  Recent work has highlighted roles for thermodynamic phase behavior in diverse
cellular processes. Proteins and nucleic acids can phase separate into
three-dimensional liquid droplets in the cytoplasm and nucleus and the plasma
membrane of animal cells appears tuned close to a two-dimensional liquid-liquid
critical point. In some examples, cytoplasmic proteins aggregate at plasma
membrane domains, forming structures such as the post-synaptic density and
diverse signaling clusters. Here we examine the physics of these surface
densities, employing minimal simulations of co-acervating polymers coupled to
an Ising membrane surface in conjunction with a complementary Landau theory. We
argue that these surface densities are a novel phase reminiscent of
pre-wetting, in which a molecularly thin three-dimensional liquid forms on a
usually solid surface. However, in surface densities the solid surface is
replaced by a membrane with an independent propensity to phase separate. We
show that proximity to criticality in the membrane dramatically increases the
parameter regime in which a pre-wetting-like transition occurs, leading to a
broad region where coexisting surface phases can form even when a bulk phase is
unstable. Our simulations naturally exhibit three surface phase coexistence
even though both the membrane and the polymer bulk can only display two phase
coexistence on their own. We argue that the physics of these surface densities
enables diverse functions seen in Eukaryotic cells.
