statistical significance based
upon λ_{g} for
any alignments between these sequences. Most often this is due to
a similar, slightly biased amino acid composition shared by the
sequences, yielding a λ′_{g} < λ_{g}. Using the standard λ_{g} then results in a smaller *E*-value than is justified, sometimes off by several
orders of magnitude. The same problem can arise with the PSI-BLAST
comparison of PSSMs to protein sequences.

It is not practical to estimate statistical parameters by random
simulation for each (query sequence, database sequence) pair for
BLAST, or each (PSSM, database sequence) pair for PSI-BLAST. Thus
we adopt the rescaling strategy introduced by the IMPALA program
(5), an inverse form of PSI-BLAST.
In brief, for a given pairwise comparison, the ungapped scale parameter λ′_{u} can
be calculated analytically (19).
By rescaling the substitution scores, this parameter may be forced
to equal λ_{u},
the ungapped scale parameter for a reference scoring system in the
context of standard amino acid frequencies. Allowing gaps, with
specified costs, changes the scale parameter for this reference
scoring system to the pre-estimated λ_{g}. We simply assume the same holds
for our rescaled substitution scores in the context of non-standard amino
acid frequencies (2,5).
This assumption is supported by random simulation (2,5). More details on the rescaling method are
given in Appendix B.

It would be possible to rescale each PSSM for each database sequence, but this would slow down the program unduly. We rescale PSSMs and then recalculate alignments only for those pairwise comparisons that have yielded near-significant alignments after a first pass that employs scores scaled assuming a standard amino acid composition. Versions of PSI-BLAST that employ these composition-based statistics will be designated with the letter ‘S’.

With PSI-BLAST, it is important to use the rescaling strategy outlined here for the initial BLAST search as well as any subsequent PSI-BLAST searches, because corruption can occur at any stage. As a result, the alignment scores produced by the initial BLAST search in general do not correspond precisely to those implied by a standard substitution matrix, and even the same alignment involving two different database sequences can receive slightly different scores because of differing amino acid compositions.

The gain in PSI-BLAST accuracy from the use of composition-based
statistics is substantial. Whereas the baseline PSI-BLAST program
FWh10^{–6} yields one corrupted PSSM and achieves
a ROC_{100} of 0.758 ± 0.005,
with composition-based statistics the program FWSh0.002 yields no
corrupted PSSMs, and the ROC_{100} rises to 0.879 ± 0.003
(Fig. (Fig.2).2). Composition-based statistics alone
go a long way toward removing the need for SEG filtering of database
sequences. For our test set, the program WSh0.0002 yielded no corruption
and achieved a ROC_{100} of 0.838 ± 0.003
(Fig. (Fig.22).

^{–6}, WSh0.0002 and FWSh0.002.

Other approaches to accounting for amino acid composition are possible when evaluating statistical significance. Mott (33) has derived an empirical formula for estimating statistical parameters that relies upon sequence composition. Employing this formula would require slightly less execution time than our rescaling procedure, but the statistics it produces may be somewhat less accurate (3).

Karplus *et al.* (34)
suggest subtracting from the optimal local alignment score the score
obtained from the best local alignment of the query with a reversed
copy of the database sequence. We designate subtracting the reverse
alignment score by the letter ‘R’; this procedure
has the advantage that it accounts for higher-order statistical
sequence properties beyond that merely of composition. A disadvantage
is that the reversed alignment score introduces some random noise.
To get a handle on this effect, one may make the rough assumption that
after suitable normalization the scores for locally aligning a random
query sequence to a database sequence and its reversed copy are
independent random variables *X*_{1} and *X*_{2} that follow a standard extreme value
distribution. Some calculus then shows that *X*_{1} – *X*_{2} has a probability density function
with right-hand tail asymptotically equal to *e ^{–x}*,
the same as the right-hand tail for

*X*

_{1}(K. Karplus, personal communication). However, the extreme value distribution is positively skewed with mean equal to Euler’s constant γ ≈ 0.577, so subtracting the reversed alignment score

*X*

_{2}tends to dilute statistical significance. The mean change in score corresponds to a multiplicative factor of e

^{γ}≈ 1.8 in significance. Incorporating the reversed-alignment score adjustment procedure decreased the ROC

_{100}score from 0.879 ± 0.003 for FWSh0.002 to 0.872 ± 0.003 for FWSRh0.01 (where the ROC score was maximum), and we have therefore not adopted this method. The R option does provide added protection against corruption arising from sequences with certain anomalous statistical properties, and may be useful in some contexts.

