Optical trap stiffness in the presence and absence of spherical aberrations

On-Axis Optical Trapping with Vortex Beams: The Role of the Multipolar Decomposition

Optical trapping is a well-established, decades old technology with applications in several fields of research. The most common scenario deals with particles that tend to be centered on the brightest part of the optical trap. Consequently, the optical forces keep the particle away from the dark zones of the beam. However, this is not the case when a focused doughnut-shaped beam generates on-axis trapping. In this system, the particle is centered on the intensity minima of the laser beam and the bright annular part lies on the periphery of the particle. Researchers have shown great interest in this phenomenon due to its advantage of reducing light interaction with trapped particles and the intriguing increase in the trapping strength. This work presents experimental and theoretical results that extend the analysis of on-axis trapping with light vortex beams. Specifically, in our experiments, we trap micron-sized spherical silica (SiO2) particles in water and we measure, through the power spectrum density method, the trap stiffness constant κ generated by vortex beams with different topological charge orders. The optical forces are calculated from the exact solutions of the electromagnetic fields provided by the generalized Lorentz-Mie theory. We show a remarkable agreement between the theoretical prediction and the experimental measurements of κ. Moreover, our numerical model gives us information about the electromagnetic fields inside the particle, offering valuable insights into the influence of the electromagnetic fields present in the vortex beam trapping scenario.

Introduction to Optical Tweezers: Background, System Designs, and Applications

Optical tweezers are a means to manipulate objects with light. With the technique, microscopically small objects can be held and steered, allowing for accurate measurement of the forces applied to these objects. Optical tweezers can typically obtain a nanometer spatial resolution, a picoNewton force resolution, and a millisecond time resolution, which makes the technique well suited for the study of biological processes from the single-cell down to the single-molecule level. In this chapter, we aim to provide an introduction to the use of optical tweezers for single-molecule analyses. We start from the basic principles and methodology involved in optical trapping, force calibration, and force measurements. Next, we describe the components of an optical tweezers setup and their experimental relevance. Finally, we will provide an overview of the broad applications in context of biological research, with the emphasis on the measurement modes, experimental assays, and possible combinations with fluorescence microscopy techniques.

Effect of aberrations on the beam quality factor of Hermite-Gauss beams

The effect of aberrations on the beam quality factor (M2) of Hermite-Gauss (HG) beams is examined. Using the method of moments, we derive closed-form analytical expressions of M2 due to astigmatism and spherical aberration. Our analysis reveals that the radius of the HG beams plays a significant role in determining the effect of the aberrations on M2. For each aberration, we establish a critical width that separates the region where M2 changes infinitesimally from the region where it changes sharply. The analytical results are validated through numerical simulations.

Differential displacement measurement of the levitated particle using D-shaped mirrors in the optical tweezers

Displacement measurement using a D-shaped mirror is a key technology in optical tweezers, which have emerged as an important tool for precision measurement. In this paper, we first study the influences of installation errors for the D-shaped mirror on the displacement measurement. The calibration factor and sensitivity of the different installation parameters are quantified. The results show that the variation of the calibration factor obeys the cosine curve with the angle error, and the sensitivity increases exponentially with the translation error. Besides, we find that the translation error will also lead to crosstalk between transverse and axial displacement. Our work will contribute to improving the performance of optical tweezers for the application in precision measurement and basic physics.

Effect of shear flow on the hydrodynamic drag force of a spherical particle near a wall evaluated using optical tweezers and microfluidics

The hydrodynamic drag force on a spherical particle in shear flow near-wall is investigated using optical tweezers and microfluidics. Simple shear flow is applied using a microfluidic channel at different volumetric flow rates. The hydrodynamic drag force exerted on the particle is detected from the displacement of the trapped particle. The effect of the wall is obtained from the force balance of the trapping and hydrodynamic drag force employing the exact solution of the theoretical model using the lubrication theory for a sphere near the wall. Here, we report the experimentally obtained hydrodynamic drag force coefficient under the influence of shear flow. The drag correction factor increases with decreasing distance from the wall due to the effect of the wall surface. We found that the calculated hydrodynamic drag force coefficient is in quantitative comparison with the theoretical prediction for a shear flow past a sphere near-wall. This study provides a straightforward investigation of the effect of the shear flow on the hydrodynamic drag force coefficient on a particle near the wall. Furthermore, these pieces of information can be used in various applications, particularly in optimizing microfluidic designs for mixing and separations of particles or exploiting the formation of the concentration gradient of particles perpendicular to flow directions caused by the non-linear hydrodynamic interactions.

