An experimentally guided umbrella sampling protocol for biomolecules.

We present a simple method for utilizing experimental data to improve the efficiency of numerical calculations of free energy profiles from molecular dynamics simulations. The method involves umbrella sampling simulations with restraining potentials based on a known approximate estimate of the free energy profile derived solely from experimental data. The use of the experimental data results in optimal restraining potentials, guides the simulation along relevant pathways, and decreases overall computational time. In demonstration of the method, two systems are showcased. First, guided, unguided (regular) umbrella sampling simulations and exhaustive sampling simulations are compared to each other in the calculation of the free energy profile for the distance between the ends of a pentapeptide. The guided simulation use restraints based on a simulated "experimental" potential of mean force of the end-to-end distance that would be measured by fluorescence resonance energy transfer (obtained from exhaustive sampling). Statistical analysis shows a dramatic improvement in efficiency for a 5 window guided umbrella sampling over 5 and 17 window unguided umbrella sampling simulations. Moreover, the form of the potential of mean force for the guided simulations evolves, as one approaches convergence, along the same milestones as the extensive simulations, but exponentially faster. Second, the method is further validated by replicating the forced unfolding pathway of the titin I27 domain using guiding umbrella sampling potentials determined from actual single molecule pulling data. Comparison with unguided umbrella sampling reveals that the use of guided sampling encourages unfolding simulations to converge faster to a forced unfolding pathway that agrees with previous results and produces a more accurate potential of mean force.

[1]  J. Kirkwood.  Statistical Mechanics of Fluid Mixtures , 1935 .

[2]  A. Cooper,et al.  Thermodynamic fluctuations in protein molecules. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[3]  D. Chandler,et al.  Theory of the hydrophobic effect , 1977 .

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

[5]  R. Levy,et al.  Protein dynamics and NMR relaxation: comparison of simulations with experiment , 1982, Nature.

[6]  M. Karplus,et al.  CHARMM: A program for macromolecular energy, minimization, and dynamics calculations , 1983 .

[7]  Bernard Pettitt,et al.  The potential of mean force surface for the alanine dipeptide in aqueous solution: a theoretical approach , 1985 .

[8]  E. Henry,et al.  Influence of vibrational motion on solid state line shapes and NMR relaxation , 1985 .

[9]  M. Mezei.  Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias , 1987 .

[10]  Robert H. Swendsen,et al.  Erratum: ``New Monte Carlo technique for studying phase transitions'' [Phys. Rev. Lett. 61, 2635 (1988)] , 1989 .

[11]  Alan M. Ferrenberg,et al.  Optimized Monte Carlo data analysis. , 1989, Physical Review Letters.

[12]  G. Ciccotti,et al.  Constrained reaction coordinate dynamics for the simulation of rare events , 1989 .

[13]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

[14]  M Karplus,et al.  Ion transport in a model gramicidin channel. Structure and thermodynamics. , 1991, Biophysical journal.

[15]  B. Berg,et al.  Multicanonical algorithms for first order phase transitions , 1991 .

[16]  Jiali Gao,et al.  A priori computation of a solvent-enhanced SN2 reaction profile in water : the Menshutkin reaction , 1991 .

[17]  Berg,et al.  Multicanonical ensemble: A new approach to simulate first-order phase transitions. , 1992, Physical review letters.

[18]  Charles L. Brooks,et al.  Conformational equilibrium in the alanine dipeptide in the gas phase and aqueous solution : a comparison of theoretical results , 1992 .

[19]  R. Swendsen,et al.  THE weighted histogram analysis method for free‐energy calculations on biomolecules. I. The method , 1992 .

[20]  R. Hooft,et al.  An adaptive umbrella sampling procedure in conformational analysis using molecular dynamics and its application to glycol , 1992 .

[21]  Yuko Okamoto,et al.  Prediction of peptide conformation by multicanonical algorithm: New approach to the multiple‐minima problem , 1993, J. Comput. Chem..

[22]  Lee,et al.  New Monte Carlo algorithm: Entropic sampling. , 1993, Physical review letters.

[23]  Wilfred F. van Gunsteren,et al.  The computation of a potential of mean force: Choice of the biasing potential in the umbrella sampling technique , 1994 .

[24]  Case,et al.  Collective NMR relaxation model applied to protein dynamics. , 1994, Physical review letters.

