Fast Force Clamp in Optical Tweezers: A Tool to Study the Kinetics of Molecular Reactions

Abstract

A dual-laser optical tweezers has been developed to study the mechanics of motor proteins or DNA filaments. A bead attached to one end of the specimen is trapped in the confocal point of the two lasers, while the other end is connected to a three-dimensional piezo-stage. The instrument can be operated under computer control either as a length clamp, applying length steps or ramps, or as a force clamp, applying abrupt changes in load of fixed magnitude and direction. The dynamic range of the instrument (0.5–75,000 nm in length and 0.5–200 pN in force) and the speed of the force feedback permit recording the kinetics of molecular and intermolecular phenomena such as the overstretching transition in double-stranded DNA (ds-DNA) or the generation of force and shortening by an ensemble of myosin motors pulling on an actin filament. We demonstrate the performance of the system in recording for the first time the transient kinetics of the ds-DNA overstretching transition, which allows the determination of the underlying reaction parameters, such as rate constants and distance to the transitions state.

Appendix: Influence of Viscosity on the Kinetics of the Elongation–Untwisting of the ds-DNA During the Overstretching Transition

1.1 Drag Produced on the Trapped Bead by the Viscosity of the Medium

A general problem with force clamp experiments made using the dual-laser tweezers apparatus is that the length change elicited by a force step is realized through movement of the piezo-stage and thus of the fluid surrounding the bead. Consequently, the change in the position of the trapped bead reliably measures the tension on the molecule only if it is not influenced by the drag due to the movement of the solution accompanying the movement of the stage. The drag on the bead is $F_{v} = 6\,\pi \,\eta \,Rv$ (Eq. 1), where η, the viscosity of the solution, is 10⁻³ Pa s, at 25 °C, R, the radius of the bead, is 1.64 or 1.09 µm, and v is the translational velocity of the bead. Considering that the stiffness of the molecule is ~60 pN µm⁻¹ and the stiffness of the trap is 150 pN µm⁻¹, a step of 2 pN complete in 2 ms implies a bead movement of ~40 nm at a velocity of ~20 µm s⁻¹. Consequently, for the bead with R = 1.09 µm, F _v attains a value of 0.5 pN and decays with a time constant of 0.5 ms (1/4 the risetime of the step). This analysis indicates that the viscous drag on the bead does not significantly influence the position of the bead during the step nor the observed elongation kinetics.

A direct test of this conclusion is obtained by comparing the r–F relations obtained with different bead diameters. In Fig. 10a, the blue points data from Fig. 7c are unpooled to identify those obtained with beads of 3.28 µm diameter (black symbols, 12 molecules) and 2.18 µm diameter (red symbols, 5 molecules). Moreover, in Fig. 10b, r is plotted on a log–log scale against the final length change (ΔL _e) induced by a step, for the same data as in a. In both cases, it is evident the absence of any effect of the bead diameter on the overstretching kinetics.

Fig. 10

a Relation of ln r versus F for the 2 pN steps (blue symbols) from Fig. 7b. Red symbols refer to data obtained with 2.18 μm bead diameter (5 molecules); black symbols to data obtained with 3.28 μm bead diameter (12 molecules). b Log–log relation between r and amount of lengthening (ΔL _e) for the same 2 pN steps as in a. c Relation of r versus the total molecular length (L) attained following 0.5 pN (green circles) and 2 pN (blue symbols) steps. 2 pN data pooled from the same molecules as in Fig. 6d. Figures close to filled symbols indicate the respective forces. From Supplementary Material in Bianco et al. [5]

1.2 Rotational Drag of the Molecule While Untwisting

A rotational drag of the ds-DNA while untwisting in response to a rise in torque has been directly measured by attaching a bead near a nick and determining the angular velocity [7]. In this way, it has been shown that the untwisting takes several minutes and the drag dominates the elongation–untwisting velocity. However, in that experiment, the drag should be several times larger than in our experiment, as it is generated by the revolutions of a large bead accompanying the untwisting of the molecule.

