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Modeling the electrostatic contribution to the line tension between lipid membrane domains using Poisson–Boltzmann theory

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Abstract

The line tension, which characterizes the excess free energy per unit length of the boundary between different lipid membrane domains, is one of the factors that determines domain size and dynamics. Consequently, experimental methods and corresponding modeling studies related to the line tension continue to attract significant interest. Considering a planar binary lipid layer with two domains consisting of neutral and anionic lipids, one with a higher and one with a lower average surface charge density, we calculate the electrostatic contribution to the line tension at the domain boundary using mean-field electrostatics. The influence of lipid mobility in each phase is studied through solutions of the Poisson–Boltzmann equation for different sets of boundary conditions that include the limiting cases of fixing the local surface charge density or surface potential, and the intermediate case of allowing the lipids to migrate subject to a demixing entropy penalty. In addition to our numerical results, we derive simple analytic expressions for the electrostatic contribution to the line tension in the linearized Debye–Hückel limit. We find the electrostatic contribution to the line tension to be negative with magnitudes on the order of piconewton close to physiological conditions. Because this is comparable to experimentally reported values of the total line tension, we conclude that electrostatic interactions generally provide an important contribution to the total line tension between differently charged domains in lipid bilayers.

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Acknowledgments

G. V. B. acknowledges a doctoral scholarship from CAPES Foundation/Brazil Ministry of Education (Grant No. 9466/13-4). M. A. B. acknowledges the Swiss National Science Foundation for an International Short Visit travel grant to North Dakota State University.

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Correspondence to Guilherme V. Bossa.

Appendix: Calculation of \(\varLambda _{el}\) using the Kontorovich–Lebedev transformation

Appendix: Calculation of \(\varLambda _{el}\) using the Kontorovich–Lebedev transformation

We solve the linearized Poisson–Boltzmann equation expressed in polar coordinates r and \(\varphi \),

$$\begin{aligned} \frac{1}{r} \frac{\partial }{\partial r} \left( r \frac{\partial \varPsi }{ \partial r}\right) +\frac{1}{r^2} \frac{\partial ^2 \varPsi }{\partial \varphi ^2}= \frac{\varPsi }{l_D^2}, \end{aligned}$$
(18)

for the dimensionless electrostatic potential \(\varPsi =\varPsi (r,\varphi )\) within the region \(0 \le r \le \infty \) and \(0 \le \varphi \le \pi \) depicted in Fig. 4. Let us first consider the mixed case where one phase is kept at constant charge density \(\sigma _1\) and the other at fixed and constant dimensionless surface potential \(\tilde{\varPsi }^{(b)}_2\). When expressed in polar coordinates, the boundary conditions specified in case II above read

$$\begin{aligned} \frac{1}{r} \left( \frac{\partial \varPsi }{\partial \varphi }\right) _{\varphi =\pi } =4 \pi l_B \frac{\sigma _1}{e}, \quad \varPsi (r,\varphi =0)=\tilde{\varPsi }^{(b)}_2 \end{aligned},$$
(19)

and \(\varPsi (r\rightarrow \infty ,\varphi )=0\). The first relation fixes the surface charge density \(\sigma _1=-e\bar{\phi }_1/a\) of phase 1, and the second relation specifies the surface potential \(\tilde{\varPhi }_2^{(b)}=k_BT\tilde{\varPsi }_2^{(b)}/e\) of phase 2. A convenient method to solve Eq. 18 and to calculate \(\varLambda _{el}\) employs the Kontorovich–Lebedev transformation [29, 30, 33, 34] \(\varPsi (r,\varphi ) \rightarrow \varUpsilon (\lambda ,\varphi )\), defined through

$$\begin{aligned} \varUpsilon (\lambda ,\varphi )= \int \limits _0^\infty \frac{dr}{r}\varPsi (r,\varphi ){\text {K}}_{i \lambda }(r/l_D) \end{aligned},$$
(20)

and its corresponding back-transformation

$$\begin{aligned} \varPsi (r,\varphi )=\frac{2}{\pi ^2}\int \limits _0^\infty d\lambda \varUpsilon (\lambda ,\varphi ){\text {K}}_{i \lambda }(r/l_D) \, \lambda \, \sinh (\pi \lambda ) \end{aligned},$$
(21)

where \(K_{i \lambda }(x)\) is the modified Bessel function of the second kind of order \(\lambda \) and i is the imaginary unit. We note that the same method has been used previously by Duplantier [35] and Kang et al. [30] for wedge-like geometries and boundary conditions of type I (yet not for type II). When expressed in terms of \(\varUpsilon =\varUpsilon (\lambda ,\varphi )\), Eq. 18 reads

