Reconstructing the equilibrium Boltzmann distribution from well‐tempered metadynamics

Metadynamics is a widely used and successful method for reconstructing the free‐energy surface of complex systems as a function of a small number of suitably chosen collective variables. This is achieved by biasing the dynamics of the system. The bias acting on the collective variables distorts the probability distribution of the other variables. Here we present a simple reweighting algorithm for recovering the unbiased probability distribution of any variable from a well‐tempered metadynamics simulation. We show the efficiency of the reweighting procedure by reconstructing the distribution of the four backbone dihedral angles of alanine dipeptide from two and even one dimensional metadynamics simulation. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009

[1]  A. Laio,et al.  Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science , 2008 .

[2]  M. Parrinello,et al.  Well-tempered metadynamics: a smoothly converging and tunable free-energy method. , 2008, Physical review letters.

[3]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[4]  J. Valleau,et al.  A Monte Carlo method for obtaining the interionic potential of mean force in ionic solution , 1975 .

[5]  G. Ciccotti,et al.  String method in collective variables: minimum free energy paths and isocommittor surfaces. , 2006, The Journal of chemical physics.

[6]  Andrew E. Torda,et al.  Local elevation: A method for improving the searching properties of molecular dynamics simulation , 1994, J. Comput. Aided Mol. Des..

[7]  P. R. ten Wolde,et al.  Sampling rare switching events in biochemical networks. , 2004, Physical review letters.

[8]  A. Laio,et al.  Equilibrium free energies from nonequilibrium metadynamics. , 2006, Physical Review Letters.

[9]  Michele Parrinello.  Eppur si muove , 2008 .

[10]  Shankar Kumar,et al.  Multidimensional free‐energy calculations using the weighted histogram analysis method , 1995, J. Comput. Chem..

[11]  A. Cavalli,et al.  Protein conformational transitions: the closure mechanism of a kinase explored by atomistic simulations. , 2009, Journal of the American Chemical Society.

[12]  Eric F Darve,et al.  Calculating free energies using average force , 2001 .

[13]  D. Landau,et al.  Efficient, multiple-range random walk algorithm to calculate the density of states. , 2000, Physical review letters.

[14]  U. Hansmann.  Parallel tempering algorithm for conformational studies of biological molecules , 1997, physics/9710041.

[15]  C. Dellago,et al.  Reaction coordinates of biomolecular isomerization. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Christian Bartels,et al.  Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations , 1997 .

[17]  A. Voter.  Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events , 1997 .

[18]  Laxmikant V. Kalé,et al.  Scalable molecular dynamics with NAMD , 2005, J. Comput. Chem..

[19]  C. Dellago,et al.  Transition path sampling and the calculation of rate constants , 1998 .

[20]  R. Elber,et al.  Computing time scales from reaction coordinates by milestoning. , 2004, The Journal of chemical physics.

[21]  Guido Tiana,et al.  Estimation of microscopic averages from metadynamics , 2007, 0711.2984.

[22]  Y. Sugita,et al.  Replica-exchange molecular dynamics method for protein folding , 1999 .

[23]  W. E,et al.  Finite temperature string method for the study of rare events. , 2002, Journal of Physical Chemistry B.

[24]  Alessandro Laio,et al.  Anisotropy of Earth's D″ layer and stacking faults in the MgSiO3 post-perovskite phase , 2005, Nature.

[25]  B. Roux.  The calculation of the potential of mean force using computer simulations , 1995 .

[26]  Martin J. Field,et al.  A temperature-dependent nudged-elastic-band algorithm , 2003 .

[27]  J. Behler,et al.  Metadynamics simulations of the high-pressure phases of silicon employing a high-dimensional neural network potential. , 2008, Physical review letters.

[28]  Alan M. Ferrenberg,et al.  New Monte Carlo technique for studying phase transitions. , 1988, Physical review letters.

[29]  Deprotonation of solvated formic acid: Car-Parrinello and metadynamics simulations. , 2006, The journal of physical chemistry. B.

[30]  Alessandra Magistrato,et al.  Dissociation of minor groove binders from DNA: insights from metadynamics simulations , 2008, Nucleic acids research.