During the overstretching transition under our conditions, the molecule of DNA elongates by 11 µm, while it reduces the number of turns from 4,500 to 1,450, that is, it untwists by 278 turns µm⁻¹. The largest elongation in response to a 2 pN step is ~5 µm attained within 0.5 s (Fig. 6). This implies a rotational speed ω of (278 × 5/0.5 =) 2,780 turns s⁻¹, so that each of the two ends of the molecule should counter-rotate at 1,390 turns s⁻¹. According to measurements of rotational drag in the experiment of Thomen et al. [39], where the two strands of DNA are attached to two independent beads and separated at different velocities, a torque of 0.6 k _B T would be necessary for the rotation at the speed of our overstretching transition. However, in that experiment, the rotating stretch is at longitudinal force zero (and therefore, the molecule is not straight), while during the overstretching transition, the longitudinal force is ~65 pN and the molecule can be assimilated to a rigid rod. Modeling a rigid rod [23] leads to a torque $\vartheta = 4\pi \eta R_{H}^{2}\,L_{\text{eff}}\,\omega$, where η is the viscosity of the solution as above, R _H the hydrodynamic radius of DNA (1.05 nm, [39]), and $L_{\text{eff}}$ the extension of the portion of DNA which rotates in order to release the torsional stress. The molecular extension in the middle of the plateau of the overstretching transition is approximately 22 µm which, under the assumption of torsional stress accumulating in the middle of the molecule, gives a $L_{\text{eff}}$ of 11 µm. With ω = 1,390 turns s⁻¹ (= 8,730 rad s⁻¹), the maximal frictional torque results to be 0.3 k _B T. If, however, the torsional stress is distributed uniformly along the whole length of the molecule, the resulting frictional torque should drop to 0.15 k _B T. A frictional reaction of 0.15–0.3 k _B T is expected to have a negligible effect on the kinetics of the B–S transition as demonstrated below.

The presence of an external torque (in this case of frictional origin) opposing the B–S transition not only modifies the free-energy difference between S and B states but also the free-energy barriers that the system must overcome for the transition (Fig. 7a). The B–S barrier will be increased (more difficult transition), while the S–B barrier will be decreased (easier transition). Let us define x _B and x _S the distances between consecutive bps in the two states and θ _B and θ _S the twist angles between consecutive bps, with numerical values: x _B = 0.34 nm, x _S = 0.58 nm θ _B = 1/10 turns = 0.63 rad, θ _S = 1/30 turns = 0.21 rad. Assuming for simplicity that the transition state is in the middle between both x _S and x _B and θ _S and θ _B, the change in barrier height due to the presence of a viscous torque can be estimated and compared with the analogous change due to the presence of an external force. The maximum torque-induced barrier change is ∆E _τ  = τ(θ _S − θ _B)/2 = 0.3 k _B T × 0.4/2 = 0.25 pN nm or 0.06 k _B T, while the barrier change caused by the external force F (~65 pN at the transition) is ∆E _F = F(x _S − x _B)/2 = 65 × 0.24/2 = 7.8 pN nm or 1.9 k _B T. Thus, since the barrier change introduced by the viscous torque is smaller by a factor of 32 relative to the barrier change caused by the external force, the torsional viscosity contribution can be disregarded.

A simple direct test of the influence of the rotational drag of the molecule on the kinetics of elongation is obtained by plotting the elongation rate r as a function of the length of the molecule L during the overstretching transition (Fig. 10c). If the elongation kinetics was dominated by the viscous friction, one would expect the relaxation rates to depend on L. Actually, the r–L relation shows a U-shaped dependence that excludes the hypothesis of a significant effect of the rotational drag on the elongation rate.

A further test is provided by the comparison of the responses to 2 and 0.5 pN force steps. For the same force, the extent of elongation (and thus ω) is smaller with the smaller step . Since the rotational drag depends linearly on ω, if it was dominating the kinetics of the process, it would have generated a reduction of the rate constant of elongation for the larger step. However, the observed rate constant is the same at the same force for either step size (Fig. 6d, blue 2 pN, green 0.5 pN).

In conclusion, both the theoretical treatment and the experimental evidences given above support the conclusion that the rate of DNA elongation following force steps applied in the overstretching transition region is not affected by viscosity and is mainly determined by the kinetics of the two-state reaction.