planar graphs
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Here, we hypothesize that, in biological systems such as cell surface receptors that relay external signals, clustering leads to substantial improvements in signaling efficiency. Representing cooperative signaling networks as planar graphs and applying Euler’s polyhedron formula, we can show that clustering may result in an up to a 200% boost in signaling amplitude dictated solely by the size and geometry of the network. This is a fundamental relationship that applies to all clustered systems regardless of its components. Nature has figured out a way to maximize the signaling amplitude in receptors that relay weak external signals. In addition, in cell-to-cell interactions, clustering both receptors and ligands may result in maximum efficiency and synchronization. The importance of clustering geometry in signaling efficiency goes beyond biological systems and can inform the design of amplifiers in nonbiological systems.