Calibration of optical tweezers with non-spherical probes via high-resolution detection of Brownian motion

Optical tweezers are commonly used and powerful tools to perform force measurements on the piconewton scale and to detect nanometer-scaled displacements. However, the precision of these instruments relies to a great extent on the accuracy of the calibration method. A well-known calibration procedure is to record the stochastic motion of the trapped particle and compare its statistical behavior with the theory of the Brownian motion in a harmonic potential. Here we present an interactive calibration software which allows for the simultaneous fitting of three different statistical observables (power spectral density, mean square displacement and velocity autocorrelation function) calculated from the trajectory of the probe to enhance fitting accuracy. The fitted theory involves the hydrodynamic interactions experimentally observable at high sampling rates. Furthermore, a qualitative extension is included in our model to handle the thermal fluctuations in the orientation of optically trapped asymmetric objects. The presented calibration methodology requires no prior knowledge of the bead size and can be applied to non-spherical probes as well. The software was validated on synthetic and experimental data. Program summary Program title: PFMCal Catalogue identifier: AEXH_v1_0 Program summary URL: http://cpc.cs.qub.ac.ukisummaries/AEXH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 206,399 No. of bytes in distributed program, including test data, etc.: 10,319,465 Distribution format: tar.gz Programming language: MatLab 2011a (Math Works Inc.). Computer: General computer running MatLab (MathWorks Inc.), using Statistics Toolbox. Operating system: Any which supports Matlab using Statistics Toolbox. RAM: 10 MB 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 VAF(exp) 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 (e.g. standard error calculated from the binning) to improve fitting efficiency. In case error values were not estimated, the third column should be filled with zeros. 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: Seconds (C) 2015 Elsevier B.V. All rights reserved.

Motion of a colloidal particle in an optical trap

László Forró, Sylvia Jeney, Andrzej Kulik, Branimir Lukic

Thermal position fluctuations of a colloidal particle in an optical trap are measured with microsecond resolution using back-focal-plane interferometry. The mean-square displacement MSD(t) and power spectral density are in excellent agreement with the theory for a Brownian particle in a harmonic potential that accounts for hydrodynamic memory effects. The motion of a particle is dominated at short times by memory effects and at longer times by the potential. We identify the time below which the particle's motion is not influenced by the potential, and find it to be approximately tau_k/20, where tau_k is the relaxation time of the restoring force of the potential. This allows us to exclude the existence of free diffusive motion (MSD(t) proprtional t) even for a sphere with a radius as small as 0.27 µm in a potential as weak as 1.5 µN/m. As the physics of Brownian motion can be used to calibrate an optical trap, we show that neglecting memory effects leads to an underestimation of more than 10% in the detector sensitivity and the trap stiffness for an experiment with a micrometer-sized particle and a sampling frequency above 200 kHz. Furthermore, these calibration errors increase in a nontrivial fashion with particle size, trap stiffness, and sampling frequency. Finally, we present a method to evaluate calibration errors caused by memory effects for typical optical trapping experiments.

2007

Developing luminescent Brownian probes for near-field investigations of the intracellular environment

