Stretching submicron biomolecules with constant-force axial optical tweezers

Forced extension of a wormlike chain in the Gibbs and Helmholtz ensembles

A semiflexible polymer can be stretched by either applying a force to it or by fixing the positions of its endpoints. The two approaches generally yield different results and correspond to experiments performed in either the Gibbs or Helmholtz statistical ensembles. Here, we derive the Helmholtz force-extension relationship for the commonly used wormlike-chain model in the strongly stretched regime. By analyzing it in comparison with the Gibbs ensemble result, we show that equivalence between the two relationships is achieved only in the long-chain thermodynamic limit.

Depth-resolved measurement of optical radiation-pressure forces with optical coherence tomography

A weakly focused laser beam can exert sufficient radiation pressure to manipulate microscopic particles over a large depth range. However, depth-resolved continuous measurement of radiation-pressure force profiles over an extended range about the focal plane has not been demonstrated despite decades of research on optical manipulation. Here, we present a method for continuous measurement of axial radiation-pressure forces from a weakly focused beam on polystyrene micro-beads suspended in viscous fluids over a depth range of 400 μm, based on real-time monitoring of particle dynamics using optical coherence tomography (OCT). Measurements of radiation-pressure forces as a function of beam power, wavelength, bead size, and refractive index are consistent with theoretical trends. However, our continuous measurements also reveal localized depth-dependent features in the radiation-pressure force profiles that deviate from theoretical predictions based on an aberration-free Gaussian beam. The combination of long-range radiation pressure and OCT offers a new mode of quantitative optical manipulation and detection with extended spatial coverage. This may find applications in the characterization of optical tractor beams, or volumetric optical manipulation and interrogation of beads in viscoelastic media.

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.

Stretching Response of Knotted and Unknotted Polymer Chains

Recent theoretical and experimental advances have clarified the major effects of knotting on the properties of stretched chains. Yet, how knotted chains respond to weak mechanical stretching and how this behavior differs from the unknotted case are still open questions and we address them here by profiling the complete stretching response of chains of hundreds of monomers and different topology. We find that the ratio of the knotted and unknotted chain extensions varies nonmonotonically with the applied force. This surprising feature is shown to be a signature of the crossover between the well-known high-force stretching regime and the previously uncharacterized low-force one. The observed differences of knotted and unknotted chain response increases with knot complexity and are sufficiently marked that they could be harnessed in single-molecule contexts to infer the presence and complexity of physical knots in micron-long biomolecules.

Tethered particle motion reveals that LacI·DNA loops coexist with a competitor-resistant but apparently unlooped conformation

The lac repressor protein (LacI) efficiently represses transcription of the lac operon in Escherichia coli by binding to two distant operator sites on the bacterial DNA and causing the intervening DNA to form a loop. We employed single-molecule tethered particle motion to observe LacI-mediated loop formation and breakdown in DNA constructs that incorporate optimized operator binding sites and intrinsic curvature favorable to loop formation. Previous bulk competition assays indirectly measured the loop lifetimes in these optimized DNA constructs as being on the order of days; however, we measured these same lifetimes to be on the order of minutes for both looped and unlooped states. In a range of single-molecule DNA competition experiments, we found that the resistance of the LacI-DNA complex to competitive binding is a function of both the operator strength and the interoperator sequence. To explain these findings, we present what we believe to be a new kinetic model of loop formation and DNA competition. In this proposed new model, we hypothesize a new unlooped state in which the unbound DNA-binding domain of the LacI protein interacts nonspecifically with nonoperator DNA adjacent to the operator site at which the second LacI DNA-binding domain is bound.

