Curvature fluctuations of fluid vesicles reveal hydrodynamic dissipation within the bilayer

Abstract

The biological function of membranes is closely related to their softness, which is often studied through the membranes' thermally driven fluctuations. Typically, the analysis assumes that the relaxation rate of a pure bending deformation is determined by the competition between membrane bending rigidity and viscous dissipation in the surrounding medium. Here, we reexamine this assumption and demonstrate that viscous flows within the membrane dominate the dynamics of bending fluctuations of nonplanar membranes with a radius of curvature smaller than the Saffman-Delbrück length. Using flickering spectroscopy of giant vesicles made of dipalmitoylphosphatidylcholine, DPPC:cholesterol mixtures and pure diblock-copolymer membranes, we experimentally detect the signature of membrane dissipation in curvature fluctuations. We show that membrane viscosity can be reliably obtained from the short time behavior of the shape time correlations. The results indicate that the DPPC:cholesterol membranes behave as a Newtonian fluid, while the polymer membranes exhibit more complex rheology. Our study provides physical insights into the time scales of curvature remodeling of biological and synthetic membranes.

Keywords: curvature fluctuations; lipid bilayer; membrane viscosity; polymersome; vesicle.

Fig. 1.

(A) Sketch of a quasi-spherical vesicle and of time-lapse vesicle contours in the equatorial plane. (B) Power spectrum of the contour fluctuations yields the bending rigidity $\kappa$. (C) The rescaled static power spectrum by $\kappa$ is a universal function of the wavenumber. The solid line is Eq. 2 with $t=0$. (D and E) Autocorrelation functions (ACF) for Fourier mode 6 of the fluctuating equatorial contours of vesicles made of SOPC (D) and DPPC:Chol (1:1) (E). Blue symbols are the experimental data. Dashed lines are the full theory Eq. 2 and the solid lines are the single exponential decay with rate given by Eq. 1. Red and black line colors correspond to dimensionless membrane viscosity ${\chi }_s=0$ and ${\chi }_s=8$, respectively. The time scale $t_{\kappa }=\eta R^3/\kappa$ is 23.3 s in (D) and 7.6 s in (E). (F) The long-time single exponential decay rate, rescaled by the bending relaxation time, obtained from the ACF as a function of the mode number. The dashed line is the theory Eq. 1.

Fig. 2.

Membrane viscosity obtained from flickering spectroscopy (FA) and electrodeformation (ED) for different DPPC:Chol mixed bilayers.

Fig. 3.

Normalized autocorrelation functions (ACF) for Fourier modes 4 to 10 of the fluctuating equatorial contour of vesicles made of (A) DPPC:Chol (1:1), (B) DPPC:Chol (7:3), and (C) PS1. Symbols are the experimental data. In (A–C) the dashed lines are the full theory Eq. 2 and the solid lines are the single exponential decay with rate given by Eq. 1. (D–F) zoom into the short-time behavior of the ACF. The solid lines are the nonviscous-membrane asymptote Eq. 4 in (D) and the viscous-membrane asymptote Eq. 5 in (E and F). $t^{\ast }$ denotes the crossover time from relaxation dominated by bulk viscosity to membrane viscosity Eq. 3.

Fig. 4.

Single-point membrane mean square displacement (dynamic roughness) of a DPPC:Chol (7:3) membrane. The solid line is the viscous asymptote Eq. 11 with ${\chi }_s=150$, same as in Fig. 3. The dashed line is the nonviscous behavior Eq. 10.

Fig. 5.

Normalized mean square displacement for Fourier modes 4 to 10 of the fluctuating equatorial contour of a polymersome made of PS1 plotted as a function of the dimensionless time multiplied by the fourth power of the mode. Inset: Same plot but for DPPC:Chol vesicle showing collapse of the data. The solid line is the viscous asymptote Eq. 5 for tensionless membranes with ${\chi }_s=150$. The time scale $t_{\kappa }=\eta R^3/\kappa$.