Bioinformatics Algorithms: Techniques and Applications

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9.5 COMPUTATIONAL RESULTS COMPUTATIONAL RESULTS 215 We have implemented a C++ software tool that reads in a set of genomes, solves one of the integer linear programming formulations presented in Sections 9.3 and 9.4 using the CPLEX optimization library [7], and outputs the list of optimal and close to optimal gene clusters. All experiments were performed on an AMD 2.2 GHz Opteron 64 bit processor with 8 GB of main memory using CPLEX 9.03. 9.5.1 Hidden Clusters in Artificial Data We generate artificial problem instances for benchmarking as follows: We randomly generate six genomes of roughly 1000 genes each (N = 2000) with 5% of 0-genes. For each D ∈{5, 10, 15, 20, 25}, we generate a cluster and hide a perturbed permutation of it in five randomly selected genomes, taking care that the different clusters do not overlap. Using w− = 2, w + = 1 and the appropriate value of D, we solve the ILP first with genome selection, setting ρ = 2D/5. We retrieve all the five clusters in 29 min, 45 min, 8 min, 113 s, and 14 s, respectively. Without genome selection, running times are much faster, but we can run into problems because of the high penalty for the genome in which the cluster is missing: We retrieve the clusters of sizes 5, 10, 15, and 25 in 17 min, 7 min, 163 s, and 4 s, respectively. For D = 20, we obtain a different cluster than the hidden one that obtains a better target function value without genome selection. While the running times vary with each instance, the times shown here are representative. This experiment indicates the importance, but also the high complexity of genome selection. 9.5.2 Comparison of Two Organisms The genomes of C. glutamicum and M. tuberculosis consist of 3057 and 3991 annotated genes, respectively. The annotated gene set is available at http://ls11www.cs.unidortmund.de/people/rahmann/research. We compute the optimal objective function value c ∗ (D) for each cluster size D ∈ [5,500] for the basic formulation with w − = w + = 1 (total CPU time: almost 25 days, on average 1:15 h per instance). Figure 9.2 shows the running time per instance as well as the optimal normalized costs c ∗ (D)/D. Local minima correspond to large approximate clusters with comparatively few errors. As Fig. 9.3 illustrates for D = 51, the ILP formulation discovers clusters that cannot be detected by any method that does not consider approximate clusters. The largest exact cluster has size 11. In a next step, we enumerate all fundamentally different close-to-optimal solutions for the local minima of Fig. 9.2, using the approach described in Section 9.3. We set the suboptimality parameter γ to 0.2. Interestingly, there are not too many different clusters with solution values within 20% of the optimum, as shown in Table 9.2. The solution lists might be an interesting starting point for further biological investigations. Figure 9.4 shows the output of our software for D = 61.
216 INTEGER LINEAR PROGRAMMING TECHNIQUES CPU time [4 hrs]; Normalized objective function c *(D) /D 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 CPU time [units of 4 h] Normalized objective function c * (D) /D 0 0 50 100 150 200 250 300 350 400 450 500 Cluster size D FIGURE 9.2 Comparison of C. glutamicum and M. tuberculosis: For each cluster size D ∈ [5,500], the running time in units of 4 h, and the normalized optimal value of the objective function is shown. Note the local minima in the objective function, for example, at D = 51. The apparent correlation between objective function and running time indicates that good approximate clusters are easier to compute than bad clusters. 9.5.3 A 28-Cluster in a 20-Genome Set We attempt to find a large approximate gene cluster in several genomes. We thank Thomas Schmidt (Bielefeld) for providing a dataset of the following twenty bacterial genomes (number of genes given in brackets): B. longum (1727), B. subtilis (4103), C. diphtheriae (2272), C. efficiens (2942), C. glutamicum (3057), E. coli K12 (4288), L. lactis (2266), L. xyli (2030), M. avium (4350), M. leprae (1605), M. tuberculosis (3991), N. farcinica (5683), P. acnes (2297), P. aeruginosa (5566), P. putida (5350), S. avermitilis (7575), S. coelicolor (7768), S. thermophilum (3337), T. whipplei (808), and W. succinogenes (2044). FIGURE 9.3 Visualization of an interesting optimal cluster in C. glutamicum and M. tuberculosis (D = 51). Differing genes are marked in gray. Three conserved regions, a, b, and c occur in the cluster.
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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
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A B C A B C SUBTREES AND SUBNETWORK
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SUBTREES AND SUBNETWORKS 153 of x a
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SUBTREES AND SUBNETWORKS 155 1. it
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SUPERTREES AND SUPERNETWORKS 157 By
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SUPERTREES AND SUPERNETWORKS 159 Go
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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 210 and 211: 9 INTEGER LINEAR PROGRAMMING TECHNI
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 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
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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
Page 258 and 259: #primers/(2x#SNPs) (%) #primers/(2x
Page 260 and 261: TABLE 11.2 (Continued ) EXPERIMENTA
Page 262 and 263: REFERENCES 257 p, discard all candi
Page 264 and 265: 12 RECENT DEVELOPMENTS IN ALIGNMENT
Page 266 and 267: 12.2 MULTIPLE SEQUENCE ALIGNMENT 12
Page 268 and 269: MULTIPLE SEQUENCE ALIGNMENT 263 The
Page 270 and 271: 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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