Double nanohole optical tweezers visualize protein p53 suppressing unzipping of single DNA-hairpins

Here we report on the use of double-nanohole (DNH) optical tweezers as a label-free and free-solution single-molecule probe for protein-DNA interactions. Using this approach, we demonstrate the unzipping of individual 10 base pair DNA-hairpins, and quantify how tumor suppressor p53 protein delays the unzipping. From the Arrhenius behavior, we find the energy barrier to unzipping introduced by p53 to be 2 × 10(-20) J, whereas cys135ser mutant p53 does not show suppression of unzipping, which gives clues to its functional inability to suppress tumor growth. This transformative approach to single molecule analysis allows for ultra-sensitive detection and quantification of protein-DNA interactions to revolutionize the fight against genetic diseases.

Invited article: a review of haptic optical tweezers for an interactive microworld exploration

This paper is the first review of haptic optical tweezers, a new technique which associates force feedback teleoperation with optical tweezers. This technique allows users to explore the microworld by sensing and exerting picoNewton-scale forces with trapped microspheres. Haptic optical tweezers also allow improved dexterity of micromanipulation and micro-assembly. One of the challenges of this technique is to sense and magnify picoNewton-scale forces by a factor of 10(12) to enable human operators to perceive interactions that they have never experienced before, such as adhesion phenomena, extremely low inertia, and high frequency dynamics of extremely small objects. The design of optical tweezers for high quality haptic feedback is challenging, given the requirements for very high sensitivity and dynamic stability. The concept, design process, and specification of optical tweezers reviewed here are focused on those intended for haptic teleoperation. In this paper, two new specific designs as well as the current state-of-the-art are presented. Moreover, the remaining important issues are identified for further developments. The initial results obtained are promising and demonstrate that optical tweezers have a significant potential for haptic exploration of the microworld. Haptic optical tweezers will become an invaluable tool for force feedback micromanipulation of biological samples and nano- and micro-assembly parts.

The molecular yo-yo method: live jump detection improves throughput of single-molecule force spectroscopy for out-of-equilibrium transitions

By monitoring multiple molecular transitions, force-clamp, and trap-position-clamp methods have led to precise determinations of the free energies and free energy landscapes for molecular states populated in equilibrium at the same or similar forces. Here, we present a powerful new elaboration of the force-clamp and force-jump methods, applicable to transitions far from equilibrium. Specifically, we have implemented a live jump detection and force-clamp algorithm that intelligently adjusts and maintains the force on a single molecule in response to the measured state of that molecule. We are able to collect hundreds of individual molecular transitions at different forces, many times faster than previously, permitting us to accurately determine force-dependent lifetime distributions and reaction rates. Application of our method to unwinding and rewinding the nucleosome inner turn, using optical tweezers reveals experimental lifetime distributions that comprise a statistically meaningful number of transitions, and that are accurately single exponential. These measurements significantly reduce the error in the previously measured rates, and demonstrate the existence of a single, dominant free energy barrier at each force studied. A key benefit of the molecular yo-yo method for nucleosomes is that it reduces as far as possible the time spent in the tangentially bound state, which minimizes the loss of nucleosomes by dissociation.

Force spectroscopy with dual-trap optical tweezers: molecular stiffness measurements and coupled fluctuations analysis

Dual-trap optical tweezers are often used in high-resolution measurements in single-molecule biophysics. Such measurements can be hindered by the presence of extraneous noise sources, the most prominent of which is the coupling of fluctuations along different spatial directions, which may affect any optical tweezers setup. In this article, we analyze, both from the theoretical and the experimental points of view, the most common source for these couplings in dual-trap optical-tweezers setups: the misalignment of traps and tether. We give criteria to distinguish different kinds of misalignment, to estimate their quantitative relevance and to include them in the data analysis. The experimental data is obtained in a, to our knowledge, novel dual-trap optical-tweezers setup that directly measures forces. In the case in which misalignment is negligible, we provide a method to measure the stiffness of traps and tether based on variance analysis. This method can be seen as a calibration technique valid beyond the linear trap region. Our analysis is then employed to measure the persistence length of dsDNA tethers of three different lengths spanning two orders of magnitude. The effective persistence length of such tethers is shown to decrease with the contour length, in accordance with previous studies.

