INTRODUCTION

TOP ABSTRACT INTRODUCTION THE TRAPPING INTERFEROMETER SAMPLE PREPARATION THEORETICAL DESCRIPTION RESULTS CONCLUSIONS REFERENCES

During the last decade the field of single molecule studies on biological systems has strongly grown in importance. A very rapidly developing part of these activities concerns force measurements on DNA molecules (Allemand et al., 1998; Bockelmann et al., 1997, 1998; Bustamante et al., 2000; Clausen-Schaumann et al., 2000; Cluzel et al., 1996; Colton et al., 1994; Essevaz-Roulet et al., 1997; Léger et al., 1999; Rief et al., 1999; Smith et al., 1992, 1996; Wang et al., 1997) and force measurements on DNA-protein complexes (Davenport et al., 2000; Léger et al., 1998; Strick et al., 2000; Wuite et al., 2000; Yin et al., 1995).

A few years ago we reported on experiments where single DNA molecules are unzipped with a soft glass microneedle (Bockelmann et al., 1997, 1998; Essevaz-Roulet et al., 1997). Force signals have been recorded that reflect the proportion of G/C compared to A/T basepairs on an average scale of ~100 basepairs. A typical force versus displacement curve consists of a series of sawtooth-shaped features. This characteristic shape is explained theoretically in the frame of equilibrium statistical mechanics. The calculated physical effect, called molecular stick-slip motion, is a reversible molecular process caused by an interplay of the energy landscape given by the genomic sequence, the elasticities of molecule and measurement device, and the Brownian motion. Theoretical papers have been published that directly relate to our experimental configuration (Cocco et al., 2001; Lubensky and Nelson, 2000; Nelson, 1999; Thompson and Siggia, 1995; Viovy et al., 1994) and different groups have reported on AFM measurements of the force to separate the two strands of the DNA double helix (Colton et al., 1994; Rief et al., 1999).

Here, we report on DNA unzipping performed with a modified molecular construction and an optical trapping interferometer. The molecular construction is optimized for high stability by the incorporation of multiple attachment points and for high molecular stiffness by the use of relatively short double-stranded linker arms. An optical trapping interferometer is chosen because it combines high measurement stiffness (the trap compliance is almost negligible compared to the molecular compliance) with a sub-pN force resolution. In this configuration DNA unzipping can occur on a significantly more local scale than in our previous work. The effect of stiffness on the basepair sensitivity and the role of fluctuations are addressed. The paper focuses on displacement velocities below 1 µm/s. For this regime we expect the highest basepair sensitivity.

The experimental configuration is schematically presented in Fig. 1. The force measurements are performed in vitro on a molecular construction that is anchored between a glass microscope slide and a silica bead. The bead is held in an optical trap and the surface is laterally displaced, which leads to a progressive opening of the double helix. The force is obtained from a measurement of the bead position within the trap.

View larger version (34K): [in this window] [in a new window]   FIGURE 1   Schematic view of the single molecule configuration for unzipping DNA with optical tweezers.

In the next section we explain the optical trapping interferometer and the calibration of the force measurements. Subsequent sections are devoted to the preparation steps of the molecular construction, the functionalization of the glass surfaces, and the theoretical description of DNA opening at thermal equilibrium. The Results section is divided into two parts. The first part is devoted to the basepair sensitivity of the unzipping signal. We show that the stiffness of the system is a key parameter for the sensitivity, as it strongly influences the amplitude of the molecular stick-slip motion. In the second part we consider fluctuations in the force signal and discuss a physical process we observed for the first time when opening or closing DNA with the trapping interferometer: sudden flips between discrete force values, occurring at specific positions in the force signal. The reader uninterested in the setup and the preparations may skip the next two sections and directly go to the Theoretical Description and Results sections.

	THE TRAPPING INTERFEROMETER

TOP ABSTRACT INTRODUCTION THE TRAPPING INTERFEROMETER SAMPLE PREPARATION THEORETICAL DESCRIPTION RESULTS CONCLUSIONS REFERENCES

Experimental setup

We have built an optical trapping interferometer for the measurement of forces up to 100 pN but with subpiconewton resolution. In brief, a gradient beam optical trap is created by tightly focusing an infrared laser beam with a high-numerical-aperture microscope objective. The force acting on the bead is derived from an interferometric measurement of the position difference between the bead and the trap center. This position measurement is based on DIC (differential interference contrast) microscopy (Allen et al., 1969). About a decade ago, Denk and Webb (1990) measured the position of microscopic beads with this interferometric technique, and since then a number of groups have adapted different types of interferometric position measurements to optical tweezer experiments on single molecules (Allersma et al., 1998; Gittes and Schmidt, 1998a; Svoboda and Block, 1994).

