Theory of Long-Range Diffusion of Proteins on a Spherical Biological Membrane: Application to Protein Cluster Formation and Actin-Comet Tail Growth

Breaking of symmetry is often required in biology in order to produce a specific function. In this work we address the problem of protein diffusion over a spherical vesicle surface towards one pole of the vesicle in order to produce ultimately an active protein cluster performing a specific biological function. Such a process is, for example, prerequisite for the assembling of proteins which then cooperatively catalyze the polymerization of actin monomers to sustain the growth of actin tails as occurs in natural vesicles such as those contained in Xenopus eggs. By this process such vesicles may propel themselves within the cell by the principle of action-reaction. In this work the physicochemical treatment of diffusion of large biomolecules within a cellular membrane is extended to encompass the case when proteins may be transiently poised by corral-like structures partitioning the membrane as has been recently documented in the literature. In such case the exchange of proteins between adjacent corrals occurs by energy-gated transitions instead of classical Brownian motion, yet the present analysis shows that long-range movements of the biomolecules may still be described by a classical diffusion law though the diffusion coefficient has then a different physical meaning. Such a model explains why otherwise classical diffusion of proteins may give rise to too small diffusion coefficients compared to predictions based on the protein dimension. This model is implemented to examine the rate of proteins clustering at one pole of a spherical vesicle and its outcome is discussed in relevance to the mechanism of actin comet tails growth.