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BIOLOGICAL SCIENCES / BIOPHYSICS
Differential detection of dual traps improves the spatial resolution of optical tweezers
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Departments of *Physics Molecular and Cell Biology and Chemistry and Howard Hughes Medical Institute, University of California, Berkeley, CA 94720
Contributed by Carlos Bustamante, April 25, 2006
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The drive toward more sensitive single-molecule manipulation techniques has led to the recent development of optical tweezers capable of resolving the motions of biological systems at the subnanometer level, approaching the fundamental limit set by Brownian fluctuations. One successful approach has been the dual-trap optical tweezers, in which the system of study is held at both ends by microspheres in two separate optical traps. We present here a theoretical description of the Brownian limit on the spatial resolution of such systems and verify these predictions by direct measurement in a Brownian noise-limited dual-trap optical tweezers. We find that by detecting the positions of both trapped microspheres, correlations in their motions can be exploited to maximize the resolving power of the instrument. Remarkably, we show that the spatial resolution of dual optical traps with dual-trap detection is always superior to that of more traditional, single-trap designs, despite the added Brownian noise of the second trapped microsphere.
single molecule | subnanometer resolution | signal-to-noise ratio
One method to increase the stability of optical tweezers is to introduce a second optical trap to hold the other end of the system, thereby decoupling it from movements of the fixed surface. Although such dual-trap optical tweezers were first envisioned more than 10 years ago (17, 18), it has not been until recently that the improvement in stability and isolation has been shown (19). As a demonstration of the possibility of such instruments, Abbondanzieri et al. (20) have observed the individual movements of RNA polymerase on dsDNA, 
3.4 Å, using an ultrastable dual-trap optical tweezers in which the position of one microsphere in a weak trap is monitored and another microsphere in a second, very stiff trap replaces the fixed surface. Alternatively, the second trap in a dual-trap optical tweezers need not be a static attachment point. Meiners and Quake (21) monitor the motion of both microspheres in traps of equal stiffness and measure the femtoNewton dynamics of dsDNA by exploiting the correlations induced into the motion of the two microspheres. Such dynamic correlations allow measurement of some of the lowest forces to date despite the much larger force noise present in the individual motion of each microsphere.
In this paper, we demonstrate that these correlations can be exploited to improve the spatial resolution of optical tweezers. By solving the equations of motion of two microspheres of arbitrary size linked by a flexible polymer held in traps of arbitrary strength, we predict the ability to resolve equilibrium changes in the polymer length based on the motions of each microsphere or from linear combinations of these motions. We find that a coordinate that exploits the correlations between the microspheres provides an increase in spatial resolution over the movements of either microsphere separately. Furthermore, we find that with the proper choice of coordinate, the spatial resolution achievable with dual optical traps is always superior to that of a single trap. Finally, we verify these predictions with a Brownian noise-limited dual-trap optical tweezers.
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eq) = dFDNA/d|eq, where we expand its force-extension behavior (22–24) about the equilibrium extension eq. The drag coefficient of each microsphere of radius r1,2, in the absence of the other microsphere, is given by 
1,2 = 6
r1,2, where is the viscosity of the solvent (9 x 10–10 pNs/nm2 for water). The small frequency dependence of this drag coefficient (25, 26) and any modification due to nearby surfaces (26) are neglected (see instrument description below); however, the hydrodynamic modifications due to the proximity of the second microsphere can provide large corrections (4, 27) and will be included. The small effects of inertial terms, the finite relaxation time of the polymer tether, and its hydrodynamic interaction with the microspheres also are neglected. See Fig. 1a for a summary of this geometry.
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where
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Off-diagonal terms in the stiffness matrix, 
, describe the coupling between the microspheres due to the DNA tether, whereas off-diagonal terms in the mobility matrix, µ, describe the hydrodynamic coupling of one microsphere with the fluid flow generated by movements of the other. This dynamic coupling is generally described by the Oseen tensor, which is approximately constant when the distance between the microspheres, R12, is much greater than their positional fluctuations (4) and the time it takes the fluid to propagate between them is sufficiently short (5). Furthermore, when R12 is much greater than the radii of the microspheres, 
4R12 (4). Finally, the fluctuating Brownian forces, F(t), have zero mean and are uncorrelated in time:
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where 
denotes the outer product of the force vectors and kB is the Boltzmann constant.
