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The aggregation or clustering of proteins and other macromolecules plays animportant role in the formation of large-scale molecular assemblies within cellmembranes. Examples include lipid rafts and postsynaptic domains (PSDs). PSDsare rich in scaffolding proteins that transiently trap transmembraneneurotransmitter receptors, thus localizing them at specific spatial positions.Hence, PSDs play a key role in determining the strength of synaptic connectionsand their regulation during learning and memory. Recently, a 2Ddiffusion-mediated aggregation model of PSD formation has been developed inwhich the spatial locations of the clusters are determined by a set of fixedanchoring sites. The system is kept out of equilibrium by the recycling ofparticles between the cell membrane and interior. This results in a stationarydistribution consisting of multiple clusters, whose average size can bedetermined using an effective mean-field description of the particleconcentration around each anchored cluster. In this paper, we derivecorrections to mean-field theory using asymptotic methods, assuming that theclusters are well separated and much smaller than the membrane surface. Theanalysis proceeds by matching inner solutions around each cluster with and anouter region where mean-field interactions occur. This generates an asymptoticexpansion of the particle concentration that sums over all logarithmic termsinvolving the small parameter $\nu=-1/\ln \epsilon$, where $\epsilon$ specifiesthe relative size of the clusters. This is then used to derive an expressionfor the radius of each cluster, which includes higher-order corrections tomean-field theory that depend on the positions of the clusters and the boundaryof the domain. Finally, motivated by a recent study of light-activated proteinoligomerization in cells, we develop the analogous theory for cluster formationin a 2D domain.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites