| Abstract | We describe a mathematical model of lamellipodial protrusion and retraction and their role in sensing the rigidity of the substrate. Several experiments have attempted to elucidate the mechanism and determined the periodicity of lamellipodium protrusions and retractions. Our goal is to use simple principles in cell mechanics and cell biology to derive a mathematical model to study how the periodicity of the protrusion and retraction arises and how it depends on, for example, the stiffness of the substrate. The variables in the model include (i) the positions of the barbed (fast polymerizing) and pointed (slow polymerizing) ends of actin filaments, (ii) the integrin density and (iii) the myosin density at the leading edge of the lamellipodium. Actin has been shown to polymerize at different rates at the barbed and pointed ends, thus the velocity of the two ends were dictated by the de/polymerization rates as well as the forces exerted by myosin and integrin on the actin filaments. The tensile strength of an actin filament was also taken into consideration, with the actin filament breaking off when the total force exerted on the actin filament exceeds a critical value. Integrin activation has been shown to result in the activation of myosin light chain kinase via the activation of Rho. Rate of change of myosin densities were thus allowed to increase with the amount of integrin present in the neighbourhood, with a time delay to account for propagation of the signal. Experiments have shown the existence of stretch-activated proteins bound to integrins which unfold to expose phosphorylation sites when stretched, allowing other proteins to assemble at the integrins to stabilize the focal contact. This is modeled by ensuring that adhesion strengthens only when the stretch exceeds a threshold. In addition, we also allow for integrin initiation when myosin starts to exert a force on the actin, therefore pulling on the leading edge of the lamellipodium. Solutions of our model are consistent with the periodic behavior observed in the experiments. We also show quantitatively how the period of the lamellipodial protrusions and retractions depends on the degree of integrin stabilization contributed by the stretch-activated proteins, the force-induced activation of integrin and the tensile strength of the actin network. Also, high tensile strength caused the system to become unstable while a low tensile strength eliminated periodicity. Periodicity was abolished when the amplitude of the force-induced activation of integrin was reduced. We also discuss how these results can be validated in further experiments. |
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