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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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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

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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)

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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)

Simple off grid model l.jpg
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

Overview l.jpg

  • Pseudo Triangulations and 1DOF mechanisms in 2D

  • Simple simulation of folding

  • Problems and future work

Pseudo triangle l.jpg
pseudo triangle

pseudo 4-gon

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Pointy Pseudo Triangulation (PT)

  • 2n-3 edges

    - Pointy

  • Planar

  • Maximal

  • Laman graph

    • Minimally rigid

  • 1dof mechanisms l.jpg
    1DOF mechanisms

    Removing a hull edge turns it into a 1DOF mechanism

    Slide10 l.jpg



    Monte carlo simulation l.jpg
    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

    System validation l.jpg
    System Validation

    • Measure (as a function of time)

      • Energy

      • Radius of gyration

    • Look for secondary structure formation

    • Can we “fold” large “proteins” ?

    Pt linkage package l.jpg
    PT Linkage Package

    • Uses:

    • PT workbench by L.Kettner

    • CGAL

    • GLUT & GLUI


    Runs on Linux

    Pt linkage package14 l.jpg

    Calculates contractive and expansive motion

    H/P Nodes

    Linkage edges

    PT Linkage Package

    Motion model l.jpg
    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

    Motion model16 l.jpg




    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

    Motion model17 l.jpg
    Motion Model

    • Take into account that nodes have radii

    • Expansive/Contractive

    • Use Newton-Raphson to solve set of equations

    • Doesn’t always work

    Pt linkage package18 l.jpg

    Rigid Components

    PT Linkage Package

    Rigid components of a pt l.jpg
    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]

    Detecting rigid components maximal convex components l.jpg
    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

    Pt linkage package21 l.jpg

    Random PT

    PT Linkage Package

    Picking a random pt l.jpg
    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

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    What Next ?

    • Understand why/when Newton-Raphson fails to find motion

    • Experiment with large proteins