1. Algorithm for Fast MC Simulation of Proteins Itay Lotan Fabian Schwarzer Dan Halperin Jean-Claude Latombe

2. MC Simulation Classic technique for studying thermodynamic properties of proteins Random walk through conformation space: –Propose random change in conformation –Accept new conformation with probability that depends on difference in energy (Metropolis criterion):

3. Our Contribution A generic algorithm for MCS which has: –Efficient incremental update of conformation given m simultaneous changes –Efficient computation of energy terms –Reuse of unchanged energy terms The algorithm supports most molecule representations, energy functions and simulation methodologies

4. Requirements 1.Small number of DOF changes per step 2.Energy function with: A.Bonded terms (bond length, bond angle, etc.) B.Pairwise non-bonded terms (vdW, elctrostatic, etc.) with cutoff distances

5. Useful Properties of MCS A protein is a long kinematic chain Small number of changes  large rigid sub-chains Cutoff distance  small subset of all pairs is needed Many energy terms can be reused

6. Challenges How to update the conformation without re-computing the position of all atoms? How to efficiently find all pairs of atoms whose distance is below the cutoff (interacting pairs)? How to efficiently discover which energy terms have changed and which can be reused?

7. Related Work The algorithm is based on a self-collision detection method for kinematic chains we presented at SoCG02 Biologists have used many tricks and heuristics to speed up their simulations. There is no general method exploiting common properties of all MCS exists

8. Molecule Representation Twofold hierarchical structure: Transformations hierarchy to approximate the kinematics of the protein at different resolutions Bounding Volume hierarchy to approximate the geometry of the protein at different resolutions

9. Transformations Hierarchy Hierarchy of “shortcut” transformations Sequence of reference frames (links) connected by rigid-body transformations (joints)

10. BV Hierarchy Chain-aligned: bottom-up, along the chain Each BV encloses its two children in the hierarchy

11. Computing distances - RSS RSS (rectangle swept sphere) - The Minkowski sum of a rectangle and a sphere (Larsen et al., ICRA 2000)

12. One Binary Tree

13. Updating and Searching

14. Complexity Updating: Finding all interacting pairs: Performs much better in practice!!! Worst-case

15. Interaction Tree Hierarchy of possible interactions between sub-chains of the protein Based on our hierarchical representation of the protein Stores partial sums of non-bonded pairwise energy terms Allows efficient reuse of unchanged terms

16. Interaction Tree

17. Results: 1-DOF change (68)(144)(374) (755)

18. Results: 5-DOF change (68)(144)(374)(755)

19. First-Pass Steric Clash Detection (68)(144)(374) (755)

20. What next? Implement a real force field: EEF1 (Lazaridis & Karplus) Based on CHARMM19 9Å cutoff distance Implicit solvent as sum of pairwise terms Run MCS of proteins with 60 – 80 residues. Run simulation of a set of small proteins – misfolding.

21. Open Problem How to find good moves to make when the conformation becomes compact and random moves are rejected with high probability?