Advanced optical traps can probe single molecules with Ångstrom-scale precision, but drift limits the utility of these instruments. To achieve Å-scale stability, a differential measurement scheme between a pair of laser foci was introduced that substantially exceeds the inherent mechanical stability of various types of microscopes at room temperature. By using lock-in detection to measure both lasers with a single quadrant photodiode, we enhanced the differential stability of this optical reference frame and thereby stabilized an optical-trapping microscope to 0.2 Å laterally over 100 s based on the Allan deviation. In three dimensions, we achieved stabilities of 1 Å over 1,000 s and 1 nm over 15 h. This stability was complemented by high measurement bandwidth (100 kHz). Overall, our compact back-scattered detection enables an ultrastable measurement platform compatible with optical traps, atomic force microscopy, and optical microscopy, including super-resolution techniques.
© 2015 Optical Society of America
Single-molecule techniques, such as optical traps  and atomic force microscopy (AFM) , have long been capable of measuring Ångstrom-scale (1 Å) displacement. Coupling this precision with Å-scale stability has opened the door to a variety of exciting biophysical applications [3–9]. For instance, the development of a dual-trap assay [Fig. 1(a)] for RNA polymerase (RNAP)  enabled real-time detection of RNAP’s fundamental step size of 1-base-pair (1 bp = 3.4 Å) . In addition to determining the step sizes of various nucleic acid enzymes [12–15], high-precision optical-trapping assays can yield insight into complex kinetic pathways by detecting intermediates in protein folding [16–18] and pauses in enzymatic motion [11, 19–22]. Notwithstanding the substantial effort invested in improving the stability of optical traps [11, 23–26], it remains challenging to detect more than a handful of 1-bp steps in register over an extended period of time (~5–30 s) due, in part, to instrumental drift . The underlying opto-mechanical stability typically needs to be ~3-fold better than the required biological precision [6, 11, 24]. In other words, a detection system with 1-Å precision and stability is needed to measure 1-bp steps along DNA. To be most useful, such performance metrics need to be complemented with high-bandwidth detection (~100 kHz); high bandwidth enables accurate stiffness calibrations based upon power spectral analysis  and detection of short-lived states (e.g., sub-ms protein-folding intermediates ). Hence, an ideal opto-mechanical measurement platform for single-molecule biophysics would have sub-Å precision and stability coupled with high bandwidth.
Sub-Å precision is critical to improving the detection of 1-bp steps RNAP , particularly in a standard surface-coupled optical-trapping assay , because the pioneering work to detect RNAP’s steps used a unique trap geometry called a passive optical force clamp . Among its benefits, the passive force clamp improves positional precision by eliminating the attenuation between biologically induced motion and detected bead motion. This reduction arises because biomolecules are elastic. Quantitatively, this attenuation is given by kbio/(kbio + ktrap)  where kbio is the effective stiffness determined by slope of the force-extension curve at a given F. For example, the attenuation is 0.7 when measuring a 1,000-nm long DNA at a moderate force (8 pN) (kbio = = 0.24 pN/nm and ktrap = 0.11 pN/nm). Hence, an opto-mechanical precision of 0.7 Å is needed in a traditional optical-trapping assay to equal the state-of-the-art results by Abbondanzieri et al. in a passive force clamp.
To be most effective, sub-Å precision needs to be complemented with equal stability. Yet over long time scales, sub-Å stability remains challenging . A key insight that led to the success of the dual-trap assay was that the mechanical drift of the microscope stage limited instrumental performance . By decoupling the assay from the coverslip [Fig. 1(a)], stability is limited by the differential stability between a pair of laser foci, which forms a local optical reference frame. Each of the two laser foci measures the position of one end of the molecule under tension. The use of such a differential reference frame eliminates common-mode noise sources, such as motion of the microscope objective or air currents in regions where the two lasers are co-linear. To improve upon the near base-pair stability of the original dual-trap RNAP assay , Abbondanzieri et al. enclosed their optics in helium to achieve 1-bp stability over tens of seconds in select RNAP records . Our goal was to develop an opto-mechanical detection system with comparable performance but without the day-to-day complexity of using a helium buffer gas or a passive force clamp. Moreover, we wanted the instrument to achieve this stability routinely to avoid convolving variability in the instrument performance with complexity of the biological assay. Finally, a wide range of single-molecule assays (optical traps [Fig. 1(b)], AFMs [Fig. 1(c)], and super-resolution techniques [Fig. 1(d)]) are coupled to surfaces, so we wanted to demonstrate these performance metrics in a surface-coupled assay by stabilizing an optical-trapping microscope.
