Thermal Undulations of Lipid Bilayers Relax by Intermonolayer Friction at Submicrometer Length Scales

The time correlation functions of the thermal undulations of a lipid membrane have been studied by molecular dynamics simulations of a coarse-grained bilayer model. We observe a double-exponential decay, with relaxation rates in good agreement with the theory by Seifert and Langer, [Europhys. Lett. 23, 71 (1993)]. Intermonolayer friction resulting from local velocity differences between the two monolayers is shown to be the dominant dissipative mechanism for fluctuations with wave lengths below $\sim 0.1\mu \mathrm{m}$.

Figure 1

The slow ${\gamma }_1$ and fast ${\gamma }_2$ relaxation rates (solid lines) of an undulating bilayer as functions of the wave number $q$, calculated using the parameters of the coarse-grained model. The slipping branches scale as $q^2$ and are damped by intermonolayer friction, the bending branches scale as $q^3$ and relax through viscous dissipation by the solvent. Dashed and dotted lines denote extrapolations of the limiting regimes at low and high $q$; the vertical distance between each set of parallel lines is $\log (\tilde{\kappa }/\kappa )$. The arrow indicates the crossover wave number $q_c$, where the relaxation characters of the slow and fast modes are gradually exchanged. Circles and squares mark the relaxation rates obtained by fitting the simulated height correlation functions of Fig. 2.

Figure 2

Time correlation functions of the Fourier components of the height of a thermally fluctuating bilayer (solid lines), for the three smallest wave numbers commensurable with the box dimension, $q_1=0.18{\sigma }^{-1}$ (top), $q_2=0.25{\sigma }^{-1}$ (middle, multiplied by 2.5), and $q_3=0.36{\sigma }^{-1}$ (bottom, multiplied by 8). The dashed lines are double-exponential fits, excluding the transient part of the simulation data; the dotted straight lines show the contributions of the slow modes to these fits.

Figure 3

The relaxation rates of the coarse-grained model before (gray) and after (black) a 100-fold increase of the intermonolayer friction, to bring this parameter on par with experimental values. Changing $b$ does not affect the relaxation rates of the bending branches, whose limits are indicated by dashed lines. The rates of the slipping branches, however, are reduced by a factor of 100, and the crossover wave number (arrows) is reduced accordingly. Atomistic and coarse-grained simulations live in the top-right corner of the plot, as these systems typically contain ${10}^3$ lipids, corresponding to $q\sim 0.1{\sigma }^{-1}$, with simulation times usually of the order of ${10}^4\tau$.