## AMINO ACID FREQUENCIES FOR COLUMNS INCLUDING GAPS

To calculate the weighted observed frequencies for the 20 amino acids in a given column of a multiple alignment (hereafter called simply the observed frequencies), the baseline PSI-BLAST program considers only sequences of the multiple alignment actually containing a residue in that column, and ignores any sequences with a gap there (2). This choice, however, may throw away important information.

To test whether integrating information from sequences with gaps in a column could improve the sensitivity of the PSSMs constructed by PSI-BLAST, we implemented the following. When calculating sequence weights for a given column (2), include all sequences that participate in the column, whether with a residue or null character (gap). A null character is treated initially as if it were simply a twenty-first amino acid. After each sequence is assigned a weight, one is faced in general with a non-zero observed frequency for the null character. However, to calculate substitution scores for the given column, the observed frequencies of the 20 amino acids should sum to 1 (2), and thus the frequency for the null character must somehow be distributed among the amino acids. Two alternative ways to do this, which we call respectively the dispersed (letter ‘D’) and concentrated (letter ‘C’) methods, distribute the null character frequency in proportion to the standard background frequencies of all 20 amino acids, and in proportion to the observed frequencies of those amino acids present in the column in question. The dispersed method captures the idea that gaps in a given position should imply a degree of indifference as to which amino acids might occur there in related proteins, while the concentrated method involves no such assumption.

When added to FWS, the dispersed method yielded a small but
not statistically significant improvement in PSI-BLAST accuracy,
partially by better suppressing the appearance of false positives
at low *E*-values, and thereby allowing the *h* parameter
to be raised. Furthermore, with the dispersed method at *h* = 0.005,
the pseudocount parameter (see below) could vary over a wide range
of values without inducing corruption, whereas for the same pseudocount
parameter range some corruption occurred with the concentrated or
original method even at *h* = 0.002. The
ROC_{100} value for FWSDh0.005 was 0.884 ± 0.002,
in contrast to 0.878 ± 0.003 for FWSCh0.002
and 0.879 ± 0.003 for FWSh0.002 (Fig. (Fig.3).3). We adopted the dispersed method for further
tests.

## DISCORDANT SEQUENCE COMPOSITIONS AND RESTRICTED SCORE RESCALING

Sometimes when two sequences, or a sequence and a PSSM are compared,
the corresponding ungapped scale parameter λ′_{u} before score rescaling is greater than
the parameter λ_{u} for sequences
with standard amino acid composition. This is a rarer event than
the case of main concern above, in which λ′_{u} < λ_{u}, and the departure from the standard parameter
tends to be of much smaller magnitude (5).
Larger than standard values for λ′_{u} usually result from sequences with discordant
rather than similar amino compositions. If λ′_{u} > λ_{u}, using the standard statistical parameters,
without rescaling the substitution scores as described above, results
in an overestimate of *E*-values. Nevertheless, there
are several reasons it may be desirable not to rescale the substitution
scores in this case.

First, discordant compositions alone provide some evidence against
biological relatedness, so rescaling is more likely to yield a chance
high-scoring and now statistically significant false positive than
a newly statistically significant true positive. Secondly, the amino
acid composition of some proteins has a mosaic structure, which
effectively implies different λs appropriate
for the comparison of different regions. Scaling scores to account
for globally discordant compositions may yield exaggerated significance
estimates for local alignments involving regions with more typical
compositions. Thus, it may be desirable to rescale substitutions
scores only when λ′_{u} < λ_{u}, and
not when λ′_{u} > λ_{u}. We designate
such restricted rescaling with the letter ‘M’.