$$\begin{aligned} \frac{\partial ^2 \varUpsilon }{\partial \varphi ^2}=\lambda ^2\varUpsilon \end{aligned},$$
(22)

and the two boundary conditions in Eq. 19 become

$$\begin{aligned} \left( \frac{\partial \varUpsilon }{ \partial \varphi }\right) _{\varphi =\pi }= \frac{\pi \tilde{\varPsi }^{(b)}_1}{2 \cosh (\frac{\pi }{2} \, \lambda )}, \end{aligned}$$
(23)
$$\begin{aligned} \varUpsilon (\lambda ,\varphi =0)=\frac{\pi \tilde{\varPsi }^{(b)}_2 }{2\lambda \sinh (\frac{\pi }{2} \, \lambda )}. \end{aligned}$$
(24)

with \(\tilde{\varPsi }_1^{(b)}=-4 \pi l_B l_D \bar{\phi }_1/a\). The condition \(\varPsi (r\rightarrow \infty ,\varphi )=0\) is consistently fulfilled due to the property \(K_n(x \rightarrow \infty )=0\) for all choices of n. The solution of Eq. 22 subject to the boundary conditions in Eqs. 23 and 24 is

$$\begin{aligned} \varUpsilon =\frac{\pi }{2\lambda \cosh (\lambda \pi )}\left[ \tilde{\varPsi }^{(b)}_2 \frac{\cosh [\lambda (\pi -\varphi )]}{\sinh (\frac{\pi }{2} \lambda )} +\tilde{\varPsi }^{(b)}_1\frac{\sinh (\lambda \varphi )}{\cosh (\frac{\pi }{2} \lambda )}\right] . \end{aligned}$$
(25)

Transforming back from \(\varUpsilon (\lambda ,\varphi )\) to \(\varPsi (r,\varphi )\) according to Eq. 21 gives rise to

$$\begin{aligned} \varPsi (r,\varphi )&= \frac{2}{\pi } \int \limits _0^\infty d\lambda \frac{{\text {K}}_{i \lambda }(r/l_D)}{\cosh (\lambda \pi )} \left[ \tilde{\varPsi }^{(b)}_2 \cosh \left( \frac{\pi }{2} \lambda \right)\right. \\&\quad\left.\times \cosh [\lambda (\pi -\varphi )] +\tilde{\varPsi }^{(b)}_1 \sinh \left( \frac{\pi }{2} \lambda \right) \sinh (\lambda \varphi )\right] . \end{aligned}$$
(26)

From the solution \(\varPsi (r,\varphi )\) in Eq. 26 we calculate the electrostatic line tension \(\varLambda _{el}\) as the excess free energy per unit length

$$\begin{aligned} \frac{\varLambda _{el}}{k_BT}&= \frac{\sigma _1}{2 e} \int \limits _0^\infty dr \left[ \varPsi (r,\pi )-4 \pi l_B l_D \frac{\sigma _1}{e}\right] \\&\quad+ \frac{\tilde{\varPsi }^{(b)}_2}{8 \pi l_B} \int \limits _0^\infty dr \left[ \frac{1}{r} \left( \frac{\partial \varPsi (r,\varphi )}{\partial \varphi }\right) _{\varphi =0}+\frac{\tilde{\varPsi }^{(b)}_2}{l_D} \right] . \end{aligned}$$
(27)

Instead of using \(\sigma _1\), it is convenient to express the fixed surface charge density in phase 1 in terms of the corresponding surface potential \(\tilde{\varPsi }^{(b)}_1=4 \pi l_B l_D \sigma _1/e\) far away from the boundary between the phases (that is \(\tilde{\varPsi }^{(b)}_1=\varPsi (r \rightarrow \infty ,\varphi =\pi )\)). From Eq. 26 we can compute both \(\varPsi (r,\pi )\) and \((\partial \varPsi /\partial \varphi )_{\varphi =0}\) and use these results to calculate the excess free energy according to Eq. 27. This leads to the expression for \(\varLambda _{el}\) as specified in Eq. 17.

