Asymmetry between pushing and pulling for crawling cells

Eukaryotic cells possess motility mechanisms allowing them not only to self-propel but also to exert forces on obstacles (to push) and to carry cargoes (to pull). To study the inherent asymmetry between active pushing and pulling we model a crawling acto-myosin cell extract as a one-dimensional layer of active gel subjected to external forces. We show that pushing is controlled by protrusion and that the macroscopic signature of the protrusion dominated motility mechanism is concavity of the force-velocity relation. In contrast, pulling is driven by protrusion only at small values of the pulling force and it is replaced by contraction when the pulling force is sufficiently large. This leads to more complex convex-concave structure of the force-velocity relation; in particular, competition between protrusion and contraction can produce negative mobility in a biologically relevant range. The model illustrates active readjustment of the force generating machinery in response to changes in the dipole structure of external forces. The possibility of switching between complementary active mechanisms implies that if necessary “pushers” can replace “pullers” and vice versa.

Figure 1

Schematic representation of an advancing lamellopodium subjected to a pushing force $q_{+}$ and a pulling force $q_{-}$.

Figure 2

The typical force-velocity relations in pure pulling (a) and pushing (b) regimes. Stress, velocity, and density profiles corresponding to points $A$, $B$, $C$, $D$, $\alpha$, and $\beta$ are shown in Figs. 5 and 4. Driving parameters are $v_{-}=1.7$ and $v_{+}=2$.

Figure 3

Schematic structure of a particle trajectory inside a cell as it approaches the steady-state TW regime. Dotted lines indicate instantaneous treadmilling of particles from the rear boundary of the cell to its front boundary.

Figure 4

Stress, velocity, and density profiles for point $\alpha$ in Fig. 2 where $\epsilon =\pm 1$, $Q=0$, $V=V^{*}=1.85$ and for point $\beta$ in Fig. 2 where $\epsilon =-1$, $Q=Q^{*}=0.45$, $V=0$. Parameters are $v_{-}=1.7$ and $v_{+}=2$.

Figure 5

Stress, velocity, and density distribution inside a cell moving with the same velocity $V=0.95$ in four different loading regimes indicated as $A$, $B$, $C$, and $D$ in Fig. 2. The dashed line shows elasticity-regularized profiles corresponding to point $C^{\prime }$ in Fig. 8.

Figure 6

(a) Various pulling regimes in the parameter space $(V_m,\Delta V)$. In the domain $V>0$ the cell moves against the load while if $V<0$ the cell is dragged by the load. Along the line XY the cell is static resisting the load (stall force conditions). The singular regimes correspond to the lines ZX (infinite localization) and XR (infinite spreading). (b) Density localization along the path indicated in (a) by the solid line which ends with the formation of a singularity at point $W$. The loading is $\epsilon =1$ and $Q=1.6$. For $\Delta V=0.3$ the singularity is located at $y_1/L_{\infty }\simeq 0.4$.

Figure 7

Sketch of particles trajectories as the steady-state (TW) mass flux approaches zero; see Fig. 6 for the related density profiles. (a) At small values of $\dot{m}$ each particle spends considerable time in a small region near a line $x=y_0+Vt$. (b) When $\dot{m}=0$ all particle trajectories converge to the line $x=y_0+Vt$, which leads to a blowup singularity.

Figure 8

Force-velocity relations in pure pushing and pulling modes with different $k_{1,2,3,4}=\{0,0.01,0.1,1\}$ and $L_0=1$. Driving parameters are $v_{-}=1.7$ and $v_{+}=2$. Internal profiles corresponding to points $C$ and $C^{\prime }$ are presented in Fig. 5(C). The minimal model is recovered at $k=0$.

Figure 9

Domain of negative mobility in the parameter space $(k,\Delta V)$. The boundary between regimes with positive and negative mobility is given by the function $k=k^{*}\left(\Delta V\right)$.

Figure 10

Force-velocity curves for the Kelvin-Voigt model with different $K_{1,2,3,4}=\{0,0.1,1,5\}$. Other parameters are ${\widehat{\rho }}_r=1$, $v_{-}=1.7$, and $v_{+}=2$. The minimal model is recovered at $K=0$.

Figure 11

Force-velocity curves for the Maxwell model with different ${\lambda }_{1,2,3}=\{0,0.1,1\}$. Other parameters are ${\rho }_0=1$, $v_{-}=1.7$, and $v_{+}=2$. The minimal model is recovered at $\lambda =0$.

Figure 12

Force velocity relations in the case of inhomogeneous friction with $\theta$ as a parameter. The minimal model is recovered at $\theta =0$.

Figure 13

Density distribution in the pulling and pushing regimes for the model with delocalized depolymerization. Parameters: $v_{+}=1$ and $\gamma =1.5$.

Figure 14

Contractile stress ${\sigma }_a$ as a function of actin density $\rho$ for different choices of parameters ${\rho }_s$ and $C$ taken from Ref. [78]. The minimal model is recovered at $C=1$, ${\rho }_s=\infty$.

Figure 15

Typical force-velocity relations for the model with actin-dependent contractility. Parameters are $v_{-}=1.7$, $v_{+}=2$. The insets show the steady-state density profiles located on the decreasing limb of the density-contractility curve; see Fig. 14. The minimal model is recovered at $C=1$, ${\rho }_s=\infty$.

Figure 16

Rate of energy consumption and efficiency as functions of the load in the cases of pure pulling and pushing loading modes. The corresponding force-velocity relation is shown in Fig. 2. Driving parameters are $v_{-}=1.7$ and $v_{+}=2$.

Figure 17

Efficiency as a function of the load in the elasticity-regularized model in pure pushing and pulling regimes with different $k_{1,2,3,4}=\{0,0.01,0.1,1\}$ and $L_0=1$. Experimental data suggest that $k=1–10$ (see Refs. [23, 24]). Parameters: $v_{-}=1.7$ and $v_{+}=2$. The corresponding force-velocity relations are shown in Fig. 8. The minimal model is recovered at $k=0$.

Figure 18

Schematic structure of the treadmilling cycle showing different densities of arriving (polymerizing) and departing (depolymerizing) material.

Figure 19

(a) Various pulling regimes in the parameter space $(\dot{m},P_c)$. The singular regimes correspond to the lines ZX (infinite localization) and to the point X(R) (infinite spreading). (b) Density localization along the path indicated in (a) by the solid line which ends with the formation of a singularity at point $W$. The loading is $\epsilon =1$ and $Q=1.6$. Analog of Fig. 6.

Figure 20

Force-velocity relations in pure pushing and pulling TW regimes when driving is performed by imposing $\dot{m}=-6.1$ and $P_c=0.3$. The insets show the ensuing dependences of $v_{+}$ and $v_{-}$ on $Q$.