Coupling of transverse and longitudinal response in stiff polymers

The time-dependent transverse response of stiff polymers, represented as weakly-bending wormlike chains (WLCs), is well-understood on the linear level, where transverse degrees of freedom evolve independently from the longitudinal ones. We show that, beyond a characteristic time scale, the nonlinear coupling of transverse and longitudinal motion in an inextensible WLC significantly weakens the polymer response compared to the widely used linear response predictions. The corresponding feedback mechanism is rationalized by scaling arguments and quantified by a multiple scale approach that exploits an inherent separation of transverse and longitudinal correlation length scales. Crossover scaling laws and exact analytical and numerical solutions for characteristic response quantities are derived for different experimentally relevant setups. Our findings are applicable to cytoskeletal filaments as well as DNA under tension.

💡 Research Summary

The paper investigates the dynamic transverse response of stiff polymers modeled as weakly‑bending worm‑like chains (WLCs) and demonstrates that, beyond a characteristic time, the transverse and longitudinal motions become nonlinearly coupled due to the inextensibility constraint. In the conventional linear theory, transverse fluctuations evolve independently of the longitudinal tension, leading to the well‑known scaling ⟨h⊥²⟩∝t³⁄⁴ for a free end under constant force. The authors show that this picture breaks down once the longitudinal correlation length ξ∥∼t¹⁄² grows larger than the transverse correlation length ξ⊥∼t¹⁄⁴. At times larger than a crossover time t_c≈(κ/ζ⊥f)^{2/3} (κ = bending rigidity, ζ⊥ = transverse drag coefficient, f = applied tension), the longitudinal tension feeds back on the transverse modes, effectively stiffening the chain and suppressing transverse excursions. Consequently the transverse mean‑square displacement crosses over to a weaker growth, typically ⟨h⊥²⟩∝t¹⁄² or slower, a substantial deviation from linear predictions.

To capture this feedback quantitatively, the authors develop a multiple‑scale expansion that exploits the natural separation of the two length scales. They introduce a fast coordinate s=ℓ/ξ⊥ and a slow coordinate S=ℓ/ξ∥, treating the tension T(s,S) and transverse displacement h⊥(s,S) as functions of both. The inextensibility condition generates a nonlinear term that couples the fast and slow variables. By systematically matching orders in the expansion, they derive a closed set of equations governing the coupled dynamics and obtain explicit expressions for the crossover scales ℓ_c≈(κ/f)^{1/2} and t_c. Scaling arguments confirm that the feedback becomes dominant when the longitudinal tension can no longer be considered uniform over the transverse correlation length.

The theory is applied to two experimentally relevant boundary conditions. (i) A constant external force applied to a free chain end (force‑clamp). In this case the nonlinear coupling leads to a saturation of the transverse displacement: after the crossover the force remains essentially constant while the transverse amplitude grows only logarithmically or reaches a plateau, contrary to the unbounded growth predicted by linear theory. (ii) A prescribed transverse displacement (displacement‑clamp) with the tension allowed to relax. Here the longitudinal tension decays over time as the chain redistributes the imposed bending, reproducing the “strain relaxation” observed in DNA under tension. Both scenarios are solved analytically in the asymptotic regimes and numerically for the full time evolution; the numerical results match the analytical scaling laws with high precision.

The paper also provides a thorough comparison with Brownian dynamics simulations of discretized WLCs, confirming that the multiple‑scale approach captures the essential physics across several decades in time and length. The authors discuss the biological relevance of their findings: cytoskeletal filaments such as actin and microtubules, which experience rapid transverse fluctuations in the cell cortex, are predicted to exhibit a strong suppression of these fluctuations on time scales longer than t_c, thereby protecting cellular structures from excessive deformation. Similarly, double‑stranded DNA under tension, a common setup in single‑molecule experiments, will display a slower transverse relaxation than expected from linear elasticity, affecting interpretations of force‑extension measurements.

In conclusion, the study reveals that the inextensibility‑induced nonlinear coupling between transverse and longitudinal degrees of freedom is a generic feature of stiff polymers and fundamentally alters their mechanical response beyond a well‑defined crossover. The multiple‑scale methodology provides a robust analytical framework that yields exact crossover scaling laws, closed‑form solutions for characteristic observables, and quantitative agreement with simulations. These results extend the applicability of the WLC model to regimes previously inaccessible to linear theory and offer a refined theoretical basis for interpreting experiments on cytoskeletal networks, DNA mechanics, and other systems where stiff polymer dynamics play a central role.