Computational Geometry, Algorithmic Robotics, and Molecular Modeling

1. Computational Geometry, Algorithmic Robotics, and Molecular Modeling Dan Halperin School of Computer Science Tel Aviv University June 2007

2. Proteins 101 (some slides from cs273/Stanford) Involved in all functions of our body: metabolism, motion, defense, etc. Michael Levitt

3. Protein Long sequence of amino-acids (dozens to thousands), also called residues from a dictionary of 20 amino-acids

4. Part I, Geometry

5. Role of Geometric Models • Represent the possible shapes of a protein (compare/classify shapes, find motifs) • Answer proximity queries: Which atoms are close to a given atom? (computation of energy) • Compute surface area (interaction with solvent) • Find shape features, e.g., cavities (ligand-protein interaction)

6. Geometric Models of Bio-Molecules • Hard-sphere model (van der Waals radii) • Van der Waals surface Van der Waals radii in Å

7. Geometric Models of Bio-Molecules • Hard-sphere model (van der Waals radii) • Van der Waals surface • Solvent- accessible surface • Molecular surface

8. Computed Molecular Surfaces Probe of 1.4Å Probe of 5Å

9. Is it art?

10. Pioneering Work on Surfaces • Lee and Richards, 1971– Solvent accessible surface • Richards, 1977– Smooth molecular surface • Connolly, 1983– First computation of smooth molecular surface

11. Computational Geometry In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. The main impetus for the development of computational geometry as a discipline was progress in computer graphics, computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature. Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), computer-aided engineering (CAE) (programming of numerically controlled (NC) machines). http://en.wikipedia.org/wiki/Computational_geometry

12. Computation of Hard-Sphere Surface (Grid method [Halperin and Shelton, 97]) • Each sphere intersects O(1) spheres • Computing each atom’s contribution to molecular surface takes O(1) time • Computation of molecular surface takes Θ(n) time Why? D. Halperin and M.H. Overmars Spheres, molecules, and hidden surface removalComputational Geometry: Theory and Applications 11 (2), 1998, 83-102. Alternative:Edelsbrunner, 1995 – Computing the molecular surface using Alpha Shapes

13. Geometric Problems (static) Krebs et al. (2003) J. Biol. Chem. 278, 50217. [Enosh et al 2004] [Yaffe et al 2007]

14. Part II, Robotics/Motion

15. Robotics RAS field of interest (ICRA, Rome, April 2007) : Robotics focuses on sensor and actuator systems that operateautonomously or semi-autonomously (in cooperation with humans) inunpredictable environments.  Robot systems emphasize intelligence andadaptability,  may be networked, and are being developed for manyapplications  such as service and personal assistants; surgery andrehabilitation; haptics; space, underwater, and remote exploration andteleoperation; education, entertainment; search  and rescue; defense;agriculture; and intelligent vehicles.

16. Algorithmic Robotics and Motion Planning

17. Proteins as Robots Long sequence of amino-acids (dozens to thousands), also called residues from a dictionary of 20 amino-acids

18. Robots with many dofs http://www.youtube.com/watch?v=k-VgI4wNyTo

19. Simulation and Predicition of Molecular Motion [Enosh-Raveh 2007]

20. Molecular SimulationsMonte Carlo Simulation (MCS) • Estimate thermodynamic quantities • Search for low-energy conformations and the folded structure Popular method for sampling the conformation space of proteins:

21. MCS: How it works • Propose random change in conformation • Compute energy E of new conformation • Accept with probability: Requires >>106 steps to sample adequately

22. D C G H A B E F I The ChainTree [Lotan,Schwarzer,H,Latombe 2004] TAIBAH TAEBAD TEIBEH TACBAB TCEBCD TEGBEF TGIBGH TABBA TBCBB TCDBC TDEBD TEFBE TFGBF TGHBG THIBH

23. Test 1-DoF change 5-DoF change [68 res.] [144 res.] [374 res.] [755 res.] [68 res.] [144 res.] [374 res.] [755 res.]

24. Dynamic Maintenance of Molecular Surfaces [Eyal-H 2005]

25. Motion Predicition [Enosh,Fleishman,Ben-Tal,H 2007]

26. q4 q5 q2 q3 q1 rigid groups of atoms T Inverse Kinematics (IK) Given a kinematic chain (serial linkage), the position/orientation of one end relative to the other (closed chain), find the values of the joint parameters

27. Relation to Robotics

28. Why is IK useful for proteins?<IK = loop closure> • Filling gaps in structure determination by X-ray crystallography • Studying the motion space of “loops” (secondary structure elements connecting α helices and β strands), which often play a key role in: • enzyme catalysis, • ligand binding (induced fit), • protein – protein interactions • Sampling conformations using homology modeling • Chain tweaking for better prediction of folded state

29. Major Goals • Dynamic maintenance of molecular properties in MD-type simulations • Simulation and prediction of motion with more dofs • Fast and accurate IK (loop closure)

30. THE END