Flavio Maurizio Mor

Life sciences have a constantly growing need for novel methodological approaches suitable to investigate the intracellular environment with increased temporal and spatial resolutions. Recently, optical near-field probes, such as laser-irradiated pointed metal, uncoated/metal-coated tapered optical fibers, as well as nano-emitters, such as single molecules or nanoparticles, have attracted increasing attention as key components of high-resolution microscopes. In parallel, super-resolution fluorescence microscopy, operating in far-field regime and overcoming the light diffraction limit, has markedly improved imaging resolution. Although both near- and far-field optical approaches drastically improve image resolution, they still represent numerous drawbacks, such as, technical difficulties concerning probe preparation (near-field) or potential damaging through very high light intensities (far-field). The aforementioned shortcomings of high-resolution optical microscopy can be overcome to a great extent by photonic force microscopy (PFM). PFM employs a strongly focused near-infrared (NIR) laser light to hold a dielectric or metallic particle as a local probe. Such an optical trap enables one to ‘cage’ a mesoscopic particle and track its three-dimensional (3D) thermal fluctuations in the surrounding environment, e.g. a viscous liquid. Therefore, besides offering near-field 3D imaging, PFM reports also on other important quantities concerning local mechanical properties, including force and viscosity as well as dynamical properties of the surrounding medium. This additional information is derived from the careful analysis of the jittery motion (so-called Brownian motion) of the trapped single-particle probe that collides with thermally-activated surrounding molecules. In this thesis, we first study in detail the Brownian motion of a single spherical particle that is confined within the harmonic potential of the optical trap. To this end, we optimize the NIR light path and electronic noise floor of our custom-built PFM set-up for detecting and quantifying resonances in the Brownian motion. These resonances arise from the coupling between the hydrodynamic memory and strong strength of the optical trap. Due to the high sensitivity of the short-time dynamics, the size of the particle can be measured by simultaneous fitting of the velocity autocorrelation function and power spectral density of its thermal fluctuations. In order to choose the best-suitable spherical probe for a given experiment in PFM, computational modeling based on a Matlab toolbox is performed. The generalized Lorenz-Mie theory is computed using the T -matrix method for various experimental conditions, including changes in the size and refractive index of the sphere, as well as different laser polarization states. The axial equilibrium position is examined to predict its location compared with the position of the laser focus. Optical forces acting on the sphere are investigated in 3D to highlight, in particular, the limits of the perfectly harmonic potential assumption. These limits are quantified and discussed. Subsequently, we identify different types of inorganic nano/micro-sized particles that can be trapped by the NIR laser of the PFM and used simultaneously as mechanical and near-field light sources. To this end, we take advantage of the visible light emission from single probes confined in the optical trap and excited by the NIR laser light. The emphasis is on particles revealing nonlinear optical properties. In particular, nonlinear optical field enhancement resulting from excitation of surface plasmon resonance (SPR) on trapped gold particles of 200 nm is demonstrated by measuring the emission of that second-harmonic generation (SHG). Furthermore, we show, under optical trapping conditions, SHG emission from single potassium niobate (KNbO3) nano/micro-sized particles. We also report on the multicolor upconversion luminescence (UCL), at the single particle level, for randomly shaped crystals of sodium yttrium fluoride codoped with ytterbium and erbium, NaYF4:Yb,Er (UCC), when they are held in the optical trap. In parallel, 3D Brownian fluctuations of a single trapped particle are quantified. Moreover, luminescence resonance energy transfer (LRET) between a KNbO3 particle or an UCC and molecules of an organic dye (rose bengal) adsorbed on their surface is highlighted and analyzed. Finally, randomly shaped UCCs and hexagonal nanoplates are characterized in detail at the single particle level. The hexagonal β phase in the polycrystalline and monocrystalline crystals is identified, in both randomly and hexagonally shaped particles by electron microscopy. UCL emission from the trapped crystal is quantified in terms of power and found to be dependent on the laser power density, size and shape of the particle as well as its orientation within the trap. Most importantly, the different colors in the UCL spectrum do not vary with the same trend for different orientations of the randomly or hexagonally shaped crystal upon trapping. This suggests the existence of crystalline anisotropy-dependent UCL. Thereby, we should be able to design the best-suited particle probe for LRET experiments with subwavelength resolution.

EPFL2013

Brownian motion studied with an optical trap

Branimir Lukic

This thesis describes measurements of Brownian motion of a colloidal particle using optical trapping. Two aspects are investigated: (i) influence of inertial effects on Brownian motion, and (ii) effect of the optical trap on Brownian motion. The first part describes the experimental setup used: in short, a focused infrared laser beam is used to create an optical trap, and also provides the light source for the position detection. A single particle is trapped by the laser if it is brought near the focus, but it still moves due to Brownian motion. By analyzing the power spectral density of the position signal, the detection limit of the experimental setup is estimated to be 2 µs. The second part contains a study of inertial effects in Brownian motion of a colloidal particle at the microsecond time scale. Measurements are performed for three cases: a particle in a bulk fluid, a particle next to a wall and a particle in a confined fluid. For a particle in a bulk fluid, power law decay of t-3/2 is observed in the particle's velocity autocorrelation function. This power law decay has its origin in the non-negligible fluid inertia. A small effect caused by the particle's inertia is also observed. For the case of a particle placed next to a wall, a faster decay with power law of t-5/2 is observed in the velocity correlations parallel to the wall. Finally, for a particle in a confined fluid, absence of power law decay is observed, and the data instead agree with the model of an exponential decay of velocity correlations. These experimental observations are in accord with the theoretical predictions and show that the effect of the fluid's inertia is most significant in the case of a particle in a bulk fluid, less significant in the case of a particle next to a wall and completely absent in the case of a particle in a confined fluid. The third part is devoted to the characterization of the Brownian motion of a colloidal particle in an optical trap. Data for the mean-square displacement 〈[Δx(t)]2〉 and power spectral density are in excellent agreement with the theory for a Brownian particle in a harmonic potential, which also accounts for inertial effects. The motion of the particle is dominated by inertial effects at short times and by the trap potential at longer times. We find the time below which the particle's motion is not influenced by the potential to be approximately τκ/20, where τκ is the relaxation time of the restoring force of the potential. This allows us to exclude the existence of free diffusive motion, 〈[Δx(t)]2〉 ∝ t , even for a spherical particle with a radius as small as 0.27 µm in a potential with a spring constant as small as 1.5 µN/m. In the experiment with a micron-sized sphere in the weakest trap potential, estimation of the time below which the particle motion is not influenced by the potential gives τκ/20 ≈ 100 µs. In the time range 2-100 µs, the experimental setup then functions as a position detector probing the motion of a free Brownian particle. Moreover, it is shown that such a detector is achieved for any sphere size in the range between 0.53 µm to 4.16 µm. Hence, the results open up a possibility for using optically trapped Brownian particle as a local reporter its environment on microsecond time scales. This technique could be applied in more complex environments, like polymer networks, cell interiors or bacterial solutions.