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Alignment of Flexible
Molecular Structures
Motivation
Proteins are flexible. One would like to
align proteins modulo the flexibility.
Hinge and shear protein domain motions
(Gerstein, Lesk , Chotia).
Conformational flexibility in drugs.
Problem definition
Flexible Geometric Hashing
Exploit the fact that neighboring parts share
the joint - accumulate mutual information at
the joint.
Achieve complexity of the same order of
magnitude as in rigid alignment.
Flexible protein alignment
without prior hinge knowledge
FlexProt - algorithm

detects automatically flexibility regions,

exploits amino acid sequence order.
Motivation
Geometric Representation
3-D Curve
{vi}, i=1…n
Experimental Results
Experimental Results
FlexProt Algorithm
Input: two protein molecules A and B, each
being represented by the sequence of the
3-D coordinates of its Ca atoms.
Task: largest flexible alignment by
decomposing the two molecules into a
minimal number of rigid fragment pairs
having similar 3-D structure.
FlexProt Main Steps
Detection of Congruent
Rigid Fragment Pairs
Joining Rigid
Fragment Pairs
Clustering
(removing ins/dels)
Rigid
Structural Comparison
Structural Similarity Matrix
Congruent Rigid
Fragment Pair
Detection of Congruent Rigid Fragment
Pairs
k+l-1
k
i-1
t
i+1
i
j-1
j
t+l-1
j+1
vi-1 vi vi+1
wj-1 wj wj+1
Fragkt(l) =
vk
wt
…
…
vi ... vk+l-1
wj … wt+l-1
RMSD (Fragkt(l) ) < e
RMSD Computation
Vi …... Vi+l
P=
Wj ...… Wj+l
Vk …... Vk+m
Q=
Wt ...… Wt+m
P UQ
RMSD( P )
RMSD( P U Q ) in O(1) time
RMSD( Q )
NOT O( |P|+|Q| )
FlexProt Main Steps
Detection of Congruent
Rigid Fragment Pairs
Joining Rigid
Fragment Pairs
Clustering
(removing ins/dels)
Rigid
Structural Comparison
How to Join Rigid
Fragment Pairs ?
Graph Representation
Graph Node
Graph Edge
Graph Representation
•The fragments are in ascending order.
•The gaps (ins/dels) are limited.
•Allow some overlapping.
a
b
+ Size of the rigid fragment pair (node b)
- Gaps (ins/dels)
Penalties
- Overlapping
Graph Representation
W_t
• DAG (directed acyclic graph)
Optimal Solution ?
W_t
•“All Shortest Paths”
O(|E|*|V|+|V|2) (for DAG)
•“Single-source shortest paths”
O(|E|+|V|)
FlexProt Main Steps
Detection of Congruent
Rigid Fragment Pairs
Joining Rigid
Fragment Pairs
Clustering
(removing ins/dels)
Rigid
Structural Comparison
Clustering (removing ins/dels)
T1
T2
If joining two fragment pairs gives small RMSD (T1 ~ T2)
then put them into one cluster.
FlexProt Main Steps
Detection of Congruent
Rigid Fragment Pairs
Joining Rigid
Fragment Pairs
Clustering
(removing ins/dels)
Rigid
Structural Comparison
Correspondence Problem
Molecular Surface
Representation
Applications to docking
Motivation
Prediction of biomolecular recognition.
Detection of drug binding ‘cavities’.
Molecular Graphics.
1. Solvent Accessible Surface – SAS
2. Connolly Surface
Connolly’s MS algorithm
A ‘water’ probe ball (1.4-1.8 A
diameter) is rolled over the van der
Waals surface.
Smoothes the surface and bridges
narrow ‘inaccessible’ crevices.
Connolly’s MS algorithm - cont.
Convex, concave and saddle patches
according to the no. of contact points
between the surface atoms and the
probe ball.
Outputs points+normals according to the
required sampling density (e.g. 10 pts/A2).
Example - the surface of crambin
Critical points based on Connolly
rep. (Lin, Wolfson, Nussinov)
Define a single point+normal for
each patch.
Convex-caps, concave-pits, saddle belt.
Critical point definition
Connolly => Shou Lin
Solid Angle local extrema
hole
knob
Chymotrypsin surface colored by solid
angle (yellow-convex, blue-concave)