Cases where λ′_{u} > λ_{u} are relatively
rare (5), so including restricted
rescaling changed the ROC_{100} score only insignificantly,
from 0.8837 ± 0.0025 for FWSDh0.005
to 0.8845 ± 0.0021 for FWSDMh0.005.
We adopted this feature because it renders the program less susceptible
to producing spurious results in the cases discussed above.

We have slightly changed two of PSI-BLAST’s default parameters.
The program’s sequence weighting scheme (2,35) may somewhat overweight closely related
sequences. Thus the original version of PSI-BLAST purged from its
multiple alignment all but one copy of aligned sequence segments ≥98% identical, and this percentage
has been changed to 94, indicated by ‘p94’. Also,
the pseudocount parameter (2),
which balances multiple alignment data with prior knowledge of amino
acid substitutability, and originally set empirically to 10, has
been changed to 9, indicated by ‘b9’. The best
values for these parameters appear to change slightly from one version
of PSI-BLAST to another, and so we have attempted to optimize them
only as a final step. Combined with the other changes discussed
above, the refined parameters yield a ROC_{100} of 0.895 ± 0.003 for FWSDMb9p94h0.005 (Fig. (Fig.44).

## IDEAS THAT DID NOT WORK

We have discussed several modifications to PSI-BLAST that in combination significantly improve search accuracy. We also tested a substantial number of modifications that did not yield improved results, several of which we describe briefly here. Most of these modifications were motivated by appealing theoretical ideas, so it is possible that some refinement or combination of these approaches could still prove fruitful.

### Adaptive pseudocount parameters

The range of evolutionary divergence optimally detected by a given amino acid substitution matrix is perhaps best captured by the matrix’s relative entropy (20,36), and the concept of average relative entropy is easily extended to PSSMs that are constructed using log odds ratios. A scoring system whose relative entropy is too high will tend to detect only closely related sequences, whereas one whose relative entropy is too low is optimized for detecting relationships so distant that they are not distinguishable from chance in any case. As a result, most substitution matrices used in database searches are tuned to distant but still detectable evolutionary distances (16–18). PSI-BLAST’s pseudocount parameter affects the average relative entropy of the PSSMs the program produces, with a higher value of the parameter yielding lower average relative entropies. Thus it may be advisable to choose the pseudocount parameter adaptively, so as to produce PSSMs with a ‘target’ average relative entropy near that associated with the most effective amino acid substitution matrices (34). However, our implementation of adaptive pseudocounts degraded search accuracy. It may be possible to parse PSI-BLAST multiple alignments so as to determine those regions most important for detecting distant relationships, and base average relative entropy calculations on those regions alone.

### Sequence weight scaling

In calculating the ‘observed’ residue frequencies
for a given column of a multiple alignment, it is important to weight
the raw counts of amino acids appearing in the column to correct for
correlations among the sequences included in the alignment (2). Many sequence weighting schemes have
been proposed, of varying sophistication and ease of implementation
(35,37–46). PSI-BLAST implements a slight modification
(2) of the method of Henikoff
and Henikoff (35), because it
is simple and rapidly computable. However, this method has no strong mathematical
foundation, and the weights it produces may tend to either overweight
or underweight highly correlated sequences. We introduced an extra
exponent parameter into the weighting scheme, transforming the weights *w*_{1},*w*_{2}, *w*_{3}, … to *cw*_{1}^{P},*cw*_{2}P, *cw*_{3}P, …, where the
coefficient *c* is chosen so the new weights sum
to 1. As the exponent *p* was varied about 1, no
clear improvement in PSI-BLAST performance could be observed. We
measured the performance of exponents 0.9, 0.95, 1.05, 1.10, as
possible enhancement to FWSDMb9p94h0.005. The ROC_{100} scores
were 0.892, 0.892, 0.894, 0.890, all slightly lower than 0.895 for FWSDMb9p94h0.005.
It is an open problem to provide theoretical justification for why *p* = 1.0 should perform near optimally.

