Lipid membranes are essential structures in cells, which comprise two separate leaflets that contain a distinct mixture of lipids and associated proteins. The constituent lipids can move rotationally and diffusively within their respective leaflets at low-energy cost, and transversally between leaflets at a much higher cost due to the need to move a hydrophilic head group through the bilayer's hydrophobic interior. Despite this barrier making transitions rare, random flip-flop still occurs on timescales between hours and days. The dynamics of this process is typically modeled using a rate equation with two linear terms representing lipids moving between the leaflets. A much faster way in which leaflets exchange lipids is through transport proteins such as flippases, floppases, and scramblases. The former two actively drive lipids into a specific leaflet, which can result in asymmetric distributions. One property that can arise from this is differential stress—a difference between the two mechanical leaflet tensions. In simulations, differentially stressed bilayers have been shown to have different rates of flip-flop compared to symmetric bilayers. The typical rate equation doesn't capture this behavior. Here, we propose a nonlinear version that includes an exponential penalty for differential stress based on leaflet area differences. This accounts for the change in flip-flop rate both in the initial relaxation phase and the fluctuations around equilibrium. Due to the Markovian nature of flip-flop, and the principle of detailed balance, an analytical form for fluctuations around the nonlinear dynamics can be found by deriving a Fokker-Planck equation in the linear noise approximation and taking its moments. In the presence of flippase activity, detailed balance is broken, and activity-dependent rates must be added by hand. We seek to clarify the resulting non-equilibrium steady state and the the noise it exhibits.
Wesnak et al. (Sun,) studied this question.