Quantifying the Precision of Single-Molecule Torque and Twist Measurements Using Allan Variance

Abstract

Single-molecule manipulation techniques have provided unprecedented insights into the structure, function, interactions, and mechanical properties of biological macromolecules. Recently, the single-molecule toolbox has been expanded by techniques that enable measurements of rotation and torque, such as the optical torque wrench (OTW) and several different implementations of magnetic (torque) tweezers. Although systematic analyses of the position and force precision of single-molecule techniques have attracted considerable attention, their angle and torque precision have been treated in much less detail. Here, we propose Allan deviation as a tool to systematically quantitate angle and torque precision in single-molecule measurements. We apply the Allan variance method to experimental data from our implementations of (electro)magnetic torque tweezers and an OTW and find that both approaches can achieve a torque precision better than 1 pN · nm. The OTW, capable of measuring torque on (sub)millisecond timescales, provides the best torque precision for measurement times ≲10 s, after which drift becomes a limiting factor. For longer measurement times, magnetic torque tweezers with their superior stability provide the best torque precision. Use of the Allan deviation enables critical assessments of the torque precision as a function of measurement time across different measurement modalities and provides a tool to optimize measurement protocols for a given instrument and application.

Single-Crystal Rutile TiO2 Nanocylinders are Highly Effective Transducers of Optical Force and Torque

“…26 (iii) The shorter angular trap relaxation time ( t c,θ = γ θ /κ θ ) results in enhanced temporal resolution. 26…”

“…The stiffness and drag are the two primary physical parameters that describe the behavior of optically trapped particles. 26 Therefore, we experimentally determined the trap stiffness (linear and angular) and the corresponding drag coefficient for each trappable rutile TiO 2 cylinder batch, with the results summarized in Figure 3. We used laser powers less than 100 mW to demonstrate the high performance of rutile TiO 2 cylinders even in the low-power regime.…”

“…Utilizing the high linear and angular stiffnesses achievable with low-drag rutile TiO 2 nanocylinders, we employ Allan deviation (AD) analysis 26 to demonstrate the ensuing improvements in measurement precision. For the primary quantities of physical interest, force, torque, position, and angle, we demonstrate in Figure 4 that one can achieve excellent precisions below 1 pN, 1 pN·nm, 1 nm, and 1 degree, respectively, on short time scales between (sub)milliseconds and seconds.…”

“…Each colored solid curve represents the normal AD obtained from a single cylinder measured at 92 mW. Colored symbols reflect the octave-sampled overlapping AD, and their fit to the theory 26 is overlaid as a dashed curve. The AD values report on the achievable measurement precision for a given averaging time, and the horizontal dashed red lines indicate the typical precisions required for measurements of individual biomolecules.…”

“…25 Alternatively, one may employ larger particles to achieve higher force and torque, but the increased frictional drag will decrease the achievable spatiotemporal resolution in these parameters (and their conjugate variables, position and angle). 26…”