Supplemental Data


Abstract
Full Text
Reversible Unfolding of Single RNA Molecules by Mechanical Force
Jan Liphardt, Bibiana Onoa, Steven B. Smith, Ignacio Tinoco Jr., and Carlos Bustamante

Supplementary Material

Molecule synthesis. RNA was synthesized from a template obtained by polymerase chain reaction (PCR) from bases 3821 to 628 of the pBR322 DNA plasmid (NEB), where the P5ab, P5abc, and P5abcΔA sequences (Operon) were cloned into the Eco RI and Hind III restriction sites and a T7 promoter was appended to the template in the course of the PCR reaction. The DNA components of the handles were prepared by PCR from pBR322. Handle A (pBR322 bases 3821 to 3) was biotinylated, and one of the primers used to amplify handle B (pBR322 bases 30 to 628) was purchased with a 5´ digoxigenin group.

Fit to two-state statistics (Fig. 2b). Assuming a constant Δx = Δx(F1/2), the energy of the hairpin-laser trap system is E(F) = ΔG(F1/2) - FΔx, where ΔG(F1/2) = F1/2Δx is the free energy change of unfolding and stretching the hairpin at the midpoint of the transition (F1/2).

Optical trap. A piezo-electric actuator controls the position of the chamber and the tip of the bead, connected to the chamber by a glass micropipette. The other bead is captured in an optical trap and the force is measured from the change in momentum of light that exits the dual-beam trap. Molecules are stretched by moving the chamber vertically. Position of the tip-bead was determined using a "light-lever" (Web fig. 1c) where a low-power diode laser coupled to a single-mode optical fiber (Thorlabs LPS-3224-635) was collimated by a positive lens (f = 1.45 mm) attached to the chamber. Deflection of the resulting beam, due to chamber movement, was detected by a distant (~1 m) position-sensitive photodiode (UDT Sensors DL-10). Position of the bead in the optical trap was inferred from the force (measured by light momentum) and the trap stiffness. Both trap stiffness and light-lever sensitivity were calibrated using video measurements of bead positions. The end-to-end length of the molecule is obtained as the difference of the light-lever (tip bead) and the photon deflection (trap bead) measurements. The analog force and lever data were smoothed with an RC circuit (5 ms time constant), digitized with 12-bit resolution, and saved to disk at 200 Hz.


Supplemental Figure 1. Optical trap. A piezo-electric actuator controls the position of the chamber and of the tip bead, which is connected to the chamber by a glass micropipette. The other bead is captured in an optical trap and the force is measured from the change in momentum of light that exits the dual beam trap. Molecules are stretched by moving the chamber vertically. Movement is monitored with a "light-lever", comprising a laser, a lens connected to the chamber, and a detector, whose signal is calibrated by video imaging of the bead positions. The end-to-end length of the molecule is obtained as the difference of the light-lever (tip bead) and the photon deflection (trap bead) measurements.


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Supplemental Figure 2. Detail of Fig. 3D. P5abc's fluctuations at 7.6 pN in Mg2+. Red arrow, 10 nm folding intermediate.


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Supplemental Figure 3. Normalized histograms of the dwell times in the open and closed states of P5ab in Mg2+ at two different forces (14.4 and 13.7 pN) obtained from the constant-force hopping measurements. The solid lines are single exponential fits to the data giving rate constants for folding and unfolding. At 14.4 pN, the unfolded state predominates, with kunfold equal to 7 ± 0.1 s-1, and kfold to 1.5 ± 0.1 s-1. At 13.7 pN the molecule is mostly folded, with kunfold equal to 0.9 ± 0.1 s-1, and kfold to 8.5 ± 0.5 s-1.


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Force drift. Our constant force reference point typically drifts 0.1 pN min-1 because of room temperature changes. Consequently, there is less force-drift noise in the data obtained from stretch and relax cycles (period: ~10 s) compared with data from constant-force hopping measurements (duration of experiment: minutes). Relatively small differences between the folding and unfolding rates (e.g., between P5ab and P5abcΔA) that cannot be easily resolved in hopping experiments (SD of measurements 10 to 20%) may thus be detected using stretch and release cycles as shown in Fig. 2e.

Mfold free energy calculations. Standard free energies were calculated at 1 M NaCl and at 298 K using mfold 2.3, or at 1 M NaCl and at 310 K using mfold 3.1, as indicated.

Comparison with bulk experiments. All thermodynamic values given here are based on reversible, equilibrium measurements. As such, we expect the ergodic hypothesis to hold and, as long as we compare processes with identical initial and final states, the thermodynamic parameters measured in reversible single-molecule unfolding will be comparable to those obtained in bulk thermal denaturation (see below for the correction of the single-molecule unfolded state for reduced entropy due to stretching). On the other hand, the unfolding/refolding kinetics of single molecules under mechanical load is expected to differ from those of conventional thermal or chemical bulk denaturation. In single-molecule force experiments, we impose a well-defined reaction coordinate (the end-to-end distance x) that is very different from the reaction coordinate of molecules undergoing thermal or chemical bulk denaturation. The reaction pathways, as well as the kinetic states visited during the denaturation process, will therefore be different.

