  By performing molecular dynamics simulations with up to 132 million
coarse-grained particles in half-micron sized boxes, we show that hydrodynamics
quantitatively explains the finite-size effects on diffusion of lipids,
proteins, and carbon nanotubes in membranes. The resulting Oseen correction
allows us to extract infinite-system diffusion coefficients and membrane
surface viscosities from membrane simulations despite the logarithmic
divergence of apparent diffusivities with increasing box width. The
hydrodynamic theory of diffusion applies also to membranes with asymmetric
leaflets and embedded proteins, and to a complex plasma-membrane mimetic.