Our setup is schematically shown in Fig. 2. It is based on a commercial inverted optical microscope (Zeiss Axiovert 100). The beam of a diode-pumped Nd:YAG cw laser (Coherent DPY321, 1W, 1064 nm) first passes a Faraday isolator to avoid back-reflection into the laser cavity that otherwise causes important intensity fluctuations. A telescope arrangement consisting of a pair of mirrors and a pair of convex lenses allows us to adjust independently the position and the angle of the beam. It also widens the parallel beam to a diameter slightly above the back plane aperture of the microscope objective (100×, N.A. = 1.25 oil immersion). The polarization of the linearly polarized laser light is rotated with a /2 plate to illuminate a first Wollaston prism with the electric field tilted by 45° with respect to the prism axis. This leads to an angular splitting of the beam and, in the focal plane of the objective, to two partially overlapping, diffraction-limited spots of equal intensity. The two spots exhibit orthogonal linear polarization and a center-to-center distance of ~200 nm.

View larger version (4K): [in this window] [in a new window]   FIGURE 2   Schematic view of the optical trapping interferometer.

With this arrangement, silica beads (mean diameter of 1 µm) are trapped in aqueous solution, close to the center of the twin spot. As the 200-nm center-to-center distance is significantly smaller than the bead diameter, the effect of the optical trap along the axis of interest (the line connecting the two spots in the sample plane) is well-described by a simple harmonic potential of stiffness, k_trap.

To measure the lateral position of the bead in the trap, the transmitted light is collected by a condenser assembly and the two polarizations are recombined using a second Wollaston prism. With the bead in the center of the trap the recombined light is linearly polarized. However, a displacement along the line connecting the two focal points leads to a difference between the optical paths of the two beams and induces an ellipticity of the recombined light. This ellipticity is measured by a modulation technique, similar to an arrangement commonly used in ellipsometry (Azzam and Bashara, 1989). Our setup involves an acousto-optic modulator that introduces a  = 50 kHz modulation of the refractive index along one direction and thus modulates the phase difference between the two polarization components. The modulated light passes a linear polarizer and is detected by a silicon photodiode. A voltage proportional to the light intensity is obtained with a custom-made current preamplifier. This voltage signal enters a lock-in amplifier with the 50 kHz signal of the modulator connected as reference. The amplifier output U is proportional to the ellipticity of the light in front of the acousto-optic modulator. The analog signal passes a frequency adjustable anti-aliasing filter and is converted by a 16-bit ADC (IDSC816 card from Microstar Laboratory). The numerical data are written to a computer hard disk and analysis of the results is done off-line.

For the horizontal sample displacement, a piezo translation table with capacitive position sensors is mounted on top of a coarse xy translation stage. Piezo stage and position sensors are operated via a computer-adjustable feedback loop, which allows controlling the relative lateral sample position with nanometer precision and imposing different cycles of displacement. The vertical position of the microscope objective relative to the sample surface is measured with an inductive position sensor. Because the setup measures lateral bead displacements with subnanometer resolution and a bandwidth of several kilohertz, it is very important to reduce as far as possible mechanical perturbations and thermal drifts between trapping beam and sample. Most importantly, we use an optical table with active vibration isolation and protect the setup against air flow. In addition, to suppress small fluctuations of the lateral trap position arising from laser beam pointing, we have introduced a feedback loop in the excitation path that includes a piezo mirror mount and a quadrant photodiode. With these arrangements we are able to reduce the drift between the trapping beam and the sample position to ~10 nm/min. Although this is sufficient for many applications, the residual drifts still induce nonnegligible shifts between the experimental curves in our high-resolution unzipping measurements (see Results).

