Lecture 14: Advanced Conformational Sampling Dr. Ronald M. Levy Statistical Thermodynamics

Spring 2013 Multidimensional Rough Energy Landscapes Interconversions between basins are infrequent at room temperature. Barriers are poorly sampled. MD ~ ns, conformational motion in macromolecules ~  s to sec

Statistical Thermodynamics Spring 2013 Biased Sampling Biasing potentials w(x) Thermodynamic properties can be “unbiased”:

Statistical Thermodynamics Spring 2013 Biased Sampling Methods Umbrella sampling Targeted/Steered MD Local elevation, conformational flooding, metadynamics, essential dynamics...

Statistical Thermodynamics Spring 2013 Generalized Ensembles Canonical ensemble Microcanonical Ensemble Multicanonical ensemble generalized weight function canonical weight Set so that: Often expressed as: E

Statistical Thermodynamics Spring 2013 Multicanonical ensemble (cont.) MD/MC with modified Hamiltonian M(  ) = M[H(  )] 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:

Statistical Thermodynamics Spring 2013 Generalized Ensembles Canonical ensemble Extended Ensemble: a parameter becomes a dynamical variable When p( ) = constant Dimensionless free energy at 

Statistical Thermodynamics Spring 2013 Extended Ensemble example: Simulated Tempering = temperature Want to sample temperature uniformly within a range: Therefore we seek: In actual applications sample a discrete set of temperatures T m Generalized Hamiltonian is:

Statistical Thermodynamics Spring 2013 Conformational sampling in Simulated Tempering MC Velocities are not considered ( Q → Z above). Generalized energy function: Sampling distribution: Two kinds of MC moves: 1. Change of coordinates at constant temperature: 2. Change of temperature at fixed coordinates:

Statistical Thermodynamics Spring 2013 Conformational sampling in Simulated Tempering MD 1. Constant temperature MD for n steps at T m with potential function Sampling distribution: 2. Attempt to temperature move at constant positions + rescaled velocities: think of it as a move: So: Same as in MC

Statistical Thermodynamics Spring 2013 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.

Statistical Thermodynamics Spring 2013 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 (can also unbias from other temperatures – WHAM/MBAR, later)

Statistical Thermodynamics Spring 2013 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.... T1T1 T2T2 T2T2 T3T3 T4T4 TnTn 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.

Statistical Thermodynamics Spring 2013 T-RE The state of the RE ensemble is specified by an ordered sequence of momenta/coordinate pairs: Two kinds of moves: 1. Change of coordinates in one replica (MC or MD): 2. Exchanges of state between a pair of replicas: 2a. MC: exchange coordinates (no velocities) 2b. MD: rescale velocities at the new temperature

Statistical Thermodynamics Spring 2013 T-RE/MC Accept or reject exchange attempt based on this quantity Recall Metropolis MC algorithm:

Statistical Thermodynamics Spring 2013 T-RE/MD Same as in MC

Statistical Thermodynamics Spring 2013 T-RE 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: and large N small N T1T1 T2T2 T1T1 T2T2 UU p(U)

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999)

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261 replica

Statistical Thermodynamics Spring 2013 Replica exchange molecular dynamics Temperature trajectory of a walker

Statistical Thermodynamics Spring 2013 Hamiltonian Replica Exchange (HREM) A RE method in which different replicas correspond to (slightly) different potential functions rather then temperature. “Base” energy “Perturbation” energy The probability of exchange between two replicas in HREM where (proposed change in energy of conformation x 1 )(proposed change in energy of conformation x 2 )

Statistical Thermodynamics Spring 2013 Two examples of HREM applications REUS: Replica Exchange Umbrella Sampling [Sugita, Kitao, Okamoto, 2000)] BEDAM: Binding Energy Distribution Analysis Method [Gallicchio, Lapelosa, Levy 2010] 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. 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]

Statistical Thermodynamics Spring 2013 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

Statistical Thermodynamics Spring 2013 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

Statistical Thermodynamics Spring 2013 Probability of a Path: Example The probability of a single path: For one dimensional overdamped Langevin dynamics, the probability of a single step is :

Statistical Thermodynamics Spring 2013 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

Statistical Thermodynamics Spring 2013 Weighted Ensemble Method (Zuckerman et al.) Schematic illustration of the Weighted Ensemble (WE) method, using N = 3 bins and M = 2 simulations per bin.