The surface-coupled optical tweezers are widely used to resolve small units of motion in biology. However, such motions could readily be interfered by the drift between the trap and surface. We present a simple and low-cost method to correct the drift both actively and passively based on video tracking the distance between the laser reflection pattern and the reference bead. As a result, we achieved sub-nanometer resolution and stability for the stuck bead over a broad range of averaging time (0.002-100 s) as demonstrated by the Allan deviation analysis. The sub-nanometer resolution was further manifested with step measurement. Finally, in double-stranded DNA and DNA hairpin stretching experiments, an extension resolution of 1-2 nm with the stability over 120 s has been demonstrated under a constant force. This work thus provides an easy way to bring the benefit of nanometer resolution and long-term stability to the surface-coupled optical tweezers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical tweezers (OT) have been applied to study a wide range of biophysical problems. The development of dual-trap optical tweezers (DOT), which decouples the experiments from the surface, enables researchers to resolve some extremely small motions in biology such as RNA polymerase’s one-base-pair step along DNA . In addition to the step size measurements, by detecting intermediates in protein/nucleic acid folding [2–4], or pauses and backtracking in enzymatic motion [5–7], high-resolution optical tweezers can also provide insights into complex kinetic pathways. However, dual-trap optical tweezers are complicated instruments to produce, requiring more equipment and precise controls of the experimental environment in order to achieve such high spatial resolutions . In contrast, the surface-coupled optical tweezers (SOT), though suffering from larger drift, are relatively simpler and easier to implement. Moreover, in comparison with the prevalent DOT assays, the SOT assay is much more efficient and convenient in forming single-molecule tethers without resort to the use of a precisely-controlled flow-cell/microfluidic system [9,10]. Therefore, in many experiments, the SOT assay is more advantageous and preferred when the extremely high spatial resolution and stability can be compromised [11–13]. Besides, for some special optical trapping techniques such as angular optical tweezers, currently they are only compatible with the surface-coupled assay [14,15]. It is thus important to further improve the performance of SOT.
So far, the main limitation of SOT assays has been the drift between the optical trap and surface. Surface-coupled optical tweezers typically have a DNA molecule attached to the surface at one end and to an optically trapped bead at the other end. These surface-coupled assays are very sensitive to mechanical drift of surface as they are physically connected to the surrounding environment either through the coverslip or micropipette [16–18].The surface can be stabilized by video tracking the reference mark or the sample itself [19,20]. Besides, the position of optical trap is usually assumed to be constant in SOT experiments. However, due to the laser pointing instability and ambient noise (mechanical vibration of the lens/mirror, acoustic noise, etc.), the trap position also varies with time. Even though the laser is coupled into an optical fiber for pointing stability improvement in conventional surface-coupled optical tweezers, a highly stable laser with the fiber output typically has a pointing instability around 1 µrad, which could still produce ∼2 nm drift on the sample plane (assuming a 100× objective with a 2 mm focal length).
To remove the limitation, Nugent-Glandorf et al. firstly introduced a local, differential measurement technique that measured 0.1-nm motion with high temporal resolution . Carter et al. further stabilized an optical microscope to 0.1 nm in three dimensions  by applying optically based reference frame , which utilized two detection lasers to detect relative movement between the trap and surface, one for monitoring the trapped bead and the other for measuring the fiducial mark position. Furthermore, Carter et al. used this technique to achieve one-base-pair resolution in surface-coupled optical tweezers . Due to limited accessibility on the side opposite to the imaging objective, instruments such as atomic force microscope or magnetic tweezers are not compatible with forward-scattered detection. To extend sub-nanometer stabilization to these instruments, the technique could be modified using back-scattered detection, enabling optical trapping experiments with sub-nm drift over hours in 3D [25,26]. Despite the success in improving the stability of optical tweezers, those techniques still require more equipment (such as extra lasers and detection systems) and procedures (such as nanofabricated fiducial marks) as well as complex software. For the majority of labs with in-use optical tweezers or limited budgets, there is still a lack of a simple, cost-effective method to improve the stability of their instruments.
