Reconstructing Circular Order from
Inaccurate Adjacency Information
Applications in NMR Data Interpretation
Ming-Yang Kao
Problem Description
(160,520)
(540,160)
160
520
(520,220)
540
220
(190,540)
(220,480)
190
480
(480,190)
Problem Description
Given
Find the correct order
(220,480)
?
(160,520)
?
(480,190)
(540,160)
(520,220)
(190,540)
?
?
?
?
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.
2.
3.
4.
5.
NMR Spectral Data generation
Peak Picking
Peak Assignment
Structural Restraint Extraction
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
Peak Assignment
• Two kinds of information available
– Distribution of spin systems for different amino acids
– The adjacency information between spin systems
Our Focus
Problem Description (Input)
(a1,b1)
b1
(a2,b2)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a5,b5)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
(a3,b3)
b2
b3
b4
b5
b6
?
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
b1
(a5,b5)
b2
b3
b4
b5
b6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1
a2
a3
a4
a5
a6
Problem Description (Output)
(a1,b1)
(a5,b5)
b5 ≤ b1 ≤ b6 ≤ b2 ≤ b3 ≤ b4
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
a1 ≤ a3 ≤ a6 ≤ a4 ≤ a2 ≤ a5
Equivalent Problem Description
(a1,b1)
v1
(a5,b5)
v2
v3
v4
v5
v6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
u1
u2
u3 u4
u5
u6
Cyclic Augmentation
(a1,b1)
v1
(a5,b5)
v2
v3
v4
v5
v6
(a3,b3)
(a4,b4)
(a2,b2)
(a6,b6)
u1
u2
u3 u4
u5
u6
A matching M is called a cyclic augmentation
if HM forms a hamiltonian cycle.
Not every matching forms a cycle
Not every matching forms a cycle
Not every matching forms a cycle
Not every matching forms a cycle
Cost of an edge in M
270
200
Cost of this edge is 70
Cost of an edge in M
100
1200
Cost of this edge is 1100
Minimum Bipartite Cyclic
Augmentation
Input: U = {u1, u2,…, un}
v1
v2
v3
v4
v5
v6
V = {v1, v2,…, vn}
H : a perfect matching
between U and V
u1
Output: A perfect matching M such that
1. HM forms a cycle
2. ∑(u,v)M|u-v| is minimized
Sum of cost of edges
u2
u3 u4
u5
u6
Bottleneck Bipartite Cyclic
Augmentation
Input: U = {u1, u2,…, un}
v1
v2
v3
v4
v5
v6
V = {v1, v2,…, vn}
H : a perfect matching
between U and V
u1
u2
Output: A perfect matching M such that
1. HM forms a cycle
2. max(u,v)M{|u-v|} is minimized
Cost of most expensive edges
u3 u4
u5
u6
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
MD : the minimum cost matching
MD : the minimum cost matching
The minimum cost matching may not be a cyclic augmentation
Exchanges
Exchanges
Exchanges
Exchanges between different cycles merges them
Cost of an Exchange
Cost of an Exchange
Cost of an Exchange
x
Cost of the exchange is 2.x
Transform MD into a minimum cost cyclic
augmentation using exchange operations
Which exchanges will yield the optimal cyclic
augmentation?
Clusters
l1
l2
l3
l4
l5
l6
l7
l8
Exchange Graph
l1
l2
l3
l4
l5
l6
l7
l8
67
12
Nodes ≡ Cycles in MD
56
45
23
34
78
Edges ≡ Adjacent Clusters
in MD
Exchange Graph
l1
l2
l3
l4
l5
l6
l7
l8
67
12
56
45
78
Weight on Edges ≡ Cost of corresponding
Exchange
.
23
34
Solution
Exchanges corresponding to the Minimum
Spanning Tree on Exchange Graph yield a
minimum cost cyclic augmentation
Results
• Minimum Bipartite Cyclic Augmentation
Ω(n log n)
• Bottleneck Bipartite Cyclic Augmentation
3 approx. algorithm
The End
Thank You