The fluctuations in x(t) are described by the correlation functions 
x(t) x(t')
or their Fourier transform, Sx(f), the noise power spectra. As detailed in Supporting Text, which is published as supporting information on the PNAS web site, we solve for this quantity by finding a suitable normal coordinate that diagonalizes Eq. 1 and find that Sx(f) is a sum of Lorentzians with the characteristic frequencies 
+/2 and –/2, associated with the two normal modes of Eq. 1. In a typical measurement, data are collected at a frequency higher than the highest characteristic frequency and then averaged to a frequency, fav, that is lower than the lowest. Thus, the power spectra are white over the measurement bandwidth, B = fav/2, and the mean squared noise in microsphere position over this bandwidth, x xB, is well approximated by the low-frequency power spectra Sx(0) multiplied by 2B. The explicit expressions for x12B, x22B, and x1x2B are given in Eqs. 16 and 17 in Supporting Text.
Measured Signals.
On this fluctuating background, we consider a process that introduces a length change, 
L, in the tether on a time scale sufficiently longer than the relaxation time of the microspheres in the optical traps, allowing the system to equilibrate before and after this length change. As a result, the first microsphere experiences an additional force kDNAL/L, and the second microsphere a force –kDNAL/L. The ratio /L converts contour length to extension; for appreciable forces (>5 pN on dsDNA), this factor is close to unity and will be neglected. Time averaging the equations of motion, Eq. 1, in response to these forces yields the changes in the equilibrium position of each microsphere x1 and x2, listed in Eqs. 20 and 21 of Supporting Text. As expected for a system with finite compliance, x1,2 
L.
The Spatial Resolution of Dual-Trap Optical Tweezers.
A DNA length change, L, is revealed by the measured signal, i.e., the change in the equilibrium displacement of each microsphere from the center of its trap, and the noise in this measurement is given by the integrated fluctuations in each microsphere position over the measurement bandwidth. Thus, the spatial resolution of the instrument (its ability to resolve the change L) is given by the ratio of the measured change in microsphere position to the root-mean-square of its fluctuations: the signal-to-noise ratio (SNR). For the microsphere in trap 1 this is
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SNR2 is found by exchanging indices, 1
2.
To better understand Eq. 3, it is useful to compare it to the SNR for single-trap optical tweezers, which can be derived from SNR1 in the limit that k2 
.
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The expression derived previously in the literature (16) ignores hydrodynamic coupling and is recovered in the –1 0 limit. The hydrodynamic coupling that appears in this equation reflects the fact that the mobility of the first microsphere is modified by the presence of the second, stationary microsphere. Similar modifications to this mobility can occur in the presence of a nearby surface (26).
Comparing Eqs. 3 and 4 for single and dual optical traps, one observes significant differences in behavior. SNRsingle is, remarkably, independent of trap stiffness and grows linearly with DNA stiffness. In contrast, for the dual optical trap, SNR1 depends on the stiffness of the second trap and becomes increasingly independent of DNA stiffness, asymptotically reaching the value
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for kDNA >> k2. This corresponds to the expected SNR for a dumbbell consisting of two microspheres connected by a rigid rod with a total drag coefficient of 1 + 2. Further inspection (see Supporting Text) shows that SNRsingle 
SNR1 for all experimental parameters, only approaching the equality in the trivial limit kDNA 0.
At first glance this appears to be a fundamental limitation of the dual-trap optical tweezers relative to the single-trap optical tweezers. However, in only considering the displacements of the microspheres individually, we have ignored the correlations between them from the coupling by the DNA tether. Significantly, this correlation x1x2B increases with DNA stiffness. If the deflections of both microspheres are monitored simultaneously, a new coordinate may be formed from a linear combination of x1 and x2 that exploits these correlations to produce a higher spatial resolution. The use of such a coordinate to measure the changes in the length of the DNA tether represents a significant advantage of a dual-trap optical tweezers in which the displacements of both microspheres are monitored.
