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

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 – Published 6 November 2020

Abstract

The unequivocal role of electrostatic forces in biological (and colloidal) systems underscores the importance of attaining accurate and rapid calculations of electrostatic forces if one wishes to faithfully simulate the electrostatic aspect of a biological system. This paper makes significant progress toward this aspect as it rigorously incorporates ionic screening at the Debye-Hückel level for an electrolyte system containing dielectric spheres of finite radii. We investigated earlier this system without mobile ions via a surface charge method. However, the need for computing a large number of Wigner rotation matrix elements per configuration can significantly slow down the numerical calculations. This difficulty was recently overcome by our Wigner-matrix-free formalism. Unfortunately, in that method ions can only be included individually, making it impractical to investigate, for example, ionic screening in a system modeled by charged dielectric spheres immersed in a solution of mobile ions. Here, we overcome this difficulty by extending the surface charge method to treat ions implicitly. Previous treatments of charged dielectric spheres in a solution of mobile ions did not emphasize the energy reciprocity of electrostatics and are largely limited to a few spheres and/or special symmetries. Our new formalism respects reciprocity and accommodates arbitrarily many dielectric spheres of different dielectric constants and sizes while being rigorous at the Debye-Hückel level. The differences, and the relationship, between our new implicit ion treatment and our previous ion-free (or explicit ion) approach are described. A closed form for the electrostatic energy with implicit ions is also provided. This new formalism speeds up the computation of the electrostatic energy in the presence of ions, and accommodates permanent and induced multipoles that are very important when the polarization effect needs to be correctly included. We also mention how the proposed method can be transformed to a numerical method for use with arbitrary nonspherical surfaces.

Received 7 August 2020
Revised 13 October 2020
Accepted 14 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052404

Published by the American Physical Society

Physics Subject Headings (PhySH)

Biomolecular interactions

Physics of Living Systems

Authors & Affiliations

Yi-Kuo Yu ^*

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

^*yyu@ncbi.nlm.nih.gov

Electrostatics of charged dielectric spheres with application to biological systems

T. P. Doerr and Yi-Kuo Yu

Phys. Rev. E 73, 061902 (2006)

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)

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Vol. 102, Iss. 5 — November 2020

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Figure 1
The reciprocity interaction energy of a system of two spheres versus their separation. Each sphere contains a unit charge at its center; sphere 1 has radius $a_{1} = a$ while sphere 2 has radius $a_{2} = 2 a$ . The other parameters are chosen to mimic two spherical proteins immersed in a rather strong ionic solution with water as solvent: $ε_{1} = ε_{2} = 4$ , $ε_{o} = 80$ , and $κ = 0.5$ . There are two forms, (E7) and (E11), of the reciprocity interaction energy, each containing two components. The $κ {\bar{Q}}_{1}^{* T} (V_{1} - ε_{o} D_{1}) {\overset{˘}{Q}}_{1}^{+}$ component of (E7) is plotted using a thick red dashed line while the $κ {\bar{Q}}_{1}^{* T} M {\overset{˘}{Q}}_{2}^{+}$ component of (E7) is plotted using thin red dashed line. The $κ {\bar{Q}}_{2}^{* T} (V_{2} - ε_{o} D_{2}) {\overset{˘}{Q}}_{2}^{+}$ component of (E11) is plotted using a thick blue dot-dashed line while the $κ {\bar{Q}}_{2}^{* T} M^{†} {\overset{˘}{Q}}_{1}^{+}$ component of (E11) is plotted using thin blue dot-dashed line. Although the components all have different values, as shown in the plot, the sums of the two components belonging to either (E7) or (E11) yield identical values shown using a solid black line. As proved in Appendix pp5, reciprocity holds for an arbitrary $l_{\max}$ cutoff. The numerical values in this plot are produced using $l_{\max} = 10$ throughout.
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Figure 2
The relative errors of the computed interaction energy, (59), between two spheres versus $l_{\max}$ . The system consists of two spheres, the first one with radius $a_{1} = a$ while the other with radius $a_{2} = 2 a$ ; each sphere has a point charge $q$ at its center. In panel (a), the relative error curves—when the two sphere centers are separated by $L = 3.1 a$ , $3.5 a$ , $4.0 a$ , $5.0 a$ , and $7.0 a$ , and for $κ = 0.1$ —are each labeled by the corresponding $L$ and shown in different colors: red ( $L = 3.1 a$ ), orange ( $L = 3.5 a$ ), green ( $L = 4.0 a$ ), magenta ( $L = 5.0 a$ ), and blue ( $L = 7.0 a$ ). In panel (b), we display the relative error versus $l_{\max}$ for various $κ$ when $L = 3.5 a$ . We find that as $κ$ increases, the relative error, for the same $l_{\max}$ value, also increases.
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Figure 3
Interaction energy (59) for a system of two spheres versus their separation distance. The first sphere has radius $a_{1} = a$ , while the second one has radius $a_{2} = 2 a$ . Each sphere has a point charge $q$ at its center. Displayed in panel (a) is the interaction energy versus separation distance for various $κ$ values. It is obvious that the magnitude of the interaction energy at a fixed separation decreases as $κ$ increases. In panel (b), the interaction energies are scaled by their values at contact distance. This allows us to see clearly the trend that the interaction energy decreases with the separation distance more rapidly for larger $κ$ values.
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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

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 – Published 6 November 2020

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

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Electrostatics of charged dielectric spheres with application to biological systems

T. P. Doerr and Yi-Kuo Yu

Phys. Rev. E 73, 061902 (2006)

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)

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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

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 – Published 6 November 2020

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

See Also

Electrostatics of charged dielectric spheres with application to biological systems

T. P. Doerr and Yi-Kuo Yu

Phys. Rev. E 73, 061902 (2006)

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)

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