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.

Kinetics and thermodynamics of phenotype: unwinding and rewinding the nucleosome

Chromatin "remodeling" is widely accepted as the mechanism that permits access to DNA by the transcription machinery. To date, however, there has been no experimental measurement of the changes in the kinetics and thermodynamics of the DNA-histone octamer association that are required to remodel chromatin so that transcription may occur. Here, we present the results of optical tweezer measurements that compare the kinetic and thermodynamic properties of nucleosomes composed of unmodified histones with those of nucleosomes that contain a mutant histone H4 (H4-R45H), which has been shown to allow SWI/SNF remodeling factor-independent transcription from the yeast HO promoter in vivo. Our measurements, carried out in a force-clamp mode, determine the force-dependent unwinding and rewinding rates of the nucleosome inner turn. At each force studied, nucleosomes containing H4-R45H unwind more rapidly and rewind more slowly than nucleosomes containing unmodified H4, indicating that the latter are the more stable. Extrapolation to forces at which the winding and unwinding rates are equal determines the absolute free energy of the nucleosome inner turn to be -32k(B)T for nucleosomes containing unmodified H4 and -27k(B)T for nucleosomes containing H4-R45H. Thus, the "loosening" or "remodeling" caused by this point mutation, which is demonstrated to be sufficient to allow transcriptional machinery access to the HO promoter (in the absence of other remodeling factors), is 5k(B)T. The correlation between the free energy of the nucleosome inner turn and the sin (SWI/SNF-independent) transcription suggests that, beyond partial unwinding, complete histone unwinding may play a role in transcriptional activation.

Force mapping of an optical trap using an acousto-optical deflector in a time-sharing regime

We suggest and study experimentally a time-sharing protocol for acousto-optical deflectors (AODs) that permits one to map the radial optical trapping force of optical tweezers without using a controllable flux control or an additional beam. Variations of the trapping potential due to modifications of the optical system are easily detected in terms of the force map. The protocol can be used in optical tweezers that already include an AOD without adding new elements in the existing optical system.

Quasiperiodic distribution of rigor cross-bridges along a reconstituted thin filament in a skeletal myofibril

Electron microscopy has shown that cross-bridges (CBs) are formed at the target zone that is periodically distributed on the thin filament in striated muscle. Here, by manipulating a single bead-tailed actin filament with optical tweezers, we measured the unbinding events of rigor CBs one by one on the surface of the A-band in rabbit skeletal myofibrils. We found that the spacings between adjacent CBs were not always the same, and instead were 36, 72, or 108 nm. Tropomyosin and troponin did not affect the CB spacing except for a relative increase in the appearance of longer spacing in the presence of Ca(2+). In addition, in an in vitro assay where myosin molecules were randomly distributed, were obtained the same spacing, i.e., a multiple of 36 nm. These results indicate that the one-dimensional distribution of CBs matches with the 36-nm half pitch of a long helical structure of actin filaments. A stereospecific model composed of three actin protomers per target zone was shown to explain the experimental results. Additionally, the unbinding force (i.e., the binding affinity) of CBs for the reconstituted thin filaments was found to be larger and smaller relative to that for actin filaments with and without Ca(2+), respectively.

