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

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22 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 150 17 (0.13) 18 (0.07) 19 (0.2) 20 (0.08) 21 (0.13) 22 (0.13) X (0.08) Y (0.08) 100 50 Quantiles of insilico fragment lengths 0 150 9 (0.1) 10 (0.09) 11 (0.1) 12 (0.11) 13 (0.06) 14 (0.1) 15 (0.08) 16 (0.15) 1 (0.12) 2 (0.08) 3 (0.09) 4 (0.06) 5 (0.09) 6 (0.08) 7 (0.09) 8 (0.09) 150 100 50 0 100 50 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Quantiles of exponential 0 1 2 3 4 5 Figure 2.1 Exponential Q-Q plot of SwaI restriction fragment lengths, grouped by chromosome, in the in silico map derived from Build 35 of the human genome sequence. The parenthesized values in the strip labels indicate rank autocorrelations. It is common to model restriction site locations by a homogeneous Poisson process, or equivalently, the fragment lengths as i.i.d. exponential variates. The Q-Q plots are roughly linear (although the mild but systematic curvature is intriguing), and the rank autocorrelations are low, suggesting only mild lack of fit. Interestingly, the slopes are not the same for all chromosomes, suggesting different rates.
23 Alternatively, it can be thought of as the realization of a random process; in particular, recognition sites along the genome have been modeled as the realizations of a homogeneous Poisson point process, or equivalently the fragment lengths as i.i.d. exponential variates. This model is supported by Figure 2.1, derived from Build 35 of the human genome sequence. The rate of this process depends on the restriction enzyme being used, as well as the genome being mapped. In some cases, it may vary across, or even within, chromosomes. Genomic differences within a species usually involve only a fraction of the genome, and corresponding restriction maps are expected to be largely similar. In any case, we are chiefly interested in modeling the generative process of data conditional on the underlying restriction map. It should be noted that the notion of a ‘true’ map is somewhat simplified. Diploid genomes have two versions of the map, largely similar but not identical. Cancer samples are usually a mixture of several cell populations that each contribute a slightly different genome. Shotgun breaks: Before they are passed into micro-channels, chromosomal DNA is randomly broken up into smaller molecules, usually by subjecting the DNA to vibration. This shearing is often referred to as a whole genome shotgun process. The origin of each observed optical map molecule is characterized by its location in the coordinate system defined by the underlying true (unknown) restriction map, as well as its length. The distribution of the location (e.g. midpoint) is assumed to be uniform over the underlying genome. It is typical to consider only optical maps longer than a predetermined threshold, usually 300 Kb. The distribution of lengths of the filtered maps is usually consistent with a truncated exponential distribution. 2.1.2 Errors Cut site errors: A restriction site in the true restriction map may fail to show up in a corresponding optical map. These missing cuts can be due to either incomplete digestion by the restriction enzyme or noise in the optical map image. Whether true cut sites are identified (success) or not (failure) is modeled as independent Bernoulli trials, with some
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: 21 Chapter 2 Modeling Optical Map D
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 and 54: 39 using optical mapping data from
Page 55 and 56: 41 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8
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
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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.
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81 will rarely be homozygous. It ma
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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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