- Technical Note
- Open Access
Protein sequence alignment with family-specific amino acid similarity matrices
© Kuznetsov et al; licensee BioMed Central Ltd. 2011
- Received: 9 June 2011
- Accepted: 16 August 2011
- Published: 16 August 2011
Alignment of amino acid sequences by means of dynamic programming is a cornerstone sequence comparison method. The quality of alignments produced by dynamic programming critically depends on the choice of the alignment scoring function. Therefore, for a specific alignment problem one needs a way of selecting the best performing scoring function. This work is focused on the issue of finding optimized protein family- and fold-specific scoring functions for global similarity matrix-based sequence alignment.
I utilize a comprehensive set of reference alignments obtained from structural superposition of homologous and analogous proteins to design a quantitative statistical framework for evaluating the performance of alignment scoring functions in global pairwise sequence alignment. This framework is applied to study how existing general-purpose amino acid similarity matrices perform on individual protein families and structural folds, and to compare them to family-specific and fold-specific matrices derived in this work. I describe an adaptive alignment procedure that automatically selects an appropriate similarity matrix and optimized gap penalties based on the properties of the sequences being aligned.
The results of this work indicate that using family-specific similarity matrices significantly improves the quality of the alignment of homologous sequences over the traditional sequence alignment based on a single general-purpose similarity matrix. However, using fold-specific similarity matrices can only marginally improve sequence alignment of proteins that share the same structural fold but do not share a common evolutionary origin. The family-specific matrices derived in this work and the optimized gap penalties are available at http://taurus.crc.albany.edu/fsm.
- Similarity Matrice
- Amino Acid Pair
- Reference Alignment
- Pairwise Sequence Alignment
- Structural Superposition
Pairwise alignment of amino acid sequences is a cornerstone sequence comparison method used in a variety of computational applications [1–4]. A mathematically rigorous and computationally efficient way of finding optimal global  and local  alignments for a given pair of sequences is provided by dynamic programming. The outcome of a dynamic programming procedure applied to align amino acid sequences critically depends on the alignment scoring function used by this procedure [7, 8]. Therefore, for a specific alignment problem one needs a way of selecting the best performing scoring function [7–9]. The traditional alignment scoring function most commonly used in dynamic programming consists of an amino acid substitution matrix and gap penalties [5, 6]. Recently, several novel sequence alignment algorithms have been developed that use scoring functions based on Hidden Markov Models (HMMs) [9–15]. On one hand, the probabilistic nature of these algorithms makes them superior to the substitution matrix-based alignment. On the other hand, they require estimation of a large number of transition and emission probabilities, which makes obtaining reliable HMMs especially tricky in the case of global alignments and small sequence datasets [16, 17]. Despite the advent of more sophisticated alignment algorithms, global pairwise alignment based on amino acid substitution matrices still remains the well-established workhorse of sequence analysis. It is widely used for basic sequence comparison tasks that include the identification of structurally equivalent positions in homology modeling, which crucially depends on the quality of alignment between the target and template sequences , and in popular multiple sequence alignment algorithms, such as CLUSTAL  and TCOFFEE . Therefore, improving the quality of substitution matrix-based global pairwise alignments is an important step in improving other more complex computational applications.
In substitution matrix-based alignment, matrix selection is the most important decision the user has to make because once a matrix is selected the values of its elements cannot be easily changed, unlike gap penalties. Most amino acid substitution matrices are similarity matrices based on the same underlying idea: they attempt to account for the similarity between two amino acids by computing how often these amino acids occur in the equivalent sequence positions in related proteins. It is assumed that if two amino acids i and j are often observed in the equivalent positions, they have similar biochemical properties and can be substituted one for another in the course of protein evolution. The differences between various similarity matrices are mainly determined by what groups of protein sequences are used to derive them and how the equivalent positions are defined. The mainstream matrices routinely utilized in sequence comparison, such as BLOSUM  and PAM , are obtained by counting the frequencies of amino acid substitutions observed in the columns of multiple sequence alignments of evolutionary related proteins. As an alternative to using multiple sequence alignments, one can count the frequencies of amino acid substitutions observed in the structurally equivalent positions of structurally similar (but not necessarily sequence-similar) proteins [22–24]. In this case, the structurally equivalent positions are identified by means of a computational technique known as structural superposition. Structural superposition detects the structurally-equivalent regions in two protein structures by using their geometrical properties only [25–28]. The amino acid sequence alignment that corresponds to the optimal structural superposition is generated after the superposition is complete.
