Identification and correction of miscalibration artifacts based on force noise for optical tweezers experiments

Force spectroscopy by optical tweezers has provided detailed insights into the mechanics and thermodynamics of nanoscale biological systems. Its high stability and precision have enabled the deciphering of energetic folding models in the folding of proteins and nucleic acids^1–7 and in concurrent binding of ligands,^8,9 as well as the observation of conformational changes in enzymes¹⁰ and the stepping of molecular motors^11–13 or of enzymatic complexes.^14,15 In a typical optical tweezers experiment, the force signal is determined from the bead displacements from the trap centers. The detectors record the bead deflection from the traps and output a signal in units of voltage that is then translated into units of length. To obtain quantitative forces from the bead deflection, the trap stiffness has to be accurately calibrated. Typically, the two necessary parameters, i.e., the detector sensitivity β (in units of nm/V) and the trap stiffness k (in units of pN/nm), are determined using the equipartition theorem by measuring the thermal noise of bead fluctuations.^16,17 When performed carefully, the calibration procedure allows for a determination of β and k with errors of <1%.¹⁷ In practice, however, the estimated accuracy of the calibration is often much worse and can exceed ≈10%. Possible sources for these larger uncertainties are diverse but mostly arise from experimental limitations, e.g., by introducing statistical errors due to a finite acquisition time of the free bead motion during calibration or systematic errors due to wrongly assumed values for temperature, viscosity, or bead size. In a dual-bead assay, further complications arise from beam depolarization, causing crosstalk between the two traps, which is typically removed using an additional calibration step.¹⁸ If these crosstalk parameters are not determined accurately or change over time (e.g., due to instrumental drift), their error propagates into all subsequent measurements. Finally, it is possible, in practice, that calibration data are not available for each individual bead pair or that the user must rely on a previously determined average.

Here, we show that miscalibration of β and k can, furthermore, cause significant misestimation of the tether persistence length or the stretch modulus by more than 100%, which, consequently, may complicate downstream analyses. We derive a framework for calculating the expected deflection noise from equilibrium force-extension curves (FECs) and use this information to identify, quantify, and reverse miscalibration artifacts. Furthermore, we introduce a force and noise model for non-harmonic trap potentials to account for trap softening at higher forces. Finally, we test our method by applying it to simulated and experimental datasets of double stranded DNA (dsDNA) and proteins coupled to dsDNA tethers.

In a dual-trap optical tweezers setup [the inset of Fig. 1(a); also see Fig. S1], the signal x_V, representing the deflection of a bead away from the trap center, is recorded for each trap in units of voltage. The deflection is then converted to a deflection in units of length, x = x_Vβ, using the calibration factor β. After this, the signal is converted to a force using the trap stiffness, k, and F = kx. In the case of miscalibration, the true calibration factors β and k are replaced by apparent calibration factors β^app = ββ^† and k^app = kk^†, respectively. Here, β^† and k^† indicate how much the calibration factors deviate from the perfectly calibrated case, in which β^† = k^† = 1.

In the following derivations, we assume that a tethered biomolecule undergoing conformational transitions is being stretched at quasi-equilibrium, i.e., the trap distance, d, is changed at speeds slower than the kinetic timescale of the system such that all components can equilibrate at any given trap distance. In particular, this requires that the pulling must be slower than the slowest timescale (e.g., folding/unfolding transitions) in the molecule of interest. For a given trap distance, d_i, we, therefore, define the average force as $F=Fdi$ and the corresponding deflection noise as $σ=Var(x)di$⁠. To unambiguously determine the trap distance d, defined here such that d = 0 when the bead surfaces touch, we employ a correlation-based strategy outlined in Fig. S1.

We first simulated force-extension curves (FECs) of a dsDNA tether (L = 360 nm, p = 25 nm, and K = 1200 pN), modeled as an extensible worm-like chain (eWLC)¹⁹ in the presence of miscalibration (see Sec. IV for further details on simulation). The tether extension ξ is given by the difference of the trap distance d and both bead deflections x₁, −x₂ [Fig. 1(a), inset]. A representative apparent force-extension curve [FEC, gray points in Fig. 1(a)] shows slight deviations from the expected behavior (dashed line). However, the shape of the apparent FEC still resembles the true FEC, and it is still well-fitted by using an eWLC model [black line in Fig. 1(a)]. As such, FECs that result from a miscalibrated trapped bead are almost indistinguishable from non-miscalibrated FECs, and inaccurate calibration parameters are hard to identify.

