Segmenting Low-resolutional Deforming Protein Structures

Yusu Wang

08/22/2005

Joint work with Leo Guibas

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Introduction

Molecular structure Crucial in understanding functionality Hard to obtain high-resolution struct. Snapshots

Low-resolution structure Cryo-EM :

Fitting detailed atomic structure Direct manipulation

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Motivation

Snapshots : Several conformations

of one protein

Goal: No atomic structure

(low resolution) Study deformation,

pathways .. etc

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Problem

Input: Two surfaces Roughly rigid sub-

components

Output: Segmentation Correspondence

identification

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Where is the Difficulty

Coarse (rigid) alignments Not too hard

Nearest neighbors Based on Euclidean /

normals not good

Inherent coordinates invariant to such motion

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Geodesic Distances

To get a coord. sys: Use landmarks: one set on each

surface A: (a1, a2, .., ak) B: (b1, b2, .., bk)

Geodesic dist to landmarks

Landmarks: A few reliable correspondences At least one from each component

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Approach Overview

Identify a few landmarks from each surface

Induce segmentation / correspondences

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Preprocessing

Given surfaces A and B: Compute elevation max for each surface

Each max: a point pair capturing a meaningful feature

P (Q): set of points from elevation max from A (B) P’ (Q’) : set of corresponding point-pairs

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Idea 1 : Every good Alignment Counts

Align any two elevation pairs p from P’ and q from Q’

Score the resulting alignments

Collect S: Set of good alignments (i.e., with high score)

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Idea 2: Majority Wins

Given an alignment: Coord.: (x, y, z, d1, d2) NN(p) : nearest neighbor of p P in Q Distance measure:

both geometric and geodesic

Given S : Vote(p, q) : number of

times q Q is NN(p) High votes indicates

reliable correspondences

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Idea 3: Point is Not Enough

Molecular surfaces: Homogeneous Point is not discriminative enough

Larger features: Segments, triangles, etc Balance between complexity

and quality

Vote for point-pairs, instead of points

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Putting Everything Together

Given A and B Compute set of elevation max P, Q, (P’, Q’) Compute S

By aligning point-pairs P’ and Q’ For each point-pair from P’, vote for its NN in Q’

Based on only good alignments S Take top votes

C: a set of reliable correspondences between point-pairs

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Select Landmarks

C : a set of good corresponding point-pairs Choose top-ranked pair Choose second pair

Output: 2 pairs (4 points) on

each surface

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Approach Overview

Identify a few landmarks

Induce segmentation / correspondences

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Segmentation

Given 4 landmarks on each surface Coord.:

(x, y, z, d1, d2, d3, d4)

Point-pairs p1(q1) and p2(q2)

Align p1 with q1 => transformation T1

Align p2 with q2 => transformation T2

align component 1

align component 2

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Segmentation -- cont.

To segment component 1 and 2: Define f : A -> R as

f (x) = Dist (x, NN(x, T (B) ) ) f (x) = Dist (x, NN(x, T (B) ) )

Classify x A to be in component 1 : if f (x) < f (x) 2 : otherwise

i1 1

1 2

1 2

2 2

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Another Example

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Next Step

Correspondence based on segmentation More correspondences based on landmarks As-rigid-as-possible deformation

Pseudo-atoms that respects the segmentation Normal modes analysis

What if geodesics also fail ? Voting + virtrual coordinates