[31]  B. Berne,et al.  Replica exchange with solute tempering: a method for sampling biological systems in explicit water. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Determining the structure of phosphorus in phase IV. , 2005, Physical review letters.

[33]  A. Laio,et al.  Simulation of structural phase transitions by metadynamics , 2004, cond-mat/0411559.

[34]  G. Torrie,et al.  Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling , 1977 .

[35]  Bernd Ensing,et al.  Perspective on the reactions between F- and CH3CH2F: the free energy landscape of the E2 and SN2 reaction channels. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[36]  Peter G. Bolhuis,et al.  A novel path sampling method for the calculation of rate constants , 2003 .

[37]  Aaron R Dinner,et al.  Automatic method for identifying reaction coordinates in complex systems. , 2005, The journal of physical chemistry. B.

[38]  P M Rodger,et al.  Metadynamics simulations of ice nucleation and growth. , 2008, The Journal of chemical physics.

[39]  Alexander D. MacKerell,et al.  All-atom empirical potential for molecular modeling and dynamics studies of proteins. , 1998, The journal of physical chemistry. B.

[40]  R. Broglia,et al.  Exploring the protein G helix free‐energy surface by solute tempering metadynamics , 2007, Proteins.

[41]  Haruki Nakamura,et al.  Flexible docking of a ligand peptide to a receptor protein by multicanonical molecular dynamics simulation , 1997 .

[42]  Simone Marsili,et al.  Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations. , 2006, The journal of physical chemistry. B.

[43]  Francesco L Gervasio,et al.  Metadynamics simulation of prion protein: beta-structure stability and the early stages of misfolding. , 2006, Journal of the American Chemical Society.

[44]  Amedeo Caflisch,et al.  Calculation of conformational transitions and barriers in solvated systems: Application to the alanine dipeptide in water , 1999 .

[45]  M. Parrinello,et al.  The unfolded ensemble and folding mechanism of the C-terminal GB1 beta-hairpin. , 2008, Journal of the American Chemical Society.

[46]  Francesco Luigi Gervasio,et al.  From A to B in free energy space. , 2007, The Journal of chemical physics.

[47]  Charles L. Brooks,et al.  Reaction paths and free energy profiles for conformational transitions: An internal coordinate approach , 1991 .

[48]  Eric Darve,et al.  Adaptive biasing force method for scalar and vector free energy calculations. , 2008, The Journal of chemical physics.

[49]  D. Marx,et al.  Glycine at the pyrite-water interface: the role of surface defects. , 2006, Journal of the American Chemical Society.

[50]  P. A. Bash,et al.  Calculation of the relative change in binding free energy of a protein-inhibitor complex. , 1987, Science.

[51]  A. Kalinichev,et al.  Dissociation of carbonic acid: gas phase energetics and mechanism from ab initio metadynamics simulations. , 2007, The Journal of chemical physics.

[52]  Grubmüller,et al.  Predicting slow structural transitions in macromolecular systems: Conformational flooding. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[53]  A. Laio,et al.  Free-energy landscape for beta hairpin folding from combined parallel tempering and metadynamics. , 2006, Journal of the American Chemical Society.

[54]  A. Laio,et al.  Escaping free-energy minima , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[55]  G. Henkelman,et al.  Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points , 2000 .

[56]  Paolo V. Giaquinta,et al.  Liquid-solid coexistence via the metadynamics approach. , 2008, The Journal of chemical physics.

[57]  G. Parisi,et al.  Simulated tempering: a new Monte Carlo scheme , 1992, hep-lat/9205018.

[58]  A. Laio,et al.  Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics. , 2006, The journal of physical chemistry. B.

[59]  A. Laio,et al.  Assessing the accuracy of metadynamics. , 2005, The journal of physical chemistry. B.

[60]  Alessandro Laio,et al.  Reconstructing the density of states by history-dependent metadynamics. , 2004, Physical review letters.

[61]  Vojtech Spiwok,et al.  Metadynamics in essential coordinates: free energy simulation of conformational changes. , 2007, The journal of physical chemistry. B.