Zhenli Xu

An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions.

In this paper, a new solvation model is proposed for simulations of biomolecules in aqueous solutions that combines the strengths of explicit and implicit solvent representations. Solute molecules are placed in a spherical cavity filled with explicit water, thus providing microscopic detail where it is most needed. Solvent outside of the cavity is modeled as a dielectric continuum whose effect on the solute is treated through the reaction field corrections. With this explicit/implicit model, the electrostatic potential represents a solute molecule in an infinite bath of solvent, thus avoiding unphysical interactions between periodic images of the solute commonly used in the lattice-sum explicit solvent simulations. For improved computational efficiency, our model employs an accurate and efficient multiple-image charge method to compute reaction fields together with the fast multipole method for the direct Coulomb interactions. To minimize the surface effects, periodic boundary conditions are employed for nonelectrostatic interactions. The proposed model is applied to study liquid water. The effect of model parameters, which include the size of the cavity, the number of image charges used to compute reaction field, and the thickness of the buffer layer, is investigated in comparison with the particle-mesh Ewald simulations as a reference. An optimal set of parameters is obtained that allows for a faithful representation of many structural, dielectric, and dynamic properties of the simulated water, while maintaining manageable computational cost. With controlled and adjustable accuracy of the multiple-image charge representation of the reaction field, it is concluded that the employed model achieves convergence with only one image charge in the case of pure water. Future applications to pKa calculations, conformational sampling of solvated biomolecules and electrolyte solutions are briefly discussed.

Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems

We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrodinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu, H. Han, Phys. Rev. E 74 (2006) 037704] to problems in multi-dimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrodinger equations in one and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.

Numerical simulation of interaction between Schrödinger field and Klein–Gordon field by multisymplectic method

Abstract We propose a multisymplectic scheme to solve the coupled Klein–Gordon–Schrodinger system. The scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law, and can simulate the original waves well in a long time. This scheme also has discrete quasi-norm conservation law. Numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results.

Effects of image charges, interfacial charge discreteness, and surface roughness on the zeta potential of spherical electric double layers.

We investigate the effects of image charges, interfacial charge discreteness, and surface roughness on spherical electric double layer structures in electrolyte solutions with divalent counterions in the setting of the primitive model. By using Monte Carlo simulations and the image charge method, the zeta potential profile and the integrated charge distribution function are computed for varying surface charge strengths and salt concentrations. Systematic comparisons were carried out between three distinct models for interfacial charges: (1) SURF1 with uniform surface charges, (2) SURF2 with discrete point charges on the interface, and (3) SURF3 with discrete interfacial charges and finite excluded volume. By comparing the integrated charge distribution function and the zeta potential profile, we argue that the potential at the distance of one ion diameter from the macroion surface is a suitable location to define the zeta potential. In SURF2 model, we find that image charge effects strongly enhance charge inversion for monovalent interfacial charges, and strongly suppress charge inversion for multivalent interfacial charges. For SURF3, the image charge effect becomes much smaller. Finally, with image charges in action, we find that excluded volumes (in SURF3) suppress charge inversion for monovalent interfacial charges and enhance charge inversion for multivalent interfacial charges. Overall, our results demonstrate that all these aspects, i.e., image charges, interfacial charge discreteness, their excluding volumes, have significant impacts on zeta potentials of electric double layers.

Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations.

An efficient method is proposed for numerical solutions of nonlinear Schrödinger equations on an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation, absorbing boundary conditions are designed to truncate the unbounded domain, which are in nonlinear form and can perfectly absorb waves outgoing from the boundaries of the truncated computational domain. The stability of the induced initial boundary value problem defined on the computational domain is examined by a normal mode analysis. Numerical examples are given to illustrate the stable and tractable advantages of the method.

Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations

We review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications, including the Poisson-Boltzmann (PB) and the generalized Born models for electrostatic solvation energy. The focus is on analytical approaches for hybrid solvation models, especially the image charge method for a spherical cavity, and also the generalized Born theory as an approximation to the PB model. This review places much emphasis on the mathematical details behind these methods.

Electrostatic interaction in the presence of dielectric interfaces and polarization-induced like-charge attraction.

Electrostatic polarization is important in many nano- and micro-scale physical systems such as colloidal suspensions, biopolymers, and nanomaterials assembly. The calculation of polarization potential requires an efficient algorithm for solving 3D Poissons equation. We have developed a useful image charge method to rapid evaluation of the Greens function of the Poissons equation in the presence of spherical dielectric discontinuities. This paper presents an extensive study of this method by giving a convergence analysis and developing a coarse-graining algorithm. The use of the coarse graining could reduce the number of image charges to around a dozen, by 1-2 orders of magnitude. We use the algorithm to investigate the interaction force between likely charged spheres in different dielectric environments. We find the size and charge asymmetry leads to an attraction between like charges, in agreement with existing results. Furthermore, we study three-body interactions and find, in the presence of an external interface, that the interaction force depends on the curvature of the interface and performs a nonmonotonic electrostatic force.

Simulation of electric double layers around charged colloids in aqueous solution of variable permittivity

The ion distribution around charged colloids in solution has been investigated intensely during the last decade. However, few theoretical approaches have included the influence of variation in the dielectric permittivity within the system, let alone in the surrounding solvent. In this article, we introduce two relatively new methods that can solve the Poisson equation for systems with varying permittivity. The harmonic interpolation method approximately solves the Greens function in terms of a spherical harmonics series, and thus provides analytical ion-ion potentials for the Hamiltonian of charged systems. The Maxwell equations molecular dynamics algorithm features a local approach to electrostatics, allowing for arbitrary local changes of the dielectric constant. We show that the results of both methods are in very good agreement. We also found that the renormalized charge of the colloid, and with it the effective far field interaction, significantly changes if the dielectric properties within the vicinity of the colloid are changed.

Image charge approximations of reaction fields in solvents with arbitrary ionic strength

In this paper, we report a new development of image charge approximations of reaction fields for a charge inside a dielectric spherical cavity immersed in an ionic solvent with arbitrary ionic strength. This new development removes the requirement of low ionic strength of the solvent in a previous result [S. Deng, W. Cai, Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: high-order image approximations for reaction fields, J. Comput. Phys. 227 (2007) 1246-1266], thus extending the applicability of the image charge approximations of reaction fields in the modeling of bio-molecular solvation.

Comparison of efficient techniques for the simulation of dielectric objects in electrolytes

We review two recently developed efficient approaches for the numerical evaluation of the electrostatic polarization potential in particle-based simulations. The first is an image-charge method that can be applied to systems of spherical dielectric objects and provides a closed-form solution of Poissons equation through multiple image-charge reflections and numerical evaluation of the resulting line integrals. The second is a boundary-element method that computes the discretized surface bound charge through a combination of the generalized minimal residual method (GMRES) and a fast Ewald solver. We compare the accuracy and efficiency of both approaches as a function of the pertinent numerical parameters. We demonstrate use of the image-charge method in a Monte Carlo simulation using the Barnes-Hut octree algorithm and the boundary-element method in a molecular dynamics simulation using the Particle-Particle Particle-Mesh (PPPM) Ewald method, and present numerical results for the ensemble-averaged induced force between two spherical colloids immersed in an electrolyte.