Active chain spirograph: Dynamic patterns formed in extensible chains due to follower activity

TL;DR

This work studies a minimal active-chain model driven by tangential follower forces in the noiseless overdamped limit to understand how geometry and local activity generate periodic and complex motions. Analytically, it provides exact results for the three-bead case, yielding a stable circular bound state with $\omega = \dfrac{f}{2-f}$ and equal bond extensions $d_1=d_2=-\dfrac{f}{2}$, and it characterizes the large-$N$ limit where the chain forms a circular center-of-mass orbit with a boundary-layer structure described by $\alpha = \tan^{-1}\left(\dfrac{f}{2-f}\right)$ and $\varepsilon_i \sim a_\pm^i$. Numerically, the authors map rich steady-state behavior for intermediate $N$, including bound and unbound COM trajectories, quasi-periodicity, and strong initial-condition dependence (multi-stability), with 3D initializations yielding helices, globules, and other nonplanar patterns. They also show that large-$N$ circular behavior is robust to different tangent definitions, highlighting a generic mechanism for periodic motion in tangentially driven chains. Overall, the findings illuminate minimal conditions for periodic, coordinated motion in active, elastic networks and offer guidance for designing active filaments and autonomous robotic swarms.

Abstract

Follower activity results in a large variety of conformational and dynamical states in active chains and filaments. These states are formed due to the coupling between chain geometry and the local activity. We study the origin and emergence of such patterns in noiseless, flexible active chains. In the overdamped limit, we observed a range of dynamical steady states for different chain lengths ($N$). The steady-state planar trajectories of the centre-of-mass of the chain include circles, periodic waves, and quasiperiodic, bound trajectories resembling spirographic patterns. In addition, out-of-plane initial configuration also leads to the formation of 3D structures, including globular and supercoiled helical structures. For the shortest chain with three segments $(N=3)$, the chain always moves in a circular trajectory. Such circular trajectories are also observed in the limit of large chain lengths $(N \gg 1)$. We analytically study the dynamical patterns in these two limiting cases, which show quantitative and qualitative matches with numerical simulations. Our analytical study also provides an estimate of the limiting $N$ where the large chain length behaviour is expected. These analyses reveal the existence of such intricately periodic patterns in active chains, arising due to the follower activity.

Figures (6)

• Figure 1: (a) Schematic of the active chain showing the activity scheme used. (b) Schematic showing the forces on different monomers for a three bead chain.
• Figure 2: (a) Variation of dimensionless chain centre-of-mass velocity $\tilde{ V} = r\omega$ (in units of $kb/\gamma$) with non-dimensional active force $\tilde{f} = f/kb$, where $k$ is the spring stiffness, $\gamma$ the damping coefficient and $b$ is the equilibrium bond length. Inset: a typical steady-state configuration in the overdamped limit. (Also see MOVIE1)(b) The variation of radii $\tilde{r}$ (in units of $b$) of monomer trajectories with the non-dimensional active force $\tilde{f}$ for the 3-bead overdamped system. The first and third monomers follow the same trajectory, while the second monomer orbits with a different radius. The solid lines in (a) and (b) indicate theoretical values. (c) Variation of bond angle $\theta$ with non-dimensional mass $\tilde{m}=mk/\gamma^2$ for the underdamped three-bead system. As $\tilde{m} \rightarrow 0$, $\theta$ converges to $\pi/2$ (dotted line). Inset: a typical configuration of an underdamped three-bead system (d) Variation monomer radii $\tilde{r}$ with mass $\tilde{m}$ for the 3-bead underdamped system for $\tilde{f}=1$. As $\tilde{m} \rightarrow 0$, the radii approaches the overdamped values (dotted lines).
• Figure 3: (a-e) Some examples of the trajectories obtained for $N\ge4$ for the overdamped system with 2D initialization (Also see MOVIE2 - MOVIE6). The individual monomer trajectories are shown in gray and the centre-of-mass(COM) trajectory is shown in black color, obtained over a brief time interval. The head monomer (passive) is marked by yellow color. Of these, trajectories (a-c) are bounded, corresponding to $N=5$ for two different initializations and $N=8$ respectively and trajectories (d-e) are unbounded, corresponding to $N=4$ and $6$ respectively. (f-g) Variation of radius and angle subtended by the COM corresponding to the bounded trajectories (b-c) respectively. (h) Variation of the signed curvature $\kappa$ of the COM corresponding to the trajectories (a-e). (i) Mean Square Displacement(MSD) of the COM for the wavelike trajectories (d-e), which goes ballistic in long term. In figures (f-i), $\tilde{t}$ represents the dimensionless time in units of $\gamma/k$.
• Figure 4: Dependence of the steady state trajectory for some values of $N$ and $f$ started from different initializations generated by varying angle $\theta$ as in (a). Parameter values are as follows. (b-c) $N=5$, $f=1$, (b) $\theta=3\pi/5$ and (c) $\theta=4\pi/5$. (d-e) $N=7$, $f=0.5$, (d) $\theta=4\pi/5$, (e) $\theta=\pi$
• Figure 5: (a-b) The circular trajectories observed, along with the configuration of a chain with the number of segments, $N=100$, for two types of tangent definitions. (a) The tangent vector is same as the bond vector (b) the tangent is the average of two consecutive bond vectors. (c) Variation of the log absolute deviation ($\log_{10}|\varepsilon_i|$) with bond index $i$ for the chain shown in (a). Circles are the simulated data points, and the solid lines are drawn to compare the slopes in log scale to the theoretical predictions. The red lines correspond to the tail solution and the black lines to the head solution. (d-e) Variation of deviations $\varepsilon_i$ and bond angles $\theta_i$ respectively with bond index. The solid green line corresponds to trajectory (a), the dashed orange line to another type of trajectory with a flexible core for comparison. The black dashed lines corresponds to theoretical predictions for the saturating values.