The advantage of structural superposition is that it allows one to obtain high quality reference sequence alignments for both distantly related homologous proteins and for proteins that share the same structural fold but no common evolutionary origin. Such structure-based reference alignments can be used not only to derive amino acid similarity matrices but also to design benchmarks for assessing the performance of sequence alignment algorithms [29–33]. In a sequence alignment benchmark, the quality of sequence alignments produced by a given alignment algorithm is assessed by comparing them to the reference alignments and calculating the percentage of correctly aligned positions. Typically, the average percentage of correctly aligned positions is used to compare two or more alignment algorithms and the algorithm with the highest average is identified as the top-performing one. One problem with such an approach based on a simple comparison of the averages is that it lacks statistical testing and a larger average can come at the expense of a larger variability, especially in smaller datasets, thus not really reflecting a statistically significant difference. Another common problem with benchmarking sequence alignment algorithms is that it is usually performed on a large pooled sequence dataset that contains many diverse protein families. The use of pooled sequences leads to the loss of family-specific information and may bias the results towards the overrepresented protein families .
Virtually all existing amino acid similarity matrices are general-purpose matrices, meaning that they were derived by averaging substitution frequencies over many diverse protein families that represent the entire protein universe. General-purpose similarity matrices are required for such task as a sequence database search because in this procedure a query sequence is aligned with millions of diverse database sequences. However, in tasks such as a global sequence alignment only related sequences are typically used and it is often known in advance which protein family these sequences belong to. In such a case, even the best general-purpose matrix may not perform equally well on all protein families, because family-specific substitution patterns were mostly averaged out. For instance, it has been shown that amino acid similarity matrix derived for a specific protein family tends to perform better on proteins from this family than BLOSUM matrices [34, 35]. The recent advances in the experimental data acquisition have provided a wealth of sequence and structural data that allow us to obtain reference sequence alignments for many diverse protein families and structural folds . These reference alignments can be used to derive protein family- and fold-specific similarity matrices and to conduct family- and fold-specific performance evaluations.
In this work, I focus on the issue of finding optimized amino acid similarity matrices for pairwise global sequence alignment. I use a comprehensive set of reference sequence alignments to design a quantitative statistical framework for evaluating the performance of alignment scoring functions on protein family and structural fold levels and apply this framework to study the utility of family- and fold-specific amino acid similarity matrices for global sequence alignment. The results of this work indicate that the quality of pairwise global sequence alignment can be significantly improved by using family-specific similarity matrices.
I used the release 1.65 of the Sequence Alignment Benchmark (SABmark)  to derive protein family-specific and fold-specific amino acid similarity matrices and to evaluate the accuracy of pairwise sequence alignments obtained using different scoring functions. SABmark is a large general-purpose "gold standard" database specifically designed to evaluate the performance of pairwise sequence alignment algorithms. It consists of groups of the reference pairwise sequence alignments obtained from the consensus structural superposition of high-quality protein structures that cover the entire SCOP database . All sequences in SABmark are divided into two sub-sets, "the Superfamily" sub-set (SUP) and "the Twilight Zone"  sub-set (TWI). Each SUP group contains homologous single-domain protein sequences with low to moderate degrees of sequence identity that belong to the same SCOP super-family. Each TWI group contains single-domain protein sequences that belong to the same SCOP fold and share no detectable sequence similarities (meaning that no significant similarities can be detected by BLAST).
Derivation of group-specific amino acid similarity matrices
Where q k (i, j) and e k (i, j) are the observed and expected frequencies of amino acid pair (i, j) in group k; f k (i, j) is the total number of amino acid pairs (i, j) in group k; p k (j) is the observed frequency of amino acid type j in group k; n k is the total number of amino acid pairs in group k; v(i, j) is the score for amino acid pair (i, j) from the VTML200 matrix . The value of the constant, 8000, in Eq.5 was selected using a grid search as a value that results in the best overall performance of group-specific matrices.
In the final matrix, all elements s k (i, j) are rounded to the nearest integer. In order to account for sparse data in small SABmark groups, each element of the final matrix S k is calculated as the weighted combination of the group-specific score and the general score from the VTML200 matrix. As the total number of aligned amino acid pairs increases, the contribution of the group-specific score increases, whereas the contribution of the VTML200 score decreases. The VTML200 matrix is used in Eq.1 because it is the best performing general-purpose matrix (see 'Analysis of general-purpose matrices' for details).