In this Letter, we present a simple method to correct the drift based on video tracking the distance between the reflection pattern of a reference beam and a reference bead. Different from previous methods of stabilizing microscope drift [20,27–29], our method can stabilize the position of microscope surface relative to a laser beam, and is suitable for correcting the drift of instruments with laser or moving object that can reflect laser, such as confocal microscope and atomic force microscope. When compared with methods especially for correcting the drift of optical tweezers [21,22,24,26], our method still offers three major advantages: (i) It is robust and has been demonstrated to be able to adapt to a relatively noisy environment. Our optical tweezers was built on a commercial inverted microscope and no extra components, such as thick posts or custom mounts, were added to enhance the stability. Moreover, the optical path was long and temporarily left open to the air for further modifications. Even without optimization, we managed to achieve high (from sub-nanometer to 1-2 nm) resolution and stability in a time frame of minutes under various experimental conditions with the proposed method. (ii) It is simple and easy to implement. The upgrade only requires a small amount of optical components such as polarizing beam-splitters and mirrors and no additional expensive equipment is needed. The reflection pattern is also insensitive to translation movement of the surface, thus no additional piezoelectric mirror is required to translate the tracking laser when the surface was moved by the piezoelectric stage during the experiment . (iii) Both the reference beam and reference bead were detected using a single camera, avoiding mechanical drift and complications in synchronization between multiple detectors . Our method offers a favorable solution for in-use setup updating or building a new cost-effective OT setup, as it can be readily integrated into most optical tweezers system with minimal hardware modifications.
The drift corrected optical tweezers was modified from our home-built single-trap optical tweezers described previously [Fig. 1(a)]. Briefly, a linearly polarized TEM00-mode 1064-nm laser beam (Changchun New Industries Optoelectronics Technology, Changchun, China) passed through acousto-optic modulator (M1080-T80L, Isomet, Virginia, USA) and was coupled to a polarization-maintaining single-mode fiber (OZ Optics, Ottawa, Canada). The laser beam was expanded to about 6 mm in diameter with two beam expanders (L1 and L2, L3 and L4, the focal lengths were as follows: L1 = 150 mm, L2 = 150 mm, L3 = 250 mm, L4 = 750 mm). Then, the beam was focused at the sample plane by a 100X, 1.45-NA, oil immersion objective mounted on an inverted microscope (CFI Plan Apo Lambda 100X oil objective and Ti-U microscope, Nikon, Tokyo, Japan). After interacting with the trapped bead, the laser beam was collected by a condenser (CON) and reflected by a dichroic mirror (DM), followed by lens (L7 = 100 mm) to image the interference pattern to the detector. The displacements of the trapped bead from the trap center were measured by a position-sensitive detector (DL100-7-PCBA3, First Sensor, Berlin, Germany) and converted into a force signal. Analog voltage signal was digitized at 10 kHz for each channel using a multiplexed analog-to-digital conversion PCI-E board (PCIe 6353, National Instruments, Austin, TX) and averaged to 1 kHz. Calibration and data conversion methods were adapted from those used by Wang et al. . The DNA and DNA hairpin samples were tethered to a 0.8-μm streptavidin-coated polystyrene bead (Spherotech, Lake Forest, IL) at one end through biotin-streptavidin interactions and to the anti-digoxigenin (Roche, Basel, Switzerland) coated cover glass surface at the other end through digoxigenin-antibody interactions. The polystyrene bead was optically trapped and the cover-glass was driven by a servo-controlled 3D piezoelectric stage (P733.3DD, Physik Instrumente, Karlsruhe, Germany). To stretch a single DNA molecule, the piezoelectric stage was moved and the force was read from the position-sensitive detector (PSD). The DNA extension was calculated based on the geometry of the experimental configuration [Fig. 1(b)] [16,30]. The 3-µm amino-coated polystyrene bead (Spherotech, Lake Forest, IL) was diluted to ∼0.1% w/v with 1M NaCl buffer and incubated for ∼30 min to form the reference bead fixed on the surface.
