We develop a mathematical model that describes key details of actin dynamics in protrusion associated with cell motility. The model is based on the dendritic-nucleation hypothesis for lamellipodial protrusion in nonmuscle cells such as keratocytes. We consider a set of partial differential equations for diffusion and reactions of sequestered actin complexes, nucleation, and growth by polymerization of barbed ends of actin filaments, as well as capping and depolymerization of the filaments. The mechanical aspect of protrusion is based on an elastic polymerization ratchet mechanism. An output of the model is a relationship between the protrusion velocity and the number of filament barbed ends pushing the membrane. Significantly, this relationship has a local maximum: too many barbed ends deplete the available monomer pool, too few are insufficient to generate protrusive force, so motility is stalled at either extreme. Our results suggest that to achieve rapid motility, some tuning of parameters affecting actin dynamics must be operating in the cell.
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