Optimization of Optical Trapping and Laser Interferometry in Biological Cells

Optical trapping and laser interferometry enable the non-invasive manipulation of colloids, which can be used to investigate the microscopic mechanics of surrounding media or bound macromolecules. For efficient trapping and precise tracking, the sample media must ideally be homogeneous and quiescent whereas such conditions are usually not satisfied in vivo in living cells. In order to investigate mechanics of the living-cell interior, we introduced (1) the in-situ calibration of optical trapping and laser interferometry, and (2) 3-D feedback control of a sample stage to stably track a colloidal particle. Investigating systematic errors that appear owing to sample heterogeneity and focal offsets of a trapping laser relative to the colloidal probe, we provide several important caveats for conducting precise optical micromanipulation in living cells. On the basis of this study, we further improved the performance of the techniques to be used in cells, by optimizing the position sensitivity of laser interferometry and the stability of the feedback simultaneously.

Optical Assembling of Micro-Particles at a Glass–Water Interface with Diffraction Patterns Caused by the Limited Aperture of Objective

Optical tweezers can manipulate micro-particles, which have been widely used in various applications. Here, we experimentally demonstrate that optical tweezers can assemble the micro-particles to form stable structures at the glass–solution interface in this paper. Firstly, the particles are driven by the optical forces originated from the diffraction fringes, which of the trapping beam passing through an objective with limited aperture. The particles form stable ring structures when the trapping beam is a linearly polarized beam. The particle distributions in the transverse plane are affected by the particle size and concentration. Secondly, the particles form an incompact structure as two fan-shaped after the azimuthally polarized beam passing through a linear polarizer. Furthermore, the particles form a compact structure when a radially polarized beam is used for trapping. Thirdly, the particle patterns can be printed steady at the glass surface in the salt solution. At last, the disadvantage of diffraction traps is discussed in application of optical tweezers. The aggregation of particles at the interfaces seriously affects the flowing of particles in microfluidic channels, and a total reflector as the bottom surface of sample cell can avoid the optical tweezers induced particle patterns at the interface. The optical trapping study utilizing the diffraction gives an interesting method for binding and assembling microparticles, which is helpful to understand the principle of optical tweezers.

New Detector Sensitivity Calibration and the Calculation of the Interaction Force between Particles Using an Optical Tweezer

We propose a new approach to calculate the sensitivity factor of the detector in optical tweezers. In this work, we used a charge-coupled device (CCD) camera and a quadrant photodiode (QPD) for the extraction of the various positions occupied by the trapped object (in this case, silica beads of different diameters). Image-J software and the Boltzmann statistical method were then used to estimate the sensitivity of the detector. Silica beads of diameter 0.8 µm, 2 µm, a system of 2 µm bead stuck to 4.5 µm one and another system of 2 µm beads stuck to 2 µm one, were studied. This work contributes significantly to making better calibration of the detector without taking into account the geometry of the object imprisoned in the optical trap. We further developed an approach to calculate the interaction force between two microbeads. This approach does not require any knowledge of solvent viscosity and works for all types of samples.

Experimental characterization and modeling of optical tweezer particle handling dynamics

We report a new framework for a quantitative understanding of optical trapping (OT) particle handling dynamics. We present a novel three-dimensional particle-based model that includes optical, hydrodynamic, and inter-particle forces. This semi-empirical colloid model is based on an open-source simulation code known as LAMMPS (large-scale atomic/molecular massively parallel simulator) and properly recapitulates the full OT force profile beyond the typical linear approximations valid near the trap center. Simulations are carried out with typical system parameters relevant for our experimental holographic optical trapping (HOT) system, including varied particle sizes, trap movement speeds, and beam powers. Furthermore, we present a new experimental method for measuring both the stable and metastable boundaries of the optical force profile to inform or validate the model's underlying force profile. We show that our framework is a powerful tool for accurately predicting particle behavior in a practical experimental OT setup and can be used to characterize and predict particle handling dynamics within any arbitrary OT force profile.

Aberration correction in holographic optical tweezers using a high-order optical vortex

Holographic optical tweezers are a powerful optical trapping and manipulation tool in numerous applications such as life science and colloidal physics. However, imperfections in the spatial light modulator and optical components of the system will introduce detrimental aberrations to the system, thereby degrading the trapping performance significantly. To address this issue, we develop an aberration correction technique by using a high-order vortex as the correction metric. The optimal Zernike polynomial coefficients for quantifying the system aberrations are determined by comparing the distorted vortex and the ideal one. Efficiency of the proposed method is demonstrated by comparing the optical trap intensity distribution, trap stiffness, and particle dynamics in a Gaussian trap and an optical vortex trap, before and after aberration corrections.

