Lecture 14: Advanced Conformational Sampling

Transcription

1 Lecture 14: Advanced Conformational Sampling Dr. Ronald M. Levy

2 Multidimensional Rough Energy Landscapes MD ~ ns, conformational motion in macromolecules ~µs to sec Interconversions between basins are infrequent at room temperature. Barriers are poorly sampled.

3 Biased Sampling Biasing potentials w(x) Thermodynamic properties can be unbiased : A w x =0 = U x dxa x e U x = dxa x e w x e U x e w x = dxe dxe w x e U x e w x Q U w A x e w x U w Q U w e w x U w = A x e w x U w e w x U w

4 Biased Sampling Methods Umbrella sampling Targeted/Steered MD Local elevation, conformational flooding, metadynamics, essential dynamics...

5 Generalized Ensembles Microcanonical Ensemble E =densityof states= d [H E] Canonical ensemble S E =k B ln E p c E Q c = de E e E e E = canonical weight Multicanonical ensemble Q mu = de E W E Set so that: p mu E = W E 1 E =e S E /k B p c E = E e E Q c generalized weight function E W E =constant Q mu Often expressed as: E p mu E M E, W E =e

6 Multicanonical ensemble (cont.) Q mu = de E e M E, = ded [H E]e M E, M [H, ] = d e MD/MC with modified Hamiltonian M(Γ) = M[H(Γ)] The M function (spline) is adjusted by trial and error until the distribution of energies is constant within a given range [Emin, Emax] The multicanonical ensemble samples both low energy and high energy conformations barrier crossing. Unbiasing: A canonical = A e [H M ] mu e [H M ] mu

7 Generalized Ensembles Canonical ensemble Q c = d e H p = e H Q c Extended Ensemble: a parameter λ becomes a dynamical variable Q= d d e f e H = d e f Q c p, = e f H e Q p = d p, = e f d e H = ef Q c Q Q When p(λ) = constant f = lnq c const. Dimensionless free energy at λ

8 Extended Ensemble example: Simulated Tempering λ = temperature Q= dtd e f T e H /k BT = dte f T Q c T Want to sample temperature uniformly within a range: p T = d p,t = ef T Q c T =constant Q Therefore we seek: f T = lnq c T const. In actual applications sample a discrete set of temperatures T m Q= m d e f m T m e mh Generalized Hamiltonian is: H m =H f m T m / m

9 Conformational sampling in Simulated Tempering MC Velocities are not considered (Q Z above). Generalized energy function: Sampling distribution: U m x =U x f m T m / m x,t m e mu m x Two kinds of MC moves: 1. Change of coordinates at constant temperature: x,t m x',t m ' =e m[u x' U x ] 2. Change of temperature at fixed coordinates: x,t m x,t m' ' =e m' m U x f m' f m

10 Conformational sampling in Simulated Tempering MD p,x,t m e m[k p U x ] f m Sampling distribution: 1. Constant temperature MD for n steps at T m with potential function U x think of it as a move: p,x,t m p',x',t m 2. Attempt to temperature move at constant positions + rescaled velocities: p,x,t m T m' /T m 1/2 p,x,t m' m' K'= m' p' 2 2M = m' m m' p 2 2M = m p 2 2M = mk So: ' =e [ m'k' m K m' m U x ] f m' f m =e [ m' m U x ] f m' f m Same as in MC

11 Simulated Tempering (cont.) The weight factors f m (T m ) a.k.a. dimensionless free energies - are adjusted by trial and error until all of the temperatures are visited approximately equally. This can be a time consuming and tedious process. Temperatures can not be spaced too far apart to keep MC acceptance probabilities at a reasonable level.

