Kirstine Berg-Sørensen

Power spectrum analysis for optical tweezers

The force exerted by an optical trap on a dielectric bead in a fluid is often found by fitting a Lorentzian to the power spectrum of Brownian motion of the bead in the trap. We present explicit functions of the experimental power spectrum that give the values of the parameters fitted, including error bars and correlations, for the best such χ2 fit in a given frequency range. We use these functions to determine the information content of various parts of the power spectrum, and find, at odds with lore, much information at relatively high frequencies. Applying the method to real data, we obtain perfect fits and calibrate tweezers with less than 1% error when the trapping force is not too strong. Relatively strong traps have power spectra that cannot be fitted properly with any Lorentzian, we find. This underscores the need for better understanding of the power spectrum than the Lorentzian provides. This is achieved using old and new theory for Brownian motion in an incompressible fluid, and new results for ...

In vivo anomalous diffusion and weak ergodicity breaking of lipid granules.

Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion.

Unintended filtering in a typical photodiode detection system for optical tweezers

We characterize the frequency-dependent response of a photo detection system based on a Si-PIN photodiode and a laser with wavelength 1064 nm, a system commonly used with optical tweezers. We chopped the laser beam with chopper frequencies from 200 Hz to 14 kHz, and found an exponentially delayed response of the detection system with a characteristic delay time of ∼20 μs. The physical mechanism causing this time delay is silicon’s transparency to 1064 nm light: Photons are absorbed and create charge carriers not only in the diode’s depletion layer, where they are detected within nano-seconds, but predominantly in the n-layer, where they remain undetected till transported out by thermal diffusion. The diode’s response is dominated by this delay which can be characterized as a first-order low-pass filter with a 3dB-frequency of 8–9 kHz, depending on laser intensity. Measurements exploiting frequencies near or above this 3dB-frequency must be corrected for this unintended filter effect. We describe how to do...

Improved axial position detection in optical tweezers measurements

We investigate the axial position detection of a trapped microsphere in an optical trap by using a quadrant photodiode. By replacing the photodiode with a CCD camera, we obtain detailed information on the light scattered by the microsphere. The correlation of the interference pattern with the axial position displays complex behavior with regions of positive and negative interference. By analyzing the scattered light intensity as a function of the axial position of the trapped sphere, we propose a simple method to increase the sensitivity and control the linear range of axial position detection.

Calibration of trapping force and response function of optical tweezers in viscoelastic media

At present, a major obstacle to the quantitative application of optical tweezers as a force spectrometer in living cells is the lack of a method to calibrate the tweezers. Calibration with approved methods such as the power spectrum method (Berg-Sorensen and Flyvbjerg 2004 Rev. Sci. Instrum. 75 594; Berg-Sorensen et al 2006 Rev. Sci. Instrum. 77 063106) is not possible as the viscoelastic properties of the bio-active medium are a priori unknown. Here, we present an approach that neither requires explicit assumptions about the size of the trapped particle nor about the viscoelastic properties of the medium. Instead, the interaction between the medium and the trapped particle is described in a general manner, through velocity and acceleration memory. Our method is applicable to general, at least locally homogeneous, viscoelastic media. The procedure combines active and passive approaches by the application of Onsagers regression hypothesis. It allows extraction of the trapping stiffness κ of the optical tweezers and of the response function χ(ω), which is the frequency-dependent effective inverse spring constant of the system. Finally, information about the viscoelastic properties of the medium may also be found. To test the method, we have performed simulations in which the system is driven sinusoidally. These simulations serve as an example of how to deal with real experimental data. For realistic parameters, we calibrate the trap stiffness κ with ~1% stochastic error.

Manipulation and Motion of Organelles and Single Molecules in Living Cells

The biomolecule is among the most important building blocks of biological systems, and a full understanding of its function forms the scaffold for describing the mechanisms of higher order structures as organelles and cells. Force is a fundamental regulatory mechanism of biomolecular interactions driving many cellular processes. The forces on a molecular scale are exactly in the range that can be manipulated and probed with single molecule force spectroscopy. The natural environment of a biomolecule is inside a living cell, hence, this is the most relevant environment for probing their function. In vivo studies are, however, challenged by the complexity of the cell. In this review, we start with presenting relevant theoretical tools for analyzing single molecule data obtained in intracellular environments followed by a description of state-of-the art visualization techniques. The most commonly used force spectroscopy techniques, namely optical tweezers, magnetic tweezers, and atomic force microscopy, are described in detail, and their strength and limitations related to in vivo experiments are discussed. Finally, recent exciting discoveries within the field of in vivo manipulation and dynamics of single molecule and organelles are reviewed.

