Electrostatics of charged dielectric spheres with application to biological systems

T. P. Doerr and Yi-Kuo Yu

Phys. Rev. E 73, 061902 – Published 2 June 2006

Abstract

Because electrostatic forces are crucial in biological systems, molecular dynamics simulations of biological systems require a method of computing electrostatic forces that is accurate and rapid. We propose a surface charge method, apply it to a system of arbitrary number of charged dielectric spheres, and obtain an exact solution for an arbitrary configuration of the spheres. The precision depends only on the number of terms kept in a series expansion and can therefore be controlled at will. It appears that the first few terms are usually adequate. The exact result exhibits a phenomenon that we call asymmetric screening. Namely, the magnitude of attractive interactions is decreased (relative to point charges in an infinite solvent) while the magnitude of repulsive interactions is increased (again, relative to point charges in an infinite solvent). This effect might aid in the adoption of correct conformations and in intermolecular recognition. Evaluation of the energy involves only matrix inversion. The surface charge method can be transformed easily to a numerical method for use with arbitrary surfaces. With modest additions, the model also describes an electrorheological fluid. Such a system provides the cleanest opportunity to apply the model.

Received 12 January 2006

DOI:https://doi.org/10.1103/PhysRevE.73.061902

Authors & Affiliations

T. P. Doerr^* and Yi-Kuo Yu^†

National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, 8600 Rockville Pike MSC 6075, Bethesda, Maryland 20894, USA

^*Electronic address: doerr@ncbi.nlm.nih.gov
^†Electronic address: yyu@ncbi.nlm.nih.gov

Electrostatics of charged dielectric spheres with application to biological systems. II. A formalism bypassing Wigner rotation matrices

Yi-Kuo Yu

Phys. Rev. E 100, 012401 (2019)

Electrostatics of charged dielectric spheres with application to biological systems. III. Rigorous ionic screening at the Debye-Hückel level

Yi-Kuo Yu

Phys. Rev. E 102, 052404 (2020)

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Vol. 73, Iss. 6 — June 2006

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Images

Figure 1
(a) The rotation that takes the local $z$ axis to the global $z$ axis. A rotation (labeled 1) about the global $z$ axis through an angle $π - φ$ is followed by a rotation (labeled 2) about the global $y$ axis through an angle $ϑ - π$ . In the text, this rotation is denoted by $R$ . (b) The rotation that takes the global $z$ axis to the local $z$ axis. A rotation (labeled 1) about the global $y$ axis through an angle $π - ϑ$ is followed by a rotation (labeled 2) about the global $z$ axis through an angle $φ - π$ . In the text, this rotation is denoted by $R^{- 1}$ . The notation for the rotation angles has been chosen for later convenience.Reuse & Permissions
Figure 2
The geometry of the two-sphere subsystem (spheres $A_{k}$ and $A_{i}$ ) with some of the useful variables indicated: $r_{k}$ is a vector from the center of $A_{k}$ to an arbitrary point, $r_{i}^{'}$ is a vector from the center of $A_{i}$ to a source on the surface of $A_{i}$ , $L n$ is a vector from the center of $A_{k}$ to the center of $A_{i}$ , ${\tilde{θ}}_{i}^{'}$ is the polar angle of $r_{i}^{'}$ in the local coordinate system, ${\tilde{θ}}_{k}$ is the polar angle of $r_{k}$ in the local coordinate system, ${\hat{θ}}_{i k}^{'}$ is the polar angle of $L n + r_{i}^{'}$ in the local coordinate system, $z$ is a unit vector parallel to the global $z$ axis, and $\tilde{z}$ is a unit vector parallel to the local $z$ axis. Two other vectors, derived from the others and prominent in the analysis, are also shown.Reuse & Permissions
Figure 3
Energy and force calculations for two identical spheres of unit radius with unit charges embedded in an infinite medium $(ϵ_{0} = 80)$ . (a) The dashed line is the energy for spheres with internal dielectric constant 4 minus the energy for spheres with internal dielectric constant 1 divided by the magnitude of the energy for spheres with internal dielectric constant 1. The solid line is calculated in the same manner as the dashed line except that in the numerator only the $L$ -independent self-energy terms are kept. The solid line therefore represents the portion of the change due to the solvation self-energy terms, which do not affect the force. (b) The force for spheres with internal dielectric constant 4 minus the force for spheres with internal dielectric constant 1 divided by the magnitude of the force for spheres with internal dielectric constant 1.Reuse & Permissions
Figure 4
The force between two spheres of unit radius, with unit interior dielectric constant, in an external dielectric with $ϵ_{0} = 80$ . (a) The force for like charges of unit magnitude. The change between $l = 7$ and $l = 8$ at $L = 2$ is 0.96%. The convergence is even better for larger $L$ . (b) The force for opposite charges of unit magnitude. Excellent convergence leads to a change between $l = 3$ and $l = 4$ of 0.018% at $L = 2$ . As in the (a), the convergence is even better for larger $L$ .Reuse & Permissions
Figure 5
Comparison of forces for unit interior dielectric constant, unit radii, and external dielectric $ϵ_{0} = 80$ . (a) The force for like charges: the dashed line is the surface charge (exact) method $(l = 2)$ , the heavy solid line is the GB model, and the light solid line is the result for point charges in the solvent without any spheres. (b) The force for opposite charges: the dashed line is the surface charge (exact) method $(l = 2)$ , the heavy solid line is the GB model, and the light solid line is the result for point charges in the solvent without any spheres.Reuse & Permissions
Figure 6
Percent deviation of the force from the force of point charges in the solvent without any spheres $(F_{pc})$ . (a) The force deviation for like charges: the dashed line is for the surface charge method $(l = 2)$ , and the heavy solid line is for the GB model. (b) The force deviation for opposite charges: the dashed line is for the surface charge method $(l = 2)$ , and the heavy solid line is for the GB model.Reuse & Permissions

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