Matrix evaluation procedure
Where n I (i, j) is the number of residue pairs aligned identically in the test and the reference alignments; l R (i, j) is the length of the reference alignment; l T (i, j) is the length of the test alignment. If the value of Q(i, j) equals 100%, it means that the test alignment and the reference alignment are identical. The test pairwise sequence alignments were obtained by using the Needlman-Wunsch algorithm  with the affine gap penalty function, G(m) = α + (m-1)*β, where m is the gap length, α is the gap initiation penalty, and β is the gap extension penalty. Gap penalties were optimized for each group/matrix as described below.
The performance of two given amino acid similarity matrices, A and B, in the pairwise alignment of sequences from a SABmark group k is analyzed by means of the two-sided paired t-test. In this approach, the pairs of sequences from group k are aligned using matrix A and the alignment quality score, Q A , is calculated for each pair using Eq.6. Then, the same sequence pairs from group k are aligned using matrix B and the alignment quality score, Q B , is calculated for each pair. After that, the distributions of alignment quality scores Q A and Q B are compared using the two-sided paired t-test. The outcome of the t-test is considered to be statistically significant if its p-value is less than 0.05. To avoid over-fitting, the alignment quality scores are obtained using a 3-fold cross-validation. In this procedure, each SABmark group is randomly partitioned into three non-overlapping sub-sets, with two sub-sets used for training and the remaining sub-set used for testing. The process is repeated three times, so that each sub-set is used for testing once. In the case of a general-purpose similarity matrix, "training" means optimizing gap penalties for this matrix. In the case of a similarity matrix specific for a SABmark group k, "training" means deriving the group-specific matrix itself and optimizing gap penalties for this matrix. "Testing" means using the matrix and optimized gap penalties obtained during the training step to align sequences from the test sub-set. During cross-validation, gap initiation, α, and gap extension, β, penalties for a given group k and a given similarity matrix A are optimized using the following grid search procedure: Sequences from the training set of group k are aligned using matrix A, and all possible combinations of integer gap penalties in range 1≤α≤50, 1≤β≤30 are tested; the combination (α, β) that results in the highest average quality score is selected as the best and is used with matrix A to align sequences from the test set of group k.
Analysis of general-purpose matrices
The pair-wise comparison of general-purpose matrices on groups from the SUP sub-set.
The pair-wise comparison of general-purpose matrices on groups from the TWI sub-set.
Analysis of family- and fold-specific matrices
The pair-wise comparison of group-specific matrices vs. general-purpose matrices.
Where SB k (i, j) is the score for amino acid pair (i, j) from the kth family-specific matrix that performs better than VTML200; SS k (i, j) is the score from the kth family-specific matrix that performs similarly to VTML200; NB is the number of matrices that perform better than VTML200; NS is the number of matrices that perform similarly to VTML200. The family-specific matrices that perform significantly worse than VTML200 were not used because there were only 5 of them. To remove the cases that show a trend toward statistical significance, matrices that perform similarly to VTML200 were defined as the ones that received the p-value greater than 0.1 during the matrix evaluation procedure (156 matrices total).
Adaptive protein family-specific alignment
The analysis of the performance of the group-specific matrices described above is based on the knowledge of the group membership for each sequence used for testing, meaning that a correct group-specific matrix is always selected for each alignment by default. However, in the real world we do not always know in advance what particular family a given pair of homologous sequences belongs to. How to select an appropriate family-specific matrix in this case? I used BLAST  sequence similarity search to design an adaptive alignment procedure that automatically selects an appropriate family-specific similarity matrix for aligning a pair of input sequences. In this approach, a given input sequence is used as a query and the entire SUP subset (except for the sequences used for testing) is used as a sequence database to run a BLASTP search. If this search returns one or more hits with E-value less than 0.05, then the family membership of the database sequence with the smallest E-value is assigned to the input sequence. For a pair of input sequences two independent BLASTP searches are run, one for each sequence, and a similarity matrix for aligning this pair is selected using the following rules. If both sequences are assigned to the same family k, then the family-specific matrix k and the optimized gap penalties for this matrix are used to align these sequences. If the sequences are assigned to different families or if at least one sequence is not assigned to any family because of the absence of significant hits, then VTML200 (the best performing general-purpose matrix) with the gap initiation penalty of -15 and the gap extension penalty of -1 is used to align these sequences. These default gap penalties were determined by the grid search procedure as the best-performing general-purpose combination for VTML200 - among all combinations tested, (-15,-1) results in the highest average alignment quality score calculated for the dataset that contains all SUP groups pooled together.
