How Spatially Modulated Activity Reshapes Active Polymer Conformations

TL;DR

This paper addresses how spatially modulated tangential activity reshapes conformations of semiflexible polymers. By formulating a continuum Rouse model with bending rigidity and performing a systematic weak-activity expansion, it derives analytical expressions for mode correlations, the gyration radius $\mathcal{R}_G$, and the end-to-end distance $\mathcal{R}_E$ under sinusoidal forcing and verifies them with Langevin simulations. The results reveal a mode-dependent shrinking-to-swelling transition, with low-mode forcing compacting the chain and higher-mode forcing creating alternating stretched and buckled segments, leading to globally swollen conformations; end-to-end distance can increase even when the gyration radius decreases, and parity effects emerge for ring-like topology. The simulations confirm the theory across a wide range of polymer lengths and forcing patterns, highlighting spatial activity as a robust mechanism to drive non-equilibrium polymer conformations. Overall, patterned activity offers a versatile handle to tune polymer shapes, with implications for designing active materials and understanding biological filaments under nonuniform driving.

Abstract

Active polymers are driven out of equilibrium by internal forces and exhibit conformational properties that differ fundamentally from those of passive chains. Here we study how spatially modulated tangential activity reshapes the conformations of semiflexible polymers. Using a continuum Rouse model with bending rigidity, we develop a systematic expansion in the limit of weak activity and derive analytical expressions for mode correlations, gyration radius, and end-to-end distance under sinusoidally varying propulsion. We show that spatially structured activity breaks self-similar scaling and induces a mode-dependent transition between polymer shrinking and swelling. Uniform or low-mode forcing produces compact, globule-like conformations, whereas higher modes generate alternating stretched and compressed segments, leading to globally swollen chains. Different polymer sizes respond differently to activity, allowing for conformations that are compact in gyration radius yet extended in end-to-end distance. Langevin dynamics simulations quantitatively confirm the theoretical predictions. Our results demonstrate that even weak, patterned activity provides a powerful mechanism to control polymer conformations far from equilibrium.

Figures (9)

• Figure 1: (a) Sketch of the continuum active polymer $\mathbf{r}(s,t)$. The self-propulsion direction is modulated along the backbone as $\cos (\pi ms/N)$ with $s$ the coordinate along the contour of the polymer, $N$ is the degree of polymerization and $m$ an integer; we name this "single-mode force" and we highlight the modulation through the verse and size of the arrows. (b) Example snapshot from numerical simulations (see Sec. \ref{['sec:numerical']}) with $m=40$; colors and other parameters as in Fig.s \ref{['fig:snapshots']}, \ref{['fig:snapshots1']}.
• Figure 2: $\langle \mathbf{r}_i\cdot \mathbf{r}_j\rangle^\infty_1$ with $l_p = 0.5$, $N=100$, and $f_m = 1$ and (a) $m=0$, (b) $m=1$, (c) $m=5$. The color codes represents the logarithm of the magnitude of the correlations, using warm colors for positive magnitudes and cool colors for negative magnitudes.
• Figure 3: $\langle \mathbf{r}_i\cdot \mathbf{r}_j\rangle^\infty_2$ with $l_p = 0.5$, $N=100$, and $f_m = 1$ and (a) $m=0$, (b) $m=1$, (c) $m=5$. The color codes represents the logarithm of the magnitude of the correlations, using warm colors for positive magnitudes and cool colors for negative magnitudes.
• Figure 4: Snapshots of active polymers, $N=200$, $\mathrm{Pe}=10$ and different values of the active force mode $m$. The chosen color map highlights both the sign and magnitude of the active force, shades of red being positive, shades of blue being negative and white being zero; arrows indicate the overall direction of the force in the section of the chain close to the arrow. (a) $m=1$, (b) $m=2$, (c) $m=3$, (d) $m=4$.
• Figure 5: More snapshots of active polymers, $N=200$, $\mathrm{Pe}=10$ and different values of the active force mode $m$. Color map as in Fig \ref{['fig:snapshots']}. (a) $m=8$, (b) $m=11$, (c) $m=14$, (d) $m=20$.