Translational Diffusion in Lipid Membranes beyond the Saffman-Delbrück Approximation

Abstract

The Saffman-Delbrück approximation is commonly used in biophysics to relate the membrane inclusion size to its translational diffusion coefficient and membrane viscosity. However, this approximation has a restricted validity range, and its application to determination of inclusion sizes from diffusion data may in certain cases lead to unreliable results. At the same time, the model by Hughes et al. (Hughes, B. D., B. A. Pailthorpe, and C. R. White. 1981. J. Fluid Mech. 110:349-372.), providing diffusion coefficients of membrane inclusions for arbitrary inclusion sizes and viscosities of the membrane and surrounding fluids, involves substantial computational efforts, which prevents its use in practical data analysis. We develop a simple and accurate analytical approximation to the Hughes et al. model and demonstrate its performance and utility by applying it to the recently published experimental data on translational diffusion of micrometer-sized membrane domains.

Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes

“…For convenience, δ R (r) is chosen to be a Gaussian. The width of δ R (r) is chosen to yield good correspondence with the SDHPW results in the infinite-system-size limit, for a cylinder with radius R. The choice δ R = 1 πb 2 e −r 2 /b 2 with b = βR and β = 0.828 494 36 reproduces the SDHPW result 19,20,24,25 to within 6% over the entire range 10 −5 ≤ R/L sd ≤ 10 5 (here, L sd = η m /2η f is the Saffman-Delbrück length). With this choice,…”

“…Expressions for D for simply shaped objects in 3D fluids are known, 17 but an exact solution for quasi-2D membrane geometry is known only for a circular disk in an infinite membrane surrounded by an infinite bulk (the well-known SaffmanDelbrück-Hughes-Pailthorpe-White, SDHPW, result [18][19][20] ) and that solution is so complex that simplified functional forms are often used to approximate the true solution. 24,25 It seems highly unlikely that this expression could be generalized to the finite periodic geometry, and even if it could, the resulting equations would almost certainly be prohibitively complicated.…”

“…Experimental measurements of L sd for lipid bilayer systems are typically in the 100 nm to micron range. 24,25,[38][39][40][41][42] In coarse-grained models, some details of lipid and solvent structure are lost and there is no reason to expect close correspondence to experimental numbers. For example, in the coarse-grained MARTINI force field, 5 η m has been determined to be 1.2 × 10 −8 P cm for DPPC bilayers, with water viscosity η f = 7 × 10 −3 P. 9 Within MARTINI simulations, we expect L sd ≈ 8.6 nm-much smaller than the experimental range.…”

“…In fact, the situation is likely even worse, since the value L sd = 78 nm 40 assumed for experimental membrane systems is conservative-most measurements find larger values for L sd . 24,25,38,39 The errors associated with (6)) and in infinite systems (Eq. (7)) for three representative proteins.…”

Dynamic label-free imaging of lipid nanodomains

“…Many theoretical models of domain diffusion (25)(26)(27) are based on rigid, cylindrical inclusions, for which thermodynamically stable and essentially noninteracting S o domains are a good approximation. Most importantly, we chose to examine S o domains as they can be kinetically trapped at a range of different sizes, and thus could be used to examine closely how diffusion and contrast vary with domain size.…”