Geometric Algorithms for Conformational Analysis of Long Protein Loops

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.

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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.

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The Homogeneous Transformation Matrix
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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.

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Part 1 Conformational Sampling
Compute random conformation achieving
loop-closure and clash avoidance constraints in
3D.
Array of dihedral angles ?1,?2,?n
A generic 3D collision detection algorithm (T.
Siméon, C. van Geem, 2001)
Sample angles randomly at random side-chain
order. Check for clashes
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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.
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Random Loop Generator (RLG) Algorithm
A standard inverse kinematics problem
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RLG Algorithm Backbone Generation
Reachable WorkSpace of Chain6-2
Closure Range of ?1
? Solving the positional-reachable problem is
simple and fast approximation to the exact
closure range
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RLG Algorithm Backbone Generation
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Polypeptide Extension (approximation)
lp length of polypeptide chain when all the
dihedral angles at p. I upper bound on the
chains length. It is the sum of the distances
between consecutive Ca atoms. The extension of a
chain is randomly sampled from a distribution
between lp and I.
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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.

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Rapidly Exploring Random Tree (RRT)

• Properties
• Expands quickly
• Unbiased relative to random walk.
• Vertices are uniformly distributed
• Short paths

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Incremental Exploration of Feasible Space
Clash-Free conformation subspace
Conformations w/ clashes
Conformations satisfying loop-closure
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Results
Motion of Loop 7 may have a pivotal rule in
facilitating molecules interactions.
Loop 7
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Results
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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.

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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.