Bioinformatics Algorithms: Techniques and Applications

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9 INTEGER LINEAR PROGRAMMING TECHNIQUES FOR DISCOVERING APPROXIMATE GENE CLUSTERS Sven Rahmann Bioinformatics for High-Throughput Technologies, Department of Computer Science 11, Technical University of Dortmund, Dortmund, Germany Gunnar W. Klau Mathematics in Life Sciences Group, Department of Mathematics and Computer Science, Free University Berlin, and DFG Research Center Matheon “Mathematics for Key Technologies,” Berlin, Germany 9.1 INTRODUCTION The chapter by Bergeron et al. in this volume introduces the concept of conserved gene clusters among several related organisms and motivates their study. We do not repeat the contents here; instead we assume that the reader is familiar with that chapter. It presents four principal combinatorial models for gene clusters, along with their algorithms: common intervals (“exact” gene clusters) versus max-gap clusters (a particular kind of “approximate” gene clusters) in permutations (assuming the same gene content in each genome, disallowing gene duplications) versus sequences (or strings, where different gene contents and duplicated or removed genes are allowed). The authors note that defining a general gene cluster model beyond max-gap clusters in strings that captures biological reality well is not an easy task. Even if we Bioinformatics Algorithms: Techniques and Applications, Edited by Ion I. Mǎndoiu and Alexander Zelikovsky Copyright © 2008 John Wiley & Sons, Inc. 203
204 INTEGER LINEAR PROGRAMMING TECHNIQUES succeeded, discovering all gene clusters according to the model definition might be a computationally hard problem. The present chapter takes a practical approach to these fundamental issues. Instead of arguing for or against one particular model, we present a class of models for approximate gene clusters that can be written as integer linear programs (ILPs) and include well-known variations, for example, common intervals, r-windows, and maxgap clusters or gene teams. While the ILP formulation does not necessarily lead to efficient algorithms, it provides a general framework to study different models and is competitive in practice for those cases where efficient algorithms are known. We show that it allows a nonheuristic study of large approximate clusters across several prokaryotic genomes. Different ways to model gene clusters are discussed in the previous chapter; alternatives have also been surveyed by [6]. Difficulties in finding a universally accepted model include the following. 1. The problem has been attacked from two sides. One philosophy is to specify an algorithm and constructively define the algorithm’s results as conserved gene clusters. The drawbacks of this approach are that it is unclear how such an algorithm maps to the biological reality, and that statistical analysis of the results becomes difficult. The other philosophy is to provide a formal specification of what constitutes a gene cluster (modeling step), and then design an algorithm that finds all clusters that satisfy the specification (solving step). 2. It is not easy to formally specify what we are looking for. Should we choose a narrow definition at the risk of missing biologically interesting gene sets, or a wide definition and browse through many biologically uninteresting sets? We believe that it is preferable to use separate modeling and solving steps. This allows us to first focus on tuning the model for biological relevance, and only then worry about efficient algorithms. Therefore, we propose a framework for defining the approximate gene cluster discovery problem (AGCDP), as defined in Section 9.2.5, as well as many variants, as an integer linear program (ILP; see [11], for a general introduction). We shall not be concerned with efficiently solving the ILPs that we define; for computational experiments, we have used the commercial solver CPLEX [7] that works reasonably well in practice. Our goal is to make as few restrictions in the model as possible. In particular, we assume that genes are represented by integers in such a way that paralogous and orthologous genes receive the same number. Genomes therefore are sequences of (unsigned) integers, not necessarily permutations. Homology detection is a delicate procedure, so we must assume that our representation contains errors. As a consequence, we need an error-tolerant formalization of the cluster concept. For certain special cases of the ILP formulations, special-purpose algorithms exist. Using them would solve the corresponding problem more efficiently than using a general ILP solver. However, a general framework has the advantage that the objective functions and constraints can be easily modified without designing and implementing
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BIOINFORMATICS ALGORITHMS
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Copyright © 2008 by John Wiley & S
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vi CONTENTS 6 A Survey of Seeding f
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PREFACE Bioinformatics, broadly def
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CONTRIBUTORS Sudha Balla, Departmen
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CONTRIBUTORS xiii Steven Hecht Orza
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1 EDUCATING BIOLOGISTS IN THE 21ST
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EDUCATING BIOLOGISTS IN THE 21ST CE
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EDUCATING BIOLOGISTS IN THE 21ST CE
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2 DYNAMIC PROGRAMMING ALGORITHMS FO
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SEQUENCE ALIGNMENT: GLOBAL, LOCAL,
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ecurrence: SEQUENCE ALIGNMENT: GLOB
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DYNAMIC PROGRAMMING ALGORITHMFOR RN
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DYNAMIC PROGRAMMING ALGORITHMFOR RN
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DYNAMIC PROGRAMMING ALGORITHMS FOR
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REFERENCES 25 the flexible structur
