Coupling of Transverse and Longitudinal Response in Stiff Polymers

The time-dependent transverse response of stiff inextensible polymers is well understood on the linear level, where transverse and longitudinal displacements evolve independently. We show that for times beyond a characteristic time $t_{\mathrm{f}}$, longitudinal friction considerably weakens the response compared to the widely used linear response predictions. The corresponding feedback mechanism is explained by scaling arguments and quantified by a systematic theory. Our scaling laws and exact solutions for the transverse response apply to cytoskeletal filaments as well as DNA under tension.

Figure 1

A transverse point force $f_{\perp }$ applied to the contour $\mathit{r}(s)$ (dark) translates, through the formation of a bulge of height ${\Delta }_{\perp }$ and width ${\ell }_{\perp }$, into a longitudinal pulling force $f_{\parallel }$ acting on the polymer’s tails. This force induces backbone tension $f(s)$ (light) that penetrates the contour within a region of size ${\ell }_{\parallel }$ where thermal undulations are straightened.

Figure 2

Numerical solutions $\overline{f}(s,t)$ to Eqs. (2b, 4, 6) for $f_{\mathrm{pre}}=0$, time is increasing from light to dark color. Inset: log-log plot of the effective longitudinal force $f_{\parallel }(t)=\overline{f}(0,t)$ (dashed line with circles), and of the nonlinear response ${\Delta }_{\perp }(t)$ from Eq. (7) (solid line). Dotted lines indicate the asymptotes of Table (b). The crossover scales are $t_{\mathrm{f}}=(\gamma f_{\perp }{)}^{-2}$ and $s_{\mathrm{f}}=({\ell }_{\mathrm{p}}/\zeta {)}^{1/2}(\gamma f_{\perp }{)}^{-1/4}$, with $\gamma =(\zeta {\ell }_{\mathrm{p}}{)}^{2/7}{f}_{\perp }^{1/7}$.

Figure 3

Regimes of intermediate asymptotics (separated by thick black lines) for the nonlinear response ${\Delta }_{\perp }(t)$ (boxed formulas); time $t/(\gamma f_{\perp }{)}^{-2}$ vs. force ratio $f_{\mathrm{pre}}/(\gamma f_{\perp })$ (log-log scale). The universal initial regime [5] (light shaded) is followed by a quasistatic regime (white) with different force scaling for asymptotically small ($<$) and large ($>$) [5] force ratio; in these limits, the respective prefactors are $b^{<}\sim [8(1+\sqrt{2}{)}^2/{\pi }^3{]}^{1/8}$ and $b^{>}\sim {\pi }^{-1/2}$. An intermediate taut-string regime (dark shaded) emerges for very small force ratio. The prefactor is $a=[3(2+\sqrt{2})/{\pi }^2{]}^{2/9}$ if $f_{\mathrm{pre}}=0$.