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Monte Carlo Simulation of Folding Processes for 2D Linkages Modeling Proteins with Off-Grid HP-Chains

Work In Progress. Monte Carlo Simulation of Folding Processes for 2D Linkages Modeling Proteins with Off-Grid HP-Chains. Ileana Streinu Smith College. Leo Guibas Rachel Kolodny Michael Levitt Stanford University. Simple Models of Proteins. Model a Protein as 2D Chain of Beads

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Monte Carlo Simulation of Folding Processes for 2D Linkages Modeling Proteins with Off-Grid HP-Chains

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  1. Work In Progress Monte Carlo Simulation of Folding Processes for 2D LinkagesModeling Proteins with Off-Grid HP-Chains Ileana Streinu Smith College Leo Guibas Rachel Kolodny Michael Levitt Stanford University

  2. Simple Models of Proteins Model a Protein as 2D Chain of Beads • Each amino acid (=bead) in the chain is polar or hydrophobic • PHHPH (still need to specify distances)

  3. Simple Exact Models • Explores what non-local interactions can create • Structure • Stability • Folding kinetics • Proposed by K. Dill (1985) From: “Principles of protein folding – A perspective from simple exact models” Dill et al. Protein Science (1995)

  4. Simple Off-Grid Model • Still HP-chains • Same energy model • Still in 2D • Simple means simple motions • Based on pseudo-triangulation mechanisms • Focus on folding

  5. Overview • Pseudo Triangulations and 1DOF mechanisms in 2D • Simple simulation of folding • Problems and future work

  6. pseudo triangle pseudo 4-gon

  7. Pointy Pseudo Triangulation (PT) • 2n-3 edges - Pointy • Planar • Maximal • Laman graph • Minimally rigid

  8. Every chain can be pseudo-triangulated by adding n-2 edges

  9. 1DOF mechanisms Removing a hull edge turns it into a 1DOF mechanism

  10. advantages disadvantages

  11. Monte-Carlo Simulation • A way to generate Boltzmann distribution on the states of the system • Need: • Transition probability between configurations satisfies detailed balance • Finite number of steps between any 2 configurations

  12. System Validation • Measure (as a function of time) • Energy • Radius of gyration • Look for secondary structure formation • Can we “fold” large “proteins” ?

  13. PT Linkage Package • Uses: • PT workbench by L.Kettner • CGAL • GLUT & GLUI • CLAPACK Runs on Linux

  14. Calculates contractive and expansive motion H/P Nodes Linkage edges PT Linkage Package

  15. Motion Model • Move mechanism until PT property is violated at an alignment event. • This guarantees chain self-avoidance throughout • Alignment can occur at any vertex • Not ones inside a rigid component • Find first one

  16. i j k Motion Model • Write a quadratic system for each vertex • 2n-3 variables • 2n-3 equations • Fixed edge lengths • 2n-4 edges • Alignment edges ik and jk at vertex k

  17. Motion Model • Take into account that nodes have radii • Expansive/Contractive • Use Newton-Raphson to solve set of equations • Doesn’t always work

  18. Rigid Components PT Linkage Package

  19. Rigid Components of a PT • Detecting rigid components in linear time • In PT: maximal convex components • with J. Snoeyink • O(n4) algorithm for general minimally rigid graphs minus one edge [SIH]

  20. Detecting Rigid ComponentsMaximal convex components • - Keep turning left (as little as possible) • Mark your path& notice when you visit twice • Backtrack if needed Linear time

  21. Random PT PT Linkage Package

  22. Picking a Random PT • Given set of points • Unknown: total number of PTs • Conjecture: Random walk on 1-Skeleton of PT polytope is rapidly mixing • Flip polynomial number of times to find random PT Known: TRUE if set is convex

  23. What Next ? • Understand why/when Newton-Raphson fails to find motion • Experiment with large proteins

  24. Thank you

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