Introduction to Optical Tweezers: Background, System Designs, and Commercial Solutions

Optical tweezers are a means to manipulate objects with light. With the technique, microscopically small objects can be held and steered, while forces on the trapped objects can be accurately measured and exerted. Optical tweezers can typically obtain a nanometer spatial resolution, a picoNewton force resolution, and a millisecond time resolution, which makes them excellently suited to study biological processes from the single-cell down to the single-molecule level. In this chapter, we will provide an introduction on the use of optical tweezers in single-molecule approaches. We will introduce the basic principles and methodology involved in optical trapping, force calibration, and force measurements. Next we describe the components of an optical tweezers setup and their experimental relevance in single-molecule approaches. Finally, we provide a concise overview of commercial optical tweezers systems. Commercial systems are becoming increasingly available and provide access to single-molecule optical tweezers experiments without the need for a thorough background in physics.

Feedback-tracking microrheology in living cells

Living cells are composed of active materials, in which forces are generated by the energy derived from metabolism. Forces and structures self-organize to shape the cell and drive its dynamic functions. Understanding the out-of-equilibrium mechanics is challenging because constituent materials, the cytoskeleton and the cytosol, are extraordinarily heterogeneous, and their physical properties are strongly affected by the internally generated forces. We have analyzed dynamics inside two types of eukaryotic cells, fibroblasts and epithelial-like HeLa cells, with simultaneous active and passive microrheology using laser interferometry and optical trapping technology. We developed a method to track microscopic probes stably in cells in the presence of vigorous cytoplasmic fluctuations, by using smooth three-dimensional (3D) feedback of a piezo-actuated sample stage. To interpret the data, we present a theory that adapts the fluctuation-dissipation theorem (FDT) to out-of-equilibrium systems that are subjected to positional feedback, which introduces an additional nonequilibrium effect. We discuss the interplay between material properties and nonthermal force fluctuations in the living cells that we quantify through the violations of the FDT. In adherent fibroblasts, we observed a well-known polymer network viscoelastic response where the complex shear modulus scales as G* ∝ (-iω)3/4. In the more 3D confluent epithelial cells, we found glassy mechanics with G* ∝ (-iω)1/2 that we attribute to glassy dynamics in the cytosol. The glassy state in living cells shows characteristics that appear distinct from classical glasses and unique to nonequilibrium materials that are activated by molecular motors.

Aberration compensation for optical trapping of cells within living mice

Optical tweezers have been used to trap and manipulate microparticles within living animals. When the optical trap is constructed with an oil-immersion objective, it suffers from spherical aberration. There have been many investigations on the influence of spherical aberration when the particles are trapped in a water medium. However, the dependence of optical force on trapping depth is still ambiguous when the trapped particles are immersed in a high refractive index medium (such as biological tissue, refractive index solution) in experiments. In this paper, the microparticles are immersed in an aqueous solution of glycerol to mimic the cells within biological tissue. As the trapping laser is focused into the specimen, spherical aberration is introduced, degrading the optical trapping performance. It is similar to trapping in water; altering the effective tube length can also compensate for the spherical aberration of the optical trap in a high refractive index medium. Finally, the cells in living mice are trapped by the optical tweezers with the help of spherical aberration compensation.

Imaging the position-dependent 3D force on microbeads subjected to acoustic radiation forces and streaming

Acoustic particle manipulation in microfluidic channels is becoming a powerful tool in microfluidics to control micrometer sized objects in medical, chemical and biological applications. By creating a standing acoustic wave in the channel, the resulting pressure field can be employed to trap or sort particles. To design efficient and reproducible devices, it is important to characterize the pressure field throughout the volume of the microfluidic device. Here, we used an optically trapped particle as probe to measure the forces in all three dimensions. By moving the probe through the volume of the channel, we imaged spatial variations in the pressure field. In the direction of the standing wave this revealed a periodic energy landscape for 2 μm beads, resulting in an effective stiffness of 2.6 nN m(-1) for the acoustic trap. We found that multiple fabricated devices showed consistent pressure fields. Surprisingly, forces perpendicular to the direction of the standing wave reached values of up to 20% of the main-axis-values. To separate the direct acoustic force from secondary effects, we performed experiments with different bead sizes, which attributed some of the perpendicular forces to acoustic streaming. This method to image acoustically generated forces in 3D can be used to either minimize perpendicular forces or to employ them for specific applications in novel acoustofluidic designs.

Axial Optical Traps: A New Direction for Optical Tweezers

Optical tweezers have revolutionized our understanding of the microscopic world. Axial optical tweezers, which apply force to a surface-tethered molecule by directly moving either the trap or the stage along the laser beam axis, offer several potential benefits when studying a range of novel biophysical phenomena. This geometry, although it is conceptually straightforward, suffers from aberrations that result in variation of the trap stiffness when the distance between the microscope coverslip and the trap focus is being changed. Many standard techniques, such as back-focal-plane interferometry, are difficult to employ in this geometry due to back-scattered light between the bead and the coverslip, whereas the noise inherent in a surface-tethered assay can severely limit the resolution of an experiment. Because of these complications, precision force spectroscopy measurements have adapted alternative geometries such as the highly successful dumbbell traps. In recent years, however, most of the difficulties inherent in constructing a precision axial optical tweezers have been solved. This review article aims to inform the reader about recent progress in axial optical trapping, as well as the potential for these devices to perform innovative biophysical measurements.

