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

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

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

Introduction Nuclear Magnetic Resonance (NMR)

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.

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

Determining Protein Structure Using NMR 1.NMR Spectral Data generation 2.Peak Picking 3.Peak Assignment 4.Structural Restraint Extraction 5.Structure Calculation

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

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

Problem Description (Input) (a 1,b 1 ) (a 2,b 2 ) (a 3,b 3 ) (a 4,b 4 ) (a 5,b 5 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1 ?

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 b5b5 b6b6 b4b4 b3b3 b2b2 b1b1

Problem Description (Output) (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) a1a1 a3a3 a6a6 a4a4 a2a2 a5a5 b3b3 b4b4 b2b2 b6b6 b1b1 b5b5 ≤≤≤≤≤ ≤≤≤≤≤

Equivalent Problem Description (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) u1u1 u2u2 u3u3 u4u4 u5u5 u6u6 v5v5 v6v6 v4v4 v3v3 v2v2 v1v1

Cyclic Augmentation (a 1,b 1 ) (a 5,b 5 ) (a 3,b 3 ) (a 4,b 4 ) (a 2,b 2 ) (a 6,b 6 ) u1u1 u2u2 u3u3 u4u4 u5u5 u6u6 v5v5 v6v6 v4v4 v3v3 v2v2 v1v1 A matching M is called a cyclic augmentation if H  M forms a hamiltonian cycle.

Not every matching forms a cycle

Cost of an edge in M Cost of this edge is 70

Cost of an edge in M Cost of this edge is 1100

Sum of cost of edges Minimum Bipartite Cyclic Augmentation Input:U = {u 1, u 2,…, u n } V = {v 1, v 2,…, v n } H : a perfect matching between U and V Output: A perfect matching M such that 1.H  M forms a cycle 2.∑ (u,v)  M |u-v| is minimized u1u1 u2u2 u3u3 u4u4 u5u5 u6u6 v5v5 v6v6 v4v4 v3v3 v2v2 v1v1

Cost of most expensive edges Bottleneck Bipartite Cyclic Augmentation Input:U = {u 1, u 2,…, u n } V = {v 1, v 2,…, v n } H : a perfect matching between U and V Output: A perfect matching M such that 1.H  M forms a cycle 2.max (u,v)  M {|u-v|} is minimized u1u1 u2u2 u3u3 u4u4 u5u5 u6u6 v5v5 v6v6 v4v4 v3v3 v2v2 v1v1

Outline M D : the minimum cost matching We will transform M D 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

M D : the minimum cost matching

The minimum cost matching may not be a cyclic augmentation

Exchanges

Exchanges between different cycles merges them

Cost of an Exchange

x Cost of the exchange is 2.x

Transform M D into a minimum cost cyclic augmentation using exchange operations Which exchanges will yield the optimal cyclic augmentation?

Clusters l1l1 l2l2 l7l7 l6l6 l5l5 l4l4 l3l3 l8l8

Exchange Graph l1l1 l2l2 l7l7 l6l6 l5l5 l4l4 l3l3 l8l8 Nodes ≡ Cycles in M D Edges ≡ Adjacent Clusters in M D

Exchange Graph l1l1 l2l2 l7l7 l6l6 l5l5 l4l4 l3l3 l8l8 Weight on Edges ≡ Cost of corresponding. Exchange

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

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

The End Thank You