The central idea of our method is that the position changes of the trapping laser on the sample plane can be tracked by the reflection pattern of the laser at the interface of cover glass and water. This reflection pattern has been found to be insensitive to the mechanical drift of surface in both x-direction and y-direction. However, once a bead is optically trapped by the laser (e.g., during optical trapping experiment), the pattern will change and become sensitive to the bead position. To avoid this complication, we adopted an idea similar to the dual-trap optical tweezers: the differential stability between two laser beams is extremely high and the reference beam has an almost identical trajectory as the trapping beam. Here, the trapping laser was split based on polarization into two beams, one for optical trapping experiment (trapping beam) and the other for tracking position changes of the trapping beam (reference beam) [Figs. 1(a) and 1(b)]. The reference beam was separated about 8 µm from the trapping beam on the sample plane by adjusting a reflection mirror (M6) conjugated to the back focal plane of objective. In this way, a good balance was achieved between maximizing the shared optical path of two beams and minimizing the interference of the reference beam to the trapping beam. A pair of lenses (L5 = 100 mm, L6 = 100 mm) with an equal focal length was inserted to adjust the focus position of the reference beam about 300 nm lower than the trapping beam in z-direction. As a result, the reflection pattern of the reference beam can be clearly imaged by the CCD camera while the trapping beam still focused inside the chamber [Fig. 1(c)]. Combining with video-tracking a reference bead (3 µm in diameter) stuck on the surface, a method which has been used to correct stage drift in magnetic tweezers  and optical tweezers , both the trap position and surface can be tracked simultaneously via the same camera mounted on the microscope. The drift between the trap and surface can then be calculated and corrected either by active feedback using the piezoelectric stage or passive subtraction of the drift after the experiment. We noted that here the laser reflection was used to track the trap position in x-direction and y-direction only, while adjusting focus or moving stage in z-direction did change the pattern. However, we found that the pattern is sensitive to the drift between the trap and surface in z-direction rather than the absolute z position of the trap. Thus, we tracked the reference bead in 3D and then kept its z position constant with the piezoelectric stage. In addition, as the intensity fluctuation of laser could also affect the reflection pattern and hence adversely influence position measurements, we took active feedback using an acousto-optic modulator to keep laser intensity constant during the whole measurement.
Firstly, we demonstrated that the position changes of the trapping beam on the sample plane can be tracked using the reflection pattern of the reference beam. An 800-nm polystyrene bead was trapped by the trapping beam while the position of the trapped bead and the reflection pattern of the reference beam were video-tracked using the QI algorithm . We assume that the x and y positions of the trapped bead were identical to those of the trapping beam as the trap stiffness was relatively high (0.1 pN/nm) and environmental disturbance has little effect on bead position comparing to optical force.
As shown in Figs. 2(c) and 2(d), the measured x position and y position of the trapped bead and reference beam exhibited similar trajectories. The standard deviations of residuals in the x-direction and y-direction in Fig. 2(e) are 1.9 nm and 2.3 nm, respectively (smoothed to 10 Hz). The pointing instability introduced drift on the sample plane can be roughly estimated using the equation $l = f\theta $, where $f$ is the focal length of the objective and $\theta $ is the pointing instability of laser beam. The differential position of the reference beam and trapping beam ideally can be assumed to be a constant, as 300 nm difference along z-direction is negligible compared to the 2 mm focal length of the objective (i.e., f is equal) and these two beams share most of the optical path (i.e., $\theta $ is equal). The fluctuation of these residuals thus more likely arose from errors in video tracking, especially the tracking errors of the trapped bead whose image could be slightly affected by the reflection pattern of trapping beam. Nevertheless, our results demonstrated that the position of the trapping beam can be tracked through the reflection pattern of the reference beam.
We further calculated the drift of the trapping beam relative to the surface using the measured positions of the reference beam and a 3-µm reference bead stuck on the surface. To verify that the drift obtained through this method was reliable, we took another independent method, i.e. using the position-sensitive detector (PSD) and back-focal-plane interferometry  to measure the drift between the trapping beam and surface. At the start of the measurement, the stage was moved separately along each (x or y) axis to translate an 800-nm polystyrene stuck bead through the trapping beam, yielding a voltage-versus-distance curve, thus allowing the beam center and linear sensitivity to be determined [16,33]. We then centered the 800-nm bead using piezoelectric stage and monitored the drift of the trapping beam relative to the 800-nm stuck bead (i.e. surface) by multiplying the voltage signal of position-sensitive detector (PSD) with the calibrated linear sensitivity. Simultaneously, the drift was detected by video tracking of the reflection pattern of the reference beam and the 3-µm reference bead [Figs. 3(a) and 3(b)]. The drift measured by video and PSD were recorded at 100 Hz and 1000 Hz, respectively. As shown in Figs. 3(c) and 3(d), the drift obtained from video exhibits an almost identical pattern with the drift measured by PSD, indicating that the movement of the trapping beam relative to the surface can be obtained using the position changes of the reference beam relative to the reference bead.
