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Diffusion is a fundamental mechanism for protein distribution in cell membranes. These membranes often exhibit complex shapes, which range from shallow domes to elongated tubular or pearl-like structures. Shape complexity of the membrane influences the diffusive spreading of proteins and molecules. Despite the importance membrane geometry plays in these diffusive processes, it is challenging to establish the dependence between diffusion and membrane morphology. We solve the diffusion equation numerically on various curved shapes representative for experimentally observed membrane shapes. Our results show that membrane necks become diffusion barriers. We determine the diffusive half time, i.e., the time that is required to reduce the amount of proteins in the budded region by one half and find a quadratic relation between the diffusive half time and the averaged mean curvature of the membrane shape. Our findings thus help to estimate the characteristic diffusive time scale based on the simple measure for membrane morphology.Significance statementDiffusion is an integral process for distributing proteins throughout biological membranes. These membranes can have complex shapes and structures, often featuring elongated shapes such as tubes and like a necklace of pearls. The diffusion process on these shapes is significantly different from the well studied planar substrate. We use numerical simulations to understand how the characteristic diffusion time is a function of membrane shape, where we find the diffusion of proteins on strongly curved shapes is significantly slower than on planar membranes. Our results provide a simple relationship to estimate the characteristic diffusion time of proteins on membranes based on its mean and Gaussian curvature.

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