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Counter-propagating dual-trap optical tweezers based on linear momentum conservation
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Image of FIG. 1.
FIG. 1.

DT vs ST setup. (a) The experimental DT setup described in this paper, based on the MiniTweezers. 13 Two optical traps are created with two objectives (Obj 1, Obj 2) from counter-propagating laser beams. The force exerted by the traps is measured by two Position Sensitive Devices (PSD 1, PSD 2) with sub-picoNewton resolution by direct measurement of light momentum, while the position of both traps is monitored with nanometer accuracy. The horizontal black arrows show the direction of light propagation. Two force signals, f 1, f 2 are measured (vertical black arrows). (b) Two optically trapped beads and the micropipette used in the ST setup in a video-microscopy image. The beads are ≃ 4 μm in diameter. (c) The ST setup introduced in Ref. 11 is used to create a single-trap and to manipulate a molecule tethered between two beads. One bead is optically trapped, while the other is immobilized on the tip of a micropipette by air suction. (d) Experimental issues in the ST setup. (Left panel) drift effects due to uncontrolled movements of the micropipette. (Right panel) the bead in the pipette is not free to rotate and the molecule can be misaligned with respect to the pulling direction. These effects are largely reduced in the DT setup. (e) The great stability of the two-trap setup allows drift free long-term measurements. Here, we show a 10-min passive mode hopping trace for a DNA hairpin as in Sec. VI A (red points: raw data acquired at 1 kHz, black points box average to 10 Hz), in which the relative position of the traps drifts less than 2 nm.

Image of FIG. 2.
FIG. 2.

Experimental Setup. The scheme of the optical setup, with the optical paths of the lasers (blue and yellow) and of the led (red). Fiber-coupled diode lasers are focused inside a fluidics chamber to form optical traps using underfilling beams in high NA objectives. All the light leaving from the trap is collected by a second objective and sent to a Position Sensitive Detector which integrates the light momentum flux, measuring changes in light momentum. 11 The laser beams share part of their optical paths and are separated by polarization. Part of the laser light (≃ 5%) is deviated by a pellicle before focusing and used to monitor the trap position (Light Lever). The trap is moved by pushing the tip of the fiber tip by piezo actuators (wiggler).

Image of FIG. 3.
FIG. 3.

Force measurement and calibration. (a) The force measurement method, based on linear momentum conservation, exploits the equality between the change in total momentum contained in a volume enclosed by a surface and the momentum flux through the surface. The total momentum change inside equals to the force acting on the bead. The PSD measures the outgoing flux ( ) and the ingoing flux ( ) is determined from measurements at zero force with a bead captured in the trap. (b) The PSD returns a current I proportional to the outgoing flux. The difference between the current measured at zero force and the current measured at a given time is proportional to the instantaneous force, . (c) PSD response during a Stokes test. (Inset) Stokes tests on beads of different size and materials and different buffer solutions. Results from five different beads were averaged in each case (error bars obtained as rmsd). Different responses are obtained because of different drag forces. (Main plot) when the response is rescaled by the viscosity η and the bead radius r all the response curves collapse and the same calibration factor is obtained under different experimental conditions.

Image of FIG. 4.
FIG. 4.

Trap shape and reflection effects in the DT setup. Main plot: force exerted by the laser beam on the bead as a function of the displacement of the bead from the center of the trap 2 (dark line). The measurement was done by immobilizing a bead on the tip of a micropipette (see text). The fair line shows the spurious force measured in the second empty trap which is due to reflected light (f ⩽ 0.2 pN). The lower left inset shows the local stiffness of the optical trap, as a function of the force, as obtained from a numerical derivative of the curve in the main plot. Trap stiffness varies from 10 pN/μm at low forces to a maximum of 25 pN/μm, due to strong nonlinear effects. In these measurements the bead is not free to move along the z axis. The upper right inset shows the reflectivity parameter (as defined in text) as a function of the trap-to-trap distance. The measured reflectivity is never above 3% and can be neglected in our force measurements.

Image of FIG. 5.
FIG. 5.

Fluctuation analysis. (a) the fluctuation spectrum of the differential coordinate obtained in the DT setup on three tethers of different lengths under 10 pN tension: a 24 kbp tether, a 3 kbp tether, and a 58 bp tether with an inserted hairpin. Data were fit to the sum of two Lorentzian (Eq. (8) ). Fit results are shown in Table I . One Lorentzian arises due to fluctuations along the pulling direction, while the other arises due to transverse fluctuations due to misalignment. 18 Transverse fluctuations decay over longer characteristic timescales ( ), their contribution being important at smaller frequencies. The total area covered by the power spectrum (the full variance of the signal) is seen to decrease with the tether's length. (b) resolution of the instrument as a function of bandwidth for different tethers (13) . Due to the large amplitude of the low frequency component in shorter tethers, in the DT setup the minimum resolvable length change is almost insensitive to averaging.