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REFERENCES 27 32. Gusfield D. Effic
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3 GRAPH THEORETICAL APPROACHES TO D
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GRAPH THEORY BACKGROUND 31 beginnin
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GRAPH THEORY BACKGROUND 33 FIGURE 3
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GRAPH THEORY BACKGROUND 35 chordal
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GRAPH THEORY BACKGROUND 37 decompos
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RECONSTRUCTING PHYLOGENIES 39 are (
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RECONSTRUCTING PHYLOGENIES 41 only
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FORMATION OF MULTIPROTEIN COMPLEXES
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3.4.1 Ribosomal Assembly FORMATION
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ACKNOWLEDGMENTS REFERENCES 51 This
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REFERENCES 53 37. Golumbic MC, Hart
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4 ADVANCES IN HIDDEN MARKOV MODELS
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HIDDEN MARKOV MODELS FOR SEQUENCE A
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ALTERNATIVES TO VITERBI DECODING 63
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Noncoding Coding Intron (a) Without
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also have this same label). We get
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change as follows: GENERALIZED HIDD
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0.00004 0.00002 0.00000 0 20000 400
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HMMS WITH MULTIPLE OUTPUTS OR EXTER
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TRAINING THE PARAMETERS OF AN HMM 8
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CONCLUSION 85 of parameters compare
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REFERENCES 87 4. Altun Y, Tsochanta
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REFERENCES 89 42. Krogh A. Using da
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REFERENCES 91 77. Xu EW, Kearney P,
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94 SORTING- AND FFT-BASED TECHNIQUE
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100 SORTING- AND FFT-BASED TECHNIQU
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6 A SURVEY OF SEEDING FOR SEQUENCE
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ALIGNMENTS 119 6.2.1 Formal Definit
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TRADITIONAL APPROACHES TO HEURISTIC
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MORE CONTEMPORARY SEEDING APPROACHE
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MORE COMPLICATED SEED DESCRIPTIONS
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SOME THEORETICAL ISSUES IN ALIGNMEN
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REFERENCES 141 6. Brown DG. Optimiz
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7 THE COMPARISON OF PHYLOGENETIC NE
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INTRODUCTION 145 known phylogeny re
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BASIC DEFINITIONS 147 The undirecte
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BASIC DEFINITIONS 149 N1 displays N
Page 160 and 161: A B C A B C SUBTREES AND SUBNETWORK
Page 162 and 163: SUBTREES AND SUBNETWORKS 153 of x a
Page 164 and 165: SUBTREES AND SUBNETWORKS 155 1. it
Page 166 and 167: SUPERTREES AND SUPERNETWORKS 157 By
Page 168 and 169: SUPERTREES AND SUPERNETWORKS 159 Go
Page 170 and 171: SUPERTREES AND SUPERNETWORKS 161 Th
Page 172 and 173: RECONCILIATION OF GENE TREES AND SP
Page 182 and 183: REFERENCES 173 21. Gòrecki P, Tiur
Page 184 and 185: 8 FORMAL MODELS OF GENE CLUSTERS An
Page 186 and 187: 8.2 GENOME PLASTICITY 8.2.1 Genome
Page 188 and 189: GENOME PLASTICITY 181 FIGURE 8.2 An
Page 190 and 191: BASIC CONCEPTS 183 “more or less
Page 192 and 193: BASIC CONCEPTS 185 of {m, o, s}. On
Page 194 and 195: MODELS OF GENE CLUSTERS 187 Definit
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Page 198 and 199: MODELS OF GENE CLUSTERS 191 FIGURE
Page 200 and 201: MODELS OF GENE CLUSTERS 193 another
Page 202 and 203: MODELS OF GENE CLUSTERS 195 of gene
Page 204 and 205: MODELS OF GENE CLUSTERS 197 The two
Page 206 and 207: REFERENCES 199 flexibility by bound
Page 208 and 209: REFERENCES 201 28. Hoberman R, Dura
Page 212 and 213: BASIC PROBLEM SPECIFICATION 205 a n
Page 214 and 215: INTEGER LINEAR PROGRAMMING FORMULAT
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Page 218 and 219: 9.4 EXTENSIONS AND VARIATIONS EXTEN
Page 220 and 221: i=1 EXTENSIONS AND VARIATIONS 213 H
Page 222 and 223: 9.5 COMPUTATIONAL RESULTS COMPUTATI
Page 224 and 225: DISCUSSION 217 TABLE 9.2 Cluster Si
Page 226 and 227: DISCUSSION 219 FIGURE 9.5 Manually
Page 228 and 229: ACKNOWLEDGMENTS REFERENCES 221 We t
Page 230 and 231: 224 EFFICIENT COMBINATORIAL ALGORIT
Page 246 and 247: 11 ALGORITHMS FOR MULTIPLEX PCR PRI
Page 248 and 249: INTRODUCTION 243 problem: given a s
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Page 254 and 255: A GREEDY ALGORITHM 249 FIGURE 11.3
Page 256 and 257: EXPERIMENTAL RESULTS 251 11.5.1 Amp
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TABLE 11.2 (Continued ) EXPERIMENTA
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REFERENCES 257 p, discard all candi
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12 RECENT DEVELOPMENTS IN ALIGNMENT
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12.2 MULTIPLE SEQUENCE ALIGNMENT 12
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MULTIPLE SEQUENCE ALIGNMENT 263 The
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MOTIF FINDING 265 Marsan and Sagot
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BIOLOGICAL NETWORK ANALYSIS 267 mul
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DISCUSSION 269 an interaction pair
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REFERENCES 271 13. Bucka-Lassen K,
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REFERENCES 273 52. Lee C, Grasso C,
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REFERENCES 275 90. Stormo GD, Hartz
