Interactions between model inclusions on closed lipid bilayer membranes

Protein inclusions in the membranes of living cells interact via the deformations they impose on that membrane. Such membrane-mediated interactions lead to sorting and self-assembly of the inclusions, as well as to membrane remodelling, crucial for many biological processes. For the past decades, theory, numerical calculations and experiments have been using simplified models for proteins to gain quantitative insights into their behaviour. Despite challenges arising from nonlinearities in the equations, the multiple length scales involved and the nonadditive nature of the interactions, recent progress now enables for the first time a direct comparison between theoretical and numerical predictions and experiments. We review the current knowledge on the biologically most relevant case, inclusions on lipid membranes with a closed surface and discuss challenges and opportunities for further progress.