Topology Based Selection and Curation of Level Sets

1. Topology Based Selection and Curation of Level Sets Andrew Gillette Joint work with Chandrajit Bajaj and Samrat Goswami

2. Problem Statement Given a trivariate function we want to select a level set L(r) = with the following properties: • L(r) is a single, smooth component. • L(r) does not have any topological or geometrical features of size less than where the size of a feature is measured in the complementary space. The value of is determined by the application domain.

3. Application: Molecular Surface Selection • We need a molecular surface model to study molecular function (charge, binding affinity, hydrophobicity, etc). • We can create an implicit solvation surface as the level set of an electron density function. • Our selected level set should be a single component and have no small features (tunnels, pockets, or voids). “The World of the Cell” 1996

4. Computational Pipeline Atomic Data (e.g. pdb files for proteins) Physical Observation Gaussian Decay Model Volumetric Data (e.g. cryo-EM for viruses) Trivariate Electron Density Function Our algorithm: Level Set (isosurface) Selection Level Set (isosurface) Curation

5. Example 1: Gramicidin A Images created from Protein Data Bank file 1MAG • Three topologically distinct isosurfaces for the molecule are shown • We need information on the topology of the complementary space to select a correct isosurface

6. Example 2: mouse Acetylcholinesterase • Two isosurfaces for the molecule are shown, with an important pocket magnified • We need information on the geometry of the complementary space to select a correct isosurface and ensure correct energetics calculations

7. Example 3: Nodavirus Data from Tim Baker, UCSD; Images generated at CVC, UT Austin • A rendering of the cryo-EM map and two isosurfaces of the virus capsid are shown • We need to locate symmetrical topological features to select a correct isosurface

8. Mathematical Preliminaries • Contour Tree • Voronoi / Delaunay Triangulation • Distance Function and Stable Manifolds

9. Prior Related Work Isosurface Selection via Contour Tree Modern application of contour trees: “Trekking in the alps without freezing or getting tired” (de Berg, van Kreveld: 1997) “Contour trees and small seed sets for isosurface traversal” (van Kreveld, van Oostrum, Bajaj, Pascucci, Schikore: 1997) Computation via split and join trees: “Computing contour trees in all dimensions” (Carr, Snoeyink, Axen: 2001) Betti numbers and augmented contour trees: “Parallel computation of the topology of level sets” (Pascucci, Cole-McLaughlin: 2003) Distance Function and Stable Manifold Computation “Shape segmentation and matching with flow discretization” (Dey, Giesen, Goswami: 2003) “Surface reconstruction by wrapping finite point sets in space” (Edelsbrunner: 2002) “The flow complex: a data structure for geometric modeling.” (Giesen, John: 2003) “Identifying flat and tubular regions of a shape by unstable manifolds” (Goswami, Dey, Bajaj: 2006)

10. Level Sets and Contours • In this talk, f(x,y,z) will denote the electron density at the point (x,y,z) • An isosurface in this context is a level set of the function f, that is, a set of the type • Each component of an isosurface is called a contour • We select an isosurface with a single component via the contour tree Isosurface with three contours

11. Contour Tree • Recall • A critical isovalue of f is a value r such that f -1(r) is not a 2-manifold • Examples: r is a value where contours emerge, merge, split, or vanish. r = 1 r = 2 r = 3 non-critical critical non-critical

12. Contour Tree • The contour tree is a tool used to aid in the selection of an isosurface • Vertices: subset of critical values of f • Edges: connect vertices along which a contour smoothly deforms Increasing isovalues  Isovalue selector

13. Isosurface  (from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase) Bar below green square indicates isovalue selection 

14. Isosurface  (from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase) Bar below green square indicates isovalue selection 

15. Isosurface  (from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase) Bar below green square indicates isovalue selection 

16. Isosurface  (from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase) Bar below green square indicates isovalue selection 

17. Isosurface  (from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase) Bar below green square indicates isovalue selection 

18. Voronoi Diagram • Let P be a finite set of points in • The set of Vp partition and “meet nicely” along faces and edges. • A 2-D example is shown 

19. Delaunay Diagram Vor P • Voronoi diagram = Vor P • Delaunay diagram = Del P • Del P is defined to be the dual of Vor P • Vertices = P • Edges = dual to Vp facets • Facets = dual to Vp edges • Tetrahedra = centered at Vor P vertices Del P

20. The distance function • Let S be a surface smoothly embedded in • Let P be a finite sampling of points on S. Then we approximate:

21. Critical points of hP by analogy

22. Minimum Saddle Maximum Flow Sample Point Orbit • Flow describes how a point x moves if it is allowed to move in the direction of steepest ascent, that is, the direction that most rapidly increases the distance of x from all points in P. • The corresponding path is called an orbit of x.

23. Stable Manifolds Given a critical value c of hP, the stable manifold of c is the set of points whose orbits end at c.

24. Algorithm and Results • Description of Algorithm • Results • Future Work

25. Algorithm in words Given an isosurface S sampled by pointset P: • Find critical points of distance function hP • Classify critical points exterior to S as max, saddle, or saddle incident on infinity • Cluster points based on stable manifolds • Classify clusters based on number of mouths • Rank clusters based on geometric significance

26. Algorithm in pictures 1 2 3 4 5 Void: Pocket: Tunnel:

27. Results

28. Results From 1RIE pdb (Rieske Iron-Sulfur Protein of the bovine heart mitochondrial cytochrome BC1-complex)

29. Results • The chaperon GroEL; generated from cryo-EM density map. • The large tunnel is used for forming and folding proteins.

30. Future Work • What makes a point set P sufficient for applying our algorithm? • How can we provide a “quick update” to the distance function for a range of isovalues? • Compare energy calculations on our pre- and post-curation surfaces.

31. Thank you! (Danke)