Image Charge Optimization for the Reaction Field by Matching to an Electrostatic Force Tensor Wei Song Donald Jacobs University of North Carolina at Charlotte

Hybrid Model and Image Charges Method The total electrostatic potential inside sphere: 1.Direct coulomb interaction from intra-molecules 2.Polarization of continuum medium by explicit charges M. Lee,et al, "An efficient hybrid explicit/implicit solvent method for biomolecular simulations,“ 2004 A. Okur et al, "Hybrid explicit/implicit salvation methods,“ 2006 S. Deng, et al, "A comparable study of image approximations to the reaction field," 2007 W. Cai, et al, “Extending the fast multipole method to charges inside or outside a dielectric sphere,” 2007

Hybrid Model and Image Charges Method a z-axis  rsrs 1 source charge Kelvin charge Infinitely long line charge that decays as a power law ii oo The total electrostatic potential inside sphere: 1.Direct coulomb interaction from intra-molecules 2.Polarization of continuum medium by explicit charges Multiple image charge method using Gauss-Rauda quadratrature: line image charge  a set of discrete image charges M. Lee,et al, "An efficient hybrid explicit/implicit solvent method for biomolecular simulations,“ 2004 A. Okur et al, "Hybrid explicit/implicit salvation methods,“ 2006 S. Deng, et al, "A comparable study of image approximations to the reaction field," 2007 W. Cai, et al, “Extending the fast multipole method to charges inside or outside a dielectric sphere,” 2007

Solving an inverse problem Image charge solvation model (ICSM) ICSM: Long-range interaction: reaction field. Short range: periodic boundary conditions Y. Lin, et al, "An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions,“ 2009 P. Qin, et al, “Image Charge Methods for a Three-Dielectric-Layer Hybrid Solvation Model of Biomolecules,” 2009

Solving an inverse problem Image charge solvation model (ICSM) Discontinuous Dielectric Model (DDM) ICSM: Long-range interaction: reaction field. Short range: periodic boundary conditions DDM: Discontinuous dielectric make it unphysical, has analytical solutions Y. Lin, et al, "An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions,“ 2009 P. Qin, et al, “Image Charge Methods for a Three-Dielectric-Layer Hybrid Solvation Model of Biomolecules,” 2009

Solving an inverse problem Image charge solvation model (ICSM) Discontinuous Dielectric Model (DDM) Continuous transition of dielectric after optimization ICSM: Long-range interaction: reaction field. Short range: periodic boundary conditions Optimization: Modifying image charges, fitting to correct electrostatic force, ignore transition details of dielectric Y. Lin, et al, "An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions,“ 2009 P. Qin, et al, “Image Charge Methods for a Three-Dielectric-Layer Hybrid Solvation Model of Biomolecules,” 2009 DDM: Discontinuous dielectric make it unphysical, has analytical solutions

Optimizing image charges PME + GROMACS W. L. Jorgensen, et al, "Comparison of simple potential functions for simulating liquid water,“ 1983 H. Berendsen, et al, “GROMACS: A message-passing parallel molecular dynamics implementation,” 1995 Using DDM to reproduce PME model

Optimizing image charges PME + GROMACS DDM (q s, r s ) (q’ image, r’ image ) W. L. Jorgensen, et al, "Comparison of simple potential functions for simulating liquid water,“ 1983 H. Berendsen, et al, “GROMACS: A message-passing parallel molecular dynamics implementation,” 1995 Using DDM to reproduce PME model

Optimized image charges deviate from DDM

The dipole moment shows the water molecules are not randomly orientated, compared with PME result. The image charges should not be too negative values

Electrostatic torques of water molecules are more important than forces for reaction field The grid points are uniform distributed, with 1 Å separation. Dipoles are made for two nearby grid points Wei Song, et al, " Effect of the Reaction Field on Molecular Forces and Torques revealed by an Image-Charge Solvation Model,“ accepted F. Finck, et al, “Advances in Moment Tensor Inversion for Civil Engineering,” 2003

Electrostatic torques of water molecules are more important than forces for reaction field The grid points are uniform distributed, with 1 Å separation. Dipoles are made for two nearby grid points Wei Song, et al, " Effect of the Reaction Field on Molecular Forces and Torques revealed by an Image-Charge Solvation Model,“ in reviewing F. Finck, et al, “Advances in Moment Tensor Inversion for Civil Engineering,” 2003

Different Boundary Conditions M. A. Kastenholz et al, "Computation of methodology-independent ionic solvation free energies from molecular simulations. I. The electrostatic potential in molecular liquids," 2006 B. Ni et al, "Effect of atom- and group-based truncations on biomolecules simulated with reaction-field electrostatics,“ 2011

Different Boundary Conditions M. A. Kastenholz et al, "Computation of methodology-independent ionic solvation free energies from molecular simulations. I. The electrostatic potential in molecular liquids," 2006 B. Ni et al, "Effect of atom- and group-based truncations on biomolecules simulated with reaction-field electrostatics,“ 2011 Atom based condition gives better optimization results than ODL based case

L/2 tau a PME, cubic box PME, TO box Dynamic water Imaged water

L/2 tau a PME, cubic box PME, TO box Dynamic water Imaged water

Conclusions Atom based boundary condition gives better optimization results The most important area is near the edge of spherical cavity, the image charges cannot be too negative values. Force tensor does not help in ODL based boundary conditions non-convergence of dielectric constant near the edge of sphere still need to be solved.