Active gel segment behaving as an active particle

We reduce a one-dimensional model of an active segment (AS), which is used, for instance, in the description of contraction-driven cell motility, to a zero-dimensional model of an active particle (AP) characterized by two internal degrees of freedom: position and polarity. Both models give rise to hysteretic force-velocity relations showing that an active agent can support two opposite polarities under the same external force and that it can maintain the same polarity while being dragged by external forces with opposite orientations. This double bistability results in a rich dynamic repertoire which we illustrate by studying static, stalled, motile, and periodically repolarizing regimes displayed by an active agent confined in a viscoelastic environment. We show that the AS and AP models can be calibrated to generate quantitatively similar dynamic responses.

Figure 1

Two interaction kernels $\varphi \left(y\right)$: Black line is the exponential kernel of this paper and blue line is the kernel used in Ref. [36]. Nondimensional length $\mathcal{L}=2$.

Figure 2

Nonlocal response of the stress field [see (9)] and the velocity field [see (10)] to a contractility distribution with no external force (first row, $F=0$ and $\mathcal{P}=1$) and to a force distribution with no contractility (second row, $\mathcal{P}=0$ and $F=1$). Nondimensional length $\mathcal{L}=2$.

Figure 3

Effective frictional viscosity ${\mu }_0$ in the AP model as a function of the motor activity $\mathcal{P}$ in the three loading configurations (for the related AS model): homogeneous loading (black line), loading localized in the middle of the segment (blue line), and loading on the segment sides (red line). Nondimensional length $\mathcal{L}=2$.

Figure 4

Spontaneous polarization in AS (thin blue line) and AP (thick black line) models when contractility increases above the critical threshold ${\mathcal{P}}_c$. We show in insets typical concentration profiles of molecular motors along the bifurcated branches in the AS model. Nondimensional length $\mathcal{L}=2$.

Figure 5

Dependence of three contractility thresholds ${\mathcal{P}}_c$, ${\mathcal{P}}_m$, and ${\mathcal{P}}_s$ on the nondimensional length $\mathcal{L}$ for the AP (thick black line) and AS (thin blue line). The value of ${\mathcal{P}}_c$ is the same in both the AS and AP models by construction. The color dots are choices of parameters related to the V-F relations shown in Fig. 6.

Figure 6

Comparison of the V-F relations in the AS and AP models. Four typical V-F relations in the AS (thick lines) and the AP (thin lines) models. The dashed parts of the V-F curves correspond to unstable regimes. Parameters $\mathcal{L}=2$ and $\mathcal{P}=5$ (black-upper left corner), $\mathcal{P}=6$ (red-upper right corner), $\mathcal{P}=7$ (green-lower left corner), and $\mathcal{P}=9$ (blue-lower right corner) are represented with the same color dots in Fig. 5.

Figure 7

Schematic representation of the single and double hysteretic V-F relations.

Figure 8

Comparison of the global polarity $C$ as a function of the force in AS (thin blue) and AP models (thick black). The dashed parts of the C-F curve correspond to unstable regimes. We show in the inset the motor concentration in the AS case at forces $F=-3$ and $F=3$. The red profiles have a positive polarity, and the blue profiles have a negative polarity. A central symmetry transforms a red (resp. blue) profile at a positive force into a blue (resp. red) profile at a negative force. The green profiles are related to the unstable branch. Nondimensional length and contractility: $\mathcal{L}=2$ and $\mathcal{P}=9$.

Figure 9

Phase diagram of an AP subjected to a viscous drag force. The inset shows that the bifurcation from a static to a motile case at a critical contractility remains of second order (supercritical). Such a transition can be also obtained by reducing the environment viscosity. The phase boundary obtained for the AS is superimposed in thin blue. Nondimensional length $\mathcal{L}=2$.

Figure 10

Initiation of motility in a viscous environment. (a) AP position $S$ as a function of time with two external viscosities corresponding a static (red) and a motile (blue) case. One of the two symmetric trajectory is chosen according to a small bias in the initial polarity. (b) Same type of dynamics for the AS model. The intensity of the coloring is proportional to the level of motor concentration. Nondimensional contractility and length $\mathcal{P}=6$ and $\mathcal{L}=2$.

Figure 11

Scheme of an elastically confined AP (in red).

Figure 12

Linear instability thresholds of the static solution in the AP and AS case. We show in insets some typical static, stalled, and oscillatory dynamics for the AP and AS case. The intensity of the coloring is proportional to the motor concentration for the AS case. Nondimensional length $\mathcal{L}=2$.

Figure 13

Stability diagram of an AP confined by harmonic springs depending on the contractility and environment stiffness. Note the region of coexistence (metastability) between the oscillations and stall phases where the two types of solutions coexist. The two thin dashed lines and associated capital letters are related to the bifurcations diagrams shown in Fig. 14. Nondimensional length $\mathcal{L}=2$.

Figure 14

Two typical bifurcation diagrams for an elastically confined AP when the stiffness $k_{\text{p}}<k_{\text{p}}^{*}$ (a) and when $k_{\text{p}}>k_{\text{p}}^{*}$ (b). The continuation of the Hopf bifurcation is shown in blue, while supercritical pitchforks (second-order phase transitions) are shown in black. In the inset of (a), we show the frequency (inverse of the period) of the stable Hopf oscillations as a function of the continuation parameter. The dashed lines are linearly unstable branches, and full lines are stable. Arrows indicate the discontinuous transitions. The capital letters are related to the phase diagram shown in Fig. 13. Nondimensional length $\mathcal{L}=2$.

Figure 15

Dynamics of the oscillations for the stiffness $k_{\text{p}}$ of the environment starting from the initial state $(S_0,C_0)$. (a) The trajectories of the oscillating particle in the phase space $(k_{\text{p}}S,\dot{S})$ for three different stiffnesses. We superimpose in black the V-F curve obtained for a fixed force (Sec. 4). (b) The related particle position steady-state oscillations. Nondimensional length and contractility $\mathcal{L}=2$ and $\mathcal{P}=6.5$.

Figure 16

Schematic representation of the harmonic confinement with a breaking threshold.

Figure 17

Stability diagram of an AP confined by breakable harmonic springs. The position of the boundary of the motile phase depends on the initial conditions of (15), which are here taken to be $(S,C)(t=0)=(S_0,C_0)$. We show in the inset some typical dynamic of the AP breaking out of the harmonic confinement. Nondimensional length and breaking limit $\mathcal{L}=2$ and $l_{\text{p}}=1.5$.

Figure 18

Effective frictional viscosity at the origin of the V-F curve, ${\mu }_0$ as a function of the nondimensional contractility $\mathcal{P}$. Nondimensional length $\mathcal{L}=2$.

Figure 19

Eigenvalues ${\theta }_0$ and ${\theta }_2$ as a function of $\mathcal{L}$ for the first motile branch $D_1$.

Figure 20

Eigenvalues ${\theta }_0$ and ${\theta }_2$ as a function of $\mathcal{L}$ for the first static branch $S_1$.