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Manuscript Title: Calibration of optical tweezers with non-spherical probes via high-resolution detection of Brownian motion
Authors: A. Butykai, F.M. Mor, R. Gaál, P. Domínguez-García, L. Forró, S. Jeney
Program title: PFMCal
Catalogue identifier: AEXH_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 196(2015)599
Programming language: MatLab 2011a (MathWorks Inc.).
Computer: General computer running MatLab (MathWorks Inc.), using Statistics Toolbox.
Operating system: Any which supports Matlab using Statistics Toolbox.
RAM: 10 MB
Keywords: Calibration of optical tweezers, non-spherical probe, Brownian motion, power spectral density (PSD), mean square displacement (MSD), velocity autocorrelation function (VAF).
PACS: 87.80.Cc, 05.40.Jc, 87.80.-y.
Classification: 3, 4.9, 18, 23.

Nature of problem:
Calibration of optical tweezers by measuring the Brownian motion of the trapped object. The voltage-to-displacement ratio of the detection system (the inverse of the sensitivity), the stiffness of the trap and the size of the bead are obtained via the simultaneous fitting of the power spectral density (PSD), mean square displacement (MSD) and velocity autocorrelation (VAF) functions calculated from the trajectory. The calibration can be performed for non-spherical probes as well.

Solution method:
Initialization points for all parameters are inferred from characteristic features of the statistical observables (PSD, MSD and VAF) based on the method developed by Grimm et al. in [1]. Theoretical functions for the PSD, the MSD and the VAF are calculated from the model of Brownian motion confined by a harmonic potential taking hydrodynamic interactions into consideration [2-4]. This calibration methodology has been successfully used in actual experiments with micro-spheres [5, 6]. Calculated functions are fitted to the measurement data via the Levenberg-Marquardt least square fitting routine available in the MatLab Optimalization Toolbox, using the nlinfit function. If the error to the measured data has been estimated, the corresponding data values can be weighted by the inverse of the standard error squared in order to eliminate bias introduced by heteroscedasticity. In order to increase robustness and avoid convergence to local minima, minimum search from multiple initial values in the vicinity of the first guess is possible.

Additional comments:
Input of the program is the experimental PSDexp, MSDexp, and VAFexp data points calculated from the measured x,y and z projections of the trajectory of the particle. The data may best be blocked and may optionally contain an error for each data point for better results. The data should be formatted into three columns: 1. independent variables (time or frequency array), 2. function values, 3. optionally, error values (eg. standard error calculated from the binning) to improve fitting efficiency. In case error values were not estimated, the third column should be filled with zeroes. The first row is reserved for a header, data is read from the second row. Data size should not be larger than a few hundreds of rows, otherwise, using larger blocks for binning is advised.
In order to observe the short time behavior of the Brownian motion, influenced by hydrodynamic effects, the sampling rate should be typically higher than 100 kHz. Total sampling time is typically tens of seconds, to achieve a good resolution in the frequency range. The optical axis of the laser, the microscope and the detector system should be co-aligned to exclude artificial crosstalk between the x,y and z channels of the position detector.

Running time:

[1] M. Grimm, T. Franosch, and S. Jeney. High-resolution detection of brow- nian motion for quantitative optical tweezers experiments. Phys. Rev. E, 86:021912, Aug 2012.
[2] R. Zwanzig and M. Bixon. Hydrodynamic theory of the velocity correlation function. Physical Review A, 2(5):2005, 1970.
[3] E. J. Hinch. Application of the langevin equation to fluid suspensions. Journal of Fluid Mechanics, 72:499-511, 12 1975.
[4] H. Clercx and P. Schram. Brownian particles in shear flow and harmonic potentials: A study of long-time tails. Physical Review A, 46(4):1942, 1992.
[5] P. Domínguez-García, F. Cardinaux, E. Bertseva, L. Forró, F. Scheffold, and S. Jeney, Accounting for inertia effects to access the high-frequency microrheology of viscoelastic fluids, Phys. Rev. E, vol. 90, p. 060301, Dec 2014.
[6] P. Domínguez-García, F. M. Mor, L. Forró, and S. Jeney, Exploiting the color of brownian motion for high-frequency microrheology of newtonian fluids, in SPIE NanoScience+ Engineering, pp. 881015-881015, Interna- tional Society for Optics and Photonics, 2013.