Calibration

We use three different approaches to calibrate the trapping interferometer. The first one, entitled "bead in a gel," allows us to relate the interferometer output to a piezo-controlled displacement of the bead in the trap and to determine the range where the interferometer output voltage and the displacement are proportional. The second configuration, "bead in an oscillating liquid," provides a direct measure of the constant of proportionality between the interferometer output and the force acting on the bead. The third configuration, entitled "Brownian motion of a captured bead," also gives this constant and, in addition, a measure of the trap stiffness. We describe the three techniques separately and then make some general remarks on the calibration of the trapping interferometer.

Bead in a gel

View larger version (1K): [in this window] [in a new window]   FIGURE 3   Bead in a gel. Interferometer output voltage U recorded with a bead moving through the optical trap. At the trap center (corresponding here to a lateral position of ~1.85 µm) the output voltage is offset from zero on this graph. This offset can be tuned by a fine position adjustment of the first Wollaston prism.

Bead in an oscillating liquid

(1)

(2)

(3)

(4)

(5)

(6)

4

(7)

(8)

View larger version (4K): [in this window] [in a new window]   FIGURE 4   Bead in an oscillating liquid. The two curves show the measured interferometer output voltage U (noisy curve) and the capacitive measurement of the sample position (smooth curve). We thus compare the displacements of the bead and the translation stage. The /2 phase shift between U and x0, characteristic of the regime 1, is apparent. The noise in the bead position is due to the low frequency components of the Brownian motion. The interferometer voltage is measured with a lock-in amplifier at the reference frequency  of the acousto-optic modulator. To extract the amplitude of the bead oscillation (Figs. 5 and 6) we use a second lock-in amplifier in series that demodulates at the frequency  of the oscillating translation stage.

View larger version (4K): [in this window] [in a new window]   FIGURE 5   Bead in an oscillating liquid. Amplitude of the output voltage (voltage demodulated at  and ) of the trapping interferometer as a function of the amplitude of the piezo stage. The liquid surrounding the bead of 1 µm in diameter is oscillated with a frequency of 20 Hz.

View larger version (3K): [in this window] [in a new window]   FIGURE 6   Bead in an oscillating liquid. Amplitude of the output voltage (voltage demodulated at  and ) of the trapping interferometer as a function of the frequency of the imposed oscillation. The liquid surrounding the bead moves due to a stage oscillation of 75 nm in amplitude.

Brownian motion of a captured bead

(9)

(10)

(11)

View larger version (0K): [in this window] [in a new window]   FIGURE 7   Brownian motion of a captured bead. Power spectral density of the interferometer signal U for a glass bead of 1 µm in diameter. The bead is captured in water with a laser power of 700 mW (70% of our maximum laser power). The fit to a Lorentzian gives two parameters: the cutoff frequency fc (here 5.7 kHz) and the area under the curve (here 4.72 mV2/s).

View larger version (0K): [in this window] [in a new window]   FIGURE 8   Cutoff frequency, obtained by a fit of the measured power spectral density to a Lorentzian, as a function of laser power (100% corresponds to 1 W at the laser).

View larger version (0K): [in this window] [in a new window]   FIGURE 9   Area under the spectral density, obtained by a fit of the measured curve to a Lorentzian, as a function of laser power (100% corresponds to 1 W at the laser).

Conclusions on the three calibration techniques

	SAMPLE PREPARATION

TOP ABSTRACT INTRODUCTION THE TRAPPING INTERFEROMETER SAMPLE PREPARATION THEORETICAL DESCRIPTION RESULTS CONCLUSIONS REFERENCES

The molecular construction

Overview

View larger version (12K): [in this window] [in a new window]   FIGURE 10   Schematic view of the molecular construction.

Preparation of modified DNA by PCR

Assembly of the DNA linker arms

Preparation of the  DNA and final ligation

Surface treatment

To attach the digoxigenin-functionalized part of the molecular construction, we have coated glass slides with anti-digoxigenin, as described below. The biotin groups are attached to commercial streptavidin-coated beads (1 µm diameter, Bangs), as described in the following section.

Standard microscope glass slides are put into an ozone atmosphere for 1 h, to oxidize organic surface contaminations. After this cleaning step, the slides are silanized. First, 10 µl of a vinyl-functionalized silane (triethoxysilyl-modified polybutadiene) are dissolved in 1 ml methyl ethyl ketone. Then 2 µl Tween (10% in H₂O) and 10 µl H₂O are added and the solution is deposited on the slides and allowed to react at room temperature for 20 min without drying. The slides are then rinsed with toluene and baked 2 h at 150°C. These slides are stored dry at room temperature and can be used for several months.

