Replica exchange methodology. John Karanicolas June 2003

Transcription

1 Replica exchange methodology John Karanicolas June 2003

2 Outline o Motivation o Theory o Practical considerations o MMTSB Tool Set

3 Why? o Sampling on rugged potential energy surfaces is difficult o Sampling is important for: o Collecting statistics (mean of some observable, free energy differences, etc.) o Finding the global minimum (e.g. structure prediction, refinement)

4 X Starting here X X or evaluate the free energy difference to here? can we get here? Doing so directly involves overcoming large barriers, and is therefore time-consuming

5 Umbrella Sampling: Strategy o Include an additional term (an umbrella ) in the Hamiltonian (potential energy function), to flatten the potential energy surface: + = Original P.E. surface + Umbrella term = Modified P.E. surface

6 Umbrella sampling: Result o Application of an umbrella flattens the potential energy surface, so that: fi All states are now (nearly) isoenergetic fi This means that they occur with similar probability fi Random walk in potential energy space fi Barrier heights are reduced fi More frequent transitions between states o We can use statistical mechanics to unbias the simulation

7 Choice of an Umbrella o Sometimes obvious: o A distance (e.g. ligand-receptor association) o An angle or dihedral (e.g. cyclohexane boat-chair transition) o Sometimes intuitive: o Radius of gyration (Rg) (e.g. protein folding) o Number of native contacts formed (e.g. protein folding)

8 A problem remains o The umbrella is not perfect. o We can t construct a perfect umbrella without knowledge of the true potential energy surface

9 Solution o Perform several independent simulations, each with a different approximate umbrella o A simple harmonic potential is often used, centered at several different places o Use statistical mechanics to stitch them together afterwards o This requires some overlap between simulations

10 Schematically: Numerous (harmonic) umbrellas: separately applied to Original P.E. surface:

11 A new problem emerges o The umbrella may not overcome all relevant barriers. o Example: Umbrellas in Rg for protein folding. o Large Rg umbrellas help sample extended states. o BUT: small Rg umbrellas do NOT induce transitions between native state and compact misfolded states.

12 Schematically: Very high barrier N M High barrier High barrier U Low Rg states High Rg states

13 Solution o Couple several simulations with different umbrellas, and periodically exchange the umbrellas o In the Rg example, may speed folding via: M (low Rg) Æ U (high Rg) Æ N (low Rg)

14 Schematically: Umbrella 1 A A A D Umbrella 2 B B D A Umbrella 3 C D B C Umbrella 4 D C C B A-D represent 4 non-interacting (MD or MC) simulations

15 Exchange probability o We can consider this set of simulations as a single system obeying equilibrium thermodynamics o This dictates that the exchanges must be accepted/rejected based on the Metropolis criterion: P(exchange) = Ï exp(-bde); DE 0 Ì Ó 1; DE < 0 b = 1 k B T

16 Exchange probability [ ] DE = E i q A where: ( ) - E ( q ) i B [ ( ) - E ( q )] j A + E j q B qa and qb represent two conformations, Ei and Ej denote the energy evaluated using different umbrellas The energy of moving from conformation A to conformation B in Hamiltonian i, and B to A in Hamiltonian j

17 Acceptance ratio o Early studies using MC find optimal sampling when the acceptance ratio is close to 20% o Indicates that the move set contains a good balance between short-range and long-range moves o What controls the acceptance ratio in replica exchange?

18 Acceptance ratio (cont d) o Force constant of (harmonic) umbrella o Distance between minima of (harmonic) umbrella

19 Move step o How do I choose which replicas to exchange? o Any can be exchanged, and Boltzmann sampling is preserved

20 Move step (cont d) o BUT: large DE results in low exchange probability o Low exchange probability means that replica exchange isn t buying you much compared with uncoupled simulations o Large DE occurs if the umbrellas are very different. o SO: we therefore exchange only replicas at neighboring conditions: o Maximizes the acceptance ratio o Allows maximally diverse conditions (ie. large spacing between umbrellas)

21 Exchange frequency o How often should I try exchanging replicas? o Unclear to the field. Some researchers try every 10 steps of MD, some every 2000 steps of MD. o We believe

22 Exchange frequency o The exchange frequency should be (at least) the relaxation time between neighboring replicas. o If the time is less than this, replicas which exchange will quickly return to their previous condition, and the intervening space will not be sampled o This time is on the order of 500 to 2000 MD steps

23 Temperature as a condition o We can write the Metropolis criterion for exchanges to different temperatures: Ï P(exchange) = Ì exp(d); D < 0 Ó 1; D 0

24 Exchange probability D = b i [ ] ( ) - E( q ) B E q A where: [ ] + b j E q B ( ) - E( q ) A or: ( ) E q A D = b i - b j [ ] ( ) - E( q ) B

25 Why exchange temperatures? o Any ideas?

26 Exchanging temperatures o Temperature acts exactly as an umbrella o Low temperatures enhance sampling in local minima o Energy dominates o High temperatures help barrier crossings o Entropy dominates o When is this useful?

27 Why exchange temperatures? o Useful when an umbrella reaction coordinate is not known (e.g. protein folding) o Useful when we re interested in temperaturedependent information (e.g. protein folding)

28 Umbrellas in multiple dimensions o Sure, why not! o Example: ligand binding to receptor o Umbrella applied to ligand-receptor distance o Umbrella applied to ligand orientation

29 Umbrellas in multiple dimensions o In multiple-dimension replica exchange, all combinations of conditions need not be represented o Example: ligand binding to receptor o When the ligand-receptor distance is large, we don t need multiple umbrellas on the orientation o No barriers here o Why is this helpful?

30 Implementation in MMTSB Tool Set o Running replica exchange: o aarex.pl, aarexamber.pl, latrex.pl o Analysis of replica exchange results: o rexinfo.pl

31 Summary o Replica exchange involves: o Defining a series of different conditions across an interesting reaction coordinate o Simultaneously running a simulation under each set of conditions o Periodically exchanging the current conformation in each condition, subject to the Metropolis criterion o This leads to enhanced sampling along this reaction coordinate o Available through the MMTSB Tool Set