We then systematically varied the miscalibration factors β^† and k^† to determine their effect on the extracted eWLC fit parameters. In the case of no miscalibration (β^† = k^† = 1), fits to the apparent FEC reproduced the true, initial parameters [Figs. 1(b) and 1(c)]. An underestimation of the trap stiffness (k^† < 1) generally led to an overestimation of the persistence length and a slight underestimation of the stretch modulus of the tether [Fig. 1(d)]. The effects were more dramatic when the sensitivity was not accurately determined (β^† ≠ 1): whereas inaccuracy of sensitivity had a similar effect on the persistence length as that of the trap stiffness, only a 10% error in the sensitivity led to misestimation of the stretch modulus by more than 100% [Fig. 1(e)].

While it is almost impossible to determine inaccuracies in calibration from fits of eWLC models to FECs alone, we reasoned that they might become apparent when the noise of the bead deflection signal, $σ=Var(x)$⁠, is considered in addition to the average force. The force noise can be quantified from either passive mode [constant trap separation in distinct steps, Fig. 2(a)] or active mode measurements [active pulling, Fig. 2(b)] and generally decreases with increasing force [Fig. 2(c)]. Since x^app = x_Vβ × β^† = x × β^†, the bead deflection will only be affected by errors in sensitivity. In contrast, the accuracy of the force depends on the amount of miscalibration present in both sensitivity and trap stiffness as F^app = k^appx^app = kx × β^†k^†. Here, we show that, under perfectly calibrated conditions, the noise level σ can be calculated directly from an FEC $F(ξ)≔Fdiξdi$⁠, but that there is a discrepancy between the measured and calculated noise levels under miscalibration conditions.

In the absence of miscalibration, the expected deflection noise at infinite bandwidth is

where k_eff = k_c + k_L is the effective tether stiffness of the system, $kc=1/k1+1/k2−1$ is the combined stiffness of the two traps, and $kL=∂F(ξ)∂ξξeq$ is the tether spring constant (see the supplementary material for further details).

We performed Langevin dynamics simulations of FECs to show that the expected noise, σ_calc [Eq. (1)], is equal to the measured noise σ_meas in the absence of miscalibration [red line and gray points in Fig. 2(d); see Sec. IV for details]. However, in the presence of miscalibration, we observed significant differences between the expected noise and the measured noise [red line and gray points in Fig. 2(e)]. Even small miscalibration artifacts are often readily identifiable in the noise, and therefore, we can use the coincidence of σ_calc and σ_meas as an indicator for miscalibration in force-extension data.

Having recognized that the difference of the experimental and calculated noise levels allows us to identify the presence of miscalibration, we next set out to devise a theoretical framework for calculating the expected noise in a realistic two-bead system and how it varies in the presence of miscalibration.

To this end, we considered the continuous-time Langevin equation of the bead deflections $x=x1x2$ in a bead–tether–bead system [see the inset of Fig. 1(a)],

where

describe the friction interactions of the beads and the spring constants in the system, respectively.²⁰ Here, k₁ and k₂ are the Hookean spring constants of the two traps, k_L is the stiffness of the tether at extension ξ, γ is the Stokes friction coefficient of a bead, and Γ describes the hydrodynamic coupling between the beads (see the supplementary material).

Thermal fluctuations are white and, with means and covariances, can be determined by

where “⊗” is the outer product. This system of equations can be solved to produce the power spectral density (PSD) of the combined bead deflection,²⁰ of which a detailed derivation can be found in Ref. 21. In brief, we diagonalize the matrix μκ such that A⁻¹μκA = λ, where λ is a diagonal matrix of the eigenvalues of μκ.²¹ The transformation matrix A can be normalized such that AA^t = μ, and hence, the covariance matrix of the experimental coordinates becomes

which can be used to derive the PSD matrix

The PSD along the experimental bead deflection coordinate x₁ − x₂ is

with $ηt=1,−1$ in the absence of miscalibration. In the more general case, in which both traps may be miscalibrated by $β1†$ and $β2†$⁠, $ηt=β1†,−β2†$⁠. An explicit form of the experimental PSD can be found in Eq. (S5). Finally, the noise of the combined bead deflection is

If both trap stiffnesses are similar (k₁ ≈ k₂ ≈ 2k_c), the PSD becomes

i.e., a Lorentzian, and we recover the noise relation expected from the equipartition theorem, $σcalc=kBT/kc+kL=kBT/keff$ [cf. Eq. (1)].

