Membranes are self-assembled bilayers of surfactants or phospholipids, which form flexible surfaces that are essentially controlled by mechanical parameters, such as membrane tension, bending modulus and spontaneous curvature. These parameters completely characterize the membrane fluctuations at equilibrium, but they are not sufficient to characterize real membranes, such as cell membranes, which are non-equilibrium systems. These systems are in general active in the sense that they are constantly maintained out of equilibrium either by active proteins (such as ATP-consuming enzymes) inside the membrane or by an energy flow due to external parameters (such as a lipid flux). In the group of P. Bassereau at Institut Curie, giant unilamellar vesicles of lipids are studied experimentally. These vesicles are activated by the inclusion of bacteriorhodopsin pumps, which transfer protons unidirectionally across the membrane by undergoing light-activated conformational changes. These experiments show that the nonequilibrium forces arising from ion pumps embedded in the membrane are able to significantly enhance the membrane fluctuations.

Figure 1:

Left Left: Sketch of a quasi-planar fluid membrane embedded in an electrolyte in the presence of an applied potential difference and concentration gradients (not represented). Right: equivalent electrical circuit, with the contribution of the two ions k = 1, 2 in parallel, and for each ion, two conductances (G_k for the membrane and H_k for the electrolyte) and an electromotive force E_k in series. When the finite thickness of the bilayer is included, there is also a capacitor C in parallel which contains contributions from Debye layers and from the membrane.

In our study [1], we consider theoretically an electrically neutral membrane containing passive ion channels in an electrolyte solution and driven out of equilibrium by the application of a DC electric field or by ion concentration gradients as shown in figure 1. We use a generalized hydrodynamic description appropriate for low Reynolds number, which is coupled to electrokinetic equations within Debye-Hückel approximation. We calculate the undulation fluctuations by expansion around the planar state of the membrane (see figure 2). We consider the case of a membrane of zero thickness, and then generalize the model to the case of a bilayer with a finite thickness and a finite dielectric constant, lower than that of the solvent. The later model is more realistic since it contains both capacitive effects and ions transport.

Figure 2:

Left: Normalized potential at zeroth order 2φ⁽⁰⁾(z)∕V as function of 2z∕L. Right: Normalized force along z at first order f_z⁽¹⁾(q,z).

There are currently two theoretical models for active membranes. The Prost-Bruinsma (PB) model takes nonequilibrium forces in the form of active noises that include diffusion and the stochastic nature (shot noise) of the pumps, but ignores the coupling between the pumps and membrane curvature. The other model proposed by Ramaswamy, Toner and Prost (RTP) incorporates this coupling but ignores the random nature of the protein activity. For steady state measurements of active membranes, the RTP model agrees quite well with experiments. In our work [2], we further explore the dynamical properties of the RTP model, argue that it is important to include the shot noise for dynamical measurements, and present two new models that include both curvature effects and pump stochasticity. In the first model, which may be an appropriate description for light-activated pumps such as bacteriorhodopsin, the magnitude of the nonequilibrium force fluctuates on a time scale that is fast compared to that of membrane fluctuations. The second model, the two-state model, which may be realized in typical ion channels, describes pumps that are able to switch from on to off state on a time scale that is long compared to membrane fluctuation time. These models represent natural generalizations of the RTP model, and they show distinct dynamical behaviors. In particular, the two-state model predicts that the mean-squared displacement (MSD) of a membrane point exhibits superdiffusion in the experimentally relevant regime, whereas the RTP model would predict subdiffusion.

[1] D. Lacoste, M. Cosentino Lagomarsino, and JF. Joanny., Europhys. Lett., 77, 18006 (2007).