### Window-based amino acid composition calculations

Because amino acid composition may vary through a protein, one
may argue that an alignment’s statistical significance should
be assessed based upon the ‘local’ composition
near the sequence segments involved in the alignment. Accordingly,
we implemented a procedure whereby the amino acid compositions used
for rescaling scores for a particular alignment are drawn from the
sequence segments constituting the alignment, plus a window of fixed
length to each side of each segment, although shorter if the end
of the protein is encountered. This procedure was executed in lieu
of the restricted score rescaling discussed in the previous section.
A Bayesian variation involved assuming a prior standard background
amino acid composition (21),
which entered the amino acid frequency calculation in the form of
pseudocounts. For window lengths 50, 100 and 200, and using Bayesian
amino acid pseudocounts or not, we observed no improvement in accuracy
using an earlier implementation of composition-based statistics.
We remeasured the performance of windows of 100 and 200 without pseudocounts
as a modification to FWSDMb9p94h0.005. The ROC_{100} score
decreased significantly from 0.895 to 0.852 for window length 100
and to 0.866 for window length 200.

### Generalized affine gap costs

Most popular sequence alignment and database search programs
employ affine gap costs, which charge a fixed penalty for the existence
of a gap, and an additional penalty for each residue within a gap
(47–50).
Altschul (51) has suggested permitting
a gap to leave an arbitrary and in general different number of residues
in both sequences unaligned; the cost of a gap involving *x* and *y* residues in the two sequences, where *x* ≤ *y*, would be *a* + *b*(*y* – *x*) + *cx*. We implemented such generalized affine gap
costs for use with BLAST and PSI-BLAST, and explored a variety of
plausible settings for the gap cost parameters, but did not observe
any improvement in PSI-BLAST accuracy vis a vis traditional affine
gap costs.

## TESTING PSI-BLAST VERSIONS WITH AN INDEPENDENT QUERY SET

To confirm that the refinements to PSI-BLAST described above yield improved accuracy in detecting distant relationships, we tested several versions of PSI-BLAST on their ability to detect relationships within three diverse protein families whose members within the nr database of 2 February 2000 have been delineated. For each family we employed two separate queries, and we pooled the results from the six runs to generate ROC scores. Each query was compared to the nr database through five rounds of PSI-BLAST searching, and the search results returned on a final sixth round against nr were evaluated.

The three protein domain families employed in this test were: (i) DHH phosphoesterases (52), a family of predicted phosphoesterases with a broad spectrum of substrates; (ii) POZ domains (53,54), a family of eukaryotic and viral domains involved in specific protein–protein interactions; (iii) metallo-β-lactamase domain proteins (55), containing a metal-dependent hydrolase domain particularly abundant in archaea, but also present in a wide variety of bacterial and eukaryotic proteins. A list of family members, and the frozen nr database are available upon request. The query sequences used respectively to seek members of these families have gi numbers 4982166 (positions 6–297) and 2498554 (7–317); 482321 (50–169) and 2315751 (95–205); and 115023 (36–257) and 1172877 (536–778).

Compared to the development query set used for refining PSI-BLAST,
this set contained many fewer queries, but many more true positives
per query. The aggregate point of diminishing returns appeared to
occur near 50 false positives, so we used ROC_{50} scores
to compare versions of PSI-BLAST. Those versions we tested were
FWh10^{–6}, WSh0.0002, FWSh0.002, FWSDh0.005, and
FWSDMb9p94h0.005. The results appear in Table Table22 along with ROC_{100} scores from
our development query set. There is broad agreement on relative PSI-BLAST
version accuracy between the two sets of queries. For the development
set, the use of the dispersed method to score columns containing
gaps led to a statistically non-significant increase in accuracy,
while for the test set it led to a statistically non-significant
decrease. However, there is strong agreement that FWSDMb9p94h0.005
is far superior to the baseline version of PSI-BLAST with which
we began.

## THE ACCURACY OF BLAST DATABASE SEARCHES

Most of the refinements discussed above concern the construction
of PSSMs, and are therefore applicable to PSI-BLAST but not to simple
BLAST searches. The exceptions are composition-based statistics,
and restricted score rescaling. To test whether these refinements
led to improved BLAST as well as PSI-BLAST accuracy, we compared
both our development and test query sets directly to their respective
target databases using the programs FW, FWS and FWSM. For the development
set, the ROC_{100} scores were 0.529 ± 0.003,
0.522 ± 0.003 and 0.525 ± 0.003,
respectively. For the test set, the ROC_{50} scores were 0.118 ± 0.002, 0.112 ± 0.002
and 0.113 ± 0.003, respectively.

