Biological Magnetometry: Torque on Superparamagnetic Beads in Magnetic Fields

Superparamagnetic beads are widely used in biochemistry and single-molecule biophysics, but the nature of the anisotropy that enables the application of torques remains controversial. To quantitatively investigate the torques experienced by superparamagnetic particles, we use a biological motor to rotate beads in a magnetic field and demonstrate that the underlying potential is $\pi$ periodic. In addition, we tether a bead to a single DNA molecule and show that the angular trap stiffness increases nonlinearly with magnetic field strength. Our results indicate that the superparamagnetic beads’ anisotropy derives from a nonuniform intrabead distribution of superparamagnetic nanoparticles.

Figure 1

(a) A single-domain NP with a (uniaxial) anisotropy axis (dashed line), magnetization $\overrightarrow{M}$ (green arrow), in an external magnetic field $\overrightarrow{B}$ (red arrow). ${\theta }_1$ (${\theta }_2$) is the angle between the magnetization $\overrightarrow{M}$ and the anisotropy axis (magnetic field $\overrightarrow{B}$). (b) A superparamagnetic bead containing multiple NPs. The net magnetic dipole moment of the bead (solid green arrow) comprises the contribution of all NPs inside it. (c)–(f) The angle ${\theta }_2$ as a function ${\theta }_{\mathrm{NP}}$. The NP is rotated from ${\theta }_{\mathrm{NP}}=0$ to $\pi$, back to $-\pi$, and again to 0. The magnetic field increases from blue to green to red. The dashed diagonal line (horizontal line) corresponds to ${\theta }_2={\theta }_{\mathrm{NP}}$ (${\theta }_2=0$, which holds for a pure paramagnet). (c) SW model. For intermediate fields (green), hysteresis occurs. (d) The limit of the SW model in which NPs behave like PDMs. (e) The special case of the SW model in which the individual NPs behave like SPMs. (f) Model in which NPs are approximated as 2SF magnets. For fields higher than the coercive field (green and red), hysteresis occurs. Details of the calculations for ${\theta }_2$ are in Sections 1–4 in the Supplemental Material [17].

Figure 2

(a) Schematic of the experimental setup (not to scale) for scanning the magnetic potential experienced by a superparamagnetic bead. The magnetic bead is connected to the flagellar motor (red) via the hook (black). The two magnets are $5\times 5\times 5\mathrm{mm}$, and the gap between the two magnets is 1 mm. (b) Top view of the experimental configuration (not to scale). (c) Camera image of the experimental configuration (scale bar $1\mu \mathrm{m}$). (d) Predictions for the different models, offset in the angular coordinate for clarity. For the SW model and the 2SF model, the predictions at high fields are plotted. (e) Sample $(x,y)$ trace of the magnetic bead position at a field strength of 15 mT. Different colors indicate data taken at different rotational positions of the magnetic trap. (f) Same data as in (e), but represented as time traces of the angular position of the bead (left) and in corresponding histograms (right). The inset is a zoom on the blue angular trace in which the $y$ axis gives the angular position in turns without backfolding after one turn.

Figure 3

(a) Schematics of the experimental configuration (not to scale) for determining the torsional stiffness experienced by a superparamagnetic bead. (b) Camera image of the experimental configuration showing the superparamagnetic bead (diameter $=1\mu \mathrm{m}$) and the nonmagnetic marker bead ($\text{diameter}=450\mathrm{nm}$). (c) Top view of the experimental configuration (not to scale). (d) Time traces of the angular orientation of a M-270 bead (left) and corresponding histograms (right). The traces are offset in the angular coordinate for clarity. The field amplitude increases from blue (2 mT) to purple (8 mT) to green (187 mT). (e) Predictions of ${\kappa }_{\theta }$ versus $|\overrightarrow{B}|$ for the different models. The inset is a zoom on the predicted values of ${\kappa }_{\theta }$ at low $|\overrightarrow{B}|$. (f) Experimentally measured ${\kappa }_{\theta }$ experienced by MyOne™ beads versus $|\overrightarrow{B}|$. (g) Experimentally measured ${\kappa }_{\theta }$ experienced by M-270 beads versus $|\overrightarrow{B}|$. The inset is a zoom on experimentally measured values of ${\kappa }_{\theta }$ at low $|\overrightarrow{B}|$. In (f) and (g), different colors correspond to different beads; the solid lines are fits to the data based on the SPM model; and data points indicate the mean and standard deviation of four experimental repeats. The dashed lines in the inset of (g) are fits based on the SW model.