AbstractWe 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.