Fast probabilistic analysis of sequence function using scoring matrices
We present techniques for increasing the speed of sequence analysis using scoring matrices. Our techniques are based on calculating, for a given scoring matrix, the quantile function, which assigns a probability, or p, value to each segmental score. Our techniques also permit the user to specify a p threshold to indicate the desired trade-off between sensitivity and speed for a particular sequence analysis. The resulting increase in speed should allow scoring matrices to be used more widely in large-scale sequencing and annotation projects.



We develop three techniques for increasing the speed of sequence analysis: probability filtering, lookahead scoring, and permuted lookahead scoring. In probability filtering, we compute the score threshold that corresponds to the user-specified p threshold. We use the score threshold to limit the number of segments that are retained in the search process. In lookahead scoring, we test intermediate scores to determine whether they will possibly exceed the score threshold. In permuted lookahead scoring, we score each segment in a particular order designed to maximize the likelihood of early termination. Our two lookahead scoring techniques reduce substantially the number of residues that must be examined. The fraction of residues examined ranges from 62 to 6%, depending on the p threshold chosen by the user. These techniques permit sequence analysis with scoring matrices at speeds that are several times faster than existing programs. On a database of 12 177 alignment blocks, our techniques permit sequence analysis at a speed of 225 residues/s for a p threshold of 10-6, and 541 residues/s for a p threshold of 10-20. In order to compute the quantile function, we may use either an independence assumption or a Markov assumption. We measure the effect of first- and second-order Markov assumptions and find that they tend to raise the p value of segments, when compared with the independence assumption, by average ratios of 1.30 and 1.69, respectively. We also compare our technique with the empirical 99. 5th percentile scores compiled in the BLOCKSPLUS database, and find that they correspond on average to a p value of 1.5 x 10-5.


AVAILABILITY
The techniques described above are implemented in a software package called EMATRIX. This package is available from the authors for free academic use or for licensed commercial use. The EMATRIX set of programs is also available on the Internet at http://motif.stanford.edu/ematrix.

title: Discovering Empirically Conserved Amino Acid Substitution Groups in Databases of Protein Families
abstract: This paper introduces a method for identifying empirically conserved amino acid substitution groups. In contrast with existing approaches that view amino acid substitution as a pairwise phenomenon, the method presented here identifies conserved groups of amino acids using a data structure called a conditional distribution matrix. The conditional distribution matrix extends the concept of a pairwise substitution matrix by changing the context of substitution from a single amino acid to a group of amino acids. The matrix tabulates information from a database of protein families that contains numerous aligned positions. Each row in the matrix contains the distribution of amino acids in those aligned positions that contain a given conditioning group of amino acids. The method converts a database of protein families into a conditional distribution matrix and then examines each possible substitution group for evidence of conservation. The algorithm is applied to the BLOCKS and HSSP databases. Twenty amino acid substitution groups are found to be conserved empirically in both databases. These groups provide insight into biochemical properties that are conserved in protein evolution.

title: Minimal-Risk Scoring Matrices for Sequence Analysis
abstract: We introduce a minimal-risk method for estimating the frequencies of amino acids at conserved positions in a protein family. Our method, called minimal-risk estimation, finds the optimal weighting between a set of observed amino acid counts and a set of pseudofrequencies, which represent prior information about the frequencies. We compute the optimal weighting by minimizing the expected distance between the estimated frequencies and the true population frequencies, measured by either a squared-error or a relative-entropy metric. Our method accounts for the source of the pseudofrequencies, which arise either from the background distribution of amino acids or from applying a substitution matrix to the observed data. Our frequency estimates therefore depend on the size and composition of the observed data as well as the source of the pseudofrequencies. We convert our frequency estimates into minimal-risk scoring matrices for sequence analysis. A large-scale cross-validation study, involving 48 variants of seven methods, shows that the best performing method is minimal-risk estimation using the squared-error metric. Our method is implemented in the package EMATRIX, which is available on the Internet at http://motif.stanford.edu/ematrix.

title: Kabat Database and its applications: 30 years after the first variability plot
abstract: The Kabat Database was initially started in 1970 to determine the combining site of antibodies based on the available amino acid sequences at that time. Bence Jones proteins, mostly from human, were aligned, using the now-known Kabat numbering system, and a quantitative measure, variability, was calculated for every position. Three peaks, at positions 24-34, 50-56 and 89-97, were identified and proposed to form the complementarity determining regions (CDR) of light chains. Subsequently, antibody heavy chain amino acid sequences were also aligned using a different numbering system, since the locations of their CDRs (31-35B, 50-65 and 95-102) are different from those of the light chains. CDRL1 starts right after the first invariant Cys 23 of light chains, while CDRH1 is eight amino acid residues away from the first invariant Cys 22 of heavy chains. During the past 30 years, the Kabat database has grown to include nucleotide sequences, sequences of T cell receptors for antigens (TCR), major histocompatibility complex (MHC) class I and II molecules and other proteins of immunological interest. It has been used extensively by immunologists to derive useful structural and functional information from the primary sequences of these proteins. An overall view of the Kabat Database and its various applications are summarized here. The Kabat Database is freely available at