Docking III:Matching via Critical Points

Yusu Wang

Joint Work with P. K. Agarwal, H. Edelsbrunner, J. Harer

Duke University

Motivation

Docking problem Partial matching Two steps

Find coarse matching Local improvement

Input: protein A and B Output: a set of coarse alignments

Matching Surfaces

Model protein As a surface instead of set of balls

Sample special points Knobs and caves

Align two sets of points Under collision-free constraint

Our Approach

Overview:

Step 1. Extract critical points Design Morse function

Step 2. Align critical points Use both topological and geometric info.

Critical Points

: manifold (closed curves/surfaces) : Morse function Critical points: min, max, saddles for

RMF :

M

F

max saddle min

Pairing

Critical points capture topological information Critical pairs, persistence of critical pairs

Some Morse Functions

Curvature Too local

Connolly function Ratio of inside/outside perimeters

Atomic Density Function

Proposed by Kuhn et al.

Best fit

cy cp )(

c

416 100

in 3D

Height Function

Atomic density function: Critical points nice Critical pairs good for removing noise But …

Height function Captures good features in vertical direction

Elevation Function

Each point critical in normal direction

Define )()()( qpkp n

Surgery

However: not continuous

MM̂

RM ˆ:

Blame the manifold! : apply surgery on Elevation function:

in 2D

~12~30

Surgery in 2D

Alignment

Input: Two proteins A and B (P and Q) Two sets of critical points/pairs

Output: Set of transformations for protein B

Sorted by score(A, T(B))

NaïveMatch

NaiveMatch Alg:

Output:

Take a pair from P, a pair from Q Align two pairs, get transformation T Compute score between A and T(B) Rank transformations by score

naiveT

PairMatch

PairMatch Alg: Take a critical pair from each set Align two critical pairs, get transformation T Rank T ’s by their scores

Output: pairT

Illustration

2D Results

NaiveMatch

PairMatch

2D Results – Cont’

: top r ranked transformations of : top s ranked transformations of How well does covers ?

sTnaiveT

rTpairT

sT rT

Future Work

Implement Elevation function in 3D Better matching algorithm in 3D?

Local improvement starting from a position with collision