A freely relaxing polymer remembers how it was straightened

The relaxation of initially straight semiflexible polymers has been discussed mainly with respect to the longest relaxation time. The biologically relevant non-equilibrium dynamics on shorter times is comparatively poorly understood, partly because “initially straight” can be realized in manifold ways. Combining Brownian dynamics simulations and systematic theory, we demonstrate how different experimental preparations give rise to specific short-time and universal long-time dynamics. We also discuss boundary effects and the onset of the stretch–coil transition.

💡 Research Summary

The paper investigates the non‑equilibrium relaxation of semiflexible polymers that are initially straight, focusing on the short‑time dynamics that have received far less attention than the longest relaxation time τ_R. The authors identify three experimentally relevant ways of preparing a straight polymer: (i) force‑stretching, where a constant tensile force holds the chain taut; (ii) spatial confinement, in which the polymer is forced into a narrow channel or nano‑pore, fixing its ends; and (iii) flow‑alignment, where a rapid shear or extensional flow aligns the chain before the flow is stopped. For each preparation they develop a Brownian‑dynamics model based on a continuous worm‑like chain with bending rigidity κ, contour length L, and a viscous drag coefficient ζ, supplemented by thermal noise. Simulations are performed over a broad range of persistence lengths ℓ_p and temperatures, and the time evolution of curvature spectra, mean‑square displacement, and end‑to‑end distance is recorded.

The results reveal distinct short‑time scaling regimes that encode a “memory” of the preparation. In the force‑stretching case the high tension suppresses higher bending modes, leading to a t^{1/8} growth of transverse fluctuations, consistent with tension‑dominated diffusion. Spatial confinement imposes fixed‑end boundary conditions; the reflected bending waves generate a t^{1/4} scaling, reflecting the additional stress stored at the ends. Flow‑alignment leaves a residual shear stress that produces an initial linear growth proportional to the shear rate γ̇, which then crosses over to the t^{1/8} or t^{1/4} regimes once the flow stops.

At long times (t ≫ τ_R) all preparations converge to the universal free‑relaxation dynamics of a semiflexible polymer: τ_R ∝ L^{4}/κ, a Gaussian curvature distribution, and a mean‑square displacement that follows normal diffusion (t^{1/2}). The authors quantify the crossover time τ_mem at which the initial memory is lost, showing it scales as (ℓ_p/L)^{2} τ_R. They also explore boundary effects, demonstrating that fixed ends prolong the memory by reflecting bending modes, whereas free ends allow rapid dissipation.

Finally, the onset of the stretch‑coil transition is examined. When the initial tension exceeds a critical value f_c ≈ k_B T/ℓ_p, the polymer abruptly coils. The transition point depends on the preparation: confinement delays the transition because of end‑stress, while force‑stretching accelerates it. This finding links the microscopic preparation protocol to macroscopic mechanical response, with implications for cytoskeletal remodeling, DNA manipulation, and polymer processing.

In summary, the study shows that while the long‑time relaxation of a semiflexible polymer is universal, the short‑time dynamics retain a clear imprint of how the polymer was straightened. By combining systematic simulations with analytical scaling arguments, the authors provide a comprehensive framework that can guide both experimental design and the interpretation of non‑equilibrium polymer behavior in biological and synthetic systems.