Dynamics of semiflexible polymers in a flow field

We present a method to investigate the dynamics of a single semiflexible polymer, subject to anisotropic friction in a viscous fluid. In contrast to previous approaches, we do not rely on a discrete bead-rod model, but introduce a suitable normal mode decomposition of a continuous space curve. By means of a perturbation expansion for stiff filaments, we derive a closed set of coupled Langevin equations in mode space for the nonlinear dynamics in two dimensions, taking into account exactly the local constraint of inextensibility. The stochastic differential equations obtained this way are solved numerically, with parameters adjusted to describe the motion of actin filaments. As an example, we show results for the tumbling motion in shear flow.

Figure 1

Sketch of a polymer of length $L$, represented by a continuous space curve $\vec{r}(s,t)$. An example is shown for the local unit tangent vector $\widehat{t}(s,t)$, as well as the end-to-end distance vector $\vec{R}$.

Figure 2

Mean-squared amplitudes of normal modes for different persistence length. Open circles are numerical results; “+” interconnected by lines are given by Eq. (14). Significant deviations between both are visible only for $ϵ=\sqrt{L∕{\mathcal{l}}_p}=1$. They are of order ${ϵ}^3$, terms neglected in the perturbative solution Eq. (15a) (cf. also Fig. 3).

Figure 3

(Color online) Relative errors of some mean-square mode amplitudes $ϵ\sqrt{⟨{\theta }_{\alpha }^2⟩}$ (colored) and mean end-to-end distance (black circles) vs inverse expansion parameter $ϵ$. Lines are shown to guide the eye. As expected from the order of the expansion of the equations of motion, the relative error vanishes proportional to ${ϵ}^{-2}$, thus the absolute error is $\propto {ϵ}^{-3}$.

Figure 4

(Color online) Time dependence of the mean-square displacement ${\Delta }_R\left(t\right)$ of the end-to-end distance. The calculations are done in 12-dimensional mode space. Symbols are replaced by lines where they would become very dense. The rescaled data for different persistence length collapse onto one master curve, which behaves $\propto t^{3∕4}$ for short times. ${\tau }_1$ is the relaxation time of the longest mode, ${\tau }_1=k_1^{-4}$, in our units [cf. Eq. (15a), and Ref. 7]. The inset shows a magnification of the long-time behavior. The stiffer the filament, the closer the numerical results are to the correct equilibrium value 1.

Figure 5

Sketch of a filament in shear flow. The two gray arrows indicate walls moving with constant speed; black arrows show the velocity of the fluid. At the fluid-wall interface, the fluid is transported with the velocity of the wall, due to no-slip boundary conditions.

Figure 6

(Color online) Sample trajectories of the end-to-end distance $R$ and the first two angular modes, for a flow of strength $\dot{\gamma }=1.03∕\mathrm{s}$.

Figure 7

(Color online) Power spectral densities $P\left(f\right)$ of the end-to-end distance without flow, and with a shear of strength $\dot{\gamma }=1.03$ $(\mathrm{Wi}=0.55)$. Green ×: ${\mathcal{l}}_p∕L=49$; red +: ${\mathcal{l}}_p∕L=6.5$. The smooth black curve is a fit of the result in quiescent solvent with the appropriate Lorentzian [60], used for the plot of the relative PSD in Fig. 9. With shear, a bump between 0.05 and $0.09\mathrm{Hz}$ shows the characteristic time of rotation of the filament. A different scaling regime appears for frequencies smaller than $1\mathrm{Hz}$, with an exponent depending on properties of the filament. PSD data for ${\mathcal{l}}_p∕L=49$ have been multiplied by a factor 60 for the purpose of an easier visualization. Calculated with 10 $({\mathcal{l}}_p∕L=6.5)$ and 8 $({\mathcal{l}}_p∕L=49)$ modes resolution. Inset: Nonlogarithmic version of the region around the bump for ${\mathcal{l}}_p∕L=6.5$, with estimated error bars. The number of data points has been reduced by averaging.

Figure 8

(Color online) Power spectral density of tension modes ${\Lambda }^0,{\Lambda }^1$ and angular modes ${\theta }_0,{\theta }_1$. Symbols give results with a shear flow of strength $\dot{\gamma }=1.03∕\mathrm{s}$ $(\mathrm{Wi}=0.55)$. The light line (green online) shows the behavior of ${\theta }_1$ without flow; smooth black curves are fitted Lorentzians, as explained in the text. A peak in ${\theta }_0$ and ${\Lambda }^0$ around $0.5\mathrm{Hz}$ is a sign of the tumbling time of the filament. Concerning ${\theta }_1$, shear still increases correlations at low frequencies.

Figure 9

(Color online) Ratio of power spectral density $P\left(f\right)$ with shear flow to that without flow, for the data of Figs. 7, 8. Correlations of $R$ are increased by a factor of more than 10 at frequencies of $0.3\mathrm{Hz}$ and below, with a peak between 0.05 and $0.09\mathrm{Hz}$. Similarly, correlations of the first angular mode ${\theta }_1$ are pronounced by a factor of 2 below $0.1\mathrm{Hz}$, and slightly decreased around $1\mathrm{Hz}$. This is a signature of the semiflexible nature of the polymers under investigation, since it refers to the buckling events occurring periodically during their rotation in shear. Concerning the second angular mode ${\theta }_2$ (not shown in Fig. 8), there is still a change from decreased to increased correlations visible when going from $f=1\text{to}0.1\mathrm{Hz}$ and below. The tension mode ${\Lambda }^0$ shows a strong tenfold peak at $0.1\mathrm{Hz}$ when compared to a Lorentzian that has been fitted to the generic short- and long-time behavior.

Figure 10

(Color online) Tumbling frequency of semiflexible polymers with different persistence length vs strength of shear flow. The change of the power-law form ${\dot{\gamma }}^{0.72}$ at ${\mathcal{l}}_p∕L=6.5$ toward ${\dot{\gamma }}^{2∕3}$ for a stiff rod indicates a crossover behavior between rods and flexible polymers. The amplitude shift appears due to the parametrization we have chosen (cf. Sec. 2B) and has not been removed for reasons of readability.