  The cellular basis for the adaptive immune response during antigen
recognition relies on a specialized protein interface known as the
immunological synapse (IS). Understanding the biophysical basis for protein
patterning by deciphering the quantitative rules for their formation and motion
is an important aspect of characterizing immune cell recognition and thence the
rules for immune system activation. We propose a minimal mathematical model for
the physical basis of membrane protein patterning in the 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 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 highlights the role of just
two dimensionless parameters that characterize the spatial and temporal
evolution of the protein pattern: a ratio of membrane elasticity to protein
elasticity, and the ratio of a hydrodynamic time scale for fluid flow relative
to the protein binding rate, and we present a simple phase diagram that
encompasses the variety of patterns that can arise.