To illustrate this point, consider the difference coordinate, x– = x1 – x2, for which we compute the small bandwidth noise, x–2B = x12B + x22B – 2x1x2B and the signal x– = x1 – x2. The coupling due to DNA leads to a positive correlation in the positions of the microspheres x1x2B, because one microsphere effectively pulls the other in the same direction via the DNA as it undergoes Brownian motion. The negative sign in the definition of x– thus implies that the small bandwidth noise x–2B is decreased by this coupling. In contrast, the signal resulting from a change in DNA length corresponds to motions of the microspheres in opposite directions, and thus the negative sign in x– increases the signal x–. The SNR in this coordinate is given by
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which, unlike SNR1, is linear in DNA stiffness and, thus, does not saturate with increasing kDNA. Unlike SNRsingle, however, SNRdiff depends on both trap stiffnesses and reaches a maximum under the condition
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in which case the SNR is optimal, and given by the simple expression
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where eff = 12/(1 + 2). For a given DNA stiffness and bandwidth, the best possible resolution is achieved with identical microspheres of the smallest possible size in traps of identical stiffness; however, even for two microspheres of unequal size, the resolution in the difference coordinate can be maximized by tuning the trap stiffnesses according to Eq. 7.
The argument outlined above is valid for any arbitrary linear combination of x1 and x2, 
1x1 – 2x2. However, in contrast to the SNR in the difference coordinate, the arbitrary amplitudes 1,2 can be chosen such that the SNR is maximized for all values of the trap stiffnesses. We compute the small bandwidth noise 2B = 12x12B + 22x22B – 212x1x2B and the signal = 1x1 – 2x2 and maximize the SNR in this coordinate, SNR, with respect to the amplitudes 1,2. Because the SNR is independent of overall scaling, SNR depends only on the ratio of the amplitudes 2/1; thus, the condition that maximizes SNR is
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Under this condition, SNR is equal to SNRopt, Eq. 8. For any set of experimental parameters, Eq. 9 defines an optimal coordinate in which the ability to resolve changes in the tether length is maximized. However, a drawback compared with the difference coordinate is that this optimal coordinate requires knowledge of parameters that may change during the course of an experiment, most notably kDNA and . An exception to this situation occurs when the experimental parameters obey the condition in Eq. 7, in which case the optimal coordinate given by Eq. 9 is the difference coordinate and is independent of kDNA. Furthermore, if identical microspheres are used, Eq. 7 is also independent of .
By exploiting the correlations between the microspheres, we have derived an SNR, SNRopt, that does not saturate with DNA stiffness and that is remarkably similar in form to SNRsingle. However, because SNRopt depends on an effective drag coefficient, eff, which is always smaller than either drag coefficient individually, it can be shown that SNRopt SNRsingle SNR1 for all values of the experimental parameters (see Supporting Text). The improvement of SNRopt over SNR1 reflects the fact that, despite the added Brownian noise due to the second trapped microsphere, only a fraction of the noise is relevant to the resolution of the signal. Because the microspheres move in a highly antisymmetric manner in response to change in the DNA length, only fluctuations that are similarly antisymmetric limit the resolution of this signal. By monitoring the motion of both microspheres, we can partition the symmetric and antisymmetric motions of the microspheres to improve the resolution. However, it may still seem surprising that the resolution of a system with two fluctuating degrees of freedom, SNRopt, is higher than that of a system with only one degree of freedom, SNRsingle. The reason is that resolution-limiting fluctuations relax faster in a system with two fluctuating microspheres, where both are involved in the dissipation of thermal energy, than in a system with only one fluctuating microsphere. The smaller effective drag coefficient eff that results from the tether-mediated collective motion of the two microspheres reflects this faster relaxation mechanism.
Hydrodynamic coupling introduces correction terms as seen in Eqs. 3
–9. Significantly, these corrections lower the spatial resolution in the difference and optimal coordinate. As demonstrated by Meiners and Quake (4), antisymmetric motions of two hydrodynamically coupled microspheres are longer-lived than symmetric motions. Thus, more noise power is contained in antisymmetric fluctuations and the net effect is to reduce the correlation x1x2B and the SNR in the difference and optimal coordinates relative to that expected in the absence of coupling. Even for reasonable experimental parameters, the hydrodynamic coupling correction can be important and should not be ignored. Despite this correction, the optimal coordinate, in which the symmetric and antisymmetric fluctuations of the microspheres are best partitioned, continues to yield a higher spatial resolution than all other coordinates.