[25]  P. Krüger,et al.  Targeted molecular dynamics: a new approach for searching pathways of conformational transitions. , 1994, Journal of molecular graphics.

[26]  C. Dobson,et al.  Comparison of MD simulations and NMR experiments for hen lysozyme. Analysis of local fluctuations, cooperative motions, and global changes. , 1995, Biochemistry.

[27]  C. Brooks,et al.  First-principles calculation of the folding free energy of a three-helix bundle protein. , 1995, Science.

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

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

[30]  Patrice Koehl,et al.  Calculation of nuclear magnetic resonance order parameters in proteins by normal mode analysis , 1996 .

[31]  M. Karplus,et al.  A Comprehensive Analytical Treatment of Continuum Electrostatics , 1996 .

[32]  Ron Elber,et al.  Calculation of classical trajectories with a very large time step: Formalism and numerical examples , 1996 .

[33]  I. Andricioaei,et al.  On Monte Carlo and molecular dynamics methods inspired by Tsallis statistics: Methodology, optimization, and application to atomic clusters , 1997 .

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

[35]  Jeremy C. Smith,et al.  Quantum Chemical and Free Energy Simulation Analysis of Retinal Conformational Energetics , 1997, J. Chem. Inf. Comput. Sci..

[36]  Krzysztof Kuczera,et al.  Multidimensional Conformational Free Energy Surface Exploration: Helical States of Alan and Aibn Peptides , 1997 .

[37]  C. Jarzynski.  Nonequilibrium Equality for Free Energy Differences , 1996, cond-mat/9610209.

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

[39]  M. Karplus,et al.  Solution conformations and thermodynamics of structured peptides: molecular dynamics simulation with an implicit solvation model. , 1998, Journal of molecular biology.

[40]  Michiel Sprik,et al.  Free energy from constrained molecular dynamics , 1998 .

[41]  A. Voter.  Parallel replica method for dynamics of infrequent events , 1998 .

[42]  Martin Karplus,et al.  Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy , 1998 .

[43]  David L. Andrews,et al.  Resonance Energy Transfer , 1999 .

[44]  Klaus Schulten,et al.  Mechanical unfolding intermediates in titin modules , 1999, Nature.

[45]  Christoph Dellago,et al.  On the calculation of reaction rate constants in the transition path ensemble , 1999 .

[46]  M Karplus,et al.  Forced unfolding of fibronectin type 3 modules: an analysis by biased molecular dynamics simulations. , 1999, Journal of molecular biology.

[47]  Ron Elber,et al.  Stochastic path approach to compute atomically detailed trajectories: Application to the folding of C peptide , 1999 .

[48]  Carlos Bustamante,et al.  Grabbing the cat by the tail: manipulating molecules one by one , 2000, Nature Reviews Molecular Cell Biology.

[49]  K. Schulten,et al.  The key event in force-induced unfolding of Titin's immunoglobulin domains. , 2000, Biophysical journal.

[50]  Lisa J. Lapidus,et al.  Measuring the rate of intramolecular contact formation in polypeptides. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[51]  Andrew B. Martin,et al.  Single-molecule protein folding: diffusion fluorescence resonance energy transfer studies of the denaturation of chymotrypsin inhibitor 2. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[52]  H Li,et al.  Atomic force microscopy reveals the mechanical design of a modular protein. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

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

[54]  Shimon Weiss,et al.  Measuring conformational dynamics of biomolecules by single molecule fluorescence spectroscopy , 2000, Nature Structural Biology.

[55]  A. Oberhauser,et al.  Mechanical design of proteins studied by single-molecule force spectroscopy and protein engineering. , 2000, Progress in biophysics and molecular biology.

[56]  K. Schulten,et al.  Steered molecular dynamics and mechanical functions of proteins. , 2001, Current opinion in structural biology.

[57]  G. Hummer,et al.  Free energy reconstruction from nonequilibrium single-molecule pulling experiments , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[58]  M. J. Pereira,et al.  In the fluorescent spotlight: global and local conformational changes of small catalytic RNAs. , 2001, Biopolymers.

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

[60]  Michael R. Shirts,et al.  Mathematical analysis of coupled parallel simulations. , 2001, Physical review letters.

