Systematic and statistical error in histogram‐based free energy calculations

A common technique for the numerical calculation of free energies involves estimation of the probability density along a given coordinate from a set of configurations generated via simulation. The process requires discretization of one or more reaction coordinates to generate a histogram from which the continuous probability density is inferred. We show that the finite size of the intervals used to construct the histogram leads to quantifiable systematic error. The width of these intervals also determines the statistical error in the free energy, and the choice of the appropriate interval is therefore driven by the need to balance the two sources of error. We present a method for the construction of the optimal histogram for a given system, and show that the use of this technique requires little additional computational expense. We demonstrate the efficacy of the technique for a model system, and discuss how the principles governing the choice of discretization interval could be used to improve extended sampling techniques. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1437–1446, 2003

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

[2]  R. Bishop,et al.  Quantum many-particle systems , 1990 .

[3]  Wilfred F. van Gunsteren,et al.  Computation of free energy , 2002 .

[4]  K. Binder,et al.  Dynamic properties of the Monte Carlo method in statistical mechanics , 1973 .

[5]  T. Lazaridis,et al.  Potentials of mean force between ionizable amino acid side chains in water. , 2003, Journal of the American Chemical Society.

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

[7]  Peter A. Kollman,et al.  FREE ENERGY CALCULATIONS : APPLICATIONS TO CHEMICAL AND BIOCHEMICAL PHENOMENA , 1993 .

[8]  Arieh Warshel,et al.  Computer Modeling of Chemical Reactions in Enzymes and Solutions , 1991 .

[9]  M. Luescher,et al.  A Portable High-quality Random Number Generator for Lattice Field Theory Simulations , 1993 .

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

[11]  H. Scheraga,et al.  Prediction of the native conformation of a polypeptide by a statistical‐mechanical procedure. I. Backbone structure of enkephalin , 1985, Biopolymers.

[12]  D. Chandler,et al.  Introduction To Modern Statistical Mechanics , 1987 .

[13]  M. Hilczer,et al.  Effect of solvent polarity on the potential of mean force between two molecular ions: MD simulation , 1998 .

[14]  F. James,et al.  RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Lüscher , 1994 .

[15]  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 .

[16]  Peter A. Kollman,et al.  Free energy calculation methods: A theoretical and empirical comparison of numerical errors and a new method qualitative estimates of free energy changes , 1997, J. Comput. Chem..

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

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

[19]  Benoît Roux,et al.  Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations , 2001 .

[20]  Berend Smit,et al.  Understanding Molecular Simulation , 2001 .

[21]  A. R. Curtis,et al.  Standard Mathematical Tables , 1971, The Mathematical Gazette.

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

[23]  M. Gilson,et al.  The statistical-thermodynamic basis for computation of binding affinities: a critical review. , 1997, Biophysical journal.

[24]  H. Scheraga,et al.  Energy parameters in polypeptides. VII. Geometric parameters, partial atomic charges, nonbonded interactions, hydrogen bond interactions, and intrinsic torsional potentials for the naturally occurring amino acids , 1975 .

[25]  Molecular dynamics simulation of Na+-DMP− and Na+-MP2− ion pair in aqueous solution , 2002 .

[26]  R. Larsen,et al.  An introduction to mathematical statistics and its applications (2nd edition) , by R. J. Larsen and M. L. Marx. Pp 630. £17·95. 1987. ISBN 13-487166-9 (Prentice-Hall) , 1987, The Mathematical Gazette.

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

[28]  R. Hall.  Many-particle systems , 1972 .

[29]  D. Frenkel.  Free-energy calculations , 1991 .