Torsional Stiffness of Single Superparamagnetic Microspheres in an External Magnetic Field

We used magnetic tweezers to measure the torsional stiffness of single micrometer-sized superparamagnetic spheres as a function of the applied magnetic field. By investigating the axial fluctuations of DNA-bound microspheres, we found that considerable rotational microsphere fluctuations can occur. Quantitative noise analysis allowed us to determine the rotational stiffness of individual microspheres, which was found to saturate at high magnetic fields. The saturation can be qualitatively explained considering the properties of the magnetic nanoparticles within the microsphere. Consequences for spatial resolution limits in single-molecule magnetic tweezer experiments and usage of DNA mechanics as a sensitive probe in magnetometry are discussed.

Figure 1

(a) Schematics of the experimental setup. (b) Time traces of the $z$ position of the magnetic microsphere at different forces. The black trace is for a fixed microsphere.

Figure 2

Noise along $z$ and force-extension relations for a microsphere with large (left) and small (right) off-center attachment $R_{\perp }$. (a, b) Measured $\sqrt{⟨z^2⟩}$ as a function of force (black squares). Solid lines are the calculated $\sqrt{⟨z^2⟩}$ with (upper line) and without correction (lower line) for low-pass filtering by the camera [Eq. (1)]. (c, d) $R_{\perp }$ measured at different forces by slowly rotating the magnetic field and following the circular track of the microsphere in the $xy$ plane. (e, f) Force-extension curves measured (black squares) and after correction by $Z_{\mathrm{corr}}$ (see inset) due to the off-center attachment (gray squares). The solid line is a fit with the extensible WLC model with a persistence length of $45\pm 5\mathrm{nm}$.

Figure 3

(a) Measured $\sqrt{⟨z^2⟩}$ at 20 pN over $R_{\perp }$. Solid lines represent the expected noise in the absence (gray) and presence (black) of rotational fluctuations of the microspheres for a $k_{\mathrm{torque}}$ of $100\mathrm{pN}\mu \mathrm{m}{\mathrm{rad}}^{-1}$. (b) $z$ deviations due to rotational displacements [see Fig. 2e inset] at 20 pN calculated from the difference in $Z_{\mathrm{corr}}$ at 20 pN and at low force where $R_{\perp }$ maximizes.

Figure 4

(a) Sketch of linear, torsional, and drag forces acting on the magnetic microsphere. (b) $k_{\mathrm{rot}}$ at 195 mT as a function of $R_{\perp }$. The red line is a fit to the data with $k_{\mathrm{rot}}=k_{\mathrm{torque}}/R^n$ with $k_{\mathrm{torque}}^{195\mathrm{mT}}=111\pm 4\mathrm{pN}\mu \mathrm{m}{\mathrm{rad}}^{-1}$ and $n=1.9\pm 0.2$. Inset: histogram of $k_{\mathrm{torque}}^{195\mathrm{mT}}$ with a mean of $94\pm 10\mathrm{pN}\mu \mathrm{m}{\mathrm{rad}}^{-1}$. (c) Overlay: all $k_{\mathrm{torque}}$ curves as a function of $B$ normalized at 195 mT (black dots). Open circles represent the mean for a given $B$. Solid lines: calculations for $C$ values of 1, 2, 4, and $7\mathrm{kJ}{\mathrm{m}}^{-3}$ [14]. Inset: $k_{\mathrm{torque}}$ curve of an individual microsphere. (d) $k_{\mathrm{torque}}$ estimated from angular displacements (see inset) over $k_{\mathrm{torque}}$ from noise measurements. Inset: angular displacement as function of the applied torque for a single microsphere. $k_{\mathrm{torque}}$ is estimated from a linear fit to the data at higher forces (solid line).