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

[62]  G. Hummer,et al.  Peptide loop-closure kinetics from microsecond molecular dynamics simulations in explicit solvent. , 2002, Journal of the American Chemical Society.

[63]  Alexander D. MacKerell,et al.  Protein-facilitated base flipping in DNA by cytosine-5-methyltransferase , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[64]  J. Berg,et al.  Molecular dynamics simulations of biomolecules , 2002, Nature Structural Biology.

[65]  Jane Clarke,et al.  Mechanical unfolding of a titin Ig domain: structure of unfolding intermediate revealed by combining AFM, molecular dynamics simulations, NMR and protein engineering. , 2002, Journal of molecular biology.

[66]  W. Eaton,et al.  Probing the free-energy surface for protein folding with single-molecule fluorescence spectroscopy , 2002, Nature.

[67]  X. Zhuang,et al.  Correlating Structural Dynamics and Function in Single Ribozyme Molecules , 2002, Science.

[68]  Rafael Brüschweiler,et al.  Contact model for the prediction of NMR N-H order parameters in globular proteins. , 2002, Journal of the American Chemical Society.

[69]  T. L. Hill,et al.  Thermodynamics of Small Systems , 2002 .

[70]  Gregory A Voth,et al.  Molecular dynamics simulation of proton transport near the surface of a phospholipid membrane. , 2002, Biophysical journal.

[71]  Eric Darve,et al.  Calculating Free Energies Using a Scaled-Force Molecular Dynamics Algorithm , 2002 .

[72]  B. Berne,et al.  Can a continuum solvent model reproduce the free energy landscape of a β-hairpin folding in water? , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[73]  H. Nymeyer,et al.  Simulation of the folding equilibrium of α-helical peptides: A comparison of the generalized Born approximation with explicit solvent , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[74]  Michele Vendruscolo,et al.  Rare fluctuations of native proteins sampled by equilibrium hydrogen exchange. , 2003, Journal of the American Chemical Society.

[75]  Gerhard Hummer,et al.  Kinetics from nonequilibrium single-molecule pulling experiments. , 2003, Biophysical journal.

[76]  D. Makarov,et al.  Theoretical studies of the mechanical unfolding of the muscle protein titin: Bridging the time-scale gap between simulation and experiment , 2003 .

[77]  A. Szabó,et al.  Single-Macromolecule Fluorescence Resonance Energy Transfer and Free-Energy Profiles , 2003 .

[78]  A. Matouschek.  Protein unfolding--an important process in vivo? , 2003, Current opinion in structural biology.

[79]  Gilles Charvin,et al.  Stretching of macromolecules and proteins , 2003 .

[80]  J. Pablo,et al.  Density of states simulations of proteins , 2003 .

[81]  E. Paci,et al.  Mechanical unfolding of a titin Ig domain: structure of transition state revealed by combining atomic force microscopy, protein engineering and molecular dynamics simulations. , 2003, Journal of molecular biology.

[82]  P. G. Debenedetti,et al.  An improved Monte Carlo method for direct calculation of the density of states , 2003 .

[83]  Mark N. Kobrak.  Systematic and statistical error in histogram‐based free energy calculations , 2003, J. Comput. Chem..

[84]  Jane Clarke,et al.  Hidden complexity in the mechanical properties of titin , 2003, Nature.

[85]  P. Bolhuis.  Transition-path sampling of β-hairpin folding , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[86]  Peter G Bolhuis,et al.  Rate constants for diffusive processes by partial path sampling. , 2004, The Journal of chemical physics.

[87]  Jordi Martí,et al.  Transition path sampling study of flip-flop transitions in model lipid bilayer membranes. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[88]  Taekjip Ha,et al.  Single-molecule three-color FRET. , 2004, Biophysical journal.

[89]  T. Schlick,et al.  Orchestration of cooperative events in DNA synthesis and repair mechanism unraveled by transition path sampling of DNA polymerase beta's closing. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[90]  T. Pan,et al.  Single-molecule studies highlight conformational heterogeneity in the early folding steps of a large ribozyme. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

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

[92]  M. Vendruscolo,et al.  Determination of protein structures consistent with NMR order parameters. , 2004, Journal of the American Chemical Society.

[93]  Rafael Brüschweiler,et al.  Prediction of methyl-side Chain Dynamics in Proteins , 2004, Journal of biomolecular NMR.