Stretching short sequences of DNA with constant force axial optical tweezers

Single-molecule techniques for stretching DNA of contour lengths less than a kilobase are fraught with experimental difficulties. However, many interesting biological events such as histone binding and protein-mediated looping of DNA, occur on this length scale. In recent years, the mechanical properties of DNA have been shown to play a significant role in fundamental cellular processes like the packaging of DNA into compact nucleosomes and chromatin fibers. Clearly, it is then important to understand the mechanical properties of short stretches of DNA. In this paper, we provide a practical guide to a single-molecule optical tweezing technique that we have developed to study the mechanical behavior of DNA with contour lengths as short as a few hundred basepairs. The major hurdle in stretching short segments of DNA is that conventional optical tweezers are generally designed to apply force in a direction lateral to the stage (see Fig. 1). In this geometry, the angle between the bead and the coverslip, to which the DNA is tethered, becomes very steep for submicron length DNA. The axial position must now be accounted for, which can be a challenge, and, since the extension drags the microsphere closer to the coverslip, steric effects are enhanced. Furthermore, as a result of the asymmetry of the microspheres, lateral extensions will generate varying levels of torque due to rotation of the microsphere within the optical trap since the direction of the reactive force changes during the extension. Alternate methods for stretching submicron DNA run up against their own unique hurdles. For instance, a dual-beam optical trap is limited to stretching DNA of around a wavelength, at which point interference effects between the two traps and from light scattering between the microspheres begin to pose a significant problem. Replacing one of the traps with a micropipette would most likely suffer from similar challenges. While one could directly use the axial potential to stretch the DNA, an active feedback scheme would be needed to apply a constant force and the bandwidth of this will be quite limited, especially at low forces. We circumvent these fundamental problems by directly pulling the DNA away from the coverslip by using a constant force axial optical tweezers. This is achieved by trapping the bead in a linear region of the optical potential, where the optical force is constant-the strength of which can be tuned by adjusting the laser power. Trapping within the linear region also serves as an all optical force-clamp on the DNA that extends for nearly 350 nm in the axial direction. We simultaneously compensate for thermal and mechanical drift by finely adjusting the position of the stage so that a reference microsphere stuck to the coverslip remains at the same position and focus, allowing for a virtually limitless observation period.

Dual-trap optical tweezers with real-time force clamp control

Single molecule force clamp experiments are widely used to investigate how enzymes, molecular motors, and other molecular mechanisms work. We developed a dual-trap optical tweezers instrument with real-time (200 kHz update rate) force clamp control that can exert 0-100 pN forces on trapped beads. A model for force clamp experiments in the dumbbell-geometry is presented. We observe good agreement between predicted and observed power spectra of bead position and force fluctuations. The model can be used to predict and optimize the dynamics of real-time force clamp optical tweezers instruments. The results from a proof-of-principle experiment in which lambda exonuclease converts a double-stranded DNA tether, held at constant tension, into its single-stranded form, show that the developed instrument is suitable for experiments in single molecule biology.

Measuring the complete force field of an optical trap

The use of optical traps to measure or apply forces on the molecular level requires a precise knowledge of the trapping force field. Close to the trap center, this field is typically approximated as linear in the displacement of the trapped microsphere. However, applications demanding high forces at low laser intensities can probe the light-microsphere interaction beyond the linear regime. Here, we measured the full nonlinear force and displacement response of an optical trap in two dimensions using a dual-beam optical trap setup with back-focal-plane photodetection. We observed a substantial stiffening of the trap beyond the linear regime that depends on microsphere size, in agreement with Mie theory calculations. Surprisingly, we found that the linear detection range for forces exceeds the one for displacement by far. Our approach allows for a complete calibration of an optical trap.

Deconvolution of dynamic mechanical networks

Time-resolved single-molecule biophysical experiments yield data that contain a wealth of dynamic information, in addition to the equilibrium distributions derived from histograms of the time series. In typical force spectroscopic setups the molecule is connected via linkers to a readout device, forming a mechanically coupled dynamic network. Deconvolution of equilibrium distributions, filtering out the influence of the linkers, is a straightforward and common practice. We have developed an analogous dynamic deconvolution theory for the more challenging task of extracting kinetic properties of individual components in networks of arbitrary complexity and topology. Our method determines the intrinsic linear response functions of a given object in the network, describing the power spectrum of conformational fluctuations. The practicality of our approach is demonstrated for the particular case of a protein linked via DNA handles to two optically trapped beads at constant stretching force, which we mimic through Brownian dynamics simulations. Each well in the protein free energy landscape (corresponding to folded, unfolded, or possibly intermediate states) will have its own characteristic equilibrium fluctuations. The associated linear response function is rich in physical content, because it depends both on the shape of the well and its diffusivity-a measure of the internal friction arising from such processes as the transient breaking and reformation of bonds in the protein structure. Starting from the autocorrelation functions of the equilibrium bead fluctuations measured in this force clamp setup, we show how an experimentalist can accurately extract the state-dependent protein diffusivity using a straightforward two-step procedure.

