Milestoning and the R to T transition in Scapharca Hemoglobin

1. Milestoning and the R to T transition in Scapharca Hemoglobin Ron Elber and Anthony West Department of Computer Science, Cornell University $ NIH

2. Time Scale Problems in Atomically Detailed Molecular Dynamics Simulation

3. Long time processes: By long range diffusion or activation

4. Divide reaction coordinate into Milestones

5. First passage time distribution: K2+

6. Complicated reaction now modeled as discrete non-Markovian K-dependent 1-D hopping t reinterpreted as incubation time (waiting time) between hops. FPT dist�n is now waiting time dist�n Process is generally non-Markovian NB: are the only input, capture all relevant details of microscopic dynamics and connect them to this abstract model. Resulting model: Continuous-time random walk

7. How to solve for

8. Equivalent to Generalized Master Equation The generalized Markov equation has time dependent rates K in the QK formulation is easier to compute than R and the Laplace transforms are related by

9. Calculations of rates / time moments (Shalloway & Vanden Eijnden) The first passage time (and all its moments) with absorbing boundary condition at the product state N

10. Time moments -- continuation

11. Rate constant: Inverse of first passage time

12. When and why it works memory loss: system equilibrates in plane much faster than between planes. Permits �fragment then glue� approach to trajectories along RC...produces effectively long-time trajectories Fragmentation gives diffusive speedup on flat energy surfaces: and exponential speed-up for systems with substantial barriers Independence gives parallelizability speedup:

13. Alanine dipeptide in water

14. Sampling Milestones

15. A sample of first passage time distribution

16. Overall time course for alpha to beta transition in alanine dipeptide

17. Total first passage time as a function of the log of number of milestones

18. Table of first passage times

19. Velocity memory is kept below 300fs

20. Free energy estimates

21. References for Milestoning Anton K. Faradjian and Ron Elber, "Computing time scales from reaction coordinates by milestoning", J. Chem. Phys. 120:10880-10889(2004) Anthony M.A. West, Ron Elber and David Shalloway, �Extending Molecular Dynamics Time Scales with MilestoningL Example of complex kinetics in a solvated peptide�, J. Chem. Phys. 126,145104(2007) Ron Elber, "A milestoning study of the kinetics of an allosteric transition: Atomically detailed simulations of deoxy Scapharca hemoglobin", Biophysical J. ,2007 92: L85-L87

22. The classical problem of the R to T transition in hemoglobin A: Four chains: two alpha, two beta, Large quaternary structural change Allosteric transition Complex kinetics Dependence on pH, other factors Tens of microsecond time scale Noble to Perutz

23. Hemoglobin A � indirect heme-heme interactions

24. Animation of the transition

25. Scapharca hemoglobin Homodimer Allosteric No large quaternary change Hemes in close contact Phenylalanine conformational transition Changes in water structure No pH effect Simpler kinetics Microsecond time scale

26. Scaphraca Hemoglobin

27. Reaction path for the R to T transition in Scapharca Hemoglobin

28. References for reaction path algorithms Pratt L., JCP. 85, 5045 1986 (global reaction path algorithm) Elber & Karplus CPL, 139, 375�(1987). (equi-distance spring) A. Ulitsky and R. Elber, JCP, 96, 1510 (1990). (exact SDP by updating planes) R. Olender and R. Elber, J.�Mol. Struct. Theochem,�398-399, 63-72 (1997).

29. Molecular dynamics simulations in explicit water

30. First passage time of milestone 7 along the reaction coordinate (computed with SPW algorithm)

31. Following distance distribution between �allosteric� phenylalanines

32. Solving the milestoning equation The overall rate is 10 microsecond, in accord with spectroscopy measurements. Identifying late barrier with global motions that follow the side chain transitions In progress: Myosin transition (Tony West)

33. Milestoning divides RC into fragments whose kinetics can be computed independently Provides factor of improvement for diffusive processes, exponential for activated processes. Uses FPTs from microscopic dynamics: System distribution given by simple integral equations that can be easily solved numerically Some care is needed in Choice of reaction coordinate Application of equilibrium assumption Summary

34. Comparison to TPS TPS provide a rate constant for a high energy barrier Rate constant exists System in equilibrium Trajectory computed sequentially (must be short) No need for a reaction coordinate

35. Comparison to Bolhuis et al. PPTIS and Milestoning use order parameter PPTIS assumes loss of memory in time (Markovian process) and a single exponential relaxation. Milestoning assumes spatial memory loss in the direction perpendicular to the order parameter PPTIS computes trajectory sequentially Milestoning allows for more efficient parallelization The two approaches can be combined!

37. Example FPT dist�n

38. Memory loss:

40. Toy model: 1D box simulation Microscopic dynamics are Brownian Simulations run at various temperatures and for 4, 8, and 16 milestones

41. 1D reaction curves (5000 trajs/MLST)

42. Extracting the rate

45. 2D simulation

47. 2D power law

48. 2D simulation

49. Memory loss demonstration

50. FREE ENERGY is a by-product (from equilibrium vector)