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 max saddlemin

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

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

Surgery However: not continuous 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

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

Illustration

2D Results NaiveMatchPairMatch

2D Results – Cont’ : top r ranked transformations of : top s ranked transformations of How well does covers ?

Future Work Implement Elevation function in 3D Better matching algorithm in 3D? Local improvement starting from a position with collision