Minimal model of directed cell motility on patterned substrates

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern geometry and the strength of actin polymerization on the directionality of the crawling cell. These results reveal that high strength of actin polymerization results in motion perpendicular to substrate stripes only when the substrate is relatively nonadhesive; in particular, this suggests potential applications in motile cell sorting and guiding on engineered substrates.

Figure 1

(a) Sketch of cell motion on substrate with patterned stripes of adhesiveness described by (1, 2, 3, 4, 5); $\theta$ measures the deflection of the initial velocity from the vertical axis. (b) For small values of ${\overline{a}}_0$ the keratocyte experiences significant oscillations in velocity, adhesion density, and substrate deformations, here ${\overline{a}}_0=0.0025$. Oscillations in number of adhesions and substrate deformations are linked to stick-slip motion. For ${\overline{a}}_0>0.8$, values of $A_0$ and $U$ tend to a stable equilibrium. Due to inhomogeneity of the substrate, the velocity $V_y$ still has oscillations although with smaller amplitude.

Figure 2

(a) Bifurcation diagram showing the onset of finite velocity cell motion (with dimensionless cell velocity $\zeta$) as the dimensionless driving force $\kappa$ increases (where $\zeta$ and $\kappa$ are defined in Sec. 2b). In the absence of substrate patterns ($k_0=0$), the minimal driving force for the onset of motion is ${\kappa }_c\approx .746$ (solid line), originally calculated in Ref. [28]. In the presence of substrate patterns ($k_0=2.5$) the minimal driving force for cell motion decreases to ${\kappa }_c\approx 3.95$, providing evidence that a smaller driving force is required for motility of cells on patterned substrates. (b) The effect $\alpha$ on the direction of motion of the cell. The parameter $\alpha$ measures the rate of advection of the cell by the actin network and is a function, e.g., of the integrin ligand bond strength and actin filament stiffness. From the same initial position $(x,y)=(0,.6)$ (approximate center of an adhesive stripe) we plot the location of the cell for various values of $\alpha$ after simulation time $t=25$. We take as initial velocity the solution of (1) and (2) closest to $(V_x,V_y)=(.3,.6)$. For small values of $\alpha (\alpha <3)$ the cell halts and is trapped in a nonadhesive stripe, for moderate values of $\alpha$ ($3<\alpha <4$) the cell exhibits perpendicular motion, and for large values of $\alpha$ ($\alpha >4$) the cell exhibits parallel motion. This qualitative change in behavior based on cell biophysics suggests a mechanism for cell sorting.

Figure 3

(a) The maximal angle of the initial velocity for which the cell tends to persistent perpendicular motion is $max\theta$. For small values of ${\overline{a}}_0$ perpendicular motion is possible ($max\theta >0$). For large values of ${\overline{a}}_0$, only parallel motion is possible. The dotted line at ${\overline{a}}_0=0.8$ represents the maximal value for which $(A_0,U)$ exhibits a limit cycle; for ${\overline{a}}_0<0.8$ the cell exhibits stick-slip behavior [i.e., $(A_0,U)$ exhibits a limit cycle]; there the cell potentially has nonzero $max\theta$. This provides evidence that perpendicular motion is facilitated by stick-slip behavior. Simulations are done using the system (1, 2, 3, 4, 5). (b) Simulation of (7) and (8) with (3)–(5) ($\varepsilon =0.0005$) shows a larger basin of attraction for perpendicular motion. Even for nearly horizontal initial conditions the cell may eventually move perpendicular to stripes, as originally observed in the full phase-field model [28]. Dependence of $max\theta$ on ${\overline{a}}_0$ is no longer continuous. Interestingly, the existence of perpendicular motion corresponds almost exactly to the regime of stick-slip behavior.

Figure 4

The value of $\alpha$ and substrate pattern affect the direction of keratocyte motion. The value of $\alpha$ measures the rate of advection of the cell by the actin network. The patterned substrate has constant period $L$ but has varying width $L_1$ of the adhesive stripe. It is seen that if $\alpha$ is small, then the cell requires a wider adhesive stripe in order to move perpendicular to the stripe and vice versa. These trends hold both for dynamics with and without inertia. For all simulations we fix ${\overline{a}}_0=0.0025$. For simulations with inertia we use $\varepsilon =0.005$.