Closing the loop: Lamellipodia dynamics from the perspective of front propagation

We develop a simple physical model that captures the large-scale lamellipodia dynamics in crawling cells and explains the observed spectrum of fish keratocytes behavior. The main ingredients in this description are the geometrical evolution of the lamellipodium leading edge, the dynamic remodeling of the actin network, and the interconnection between them. We deviate from existing theoretical works and consider the lamellipodium leading edge as a propagating front. The agreement of our model with experimental works suggests that the large-scale morphological and migration features exhibited by keratocyte cells are a direct consequence of the closed feedback loop between the shape of the leading edge and the density of the actin network.

Figure 1

Experiments show that substrate properties significantly affect morphological features and dynamics behavior.

Figure 2

Schematic description of the model. The left inset shows the graded protrusion velocity (arrows) and actin density (gray shade) for a constant shape. A location on the leading edge can be expressed in terms of a laboratory coordinate system $(x,y)$ or by the arclength $s$; see right inset.

Figure 3

Membrane tension $T$ affects the lamellipodia dynamics by altering the protrusion-density relation.

Figure 4

Schematic illustration of the leading edge geometry for two different times separated by $dt$ (exaggerated). Blue, red, and dark arrows represent the normal velocity $U$, the lateral velocity $V$, and the velocity at constant $s$, $\partial \vec{x}/\partial t$. At steady state the black arrows are parallel and indicate the direction of propagation. This is not the case in general (see the inset). For clarity, only part of the leading edge is shown.

Figure 5

Steady-state configurations: (a) shape of the leading edge and (b) distribution of actin density along the leading edge. Crawling velocity is indicated by the color of the curve, where slower (faster) crawling velocities are lighter (darker). Due to symmetry, only the right half of the leading edge is shown.

Figure 6

Comparison between the dynamic response (leading edge geometry at different times) of two stable steady-state configurations subjected to random fluctuations in the actin density at the end of the leading edge: (a) $U_0=0.2U_m$ and (b) $U_0=0.6U_m$. The maximum magnitude of density fluctuations is $\rho /{\rho }_s=0.2$, with a correlation length of $10\text{s}$.

Figure 7

Geometry of the leading edge (only the right half is shown) at different times when subjected to random fluctuations in the actin density at the end of the leading edge: $U_0=0.2U_m$, with a simulation length of $40\text{s}$.

Figure 8

Deviation $D\left(t\right)={\int }_0^L|\rho -{\rho }^{*}|ds$ from the steady-state solution ${\rho }^{*}$ as a function of $U_0$ for perturbations types (i)–(iv) in Eq. (B1). The graphs on the right are identical to those on the left, but focused on the region $0.8U_m\le U_0\le 0.9U_m$. In each plot, the different lines correspond to a different time $t=0,250,500,750,1000$ s.

Figure 9

Relation between pairs of motility features of stable steady-state configurations. Front radius, area, and speed are measured in $\mu \text{m}$, $\mu {\text{m}}^2$, and $\mu \text{m}{\text{s}}^{-1}$, respectively. Aspect ratio and actin ratio are nondimensional quantities.