Harmonic Fourier beads method for studying rare events on rugged energy surfaces.

We present a robust, distributable method for computing minimum free energy paths of large molecular systems with rugged energy landscapes. The method, which we call harmonic Fourier beads (HFB), exploits the Fourier representation of a path in an appropriate coordinate space and proceeds iteratively by evolving a discrete set of harmonically restrained path points-beads-to generate positions for the next path. The HFB method does not require explicit knowledge of the free energy to locate the path. To compute the free energy profile along the final path we employ an umbrella sampling method in two generalized dimensions. The proposed HFB method is anticipated to aid the study of rare events in biomolecular systems. Its utility is demonstrated with an application to conformational isomerization of the alanine dipeptide in gas phase.

[1]  M. Fixman.  Classical statistical mechanics of constraints: a theorem and application to polymers. , 1974, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Osamu Miyashita,et al.  Simple energy landscape model for the kinetics of functional transitions in proteins. , 2005, The journal of physical chemistry. B.

[3]  J. Onuchic,et al.  Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins , 2003, Proceedings of the National Academy of Sciences of the United States of America.

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

[5]  J. Straub,et al.  The MaxFlux algorithm for calculating variationally optimized reaction paths for conformational transitions in many body systems at finite temperature , 1997 .

[6]  Ron Elber,et al.  A method for determining reaction paths in large molecules: application to myoglobin , 1987 .

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

[8]  Harry A. Stern,et al.  Bridging the gap between long time trajectories and reaction pathways , 2003 .

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

[10]  M. Karplus,et al.  Simulation of activation free energies in molecular systems , 1996 .

[11]  F. Tama,et al.  Normal mode based flexible fitting of high-resolution structure into low-resolution experimental data from cryo-EM. , 2004, Journal of structural biology.

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

[13]  C. Brooks,et al.  Constant-temperature free energy surfaces for physical and chemical processes , 1993 .

[14]  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.

[15]  G. Ciccotti,et al.  Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes , 1977 .

[16]  Roberto Car,et al.  Free energy profile along a discretized reaction path via the hyperplane constraint force and torque. , 2005, The Journal of chemical physics.

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

[18]  M. Karplus,et al.  Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom , 1992 .

[19]  M. Fixman,et al.  Simulation of polymer dynamics. I. General theory , 1978 .

[20]  Harold A. Scheraga,et al.  On the Use of Classical Statistical Mechanics in the Treatment of Polymer Chain Conformation , 1976 .

[21]  Tamar Schlick,et al.  Deoxyadenosine sugar puckering pathway simulated by the stochastic difference equation algorithm , 2003 .

[22]  Ron Elber,et al.  Long-timescale simulation methods. , 2005, Current opinion in structural biology.

[23]  M. I. Bán,et al.  A comparison of some global path-following methods, I.☆ , 2000 .

[24]  Charles L. Brooks,et al.  Molecular dynamics with internal coordinate constraints , 1988 .

[25]  E. Vanden-Eijnden,et al.  String method for the study of rare events , 2002, cond-mat/0205527.

[26]  H. Jónsson,et al.  Optimization of hyperplanar transition states , 2001 .

[27]  M. Karplus,et al.  Molecular dynamics and protein function. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Weiqing Ren,et al.  Higher Order String Method for Finding Minimum Energy Paths , 2003 .

[29]  R. Byrd,et al.  A line integral reaction path approximation for large systems via nonlinear constrained optimization: application to alanine dipeptide and the beta hairpin of protein G. , 2006, The Journal of chemical physics.

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

[31]  D. L. Freeman,et al.  The construction of double-ended classical trajectories , 1994 .

[32]  Ron Elber,et al.  Calculation of the potential of mean force using molecular dynamics with linear constraints: An application to a conformational transition in a solvated dipeptide , 1990 .

[33]  Aatto Laaksonen,et al.  Bead-Fourier path integral molecular dynamics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  A. Chakraborty,et al.  A growing string method for determining transition states: comparison to the nudged elastic band and string methods. , 2004, The Journal of chemical physics.

[35]  David Chandler,et al.  Transition path sampling: throwing ropes over rough mountain passes, in the dark. , 2002, Annual review of physical chemistry.

[36]  P. Schleyer.  Encyclopedia of computational chemistry , 1998 .

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