Inverse Kinematics for Molecular World Sadia Malik April 18, 2002 CS 395T U.T. Austin

Protein Docking

Goals Determine whether ligand and receptor can associate. Predict the geometric structure of the combined complex.

Solution Requirements Solve geometric problem of computing “reasonable” relative configurations. –Inverse Kinematics can be used to solve this problem. Solve chemical problem of evaluating the free energies of tentative molecules.

Constraints Rotational –Torsional degrees of freedom and the allowed values. Distance –Relative position between feature atoms.

Does Inverse Kinematics Apply? For a given desired position of the end effector, what set of angle values for the joints will achieve this goal?

Robots and Bio-Molecules? Both have three-dimensional geometry. Structure of both depends on the geometry. –Most of the chemical and biological function of molecules can be explained as a function of their geometric conformations. –Synthesis of robots and robotic algorithms is tightly bound with geometric analysis.

Molecules as Robots? Molecules can be modeled as tiny robots. –Treat molecule as rigid body –Each atom acts as a point, or, in some cases, can group atoms such that none of the bonds among the atoms rotate. –Replace each link of the robot by a section of molecular chain with fixed dihedral angles and each joint with variable dihedral angle.

Inverse Kinematics Can use Inverse Kinematics –For robotics, dihedral angles used to solve IK problems.. –For molecules, torsional Angles used to solve IK problems. Position of any point can be represented as polynomial or trigonometric equation.

Tree Structure

Information Per Atom Each atom has –Van der Waals radius –Bond length –Bond angle –Set of possible torsion angles In most studies, only variations in torsion angles are considered.

Denavit Hartenburg

Hypothetical Molecule

Post Kinematics Calculations Kinematics Error Function –Expresses how closely the feature atoms’ constraints are satisfied. Minimum Energy Function –Measures the likelihood that the computed conformation can exist in nature. –Inverse Kinematics can generate several “false positive” geometries.

Other Molecular Problems Protein Folding –Predict native protein structure using amino- acid sequence knowledge only. –Requires molecular dynamics algorithms Ring Closure –Conformation of cyclic molecules in which the cyclic covalent structure of the molecule is maintained. –Can use inverse kinematics to solve this.

Our Molecular World Algorithms used to compute geometric configurations can be used for molecular world project. –Information given in a PDB file can be used to solve inverse kinematics problems for molecules.

Plans for Animation Work with the Human Body group to implement software that will solve molecular configuration for our project. –Similar co-ordinate system. –Both projects can use Inverse Kinematics to compute animation frames.

Conclusion Molecular structure, though much smaller in size, can be approximated to robotic structure. Inverse Kinematics is a very useful mechanism to compute geometrical structures.

References [1] Zhang, M. and Kavraki, L. “Finding Solutions of the Inverse Kinematics Problems in Computer-aided Drug Design”, [2] Parsons, D. and Canny, J “Geometric Problems in Molecular Biology and Robotics”, [3] Lavalle, S.,Finn, P., Kavraki, L., Latombe, J “A Randomized Kinematics-Based Approach to Pharmacophore-Constrained Conformational Search and Database Screening”,