Passive All-Optical Force Clamp for High-Resolution Laser Trapping

Optical traps are useful for studying the effects of forces on single molecules. Feedback-based force clamps are often used to maintain a constant load, but the response time of the feedback limits bandwidth and can introduce instability. We developed a novel force clamp that operates without feedback, taking advantage of the anharmonic region of the trapping potential where the differential stiffness vanishes. We demonstrate the utility of such a force clamp by measuring the unfolding of DNA hairpins and the effect of trap stiffness on opening distance and transition rates.

Figure 1

(a) Experimental geometry for the dual trap apparatus (not to scale). Force-displacement ($F\mathrm{\text{−}}x$) curves for beads held in the two optical traps are drawn schematically; the beads (blue) are connected by a dsDNA tether. Near the trap center, force rises linearly with displacement; farther out, force rolls over and falls to zero. The beam intensity in T1 is weaker than that in T2, so that at equilibrium the bead in T1 sits at the maximum of its $F\mathrm{\text{−}}x$ curve, where the stiffness is zero. The bead in T2 remains inside the calibrated linear region, permitting measurement of the applied force. (b) $F\mathrm{\text{−}}x$ curves for T1. The force is measured in T2 as a function of displacement in T1 (black diamonds) and fit to the derivative of a Gaussian (solid red line). An independent measurement of the $F\mathrm{\text{−}}x$ curve based on the drag force on an untethered bead (open blue circles) is obtained by moving the microscope stage and sample at predetermined rates while measuring the displacement from the trap center.

Figure 2

(a) A DNA tetraloop hairpin with 20 bp stem (inset) attached to dsDNA handles is unfolded near equilibrium by a force of $\sim 14\mathrm{pN}$. The extension alternates between folded and unfolded states of the hairpin. A histogram of the extension of this hairpin collected over 30 s and fit to two Gaussians (red) shows two peaks separated by an opening distance of 17.9 nm. (b) The change in extension upon hairpin opening (black circles) increases as a function of the average displacement from the trap center as the trap stiffness (blue line) decreases. Data are fit to Eq. (1) (red line), using parameters from the $F\mathrm{\text{−}}x$ curve in Fig. 1.

Figure 3

(a) The presence of the optical trap modifies the potential landscape of the unloaded hairpin, reducing it by the work performed by the trap. Positive effective stiffness of the trap and dsDNA handles reduces the height of the energy barrier, while negative stiffness increases the barrier height. (b) The unfolding/folding rate at equilibrium decreases as a function of the average displacement from the trap center (black circles). The data are fit to Eq. (2) (solid line), using parameters from the $F\mathrm{\text{−}}x$ curve in Fig. 1.