Equation (8) describes the noise only at infinite bandwidth, which is experimentally inaccessible. In a real experiment, the expected noise is modified by a filter cascade that depends on the instrumental setup [Fig. 3(a)]. Possible filters include parasitic filtering by using detectors with a heterogeneous frequency response,¹⁶ anti-aliasing filters that are applied before signal sampling,²² and/or down-sampling before signal storage. The effect of filtering can be accounted for using

where $G(ν)$ is the frequency-dependent filter gain. We list several commonly used models of filters in the supplementary material. The calculated noise $σcalcfilt$ [Eq. (10)] differs from the previously derived approximation for σ°_calc [Eq. (1)] in that it correctly describes the filtering and sampling-induced noise modifications that are present in all experimental data.

Miscalibration artifacts can be reversed when β^† and k^† are known. Under miscalibration, the apparent deflection is x^app = xβ^†. Similarly, the true force and combined spring constant are replaced by F^app = Fβ^†k^† and $kcapp=kck†$⁠, respectively. Consequently,

which is only applicable in the case of identical traps, i.e., k₁ = k₂, $k1†=k2†=k†$⁠, and $β1†=β2†=β†$⁠. In the more general case of all these values being different between traps, the miscalibration factors in Eq. (11) become

The apparent noise in the presence of miscalibration $σcalcapp(β1†,β2†,k1†,k2†)$ is then calculated using Eq. (10) and $ηt=β1†,−β2†$⁠, $k1=k1app/k1†$⁠, and $k2=k2app/k2†$⁠. For example, in the simplified case of identical traps and in the limit of infinite bandwidth, the calculated noise becomes

Next, we performed simulations to verify that the calculated noise $σcalcapp(β1†,β2†,k1†,k2†)$ in different filtering scenarios, indeed, coincides with the measured apparent noise σ_meas. Therefore, we can find the correction factors of miscalibration β^† and k^† by matching the measured and calculated noise using

for identical traps, or in the general case,

A solution to Eqs. (15) and (16) can be found by non-linear least squares.

To verify the effectiveness of the proposed correction procedure [Fig. 3(b)], we simulated FECs with varying values for β^† and k^† and applied Eq. (16) to estimate the miscalibration factors $β†̂$ and $k†̂$⁠. We were then able to calculate the corrected forces and bead deflections $F=Fapp/β†̂k†̂$ and $x=xapp/β†̂$⁠, leading to the recovery of the true FEC [dashed line in Fig. 1(a)] and, therefore, the true eWLC fit parameters [blue points in Figs. 1(d) and 1(e)].

The post-measurement correction of data is only unbiased when both the experimental noise σ_meas(d) and the compliance of the tether k_L(d) are accurate. We tested the data recovery performance using two realistic, but non-ideal experimental scenarios: an FEC and its corresponding noise can be sourced either from discrete steps of the trap distance d [Fig. 2(a)] or from an equilibrium force ramp in which the noise is estimated based on fluctuations of the deflection relative to a fitted baseline [Fig. 2(b)]. In both situations, there is a trade-off between accurately estimating σ_meas(d) (requiring few, but long bins) and $kL(d)=∂F∂ξ$ (usually determined by finite differences, i.e., requiring many short bins). In the scenario of a 360 nm tether, simulations show that σ_meas(d) can be calculated to sufficient accuracy if the number of points in a bin is ≳300 (i.e., ≳10 ms at 30 kHz, Fig. S2). For data with positive compliance, k_L is adequately estimated if an FDC consists of ≳50 data points. In practice, a single curve pulled at 500 nm/s is sufficient.

