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8.4 How to Select a Scoring Matrix? 157 8.4 How to Select a Scoring Matrix? From the discussion in Section 8.3, given a protein sequence model in which amino acids occur by chance with background frequencies p i (1 ≤ i ≤ 20), a scoring matrix s ij with negative expected score and at least one positive entry is only optimal to the alignments in which the amino acids a i and a j are aligned with target frequencies q ij that satisfy equation (8.2). In other words, different scoring matrices are optimal for detecting different classes of alignments. This raises the problem of selecting scoring matrix for best distinguishing true protein alignment from chance. Multiplying a scoring matrix by a positive constant has no effect on the relative scores of different MSPs. Therefore, two matrices related by such a constant factor are said to be equivalent. By (8.2), any scaling corresponds merely to taking logarithm to a different base. When λ is set to 1, the scores are natural logarithms of the odds ratios; when λ is set to ln2, the scores become logarithms to base 2. Let N denote the expected number of MSPs with score s or more obtained in the alignment of two sequences of length n 1 and n 2 . Setting λ to ln2 in (7.45), we obtain ( ) K s = log 2 + log N 2 (n 1 n 2 ). (8.10) The parameter K is usually less than 1 for a typical scoring matrix. An alignment may be considered significant when N is 0.01. As a result, the right-hand side of equation (8.10) is dominated by the term log 2 (mn), and alignment score needed to distinguish a MSP from chance is approximately log 2 (mn). Thus, for comparing two proteins of 250 amino acid residues, a MSP is statistically significant only if its score is 16 (≈ 2log 2 250) or more; if such a protein is searched against a database of 10,000,000 residues, a significant MSP should have then score 31 (≈ log 2 2500000000) or more. The PAM and BLOSUM matrices given in Sections 8.1 and 8.2 have different λs (see Table 8.3). By multiplying λ/ln2, a scoring matrix is normalized into the logarithms of odds ratios to base 2. Scores obtained with such a normalized scoring matrix can be considered as bit information. As a result, we may say that 16-bit information is required to distinguish a MSP from chance in comparing two protein sequences of 250 amino acid residues. Given a protein model (which specifies the background frequencies p i ) and a normalized scoring matrix (s ij ), one can calculate the target frequencies q ij , which characterize the alignments on which the scoring matrix is optimal, as Define H = ∑ i, j q ij = p i p j e (ln2)s ij . ( ) qij q ij s ij = ∑q ij log 2 . (8.11) i, j p i p j H is the expected score (or bit information) per residue pair in the alignments characterized by the target frequencies. From theoretic-information point of view, H is
158 8 Scoring Matrices Table 8.3 The values of the parameter λ in equation (8.2) and the relative entropy of PAM and BLOSUM matrices listed in Sections 8.1 and 8.2. PAM30 PAM70 PAM120 PAM250 BLOSUM45 BLOSUM62 BLOSUM80 λ ln2/2 ln2/2 ln2/2 ln10/10 ln10/10 ln2/2 ln10/10 Entropy 2.57 1.60 0.979 0.354 0.3795 0.6979 0.9868 the relative entropy of the target frequency distribution with respect to the background distribution (see Section B.6). Hence, we call H the relative entropy of the scoring matrix (s ij ) (with respect to the protein model). Table 8.3 gives the relative entropy of popular scoring matrices with respect to the implicit protein model. Intuitively, if the value of H is high, relatively short alignments with the target frequencies can be distinguished from chance; if the value of H is low, however, long alignments are necessary. Recall that distinguishing an alignment from chance needs 16 bits of information in comparison of two protein sequences of 250 amino acids. Using this fact, we are able to estimate the length of a significant alignment of two sequences that are x-PAM divergent. For example, at a distance of 120 PAMs, there is on average 0.979 bit of information in every aligned position as shown in Table 8.3. As a result, a significant alignment has at least 17 residues. For database search, the situation is more complex. In this case, alignments are unknown and hence it is not clear which scoring matrix is optimal. It is suggested to use multiple PAM matrices. The PAMx matrix is designed to compare two protein sequences that are separated by x PAM distance. If it is used to compare two protein sequences that are actually separated by y PAM distance, the average bit information achieved per position is smaller than its relative entropy. When y is close to x, the average information achieved is near-optimal. Assume we are satisfied with using a PAM matrix that yields a score greater than 93% of the optimal achievable score. Because a significant MSP contains about 31 bits of information in searching a protein against a protein database containing 10,000,000 residues, the length range of the local alignments that the PAM120 matrix can detect is from 19 to 50. As a result, when PAM120 is used, it may miss short but strong or long but weak alignments that contain sufficient information to be found. Accordingly, PAM40 and PAM250 may be used together with PAM120 in database search. 8.5 Compositional Adjustment of Scoring Matrices We have showed that a scoring matrix is only valid in one unique context in Section 8.3. Thus, it is not ideal to use a scoring matrix that is constructed for a specific set of target and background frequencies in a different context. To compare proteins having biased composition, one approach is to repeat Henikoff and Henikoff’s procedure of construction of the BLOSUM matrices. A set of true alignments for
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Computational Biology Editors-in-Ch
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Kun-Mao Chao·Louxin Zhang Sequence
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KMC: To Daddy, Mommy, Pei-Pei and L
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viii Foreword I invite you to study
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x Preface Chapters 2 to 5 form the
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Acknowledgments We are extremely gr
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Contents Foreword .................
