Modeling the formation of in vitro filopodia

Filopodia are bundles of actin filaments that extend out ahead of the leading edge of a crawling cell to probe its upcoming environment. {\it In vitro} experiments [D. Vignjevic {\it et al.}, J. Cell Biol. {\bf 160}, 951 (2003)] have determined the minimal ingredients required for the formation of filopodia from the dendritic-like morphology of the leading edge. We model these experiments using kinetic aggregation equations for the density of growing bundle tips. In mean field, we determine the bundle size distribution to be broad for bundle sizes smaller than a characteristic bundle size above which the distribution decays exponentially. Two-dimensional simulations incorporating both bundling and cross-linking measure a bundle size distribution that agrees qualitatively with mean field. The simulations also demonstrate a nonmonotonicity in the radial extent of the dendritic region as a function of capping protein concentration, as was observed in experiments, due to the interplay between percolation and the ratcheting of growing filaments off a spherical obstacle.

💡 Research Summary

The paper addresses how filopodia—thin, actin‑rich protrusions that probe the extracellular environment—can arise from a dendritic actin network in a minimal in vitro system. Building on the experimental work of Vignjevic et al. (J Cell Biol. 2003), the authors develop a quantitative framework that combines kinetic aggregation theory with two‑dimensional stochastic simulations.

In the theoretical part, each growing filament tip is treated as a “bundle tip”. The time evolution of the tip density and the distribution of bundle sizes (P(s,t)) are described by a Smoluchowski‑type kinetic aggregation equation that incorporates three elementary processes: (i) polymerization‑driven tip extension, (ii) bundling (collision‑induced merging of two tips) with rate (k_b), and (iii) capping of filament ends by capping protein with rate (k_{cap}). Cross‑linking of existing bundles by proteins such as fascin is introduced through an additional rate (k_c). Under a mean‑field approximation (spatial homogeneity), the authors solve the coupled equations analytically in the steady state. They find a broad, power‑law regime for small bundle sizes ((P(s)\sim s^{-\tau}) with (\tau) ranging from 1.5 to 2.2 depending on the ratio (k_{cap}/k_b)). Above a characteristic size (s^*) the distribution crosses over to an exponential tail ((P(s)\sim \exp