We propose a minimal mathematical model for the physical basis of membrane protein patterning in the immunological synapse (IS), which encompass membrane mechanics, protein binding kinetics and motion, and fluid flow in the synaptic cleft. Our theory leads to
simple predictions for the spatial and temporal scales of protein cluster formation, growth
and arrest as a function of membrane stiffness, rigidity and kinetics of the adhesive proteins,
and the fluid flow in the synaptic cleft. Numerical simulations complement these scaling
laws by quantifying the nucleation, growth and stabilization of proteins domains on the size
of the cell. Direct comparison with experiment shows that passive elastohydrodynamics
and kinetics of protein binding in the synaptic cleft can describe the short-time formation
and organization of protein clusters, without evoking any active processes in the cytoskeleton. Despite the apparent complexity of the process, our analysis shows that just two dimensionless parameters characterize the spatial and temporal evolution of the protein pattern: a
ratio of membrane elasticity to protein stiffness, and the ratio of a hydrodynamic time scale
for fluid flow relative to the protein binding rate. A simple phase diagram encompasses the
variety of patterns that can arise