Magnetic Tweezers - Force Calibration

Force Calibration

The determination of the force that is exerted by the magnetic field on the magnetic beads can be calculated considering thermal fluctuations of the bead in the horizontal plane: The problem is rotational symmetric with respect to the vertical axis; hereafter one arbitrarily picked direction in the symmetry plane is called . The analysis is the same for the direction orthogonal to the x-direction and may be used to increase precision. If the bead leaves its equilibrium position on the -axis by due to thermal fluctuations, it will be subjected to a restoring force that increases linearly with in the first order approximation. Considering only absolute values of the involved vectors it is geometrically clear that the proportionality constant is the force exerted by the magnets over the length of the molecule that keeps the bead anchored to the tethering surface:

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The equipartition theorem states that the mean energy that is stored in this "spring" is equal to per degree of freedom. Since only one direction is considered here, the potential energy of the system reads: . From this, a first estimate for the force acting on the bead can be deduced:

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For a more accurate calibration, however, an analysis in Fourier space is necessary. The power spectrum density of the position of the bead is experimentally available. A theoretical expression for this spectrum is derived in the following, which can then be fitted to the experimental curve in order to obtain the force exerted by the magnets on the bead as a fitting parameter. By definition this spectrum is the squared modulus of the Fourier transform of the position over the spectral bandwidth :

can be obtained considering the equation of motion for a bead of mass :

The term corresponds to the Stokes friction force for a spherical particle of radius in a medium of viscosity and is the restoring force which is opposed to the stochastic force due to the Brownian motion. Here, one may neglect the inertial term, because the system is in a regime of very low Reynolds number .

The equation of motion can be Fourier transformed inserting the driving force and the position in Fourier space:

This leads to:

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The power spectral density of the stochastic force can be derived by using the equipartition theorem and the fact that Brownian collisions are completely uncorrelated:

This corresponds to the Fluctuation-dissipation_theorem. With that expression, it is possible to give a theoretical expression for the power spectrum:

The only unknown in this expression, can be determined by fitting this expression to the experimental power spectrum. For more accurate results, one may subtract the effect due to finite camera integration time from the experimental spectrum before doing the fit.

Another force calibration method is to use the viscous drag of the microbeads: Therefore the microbeads are pulled through the viscous medium while recording their position. Since the Reynolds number for the system is very low, it is possible to apply Stokes law to calculate the friction force which is in equilibrium with the force exerted by the magnets:

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The velocity can be determined by using the recorded velocity values. The force obtained via this formula can then be related to a given configuration of the magnets, which may serve as a calibration.