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

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.