Improved axial trapping with holographic optical tweezers

Conventional optical tweezers suffer from several complications when applying axial forces to surface-tethered molecules. Aberrations from the refractive-index mismatch between an oil-immersion objective's medium and the aqueous trapping environment both shift the trap centre and degrade the trapping strength with focal depth. Furthermore, interference effects from back-scattered light make it difficult to use back-focal-plane interferometry for high-bandwidth position detection. Holographic optical tweezers were employed to correct for aberrations to achieve a constant axial stiffness and modulate artifacts from backscattered light. Once the aberrations are corrected for, the trap height can be precisely determined from either the back-scattered light or Brenner's formula.

Bridging cells of three colors with two bio-orthogonal click reactions

Cell-cell interactions play a crucial role in the development and function of multicellular organisms. To study cell-cell interactions in vitro, it is a big challenge for researchers to artificially build up cell junctions to bridge different types of cells for this purpose. Herein, by employing two orthogonal click reactions, we rationally designed four click reagents Mal-CBT, Mal-Cys, Mal-Alkyne, and Mal-N₃ and successfully applied them to bridge cells of three colors. Orthogonality between these two click reactions was validated in solution and characterized with HPLC and ESI-MS analyses. After modifications of fluorescent protein-expressing prokaryotic Escherichia coli (E. coli) cells (or eukaryotic HEK 293T cells) of three colors with the reagents Mal-Cys, Mal-CBT and Mal-Alkyne, or Mal-N₃ , the cells were sequentially bridged. The HEK 293T cells showed a higher efficiency of cell bridging than the E. coli cells. Finally, using optical tweezers, we quantitatively measured the bridging probability between Mal-Cys-modified and Mal-CBT-modified HEK 293 cells, as well as the rupture force between two bridged cells. We found that the CBT-Cys click reaction markedly improved the efficiency of cell bridging and the rupture force between two bridged cells was measured to be 153.8 pN at a force-loading rate of 49 pN s^-1. Our results demonstrate that it is possible to use two (or n) orthogonal click reactions to bridge three (or n + 1) types of cells. Taking the biological importance of cell junctions into consideration, we anticipate that our method of bridging three types of cells with two bio-orthogonal click reactions will be a useful tool for biologists to study cell-cell interactions with more convenience and efficiency.

Direct 2D measurement of time-averaged forces and pressure amplitudes in acoustophoretic devices using optical trapping

Ultrasonic standing waves are increasingly applied in the manipulation and sorting of micrometer-sized particles in microfluidic cells. To optimize the performance of such devices, it is essential to know the exact forces that the particles experience in the acoustic wave. Although much progress has been made via analytical and numerical modeling, the reliability of these methods relies strongly on the assumptions used, e.g. the boundary conditions. Here, we have combined an acoustic flow cell with an optical laser trap to directly measure the force on a single spherical particle in two dimensions. While performing ultrasonic frequency scans, we measured the time-averaged forces on single particles that were moved with the laser trap through the microfluidic cell. The cell including piezoelectric transducers was modeled with finite element methods. We found that the experimentally obtained forces and the derived pressure fields confirm the predictions from theory and modeling. This novel approach can now be readily expanded to other particle, chamber, and fluid regimes and opens up the possibility of studying the effects of the presence of boundaries, acoustic streaming, and non-linear fluids.

Calibration of optical tweezers based on an autoregressive model

The power spectrum density (PSD) has long been explored for calibrating optical tweezers stiffness. Fast Fourier Transform (FFT) based spectral estimator is typically used. This approach requires a relatively longer data acquisition time to achieve adequate spectral resolution. In this paper, an autoregressive (AR) model is proposed to obtain the Spectrum Density using a limited number of samples. According to our method, the arithmetic model has been established with burg arithmetic, and the final prediction error criterion has been used to select the most appropriate order of the AR model, the power spectrum density has been estimated based the AR model. Then, the optical tweezers stiffness has been determined with the simple calculation from the power spectrum. Since only a small number of samples are used, the data acquisition time is significantly reduced and real-time stiffness calibration becomes feasible. To test this calibration method, we study the variation of the trap stiffness as a function of the parameters of the data length and the trapping depth. Both of the simulation and experiment results have showed that the presented method returns precise results and outperforms the conventional FFT method when using a limited number of samples.

