Modeling the dynamics of a tracer particle in an elastic active gel

The internal dynamics of active gels both in artificial (in vitro) model systems and inside the cytoskeleton of living cells has been extensively studied with experiments of recent years. These dynamics are probed using tracer particles embedded in the network of biopolymers together with molecular motors, and distinct nonthermal behavior is observed. We present a theoretical model of the dynamics of a trapped active particle, which allows us to quantify the deviations from equilibrium behavior, using both analytic and numerical calculations. We map the different regimes of dynamics in this system and highlight the different manifestations of activity: breakdown of the virial theorem and equipartition, different elasticity-dependent “effective temperatures,” and distinct non-Gaussian distributions. Our results shed light on puzzling observations in active gel experiments and provide physical interpretation of existing observations, as well as predictions for future studies.

Figure 1

Distribution of position and velocity for particle trapped within a harmonic trap of various stiffness ($k=1,1000$ for a, b and c, d, respectively) and for different regimes of friction ($\lambda =50,0.1$ for a, c and b, d, respectively). The time scale of the active bursts is $\mathrm{\Delta }\tau =0.1$, the amplitude of the active force $f_0=1$, and the waiting time $\tau =1$ (so that $p_{on}\approx 0.1$). For simplicity we plot the behavior for the case of a single motor with a constant burst duration. The insets compare the simulated distribution (blue line) to the analytic approximation (red dashed line), in log-linear scale, as simple Gaussians or as a sum of shifted thermal Gaussians.

Figure 2

(a) Distribution of particle displacements $P\left[\mathrm{\Delta }x\left({\tau }_{\omega }\right)\right]$, for various lag time duration ${\tau }_{\omega }$ (blue lines), for a single motor and constant burst duration. The traces correspond to increasing time lag durations (black arrow), in the range $1>{\tau }_{\omega }>5\times {10}^{-4}$. The red dashed line denotes the spatial distribution $P\left(x\right)$, and the black dashed lines denote $P\left(\mathrm{\Delta }x\right(\infty \left)\right)$ [Eq. (6)] for the approximation of $P\left(x\right)$ as a sum of three Gaussians. Inset shows the displacement distribution for very short lag times ${\tau }_{\omega }$. Parameters as in Fig. 1. (b) Calculated NGP for the $P\left[\mathrm{\Delta }x\left({\tau }_{\omega }\right)\right]$, for various number of motors ($N_m$), and confinement strength. The short horizontal black lines denote (left) the NGP of $P\left(x\right)$, and (right) of $P\left[\mathrm{\Delta }x\right(\infty \left)\right]$ [Eq. (6)], for the $k=1000,N_m=1$ case. (c) Displacement distributions [as in (a)] for a calculation without the inertia term [Eq. (2), using $k=1000$, Poissonian $\langle \mathrm{\Delta }\tau \rangle =0.1$, and increasing lag time indicated by the arrow ${\tau }_{\omega }={10}^{-3},2.5\times {10}^{-3},5\times {10}^{-3},{10}^{-2},1]$, and (d) the corresponding NGP, comparing the simulation (solid gray lines) to the analytical result (see Appendix for details, dashed lines), for $k=300,1000$ (top, bottom). The NGP with inertia is given by the solid black lines.

Figure 3

Calculated mean-square position fluctuations (plotted as a mean potential energy) for the trapped particle: Brown line, full solution; purple line, approximate solution [Eq. (B8)]; blue line, approximate expression $T_x^{\prime }$ for the limit $\lambda \mathrm{\Delta }\tau \gg 1$ [Eq. (B9)]. In both panels we used $\mathrm{\Delta }\tau =1$, and (a) $\lambda =10$, (b) $\lambda =0.01$. In (a) the blue line agrees perfectly with the full solution, while in (b) it has a discrepancy at intermediate confinements.

Figure 4

Simulated mean-square particle displacements (a) and velocity (b) in the limit of $\lambda \mathrm{\Delta }\tau \gg 1$, using $\mathrm{\Delta }\tau =1, \lambda =50, f_0=1, p_{on}=0.1$. The dashed lines indicate the power laws with exponents $-1,-2$ in (a) and $-3/2$ in (b).

Figure 5

Calculated mean-square velocity fluctuations for the trapped particle: brown line, full solution; purple line, approximate solution [Eq. (C3)]; blue line, highly damped limit [Eq. (C4)]; and the dashed blue line is the free-particle value [11]. In both panels we used $\mathrm{\Delta }\tau =1$ and (a) $\lambda =10$, (b) $\lambda =0.01$.

Figure 6

Simulated particle position distribution $P\left(x\right)$ (red lines) and displacement distributions $P\left(\mathrm{\Delta }x\left({\tau }_{\omega }\right)\right)$ (blue lines), for increasing number of motors: $N_m=5,10,20$ (left to right), using $\mathrm{\Delta }\tau =1, \lambda =50, f_0=1, p_{on}=0.1$.