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Efficient Energy Computation for Monte Carlo Simulation of Proteins

Efficient Energy Computation for Monte Carlo Simulation of Proteins. Itay Lotan Fabian Schwarzer Jean-Claude Latombe. Stanford University. Monte Carlo Simulation (MCS). Estimation of thermodynamic quantities over the space

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Efficient Energy Computation for Monte Carlo Simulation of Proteins

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  1. Efficient Energy Computation for Monte Carlo Simulation of Proteins Itay Lotan Fabian Schwarzer Jean-Claude Latombe Stanford University

  2. Monte Carlo Simulation (MCS) • Estimation of thermodynamic quantities over the space • Search for low-energy conformations, in particular the native (folded) state Popular method for studying the conformation space of proteins:

  3. Preview of What’s to Come • Method for speeding up MCS of proteins • Exploits the fact that a protein backbone is a kinematic chain • Avoids the combinatorial explosion of atomic interactions • Gives as much as 12X speed-up for proteins we tested

  4. MCS: What It Is Random walk through the conformation space of a protein that samples conformations on its path. Converges to the underlying distribution of conformations after enough time.

  5. MCS: How It Works • Propose random change in conformation • Compute energy E of new conformation • Accept new conformation with probability:

  6. Energy Function • Bonded terms: Bond length, Bond angle, etc.. • Non-bonded terms Van der Waals, Electrostatic and heuristic Non-bonded terms depend on distances between pairs of atoms O(n2),expensive to compute

  7. Pairwise Interactions Use cutoff distance (6 - 12Å) Only O(n) interactions(Halperin & Overmars ’98) O(1) interactions per atom Find interacting pairs without enumerating all pairs!

  8. Reusing Energy Terms Only few DOFs are changed at each step 1) 2) • Large sub-chains remain rigid between steps • Many energy terms unaffected by change

  9. Our Goal Improve computational efficiency of MCS by reducing average time to accept/reject a new conformation Independent of: • Energy function • Step generator • Acceptance criterion Exploiting: protein backbone is kinematic chain

  10. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  11. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  12. Grid Method • Subdivide space into cubic cells • Compute cell that contains each atom center • Store results in hash table dcutoff

  13. Grid Method – cont. • Θ(n) time to recompute • O(1) time to find interactions for each atom • Θ(n) to find all interactions in all cases • No way of detecting unchanged interactions Asymptotically optimal in worst-case!

  14. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  15. BV(A,B) BV(C,D) The ChainTree TNO= TJK*TKL TJK TKL

  16. Updating the ChainTree Update path to root: • Recompute transforms that shortcut change • Recompute BVs that contain change

  17. Finding Interacting Pairs Test the ChainTree against itself

  18. Finding Interacting Pairs • Do not search inside rigid sub-chains (unmarked nodes) • Do not test two nodes with no marked node in between

  19. Finding Interacting Pairs

  20. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  21. Summing the Interactions At each step need to sum contribution of: • New interactions • Changed interactions • Unchanged interactions (1) & (2) are found by ChainTree search How to retrieve (3) efficiently?

  22. The EnergyTree A caching scheme for partial energy sums: • Efficient to update • Efficient to query

  23. E(N,N) E(J,L) E(L,L) E(K,L) E(M,M) Using the EnergyTree

  24. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  25. Test Setup • Energy function: • Van der Waals • Electrostatic • Attraction between native contacts • Cutoff at 12Å • 300,000 steps MCS • Early rejection for large vdW terms

  26. (755) (68) (144) (374) Results: 1-DOF change

  27. (68) (144) (374) (755) Results: 5-DOF change

  28. Outline • Related work • The ChainTree • Energy maintenance • Tests • Conclusion

  29. Conclusion • Novel method to reduce average time per step in MCS of proteins • Exploits kinematic chain nature of protein • Significant speed-up for small number of simultaneous DOF changes • Better for larger proteins

  30. MCS Software • EEF1 force field (Lazaridis & Karplus ’99) • Backbone DOFs (Φ,Ψ) and fixed rotamers for side-chains (Dunbrack & Cohen ’97) • Classical MCS with simple move-set • Download and customize http://robotics.stanford.edu/~itayl/mcs

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