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Efficient Energy Computation for Monte Carlo Simulation of Proteins

This study presents a method to speed up Monte Carlo Simulation of proteins by exploiting the kinematic chain nature of protein backbones, achieving up to 12X speed-up. The approach reduces the average time to accept/reject new conformations, independent of energy function or step generator. Key techniques include Pairwise Interactions with cutoff distance to reduce computational cost and Reusing Energy Terms to efficiently handle DOF changes. The proposed method shows significant speed improvements for protein simulations with small simultaneous DOF changes, making it preferable for larger proteins.

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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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