The fact that the trap stiffnesses do not appear in the optimal spatial resolution, Eq. 8, is significant and hints at the generality of this analysis. For example, it is common to run optical traps in force feedback, displacing the position of one trap relative to the other to maintain a constant force (28) or exploiting nonlinear trapping regimes in which the stiffness of the trap disappears (29). We expect, by the independence of SNRopt on the trap stiffnesses, that these systems should have spatial resolutions identical to those derived above. It is possible to verify this prediction directly by replacing the position of one of the microspheres with the position of the feedback trap, setting the stiffness of that trap to zero in Eq. 1, and deriving the appropriate coupled Langevin equations. An identical analysis yields the same SNR as derived above with the feedback trap stiffness set to zero, as expected. Thus, the best resolution, SNRopt, is still achieved by monitoring the movements of both microspheres, with no improvement to spatial resolution provided by force feedback.
Brownian Noise-Limited Dual-Trap Instrument.
To test the validity of this model, we have constructed a Brownian noise-limited dual-trap optical tweezers instrument with a dual-trap detection system (see Fig. 1b for details). The dual traps are formed from two orthogonally polarized beams generated by a single laser. One beam is steered relative to the other via a two-axis piezo mirror capable of rotating through 2 mrad with stability better than 20 nrad, corresponding to trap movements of 6 µm full range with 0.6-Å stability in the sample plane. Subangstrom determination of each microsphere position relative to its trap center is accomplished by back focal plane interferometry (30), in which the forward scattered light from each trapped microsphere is collected, separated by polarization, and imaged onto two separate position-sensitive photodetectors. With careful alignment, cross-talk between polarizations is maintained below 2%.
Despite the increased isolation due to the dual-trap geometry, significant efforts are also taken to isolate this instrument from different forms of environmental noise. The instrument is located in a basement room on a thick optical table levitated on air, and placed in a temperature controlled room with a typical stability of ±0.2°C, with all fanned cooled electronics placed outside the room. During experiments, the instrument is operated entirely remotely to eliminate thermal fluctuations due to human contact. All optical paths are isolated from background air currents via plastic enclosures. Abbondanzieri et al. (20) fill similar enclosures with helium gas, which has a lower index of refraction than air, to reduce Schlieren noise and improve spatial resolution. In contrast, in our instrument, differential movements of the two traps due to air density fluctuations have been reduced by minimizing the total differential path between the beams. We observe no improvement in spatial resolution by the introduction of helium, within the range of parameters probed here.
Spatial Resolution Measurements.
To test the above theory, a single flexible tether of dsDNA 5 kb in length, 1.7 µm, is formed between two polystyrene microspheres held in traps of variable stiffness (see Supporting Text). The observed noise spectra for these tethers are well described by the expected Lorentzian power spectra above 1 Hz for a large range of DNA tensions (see Fig. 5, which is published as supporting information on the PNAS web site). This result indicates that the predicted spatial resolution can be directly tested on bandwidths larger than 1 Hz. By modifying the distance between the traps, we produce equilibrium movements in the position of the microspheres that well approximate the expected signal for an identical change in the length of the DNA tether. By directly measuring these equilibrium changes and comparing them with measured noise values on a given bandwidth, we compute the SNR. To facilitate comparison to movements of different sizes and durations, we scale these SNR to that of a 1-nm step in a 1-Hz bandwidth.
In the first test of our predicted SNR, a DNA tether is formed between two 860-nm microspheres in traps of equal stiffness, 0.13 pN/nm. By stretching the tether to different tensions, thereby increasing its stiffness, we probe directly the dependence of the different SNRs on kDNA. In Fig. 2, we plot this dependence for the measured SNR for a single trap and the optimal coordinate (also the difference coordinate for a symmetric system) from six different DNA–microsphere dumbbells. Fig. 2 also contains the predicted values for SNR1 and SNRopt plotted as 1-
confidence intervals to reflect the uncertainties in the experimental parameters. The measured values of SNR support the above predictions in both the single trap and the optimal coordinate for low values of DNA stiffness and demonstrate the improvement in SNR in the latter coordinate. As the tether is pulled to higher forces and, thus, higher stiffness, these measurements become increasingly sensitive to nonlinearity in the force and light deflection from each trap; thus, even the optimal coordinate effectively saturates to a maximum resolution. However, even at saturation, this coordinate still shows a 3-fold improvement in resolution over the single-trap coordinate.