A further potential source of bias arises from variations in bead size. In our derivations above, we assumed that both beads have the same size and, therefore, the same Stokes friction coefficient. Large differences (≳30%) in bead size will, indeed, affect the PSD. However, a typical optical tweezers setup includes a bright-field camera to visualize trapped beads, and hence, unusual bead sizes and shapes are immediately apparent. In contrast, small differences in bead size are likely to remain undetected, and therefore, we performed simulations in which we assumed a variability in the bead diameter of 10%. The uncertainty in β^† increased only slightly from 1.8% to 2.0%, whereas the uncertainty in k^† remained unaffected at 2.6%. Hence, small differences in the bead diameter are negligible, and the assumption of representing two beads with the same diameter is met.

We note that the fluctuation noise model only describes the thermodynamic fluctuation noise of the mechanical system itself. Significant external noise (for example, electronic noise or mechanical vibrations) will increase the apparent signal noise, usually in a narrow frequency range, and may lead to erroneous results if unidentified. It is, therefore, still important to inspect the signal PSDs for excess noise.

Quasi-equilibrium fluctuations of proteins have previously been used to determine their free energy of folding.^23,24 Here, we simulated the effect of miscalibration artifacts on a protein with an unfolded contour length of 30 nm and a free energy ΔG = −16.1  k_BT. The protein modeled was connected to a tether of length 360 nm, and the trap distance was increased in 80 steps from 200 to 500 nm. Similar to real experiments, trajectories were 8-pole Bessel filtered at 75 kHz, acquired at 150 kHz, and down-sampled using a 5-point boxcar filter before storage. Figure 4(a) shows FECs with different, random miscalibration of trap stiffness and sensitivity. The trajectories scatter significantly, and consequently, there are large variations in the obtained free energy. After miscalibration correction, the FECs collapse onto the same curve [Fig. 4(b)] and the precision of free energy estimation increases significantly, effectively removing the substantial artifacts.

Until this point, the traps were treated as Hookean springs as this is the simplest approximation mathematically for any attractive potential and is widely used for optical traps. However, optical traps become decidedly non-Hookean at higher forces, especially when measuring with silica beads that have a lower refractive index than polystyrene beads. Therefore, it is necessary to account for deviations from the ideal case of a harmonic trap potential.

In the low-force regime, the trap potential is well-approximated by a constant trap stiffness k,

resulting in the linear dependence of force on the deflection, x. However, a realistic trap potential does not extend to infinity, and consequently, the trap must soften. Therefore, we consider the following descriptive model of an attractive trap potential that has the same stiffness at the equilibrium point but softens as the deflection increases [Figs. 5(a) and S3]:

where w describes the harmonicity of the potential and k is the stiffness around the equilibrium point x = 0. A fully Hookean trap is recovered as w → ∞. In addition to the softening of the trap, for back-focal plane detection, where the trapping laser also detects the bead deflection, also the deflection signal is affected. Hence, when the instrument becomes nonlinear at high force and this is unknown to the experimenter (i.e., they work with a Hookean trap assumption), they will observe a bent FEC, i.e., wrongly interpret it as a softening of the tether [Fig. 5(b)].

The apparent deflection signal is lower than the true deflection to the same extent that a realistic force is lower than a Hooke force,

Consequently, the apparent force experienced by beads in a softening trap in a setup with back-focal plane detection,

is, indeed, the true force, and only the bead deflections themselves have to be adjusted.

To correct for non-harmonicity of the traps, we can obtain the true deflection of trap i (with i = 1, 2 in a dual-trap assay) by inverting Eq. (19),

and an FEC can thus be corrected using the transformations

which, in the more general case of also including inaccuracies in the zero-deflection calibration factors (i.e., $ki†≠1,βi†≠1$⁠), become

Here, ψ_i of trap i is defined as

In summary, an FEC recorded on an instrument with non-harmonic and miscalibrated traps can be corrected when the user has access to the values of $ki†$⁠, $βi†$⁠, and w_i. As described above, the miscalibration parameters can be extracted by matching using the theoretical and experimental noise profiles [Eq. (16)]. Here, we set k_L = ∂F/∂ξ [Eqs. (24) and (25)], describe the softening of the traps with the transformation

and take into account the additional noise reduction due to the softening traps [Var(f(x)) ≈ (f′(U(x)))² Var(x)],

We verified in simulations that we can efficiently restore true FECs from miscalibrated non-harmonic FECs for a wide range of miscalibration factors and trap non-harmonicities [Figs. 5(c)–5(e)].