Success in applying an optical reference frame to surface-coupled assays requires precise measurement of unwanted stage motion. Such motion can be detected by imaging a bead stuck to a coverslip [30, 31]. Although the bandwidth of the video detection continues to improve , laser-based detection offers higher bandwidth and precision than image-based techniques [1, 33]. In the short term, unwanted stage motion can be subtracted out with 1-Å precision in 1 ms . However, stuck beads move relative to the coverslip on the sub-nm scale . To overcome this problem, we developed firmly attached, nanofabricated fiducial marks that accurately reflect coverslip position [24, 35]. Stability is achieved by active feedback through a 3-axis piezo-electric (PZT) stage . Such active feedback enabled us to achieve 1-bp stability in a surface-coupled optical-trapping assay [Fig. 1(b)] when detecting forward-scattered light . To extend optical stabilization to AFM [Fig. 1(c)] , we enhanced back-scattered detection (BSD)  to achieve 1 Å in 3D  and thereby stabilize tip-sample lateral position to 4 Å over 80 min of imaging . Such success relied upon reducing low-frequency (f) noise by enhancing laser stability and minimizing non-common mode noise . Laser stability was enhanced by active techniques (see Methods). Decreased non-common mode noise was achieved by launching both lasers from the same fiber so that the pointing noise associated with the fiber was suppressed in our differential measurement . To avoid the use of helium, the beam paths—especially for non-common mode beam paths—were minimized. However, our previous designs still incorporated separate quadrant photodiodes (QPDs) for position detection. Differential motion of these separate detectors degrades instrumental performance. Additionally, small thermal variations in the electronics, even in a temperature-regulated room ( ± 0.3 C), decrease long-term stability.
In this paper, we significantly enhanced the long-term stability of BSD while achieving high-temporal bandwidth (100 kHz). To do so, we used a single QPD to detect both laser beams, suppressing the residual motion of the QPD. Each laser was modulated at a separate frequency (1 and 2.5 MHz) using an acousto-optic modulator (AOM). The resulting QPD signal was deconvolved using lock-in amplifiers . Besides enabling the separation of two signals on a single detector, lock-in amplification excels at suppressing low-frequency noise, including a recent application to optical traps . Ångstrom-scale vertical sensitivity, which relies upon excellent intensity stability, was preserved by implementing active intensity control of the modulated laser beams. To demonstrate the performance of this enhanced system, we stabilized the sample of an optical-trapping microscope in 3D with one laser while measuring the resulting stability with a second laser as an out-of-loop monitor. Lateral stability is a key metric for molecular-motor and protein-folding assays, and we achieved a 0.2-Å lateral stability over 100 s based on an Allan deviation analysis (see Fig. 6). Moreover, sub-Å stability was common and reproducible; analysis of a 28-h record showed 100% of sequential 100-s segments achieved a 0.7-Å lateral stability. Our enhanced BSD with lock-in detection improved 3D stabilities as well. We achieved 1-Å stability in 3D over 1,000 s and 1 nm over ~15 h, primarily limited by fluctuations in room temperature. A variety of high-precision single-molecule assays, including optical traps, AFM, magnetic tweezers, and even super resolution techniques , can immediately benefit from this enhanced performance.
Using a single QPD to simultaneously measure two separate signals required modulated lasers and lock-in amplifiers. We first discuss the optical apparatus (§2.1), the specifics of intensity stabilization of modulated lasers (§2.2), the details of high-bandwidth detection using lock-in amplifiers (§2.3), and the data acquisition and position calibration procedures (§2.4). Overall system performance was tested by stabilizing an optical-trapping microscope and quantifying the residual motion with a second laser as an out-of-loop monitor (§2.5).
2.1 Experimental layout
The design strategy and overall optical setup (Fig. 2) was similar to our prior work on BSD . Briefly, two diode lasers (845 and 945 nm, Lumics) were actively stabilized (see §2.2) by the combination of optics shown in the gray-dashed box in Fig. 2. This set of optics enhanced stability by converting a variety of noise sources (pointing, mode, and polarization) into intensity noise that, in turn, was minimized using an AOM inside an analog feedback loop . Each laser was independently translated in the imaging plane by mirrors conjugate to the objective’s back aperture. The diameters of the beams at the back aperture were 3.3 mm (FWHM), purposely too small a diameter for trapping, and had a laser power of ~1 mW at the sample plane. The foci of the two lasers were aligned to each other in 3D and then, via a three axis PZT stage (P561.3DD, PI) to a fiducial mark on the surface. The fiducial marks consisted of a two-dimensional array of silicon posts (650-nm dia., 80-nm high; for a detailed protocol on parallel fabrication of metallic posts, see ). Finally, the combination of the polarizing beam splitter (PBS) and quarter-waveplate (λ/4) led to highly efficient BSD.