12 Simulated Tempering (cont.) When the system is visiting high temperatures, barrier crossings are more likely. Then new conformations may cool down and reach the temperature of interest. The samples at the temperature of interest can be used directly to compute thermodynamic averages; each T-ensemble is canonical A T = 1 n T samplesatt A x k (can also unbias from other temperatures WHAM/MBAR, later)

13 Temperature Replica Exchange (a.k.a. Parallel Tempering) In Simulated Tempering equal visitation of temperatures is ensured by the free energy weights f m In Parallel Tempering the same is ensured by having each temperature correspond to an individual replica of the system. T 1 T 2 T 2 T 3 T 4... T n We consider the generalized canonical ensemble of the collection of replicas. Because the replicas are not interacting, the partition function of the RE ensemble is the product of the individual partition functions. Q RE =Q 1 Q 2 Q 3 Q n Q i = d i e ih i Q RE = d 1 d n e i i H i

14 T-RE The state of the RE ensemble is specified by an ordered sequence of momenta/coordinate pairs: Two kinds of moves: 1, 2, n 1. Change of coordinates in one replica (MC or MD): 1,, i,, n 1,, ' i,, n 2. Exchanges of state between a pair of replicas: 1,, i,, j,, n 1,, j,, i,, n 2a. MC: exchange coordinates (no velocities) 2b. MD: rescale velocities at the new temperature

15 Recall Metropolis MC algorithm: probability of accepting move=min ( 1, ρ j ρ i ) =min (1,e βδu ij ) T-RE/MC RE before exp 1 U 1 i U i j U j n U n RE after exp 1 U 1 i U j j U i n U n RE after RE before exp[ j i U i U j ] Accept or reject exchange attempt based on this quantity

16 T-RE/MD R.E. before exp[ i K i U i j K j U j ] R.E. after exp[ i j i K j U j j i j K i U i ] R.E. after R.E. before exp[ j i U i U j ] Same as in MC

17 T-RE R.E. after R.E. before exp[ j i U i U j ] Exchange will be accepted with 100% probability if lower temperature gets the lower energy. Otherwise the exchange has some probability to succeed if either the temperature difference is small or if the energy difference is small, or both. On average larger systems require smaller spacing of temperatures: U i U j U T T j T i =C v T j T i and C v N T 1 T 2 T 1 T 2 p(u) small N p(u) large N U U

18 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

19 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

20 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

21 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

22 Replica exchange molecular dynamics replica Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261

23 Replica exchange molecular dynamics 650 Temperature trajectory of a walker Step number (fsec)

24 Hamiltonian Replica Exchange (HREM) A RE method in which different replicas correspond to (slightly) different potential functions rather then temperature. U(x,λ)=U 0 (x)+v(x,λ) Base energy Perturbation energy HRE exp{ [U x 1, 1 U x n, n ]} The probability of exchange between two replicas in HREM U before =U x 1, 1 U x 2, 2 U after =U x 2, 1 U x 1, 2 where U 1 =U x 1, 2 U x 1, 1 after before =exp[ U 1 U 2 ] U 2 =U x 2, 1 U x 2, 2 (proposed change in energy of conformation x 1 ) (proposed change in energy of conformation x 2 )

25 Two examples of HREM applications REUS: Replica Exchange Umbrella Sampling [Sugita, Kitao, Okamoto, 2000)] ( x, ) = w( x ) V λ ; d λ w x =biasing potential Originally proposed to compute the end-to-end distance PMF of a peptide, in which case the biasing potentials are harmonic restraining potentials of the end-to-end distance d. BEDAM: Binding Energy Distribution Analysis Method [Gallicchio, Lapelosa, Levy 2010] V x, = V RL x V RL x =ligand-receptor interaction energy Replicas are distributed from λ=0 (unbound state) to λ=1 (bound state). Replicas at small λ provide good sampling of ligand conformations whereas replicas at larger λ s provide good statistics for binding free energy estimation. In either case the Weighted Histogram Analysis Method (WHAM) is used to merge the data from multiple replicas [Gallicchio, Andrec, Felts, Levy, 2005]

26 Transition Path Sampling: Simple Picture The trajectory of a Brownian particle moving in a double well First passage (waiting) time and the transition event Large conformational change is a rare and fast event

27 Probability of a Path Classical Mechanics: one dominant path and deterministic Path integral: famous in Quantum Mechanics Stochastic Process: random, statistical result from path ensemble

28 Probability of a Path: Example The probability of a single path: For one dimensional overdamped Langevin dynamics, the probability of a single step is :

29 Transition Path Sampling (Chandler and Bolhuis) Monte Carlo simulation in path space Get one successful path from state A to B From the middle of the original path, shoot to A or B shoot to the other state

30 Weighted Ensemble Method (Zuckerman et al.) Schematic illustration of the Weighted Ensemble (WE) method, using N = 3 bins and M = 2 simulations per bin.