Inverse Kinematics for Molecular World

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Inverse Kinematics for Molecular World

• Sadia Malik
• April 18, 2002
• CS 395T
• U.T. Austin

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Protein Docking
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Goals

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

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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.

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Constraints

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

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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?

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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.

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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.

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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.

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Tree Structure
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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.

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Denavit Hartenburg
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Hypothetical Molecule
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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.

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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.

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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.

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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.

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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.

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References

• 1 Zhang, M. and Kavraki, L. Finding Solutions
  of the Inverse Kinematics Problems in
  Computer-aided Drug Design,
• http//cs-tr.cs.rice.edu/Dienst/UI/2.0/Describe/n
  cstrl.rice_cs/TR02-385
• 2 Parsons, D. and Canny, J. 1994. Geometric
  Problems in Molecular Biology and Robotics,
  http//citeseer.nj.nec.com/parsons94geometric.html
  
• 3 Lavalle, S.,Finn, P., Kavraki, L., Latombe,
  J. 2000. A Randomized Kinematics-Based Approach
  to Pharmacophore-Constrained Conformational
  Search and Database Screening,
  http//robotics.stanford.edu/latombe/papers/jcc/p
  aper.pdf