1. 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

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. 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

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. 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

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

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. 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

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(n 4 ) algorithm for general minimally rigid graphs minus one edge [SIH]

20. 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

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