Geometric Algorithms for Conformational Analysis of Long Protein Loops J. Cortess, T. Simeon, M. Remaud- Simeon, V. Tran

Motivation Filter unfeasible loop conformations to aid searching conformational space for various application: –Protein loop modeling –Molecular simulations: conformational changes under environmental conditions.

Structural Constraints Loop-closure Steric clash – internal segment clashes (self- clashes), external clashes, VdW radii.

Loop Closure Approaches Analytical – IK techniques Optimization – e.g. CCD Database based methods

Clash Filtering Approaches Energetic – accepting/rejecting a conformation according to some energetic (repulsive VdW energy) cutoff. Geometric – “clash grids”. Robotics – motion planning.

Robotics – collision avoidance Exploration of the conformations space, searching for feasible conformations. Existing techniques capture the topology of the feasible space within a data-structure (graph or a tree) by performing random exploration.

Outline Part 1: presents conformational sampling technique satisfying loop-closure and clash avoidance constraints. Part 2: presents a data structure capturing the connectivity of the geometrically feasible conformations sub-space.

Problem Formulation: Geometric Model Van der Waals molecule model Standard Phi-Psi model Conformation q is a an array of dihedral angles of the backbone and side-chains.

The Homogeneous Transformation Matrix

Loop Closure Constraint Clash avoidance – distance between non- bonded atoms must not be shorter than the sum of their VdW radii. Condition must be satisfied between atoms of the articulated segment and between atoms of the rest of the molecule. Problem Formulation: Geometric Constraint

Part 1: Conformational Sampling Compute random conformation achieving loop- closure and clash avoidance constraints in 3D. Array of dihedral angles: θ 1,θ 2, … θ n A generic 3D collision detection algorithm (T. Siméon, C. van Geem, 2001) Sample angles randomly at random side-chain order. Check for clashes

Random Backbone Conformation Generation Passive sub-chain: dependent variables J 3, J 4, J 5. (Corresponding to three residues and six dihedral angles) Active sub-chain: independent variables J 1, J 2, J 6. Closed Loop

Random Loop Generator (RLG) Algorithm A standard inverse kinematics problem

RLG Algorithm: Backbone Generation Reachable WorkSpace of Chain6-2 Closure Range of θ 1  Solving the positional-reachable problem is simple and fast approximation to the exact closure range

RLG Algorithm: Backbone Generation

Polypeptide Extension (approximation) l π – length of polypeptide chain when all the dihedral angles at π. Ĩ – upper bound on the chain ’ s length. It is the sum of the distances between consecutive C α atoms. The extension of a chain is randomly sampled from a distribution between l π and Ĩ.

Part 2: Conformational Space Exploration Apply Sampling-based Motion Planning Techniques to the Protein Loop Problem. In particular, the Probabilistic RoadMap (PRM) approach. Rapidly-exploring Random Tree (RRT) is a data structure and a sampling scheme to quickly search high-dimensional constrained spaces.

Rapidly Exploring Random Tree (RRT) Properties: Expands quickly Unbiased relative to random walk. Vertices are uniformly distributed Short paths

Clash-Free conformation subspace Conformations satisfying loop- closure Random conf. Or from DB Sample q a Linear Inter. and solving the closure eq. for q p Gaussian smpl Believed to be an estimate to coverage Conformations w/ clashes Incremental Exploration of Feasible Space

Results Loop 7 Motion of Loop 7 may have a pivotal rule in facilitating molecules interactions.

Results

CCD vs. RLG Similar performance in terms of finding conformations close to the wild-type. RLG computes exact solutions while CCD outputs approximated solutions. CCD may favor large changes in the first residues. RLG produces a more uniformly distributed samples.

Future Directions Check clashes at each stage. Tailor a collision detection algorithm for the molecular application (Collision detection is by far the most computation expensive task) Incorporate energetic analysis (constraints) into the incremental search technique.