Extreme Elevation on a 2-Manifold

Pankaj K. Agarwal

Joint Work with H. Edelsbrunner, J. Harer, Y. Wang

Duke University

Protein-Protein Docking

Sample Protein Complexes

1A22 1JLT 3HLA

Docking

Earlier methods: Lenhof, Nussinov and Wolfson (Tel Aviv ), Kuntz

et al (UC San Diego), Singh and Brutlag (Stanford)

Famous softwares: DOCK (Kuntz), AutoDock (Olson), Zdock (Wen)

More Protein-protein docking: 3D-Dock, HEX, DOT, GRAMM, PPD, BIGGER

Docking: Geometric Approach

Shape Complementarity Partial matching Rigid motion

Extract features Locate feature points More global features?

Matching Features

Our goal: more than good feature points

Related Work Curvature

Too local

Connolly function [Connolly, 83]

Ratio of inside/outside perimeters

Atomic density [Mitchell et al., 01]

FADE, PADRE

Our Results

Elevation function Maxima of elevation Associated with feature

Results being applied to docking Shape-matching algorithms Plugging into local-improvement algorithms

Elevation Function

Elevation on Earth Fit an ellipsoid Choose the center to measure

elevation

For general manifolds No good choice for origin

Function invariant under rigid motion

Persistence Critical points [Munkres, 84]

Critical pairs, persistence [Edelsbrunner et al. 02]

Critical pairs identify topological features

Miss non-vertical features

Family of Height Functions

Fix an arbitrary origin

Given u S2, define hu : M R , as hu(x) = <x, u>

H : M x S2 R : a family of height functions

H(x, u) = hu(x)

Height in All directions

Each x M critical in normal direction

Pair (x,y)

Persistence p(x) = p(y) = |hu(x) – hu(y)|

Elevation

Define E : M R as E(x) = p(x)

Interested in max of E Each x M captures feature in its normal direction E(x) indicates size of feature

However, E is not everywhere continuous

Surgery

Reason: singular tangency Inflexion pt (birth-death pt) Double tangency (interchange)

Blame the manifold! M’ : apply surgery on M Elevation function:

E : M’ R

2D Example

One Component

Pedal Surface

Goal: help understand E

Definition for P : M R3

Singularities Inflexion pt cusp Double tangency xing

Relation

Critical pts for hu P l(u) P = P(M)

Singularities of P along l(u)

Dictionary of Singularities

M Height P

inflexion pt birth-death pt cusp

double tangency interchange xing

Jacobi pt 2 birth-death pts dovetail pt

triple tangency 3 interchanges triple pt

bd-pt + interchange

cusp intersection

2 birth-death pts cusp-cusp crossing

2 interchanges xing-xing crossing

bd-pt + interchange

cusp-xing crossing

Examples

Examples of triple pt and cusp + intersection

Dictionary of Singularities

M Height P

inflexion pt birth-death pt cusp

double tangency interchange xing

Jacobi pt 2 birth-death pts dovetail pt

triple tangency 3 interchanges triple pt

bd-pt + interchange

cusp intersection

2 birth-death pts cusp-cusp crossing

2 interchanges xing-xing crossing

bd-pt + interchange

cusp-xing crossing

Another Example

dovetail

Max of E

An inflexion pt x cannot be a maximum

Dictionary of Singularities

M Height P

inflexion pt birth-death pt cusp

double tangency interchange xing

Jacobi pt 2 birth-death pts dovetail pt

triple tangency 3 interchanges triple pt

bd-pt + interchange

cusp intersection

2 birth-death pts cusp-cusp crossing

2 interchanges xing-xing crossing

bd-pt + interchange

cusp-xing crossing

Classification of Max for E

Four types of max x No singularity involved:

1-legged : x regular, paired w/ regular pt

Singularity involved: 2-legged : x regular, paired w/ double pt 3-legged : x regular, paired w/ triple pt 4-legged : x double pt, paired w/ double pt

New Manifold M´

Point x M´ May correspond to multiple pts in M, and

vice versa regular pt double pt : double tangency triple pt : triple tangency

A max can be: k-legged, k = 1, 2, 3, or 4.

Illustration in M

3-legged max

x

2-Legged

3-Legged

4-Legged

Elevation: Example 1

Elevation: Example 2

Elevation vs. Atomic Density

Elevation Atomic density

Summary

Different feature information Positions (points) Shape (k-legs) Size (persistence) Direction (normal)

Feature pairs: A max and (one of) its leg

Two-steps Coarse align via features Local improvement

Input: Protein A, B

Output: A set of potential alignments

Docking

Conclusion

Elevation function Relation w/ pedal function Classification of max of Elevation function

Compute max for PL-case

More efficient alg. to compute max Matching (docking)

Height Function

Height function h : M R Morse function

No degenerate critical pts No two critical pts have same

function value Critical points

Capture topological features Pairs of critical pts

Persistence Alg. (ELZ01) Pairing decided by order of

heights

x

Characterization of Max

PointMatch

PointMatch Alg:

Output: Tpoint

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

PairMatch PairMatch Alg:

Take a feature pair from each set Align two feature pairs, get transformation T Rank T ’s by their scores

Output: Tpair

Docking for 1BRSRank RMSD

1 1.516

2 2.313

3 35.947

4 21.598

5 21.599

6 17.995

7 34.582

8 34.774

9 45.726

10 22.656

Docking Results

pdb-id Rank RMSD

1BRS 1 1.516

1A22 5 2.745

2PTC 4 6.18

1MEE 1 1.333

1CHO 1 2.712

1JLT 1 2.375

1CSE 2 2.211

3SGB 1 3.209

3HLA 1 3.492