So far, we have reported results based on simulated datasets. To address the performance and limitations of the proposed correction methods, we next applied it to experimental data.

We first examined a simple construct consisting of only two short dsDNA handles, each 545 bp in length, that were bridged by a dimerized oligonucleotide (see Sec. IV for details). These tethers were stretched using 1 µm silica beads, and traps set to each have a stiffness of k_1,2 ≈ 0.3 pN/nm. One of our first observations was that the experimental FECs often deviated from the expected dsDNA behavior at high force [orange points in Fig. 6(a)]. Our hypothesis that this deviation was caused by a softening of the traps instead of a softening of the DNA tether was confirmed by an analysis of the noise: When we calculated the expected noise from the apparent tether stiffness assuming harmonic traps, we found significant deviations from the experimentally determined noise [Fig. 6(b)]. Next, we used our correction method to identify and account for trap non-harmonicity such that the calculated noise matched the experimental noise [Fig. 6(c)]. The resulting corrected FEC followed the expected eWLC behavior without the artificial curvature at high forces [purple points in Fig. 6(a)]. When applied to data of multiple dsDNA molecules, our correction procedure robustly removed variations at both low and high forces [Figs. 6(d) and 6(e)].

In a common experimental setup, a biomolecule of interest is tethered between dsDNA handles. Here, we chose to reanalyze published data of CTPRrv5, a tetratricopeptide repeat protein with five helix-turn-helix repeats.²⁵ FECs were recorded on the same instrument at different times over the course of almost three years. When overlaying FECs from different time periods, we noticed minor but noticeable differences in their shapes [Fig. 7(a)]. In our previous study, we developed a model to fit equilibrium FEC to extract the mechanical parameters of the dsDNA linker and the energetic parameters of the protein. After fitting this model to our data, we found that the resulting values for the stretch modulus of the linker scattered significantly between different molecules [the inset of Fig. 7(a)]. We verified that the observed heterogeneity was not a consequence of physical modifications to the protein or dsDNA linker using, e.g., mass spectrometry and electrophoresis. Instead, we were able to trace back larger variations in the data to hardware modifications of the instrument between measurement cycles. Therefore, we compared the expected and experimental noise for a molecule [Fig. 7(b)] measured before and after one particular instrument modification and observed that the noise ratio, $σcalcapp/σmeas$⁠, differed significantly. This led us to assume that the physical shape of the trap potential and/or the calibration factors changed due to the instrument reconfiguration. After applying our correction procedure, all FECs superimposed clearly [Fig. 7(c)], the deflection noise fulfilled the theoretical thermodynamics expectations [Fig. 7(d)], and the scatter was removed from the fitted stretch modulus [the inset of Fig. 7(c)].

Force spectroscopy experiments have been used successfully and extensively to characterize the thermodynamics and kinetics of nanoscale biological systems. However, the accuracy and precision of such measurements strongly depend on the correct calculation of parameters that describe all mechanical components present in the system. Particularly in systems in which the energy of a conformational change within the biomolecule is much smaller than the energy stored in the stretched handles, it is crucial to describe the tether properties accurately and precisely. As with any other type of experiment, artifacts can be present in force measurements but are difficult to identify solely from force signals (Fig. 1). Here, we showed that small inaccuracies in instrument calibration lead to large discrepancies in the tether properties. Consequently, as miscalibration artifacts are propagated, they also influence the thermodynamic description of the system under investigation and, hence, may alter down-stream conclusions.

The methodological framework presented here provides researchers with tools to identify and correct these artifacts by comparing and fitting theoretical and experimental noise fluctuations. This procedure is not limited to dual-beam optical tweezers but can also be adapted to other types of nanomechanical experiments, such as atomic force microscopy, acoustic force spectroscopy, and magnetic tweezers, where the relevance of registering noise fluctuations has also been recognized by other groups.²⁶ Furthermore, it can be used to verify and re-calibrate data post-analysis even if they were previously recorded with uncertain calibration parameters. We would like to point out that analyzing noise fluctuations can also be an integral part of characterizing the system of interest,²⁶ and hence, our method can be streamlined with other thermodynamic investigations. In summary, we anticipate that our work will aid in the reduction and removal of commonly encountered calibration artifacts and hope that such a method has the potential to enable a world-wide benchmark study of the custom-built and rapidly increasing number of commercial instruments.