Protein-mediated DNA loop formation and breakdown in a fluctuating environment

Living cells provide a fluctuating, out-of-equilibrium environment in which genes must coordinate cellular function. DNA looping, which is a common means of regulating transcription, is very much a stochastic process; the loops arise from the thermal motion of the DNA and other fluctuations of the cellular environment. We present single-molecule measurements of DNA loop formation and breakdown when an artificial fluctuating force, applied to mimic a fluctuating cellular environment, is imposed on the DNA. We show that loop formation is greatly enhanced in the presence of noise of only a fraction of k_{B}T, yet find that hypothetical regulatory schemes that employ mechanical tension in the DNA-as a sensitive switch to control transcription-can be surprisingly robust due to a fortuitous cancellation of noise effects.

Single-molecule approaches to probe the structure, kinetics, and thermodynamics of nucleoprotein complexes that regulate transcription

Single-molecule experimentation has contributed significantly to our understanding of the mechanics of nucleoprotein complexes that regulate epigenetic switches. In this minireview, we will discuss the application of the tethered-particle motion technique, magnetic tweezers, and atomic force microscopy to (i) directly visualize and thermodynamically characterize DNA loops induced by the lac, gal, and lambda repressors and (ii) understand the mechanistic role of DNA-supercoiling and DNA-bending cofactors in both prokaryotic and eukaryotic systems.

Revealing the base pair stepping dynamics of nucleic acid motor proteins with optical traps

Nearly all aspects of nucleic acid metabolism involve motor proteins. This diverse group of enzymes, which includes DNA and RNA polymerases, the ribosome, helicases, and other translocases, converts chemical energy in the form of bond hydrolysis into concerted motion along nucleic acid filaments. The direct observation of this motion at its fundamental distance scale of one base pair has required the development of new ultrasensitive techniques. Recent advances in optical traps have now made these length scales, once the exclusive realm of crystallographic techniques, accessible. Several new studies using optical traps have revealed for the first time how motor proteins translocate along their substrates in a stepwise fashion. Though these techniques have only begun to be applied to biological problems, the unprecedented access into nucleic acid motor protein movement has already provided important insights into their mechanism. In this perspective, we review these advances and offer our view on the future of this exciting development.

Femtonewton entropic forces can control the formation of protein-mediated DNA loops

We show that minuscule entropic forces, on the order of 100 fN, can prevent the formation of DNA loops-a ubiquitous means of regulating the expression of genes. We observe a tenfold decrease in the rate of LacI-mediated DNA loop formation when a tension of 200 fN is applied to the substrate DNA, biasing the thermal fluctuations that drive loop formation and breakdown events. Conversely, once looped, the DNA-protein complex is insensitive to applied force. Our measurements are in excellent agreement with a simple polymer model of loop formation in DNA, and show that an antiparallel topology is the preferred LacI-DNA loop conformation for a generic loop-forming construct.

Entropic boundary effects on the elasticity of short DNA molecules

We have measured the entropic elasticity of double-stranded-DNA molecules ranging from 247 to 1298 bp in length using axial force-clamp optical tweezers. We show that entropic end effects and excluded-volume forces from surface attachments become significant for such short molecules. The effective persistence length of the shortest molecules decreases by a factor of 2 compared to the established value for long molecules, and excluded-volume forces extend the molecules to about one third of their nominal contour length. We interpret these results in the framework of an inextensible semiflexible rod model.