We now proceed to determine the surface charge induced on the phase kept at constant potential. Defining this charge density by \(\sigma _2^{ind}\), we take the first angular derivative of Eq. 26 at phase 2 (i.e., at \(\varphi =0\)) and find:

$$\begin{aligned} &\frac{\sigma _2^{ind}}{e}=-\frac{1}{4\pi l_Br}\left( \frac{\partial \varPsi }{\partial \varphi }\right) _{\varphi =0}\\&\quad=-\frac{1}{4\pi l_Br}\int _0^\infty d\lambda \frac{{\text {K}}_{i \lambda }(r/l_D)}{\cosh (\lambda \pi )} \left[ -\tilde{\varPsi }_2 \cosh (\pi \lambda /2)\lambda \sinh (\lambda \pi )\right.\\&\quad\quad\quad\left.+\tilde{\varPsi }_1\sinh (\pi \lambda /2)\sinh (\lambda \phi )\right] . \end{aligned}$$
(28)

We note that the form how the \(\lambda \) integrals appear in this equation does not allow a direct analytical solution. However, we can compute the effective charge per unit length on phase 2 using

$$\begin{aligned} \frac{\varDelta Q_2}{e}=\int _0^\infty (\sigma _2^{ind}/e) dr-\int _0^\infty \frac{\tilde{\varPsi }_2}{4\pi l_B l_D}dr, \end{aligned}$$
(29)

where the first term accounts for the induced charge density and the second one subtracts the charge in the bulk of the phase (i.e., far away from the boundary \(r=0\)). Then, inserting Eq. 28 into 29 and performing the r integration prior to the \(\lambda \) integration, we find

$$\begin{aligned} \frac{\varDelta Q_2^{ind}}{e}=-\frac{(\tilde{\varPsi }_1-\tilde{\varPsi }_2)}{8\pi l_B}. \end{aligned}$$
(30)

The expression obtained Eq. 30 implies that for any \(\tilde{\varPsi }_2<\tilde{\varPsi }_1\), charges on phase 1 will induce a positive charge on phase 2 (recall that we only have negative and neutral lipids in the system). This fact explains why the values obtained for the line tension can approach negative values of large magnitude when \(\bar{\phi }_2\) decreases.

When each of the two phases carries a fixed surface charge density (\(\sigma _1\) in phase 1 and \(\sigma _2\) in phase 2, as specified in boundary condition type I above), Eq. 19 reads

$$\begin{aligned} \frac{1}{r}\left( \frac{\partial \varPsi }{\partial \varphi }\right) _{\varphi =\pi }&= 4 \pi l_B \frac{\sigma _1}{e},\quad \frac{1}{r} \left( \frac{\partial \varPsi }{\partial \varphi }\right) _{\varphi =0} =-4 \pi l_B \frac{\sigma _2}{e}. \end{aligned}$$
(31)

The derivation of \(\varLambda _{el}\) follows the same steps as for the mixed boundary conditions, leading to the final result specified in Eq. 16.

We note that the result in Eq. 16 also appears as a special case of calculating \(\varLambda _{el}\) in a wedge-like geometry where the wedge subtends an angle \(\alpha \). For this case, \(\varLambda _{el}\) is given by

$$\begin{aligned} \varLambda _{el}(\alpha )=\frac{ (\tilde{\varPsi }^{(b)2}_1+\tilde{\varPsi }^{(b)2}_2) M_1(\alpha )-(\tilde{\varPsi }^{(b)}_1-\tilde{\varPsi }^{(b)}_2)^2 M_2(\alpha )}{4 \pi l_B}. \end{aligned}$$
(32)

The two functions \(M_1(\alpha )\) and \(M_2(\alpha )\) have been calculated previously by Kang et al. [30]

$$\begin{aligned} M_1(\alpha )=\int \limits _0^\infty d\omega \, \left[ \frac{\tanh {(\omega \alpha )}}{ \tanh {(\omega \pi )}}-1\right] \end{aligned}$$
(33)

and

$$\begin{aligned} M_2(\alpha )=\int \limits _0^\infty d\omega \, \left[ \frac{\tanh {(\omega \alpha )}}{ \sinh {(2 \omega \pi )}}-c\right] \end{aligned}$$
(34)

with \(c=0\). We note a typo in the work of Kang et al. [30], who state Eq. 34 with \(c=1\). For \(\alpha =\pi \) we find \(M_1(\alpha ) =0\) and \(M_2(\pi )=1/(2 \pi )\), and Eq. 32 becomes the same as Eq. 16.

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Bossa, G.V., Brown, M.A., Bohinc, K. et al. Modeling the electrostatic contribution to the line tension between lipid membrane domains using Poisson–Boltzmann theory. Int J Adv Eng Sci Appl Math 8, 101–110 (2016). https://doi.org/10.1007/s12572-015-0158-6

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