Active elastic dimers: Cells moving on rigid tracks

Experiments suggest that the migration of some cells in the three-dimensional extracellular matrix bears strong resemblance to one-dimensional cell migration. Motivated by this observation, we construct and study a minimal one-dimensional model cell made of two beads and an active spring moving along a rigid track. The active spring models the stress fibers with their myosin-driven contractility and $\alpha$-actinin-driven extendability, while the friction coefficients of the two beads describe the catch and slip-bond behaviors of the integrins in focal adhesions. In the absence of active noise, net motion arises from an interplay between active contractility (and passive extendability) of the stress fibers and an asymmetry between the front and back of the cell due to catch-bond behavior of integrins at the front of the cell and slip-bond behavior of integrins at the back. We obtain reasonable cell speeds with independently estimated parameters. We also study the effects of hysteresis in the active spring, due to catch-bond behavior and the dynamics of cross linking, and the addition of active noise on the motion of the cell. Our model highlights the role of $\alpha$-actinin in three-dimensional cell motility and does not require Arp2/3 actin filament nucleation for net motion.

Figure 1

Side-view schematic of a two-bead-spring model of a cell crawling along a narrow track.

Figure 2

Schematic of contractile units in a stress fiber in extended mode (top) and contracted mode (bottom). The blue filaments represent actin filaments, red rectangles represent $\alpha$-actinin, and the green shapes represent myosin minifilaments. For simplicity, we have not shown any contractile units in parallel, only three units in series.

Figure 3

(a) Plot of the equilibrium spring length $x_{\mathrm{eq}}$ as a function of $x_1-x_2$. (b) Plot of friction coefficient ${\gamma }_1$ as a function of $x_1-x_2$. The parameters used are listed in Table 1.

Figure 4

(a) Plot of cell length $x=x_1-x_2$ as a function of time for the parameters given in Table 1. (b) Plot of position of the center of mass, $x_{\mathrm{cm}}$, as a function of time. (c) Plot of velocity of the center of mass, $v_{\mathrm{cm}}$, as a function of time.

Figure 5

Plots of center of mass for cells as a function of time for different spring constants $k$. The parameters are from Table 1 (unless stated otherwise).

Figure 6

(a) Plot of ${\overline{v}}_{\mathrm{cm}}\left(w\right)$. The inset plots $x_{\mathrm{cm}}\left(t\right)$ for different widths. (b) Plot of ${\overline{v}}_{\mathrm{cm}}\left(h\right)$ for $w=0.5\mu \text{m}$. The inset plots $x_{\mathrm{cm}}\left(t\right)$ for different heights of hysteresis loop.

Figure 7

(a) Plot of $x_{\mathrm{cm}}\left(t\right)$ for different friction coefficients. (b) Plot of ${\overline{v}}_{\mathrm{cm}}\left({\gamma }_{11}\right)$.

Figure 8

Plot of $x_{\mathrm{cm}}\left(t\right)$ for different values of the noise with $A=A_1=A_2$.

Figure 9

Plot of $\langle v_{\mathrm{cm}}\rangle \left(A\right)$ for finite slope case with a slope of 5/2. (Inset) Plot of $\langle x_{\mathrm{cm}}\rangle \left(t\right)$ for $A=0$ and $A=0.01{\text{nN}}^2\text{s}$.