Free energy of unfolding. Pulling experiments on a single molecule can be treated using standard thermodynamic theory with force and length being the one-dimensional equivalents of the fundamental thermodynamic parameters of pressure and volume. The free energy change to convert one molecule of hairpin to one molecule of single-strand at zero force is

(1)

where Keq(F) is the equilibrium constant for the folded ( unfolded process. At constant temperature and pressure, the reversible work is

(2)

where ΔG(F)stretching accounts for stretching of the unfolded state. The integral under the molecule's reversible force extension curve is the potential of mean force; at constant temperature and pressure the integral is equal to the Gibbs free energy required to unfold the molecule, to tilt the folded unfolded equilibrium towards the unfolded state, and to stretch the unfolded state. In particular, at the mid-point of the transition (i.e., F = F1/2), kBTln Keq(F) = 0 and thus

(3)

WLC correction. To compare free energies obtained in stretching and hopping experiments to the free energy of unfolding an untethered molecule, the reduced entropy of the unfolded state due to tethering must be subtracted from the free energies. An estimate for the entropy correction may be obtained from the worm-like-chain (WLC) interpolation formula (1),

(4)

which relates the force F of entropic origin to the end-to-end distance (x) of a polymer, where P is the persistence length of the polymer and L is the contour length. For RNA we use L = 0.59 nm per nucleotide, the interphosphate distance, if we assume a C3´ endo sugar pucker (2). We obtain x per nucleotide from the P5ab unfolding plateau by increasing the plateau length (18 nm) by the length of the folded state (helix diameter = 2.2 nm) and dividing the total length by 49 bases in the hairpin. Thus x = 0.42 nm per nucleotide at 14.5 pN force. Using x/L = 0.42/0.59 in the WLC formula gives a value P = 1.0 nm, well within a range of earlier estimates (0.7 nm to 5 nm) (3-5). The entropic energy difference, -TΔS, between a tethered and a free single strand is given by the calculated area under the WLC force-extension curve integrated from zero to the RNA's extension at the unfolding force. Assuming P = 1.0 nm, -TΔS = 50 kJ mol-1 for P5ab; other estimates for P and L give a range of values from 34 to 53 kJ mol-1, with an average of 44 kJ mol-1.


Supplemental Figure 4. Predicted free energy surface of P5ab's folding/unfolding under an initial external load of 14.5 pN. Free energies for P5ab were predicted as a function of number of base pairs sequentially unzipped by using the mfold 3.1 program (16) with parameters 1 M NaCl and 310 K. Energies were subsequently reduced by 24% to scale them to the measured free energy of unfolding P5ab in our buffer (250 mM NaCl, 10 mM MgCl2 at 298 K) and by 3.42 pN nm per broken base pair due to stretching at 14.5 pN. The energy required to unfold a G-A base pair is assumed to be the same as to unfold an A-U base pair. The free energy change of opening the last base pair (G27-C32) and adjacent tetraloop was taken to be -20.8 pN nm. With each base pair opened, the end-to-end distance increases by 0.84 nm, assuming WLC behavior where L = 1.18 nm, P = 1 nm, and a constant force of 14.5 pN. The resulting energy versus distance curve is then tilted by addition of the bead/trap potential, (Fdx. In our optical trap, F = κ(x - x0), where κ = 0.1 pN nm-1 is the trap stiffness. The increase in free energy to the right of the unfolded state (U), reflects decrease in entropy of the unfolded RNA due to stretching (WLC behavior). The increase of energy to left of the folded state (F) reflects trap/bead potential overpowering a slight entropic gain of the slack handles, assuming L = 320 nm and P = 50 nm for A-form handles. The width of the pink band corresponds to ± 1 kT (± 4.1 pN nm), illustrating the effect of a 298 K thermal bath on the smoothness of the energy landscape. Features of the free energy surface (black line) within the band do not represent significant barriers at this temperature. Dashed curve represents computed free energy curve for P5abΔU hairpin at its equilibrium force (16.3 pN).


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Supplemental Figure 5. The ln k versus F for P5ab in Mg2+. Only processes with rates between 0.05 and 20 Hz may be followed in our instrument. This "operational range" is illustrated with the gray bar. P5ab's folding/unfolding rate constants of ~5 Hz at the equilibrium force are within the window, and hopping is consequently detected. By contrast, molecules with particularly slow kinetics at equilibrium (such as P5abc in Mg2+ and P5abΔ U in both Mg2+ and EDTA) will be below the window, and hence, their transitions between the folded and unfolded states at F1/2 will be too rare to follow reliably. The faint line shows the ln kunfold versus F curve for P5abc in Mg2+, and the pink band an estimate for the intersection location with the ln kfold versus F curve (i.e., the force at which Keq = 1).