The current setup has both beams detected by a common QPD, distinct from our prior work where the signals were split by wavelength and detected on separate QPDs . However, both BSD signals still needed to be independently centered on the QPD to achieve the maximum spatial sensitivity. This centering was done using a custom-built mount designed for stability and minimizing the non-common mode beam path, identical in design to the mount used to translate the beams in the imaging plane. Wavelength separation by dichroic mirrors is not 100%, so optical bandpass filters were added to each arm to block the stray leaked light. The recombined beams were detected with a larger area QPD [dia. = 11.2 mm (YAG-444-4A, EG&G)] reverse biased at −105 V to increase bandwidth . This QPD, and the photodiode for intensity control, were connected to a heat sink to minimize the adverse thermal effects arising from the relatively high reverse bias.
2.2 Intensity stabilization of modulated lasers
When using BSD to detect motion in 3D , fluctuations in laser intensity directly appear as spurious variations in z. Hence, we needed to actively stabilize the laser intensity to achieve 1-Å vertical precision and stability . In our earlier work, the bandwidth of the intensity stabilization feedback loop was ~250 kHz. In the present work, the bandwidth of the demodulated signal was limited by the modulation frequency and the accompanying electronics. Initially, we used modulation frequencies of 65 and 550 kHz, but the resulting bandwidths of the intensity servos were too slow (~1 and ~10 kHz, respectively). As a result, we achieved relatively modest reductions in intensity noise. To increase the bandwidth of the demodulated signal, we raised the modulation frequencies to 1 and 2.5 MHz, relatively high frequencies for our large area (11.2 mm dia.) photodiode. This large area, in combination with known signal filtering of wavelengths beyond the silicon band gap when using a silicon photodiode , necessitated a substantial reverse bias (180 V) to achieve high-bandwidth. This increased bandwidth significantly improved our intensity stabilization as well as allowing for ~100 kHz of bandwidth for conventional AFM and optical-trapping applications (see §3.1).
2.3 Multiplexed position detection using lock-in amplifiers
Position measurements were deduced from voltage signals on the QPD. We first demodulated each quadrant. The resulting lateral signals (Vx and Vy) for each laser were amplified from the normalized difference signals from each half of the QPD, and the raw vertical signals were deduced from variations in the total light falling upon the four quadrants of the QPD. To improve precision, the initial vertical signals were offset amplified to yield final vertical signals (Vz) that better matched to the input voltage range of the data acquisition system .
Thermal stability of the electronics was critical to attaining long-term stability. While the lock-in design was standard, component selection for our custom-built lock-in amplifiers and associated electronics focused on minimizing the effects of thermal variation. For instance, we used thermally stable voltage references (LM399, Linear Technology). We also minimized thermal heating of the electronics by improving the passive air cooling to the pre-amplifier housed on the back of the QPD and to the main electronic boards (amplifiers, filters, lock-in amplifiers, etc). Finally, we lowered the reverse bias voltage on the QPD—but not the PD used for intensity stabilization—from −180 V to −105 V to improve long-term stability without significant loss of temporal bandwidth. Bandwidth for the two separate lock-in signals was determined by measuring normalized peak-to-peak amplitude after demodulation of a blinking light-emitting diode (LED) placed in front of the QPD.
2.4 Data acquisition and position calibration
Precise position determination required compensating for crosstalk in the BSD signals. In other words, a motion of the sample x led not only to a change in Vx, but also Vy and Vz. To quantify this crosstalk, we scanned the fiducial mark in a 3D volume (e.g., 25 × 25 × 25 nm) while measuring the BSD signals (Vx, Vy, and Vz) at each stage position (xstage, ystage, and zstage). The resulting data set was analyzed using an algorithm adapted from optical trapping  to create a 3D position calibration and a set of 35 calibration coefficients () per axis of the form
To rapidly compensate for this optical crosstalk, we used a field programmable gate array [FPGA (PXI-7854R, National Instruments)] to digitize the signal, linearize the response, and move the sample via a 3D PZT stage to hold the sample stable with respect to laser focus. For this experiment, we digitized the signal at 5 kHz after using a 2.4 kHz adjustable antialiasing filter (828L8E-Y, Frequency Devices). A standard PI control loop (Labview 2012, National Instruments) implemented on the FPGA then maintained a constant position. The feedback loop communicated at 5 kHz with the stage controller using a 16-bit parallel digital interface. We limited position updates to 500 Hz since the PZT stage took ~2 ms to respond to nm-sized steps. The position data from both lasers were transferred from the FPGA to the computer using a FIFO (first in, first out) memory structure.
