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 two protein molecules A and B, each being represented by the sequence of the being represented by the sequence of the 3-D coordinates of its 3-D coordinates of its CC atoms.atoms.

Task: largest flexible alignment by largest flexible alignment by decomposing the two molecules into a decomposing the two molecules into a minimalminimal number of rigid fragment pairs number of rigid fragment pairs having similar 3-D structure.having similar 3-D structure.

Detection of Congruent Detection of Congruent Rigid Fragment PairsRigid Fragment Pairs

Joining Rigid Joining Rigid Fragment PairsFragment Pairs

Rigid Rigid Structural ComparisonStructural Comparison

ClusteringClustering(removing ins/dels)(removing ins/dels)

FlexProt Main Steps

Structural Similarity Matrix

Congruent Rigid Fragment Pair

j

i+1

j+1

i

j-1

i-1

vi-1 vi vi+1

wj-1 wj wj+1

Fragkt(l) = vk … vi ... vk+l-1

wt … wj … wt+l-1

RMSD (Fragkt(l) ) <

Detection of Congruent Rigid Fragment Pairs

k

t

k+l-1

t+l-1

RMSD Computation

VVii …...…... VVi+li+l

WWjj ...…...… WWj+lj+l

VVkk …...…... VVk+mk+m

WWtt ...…...… WWt+mt+mPP== Q=Q=

P U Q

RMSD( P U Q ) in O(1) time

NOT O( |P|+|Q| )

RMSD( P )RMSD( P )

RMSD( Q )RMSD( Q )

Detection of Congruent Detection of Congruent Rigid Fragment PairsRigid Fragment Pairs

Joining Rigid Joining Rigid Fragment PairsFragment Pairs

Rigid Rigid Structural ComparisonStructural Comparison

ClusteringClustering(removing ins/dels)(removing ins/dels)

FlexProt Main Steps

How to Join Rigid Fragment Pairs?

Graph Representation

Graph NodeGraph Node

Graph EdgeGraph Edge

Graph Representation •The fragments are in ascending order.The fragments are in ascending order.

•The gaps (ins/dels) are limited.The gaps (ins/dels) are limited.

•Allow some overlapping.Allow some overlapping.

W

+ Size of the rigid fragment pair (node b)

- Gaps (ins/dels)

- OverlappingPenalties

a b

Graph Representation

W_i

W_k

W_t

W_m

W_n

• DAG DAG (directed acyclic graph)(directed acyclic graph)

Optimal Solution?

•““All Shortest Paths” All Shortest Paths” O(|E|O(|E|**|V|+|V||V|+|V|22) (for DAG) ) (for DAG)

W_i

W_k

W_t

W_m

W_n

•““Single-source shortest paths”Single-source shortest paths” O(|E|+|V|) O(|E|+|V|)

Detection of Congruent Detection of Congruent Rigid Fragment PairsRigid Fragment Pairs

Joining Rigid Joining Rigid Fragment PairsFragment Pairs

Rigid Rigid Structural ComparisonStructural Comparison

ClusteringClustering(removing ins/dels)(removing ins/dels)

FlexProt Main Steps

Clustering (removing ins/dels)

T1

T2

If joining two fragment pairs gives small RMSD (T1 ~ T2) then put them into one cluster.

Detection of Congruent Detection of Congruent Rigid Fragment PairsRigid Fragment Pairs

Joining Rigid Joining Rigid Fragment PairsFragment Pairs

Rigid Rigid Structural ComparisonStructural Comparison

ClusteringClustering(removing ins/dels)(removing ins/dels)

FlexProt Main Steps

Correspondence Problem

Molecular Surface Representation

Applications to docking

Motivation

Prediction of biomolecular recognition.

Detection of drug binding ‘cavities’.

Molecular Graphics.

1. Solvent Accessible Surface – SAS2. 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

knob

hole

Chymotrypsin surface colored by solid angle (yellow-convex, blue-concave)