Topological localisation and motility of active knots

Abstract

Nonequilibrium active polymers provide a minimal framework to investigate biopolymers such as DNA and chromatin under the action of molecular motors. Here we study active ring polymers with controlled topology and show that knot type qualitatively determines their nonequilibrium behaviour. We find that activity induces opposite localisation responses in different topological families: torus knots systematically delocalise and inflate, whereas twist knots tighten and remain localised. We trace this divergent behaviour to the distinct symmetry properties of their tangent fields, which control the alignment of active forces along the chain. We show that topology also governs internal and emergent dynamics. Active torus knots behave as soft chiral self-propelled particles exhibiting persistent motion with a well-defined handedness fixed by their topological chirality. In contrast, achiral knots show no net handedness. The knot thus acts as a deformable topological quasiparticle whose morphology and propulsion are selected by topology. These results suggest potential routes toward programmable soft chiral particles with controllable morphology and emergent motility modes.

Figures (4)

• Figure 1: Localization. Activity can induce a localization or a delocalization for torus knots.(a-f) Snapshots from simulations of active knots of different knot types $\pi$, $5_1$ ((a)-(d)), $5_2$ ((b)-(e)) and $7_2$ ((c)-(f)) at two activity strengths, $\text{Pe}=8$ ((a)-(b)-(c)) and $\text{Pe}=20$ ((d)-(e)-(f)). (g) Average fraction of beads in the knotted region as a function of activity for different knot types, averaged at steady state, over different replicas. (h) Beads velocity-velocity correlation function as a function of time for $5_1$ and $5_2$ knots simulated at low ($\text{Pe}=8$) and high ($\text{Pe}=20$) activity. i Snapshot of quasi-ideal conformation of a $5_1$, a $5_2$ and a $7_2$ knot from simulations at $\text{Pe}=14$. The knots are decorated with arrows indicating the orientation of the activity.
• Figure 2: Persistence and reptation.(a-b) Trajectories of the center of mass of a $5_1$ and a $5_2$ knot simulated at $F_a=10$. (c) Correlation function (for $\text{Pe}=20$) and persistence length (d) of the tangents to the trajectory of the center of mass of simulated active knots. The inset in panel (d) shows the average velocity of the beads as a function of activity, for different topologies. (e) Sequence of subsequent snapshots from simulations of a $5_2$ knot revolving in space. Highlighted in red is a given selected bead, which quickly traverses the whole contour before the knot shape changes significantly. (f) Trajectory of the crossing (blue) of a delocalized torus knot ($5_1$ for $\text{Pe}=20$) , and of one of its beads (red), over a time interval of $10^3\tau$. (g) Reptation period for the same case displayed in panel (f).
• Figure 3: Chirality.(a-b) Trajectory of a crossing of a $5_1^{-}$ (a) and $5_1^{+}$ (b) knot, over an interval of $100\tau$, simulated at $\text{Pe}=20$. (c) Average handedness of the trajectories of a crossing of a simulated $5_1^{-}$ (purple), and a $5_1^{+}$ (orange) knots, and of the trajectory of a comoving point to a simulated $4_1$ knot (black bar). d Average handedness of comoving trajectories to $5_1^{-}$ knots simulated at $\text{Pe}=14$ with different values of the stiffness parameter $K_p$.
• Figure 4: Topology and efficiency. Average radial velocity for various knot types $\pi$ relative to the unknot ($0_1$) at $\text{Pe}=20$. As shown in the top-left inset, this is calculated from the first passage time out of a spherical boundary of radius $r=25\sigma$. For comparison, the result for the unknots at $\text{Pe}=0$ is reported too (black bar). Note that the composite knots are twice as long ($N_b=128$) and stiffer ($K_p=10 k_bT$) than the prime knots.