Composition-based statistics do not appear to improve BLAST
search accuracy, even when restricted score rescaling is also incorporated.
One reason for the discrepancy in these results with those for PSI-BLAST
is that a rare unrelated database sequence with an anomalously low *E*-value has only a minor effect on the BLAST ROC
score, but by corrupting the resulting PSSM it can have a major
negative effect on that for PSI-BLAST. Some users may find it desirable
to use composition-based statistics and restricted score rescaling
even for BLAST, to avoid rare spuriously low *E*-values.
These changes have the disadvantage of frequently returning different
scores for identical alignments, due to the varying compositions
of database sequences, and of rarely returning scores that correspond
precisely to those implied by the standard, unscaled substitution
matrix used. For these reasons, we recommend the changes described
above primarily in the context of PSI-BLAST.

We note that using our testing protocol, the ROC_{100} for
the development set improves from ~0.53
using one round of searching with BLAST to 0.89 using six rounds
(five versus nr and one versus yeast) of searching with PSI-BLAST.
This reinforces the conclusions of previous studies (56–60) that most biologists who still commonly
use only BLAST for protein queries would benefit substantially by
switching to PSI-BLAST.

## THE PRECISION OF BLAST AND PSI-BLAST
*E*-VALUES

The ROC method we have used to compare various versions of BLAST
and PSI-BLAST relies on reported *E*-values only
to sort into one list the matches for different queries. The accuracy
of *E*-values in describing statistical significance
is relevant within the program only for setting the *h* parameter, whose
values have in any case been chosen empirically. However, a user
of BLAST or PSI-BLAST may wish to rely upon reported *E*-values
in evaluating the potential relevance of similarities found. Accordingly,
we test here how well the *E*-values returned
by these programs reflect the distribution of scores produced by
chance.

To obtain a large number of relatively independent query sequences,
we selected 1000 *Escherichia coli* proteins from GenBank
(8) with GI numbers ending in
1, 2 or 3. For BLAST, we compared these sequences to a ‘shuffled’ version
of our yeast database, in which the letters of each sequence were randomly
permuted. For PSI-BLAST, we compared each query to the unscrambled
nr database through five iterations or until convergence, and then
compared the resulting PSSM to the scrambled yeast database. Because
we were unable to assess theoretically the magnitude of the effect
restricted score rescaling has on random score distributions, we
studied versions of BLAST and PSI-BLAST that include and exclude this
feature.

Specifically, we tested versions of FWSDh0.005 and FWSDMb9p94h0.005
for PSI-BLAST, and the corresponding versions FWS and FWSM for BLAST.
For the 1000 query sequences, we recorded the total number or alignments returned
with *E*-value ≤ 1.0,
0.1 and 0.01, which by theory are expected to be near 1000, 100
and 10 respectively; the results are shown in Table Table3.3. In all cases, the number of alignments observed
at various *E*-values are within a factor of two
of the number predicted. The reported *E*-values
for BLAST appear slightly high (i.e. conservative), while those
for the version of PSI-BLAST without restricted score rescaling
appear slightly low.

## IMPLEMENTATION

We use the terms BLAST and PSI-BLAST throughout to refer to versions of the command-line program more precisely called blastpgp. There is also a Web-based version (http://www.ncbi.nlm.nih.gov/BLAST/, follow hyperlinks there), which differs primarily in how the options are set, and in that the user can control which matching sequences get used in constructing the PSI-BLAST PSSM. Since it is impossible to synchronize the public releases of the command-line blastpgp (because it is part of the larger NCBI software toolkit), the Web version and papers, we summarize how the modifications described herein are encapsulated in different numbered versions of the released code.