The results of the assessment of the performance of the adaptive alignment procedure on the SUP sub-set, as determined by the 3-fold cross-validation, are shown in Table 3. The procedure was not tested on the TWI sub-set because sequences in this sub-set do not share similarities that can be detected by BLASTP. Overall, the adaptive procedure provides a correct family assignment for 53.68%, no assignment for 46.31%, and incorrect assignment for 0.01% of all sequence pairs. One potential direction for further improving the matrix selection procedure is to use a sequence comparison program which is more sensitive and specific than BLASTP. Nevertheless, even despite the observed incomplete family assignments, the adaptive alignment procedure significantly outperforms general-purpose matrices. For instance, the adaptive procedure significantly outperforms VTML200 on 35 (14.3%) families and BLOSUM50 on 50 (20.5%) families. Thus, it can be used for a completely automated alignment of homologous protein sequences using the proposed family-specific similarity matrices in the absence of a prior knowledge of what specific protein family these sequences belong to.
1. I utilized a large set of reference alignments obtained from the structural superposition of homologous and analogous proteins to design a quantitative statistical framework for the comparative evaluation of the performance of alignment scoring functions in the pairwise global sequence alignment. This framework was applied to study how the existing general-purpose amino acid similarity matrices perform on individual protein families and structural folds and to compare them to the family-specific and fold-specific similarity matrices derived in this work.
2. Among all the general-purpose matrices tested, VTML200 is the best-performing one. It produces global pair-wise sequence alignments most similar to the reference alignments derived from structural superposition. However, the results of this work suggest that the performance of each general-purpose matrix may be significantly biased towards a certain set of protein families. Even the top-performing general-purpose matrix cannot universally outperform other matrices on all protein families and all folds.
3. Using protein family-specific similarity matrices and optimized gap penalties can significantly improve the quality of alignment of homologous sequences compared to the traditional sequence alignment based on a single general-purpose similarity matrix. However, using fold-specific matrices can only marginally improve sequence alignment of proteins that share the same structural fold but do not share a well-defined common evolutionary origin.
4. I presented an adaptive alignment procedure that automatically selects an appropriate amino acid similarity matrix and optimized gap penalties based on the properties of the sequences being aligned. This procedure does not require a manual assignment of evolutionary relationships and can be used for an automated optimized alignment of homologous protein sequences in the absence of a prior knowledge of what protein family these sequences belong to. The current limitation of the presented adaptive alignment procedure is that it is directly applicable only to single-domain proteins or to individual domains from multi-domain proteins. This limitation will be addressed in the future.
The protein family-specific amino acid similarity matrices derived in this work and the associated optimized gap penalties, along with the optimized family-specific gap penalties for the VTML200 and BLOSUM50 matrices, are available on-line at http://taurus.crc.albany.edu/fsm.
Acknowledgements and Funding
This work was supported by grant R15GM090228 from the National Institute of General Medical Sciences.
