Integral Representation of Electrostatic Interactions inside a Lipid Membrane

Abstract

Interactions between charges and dipoles inside a lipid membrane are partially screened. The screening arises both from the polarization of water and from the structure of the electric double layer formed by the salt ions outside the membrane. Assuming that the membrane can be represented as a dielectric slab of low dielectric constant sandwiched by an aqueous solution containing mobile ions, a theoretical model is developed to quantify the strength of electrostatic interactions inside a lipid membrane that is valid in the linear limit of Poisson-Boltzmann theory. We determine the electrostatic potential produced by a single point charge that resides inside the slab and from that calculate charge-charge and dipole-dipole interactions as a function of separation. Our approach yields integral representations for these interactions that can easily be evaluated numerically for any choice of parameters and be further simplified in limiting cases.

Keywords: Bessel function; Debye-Hückel; dielectric slab; screened Coulomb potential.

Figure A1

The origin of a cylindrical coordinate system $\{r,\varphi ,z\}$ is placed at the location of a point charge q, with distances $d_1$ and $d_2$ away from the two interfaces of the dielectric slab, which has a thickness $d=d_1+d_2$. The dielectric constant inside the dielectric slab is ${ϵ}_l$, that inside the sandwiching media is ${ϵ}_w$. We denote the electric potential in the region $z>d_1$ by ${\mathsf{\Phi }}_1$, that in the region $d_1>z>0$ by ${\mathsf{\Phi }}_2$, that in the region $0>z>-d_2$ by ${\mathsf{\Phi }}_3$, and that in the region $-d_2>z>$ by ${\mathsf{\Phi }}_4$. Salt is present in the two sandwiching media with bulk concentration $n_0$.

Figure A2

Diagram (a) Two point charges, $q_1$ and $q_2$ located inside a lipid membrane produce an electrostatic potential $\mathsf{\Phi }$. Diagram (b) We remove the charge $q_2$ and denote the potential due to the presence of the remaining charge $q_1$ by ${\mathsf{\Phi }}_1$. Diagram (c) We remove the charge $q_1$ and denote the potential due to the presence of the remaining charge $q_2$ by ${\mathsf{\Phi }}_2$.

Figure 1

(a) Two interacting point charges, $q_1$ and $q_2$, separated by a distance r in the middle of a dielectric slab of dielectric constant ${ϵ}_l$ and thickness d. The dielectric constant of the sandwiching media is ${ϵ}_w$. (b) Salt ions with bulk concentration $n_0$ are present in the two sandwiching media. (c) The two interacting charges are moved up or down so that their mutual distance is $\sqrt{d_{12}^2+r^2}$ and their distances to the dielectric interfaces are $d_1$ and $d_2$, with $d_1+d_2+d_{12}=d$. (d) The two charges in diagram b are replaced by two dipoles, both located at the middle of and oriented normal to the dielectric slab, either parallel (as shown) or anti-parallel (not shown). (e) Two dipoles as in diagram d, yet with arbitrary orientations. (f) The two interacting dipoles shown in diagram e are jointly moved up or down so that their distances to the dielectric interfaces are $d_1$ and $d_2$, with $d_1+d_2=d$.

Figure 2

Scaled interaction energy $\tilde{U}=(4\pi {ϵ}_0{ϵ}_{l}d/q_1{q}_2)\times U$ between two point charges $q_1$ and $q_2$ as function of the scaled separation $r/d$ according to Equation (4). Different curves corresponds to: $w=-1$ (red line); $w=0$ and $l_D\to \infty$ (green dotted line); $w=0$ and $l_D=d$ (green dashed line); $w=-78/82$ and $l_D\to \infty$ (blue dotted line); $w=-78/82$ and $l_D=d$ (blue dashed line). The gray line shows the approximation $\tilde{U}=e^{-\pi r/d}$ of the red line, valid in the limit $r\gg d$ [38]. Note that in all cases the two charges are located in the middle of the dielectric slab, $d_1=d_2=d/2$.

Figure 3

Scaled interaction energy $\tilde{U}=(4\pi {ϵ}_0{ϵ}_l{l}_D/q_1{q}_2)\times U$ between two point charges $q_1$ and $q_2$ that are both attached to the same interface ($d_1=0$ and $d_2=d$) as function of the scaled separation $r/l_D$ for $w=-78/82$. The blue lines correspond to $d\to \infty$ (dotted line), $d=l_D$ (dashed line), and $d\to 0$ (solid line), all calculated according to Equation (6). The gray dotted line is Hurd’s [32] decomposition specified in Equation (10).

Figure 4

Scaled interaction energy $\tilde{U}=(4\pi {ϵ}_0{ϵ}_l{d}^3/{\mu }_1{\mu }_2)\times U$ between two dipoles as function of the scaled separation $r/d$, calculated according to Equation (11). The green dotted line refers to $w=0$ and $l_D\to \infty$, the green dashed line to $w=0$ and $l_D=d$, the blue dotted line to $w=-0.5$ and $l_D\to \infty$, and the blue dashed line to $w=-0.5$ and $l_D=d$. The “metallic” case ($l_D\to 0$) is shown by the red line, which is independent of w.