Extending the range for force calibration in magnetic tweezers

Magnetic tweezers are a wide-spread tool used to study the mechanics and the function of a large variety of biomolecules and biomolecular machines. This tool uses a magnetic particle and a strong magnetic field gradient to apply defined forces to the molecule of interest. Forces are typically quantified by analyzing the lateral fluctuations of the biomolecule-tethered particle in the direction perpendicular to the applied force. Since the magnetic field pins the anisotropy axis of the particle, the lateral fluctuations follow the geometry of a pendulum with a short pendulum length along and a long pendulum length perpendicular to the field lines. Typically, the short pendulum geometry is used for force calibration by power-spectral-density (PSD) analysis, because the movement of the bead in this direction can be approximated by a simple translational motion. Here, we provide a detailed analysis of the fluctuations according to the long pendulum geometry and show that for this direction, both the translational and the rotational motions of the particle have to be considered. We provide analytical formulas for the PSD of this coupled system that agree well with PSDs obtained in experiments and simulations and that finally allow a faithful quantification of the magnetic force for the long pendulum geometry. We furthermore demonstrate that this methodology allows the calibration of much larger forces than the short pendulum geometry in a tether-length-dependent manner. In addition, the accuracy of determination of the absolute force is improved. Our force calibration based on the long pendulum geometry will facilitate high-resolution magnetic-tweezers experiments that rely on short molecules and large forces, as well as highly parallelized measurements that use low frame rates.