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Dual-trap detection should make it possible to detect smaller steps at larger bandwidths and lower DNA tensions than would be possible by only monitoring the motions of one microsphere. To demonstrate this point, we pull a 3.4-kb molecule of DNA tethered between two 860-nm microspheres in traps of equal stiffness to a tension of 7.3 ± 0.1 pN. The distance between the traps is then increased by 3.4 Å every 2 s, corresponding to a length change of dsDNA by 1 bp. Fig. 4a shows the displacements for a single trap and the optimal coordinate filtered to 100 Hz and 2 Hz. Whereas each of the seven steps can be seen clearly in the optimal coordinate, the presence of steps is not apparent in the displacements of a single microsphere. The pairwise distributions for the optimal coordinate and the single microsphere coordinate at 2 Hz (Fig. 4 b and c) confirm this result. Although a peak for all seven steps can be seen clearly for the optimal coordinate, there is no evidence of such peaks for the single microsphere. Because of finite trap stiffness, the peaks in Fig. 4b are separated by only 2.3 Å, the expected signal in the optimal coordinate for a 3.4-Å change in trap separation with the above experimental parameters. Comparable demonstrations of high spatial resolution have already been reported for dual-trap optical tweezers (20), although only one microsphere was monitored in those studies. The results presented here show, however, that monitoring the optimal coordinate makes it possible to resolve these movements with larger microspheres, longer DNA, and 2.5-fold less DNA tension than previously reported.
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Despite the good agreement between theory and experiment in the parameter range studied, it is useful to remember that several higher-order effects are ignored in the derivation of the above SNR. First, the DNA has a finite relaxation time, degrading the instantaneous correlation between microspheres that gives rise to the improvement in SNRopt. Using a simple Rouse model (31) with a drag coefficient extrapolated from experimental measurements (21), we estimate that this effect contributes no more than a 1% correction to the predicted SNR in Fig. 3. However, as the length of the DNA increases, this relaxation time increases and corrections due to DNA dynamics should become more important. Second, fluctuations of the microspheres in directions orthogonal to the DNA tether are coupled into the tether direction and become increasingly important as the DNA length decreases. The Brownian rotation of the microspheres also couple noise into this direction and become increasingly important as the microsphere diameter increases and DNA length decreases. We estimate that orthogonal movements and Brownian rotation degrade the SNR in Fig. 3 by 0.1% and 1%, respectively. Finally, we see no evidence in our measurements for partial shielding of the hydrodynamic coupling between the microspheres by the DNA as reported by Meiners and Quake (21). Our data agree well with the full 14% hydrodynamic correction in the predicted SNR, as given by Eq. 8. The absence of noticeable shielding is not surprising because the measurements of Meiners and Quake (21) are conducted with DNA that was 10 times longer than that used here, and we expect this effect to increase as the DNA length increases.
Practically, it has been shown that the greatest improvement provided by the addition of a second optical trap is the dramatic increase in isolation from the macroscopic environment (19, 20). Beyond this point, we have shown that there is a significant added benefit to monitoring the motions of both microspheres. In particular, the ability of the optimal coordinate to maximize the spatial resolution for a range of experimental parameters represents a significant increase in versatility of dual-trap systems. For example, the freedom to modulate the stiffnesses of the traps without adversely affecting the spatial resolution may be useful in certain applications that require smaller laser fluxes to minimize photoinduced damage or laser heating. Furthermore, many of the biological systems that manipulate nucleic acids or proteins on the subnanometer scale operate only in a limited range of forces. By exploiting correlations in the fluctuations of the microspheres, we have achieved subnanometer resolution at lower forces than previously reported, allowing the study of such systems under more permissive conditions.
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Abbreviations: SNR, signal-to-noise ratio.
¶To whom correspondence should be addressed. E-mail: carlos{at}alice.berkeley.edu
J.R.M. and Y.R.C. contributed equally to this work. 
Author contributions: C.B. designed research; J.R.M., Y.R.C., and D.I. performed research; J.R.M., Y.R.C., and D.I. analyzed data; and J.R.M. and Y.R.C. wrote the paper.
Conflict of interest statement: No conflicts declared.
© 2006 by The National Academy of Sciences of the USA
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