Aurora A orchestrates entosis by regulating a dynamic MCAK-TIP150 interaction

Entosis, a cell-in-cell process, has been implicated in the formation of aneuploidy associated with an aberrant cell division control. Microtubule plus-end-tracking protein TIP150 facilitates the loading of MCAK onto the microtubule plus ends and orchestrates microtubule plus-end dynamics during cell division. Here we show that TIP150 cooperates with MCAK to govern entosis via a regulatory circuitry that involves Aurora A-mediated phosphorylation of MCAK. Our biochemical analyses show that MCAK forms an intra-molecular association, which is essential for TIP150 binding. Interestingly, Aurora A-mediated phosphorylation of MCAK modulates its intra-molecular association, which perturbs the MCAK-TIP150 interaction in vitro and inhibits entosis in vivo. To probe if MCAK-TIP150 interaction regulates microtubule plasticity to affect the mechanical properties of cells during entosis, we used an optical trap to measure the mechanical rigidity of live MCF7 cells. We find that the MCAK cooperates with TIP150 to promote microtubule dynamics and modulate the mechanical rigidity of the cells during entosis. Our results show that a dynamic interaction of MCAK-TIP150 orchestrated by Aurora A-mediated phosphorylation governs entosis via regulating microtubule plus-end dynamics and cell rigidity. These data reveal a previously unknown mechanism of Aurora A regulation in the control of microtubule plasticity during cell-in-cell processes.

Comparative study of methods to calibrate the stiffness of a single-beam gradient-force optical tweezers over various laser trapping powers

Optical tweezers have become an important instrument in force measurements associated with various physical, biological, and biophysical phenomena. Quantitative use of optical tweezers relies on accurate calibration of the stiffness of the optical trap. Using the same optical tweezers platform operating at 1064 nm and beads with two different diameters, we present a comparative study of viscous drag force, equipartition theorem, Boltzmann statistics, and power spectral density (PSD) as methods in calibrating the stiffness of a single beam gradient force optical trap at trapping laser powers in the range of 0.05 to 1.38 W at the focal plane. The equipartition theorem and Boltzmann statistic methods demonstrate a linear stiffness with trapping laser powers up to 355 mW, when used in conjunction with video position sensing means. The PSD of a trapped particle's Brownian motion or measurements of the particle displacement against known viscous drag forces can be reliably used for stiffness calibration of an optical trap over a greater range of trapping laser powers. Viscous drag stiffness calibration method produces results relevant to applications where trapped particle undergoes large displacements, and at a given position sensing resolution, can be used for stiffness calibration at higher trapping laser powers than the PSD method.

Calibrating optical tweezers with Bayesian inference

We present a new method for calibrating an optical-tweezer setup that does not depend on input parameters and is less affected by systematic errors like drift of the setup. It is based on an inference approach that uses Bayesian probability to infer the diffusion coefficient and the potential felt by a bead trapped in an optical or magnetic trap. It exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy.

Optical trapping of red blood cells in living animals with a water immersion objective

We demonstrate optical trapping of red blood cells (RBCs) in living animals by using a water immersion objective. First, the cells within biological tissue are mimicked by the particles immersed in aqueous solutions of glycerol. The optical forces depending on trapping depth are investigated when a parallel laser beam enters the water immersion objective. The results show that the optical forces vary with trapping depth, and the optimal trapping depth in aqueous solutions of glycerol (n=1.39) is 50 μm. Second, the optimal trapping depth in aqueous solutions of glycerol can be changed by altering the actual tube length of the water immersion objective. Finally, we achieved optical trapping and manipulation of RBCs in living mice.

A high-speed vertical optical trap for the mechanical testing of living cells at piconewton forces

Although atomic force microscopy is often the method of choice to probe the mechanical response of (sub)micrometer sized biomaterials, the lowest force that can be reliably controlled is limited to ≈0.1 nN. For soft biological samples, like cells, such forces can already lead to a strain large enough to enter the non-elastic deformation regime. To be able to investigate the response of single cells at lower forces we developed a vertical optical trap. The force can be controlled down to single piconewtons and most of the advantages of atomic force microscopy are maintained, such as the symmetrical application of forces at a wide range of loading rates. Typical consequences of moving the focus in the vertical direction, like the interferometric effect between the bead and the coverslip and a shift of focus, were quantified and found to have negligible effects on our measurements. With a fast responding force feedback loop we can achieve deformation rates as high as 50 μm/s, which allow the investigation of the elastic and viscous components of very soft samples. The potential of the vertical optical trap is demonstrated by measuring the linearity of the response of single cells at very low forces and a high bandwidth of deformation rates.