Moreover, our method can measure both position changes of the reference beam and reference bead, which have been demonstrated to reflect the position changes of the trap and surface, respectively. Therefore, different from previous reports [22,24,25], in our experiment, the mechanical drift of the surface and the pointing drift of the trapping beam can both be obtained separately [Figs. 3(c) and 3(d)]. We noted that the drifting pattern of the surface is different from that of the trapping beam. The drift of surface increased along a certain direction, while the position of the trapping beam dithered around a certain position. During 10 min of measurement, the surface moved about 10 nm along x-direction, while the trapping beam still stayed close to the original position, but it swung with a peak-to-peak value around 40 nm and a standard deviation of 5.4 nm, which is consistent with the 7 µrad pointing instability measured after fiber launch position [Fig. 3(c)]. The surface moves about 45 nm along y-direction, and the trapping beam drifted ∼20 nm and swung with a peak-to-peak value close to 40 nm [Fig. 3(d)]. The difference in the stability of x-direction and y-direction possibly came from the mounting of the piezoelectric stage in our set-up. Notice that many previous improvements on optical tweezers drift focused on eliminating surface drift. Our experiment highlights that the pointing stability of trapping beam is also crucial to the stability of optical tweezers. Although trapping beam does not exhibit a long-term drift in one direction, it could have large dithers in a short period (30 nm, 1 s), which coincides with the time-scale of many biological events  such as enzyme movement and protein folding, thus brings extra noise to the measurement and certainly needs to be carefully solved.
To further test the performance of our method, we stabilized the surface by measuring the drift with video tracking and taking active feedback at 100 Hz with piezoelectric stage to minimize the drift (i.e. keep the distance between the reference beam and 3-µm reference bead constant). Meanwhile, we quantified this stability with the trapping beam and an 800-nm stuck bead using PSD signals. The position changes of the reference beam relative to the reference bead (measured by video) and the trapping beam relative to the 800-nm stuck bead on the surface (measured by PSD) were recorded at 100 Hz and 1000 Hz, respectively. In 10 min of measurement with the active feedback, the standard deviations of position changes in the x-direction and y-direction of the trapping beam relative to the 800-nm stuck bead (measured by PSD) were 1.6 nm and 2.3 nm, respectively, which were 3.4 folds and 8.3 folds smaller than 5.4 nm in x-direction and 19.2 nm in y-direction without feedback (data were all smoothed to 10 Hz) [Figs. 3(e) and 3(f)]. This demonstrated that our method can effectively reduce the drift between the trap and surface.
We also used the Allan deviation (σ)  to quantify the instrumental performance, which was determined using
To demonstrate the precision of our method, we moved the surface in a series of 2 nm steps and detected the resulting motion. More specifically, we actively stabilized the distance between the reference beam and reference bead in 3D at 100 Hz and then increased the x-direction set-point by 2 nm every 2 s while measuring resulting position changes with an 800-nm polystyrene stuck bead and the trapping beam through PSD. The steps were well resolved in the measured trace with the standard deviation of the position at each step being ∼0.4 nm on average when the data was smoothed to 10 Hz [Fig. 4(b)]. This standard deviation became ∼0.6 nm when the data was smoothed to 100 Hz (data not shown). Thus, by reducing mechanical perturbations between the trap and surface, we achieved sufficient stability and even sub-nanometer precision to measure step motions with a relatively high temporal resolution, consistent with Allan deviation analysis in Fig. 4(a). To better illustrate the steps, we also calculated the pair-wise distance difference (PDD)  that computed the position difference between pairs of data points. Ideally, every such difference would be a multiple of the step (in our case, 2 nm) and a histogram of PDD data would then show peaks at 2 nm intervals [Fig. 4(c)]. A Fourier transform of the PDD histogram showed that the primary spatial frequency component was 0.5 ± 0.03 nm-1 (peak ± HWHM), corresponding to a step size of 2 ± 0.12 nm. Consequently, a signal-to-noise ratio about 17 was obtained based on 1 σ = FWHM/2.35 (assuming Gaussian function) for data smoothed to 10 Hz [Fig. 4(d)]. Noted that HWHM stands for half width at half-maximum and FWHM stands for full width at half-maximum.
