Lecture 14: Advanced Conformational Sampling Molecular Simulations in Computational Biology 2/4/08 Statistical Thermodynamics Lecture 14: Advanced Conformational Sampling Dr. Ronald M. Levy ronlevy@temple.edu Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs)

Molecular Simulations in Computational Biology 2/4/08 Multidimensional Rough Energy Landscapes MD ~ ns, conformational motion in macromolecules ~ms to sec Interconversions between basins are infrequent at room temperature. Barriers are poorly sampled. Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 2

Molecular Simulations in Computational Biology 2/4/08 Biased Sampling Biasing potentials w(x) Thermodynamic properties can be “unbiased”: Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 3

Molecular Simulations in Computational Biology 2/4/08 Biased Sampling Methods Umbrella sampling Targeted/Steered MD Local elevation, conformational flooding, metadynamics, essential dynamics ... Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 4

Molecular Simulations in Computational Biology 2/4/08 Generalized Ensembles Microcanonical Ensemble Canonical ensemble canonical weight Multicanonical ensemble E generalized weight function Set so that: Often expressed as: Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 5

Molecular Simulations in Computational Biology 2/4/08 Multicanonical ensemble (cont.) MD/MC with modified Hamiltonian M(G) = M[H(G)] 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: Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 6

Molecular Simulations in Computational Biology 2/4/08 Generalized Ensembles Canonical ensemble Extended Ensemble: a parameter l becomes a dynamical variable When p(l) = constant Dimensionless free energy at l Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 7

Molecular Simulations in Computational Biology 2/4/08 Extended Ensemble example: Simulated Tempering l = temperature Want to sample temperature uniformly within a range: Therefore we seek: In actual applications sample a discrete set of temperatures Tm Generalized Hamiltonian is: Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 8

Molecular Simulations in Computational Biology 2/4/08 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: Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 9

Molecular Simulations in Computational Biology 2/4/08 Conformational sampling in Simulated Tempering MD Sampling distribution: 1. Constant temperature MD for n steps at Tm with potential function think of it as a move: 2. Attempt to temperature move at constant positions + rescaled velocities: So: Same as in MC Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 10

Molecular Simulations in Computational Biology 2/4/08 Simulated Tempering (cont.) The weight factors fm(Tm) – 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. Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 11

Molecular Simulations in Computational Biology 2/4/08 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) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 12

Molecular Simulations in Computational Biology 2/4/08 Temperature Replica Exchange (a.k.a. Parallel Tempering) In Simulated Tempering equal visitation of temperatures is ensured by the free energy weights fm In Parallel Tempering the same is ensured by having each temperature correspond to an individual replica of the system. T1 T2 T2 T3 T4 ... Tn 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. Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 13

Molecular Simulations in Computational Biology 2/4/08 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 Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 14

Molecular Simulations in Computational Biology 2/4/08 Recall Metropolis MC algorithm: T-RE/MC Accept or reject exchange attempt based on this quantity Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 15

Molecular Simulations in Computational Biology 2/4/08 T-RE/MD Same as in MC Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 16

Molecular Simulations in Computational Biology 2/4/08 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 T1 T2 T1 T2 small N large N p(U) p(U) U U Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 17

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 18

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 19

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 20

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics Y. Sugita, Y. Okamoto Chem. Phys. Let., 314, 261 (1999) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 21

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics replica Y. Sugita, Y. Okamoto (1999) Chem. Phys. Let., 314:261 Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 22

Molecular Simulations in Computational Biology 2/4/08 Replica exchange molecular dynamics Temperature trajectory of a walker Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 23

Molecular Simulations in Computational Biology 2/4/08 Hamiltonian Replica Exchange (HREM) A RE method in which different replicas correspond to (slightly) different potential functions rather then temperature. “Perturbation” energy “Base” energy The probability of exchange between two replicas in HREM where (proposed change in energy of conformation x1) (proposed change in energy of conformation x2) Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 24

Molecular Simulations in Computational Biology 2/4/08 Two examples of HREM applications REUS: Replica Exchange Umbrella Sampling [Sugita, Kitao, Okamoto, 2000)] 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] 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] Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs) 25