Following the mechanisms of bacteriostatic versus bactericidal action using Raman spectroscopy

Antibiotics cure infections by influencing bacterial growth or viability. Antibiotics can be divided to two groups on the basis of their effect on microbial cells through two main mechanisms, which are either bactericidal or bacteriostatic. Bactericidal antibiotics kill the bacteria and bacteriostatic antibiotics suppress the growth of bacteria (keep them in the stationary phase of growth). One of many factors to predict a favorable clinical outcome of the potential action of antimicrobial chemicals may be provided using in vitro bactericidal/bacteriostatic data (e.g., minimum inhibitory concentrations—MICs). Consequently, MICs are used in clinical situations mainly to confirm resistance, and to determine the in vitro activities of new antimicrobials. We report on the combination of data obtained from MICs with information on microorganisms’ “fingerprint” (e.g., DNA/RNA, and proteins) provided by Raman spectroscopy. Thus, we could follow mechanisms of the bacteriostatic versus bactericidal action simply by detecting the Raman bands corresponding to DNA. The Raman spectra of Staphylococcus epidermidis treated with clindamycin (a bacteriostatic agent) indeed show little effect on DNA which is in contrast with the action of ciprofloxacin (a bactericidal agent), where the Raman spectra show a decrease in strength of the signal assigned to DNA, suggesting DNA fragmentation.

Long-distance axial trapping with focused annular laser beams

Focusing an annular laser beam can improve the axial trapping efficiency due to the reduction of the scattering force, which enables the use of a lower numerical aperture (NA) objective lens with a long working distance to trap particles in deeper aqueous medium. In this paper, we present an axicon-to-axicon scheme for producing parallel annular beams with the advantages of higher efficiency compared with the obstructed beam approach. The validity of the scheme is verified by the observation of a stable trapping of silica microspheres with relatively low NA microscope objective lenses (NA = 0.6 and 0.45), and the axial trapping depth of 5 mm is demonstrated in experiment.

Practical axial optical trapping

We describe a new method for calibrating optical trapping measurements in which tension is applied in the direction of the laser beam to a molecule tethered between a surface and an optically trapped bead. Specifically, we present a generally-applicable procedure for converting from the measured scattering intensity and the measured stage displacement to applied tension and bead-coverslip separation, using measurements of the light intensity scattered from an untethered, trapped bead. Our calibration accounts for a number of effects, including aberrations and the interference of forward-reflected bead-scattered light with the trapping beam. To demonstrate the accuracy of our method, we show measurements of the DNA force-versus-extension relation using a range of laser intensities, and show that these measurements match the expected extensible wormlike-chain (WLC) behavior. Finally, we also demonstrate a force-clamp, in which the tension in a tether is held fixed while the extension varies as a result of molecular events.

Cell visco-elasticity measured with AFM and optical trapping at sub-micrometer deformations

The measurement of the elastic properties of cells is widely used as an indicator for cellular changes during differentiation, upon drug treatment, or resulting from the interaction with the supporting matrix. Elasticity is routinely quantified by indenting the cell with a probe of an AFM while applying nano-Newton forces. Because the resulting deformations are in the micrometer range, the measurements will be affected by the finite thickness of the cell, viscous effects and even cell damage induced by the experiment itself. Here, we have analyzed the response of single 3T3 fibroblasts that were indented with a micrometer-sized bead attached to an AFM cantilever at forces from 30–600 pN, resulting in indentations ranging from 0.2 to 1.2 micrometer. To investigate the cellular response at lower forces up to 10 pN, we developed an optical trap to indent the cell in vertical direction, normal to the plane of the coverslip. Deformations of up to two hundred nanometers achieved at forces of up to 30 pN showed a reversible, thus truly elastic response that was independent on the rate of deformation. We found that at such small deformations, the elastic modulus of 100 Pa is largely determined by the presence of the actin cortex. At higher indentations, viscous effects led to an increase of the apparent elastic modulus. This viscous contribution that followed a weak power law, increased at larger cell indentations. Both AFM and optical trapping indentation experiments give consistent results for the cell elasticity. Optical trapping has the benefit of a lower force noise, which allows a more accurate determination of the absolute indentation. The combination of both techniques allows the investigation of single cells at small and large indentations and enables the separation of their viscous and elastic components.

High-resolution microrheology in the pericellular matrix of prostate cancer cells

Many cells express a membrane-coupled external mechanical layer, the pericellular matrix (PCM), which often contains long-chain polymers. Its role and properties are not entirely known, but its functions are believed to include physical protection, mechanosensing, chemical signalling or lubrication. The viscoelastic response of the PCM, with polysaccharides as the main structural components, is therefore crucial for the understanding of its function. We have here applied microrheology, based on optically trapped micrometre-sized colloids, to the PCM of cultured PC3 prostate cancer cells. This technology allowed us to measure the extremely soft response of the PCM, with approximately 1 µm height resolution. Exogenously added aggrecan, a hyaluronan-binding proteoglycan, caused a remarkable increase in thickness of the viscoelastic layer and also triggered filopodia-like protrusions. The viscoelastic response of the PCM, however, did not change significantly.