Finally, we demonstrated the resolution and stability of our optical tweezers using DNA hairpin hopping experiment. DNA hairpin is an important structure and may occur in many molecular biological processes such as replication and DNA repair . It also plays essential roles in various applications including DNA amplification (through polymerase chain reaction, PCR), DNA computing, bio-sensing, etc [40–42]. Measuring DNA hairpin hopping can thus provide important information about its stability and folding/unfolding dynamics, helping people gain more insights into its biological roles as well as providing guidance for designing an optimized DNA primer or DNA sensor. We achieved nanometer-scale stability in a typical DNA hopping experiment with the surface-coupled assay. The drift in y-direction and z-direction were actively stabilized using corresponding axes of piezoelectric stage. The drift in x-direction was not fed back but it was subtracted in data processing after the experiment. The x-axis of piezoelectric stage was then used for feedback to keep the force constant. We noted that our current optical tweezers suffered from various sources of noise, including mechanical drift of objective induced by immersion oil and laser heating, which usually takes more than 30 mins for the whole system to reach equilibrium. Thus the measurement of 2.8-kbp double-stranded DNA (dsDNA) was taken at a moderate force of 10 pN under two conditions, one at 10 min after the sample was placed on microscope (i.e., the system was not in equilibrium) and the other at 2 h after the sample was placed on microscope (i.e., the system was in equilibrium), to test the robustness of our method.
Combining active stabilization and post drift correction, we were able to measure a constant dsDNA extension within 1.7 nm (S.D.) at a constant force (clamped at 10 pN) for 120 s (data were smoothed to 10 Hz) under a non-equilibrium condition, which was 4 fold lower compared to 7.0 nm (S.D.) without stabilization. Remarkably, we achieved a 1.2 nm uncertainty (S.D.) of dsDNA extension in 120 s (smoothed to 10 Hz) under an equilibrium condition, which was still 2.8 folds lower compared to 3.3 nm (S.D.) without stabilization and drift correction. Similarly, a significant improvement was also achieved in DNA hairpin folding/unfolding experiment. Two hopping states of DNA hairpin were clearly identified in the data with drift correction compared to more ambiguous peaks for the data without drift correction. The extension standard deviation of each state with drift correction was 1.7 nm and 1.8 nm, respectively, in contrast with 4.2 and 4.0 nm without drift correction under a non-equilibrium condition (data were smoothed to 10 Hz) [Fig. 5(c)]. Similar improvement (i.e. 1.8 nm for the standard deviation of both states with correction) has also been found in the experiment under an equilibrium condition [Fig. 5(d)]. Therefore, our results showed that even under an equilibrium condition for the setup, the improvement of the data quality with our drift correction method is still substantial. The extension changes of DNA hairpin folding/unfolding measured by Gaussian fitting were 13.2 ± 0.7 (S.E.) nm with correction and 12.4 ± 1.5 (S.E.) nm without correction under a non-equilibrium condition. And these values changed to 13.0 ± 0.7 (S.E.) nm with correction and 12.8 ± 1.1 (S.E.) nm without correction under an equilibrium condition. The result with drift correction was thus closer to the theoretically predicted 13.5 nm from worm-like-chain (WLC) model compared to that without drift correction in both cases.
We have corrected the drift between the trap and surface using laser reflection patterns to achieve high (from sub-nanometer to 1-2 nm) positional precision and stability in surface-coupled optical tweezers over minutes under various experimental conditions. By video tracking the reference beam and reference bead, we can directly measure both mechanical drift and laser pointing drift separately. Using active feedback to stabilize the surface, we were able to measure 2 nm steps for a stuck bead with sub-nanometer precision and relatively high (e.g. 100 Hz) temporal resolution. The stability (i.e. typically around 1-2 nm over a few minutes) was further demonstrated in dsDNA and DNA hairpin force-clamp experiments, which showed a significant improvement over data without correction. Notice that, although here we used the DNA hairpin hopping experiment to demonstrate the drifting correction, our method itself is general and can be easily extended to study protein or RNA folding/unfolding kinetics.
Our current short-term positional precision is limited by the accuracy of tracking algorithm. The laser reflection pattern is not perfectly rotationally symmetric, thus tracking accuracy of reference beam may not be as good as that of reference bead when using QI algorithm. Further improvements to increase surface stability could be achieved by including a new tracking algorithm for laser reflection pattern. The long-term stability is limited by the differential pointing instability of trapping beam and reference beam, which is affected by environmental disturbances in their non-common optical paths, such as air flow, mechanical vibration, etc. To improve long-term stability, the differential laser motion could be minimized with a more compact optical design (minimize the non-common optical paths) and a closed optical path (minimize the environmental disturbances) , which shall further reduce the differential pointing instability between the trapping and reference beams.
National Natural Science Foundation of China (11674403, 12074445); Science and Technology Planning Project of Guangdong Province (2018A050506034).
We thank members of the J.M. laboratory for helpful discussions. We also wish to thank for the support from the Physical Research Platform in School of Physics, Sun Yat-sen University (PRPSP, SYSU).
The authors declare no conflicts of interest.
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