Experimental surveys of the protein content in cell membranes suggest that membrane proteins exhibit great diversity in their oligomeric state and transmembrane shape. Furthermore, membrane proteins are often observed to form clusters, which can be composed of thousands of proteins. Recent breakthroughs in high-resolution imaging techniques have shown that the arrangement of proteins in membrane protein clusters is often far from random, with spatial ordering of proteins into regular two-dimensional lattices and, depending on the protein oligomeric state and transmembrane shape, distinct orientational ordering of neighboring membrane proteins. These observations have led, on the one hand, to the proposal that the regular arrangement of membrane proteins may play a role in the biological function of cell membranes. On the other hand, the question arises as to what are the physical mechanisms responsible for the self-assembly, symmetry, and stability of membrane protein lattices. Based on experimental observations and physical models, membrane-mediated interactions between proteins, which result from protein-induced lipid bilayer deformations, have been advanced as a general mechanism for long-ranged interactions between membrane proteins. Here we develop a computational framework for the calculation of membrane-mediated interactions in the large and complicated membrane protein lattices observed in experiments. We find that, depending on the specific shape and oligomeric state of the protein under consideration, membrane-mediated interactions can be attractive or repulsive, several kBT in strength, and depend crucially on the spatial and orientational symmetry in membrane protein lattices. Combining our approach with the theory of regular lattices of polygons, we carry out a systematic survey of the connection between the shape of membrane proteins, and the symmetry and stability of membrane protein lattices. Our results suggest direct experimental tests of how the self-assembly and biological function of membrane protein lattices is influenced by membrane-mediated interactions.
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