Ensemble method to measure the potential energy of nanoparticles in an optical trap

A method is described for measuring the potential energy of nanoparticles in an optical trap by trapping an ensemble of particles with a focused laser beam. The force balance between repulsive osmotic and confining gradient-force pressures determines the single-particle trapping potential independent of interactions between the particles. The ensemble nature of the measurement permits evaluation of single-particle trapping energies much smaller than k_BT. Energies obtained by this method are compared to those of single-particle methods as well as to theoretical calculations based on classical electromagnetic optics.

Introduction to optical tweezers: background, system designs, and commercial solutions

Optical tweezers are a means to manipulate objects with light. With the technique, microscopically small objects can be held and steered while forces on the trapped objects can be accurately measured and exerted. Optical tweezers can typically obtain a nanometer spatial resolution, a piconewton force resolution, and a millisecond time resolution, which make them excellently suited to study biological processes from the single-cell down to the single-molecule level. In this chapter, we provide an introduction on the use of optical tweezers in single-molecule approaches. We introduce the basic principles and methodology involved in optical trapping, force calibration, and force measurements. Next, we describe the components of an optical tweezers setup and their experimental relevance in single-molecule approaches. Finally, we provide a concise overview of commercial optical tweezers systems. Commercial systems are becoming increasingly available and provide access to single-molecule optical tweezers experiments without the need for a thorough background in physics.

Parameter exploration of optically trapped liquid aerosols

When studying the motion of optically trapped particles on the microsecond time scale, in low-viscosity media such as air, inertia cannot be neglected. Resolution of unusual and interesting behavior not seen in colloidal trapping experiments is possible. In an attempt to explain the phenomena we use power-spectral methods to perform a parameter study of the Brownian motion of optically trapped liquid aerosol droplets concentrated around the critically damped regime. We present evidence that the system is suitably described by a simple harmonic oscillator model which must include a description of Faxén's correction, but not necessarily frequency dependent hydrodynamic corrections to Stokes' law. We also provide results describing how the system behaves under several variables and discuss the difficulty in decoupling the parameters responsible for the observed behavior. We show that due to the relatively low dynamic viscosity and high trap stiffness, it is easy to transfer between over- and underdamped motion by experimentally altering either trap stiffness or damping. Our results suggest stable aerosol trapping may be achieved in underdamped conditions, but the onset of deleterious optical forces at high trapping powers prevents the probing of the upper stability limits due to Brownian motion.

Dynamic axial stabilization of counter-propagating beam-traps with feedback control

Optical trapping in a counter-propagating (CP) beam-geometry provides unique advantages in terms of working distance, aberration requirements and intensity hotspots. However, its axial performance is governed by the wave propagation of the opposing beams, which can limit the practical geometries. Here we propose a dynamic method for controlling axial forces to overcome this constraint. The technique uses computer-vision object tracking of the axial position, in conjunction with software-based feedback, for dynamically stabilizing the axial forces. We present proof-of-concept experiments showing real-time rapid repositioning coupled with a strongly enhanced axial trapping for a plurality of particles of varying sizes. We also demonstrate the technique's adaptability for real-time reconfigurable feedback-trapping of a dynamically growing structure that mimics a continuously dividing cell colony. Advanced implementation of this feedback-driven approach can help make CP-trapping resistant to a host of perturbations such as laser fluctuations, mechanical vibrations and other distortions emphasizing its experimental versatility.

Quantitative measurements of absolute dielectrophoretic forces using optical tweezers

Optical tweezers were used for quantitative measurement of the absolute dielectrophoresis (DEP) forces acting on polystyrene microparticles. The electrodes and tweezers were configured to create one-dimensional DEP forces acting perpendicular to the tweezers' beam. The influences of various external factors, such as applied voltage frequency, conductivity of the medium, and particle size on the measurement were estimated. By accounting for these factors, actual measurements were in close agreement with theoretical predictions. Our results show that the optical tweezers may serve as a unique tool for the measurement of DEP forces in various applications.

Combining optical tweezers, single-molecule fluorescence microscopy, and microfluidics for studies of DNA-protein interactions

The technically challenging field of single-molecule biophysics has established itself in the last decade by granting access to detailed information about the fate of individual biomolecules, unattainable in traditional biochemical assays. The appeal of single-molecule methods lies in the directness of the information obtained from individual biomolecules. Technological improvements in single-molecule methods have made it possible to combine optical tweezers, fluorescence microscopy, and microfluidic flow systems. Such a combination of techniques has opened new possibilities to study complex biochemical reactions on the single-molecule level. In this chapter, we provide general considerations for the development of a combined optical trapping, fluorescence microscopy, and microfluidics instrument, along with methods to solve technical issues that are critical for designing successful experiments. Finally, we present several experiments to illustrate the power of this combination of techniques.

