Diffusion of proteins on lipid membranes plays a central role in cell signaling processes. From a mathematical perspective, most membrane diffusion processes are explained by the Saffman-Delbrück theory. However, recent studies have suggested a major limitation in the theoretical framework, the lack of complexity in the modeled lipid membrane. Lipid domains (sometimes termed membrane rafts) are known to slow protein diffusion, but there have been no quantitative theoretical examinations of how much diffusion is slowed in a general case. We provide an overall theoretical framework for confined-domain (corralled) diffusion. Further, there have been multiple apparent contradictions of the basic conclusions of Saffman and Delbrück, each involving cases in which a single protein or an oligomer has multiple transmembrane regions passing through a lipid phase barrier. We present a set of corrections to the Saffman-Delbrück theory to account for these experimental observations. Our corrections are able to provide a quantitative explanation of numerous cellular signaling processes that have been considered beyond the scope of the Saffman-Delbrück theory, and may be extendable to other forms of subdiffusion.
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