Axial Thermal Rotation of Slender Rods

Axial rotational diffusion of rodlike polymers is important in processes such as microtubule filament sliding and flagella beating. By imaging the motion of small kinks along the backbone of chains of DNA-linked colloids, we produce a direct and systematic measurement of axial rotational diffusivity of rods both in bulk solution and near a wall. The measured diffusivities decrease linearly with the chain length, irrespective of the distance from a wall, in agreement with slender-body hydrodynamics theory. Moreover, the presence of small kinks does not affect the chain’s axial diffusivity. Our system and measurements provide insights into fundamental axial diffusion processes of slender objects, which encompass a wide range of entities including biological filaments and linear polymer chains.

Figure 1

Snapshots of a 12-bead rigid DNA-linked chain under Brownian motion in aqueous solution near the bottom substrate. The snapshots (a)–(h) are taken 10 seconds apart, and the red dashed circles are highlighting the kink formed by the 7th, 8th, and 9th beads from the left. The unusual large size ($\varphi \approx 75°$) of this kink is only for demonstration, whereas the kinks used to measure the axial rotational diffusivities in this Letter are much smaller than this ($14°\le \varphi \le 25°$).

Figure 2

Measuring axial rotation of a 15-bead DNA-linked chain by tracking motions of kinks. (a) Snapshot of a 15-bead DNA-linked chain undergoing Brownian motion. (b) Geometry of a kink in 3D configuration and its projection. (c) Definitions of position coordinates, bond angle $\varphi (s_i)$, and tangent angle $\theta (s_i)=\pi -\varphi (s_i)$ along the chain.

Figure 3

(a) Overlay of bond angles $\varphi$ of all kinks in a 10-bead chain within 800 seconds. (b) Axial rotational angle $\Phi (t)$ measured from the sixth kink of the same chain in (a). (c) Histogram of the same angles in (b). (d) Typical plot of axial rotational angular MSD as a function lag time of 10-bead (black circle), 16-bead (blue circles), 30-bead (red circles), and 40-bead (green circles) chains in bulk.

Figure 4

Axial rotational diffusivities of rigid rods. Slender-body hydrodynamics theory predictions of axial rotational diffusivities in bulk (red line) and near a wall (black line) as a function of rod lengths. Experimental measured axial rotational diffusivities of rods in bulk (green circles) and near a wall (blue circles). Error bars are standard deviations of axial rotational diffusivities obtained from different kinks of the same chains. (Insets) Schematic illustrations of a chain near a wall (bottom left, where $h$ is the distance between the center of the beads and the wall) and a chain elevated by a magnetic field (top right). Distances are not to scale.