Using Rigidity to Bias Sampling

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Using Rigidity to Bias Sampling

• Shawna Thomas and Nancy Amato
• Texas AM University

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Mapping the Folding Landscape

• Use probabilistic roadmap method from robotics to
  build roadmap
• Generate samples
• Connect neighboring samples
• Assign edge weights to reflect energetic
  feasibility
• Roadmap approximates the proteins folding
  landscape
• Characterizes the main features of the landscape
• Can extract multiple folding pathways from roadmap

Native state
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Previous Sampling Techniques
UniformSampling
GaussianSampling
Iterative GaussianSampling
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Keys to Studying Larger Proteins

• As poteins get larger, the search space grows so
  we need smarter techniques to build maps
• Primatives
• Sampling Methods
• Distance Metric to identify neighboring samples
• Connection strategy to connect neighboring samples

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Rigidity-Based Sampling

• We use rigidity analysis to identify independent
  hinges, rigid clusters, and dependent hinge sets
• Perturb independent hinges with probability Pflex
• Perturb rigid dof with probability Prigid
• For each dependent hinge set
• Randomly select 1 hinge to perturb with
  probability Pflex
• Perturb remaining hinges with probability Prigid
• Accept the conformation with the probability

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Our Model

• Each amino acid has 2 dof
• Use body-bar version of pebble game
• Model each Ca as rigid body
• Model peptide bonds with 5 bars
• Model hydrogen bonds with 1 bar

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Rigidity-Based Sampling Example

• Identify indep. hinges, rigid clusters, and dep.
  hinge sets
• Perturb independent hinges with probability Pflex
• Perturb rigid dof with probability Prigid
• For each dependent hinge set
• Randomly select 1 hinge to perturb with
  probability Pflex
• Perturb remaining hinges with probability Prigid

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Initial Findings

• Goal capture main features of the landscape with
  a smaller roadmap

a. b1b2, b3b4, b1b4 gt 100
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Initial Findings

• Goal capture main features of the landscape with
  a smaller roadmap

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Initial Findings

• Goal capture main features of the landscape with
  a smaller roadmap

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Initial Findings
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Initial Findings
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Initial Findings
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Current Work

• Mapping between Conformations
• Maintaining Closure Constraints for Dependent
  Hinge Sets

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Current WorkMapping between Conformations

• For extensive study between two conformations
• Generate rigidity samples biased to 1st
  conformation and connect them (set A)

A
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Current WorkMapping between Conformations

• For extensive study between two conformations
• Generate rigidity samples biased to 1st
  conformation and connect them (set A)
• Generate rigidity samples biased to 2nd
  conformation and connect them (set B)

B
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Current WorkMapping between Conformations

• For extensive study between two conformations
• Generate rigidity samples biased to 1st
  conformation and connect them (set A)
• Generate rigidity samples biased to 2nd
  conformation and connect them (set B)

B
A
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Initial Findings Calmodulin
B
A
Path profile
movie
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Closed Chain Systems
Robots cooperatively manipulating an object
Digital Actors
http//www.robotics.is.tohoku.ac.jp/lab/robot
http//robotics.stanford.edu/
Reconfigurable Robots
Molecular Chains
http//www.isi.edu/conro/conro2.jpg
http//www.schrodinger.com
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Randomized Motion Planning for Closed Chain
Systems

• Randomized Gradient Descent Approaches
• S. LaValle, J. Yakey and L. Kavraki (ICRA99)
• PRM-based planner
• S. LaValle, J. Yakey and L. Kavraki (TRA01)
• RRT based planner
• Forward Inverse Kinematics-Based Approaches
• L. Han and N.M. Amato (WAFR00)
• PRM sampling for some links inverse kinematics
  for others
• J. Cortes, T. Simeon and J. Laumond (ICRA02)
• ala Han Amato, but sample one dof at a time
• Apply to protein loops (Cortes PhD03, JCC04,
  WAFR04)
• D. Xie N.M. Amato (ICRA04)
• Decomposition and hierarchical solution of
  linkages with multiple and long closed chains

lt 10 links hours
lt 10 links minutes
10-15 links seconds
20-25 links seconds Protein loops
100 links Multiple chains seconds
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Kinematics-Based PRM Overview(Xie Amato, ICRA
2004)

• Use Ear Decomposition technique to partition the
  linkage into ordered set of chains
• one closed ear (the first)
• open ears have end points on previous ears
• Process the chains (ears) in order
• If the ear is small (lt 10-15 links) then
• Sample joint angles for some links (e.g., all but
  3)
• Use inverse kinematics on the rest to close the
  chain
• If the ear has too many links to be solved
  efficiently with inverse solver then
• coarsen by freezing some of the links
• apply KB-PRM to the coarsened problem (recurse?)
• apply KB-PRM to the sub-problems after unfreezing

sub problems
Coarse problem
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Ear Decomposition
Theorem
For a given graph G, Ear Decomposition exists ?
G is 2-edge connected (a connected graph that is
not broken into disconnected pieces by deleting
any single edge)
Bridge an edge of G whose removal disconnects
G 2-edge connected component a maximal induced
subgraph of G which is 2-edge connected Idea
Apply ear decomposition to each 2-edge connected
component
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Experimental Results(Xie Amato, ICRA 04)
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Current WorkMaintaining Closure Constraints for
Dep. Hinges

• Adapting this approach for proteins
• Has been done by J. Cortes et al
• to generate more realistic structures for protein
  folding work
• to model molecular interactions
• Currently we randomly pick one hinge/set, treat
  it as flexible and treat the rest as rigid
• Dependent hinge sets contain closed
  chains/closure constraints
• Treat set as a closed chain, randomly pick one to
  perturb, compute what the remaining hinges need
  to be to satisfy closure constraints

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Current WorkMaintaining Closure Constraints for
Dep. Hinges

• Currently we randomly pick one hinge/set, treat
  it as flexible and treat the rest as rigid
• Dependent hinge sets contain closed
  chains/closure constraints
• Treat set as a closed chain, randomly pick one to
  perturb, compute what the remaining hinges need
  to be to satisfy closure constraints

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Contact Information

• For more information, check out our website
• http//parasol.tamu.edu/folding/
• or email
• amato_at_cs.tamu.edu
• sthomas_at_cs.tamu.edu

Folding Server http.//parasol.tamu.edu/foldingse
rver