Fully coupled numerical model of actin treadmilling in the lamellipodium of the cell

Cells rely on an interplay of subcellular elements for motility and migration. Certain regions of motile cells, such as the lamellipodium, are made of a complex mixture of actin monomers and filaments, which polymerize at the front of the cell, close to the cell membrane, and depolymerize at the rear. The dynamic actin turnover induces the so-called intracellular retrograde flow, and it is a fundamental process for cell motility. Apart from some comprehensive mathematical models, the computational modelling of actin treadmilling has been based on simpler biophysical models. Here, we adopt a highly detailed theoretical model of the actin treadmilling process and develop a coupled unsteady finite element formulation. We clearly describe the structure and implementation of the coupled problem within the finite element method. Our numerical results show an excellent correlation with experimental results from literature and with previous models. We include time dependent effects and convective transport terms, which expose puzzling dynamics in the retrograde flow. We propose several biological scenarios to analyze the behavior of the actin treadmilling along space and time. We observed response times of the main density variables in the order of seconds. Compared with previous analytical solutions, which make assumptions related to convective transport, transient dynamics, and actin fluxes, the generic solution can have significant influence on the retrograde flow. All together, our results unveil a promising applicability of classical finite element methods to derive an in silico testing platform for the actin treadmilling processes in motile cells, which could allow for an extension to other biophysical effects.