Role of nonlocal electrostatic effects in the stabilization of monovalent cations in an aqueous cavity surrounded by a weakly polar environment

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Abstract

We developed earlier (Russ. J. Electrochem., 2018, vol. 54, p. 879), a new nonlocal electrostatic (NE) method for calculating electric field distributions in systems where there are spatially limited regions filled with polar media with nonlocal dielectric properties. This method was used for NE analysis of the stabilization of a monovalent cation in a spherical cavity filled with water and surrounded by a local dielectric. For one- and three-mode models of the dielectric function, NE formulas were obtained for the field distribution inside such a cavity, provided that the ion is located at its center. NE relations have been derived for the change in cation solvation energy ΔW during its transition from solution to the center of such a cavity. It is shown that when the correlation length of water in the cavity decreases compared to the solution (at the same values of the dielectric constant of water in the cavity and in the volume of the solution), the amount of work to transfer an ion from the solution into the cavity (–ΔW) decreases significantly compared to the calculation using the local theory used in the work (B. Roux, R. MacKinnon, Science, 1999, vol. 285, p. 100).

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About the authors

A. A. Rubashkin

Institute of Cytology RAS

Author for correspondence.
Email: andrey.rubashkin@gmail.com
Russian Federation, St. Petersburg

V. A. Vigont

Institute of Cytology RAS

Email: andrey.rubashkin@gmail.com
Russian Federation, St. Petersburg

M. A. Vorotyntsev

Institute of Physical Chemistry and Electrochemistry named after A.N. Frumkin RAS

Email: mivo2010@yandex.com
Russian Federation, Moscow

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Supplementary files

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2. Appendix 1
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3. Appendix 2
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4. Fig. 1. (a) Schematic representation of the system: an ion of radius a in the center of a spherical cavity of radius R (its permittivity is indicated – in the general case described by the nonlocal permittivity function εcav(k)), surrounded by a dielectric medium with permittivity εp. (b) Change in the Gibbs free energy (ΔG = –ΔW, in units of kT) upon the transition of a K+ ion from solution to the center of the cavity depending on the permittivity of the protein environment of the aqueous cavity εp. Calculation (ΔG = ΔGLE) using formulas (1.1)–(1.2) of the local electrostatics model. The permittivity inside the cavity εcav is equal to 80 (curve 1) or 4.9 (curve 2). The radius of the cavity R = 5Å. The radius of the K+ cation (a = 1.49Å) is taken according to the Gurary and Adrian scale.

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5. Fig. 2. Change in the Gibbs free energy (ΔG = –ΔW, in kT units) during the transition of a K+ ion to the center of the cavity depending on the permittivity of the protein environment of the aqueous cavity. All calculations were performed using nonlocal electrostatics (NE) and a single-mode model of the dielectric function of water. Solid curves 1 and 2 were calculated using NE formulas (23)–(24), derived in accordance with the inverse dielectric approximation (IDA) [38]. Circles and rectangles are calculations using NE formulas (14)–(16), obtained based on the dielectric approximation (DA) approach [23]. For curve 2 and rectangles, the permittivity in the cavity is εcav = 4.9, for curve 1 and circles εcav = 80. For all cases, the parameter εo = 4.9, the correlation lengths in the free solution (Λ) and in the cavity (Λcav) are equal to each other: Λ = Λcav = 5 Å; the radius of the cavity R = 5Å.

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6. Fig. 3. Change in the Gibbs free energy (ΔG = –ΔW, in kT units) upon the transition of a K+ ion to the center of the cavity depending on the permittivity of the protein environment of the aqueous cavity. Solid curves 1 and 2 are calculations using formulas (23)–(24) of the single-mode model (1M) of nonlocal electrostatics (NE), dashed curves 3 and 4 are calculations (ΔG = ΔGLE) using formulas (1.1)–(1.2) of local electrostatics (LE). For curves 1 and 3, the value εcav = ε = 80, for curves 2 and 4 εcav = εo = 4.9. The cavity radius R = 5Å, other parameters are given in Table 1.

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7. Fig. 4. Nonlocal electrostatic (NE) calculations of the change in Gibbs free energy (ΔG = –ΔW, in kT units) during the transition of a K+ ion to the center of the cavity, depending on the permittivity of the protein environment of the aqueous cavity. Curves 2 and 4 are calculations using formulas (23)–(24) of the single-mode (1M) NE model. Curves 1 and 3 are calculations using formulas (34.1)–(37) of the three-mode (3M) NE model. For solid curves 1 and 2, the correlation length of water in the cavity is equal to λ3(cav) = 5Å, for dashed curves 3 and 4, the value λ3(cav) = 2Å. The cavity radius R = 5Å, other parameters are given in Table 2.

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8. Fig. 5. Change in free energy (ΔG = –ΔW, in kT units) during the transition of a K+ ion to the center of the cavity depending on the correlation length λ3(cav) in the aqueous cavity. Curves 1, 1' – calculation using the formulas of the single-mode model (1M) of nonlocal electrostatics (NE). Curves 2, 2' – calculation using the formulas of the three-mode model (3M) of NE. Horizontal straight lines 3, 3' – calculation using local electrostatics (LE). For solid curves 1, 2, 3 the cavity radius R = 5Å, for dashed curves 1', 2', 3' the cavity radius R = 6Å, other parameters are given in Table 3.

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