[94]  W. Im,et al.  Theoretical and computational models of biological ion channels , 2004, Quarterly Reviews of Biophysics.

[95]  Anthony K. Felts,et al.  Temperature weighted histogram analysis method, replica exchange, and transition paths. , 2005, The journal of physical chemistry. B.

[96]  Jane Clarke,et al.  What contributions to protein side-chain dynamics are probed by NMR experiments? A molecular dynamics simulation analysis. , 2005, Journal of molecular biology.

[97]  T. Endo,et al.  Comparison of the protein-unfolding pathways between mitochondrial protein import and atomic-force microscopy measurements. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[98]  James R. Macfadyen,et al.  A skewed-momenta method to efficiently generate conformational-transition trajectories. , 2005, The Journal of chemical physics.

[99]  Lennart Nilsson,et al.  Molecular dynamics simulations and free energy calculations of base flipping in dsRNA. , 2005, RNA.

[100]  Serdal Kirmizialtin,et al.  Topography of the free-energy landscape probed via mechanical unfolding of proteins. , 2005, The Journal of chemical physics.

[101]  Guohui Li,et al.  Direct determination of reaction paths and stationary points on potential of mean force surfaces. , 2005, Journal of molecular graphics & modelling.

[102]  Repetitive pulling catalyzes co-translocational unfolding of barnase during import through a mitochondrial pore. , 2005, Journal of molecular biology.

[103]  Jean-Pierre Clamme,et al.  Three-color single-molecule fluorescence resonance energy transfer. , 2005, Chemphyschem : a European journal of chemical physics and physical chemistry.

[104]  F. Ritort,et al.  The nonequilibrium thermodynamics of small systems , 2005 .

[105]  Emanuele Paci,et al.  Prediction of the translocation kinetics of a protein from its mechanical properties. , 2006, Biophysical journal.

[106]  D. Herschlag,et al.  Direct Measurement of the Full, Sequence-Dependent Folding Landscape of a Nucleic Acid , 2006, Science.

[107]  J. Straub,et al.  Statistical-temperature Monte Carlo and molecular dynamics algorithms. , 2006, Physical review letters.

[108]  Jeff Wereszczynski,et al.  On structural transitions, thermodynamic equilibrium, and the phase diagram of DNA and RNA duplexes under torque and tension , 2006, Proceedings of the National Academy of Sciences.

[109]  Gerhard Hummer,et al.  Intrinsic rates and activation free energies from single-molecule pulling experiments. , 2006, Physical review letters.

[110]  M. Rief,et al.  Single-molecule unfolding force distributions reveal a funnel-shaped energy landscape. , 2006, Biophysical journal.

[111]  Aaron R Dinner,et al.  Dynamic coupling between coordinates in a model for biomolecular isomerization. , 2006, The Journal of chemical physics.

[112]  Emanuele Paci,et al.  Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality. , 2006, The Journal of chemical physics.

[113]  I. Andricioaei,et al.  Exact low-force kinetics from high-force single-molecule unfolding events. , 2007, Biophysical journal.

[114]  C. Kiang,et al.  Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski's equality. , 2007, Physical review letters.

[115]  Gregory A Voth,et al.  Proton transport behavior through the influenza A M2 channel: insights from molecular simulation. , 2007, Biophysical journal.

[116]  David D L Minh.  Multidimensional potentials of mean force from biased experiments along a single coordinate. , 2007, The journal of physical chemistry. B.

[117]  W. Eaton,et al.  Protein folding studied by single-molecule FRET. , 2008, Current opinion in structural biology.

[118]  Gerhard Hummer,et al.  Pulling direction as a reaction coordinate for the mechanical unfolding of single molecules. , 2008, The journal of physical chemistry. B.

[119]  I. Andricioaei,et al.  Entropy-energy decomposition from nonequilibrium work trajectories. , 2008, The Journal of chemical physics.

[120]  Christina L. Ting,et al.  Two-dimensional fluorescence resonance energy transfer as a probe for protein folding: a theoretical study. , 2008, The Journal of chemical physics.

[121]  Hui Lu,et al.  Solvent molecules bridge the mechanical unfolding transition state of a protein , 2008, Proceedings of the National Academy of Sciences.