2.5 Stabilizing an optical-trapping microscope in 3D
Our preferred metric for characterizing the opto-mechanical performance of our system is to stabilize an optical-trapping microscope with one laser and use the other laser as an out-of-loop monitor [Fig. 3]. This out-of-loop detector helps identify problems hidden when only analyzing the in-loop error signal. For example, motion of the 845-nm laser focus in the imaging plane—the laser used for stabilizing—is indistinguishable from stage motion. Hence, in such a scenario, the feedback loop compensates for this false displacement, leading to unwanted sample motion. By using a sensor outside of the feedback loop (the out-of-loop sensor), we detected such spurious motion. The differential stability reported by this assay is the same stability needed in subsequent optical-trapping applications : each laser detects one end of the molecule, and the difference reports on the extension of the molecule.
To test our performance using this metric, we stabilized an optical-trapping microscope in 3D by measuring the position of a silicon post with one laser (845 nm modulated at 2.5 MHz), while quantifying this stability with the second laser (945 nm modulated at 1 MHz). The resulting position measurements by both lasers were recorded at 5 kHz. For presentation, the data were filtered to 10 Hz unless otherwise noted. For quantification, we used the Allan deviation (σ), which was determined using
State-of-the-art single-molecule experiments are advanced by providing high-bandwidth detection coupled with atomic-scale stability and precision. After first proving high-bandwidth detection (§3.1), we stabilized a microscope in 3D over 28 h. Analysis shows that lateral stability over any 100-s period was better than 0.7 Å. Moreover, we achieved 3D stabilities of 1 Å over 1,000 s and 1 nm over 15 h (§3.2). Finally, atomic-scale sensitivity was demonstrated by generating and then detecting 1-Å steps (§3.3).
3.1 High-bandwidth, multiplexed detection
High-bandwidth position detection is critical to biophysical studies using optical traps and atomic force microscopy. Both techniques rely on power spectral analysis of thermal fluctuations to deduce the stiffness of the force probe. High bandwidth also allows for detection of briefly populated states, including short-lived (<1 ms) protein-folding intermediates  and more effective averaging of Brownian motion for Å-scale precision.
The bandwidth of our present system was limited by the lock-in detection used to separate the two position signals detected on a common QPD. To measure the performance of our system, we rapidly turned a LED on and off and plotted the peak-to-peak amplitude, normalized by the low-frequency response, as a function of blinking frequency [Fig. 4]. The data shows ~100 kHz of bandwidth, with the more rapidly modulated signal showing slightly faster response, as expected. This relatively high bandwidth is sufficient for typical optical-trapping and AFM-based single-molecule force spectroscopy applications.
Besides this signal separation, our application benefited from lock-in detection’s suppression of low-frequency noise. Lock-in detection—developed in the 1940s —is ubiquitously used in a relatively narrow bandwidth around its modulation frequency to isolate a weak signal in the presence of low-frequency noise. For instance, in AMO physics, it is common to modulate a signal detected on a photodiode at a moderately high frequency (e.g., 50 kHz) to suppress spurious signals, including room lights and thermal variation. In contrast, we optimized our lock-in electronics for stability and responsivity over a broad frequency range.
3.2 Sub-nanometer stability in 3D over multiple hours
Our primary goal in developing multiplexed back-scattered detection was to assure that the smallest fundamental motions in biology [e.g., 1 bp (3.4 Å)] would not be masked by instrumental instability over a long period (~100 s). In other words, we strived for Å-scale stability not just over occasional 100-s periods but overall 100-s periods. To demonstrate our resulting instrumental performance, we stabilized a microscope in 3D over 28 h and analyzed the resulting data. More specifically, we used one laser beam to stabilize the sample and a second laser as an out-of-loop detector to quantify performance, as outlined in Fig. 3. For clarity, we show the full record averaged and decimated to 0.1 Hz [Fig. 5(a)]. Over the first ~15 hours, all 3 dimensions showed less than 1 nm of drift, with slightly increased drift over the last 13 h. While sections of this 28-h record show more drift, a more detailed inspection of different 100-s periods shows excellent Å-scale stability over 100 s [Figs. 5(b)–5(d)]. We note that the y-axis consistently outperformed the other two axes. The exact origin of this increased performance is not known, but we attribute it to the higher sensitivity to motion in the y-axis than the x-axis.
