Osmotic forces modify lipid membrane fluctuations

Abstract

In hydrodynamic descriptions of lipid bilayers, the membrane is often approximated as being impermeable to the surrounding, solute-containing fluid. However, biological and in vitro lipid membranes are influenced by their permeability and the resultant osmotic forces -- whose effects remain poorly understood. Here, we study the dynamics of a fluctuating, planar lipid membrane that is ideally selective: fluid can pass through it, while the electrically-neutral solutes cannot. We find that the canonical membrane relaxation mode, in which internal membrane forces are balanced by fluid drag, no longer exists over all wavenumbers. Rather, this mode only exists when it is slower than solute diffusion -- corresponding to a finite range of wavenumbers. The well-known equipartition result quantifying the size of membrane undulations due to thermal perturbations is consequently limited in its validity to the aforementioned range. Moreover, this range shrinks as the membrane surface tension is increased, and above a critical tension the membrane mode vanishes. Our findings are relevant when interpreting experimental measurements of membrane fluctuations, especially in vesicles at moderate to high tensions.

Figures (3)

• Figure 1: Plot of the two impermeable frequency solutions as a function of wavenumber. Blue dashed and red dotted lines are respectively the exact inertial and membrane branches, obtained numerically. The approximate solutions in Eq. \ref{['eq_imp_tilomega']} are shown as thick, transparent bands. Relevant parameters are $\rho^{}_{\mkern-1mu\text{f}} = 10^{-9}$ pg$/$nm$^3$, $\nu^{}_{\mkern-1mu\text{f}} = 9 \! \cdot \! 10^{-5}$ nm$^2$/µ sec, $\rho^{}_{\mkern-1mu\text{m}} = 10^{-8}$ pg$/$nm$^2$, $\lambda^{}_{\text{c}} = 2 \! \cdot \! 10^{-3}$ pN$/$nm, and $k_{\text{b}} = 10^2$ pN$\cdot$nm.
• Figure 2: Relevance of the diffusive time scale to the dynamics of the semipermeable system. (a) An overlay of $\tilde{\omega}^{}_{\mkern-2mu D}$ (solid green line) on top of the impermeable frequencies in Fig. \ref{['fig_imp_omega']}. There are two wavenumbers for which the impermeable membrane frequency $\omega^{\text{imp}}_{\mkern-1mu\text{m}}$ equals $\tilde{\omega}^{}_{\mkern-2mu D}$; approximating the former as $\tilde{\omega}^{}_{\mkern-1mu\text{m}}$ yields the two crossover wavenumbers $q_{0}^-$ and $q_{0}^+$\ref{['eq_semi_qzpm']}---shown as vertical, dashed lines. (b) Plot of $q_{0}^\pm$ over a range of the base membrane tension $\lambda^{}_{\text{c}}$, which varies across systems. The surface tension from (a) of $\lambda^{}_{\text{c}} = 2 \! \cdot \! 10^{-3}$ pN$/$nm is shown as the horizontal dashed line. There exists a critical tension $\lambda_{\text{c}}^{\mkern-1mu\ast}$\ref{['eq_semi_lambdacast']} above which $\tilde{\omega}^{}_{\mkern-1mu\text{m}} > \tilde{\omega}^{}_{\mkern-2mu D}$ for all wavenumbers. (c) Plot of the real parts of the semipermeable frequencies (dotted or dashed solid lines), overlaid on top of the impermeable frequencies (thick, transparent bands) for comparison. The inertial branch is essentially unchanged. The membrane branch only exists when $\omega^{}_{\mkern-1mu\text{m}} < \tilde{\omega}^{}_{\mkern-2mu D}$ ---and is thus confined to the range $q \in (q_{0}^-, q_{0}^+)$. For base membrane tensions $\lambda^{}_{\text{c}} > \lambda_{\text{c}}^{\mkern-1mu\ast}$, the membrane branch vanishes altogether.
• Figure 3: Analysis of GUV contour fluctuations, modified due to membrane semipermeability. (a) Canonical protocol to calculate the bending modulus and surface tension of a fluctuating GUV. In a spherical vesicle of radius $r^{}_{\mkern-1mu\mkern-1mu \text{v}}$ (left), the equatorial cross-section is imaged (center) at successive time intervals. Radial deviations from a circle are recorded as $h(x, t)$, where $x$ is the arclength along the unperturbed circle. The shape disturbances $h(x, t)$ are decomposed into one-dimensional Fourier modes, whose thermally-averaged amplitudes (right; adapted from Ref. takatori-prl-2020) are fit to Eq. \ref{['eq_exp_hqxsq']} by tuning the values of $k_{\text{b}}$ and $\lambda^{}_{\text{c}}$ (right; solid orange line) pecreaux-epje-2004. Theory and experiment deviate at large $q^{}_x$ due to finite optical resolution in the latter. (b) Limitations of the equipartition result \ref{['eq_exp_equipartition']} due to solutes in the surrounding fluid. (left) Modes inside the dome (red lines) satisfy Eq. \ref{['eq_exp_equipartition']}; those outside the dome (green dots) do not. Membrane undulations cannot be visualized above $q^{}_{\text{opt}} \approx 0.025$ nm$^{-1}$ (gray shading) due to limitations in the optical resolution ipsen-ch14. (right) For a GUV with surface tension $\lambda^{}_{\text{c}} \in (0, \lambda_{\text{c}}^{\mkern-1mu\ast})$, modes are characterized in the $q^{}_x$--$q^{}_y$ plane. The equipartition result \ref{['eq_exp_equipartition']} is limited to modes with amplitude $q > q_{0}^-$, and large-$q$ modes cannot be visualized. The domains of integration in Eq. \ref{['eq_exp_hqxsq']}, corresponding to two different values of $q^{}_x$, are shown as dashed vertical lines. The result of Eq. \ref{['eq_exp_hqxsq']} is invalid when $q^{}_x < q_{0}^-$, as in such cases the integral contains modes outside the dome. (c) Plot of experimental membrane fluctuations, over a range of reported surface tensions $\lambda^{}_{\text{c}}$, from prior studies: Faizi et al. faizi-sm-2020 ($3.1 \! \cdot \! 10^{-6}$ pN$/$nm), Rautu et al. rautu-sm-2017 ($1.4 \! \cdot \! 10^{-5}$ pN$/$nm), Pécréaux et al. pecreaux-epje-2004 ($1.7 \! \cdot \! 10^{-4}$ pN$/$nm), Park et al. park-sm-2022 ($7 \! \cdot \! 10^{-4}$--$3 \! \cdot \! 10^{-3}$ pN$/$nm), and Takatori & Sahu takatori-prl-2020 ($4 \! \cdot \! 10^{-3}$ pN$/$nm). The $y$-axis is chosen to be independent of the vesicle radius, and depends only on $\lambda^{}_{\text{c}}$ and $k_{\text{b}}$ [see Eq. \ref{['eq_exp_hqxsq']}]. At large $q^{}_x$, the data saturates due to optical resolution limitations. All data points below the dashed black line are outside the dome, and should be excluded when determining membrane parameters. Fluctuation data from the high-tension vesicle in Ref. vutukuri-n-2020 is not publicly available, and so cannot be plotted ---though the first 88 modes should be excluded based on the reported value of $\lambda^{}_{\text{c}} = 0.025$ pN$/$nm supplemental.