MatLab program for precision calibration of optical tweezers

Optical tweezers are used as force transducers in many types of experiments. The force they exert in a given experiment is known only after a calibration. Computer codes that calibrate optical tweezers with high precision and reliability in the (x,y)-plane orthogonal to the laser beam axis were written in MatLab (MathWorks Inc.) and are presented here. The calibration is based on the power spectrum of the Brownian motion of a dielectric bead trapped in the tweezers. Precision is achieved by accounting for a number of factors that affect this power spectrum. First, cross-talk between channels in 2D position measurements is tested for, and eliminated if detected. Then, the Lorentzian power spectrum that results from the Einstein-Ornstein-Uhlenbeck theory, is fitted to the low-frequency part of the experimental spectrum in order to obtain an initial guess for parameters to be fitted. Finally, a more complete theory is fitted, a theory that optionally accounts for the frequency dependence of the hydrodynamic drag force and hydrodynamic interaction with a nearby cover slip, for effects of finite sampling frequency (aliasing), for effects of anti-aliasing filters in the data acquisition electronics, and for unintended “virtual” filtering caused by the position detection system. Each of these effects can be left out or included as the user prefers, with user-defined parameters. Several tests are applied to the experimental data during calibration to ensure that the data comply with the theory used for their interpretation: Independence of x- and y-coordinates, Hooke's law, exponential distribution of power spectral values, uncorrelated Gaussian scatter of residual values. Results are given with statistical errors and covariance matrix.

Program summary

Title of program: tweezercalibCatalogue identifier: ADTVProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland.Program Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTVComputer for which the program is designed and others on which it has been tested: General computer running MatLab (MathWorks Inc.).Programming language used: MatLab (MathWorks Inc.). Uses “Optimization Toolbox” and “Statistics Toolbox”.Memory required to execute with typical data: Of order 4 times the size of the data file.High speed storage required: NoneNo. of lines in distributed program, including test data, etc.: 133 183No. of bytes in distributed program, including test data, etc.: 1 043 674Distribution format: tar gzip fileNature of physical problem: Calibrate optical tweezers with precision by fitting theory to experimental power spectrum of position of bead doing Brownian motion in incompressible fluid, possibly near microscope cover slip, while trapped in optical tweezers. Thereby determine spring constant of optical trap and conversion factor for arbitrary-units-to-nanometers for detection system.Method of solution: Elimination of cross-talk between quadrant photo-diode's output channels for positions (optional). Check that distribution of recorded positions agrees with Boltzmann distribution of bead in harmonic trap. Data compression and noise reduction by blocking method applied to power spectrum. Full accounting for hydrodynamic effects: Frequency-dependent drag force and interaction with nearby cover slip (optional). Full accounting for electronic filters (optional), for “virtual filtering” caused by detection system (optional). Full accounting for aliasing caused by finite sampling rate (optional). Standard non-linear least-squares fitting. Statistical support for fit is given, with several plots suitable for inspection of consistency and quality of data and fit.Restrictions on the complexity of the problem: Data should be positions of bead doing Brownian motion while held by optical tweezers. For high precision in final results, data should be time series measured over a long time, with sufficiently high experimental sampling rate: The sampling rate should be well above the characteristic frequency of the trap, the so-called corner frequency. Thus, the sampling frequency should typically be larger than 10 kHz. The Fast Fourier Transform applied requires the time series to contain 2ⁿ data points, and long measurement time is obtained with n>12-15. Finally, the optics should be set to ensure a harmonic trapping potential in the range of positions visited by the bead. The fitting procedure checks for harmonic potential.Typical running time: (Tens of) minutesUnusual features of the program: NoneReferences: The theoretical underpinnings for the procedure are found in [K. Berg-Sørensen, H. Flyvbjerg, Rev. Sci. Instrum. 75 (3) (2004) 594].