The improved statistical parameters λ and *K* were first made available in version 2.1.3. The
new ‘edge-effect’ parameters (3)
accounting for the fact that alignments are unlikely to begin near
the ends of sequences, are in version 2.2.1. The improved precision
and use of proper amino acid pair frequency ratios started in version
2.1.1. Version 2.1.1 made available Smith–Waterman alignments
(W) as a command-line option only, but disallowed its use on the
Web page because of worst-case running time. We evaluated the effect
on search accuracy of turning off W on versions of PSI-BLAST. The
resulting ROC_{100} scores for our development query set
improve from 0.780 ± 0.005 for Fh10^{–6} to
0.884 ± 0.004 for FSDMb9p94h0.005.

Version 2.1.1 first made available an option that combines filtering
of database sequences (F), a preliminary implementation of composition-based
statistics (S) and restricted score rescaling (M), under the combined
name ‘composition-based statistics’. Version 2.2.1
includes a better implementation of S. Version 2.1.1 and beyond
implement the dispersed method for inferring frequencies from gaps
(D), but it cannot be turned off. No released version implements
reversed sequence score normalization (R). Among the three numerical
parameters, the user can control the settings of the pseudocount
parameter (b) and the *E*-value threshold for inclusion
in a PSI-BLAST multiple alignment (h), but not the purging percentage
(p). Versions 2.0.* used b10p98h0.001 by default. Version
2.1.1 changed the default to b7p98h0.002, which is conservative, since
we showed that one can safely raise *h* to 0.005.
Version 2.2.1 changes the default again to b9p94h0.005.

## DISCUSSION AND CONCLUSION

PSI-BLAST is a widely-used extension to BLAST that permits iterative
searching, and is particularly good at finding distant relationships.
There have been several evaluations of PSI-BLAST accuracy published
by other groups (56–60) which show there was substantial room
for improvement in the accuracy of versions 2.0.*. A particularly
frustrating limitation to using PSI-BLAST for large-scale automated
protein analysis was that on a small, but certainly not negligible percentage
of queries, false positives could enter the list of matches at one
iteration with an *E*-value low enough to corrupt the
PSSMs constructed for searching in subsequent iterations. In some
cases the corruption got so bad that the program would exhaust all
virtual memory and crash, or produce tens of megabytes of worthless
output.

We have described a long list of modifications that were implemented in a large-scale attempt to improve PSI-BLAST accuracy in protein database searching. Of course, there are more ideas left to test, especially involving methods for generating the multiple alignments used to construct the PSI-BLAST PSSMs. However, the improvements described herein are substantial enough that it seemed desirable to suspend the testing of new ideas long enough to quantify these improvements and make them publicly available. The modifications that improved performance are retained in PSI-BLAST version 2.2.1. Most of these were introduced, at least in preliminary implementations in version 2.1.1. Version 2.1.1 appears to have nearly eliminated the corruption problem in PSI-BLAST.

The change that yielded the majority of the improvement is the use of ‘composition-based’ statistics to re-evaluate candidate alignments. Composition-based statistics take into account the letter frequencies in database sequences, and adjust the scale of the query PSSM accordingly. The use of composition-based statistics represents a more careful interpretation of the statistical theory behind BLAST (19,27), initially assessing the significance of a local alignment between two sequences, and then extending the results to database searching. The original BLAST implementation of that extension codified the questionable assumption that all database sequences should be treated as if they have the same average letter frequencies. This assumption is dangerous in the iterative context of PSI-BLAST, where allowing false positives to enter the set of matches used to construct a PSSM can lead to corruption. We implemented composition-based statistics efficiently by using them only to re-evaluate candidate matches identified using the average composition assumption.

Incorporating composition-based statistics substantially improved the accuracy of PSI-BLAST searches, primarily by decreasing the retrieval of false positives, and thereby suppressing the corruption of constructed PSSMs. This modification of PSI-BLAST is most important for large-scale searches, for example during genome annotation, and for all searches with compositionally-biased queries. However, our results are averaged over many cases, and for individual queries the use of composition-based statistics, or indeed any of the other refinements we have introduced, may degrade performance.

