Rigorous treatment of pairwise and many-body electrostatic interactions among dielectric spheres at the Debye-Hückel level.

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin-Landau-Verwey-Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye-Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye-Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium.

Electrical interactions in the cell: Asymmetric screening in a watery antiverse.

The problem of electrostatics in biomolecular systems presents an excellent opportunity for cross-disciplinary science and a context in which fundamental physics is called for to answer complex questions. Due to the large density in biological cells of charged biomacromolecules such as protein factors and DNA, it is challenging to understand quantitatively the electric forces in these systems. Two questions are especially puzzling. First, how is it that such a dense system of charged molecules does not simply aggregate in random and non-functional ways? Second, since some mechanism apparently prevents such aggregation, how is it that binding of biomolecules still occurs so reliably? Recognizing the role of water as a universal solvent in living systems is key to understanding these questions. We present a simplified physical model in which water is regarded as a medium of high dielectric constant that nevertheless exhibits the key features essential for answering the two questions presented. The answer to the first question lies in the strong screening ability of water, which reduces the energy scale of the electrostatic interactions. Furthermore, our model reveals the existence of asymmetric screening, a pronounced asymmetry between the screening for a system with like charges and that for a system with opposite charges, and this provides an answer to the second question.

Rigorous treatment of electrostatics for spatially varying dielectrics based on energy minimization.

An energy minimization formulation of electrostatics that allows computation of the electrostatic energy and forces to any desired accuracy in a system with arbitrary dielectric properties is presented. An integral equation for the scalar charge density is derived from an energy functional of the polarization vector field. This energy functional represents the true energy of the system even in nonequilibrium states. Arbitrary accuracy is achieved by solving the integral equation for the charge density via a series expansion in terms of the equation's kernel, which depends only on the geometry of the dielectrics. The streamlined formalism operates with volume charge distributions only, not resorting to introducing surface charges by hand. Therefore, it can be applied to any spatial variation of the dielectric susceptibility, which is of particular importance in applications to biomolecular systems. The simplicity of application of the formalism to real problems is shown with analytical and numerical examples.

Improving series convergence: the simple pendulum and beyond.

A simple and easy to implement method for improving the convergence of a power series is presented. We observe that the most obvious or analytically convenient point about which to make a series expansion is not always the most computationally efficient. Series convergence can be dramatically improved by choosing the center of the series expansion to be at or near the average value at which the series is to be evaluated. For illustration, we apply this method to the well-known simple pendulum and to the Mexican hat type of potential. Large performance gains are demonstrated. While the method is not always the most computationally efficient on its own, it is effective, straightforward, quite general, and can be used in combination with other methods.

Extending electrostatics of dielectric spheres to arbitrary charge distributions with applications to biosystems.

A previously developed classical model of electrostatic interactions, based on a formalism of dielectric spheres, which has been found to have surprising accuracy for S state atoms, is extended by allowing higher-order moments of the intrinsic charge distribution. Two methods to introduce the charge distribution (point moments at the center vs surface charge) are shown to be equivalent and are compared with another common model for polarizable atoms that utilizes polarizable point dipoles. Unlike the polarizable point dipole model, the polarizable spheres models do not suffer from a divergence at small separation of atoms and are easily generalized to higher multipoles.

Electrostatics of charged dielectric spheres with application to biological systems.

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.

Can dielectric spheres accurately model atomic-scale interactions?

We calculate the polarization portion of electrostatic interactions at the atomic scale using quantum mechanical methods such as density functional theories (DFT) and the coupled cluster approach, and using classical methods such as a surface charge method and a polarizable force field. The agreement among various methods is investigated. Using the coupled clusters method CCSD(T) with large basis sets as the reference, we find that for systems comprising two to six atoms and ions in S-states the classical surface charge method performs much better than commonly used DFT methods with moderate basis sets such as B3LYP/6-31G(d,p). The remarkable performance of the classical approach comes as a surprise. The present results indicate that the use of a rigorous formalism of classical electrostatics can be better justified for determining molecular interactions at intermediate distances than some of the widely used methods of quantum chemistry. PACS numbers: 41.20.Cv,32.10.Dk, 87.10.Tf.

Simple electrostatic model applicable to biomolecular recognition.

An exact, analytic solution for a simple electrostatic model applicable to biomolecular recognition is presented. In the model, a layer of high-dielectric constant material (representative of the solvent, water), whose thickness may vary separates two regions of low-dielectric constant material (representative of proteins, DNA, RNA, or similar materials), in each of which is embedded a point charge. For identical charges, the presence of the screening layer always lowers the energy compared to the case of point charges in an infinite medium of low-dielectric constant. Somewhat surprisingly, the presence of a sufficiently thick screening layer also lowers the energy compared to the case of point charges in an infinite medium of high-dielectric constant. For charges of opposite sign, the screening layer always lowers the energy compared to the case of point charges in an infinite medium of either high or low dielectric constant. The behavior of the energy leads to a substantially increased repulsive force between charges of the same sign. The attractive force between charges of opposite signs is weaker than in an infinite medium of low dielectric constant material but stronger than in an infinite medium of high dielectric constant material. The presence of this behavior, which we name asymmetric screening, in the simple system presented here confirms the generality of the behavior that was established in a more complicated system of an arbitrary number of charged dielectric spheres in an infinite solvent.