Direct Observation of the Dynamics of Semiflexible Polymers in Shear Flow

The flow behavior of polymeric liquids can be traced back to the complex conformational dynamics of polymer molecules in shear flow, which poses a major challenge to theory and experiment alike due to the inherently large number of degrees of freedom. Here we directly determine the configurational dynamics of individual actin filaments with varying lengths in a well defined shear geometry by combining microscopy, microfluidics, and a semiautomated moving stage. This allows the identification of the microscopic mechanisms and the derivation of an analytical model for the dynamics of individual filaments based on the balance of drag, bending, and stochastic forces.

Figure 1

(a) Successive images of two fluorescently labeled actin filaments of length $8\mu \mathrm{m}$ (left) and $16\mu \mathrm{m}$ (right) experiencing a mean shear rate of $\dot{\gamma }=6{\mathrm{s}}^{-1}$. The fluid motion in the filaments’ center of mass frame is sketched in the lower left corner (scale bar: $10\mu \mathrm{m}$). (b) The time course of the orientational angle $\varphi$ exhibits characteristic steps of $\pi$ for every interchange of filament orientation. The steps are followed by extended plateaus around multiples of $\pi$ of varying length which are identified with extended periods of flow alignment. (c) The mean curvature radius $R_U$ of the filament’s $U$-turn-like conformation does not change significantly for filaments of different contour lengths $L_c$. (filled circle $\dot{\gamma }\sim 6{\mathrm{s}}^{-1}$, open square $\dot{\gamma }\sim 12{\mathrm{s}}^{-1}$, open triangle $\dot{\gamma }\sim 19{\mathrm{s}}^{-1}$). (Inset) The end-to-end distance of a $16\mu \mathrm{m}$ filament shows alternating dips and plateaus, which reflect the stages of bending and full extension for every tumbling cycle. (d) Overlay of the conformations shown in (a). Both filaments follow a $U$-turn-like motion in which most of the filament segment remains flow aligned (scale bar: $10\mu \mathrm{m}$). See the movies in the Supplemental Material [28].

Figure 2

(a) Angular velocity $\dot{\varphi }$ vs orientation angle $\varphi$ of actin filaments with lengths of $4\mu \mathrm{m}$ (diamonds), $8\mu \mathrm{m}$ (crosses), and $16\mu \mathrm{m}$ (open circles). All filaments show quite similar behavior, supporting our prediction, Eq. (2) (black line). (b) The orientation angle $\varphi$ shows a predominantly deterministic motion during the advective phase, namely the passage between ${\varphi }_c$ and $\pi -{\varphi }_c$, followed by a diffusive phase between $\pi -{\varphi }_c$ and $\pi +{\varphi }_c$ where the realignment is superimposed by a rather diffusive motion. (c) and (d) Probability distributions of the orientational angle are shown for a quite stiff $4\mu \mathrm{m}$ actin (c) and $16\mu \mathrm{m}$ filament (d) at a shear rate $\dot{\gamma }\approx 4{\mathrm{s}}^{-1}$. The theoretical prediction for a rodlike polymer are overlayed as a solid line.

Figure 3

Dimensionless tumbling times ${\tau }_{\mathrm{T}}/{\tau }_r$ multiplied by $(D_{\mathrm{rod}}/D_r{)}^{1/3}$ of actin polymers at various shear rates plotted vs $\mathrm{Wi}=\dot{\gamma }{\tau }_r$. Contour length ranges are given in the legend. Tumbling times for the different shear rates decay as described by Eq. (4) (solid line); at strong flows a deviation can eventually be observed. Open squares show data points of nonequilibrium Brownian dynamics (NEBD) simulations for semiflexible polymers.

Figure 4

Tumbling events of a (a) semiflexible $34\mu \mathrm{m}$ and (b) quite stiff $4\mu \mathrm{m}$ filament in steady shear with $\dot{\gamma }=5{\mathrm{s}}^{-1}$. Multiple bending events occur during one tumbling event (see the Supplemental Material [28]). The fluid motion in the center of mass frame is sketched on the left [scale bar: $10\mu \mathrm{m}$ in (a) and $3\mu \mathrm{m}$ in (b)]. (c) Time course of orientational angle $\varphi$. (d) Time course of end-to-end-distance $R_{\mathrm{ee}}$.