To quantify instrumental performance, we used the Allan variance . The Allan variance analysis is a particularly useful metric for single-molecule applications like optical trapping , since it shows how effectively one can trade off temporal resolution for improved spatial precision by averaging the signal until the instrumental performance is limited by low-frequency drift. We computed the Allan variance—or, more technically, the Allan deviation as described in the methods—from data shown in Fig. 5. The resulting plot of Allan deviation versus averaging time shows sub-Å precision over approximately four decades of averaging time (~0.1–1,000 s). More generally, on short times scales (< 1 s), the spatial precision improves with longer averaging times for all 3 axes, following a t-1/2 dependence that is typical for averaging noise [Fig. 6, dashed line)]. Over intermediate time scales (1–100 s), the Allan deviation stops decreasing and remains near 0.2, 0.1, and 0.4 Å for x, y, and z, respectively. On long time scales (>100 s), the Allan deviation increases with increasing averaging time. Notwithstanding this decreased stability over such extended periods, the Allan deviation stayed below 1 Å up to approximately 2500, 6600, and 2500 s for x, y, and z, respectively. Moreover, the Allan deviation is below 1 nm for all 3 spatial dimensions up to 50,000 s (~14 h), as expected from visual inspection of Fig. 5(a).
Our lab’s thermal stability limited the instrumental performance over extended periods (hours). Variations in temperature often, but not always, correlated with both lateral and vertical drift. While our lab temperature was stabilized, the temperature could vary from ~0.1–1 °C over an 8-h period, depending on outside ambient conditions and other external parameters. For the traces shown in Fig. 5(a), the room temperature slowly increased by ~0.2 °C over 15 h but changed sign and then decreased by 0.4 °C over the last 6 h. Hence, improved long-term stability could be attained by better control of room temperature. Notwithstanding such drift over hours, we emphasize that even during relatively rapid changes in the out-of-loop position shown in Fig. 5 (t = 20–28 h), the stabilization over any 100-s period remained excellent [Figs. 5(b)–5(d)].
3.3 Detecting 1-Å steps
To demonstrate Å-scale precision of our multiplexed detection scheme, we moved the sample in a series of 1-Å steps and detected the resulting motion. More specifically, we actively stabilized stage position in 3D at 500 Hz (see §2.5) and then increased the lateral set point by 1 Å every 2 s while measuring the resulting motion with the out-of-loop detection laser. We note that this step size is 2-fold lower than the manufacturer’s specified precision of our 3-axis PZT stage.
To highlight the importance of an out-of-loop measurement, we plot both the in-loop and out-of-loop signal [Fig. 7(a)]. The in-loop signal is much quieter, as expected. Care should be taken in interpreting this in-loop result, since a properly functioning feedback loop necessarily drives the signal to the set point. Hence, we based our analysis on the out-of-loop measurement. Albeit noisier, this measurement gives a more accurate representation of what to expect in a typical biophysical application. The statistical significance of the detected steps is computed from the Fourier transform of pair-wise distance difference (PDD)  between all pairs of points [Fig. 5(b)]. The primary spatial frequency component was 1 ± 0.25 Å−1 (peak ± HWHM), implying a SNR ≅ 5 based on 1 σ = FWHM/2.35 for data smoothed to 5 Hz.
We have developed a versatile ultrastable measurement platform that has sub-Å stability in 3D over any 100-s period and sub-nm stability over hours for a surface-coupled assay. This platform is compatible with a variety of applications, including AFM, optical trapping, and super-resolution microscopy. We expect this ultrastable platform to enable more robust and routine detection of the smallest unitary steps along DNA (1 bp), allowing researchers to focus on their biological application rather than variability in instrumental performance. In particular, extension of this detection scheme to dual-beam optical-trapping assay [Fig. 1(a)] should yield even better performance, since mechanical noise from stage motion does not degrade the assay. Finally, this extreme stability should open the door to studying a wide range of Å-scale dynamics of single proteins beyond those that show discrete, repeatedly sized motions.
This work was supported by a fellowship from the National Research Council (R.W.), the National Science Foundation (DBI-1353987, Phys-1125844) and NIST. Mention of commercial products is for information only; it does not imply NIST’s recommendation or endorsement. TTP is a staff member of NIST’s Quantum Physics Division.
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