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

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

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

Part I, Geometry

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)

Geometric Models of Bio-Molecules  Hard-sphere model (van der Waals radii)  Van der Waals surface HCNOFPSCl Van der Waals radii in Å

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

Computed Molecular Surfaces Probe of 1.4Å Probe of 5Å

Is it art?

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

Computational Geometry In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry.computer sciencegeometry 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.computer graphicsCADCAM 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). roboticsgeographic information systemsintegrated circuit

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 removal Computational Geometry: Theory and Applications 11 (2), 1998, Spheres, molecules, and hidden surface removal Alternative: Edelsbrunner, 1995 – Computing the molecular surface using Alpha Shapes

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

Part II, Robotics/Motion

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

Algorithmic Robotics and Motion Planning

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

Robots with many dofs

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

Molecular Simulations Monte Carlo Simulation (MCS) Popular method for sampling the conformation space of proteins:  Estimate thermodynamic quantities  Search for low-energy conformations and the folded structure

MCS: How it works 2. Compute energy E of new conformation 3. Accept with probability: Requires >>10 6 steps to sample adequately 1.Propose random change in conformation

The ChainTree [Lotan,Schwarzer,H,Latombe 2004] T AB B A T BC B B T CD B C T DE B D T EF B E T FG B F T GH B G T HI B H T AC B AB T CE B CD T EG B EF T GI B GH T AE B AD T EI B EH T AI B AH A B C D E F G H I

Test [68 res.][144 res.][374 res.][755 res.] [68 res.][144 res.][374 res.][755 res.] 1-DoF change5-DoF change

Dynamic Maintenance of Molecular Surfaces [Eyal-H 2005]

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

Inverse Kinematics (IK) T q1q1 q2q2 q3q3 q4q4 q5q5 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 rigid groups of atoms

Relation to Robotics

Why is IK useful for proteins?  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

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)

THE END