Rotation of birefringent particles in optical tweezers with spherical aberration

Birefringent particles rotate when trapped in elliptically polarized light. When an infinity corrected oil-immersion objective is used for trapping, rotation of birefringent particles in optical tweezers based on an infinity optical microscope is affected by the spherical aberration at the glass-water interface. The maximum rotation rate of birefringent particles occurs close to the coverslip, and the rotation rate decreases dramatically as the trapped depth increases. We experimentally demonstrate that spherical aberration can be compensated by using a finite-distance-corrected objective to trap and rotate the birefringent particles. It is found that the trapped depth corresponding to the maximum rotation rate is 50 microm, and the rotation rates at deep trapped depths are improved.

Brownian vortexes

Mechanical equilibrium at zero temperature does not necessarily imply thermodynamic equilibrium at finite temperature for a particle confined by a static but nonconservative force field. Instead, the diffusing particle can enter into a steady state characterized by toroidal circulation in the probability flux, which we call a Brownian vortex. The circulatory bias in the particle's thermally driven trajectory is not simply a deterministic response to the solenoidal component of the force but rather reflects interplay between advection and diffusion in which thermal fluctuations extract work from the nonconservative force field. As an example of this previously unrecognized class of stochastic heat engines, we consider a colloidal sphere diffusing in a conventional optical tweezer. We demonstrate both theoretically and experimentally that nonconservative optical forces bias the particle's fluctuations into toroidal vortexes whose circulation can reverse direction with temperature or laser power.

Viscoelastic response of a model endothelial glycocalyx

Many cells cover themselves with a multifunctional polymer coat, the pericellular matrix (PCM), to mediate mechanical interactions with the environment. A particular PCM, the endothelial glycocalyx (EG), is formed by vascular endothelial cells at their luminal side, forming a mechanical interface between the flowing blood and the endothelial cell layer. The glycosaminoglycan (GAG) hyaluronan (HA) is involved in the main functions of the EG, mechanotransduction of fluid shear stress and molecular sieving. HA, due to its length, is the only GAG in the EG or any other PCM able to form an entangled network. The mechanical functions of the EG are, however, impaired when any one of its components is removed. We here used microrheology to measure the effect of the EG constituents heparan sulfate, chondroitin sulfate, whole blood plasma and albumin on the high-bandwidth mechanical properties of a HA solution. Furthermore, we probed the effect of the hyaldherin aggrecan, a constituent of the PCM of chondrocytes, and very similar to versican (present in the PCM of various cells, and possibly in the EG). We show that components directly interacting with HA (chondroitin sulfate and aggrecan) can increase the viscoelastic shear modulus of the polymer composite.

Comparison of Faxén's correction for a microsphere translating or rotating near a surface

Boundary walls in microfluidic devices have a strong influence on the fluid flow and drag forces on moving objects. The Stokes drag force acting on a sphere translating in the fluid is increased by the presence of a neighboring wall by a factor given by Faxén's correction. A similar increase in the rotational drag is expected when spinning close to a wall. We use optical tweezers to confirm the translational drag correction and report the hitherto unmeasured rotational equivalent. We find that the corrections for the rotational motion is only required for particle-wall separations an order of magnitude shorter than that for the translational cases. These results are particularly significant in the use of optical tweezers for measuring viscosity on a picolitre scale.

Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability

We demonstrate both experimentally and theoretically that a colloidal sphere trapped in a static optical tweezer does not come to equilibrium, but rather reaches a steady state in which its probability flux traces out a toroidal vortex. This nonequilibrium behavior can be ascribed to a subtle bias of thermal fluctuations by nonconservative optical forces. The circulating sphere therefore acts as a Brownian motor. We briefly discuss ramifications of this effect for studies in which optical tweezers have been treated as potential energy wells.

See me, feel me: methods to concurrently visualize and manipulate single DNA molecules and associated proteins

Direct visualization of DNA and proteins allows researchers to investigate DNA–protein interactions with great detail. Much progress has been made in this area as a result of increasingly sensitive single-molecule fluorescence techniques. At the same time, methods that control the conformation of DNA molecules have been improving constantly. The combination of both techniques has appealed to researchers ever since single-molecule measurements have become possible and indeed first implementations of such combined approaches have proven useful in the study of several DNA-binding proteins in real time. Here, we describe the technical state-of-the-art of various integrated manipulation-and-visualization methods. We first discuss methods that allow only little control over the DNA conformation, such as DNA combing. We then describe DNA flow-stretching approaches that allow more control, and end with the full control on position and extension obtained by manipulating DNA with optical tweezers. The advantages and limitations of the various techniques are discussed, as well as several examples of applications to biophysical or biochemical questions. We conclude with an outlook describing potential future technical developments in combining fluorescence microscopy with DNA micromanipulation technology.