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Contents xix Part II. Theory ......
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Chapter 1 Introduction 1.1 Biologic
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1.2 Alignment: A Model for Sequence
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1.2 Alignment: A Model for Sequence
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1.3 Scoring Alignment 7 ( ) k m a k
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1.4 Computing Sequence Alignment 9
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1.5 Multiple Alignment 11 1.5 Multi
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1.8 Bibliographic Notes and Further
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PART I. ALGORITHMS AND TECHNIQUES 1
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18 2 Basic Algorithmic Techniques 2
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20 2 Basic Algorithmic Techniques F
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22 2 Basic Algorithmic Techniques s
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28 2 Basic Algorithmic Techniques P
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30 2 Basic Algorithmic Techniques a
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32 2 Basic Algorithmic Techniques O
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Chapter 3 Pairwise Sequence Alignme
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3.3 Global Alignment 37 3.2 Dot Mat
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3.3 Global Alignment 39 ⎧ ⎨ S[i
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3.3 Global Alignment 41 ( ai b j )
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3.4 Local Alignment 43 ⎧ 0, ⎪
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3.4 Local Alignment 45 Algorithm LO
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3.5 Various Scoring Schemes 47 Fig.
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3.6 Space-Saving Strategies 49 Fig.
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3.6 Space-Saving Strategies 51 Algo
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3.6 Space-Saving Strategies 53 scor
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3.7 Other Advanced Topics 55 ning,
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3.7 Other Advanced Topics 57 (0,0).
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3.7 Other Advanced Topics 59 3.7.4
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Chapter 4 Homology Search Tools The
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4.1 Finding Exact Word Matches 65 F
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4.1 Finding Exact Word Matches 67 F
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4.3 BLAST 69 SALSDLHAHKLRVDPVNFKLLS
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4.3 BLAST 71 length w, whereas for
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4.3 BLAST 73 Fig. 4.9 A scenario of
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4.5 PatternHunter 75 BLAT identifie
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4.5 PatternHunter 77 can develop an
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82 5 Multiple Sequence Alignment S
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84 5 Multiple Sequence Alignment S
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86 5 Multiple Sequence Alignment Fi
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Chapter 6 Anatomy of Spaced Seeds B
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6.2 Basic Formulas on Hit Probabili
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6.3 Distance between Non-Overlappin
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Page 126 and 127: 6.6 Generalizations of Spaced Seeds
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Page 160 and 161: Chapter 8 Scoring Matrices With the
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Page 172 and 173: 8.6 DNA Scoring Matrices 161 This i
Page 174 and 175: 8.7 Gap Cost in Gapped Alignments 1
Page 184 and 185: Appendix A Basic Concepts in Molecu
Page 186 and 187: A.4 The Genomes 175 ondary structur
Page 188 and 189: Appendix B Elementary Probability T
Page 190 and 191: B.3 Major Discrete Distributions 17
Page 192 and 193: B.3 Major Discrete Distributions 18
Page 194 and 195: B.5 Mean, Variance, and Moments 183
Page 196 and 197: B.5 Mean, Variance, and Moments 185
Page 198 and 199: B.6 Relative Entropy of Probability
Page 200 and 201: B.7 Discrete-time Finite Markov Cha
Page 202 and 203: B.8 Recurrent Events and the Renewa
Page 204 and 205: B.8 Recurrent Events and the Renewa
Page 206 and 207: 196 C Software Packages for Sequenc
Page 208 and 209: 198 References 19. Bafna, V. and Pe
Page 210 and 211: 200 References 71. Fitch, W.M. and
Page 212 and 213: 202 References 122. Letunic, I., Co
Page 214 and 215: 204 References 173. Robinson, A.B.
Page 216 and 217: Index O-notation, 18 P-value, 139 a
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Index 209 heuristic, 85 progressive
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