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 Work In Progress

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)

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 Exact Models

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

pseudo triangle pseudo 4-gon

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

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

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

2D GridOff-Grid by PT Can explore exhaustively (exponential time) Tighter sphere packing Varying bond lengths Every compact state can be reached Fixed bond length Fixed bond angles More complicated Need Monte-Carlo simulations to explore advantages disadvantages

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

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

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

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 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 k i j

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

Rigid Components PT Linkage Package

Rigid Components of a PT Detecting rigid components in linear time –In PT: maximal convex components –with J. Snoeyink O(n 4 ) algorithm for general minimally rigid graphs minus one edge [SIH]

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

Random PT PT Linkage Package

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

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

Thank you