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Reconstructing Circular Order from Inaccurate Adjacency Information

Reconstructing Circular Order from Inaccurate Adjacency Information. Applications in NMR Data Interpretation. Ming-Yang Kao. 160. 520. 540. 220. 190. 480. Problem Description. (160,520). (540,160). (520,220). (190,540). (220,480). (480,190). ?. ?. ?. ?. ?. ?.

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Reconstructing Circular Order from Inaccurate Adjacency Information

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  1. Reconstructing Circular Order from Inaccurate Adjacency Information Applications in NMR Data Interpretation Ming-Yang Kao

  2. 160 520 540 220 190 480 Problem Description (160,520) (540,160) (520,220) (190,540) (220,480) (480,190)

  3. ? ? ? ? ? ? Problem Description Given Find the correct order (220,480) (160,520) (480,190) (540,160) (520,220) (190,540)

  4. Introduction • Nuclear Magnetic Resonance (NMR)

  5. Introduction • Nuclear Magnetic Resonance (NMR) • Use the strong magnetic wave to align nuclei (isotopes). • When this spin transition occurs, the nuclei are said to be in resonance with the applied radiation.

  6. NMR Measurement • Chemical Shift • ppm • Electrons in the molecule have small magnetic fields • When magnetic field is applied, electrons tend to oppose the applied field • NMR Spectrum

  7. Determining Protein Structure Using NMR • NMR Spectral Data generation • Peak Picking • Peak Assignment • Structural Restraint Extraction • Structure Calculation

  8. NMR Data Interpretation • Peak Assignment. • Map resonance peaks from different NMR spectra to same residue • Identify adjacency relationship • Assign the segments to the protein sequence • Currently done manually • Bottleneck for high throughput structure determination

  9. Our Focus Peak Assignment • Two kinds of information available • Distribution of spin systems for different amino acids • The adjacency information between spin systems

  10. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Input) (a1,b1) (a2,b2) (a3,b3) (a4,b4) (a5,b5) (a6,b6)

  11. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6) ?

  12. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  13. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  14. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  15. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  16. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  17. b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  18. b5 b1 b6 b2 b3 b4 a1 a3 a6 a4 a2 a5 Problem Description (Output) (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6) ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

  19. v1 v2 v3 v4 v5 v6 u1 u2 u3 u4 u5 u6 Equivalent Problem Description (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6)

  20. v1 v2 v3 v4 v5 v6 u1 u2 u3 u4 u5 u6 Cyclic Augmentation (a1,b1) (a5,b5) (a3,b3) (a4,b4) (a2,b2) (a6,b6) A matching M is called a cyclic augmentationif HM forms a hamiltonian cycle.

  21. Not every matching forms a cycle

  22. Not every matching forms a cycle

  23. Not every matching forms a cycle

  24. Not every matching forms a cycle

  25. Cost of an edge in M 270 200 Cost of this edge is 70

  26. Cost of an edge in M 100 1200 Cost of this edge is 1100

  27. v1 v2 v3 v4 v5 v6 u1 u2 u3 u4 u5 u6 Sum of cost of edges Minimum Bipartite Cyclic Augmentation Input: U = {u1, u2,…, un} V = {v1, v2,…, vn} H : a perfect matching between U and V • Output: A perfect matching M such that • HM forms a cycle • ∑(u,v)M|u-v| is minimized

  28. v1 v2 v3 v4 v5 v6 u1 u2 u3 u4 u5 u6 Cost of most expensive edges Bottleneck Bipartite Cyclic Augmentation Input: U = {u1, u2,…, un} V = {v1, v2,…, vn} H : a perfect matching between U and V • Output: A perfect matching M such that • HM forms a cycle • max(u,v)M{|u-v|} is minimized

  29. Outline • MD : the minimum cost matching • We will transform MD to an optimal cost matching using exchange operations • Some properties of an optimal matching to prune down the space of exchanges required • Exchange graph • Optimal matching – MST in exchange graph

  30. MD : the minimum cost matching

  31. MD : the minimum cost matching The minimum cost matching may not be a cyclic augmentation

  32. Exchanges

  33. Exchanges

  34. Exchanges Exchanges between different cycles merges them

  35. Cost of an Exchange

  36. Cost of an Exchange

  37. Cost of an Exchange x Cost of the exchange is 2.x

  38. Transform MD into a minimum cost cyclic augmentation using exchange operations Which exchanges will yield the optimal cyclic augmentation?

  39. Clusters l1 l2 l3 l4 l5 l6 l7 l8

  40. Exchange Graph l1 l2 l3 l4 l5 l6 l7 l8 67 Nodes ≡ Cycles in MD 12 56 45 78 Edges ≡ Adjacent Clusters in MD 23 34

  41. Exchange Graph l1 l2 l3 l4 l5 l6 l7 l8 67 12 56 Weight on Edges ≡ Cost of corresponding . Exchange 45 78 23 34

  42. Solution Exchanges corresponding to the Minimum Spanning Tree on Exchange Graph yield a minimum cost cyclic augmentation

  43. Results • Minimum Bipartite Cyclic Augmentation • Bottleneck Bipartite Cyclic Augmentation Ω(n log n) 3 approx. algorithm

  44. The EndThank You

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