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Geometric Algorithms for Conformational Analysis of Long Protein Loops

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

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Geometric Algorithms for Conformational Analysis of Long Protein Loops

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

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

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

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

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

  6. 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.

  7. 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.

  8. 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.

  9. The Homogeneous Transformation Matrix

  10. Problem Formulation: Geometric Constraint • 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.

  11. Part 1: Conformational Sampling Compute random conformation achieving loop-closure and clash avoidance constraints in 3D. Array of dihedral angles: θ1,θ2,…θn A generic 3D collision detection algorithm (T. Siméon, C. van Geem, 2001) Sample angles randomly at random side-chain order. Check for clashes

  12. Closed Loop Random Backbone Conformation Generation Passive sub-chain: dependent variables J3, J4, J5. (Corresponding to three residues and six dihedral angles) Active sub-chain: independent variables J1, J2, J6.

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

  14. RLG Algorithm: Backbone Generation Reachable WorkSpace of Chain6-2 Closure Range of θ1  Solving the positional-reachable problem is simple and fast approximation to the exact closure range

  15. RLG Algorithm: Backbone Generation

  16. Polypeptide Extension (approximation) lπ – 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 Cα atoms. The extension of a chain is randomly sampled from a distribution between lπ and Ĩ.

  17. 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.

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

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

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

  21. Results

  22. 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.

  23. 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.

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