We measured the performance of our modifications on a set of
103 query sequences, with lists of true positives in yeast curated
by human experts. On this set the ROC_{100} score improved
from 0.758 ± 0.005 to 0.895 ± 0.003,
even without accounting for four improvements implemented in the
baseline version. Nevertheless the gap between our current best
ROC_{100} score of 0.895 and the best possible score of
1.0 is substantial. More research is needed to identify new algorithmic
and/or statistical refinements to narrow this gap further.

Our testing protocol involved saving a checkpoint PSSM after
five PSI-BLAST rounds versus the nr database, and then comparing
that checkpoint to an annotated yeast database. The choice of five
rounds versus nr was based on early experiments which suggested
that five initial rounds of searching produced results comparable
to 10. We performed a post facto evaluation of this choice by saving
the PSSM generated by PSI-BLAST version FWSDMb9p94h0.005 after one
to 10 rounds versus nr. The resulting ROC_{100} scores were:
0.754, 0.829, 0.861, 0.886, 0.895, 0.896, 0.894, 0.893, 0.891 and
0.895. From round 5 through round 10, the scores move slightly up
and down, but there is no corruption with any query in any round.
We also recorded the number of queries that converged after each
round versus nr, meaning that no new matches were found with an *E*-value ≤ *h* = 0.005. The numbers of converged queries
out of 103 were: 2, 19, 27, 43, 55, 60, 67, 72, 75 and 78. These
experiments suggest that for large-scale, automated applications, running
PSI-BLAST for five to six rounds (which corresponds to saving the
checkpoint computed after four to five rounds) may reveal most of
the matches that could be found by running until convergence.

## APPENDIX A

### Estimating the standard deviation of ROC scores

Fix a query sequence and consider any database search method that
ranks the relatedness of database sequences to the query (e.g.,
by using *E*-values). For simplicity, classify each
database sequence as either a true positive or a false positive
with respect to the query. Let *i* = 1,2,3 … index
the rank of the false positives, and let *t** _{i}* be
the number of true positives ranked ahead of the

*i*th false positive. For the Appendix, we assume for simplicity that there are no ties in rank, although ties can be accounted for easily in the estimates below, and were accounted for in the ROC scores and standard deviations reported above. The accuracy of the search method can be assessed by the ‘receiver operator characteristic’ for

*n*false positives, defined as:

**4**

Here *T* is total number of true positives in
the database, and we used either *n* = 50
or 100 in the results reported. The sum alone in equation **4** we
denote

Given two search methods, resampling with a bootstrap (described
in detail below) can assign an empirical *p*-value
to the difference between their corresponding ROC* _{n}* values.
For each bootstrap sample, calculate the difference between the sample
ROC

*’s and then note the position of the actual difference within the bootstrap distribution. This appendix announces some simple analytic results that can be used to decide whether the difference between two ROC*

_{n}*values is statistically significant under bootstrap resampling. The mathematical details will be published elsewhere (Czabarka and Spouge, manuscript in preparation).*

_{n} Two bootstrap resampling schemes could be used: either resample
all sequences or resample only from the false positives, leaving
true positives fixed. Usually, the set of true positives is well
characterized, so the false positives generate the real ‘noise’ in
the ROC_{n} measurement. When
only the false positive sequences are resampled, the analytic results
simplify because the denominator (*nT*) in equation **4** remains constant. The following presents results
on resampling only the false positive sequences.

Let *F* be the total number of false positives
in the database. A bootstrap sample consists of *F* false
positives sampled uniformly, independently and with replacement
from the entire set of false positives. Let *F** _{i}* be the (random) number of times that
the false positive of retrieval rank

*i*is resampled. Define the random variable

*N*

*to be the smallest integer satisfying*

_{n}

** ** **5**

i.e., the false positive ranked *n* in the resampling
has original rank *N*_{n}.

The false positive sequences contributing to ROC_{n},
for any reasonable choice of *n*, form a very small
proportion of the database. Thus, the distribution of the ROC_{n} under resampling can be approximated
by taking each false positive in its rank order and sampling it *F*_{i} times,
where *F*_{i} is
chosen from a Poisson distribution with mean 1. One can stop when
the total of the Poisson sample counts reaches *n*.