Active-passive calibration of optical tweezers in viscoelastic media

In order to use optical tweezers as a force measuring tool inside a viscoelastic medium such as the cytoplasm of a living cell, it is crucial to perform an exact force calibration within the complex medium. This is a nontrivial task, as many of the physical characteristics of the medium and probe, e.g., viscosity, elasticity, shape, and density, are often unknown. Here, we suggest how to calibrate single beam optical tweezers in a complex viscoelastic environment. At the same time, we determine viscoelastic characteristics such as friction retardation spectrum and elastic moduli of the medium. We apply and test a method suggested [M. Fischer and K. Berg-Sørensen, J. Opt. A, Pure Appl. Opt. 9, S239 (2007)], a method which combines passive and active measurements. The method is demonstrated in a simple viscous medium, water, and in a solution of entangled F-actin without cross-linkers.

Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells

With the success of in vitro single-molecule force measurements obtained in recent years, the next step is to perform quantitative force measurements inside a living cell. Optical traps have proven excellent tools for manipulation, also in vivo, where they can be essentially non-invasive under correct wavelength and exposure conditions. It is a pre-requisite for in vivo quantitative force measurements that a precise and reliable force calibration of the tweezers is performed. There are well-established calibration protocols in purely viscous environments; however, as the cellular cytoplasm is viscoelastic, it would be incorrect to use a calibration procedure relying on a viscous environment. Here we demonstrate a method to perform a correct force calibration inside a living cell. This method (theoretically proposed in Fischer and Berg-Sørensen (2007 J. Opt. A: Pure Appl. Opt. 9 S239)) takes into account the viscoelastic properties of the cytoplasm and relies on a combination of active and passive recordings of the motion of the cytoplasmic object of interest. The calibration procedure allows us to extract absolute values for the viscoelastic moduli of the living cell cytoplasm as well as the force constant describing the optical trap, thus paving the way for quantitative force measurements inside the living cell. Here, we determine both the spring constant of the optical trap and the elastic contribution from the cytoplasm, influencing the motion of naturally occurring tracer particles. The viscoelastic moduli that we find are of the same order of magnitude as moduli found in other cell types by alternative methods.

The colour of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation

One hundred years after Einstein modelled Brownian motion, a central aspect of this motion in incompressible fluids has not been verified experimentally: the thermal noise that drives the Brownian particle, is not white, as in Einsteins simple theory. It is slightly coloured, due to hydrodynamics and the fluctuation?dissipation theorem. This theoretical result from the 1970s was prompted by computer simulation results in apparent violation of Einsteins theory. We discuss how a direct experimental observation of this colour might be carried out by using optical tweezers to separate the thermal noise from the particles dynamic response to it. Since the thermal noise is almost white, very good statistics is necessary to resolve its colour. That requires stable equipment and long recording times, possibly making this experiment one for the future only. We give results for experimental requirements and for stochastic errors as functions of experimental window and measurement time, and discuss some potential sources of systematic errors.

Optical manipulation of single molecules in the living cell

Optical tweezers are the only nano-tools capable of manipulating and performing force-measurements on individual molecules and organelles within the living cell without performing destructive penetration through the cell wall and without the need for inserting a non-endogenous probe. Here, we describe how optical tweezers are used to manipulate individual molecules and perform accurate force and distance measurements within the complex cytoplasm of the living cell. Optical tweezers can grab individual molecules or organelles, if their optical contrast to the medium is large enough, as is the case, e.g., for lipid granules or chromosomes. However, often the molecule of interest is specifically attached to a handle manipulated by the optical trap. The most commonly used handles, their insertion into the cytoplasm, and the relevant micro-rheology of the cell are discussed here and we also review recent and exciting results achieved through optical force manipulation of individual molecules in vivo.