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Supplemental Figure 6. Sequence and predicted structure of the P5abΔ U RNA, and its force-extension curves in Mg2+ and EDTA. The bases highlighted in red have been changed with respect to P5ab. Note that the bulged U has been deleted, and the replacement of the weak G-A with G-C base pairs. Folding/unfolding is hysteretic in both ionic conditions, with an average hysteresis of ~4 pN.


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Hopping. Whether hopping can be observed with a particular type of RNA depends on the time resolution of the instrument, its drift rate, and the kinetic barrier to unfolding as determined by the free energy curve of the molecule and the stiffness of the trap. Web fig. 5 illustrates the maximum and minimum hopping rates that can be detected with the instrument (0.05 Hz to 20 Hz) and that thus define its operational range (gray bar). Superimposed are seen the ln k versus F plots for the folding and unfolding reactions of P5ab. The two lines intersect at the force that equalizes the two rates. Hopping is only seen if the ln k versus F curves for folding and unfolding intersect within the detection window, as is the case for P5ab. Nonetheless, a force can always be applied at which either folding or unfolding, but not both, will be observed. Whether the ln k versus F curves intersect at detectable rates depends on the features of the molecule's free energy surface along the reaction coordinate in the presence of an external force. Web fig. 4 shows the free energy curve of P5ab at the equilibrium force (14.5 pN) obtained by subtracting the FΔx potential of the trap from the molecular energy surface. This curve has two minima separated by a maximum, explaining the cooperativity of P5ab's folding/unfolding. To illustrate how the details of an RNA's sequence affects its folding/unfolding kinetics, reversibility, and cooperativity, we synthesized a molecule similar to P5ab but in which the bulged U at the base was deleted and the weak G-A base pairs halfway up the helix were replaced with stable G-Cs (P5abΔ U, Web fig. 6). The calculated free energy surface along the reaction coordinate of P5abΔU at the equilibrium force (16.2 pN) shows that the barrier height is increased by ~10 pN nm (Web fig. 4, dashed line) relative to that of P5ab. These modifications are thus expected to decrease the rate of transition barrier crossing 10- to 20-fold at the equilibrium force. Indeed, no hopping is observed when P5abΔU is held at constant force, and its pulling curves are not reversible, showing hysteresis in both Mg2+ and EDTA (Web fig. 6). The ability of an RNA to hop at detectable rates can therefore be turned on or off by modifying specific features of the RNA hairpin's free energy surface.

Absolute hopping rates. Although the absolute values of the rate constants are likely to be affected by the spring constant of the laser trap, the size of the beads, and the mechanical properties of the RNA/DNA hybrid handles, we did not observe any statistically significant effects on hopping rates when several machine parameters were varied to the maximum extent possible. In particular, we reduced the bead size by a factor of 3 (3 to 0.9 micron diameter) and the spring constant of the laser trap in constant force mode by a factor of 5 (0.15 to 0.03 pN nm-1).

Simulating the activation energy for hopping. A simulation was made of the potential energy of a hairpin molecule in a laser trap where the molecular free energy was added to the trap potential energy. The molecular energy was calculated from mfold 3.1 as described. The end-to-end distance was calculated from the WLC assuming L = 0.59 nm per nucleotide and P = 1 nm. Thus, an extra 0.84 nm accrues from every base pair that is unfolded at 14.5 pN. Beyond complete unfolding (curve to the right of U in Web fig. 4), the potential of the chain (composed of 56 bases of single-stranded RNA and 1141 bases of double-stranded RNA:DNA handles) rises as predicted from the WLC model. Prior to hairpin fraying (curve to the left of origin in Web fig. 4), the potential rises due to the imbalance of WLC force from the handles versus the trap force.

Variability of measurements. Several types of behaviors in addition to "good" hookups are observed when pulling a particular type of molecule. These correspond to simultaneous multiple-molecule hookups between the two beads, early breaking of the attachment of the RNA to the beads, defective handles, and direct bead-to-bead bonding. Typically, over 1/3 of the tethers gave the repeatable and distinctly recognizable single-molecule behavior described in the paper. The behavior of any given molecule in this class was almost always (>95% of experiments) indistinguishable from the average behavior over all molecules of that type. The largest source of variability and instrumental noise were air currents and building vibrations (e.g., due to building construction and elevators). However, since each tether was pulled many times (10 to 40 stretches), we were able to choose a representative subset of force curves displaying low noise. The variations in transition length, force, thermodynamic, and kinetic constants observed for these subsets, were used to determine the standard deviations quoted in the text. The total number of stretches performed per type of molecule (e.g., P5ab, P5abc) was about 500.

References and Notes

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