Simulations of the tether fluctuations at a trap distance d were performed by integrating the time-discretized Langevin equations of the bead displacements [Eq. (2)] as described previously,²³ 

where ξ = d − x₁ − x₂, Δt = 1 × 10⁻⁸ s, and Γ_i describes the uncorrelated noise with $Γi(t)=0,Γi(t)Γi(t′)=δ(t−t′)$⁠. Note that the sign of x₂ is opposite to the description in the main text.

The force response of the tether was modeled by using an extensible worm-like chain model,¹⁹ 

For simulations of equilibrium transitions, an additional worm-like chain compliance was introduced.²⁷ Its contour length was determined using an additional Monte Carlo step, and the kinetics were chosen such that transitions occurred at a much faster timescale than that of pulling. Unless indicated otherwise, trajectories were filtered using an 8-pole Bessel filter at 75 kHz, sampled at 150 kHz, subjected to a five-point boxcar filter, and stored after down-sampling to 30 kHz, thereby mimicking typical experimental conditions. Folding free energies of simulated FDCs were then obtained using an equilibrium model as described previously.²³ 

Bridged DNA oligos were generated by linking two 3′-maleimide-modified oligos overnight at 4° C using DTT at an equimolar ratio. Linked oligos were subsequently purified by gel filtration using a S200 increase 10/300 GL size exclusion column (GE Healthcare) equilibrated in 25 mM Tris-HCl pH7.0, 150 mM NaCl. CTPRrv5-oligo chimeras were generated using cysteine–maleimide- or ybbR-CoA-based coupling strategies, which are described in detail elsewhere.²⁵ Oligo-dimers were hybridized to 545 bp dsDNA handles modified with dual-biotin/dual-digoxigenin modifications (Biomers, custom synthesis), and protein-oligo chimeras were hybridized as described previously.²⁸ Measurements were performed in 25 mM (short DNA) or 50 mM (CTPRrv5) Tris-HCl pH 7.5, 150 mM NaCl using the glucose-oxidase oxygen scavenging system [0.65% (w/v) glucose (Sigma), 13 U ml⁻¹ glucose oxidase (Sigma), and 8500 U ml⁻¹ catalase (Calbiochem)].

Data of short dsDNA handles were acquired on a LUMICKS C-trap at a sampling rate of 78.125 kHz. Parasitic filtering due to detection using QPD devices was modeled as described previously.²⁹ Signals were passed through an on-board anti-aliasing filter (National Instruments) prior to storage, and its gain was modeled according to manufacturer publications.

CTPRrv5 data were acquired on a custom-built optical tweezers instrument, and detection filtering by using QPD devices was modeled again as described.²⁹ Data were passed through an 8-pole Bessel filter set to 50 kHz, sampled at 100 kHz, and boxcar-filtered down to 20 kHz to save storage space.

In both cases, the correction procedure modeled the respective filter cascade.

See the supplementary material for detailed information on deflection noise, PSDs, hydrodynamic coupling, and commonly used filter models, as well as additional figures about the length convention in a two-bead dumbbell assay, the estimator accuracy, and the extent of non-harmonic trap potentials.

We thank Andreas Weißl and members of the Stigler and Hopfner labs for critical reading of the manuscript. M.S. acknowledges the funding from the BBSRC Doctoral Training Partnerships, Cambridge, and a travel grant provided by the Non-globular Protein Network (COST Action BM1405-39176). J.S. acknowledges the support from the LMU Center for Nanoscience CeNS, a DFG Emmy Noether grant (No. STI673/2-1), and an ERC Starting Grant (No. 758124).

The authors have no conflicts to disclose.

M.F. and D.K. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

An implementation of the correction procedure (Correction of Tweezers Calibration Factor, CTCF) is available as a Python script on http://github.com/StiglerLab/CTCF.