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Optical tweezers and AFM cantilevers are often calibrated by fitting theirexperimental powerspectra of Brownian motion. We demonstrate here that if thisis done with typical weighted least-squares methods the result is a bias ofrelative size between -2/n and +1/n on the value of the fitted diffusioncoefficient. Here n is the number of power-spectra averaged over, so typicalcalibrations contain 10-20% bias. Both the sign and the size of the biasdepends on the weighting scheme applied. Hence, so do length-scale calibrationsbased on the diffusion coefficient. The fitted value for the characteristicfrequency is not affected by this bias. For the AFM then, force measurementsare not affected provided an independent length-scale calibration is available.For optical-tweezers there is no such luck, since the spring constant is foundas the ratio of the characteristic frequency and the diffusion coefficient. Wegive analytical results for the weight-dependent bias for the wide class ofsystems whose dynamics is described by a linear (integro-)differential equationwith additive noise, white or colored. Examples are optical tweezers withhydrodynamic self-interaction and aliasing, calibration of Ornstein-Uhlenbeckmodels in finance, models for cell-migration in biology, etc. Because the biastakes the form of a simple multiplicative factor on the fitted amplitude (e.g.the diffusion coefficient) it is straightforward to remove, and the user willneed minimal modifications to his or her favorite least-square fittingprograms. Results are demonstrated and illustrated using synthetic data, so wecan compare fits with known true values. We also fit some commonly occurringpower spectra once-and-for-all in the sense that we give their parameter valuesand associated error-bars as explicit functions of experimental power-spectralvalues.

review of scientific instruments online/review of scientific instruments

Power spectrum analysis with least-squares fitting: Amplitude bias and  its elimination, with application to optical tweezers and atomic force  microscope cantilevers