Conditions for extreme sensitivity of protein diffusion in membranes to cell environments

Abstract

We study protein diffusion in multicomponent lipid membranes close to a rigid substrate separated by a layer of viscous fluid. The large-distance, long-time asymptotics for Brownian motion are calculated by using a nonlinear stochastic Navier-Stokes equation including the effect of friction with the substrate. The advective nonlinearity, neglected in previous treatments, gives only a small correction to the renormalized viscosity and diffusion coefficient at room temperature. We find, however, that in realistic multicomponent lipid mixtures, close to a critical point for phase separation, protein diffusion acquires a strong power-law dependence on temperature and the distance to the substrate H, making it much more sensitive to cell environment, unlike the logarithmic dependence on H and very small thermal correction away from the critical point.

Fig. 1.

Schematic of a flat two-dimensional lipid membrane of thickness h separated by a layer of a bulk liquid of depth H from a parallel rigid substrate. Since realistic lipid bilayers do not permit shear across the membrane, the velocity field in the membrane is taken to be strictly two-dimensional, characterized by a (two-dimensional) dynamic viscosity μh. The liquid is described by a dynamic viscosity μ′ (for water, μ′ ≪ μ). No-slip boundary conditions are assumed at both the substrate and membrane interface. Two scenarios for thermally driven protein diffusion are considered in the text. We first discuss Brownian motion of proteins of lateral size a within a uniform lipid background. In the second model, proteins segregate in lipid rafts of (predominantly) a particular chemical composition above the critical temperature T_c of a binary lipid mixture. The large-scale diffusion behavior of proteins is then reformulated in terms of the raft diffusion of size ξ. Physically, ξ is the correlation length of the binary mixture, which diverges as a power law near the critical temperature. As ξ grows near T_c, protein diffusion becomes extremely sensitive to temperature and substrate depth H (see Eq. 23).

Fig. 2.

The solid line shows the undimensionalized diffusion coefficient (Eq. 3) in units of k_BT/(4πμh) as a function of δ/a. The dashed (dotted) line represents the large (small) δ/a limit discussed in the text.