- Pearson WR, Lipman DJ: Improved tools for biological sequence comparison. Proc Natl Acad Sci USA. 1988, 85: 2444-2448. 10.1073/pnas.85.8.2444.PubMedPubMed CentralView ArticleGoogle Scholar
- Altschul SF, Madden TL, Schäffer AA, Zhang J, Zhang Z, Miller W, Lipman DJ: Gapped BLAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res. 1997, 25: 3389-3402. 10.1093/nar/25.17.3389.PubMedPubMed CentralView ArticleGoogle Scholar
- Cavasotto CN, Phatak SS: Homology modeling in drug discovery: current trends and applications. Drug Discov Today. 2009, 14: 676-683. 10.1016/j.drudis.2009.04.006.PubMedView ArticleGoogle Scholar
- Sierk ML, Smoot ME, Bass EJ, Pearson WR: Improving pairwise sequence alignment accuracy using near-optimal protein sequence alignments. BMC Bioinformatics. 2010, 11: 146-10.1186/1471-2105-11-146.PubMedPubMed CentralView ArticleGoogle Scholar
- Needleman SB, Wunsch CD: A general method applicable to the search for similarities in the amino acid sequence of two proteins. J Mol Biol. 1970, 48: 443-453. 10.1016/0022-2836(70)90057-4.PubMedView ArticleGoogle Scholar
- Smith TF, Waterman MS: Identification of common molecular subsequences. J Mol Evol. 1981, 147: 195-197.Google Scholar
- Vogt G, Etzold T, Argos P: An assessment of amino acid exchange matrices in aligning protein sequences: the twilight zone revisited. J Mol Biol. 1995, 249: 816-831. 10.1006/jmbi.1995.0340.PubMedView ArticleGoogle Scholar
- Edgar RC: Optimizing substitution matrix choice and gap parameters for sequence alignment. BMC Bioinformatics. 2009, 10: 396-10.1186/1471-2105-10-396.PubMedPubMed CentralView ArticleGoogle Scholar
- Edgar RC, Sjölander K: A comparison of scoring functions for protein sequence profile alignment. Bioinformatics. 2004, 20: 1301-1308. 10.1093/bioinformatics/bth090.PubMedView ArticleGoogle Scholar
- Edgar RC, Sjölander K: SATCHMO: sequence alignment and tree construction using hidden Markov models. Bioinformatics. 2003, 19: 1404-1411. 10.1093/bioinformatics/btg158.PubMedView ArticleGoogle Scholar
- Do CB, Mahabhashyam MS, Brudno M, Batzoglou S: ProbCons: Probabilistic consistency-based multiple sequence alignment. Genome Res. 2005, 15: 330-340. 10.1101/gr.2821705.PubMedPubMed CentralView ArticleGoogle Scholar
- Pei J, Grishin NV: MUMMALS: multiple sequence alignment improved by using hidden Markov models with local structural information. Nucleic Acids Res. 2006, 34: 4364-4374. 10.1093/nar/gkl514.PubMedPubMed CentralView ArticleGoogle Scholar
- Liu Y, Schmidt B, Maskell DL: MSAProbs: multiple sequence alignment based on pair hidden Markov models and partition function posterior probabilities. Bioinformatics. 2010, 26: 1958-1964. 10.1093/bioinformatics/btq338.PubMedView ArticleGoogle Scholar
- Söding J: Protein homology detection by HMM-HMM comparison. Bioinformatics. 2005, 21: 951-960. 10.1093/bioinformatics/bti125.PubMedView ArticleGoogle Scholar
- Eddy SR: Multiple alignment using hidden Markov models. Proc Int Conf Intell Syst Mol Biol. 1995, 3: 114-120.PubMedGoogle Scholar
- Ahola V, Aittokallio T, Uusipaikka E, Vihinen M: Efficient estimation of emission probabilities in profile hidden Markov models. Bioinformatics. 2003, 19: 2359-2368. 10.1093/bioinformatics/btg328.PubMedView ArticleGoogle Scholar
- Bernardes JS, Dávila AM, Costa VS, Zaverucha G: Improving model construction of profile HMMs for remote homology detection through structural alignment. BMC Bioinformatics. 2007, 8: 435-10.1186/1471-2105-8-435.PubMedPubMed CentralView ArticleGoogle Scholar
- Thompson JD, Higgins DG, Gibson TJ: CLUSTAL W: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic Acids Res. 1994, 22: 4673-4680. 10.1093/nar/22.22.4673.PubMedPubMed CentralView ArticleGoogle Scholar
- Notredame C, Higgins DG, Heringa J: T-Coffee: A novel method for fast and accurate multiple sequence alignment. J Mol Biol. 2000, 302: 205-217. 10.1006/jmbi.2000.4042.PubMedView ArticleGoogle Scholar
- Henikoff S, Henikoff JG: Amino acid substitution matrices from protein blocks. Proc Natl Acad Sci USA. 1992, 89: 10951-10919.View ArticleGoogle Scholar
