On the Analysis of Optical Mapping Data - University of Wisconsin ...

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40 is rejected at level α if S > c α where F 0 (c α |M) = 1 − α. The advantage of this formulation is that given a choice of P G , we can in principle simulate from F 0 (·|M) to obtain a suitable cutoff, without requiring any probabilistic model for the optical map M. An effective choice of P G is given by random permutations of the reference ˜G. This preserves characteristics of the reference that are known to affect the spurious score distribution, namely the number and lengths of fragments. Permuting the order of fragments is also reasonable given the additive nature of score functions, which essentially reward matches in order. Formally, if we assume that the fragment lengths defining G are i.i.d. from some distribution in a family F, permutation can be viewed as sampling from P G conditional on the set of fragment lengths in ˜G, which is sufficient for F. Such tests are often called permutation tests (Cox and Hinkley, 1979, Chapter 6). See Figure 3.1 for a graphical justification of the i.i.d. assumption. 3.3 Results 3.3.1 Exploration We use optical map data from GM07535, a diploid normal human lymphoblastoid cell line, for illustration. The data consists of 206796 optical maps longer than 300 Kb. These maps are aligned against an in silico reference map derived from Build 35 of the human genome sequence (International Human Genome Sequencing Consortium, 2004), with sequence gaps replaced by their estimated length. We use a score function implemented in the SOMA software suite with parameters that have been extensively used with optical map data. The actual score function, henceforth referred to as the SOMA score, is described in Appendix A. In addition to the best alignment scores against the in silico reference, we consider best scores for each map against several independent random permutations of the reference. The permutations are done separately for every chromosome, thus retaining the total length and number of fragments within each. For the most part, we restrict our attention to ungapped global alignments. In theory, we can approximate the conditional null distribution F 0 (·|M) by sampling from it an arbitrary number of times. In practice, each such sample involves a permutation of
41 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 Pairs of successive ranks 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 chr19 ( 0.2 ) chr20 ( 0.08 ) chr21 ( 0.13 ) chr22 ( 0.13 ) chrX ( 0.08 ) chrY ( 0.08 ) chr13 ( 0.06 ) chr14 ( 0.1 ) chr15 ( 0.08 ) chr16 ( 0.15 ) chr17 ( 0.13 ) chr18 ( 0.07 ) chr7 ( 0.09 ) chr8 ( 0.09 ) chr9 ( 0.1 ) chr10 ( 0.09 ) chr11 ( 0.1 ) chr12 ( 0.11 ) chr1 ( 0.12 ) chr2 ( 0.08 ) chr3 ( 0.09 ) chr4 ( 0.06 ) chr5 ( 0.09 ) chr6 ( 0.08 ) 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3.1 Independence of in silico fragment lengths. Our method presumes that restriction fragment lengths in the reference copy are i.i.d.. Here, we plot successive pairs of ranks separately for each chromosome. The scatter is fairly uniform, supporting the i.i.d. assumption. The figures in parentheses indicate rank autocorrelation. The average fragment length is known to differ across chromosomes, so we restrict all permutations to within chromosomes.
Page 1 and 2:
ON THE ANALYSIS OF OPTICAL MAPPING
Page 3 and 4: To my parents. i
Page 5 and 6: DISCARD THIS PAGE
Page 7 and 8: iv Page 3.3 Results . . . . . . . .
Page 9 and 10: v LIST OF TABLES Table Page 1.1 Sum
Page 11 and 12: vi LIST OF FIGURES Figure Page 1.1
Page 13 and 14: ON THE ANALYSIS OF OPTICAL MAPPING
Page 15 and 16: 1 Chapter 1 Overview of Optical Map
Page 17 and 18: 3 hard, do not always have a unique
Page 19 and 20: 5 for microbial and other small gen
Page 21 and 22: 7 Figure 1.2 Close-up of a typical
Page 23 and 24: 9 0.96 0.98 1.00 1.02 1.04 Offset a
Page 25 and 26: 11 direct glimpse at the underlying
Page 27 and 28: 13 which is not surprising since we
Page 29 and 30: 15 Gapped alignments: The above des
Page 31 and 32: Figure 1.5 A visualization of align
Page 33 and 34: 19 Assembly: For these examples, th
Page 35 and 36: 21 Chapter 2 Modeling Optical Map D
Page 37 and 38: 23 Alternatively, it can be thought
Page 39 and 40: 25 and V (X i ) = E(V (Y i R i |R i
Page 41 and 42: 27 affect inference. If necessary,
Page 43 and 44: 29 Quantiles of fragment lengths (K
Page 45 and 46: 31 as a function of the parameters.
Page 47 and 48: 33 by rejecting maps that do not al
Page 49 and 50: 35 30 0.700 − 0.005 0 50 100 150
Page 51 and 52: 37 Chapter 3 Significance of Optica
Page 53: 39 using optical mapping data from
Page 57 and 58: 43 3.3.2 Simplifications Direct app
Page 59 and 60: 45 Mean spurious score 0 −10 −2
Page 61 and 62: 47 3.3.3 Simulation Given a generat
Page 63 and 64: 49 3.4 Discussion 3.4.1 Uses Alignm
Page 65 and 66: 51 The ability to simulate from the
Page 67 and 68: 53 maps, where the separation betwe
Page 69 and 70: Figure 3.10 Schematic representatio
Page 71 and 72: 57 especially a short noisy one, to
Page 73 and 74: 59 Test statistics: Variability due
Page 75 and 76: 61 in sequence assembly and validat
Page 77 and 78: 63 1988). However, due to sampling
Page 79 and 80: 65 and rate parameters Λ i = E(N i
Page 81 and 82: 67 with mean µ k for the k th stat
Page 83 and 84: 69 Estimated Copy Number in simulat
Page 85 and 86: 71 Posterior probabilities 1.0 0.8
Page 87 and 88: 73 (a) Observed counts and decoded
Page 89 and 90: 75 Conclusion: Copy number alterati
Page 91 and 92: 77 well in its current form, but th
Page 93 and 94: 79 Change in score 15 10 5 0 0.9 1.
Page 95 and 96: 81 will rarely be homozygous. It ma
Page 97 and 98: 83 E.T. Dimalanta, A. Lim, R. Runnh
Page 99 and 100: 85 Appendix A: Score functions for
Page 101 and 102: 87 Appendix B: Hidden Markov Model
Page 103 and 104: 89 which can be shown to have highe
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