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PART III MICROARRAY DESIGN AND DATA
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280 ALGORITHMS FOR OLIGONUCLEOTIDE
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14 CLASSIFICATION ACCURACY BASED MI
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INTRODUCTION 305 Decomposition (SVD
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METHODS 307 Note that in most of th
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estimated as K� 1 ai,j = aik,j. d
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METHODS 311 [7]. The KNN-classifier
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ROWimpute-KNN ROWimpute-SVM KNNimpu
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Classification accuracies of SRBCT
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Classification accuracies of SRBCT
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REFERENCES 325 From these two plots
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REFERENCES 327 18. Troyanskaya OG,
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330 META-ANALYSIS OF MICROARRAY DAT
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16 PHASING GENOTYPES USING A HIDDEN
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A HIDDEN MARKOV MODEL FOR RECOMBINA
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LEARNING THE HMM FROM UNPHASED GENO
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EXPERIMENTAL RESULTS 365 It is also
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DISCUSSION 367 TABLE 16.1 Phasing A
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GERBIL PHASE fastPHASE 0.4 0.35 0.3
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REFERENCES 371 however, that direct
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17 ANALYTICAL AND ALGORITHMIC METHO
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INTRODUCTION 375 The use of real ha
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follows: X11 = 2N11 + N12 + N21 X21
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METHODS 379 FIGURE 17.1 The likelih
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METHODS 381 TABLE 17.3 Tests for Ha
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METHODS 383 The sixth stochastic al
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RESULTS 385 TABLE 17.4 The Distribu
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RESULTS 387 TABLE 17.6 The Distribu
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DISCUSSION 389 2SNP also produced r
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ACKNOWLEDGMENTS 391 haplotypes need
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REFERENCES 393 16. Hill WG. Estimat
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18 OPTIMIZATION METHODS FOR GENOTYP
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Tag-restricted haplotype n Complete
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INFORMATIVE SNP SELECTION 399 from
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DISEASE ASSOCIATION SEARCH 401 18.2
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DISEASE ASSOCIATION SEARCH 403 18.3
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Below is the formal description of
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RESULTS AND DISCUSSION 407 to decid
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RESULTS AND DISCUSSION 409 � Comp
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RESULTS AND DISCUSSION 411 TABLE 18
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RESULTS AND DISCUSSION 413 nonindex
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REFERENCES 415 20. Lee PH, Shatkay
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19 TOPOLOGICAL INDICES IN COMBINATO
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TOPOLOGICAL INDICES 421 The quantit
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Theorem 19.2 Let T = (V, E) be a tr
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HOSOYA POLYNOMIAL 425 The Laplacian
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H2(G, x) = � {u,v}⊆V INVERSE WI
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HEXAGONAL SYSTEMS 429 hexagonal sys
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C 2 HEXAGONAL SYSTEMS 431 FIGURE 19
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THE WIENER INDEX OF PEPTOIDS 433 Th
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if R ≥ L, then π(Lp) = i; Lp = L
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REFERENCES 437 19. Entringer RC, Me
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20 EFFICIENT ALGORITHMS FOR STRUCTU
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COMPOUND REPRESENTATION 441 FIGURE
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COMPOUND REPRESENTATION 443 breakag
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TABLE 20.1 Bond List of Aspirin Bon
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COMPOUND REPRESENTATION 447 20.2.5
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Initial class value for node A A 3
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CHEMICAL COMPOUND DATABASE 451 In c
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CHEMICAL COMPOUND DATABASE 453 taki
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CHEMICAL COMPOUND DATABASE 455 Othe
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REFERENCES 457 lab may take months
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REFERENCES 459 22. Curco D, Rodrigu
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REFERENCES 461 61. An J, Nakama T,
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REFERENCES 463 101. Shen J. HAD An
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466 COMPUTATIONAL APPROACHES TO PRE
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INDEX 2SNP computer program 383, 38
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degeneracy 101-104, 112 degenerate
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lowest p-value method 484-486 max-g
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pseudoknots 20 p-value 339-343, 347
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