The equivalence of an experiment resampling false positives and
an experiment sampling from a Poisson distribution implies that
many important quantities concerning ROC_{n} can be
approximated analytically. For example, by definition, the resampled,
non-normalized ROC_{n} is:

** 6**

The mean µ(*R** _{n}*)
and the variance σ

^{2}(

*R*

_{n}) have analytic forms that can be practically computed under all conditions (although they are not shown here because they are too complex as formulas). Mathematical theorems show that under typical conditions µ(

*R*

_{n}) ≈ Σ

_{1 ≤ i ≤ n}

*t*

_{i}=

and σ^{2}(*R*_{n}) ≈ Σ_{1 ≤ i ≤ n} (*t*_{n+1} – *t*_{i})^{2}. Moreover,
the distribution of *R*_{n} is
close to normal, so the approximate standard deviation can be used
to provide an approximate *p*-value.

These distributional results can be extended to differences between
resampled ROC_{n}’s,
which is of use when comparing a pair of retrieval methods. Let
us use the prime symbol (′) to denote
quantities pertaining to a second search method. Then:

and

**7**

The summation condition *D*_{i} = *D*′_{j} indicates
a sum over sequences common to the first *n* false
positives in both retrieval lists. The notation expresses the condition
that false positive *D*_{i} in
the first list is the same as false positive *D*′_{j} in
the second list.

The comparisons of experimental data above use the estimate σ^{2}(*R** _{n}* –

*R*′

_{n}) ≈ σ

^{2}(

*R*

_{n}) + σ

^{2}(

*R*′

_{n}) and ignore the final correlation term in equation

**7**, so that we can assign a standard deviation to each version rather than each pair. The omission of the third term, which is subtracted, leads to an overestimate of the standard deviation.

## APPENDIX B

### Matrix rescaling

The method of rescaling matrices can be be summarized as follows.
Let λ* _{u}* be
the ungapped scale parameter (19)
for the reference substitution matrix (e.g., BLOSUM62) in the context of
standard amino acid frequencies (21).
Suppose that

*Q*is the query sequence and

*D*

_{init}is the initial matching database sequence.

(i) Multiply the gap costs by a scaling factor *f*,
and divide λ_{u} by *f*.
We currently set *f* = 32 for five extra
bits of precision.

(ii) Compute the residue frequencies in *Q*.

(iii) If filtering (modification F) is turned on, let *D* be
the output of filtering *D*_{init} with SEG
(currently using parameters 10, 1.8, 2.1). If F is off, let *D* = *D*_{init}. Compute the residue frequencies
for *D*, ignoring all segments that were replaced with
X’s by SEG.

(iv) Given frequency ratios *R** _{ij}* for
the current score matrix, compute scaled-up scores

*S*

_{ij}as the nearest integer to log(

*R*

_{ij})/λ

_{u}. Since λ

_{u}has been previously divided by

*f*, this will have the effect of multiplying each score by roughly

*f*.

(v) Using the residue frequencies for *Q* and *D* and the scaled up matrix scores *S*_{ij} compute a match-specific λ′_{u} (19).

(vi) Let the ratio *r* = λ′_{u}/λ_{u}. If restricted score rescaling (modification
M) is turned on, change *r* to min(*r*,1).

(vii) For each position *i*, *j* in
the matrix, compute the rescaled score *S*′_{ij} as the nearest integer to *r* · log(*R*_{ij})/λ_{u}.

Since the resulting rescaled score matrix *S*′ and gap costs are both scaled up by
a factor of *f*, we divide the final raw alignment
scores by *f* before printing the output. Note that
at steps 4 and 7, the scaling is done in floating point first, and
the nearest integer score is computed at the end.

## ACKNOWLEDGEMENTS

Thanks to Drs Jorja and Steven Henikoff for providing the frequency ratios used to derive the BLOSUM score matrices. Thanks to two referees for some very helpful suggestions.

## References

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*Atlas of Protein Sequence and Structure*. National Biomedical Research Foundation, Washington, DC, Vol. 5, suppl. 3, pp. 345–352.

*Atlas of Protein Sequence and Structure*. National Biomedical Research Foundation, Washington, DC, Vol. 5, suppl. 3, pp. 353–358.

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