We model a cylindrical inclusion (lipid or membrane protein) translating with velocity $U$ in a thin planar membrane (phospholipid bilayer) that is supported above and below by Brinkman media (hydrogels). The total force $F$, membrane velocity, and solvent velocity are calculated as functions of three independent dimensionless parameters: $\Lambda = \eta a/ ({\eta }_{m} h)$, ${\ell }_{1} / a$ and ${\ell }_{2} / a$. Here, $\eta $ and ${\eta }_{m} $ are the solvent and membrane shear viscosities, $a$ is the particle radius, $h$ is the membrane thickness, and ${ \ell }_{1}^{2} $ and ${ \ell }_{2}^{2} $ are the upper and lower hydrogel permeabilities. As expected, the dimensionless mobility $4\mathrm{\pi} \eta aU/ F= 4\mathrm{\pi} \eta aD/ ({k}_{B} T)$ (proportional to the self-diffusion coefficient, $D$) decreases with decreasing gel permeabilities (increasing gel concentrations), furnishing a quantitative interpretation of how porous, gel-like supports hinder membrane dynamics. The model also provides a means of inferring hydrogel permeability and, perhaps, surface morphology from tracer diffusion measurements.

A variety of observations—sometimes controversial—have been made in recent decades when attempting to elucidate the roles of interfacial slip on tracer diffusion in phospholipid membranes. Evans–Sackmann theory (1988) has furnished membrane viscosities and lubrication-film thicknesses for supported membranes from experimentally measured lateral diffusion coefficients. Similar to the Saffman and Delbrück model, which is the well-known counterpart for freely supported membranes, the bilayer is modelled as a single two-dimensional fluid. However, the Evans–Sackman model cannot interpret the mobilities of monotopic tracers, such as individual lipids or rigidly bound lipid assemblies; neither does it account for tracer–leaflet and inter-leaflet slip. To address these limitations, we solve the model of Wang and Hill, in which two leaflets of a bilayer membrane, a circular tracer and supports are coupled by interfacial friction, using phenomenological friction/slip coefficients. This furnishes an exact solution that can be readily adopted to interpret the mobilities of a variety of mosaic elements—including lipids, integral monotopic and polytopic proteins, and lipid rafts—in supported bilayer membranes.

Tools to measure transmembrane-protein diffusion in lipid bilayer membranes have advanced in recent decades, providing a need for predictive theoretical models that account for interleaflet leaflet friction on tracer mobility. Here we address the fully three-dimensional flows driven by a (nonprotruding) transmembrane protein embedded in a dual-leaflet membrane that is supported above and below by soft porous supports (e.g., hydrogel or extracellular matrix), each of which has a prescribed permeability and solvent viscosity. For asymmetric configurations, i.e., supports with contrasting permeability, as realized for cells in contact with hydrogel scaffolds or culture media, the diffusion coefficient can reflect interleaflet friction. Reasonable approximations, for sufficiently large tracers on low-permeability supports, are furnished by a recent phenomenological theory from the literature. Interpreting literature data, albeit for hard-supported membranes, provides a theoretical basis for the phenomenological Stokes drag law as well as strengthening assertions that nonhydrodynamic interactions are important in supported bilayer systems, possibly leading to overestimates of the membrane/leaflet viscosity. Our theory provides a theoretical foundation for future experimental studies of tracer diffusion in gel-supported membranes.

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