Image of FIG. 6.
FIG. 6.

Elasticity of dsDNA tethers. (a) Linear model of the dumbbell shown in Fig. 1(a) , where three elastic elements with different stiffnesses are arranged in series: Trap 1 (k 1), Trap 2 (k 2), and the tether (k m ). (b) Probability distribution and variance of the generalized force signal, f ϕ, defined in Eq. (5) . Note that the slight shift in the value of the mean force shown in the main plot is due to small force calibration errors (≃3%). In the inset we show the variance computed from the probability distribution at different values of ϕ (solid symbols) and the parabolic fit used to measure the stiffness of both traps and the tether through Eq. (6) (dashed line). (b) Force-distance curves (f 1, f 2) for dsDNA half-λ tethers measured in the DT setup. Note that the two forces have equal averages and opposite sign. Data for two different molecules are shown. (d) Main plot: molecular stiffness (k m ) measured for 5 different molecules. The continuous line shows a fit to the WLC model (18) , giving P = 52 ± 4 nm, S = 1000 ± 200 pN. In the smaller plots we compare the stiffness values of the two traps, k 1 and k 2, measured through Eq. (6) (open symbols) with those measured by immobilizing the bead on the micropipette (solid symbols), see Sec. III . Measurements agree within experimental errors.

Image of FIG. 7.
FIG. 7.

Hopping experiments in DNA hairpins. (a) Signal optimization of the signal in the DT setup. The signal coming from one trap (for example, f 1) has too large a variance to distinguish between the folded and unfolded states which do not appear in the force distribution (fair line, right panel). However, using we can resolve the two peaks (dark line, right panel). The folding/unfolding transition can be observed by using instead of f 1 or f 2 (background noisy trace). (b) Force dependent PM hopping rates of the hairpin measured in two different setups: the DT setup described in this paper (folding/unfolding rates are open/solid triangles) and the ST setup described in Ref. 13 (folding/unfolding rates are open/solid circles). Lines are exponential fits to the data using the Bell–Evans model. 19 Kinetic parameters extracted from the fits are reported in Table II . (Inset) free energy difference between the folded and unfolded states as measured in the two setups via Eq. (20) (DT: solid triangles, ST: open circles). Thermodynamic parameters are reported in Table III .

Image of FIG. 8.
FIG. 8.

Stiffness measurements on short tethers. (a) Stiffness of a molecular construct consisting of two 29 bp handles interspaced by a molecular hairpin. The stiffness was measured from fluctuations, removing the contribution due to misalignment, as detailed in Ref. 18 . The hairpin stays closed in the force range explored. (b) Data points show the force extension curve obtained form the molecular stiffness by integration (Eq. (21) ), the continuous line shows a freely jointed chain fit to the data. Fit parameters b = 1 ± 0.1 nm S = 20 ± 2 pN. (c) Comparison of measured stiffness per basepair on dsDNA tethers of different length. Open symbols: data for a 24 kbp (diamonds) and 3 kb (squares) tethers. Solid symbols data for the 58 bp tether. Stiffness per basepair in the shortest tether is an order of magnitude smaller than in longer tethers. Data obtained from 5 different molecules are shown in the three cases.


Generic image for table
Table I.

Results of the double Lorentzian fits (Eq. (8) ) to the measured spectra in Fig. 5 . and are the amplitude of the slow and fast component, respectively, while and are the corresponding corner frequencies. It might look surprising that the corner frequency is higher for the 3 kbp tether than for the 58 bases tether as the latter is stiffer. This is due to the fact that hydrodynamic interactions are much bigger in the case of the shortest tether.

Generic image for table
Table II.

Kinetic parameters obtained by fitting Bell–Evans model rates 19 to data in Fig. 7(c) . The co-existence kinetic rate k c depends on the setup, but thermodynamic quantities and free energy landscape parameters, , do not. The results are averaged over 5 different molecules and errors are standard error over different molecules.

Generic image for table
Table III.

Thermodynamic parameters of the hairpin obtained by fitting Eq. (20) to the data in the inset of Fig. 7(c) . The results are consistent within experimental error. The results are averaged over 5 different molecules. Errors are standard error over measured over different molecules.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Counter-propagating dual-trap optical tweezers based on linear momentum conservation