- Dayhoff MO, Schwartz RM, Orcutt BC: Establishing homologies in protein sequences. Methods Enzymol. 1983, 91: 524-545.PubMedView ArticleGoogle Scholar
- Johnson MS, Overington JP: A structural basis for sequence comparison. An evaluation of scoring methodologies. J Mol Biol. 1993, 233: 716-738. 10.1006/jmbi.1993.1548.PubMedView ArticleGoogle Scholar
- Prlić A, Domingues FS, Sippl MJ: Structure-derived substitution matrices for alignment of distantly related sequences. Protein Eng. 2000, 13: 545-550. 10.1093/protein/13.8.545.PubMedView ArticleGoogle Scholar
- Blake JD, Cohen FE: Pairwise sequence alignment below the twilight zone. J Mol Biol. 2001, 307: 721-35. 10.1006/jmbi.2001.4495.PubMedView ArticleGoogle Scholar
- Holm L, Sander C: Touring protein fold space with Dali/FSSP. Nucleic Acids Res. 1998, 26: 316-319. 10.1093/nar/26.1.316.PubMedPubMed CentralView ArticleGoogle Scholar
- Shindyalov IN, Bourne PE: Protein structure alignment by incremental combinatorial extension (CE) of the optimal path. Protein Eng. 1998, 11: 739-747. 10.1093/protein/11.9.739.PubMedView ArticleGoogle Scholar
- Taylor WR: Protein structure comparison using SAP. Methods Mol Biol. 2000, 143: 19-32.PubMedGoogle Scholar
- Salem S, Zaki MJ, Bystroff C: FlexSnap: flexible non-sequential protein structure alignment. Algorithms Mol Biol. 2010, 4: 5-12.Google Scholar
- Sauder JM, Artur JW, Dunbrack RL: Large-scale comparison of protein sequence alignment algorithms with structural alignments. Proteins. 2000, 40: 6-22. 10.1002/(SICI)1097-0134(20000701)40:1<6::AID-PROT30>3.0.CO;2-7.PubMedView ArticleGoogle Scholar
- Panchenko AR, Bryant SH: A comparison of position-specific score matrices based on sequence and structure alignments. Prot Sci. 2002, 11: 361-370.View ArticleGoogle Scholar
- Mizuguchi K, Deane CM, Blundell TL, Overington JP: HOMSTRAD: a database of protein structure alignments for homologous families. Protein Sci. 1998, 7: 2469-2471. 10.1002/pro.5560071126.PubMedPubMed CentralView ArticleGoogle Scholar
- Raghava GP, Searle SM, Audley PC, Barber JD, Barton GJ: OXBench: a benchmark for evaluation of protein multiple sequence alignment accuracy. BMC Bioinformatics. 2003, 4: 47-10.1186/1471-2105-4-47.PubMedPubMed CentralView ArticleGoogle Scholar
- Van Walle I, Lasters I, Wyns L: SABmark - a benchmark for sequence alignment that covers the entire known fold space. Bioinformatics. 2005, 21: 1267-1268. 10.1093/bioinformatics/bth493.PubMedView ArticleGoogle Scholar
- Vilim RB, Cunningham RM, Lu B, Kheradpour P, Stevens FJ: Fold-specific substitution matrices for protein classification. Bioinformatics. 2004, 20: 847-853. 10.1093/bioinformatics/btg492.PubMedView ArticleGoogle Scholar
- Agrawal A, Huang X: Pairwise statistical significance of local sequence alignment using sequence-specific and position-specific substitution matrices. IEEE/ACM Trans Comput Biol Bioinform. 2011, 8: 194-205.PubMedView ArticleGoogle Scholar
- Lo Conte L, Ailey B, Hubbard TJ, Brenner SE, Murzin AG, Chothia C: SCOP: a structural classification of proteins database. Nucleic Acids Res. 2000, 28: 257-259. 10.1093/nar/28.1.257.PubMedPubMed CentralView ArticleGoogle Scholar
- Rost B: Twilight zone of protein sequence alignments. Protein Eng. 1999, 12: 85-94. 10.1093/protein/12.2.85.PubMedView ArticleGoogle Scholar
- Müller T, Spang R, Vingron M: Estimating amino acid substitution models: a comparison of Dayhoff's estimator, the resolvent approach and a maximum likelihood method. Mol Biol Evol. 2002, 19: 8-13.PubMedView ArticleGoogle Scholar
- Qian B, Goldstein RA: Optimization of a new score function for the generation of accurate alignments. Proteins. 2002, 48: 605-610. 10.1002/prot.10132.PubMedView ArticleGoogle Scholar
- Gonnet GH, Cohen MA, Benner SA: Exhaustive matching of the entire protein sequence database. Science. 1992, 256: 1443-1445. 10.1126/science.1604319.PubMedView ArticleGoogle Scholar
- Brenner SE, Chothia C, Hubbard TJP: Assessing sequence comparison methods with reliable structurally identified distant evolutionary relationships. Proc Natl Acad Sci USA. 1998, 95: 6073-6078. 10.1073/pnas.95.11.6073.PubMedPubMed CentralView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.