Molecular Distance Geometry

1
Molecular Distance Geometry
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

2
Exact Distance Geometry Problem
Given exact distances between all pairs of
points, find the locations of the points.
Given exact distances between all pairs of atoms
in a molecule, find the coordinates of the atoms.
3
Sparse Sets of Distances
Given exact distances for a subset of pairs of
points, find the locations of the points.
Given exact distances for a subset of pairs of
atoms in a molecule, find the coordinates of the
atoms.
4
Graph Embedding
Given a weighted graph G (V, E, W), where Vvi
i1,,n, E(vi,vj) (i, j) in S, and
Wwi,jw (vi,vj) (vi,vj) in E,
v2
x2
5
3
5
3
v1
v3
x1
x3
4
4
5
One-Dimensional Problem
Given a weighted graph G (V, E, W), where Vvi
i1,,n, E(vi,vj) (i, j) in S, and
Wwi,jw (vi,vj) (vi,vj) in E,
v2
x1
3
7
3
4
7
x2
x3
v1
v3
4
6
Two Dimensional Problem
Given a weighted graph G (V, E, W), where Vvi
i1,,n, E(vi,vj) (i, j) in S, and
Wwi,jw (vi,vj) (vi,vj) in E,
v2
x2
5
3
5
3
v1
v3
x1
x3
4
4
7
Three Dimensional Problem
Given a weighted graph G (V, E, W), where Vvi
i1,,n, E(vi,vj) (i, j) in S, and
Wwi,jw (vi,vj) (vi,vj) in E,
v2
x2
4
x4
5
3
5
3
3
4
v4
d
3
d
v1
v3
x1
x3
4
4
8
Under-Determined System
kn -- total number of coordinates
k(k1)/2 -- k translations, k(k-1)/2 rotations
9
Over-Determined System
10
Inconsistent Data
v2
x2
8
3
8
3
x3
v1
v3
x1
x3
4
4
Triangular inequality, c a b, may be
violated!
11
Flexible Structures
v2
v4
x2
x4
v1
v3
x1
x3
v2
v4
x2
x4
x2
x4
v1
v3
x1
x3
The structure can be deformed continuously
without violating any distance constraints.
12
Rigid Structures
v2
v4
x2
x4
v1
v3
x1
x3
v2
v4
x2
x4
x2
x4
v1
v3
x1
x3
The structures cannot be deformed any more!
13
Unique Structure
x2
x4
x1
x3
x2
14
Reflections
x2
4
x4
rigid unique?
5
3
3
d
v2
x1
x3
4
3
5
4
v4
3
d
v1
v3
4
x2
x4
4
5
3
3
d
x1
x3
4
15
Inexact Distance Data
16
Lower / Upper Bounds