First passage times in multi-protein self-assembly

Self-assembly of proteins is essential for various cellular processes such as signalling, and clathrin-mediated endocytosis. In general, proteins diffuse in a 3D solution and recruit other proteins. But they can also localize to a 2D membrane surface via lipids. In a recent study, we quantified how localization of proteins to a 2D surface from 3D solution reduces their search space and proteins can exploit this dimensionality reduction to trigger self-assembly. Here we show how localization can in many cases accelerate the assembly process, despite significantly slower diffusion on the 2D surface. We formulate the self-assembly of protein binding pairs as first-passage problems and calculate the mean time to approach the thermodynamic equilibrium. We can theoretically approximate the role of localization in slowing or accelerating the overall assembly process and the regimes where diffusion becomes a limiting factor. We validate these predictions against numerical solutions using reaction-diffusion simulations. These results highlight the role of diffusion versus binding rates and concentration in controlling time-scales of assembly.