At this stage, a small plastic ring is glued on each slide. In the following surface reactions, liquid volumes of the order of 10 µl are deposited inside the ring, and this sample is covered with a circular glass slide to avoid evaporation. We then proceed to a statistical polymerization based on a mixture of acrylic and maleic acids. Covalent binding is obtained by a reaction between the carbon double bonds of the silane and polymer molecules. The linear polymers introduce amide and carboxyl groups and lead to a negatively charged, hydrophilic surface. The preparation is based on standard protocols of acrylamide gels for electrophoresis (Sambrook et al., 1989). A typical polymerization reaction contains 3% (w/v) acrylic acid, 0.05% (w/v) maleic acid and, as catalyst and initiator, 0.1% (w/v) persulfate and 0.1% (v/v) TEMED (N,N,N',N'-tetramethylethylenediamine) in deoxygenated water. Polymerization between the silanized slide and a covering slide is allowed for 1 h at room temperature. The covered slides are stored in a refrigerator in a humid environment and can be used for several months.

For the subsequent coupling of the anti-digoxigenin antibody, the covering slide is removed and the surface rinsed with H₂O. The carboxyl groups of the polymer layer are activated with an aqueous solution containing 200 mM EDC (1-ethyl-3-(3-dimethylaminopropyl)carbodiimide) and 50 mM NHSS (N-hydroxysulfosuccinimide) for 5 min. The surface is rinsed again and a solution (0.5 mg/ml in PBS) of an antibody directed against digoxigenin is incubated for 5 min. The surface is deactivated by incubating 7 min with an ethanolamine solution (1 M, pH 8.5). The anti-digoxigenin-coated surface is then rinsed with H₂O and stored under a blocker solution (casein in PBS, Pierce).

Final steps before force measurement

A small volume of the dilution containing typically 10¹⁷ mol of the molecular construction is deposited on an anti-digoxigenin-coated slide, the surface is covered, and the sample incubated for 10 min at room temperature. This gives enough correctly attached molecular constructions for multiple force measurements on a given surface, and the DNA concentration is still sufficiently low to avoid nonspecific DNA-surface interactions and attachment of beads via more than a single DNA molecule. About 150 µl PBS (pH 7, 10 mM phosphate, 150 mM NaCl), containing the streptavidin-coated silica beads, are added. The sample is mounted on the translation stage of the trapping interferometer. After ~10 min, we induce a rotation of the circular slide covering the sample by applying an air current at an oblique angle on one edge of the circular slide. This entails a rotation of the slide and of the liquid, and we exploit the following two effects. First, the flow induces a viscous force that allows us to identify the correctly attached beads. The latter display superposed on Brownian motion, a characteristic displacement that corresponds to the total length of the linker arms in the molecular construction (~5 µm in the present case). The second effect of the liquid rotation is that the nonattached beads on the bottom of the sample progressively concentrate close to the rotation axis. This way, we can significantly reduce the number of free beads in the vicinity of the measurement position and we avoid capturing more than a single bead in the force measurement. Once a correctly attached bead is identified, the rotation is stopped, the trap is laterally positioned, and the bead is captured by switching on the laser. Immediately afterward, force curves can be measured by laterally displacing the sample with the piezo stage and recording the interferometer output. The axial sample position is kept constant; typically the bead is held in the liquid at a height of 2-4 µm above the glass slide.

	THEORETICAL DESCRIPTION

TOP ABSTRACT INTRODUCTION THE TRAPPING INTERFEROMETER SAMPLE PREPARATION THEORETICAL DESCRIPTION RESULTS CONCLUSIONS REFERENCES

We have developed a theoretical description of DNA unzipping with a force measurement device. It is based on equilibrium statistical mechanics and involves the following four energy contributions.

The first energy, called E_DNA(j), describes the work necessary to separate the two strands of the DNA double helix from the basepair of index 1 to the base pair of index j. This energy is derived from the work of SantaLucia (1998), who compiled data from melting experiments on oligonucleotides and DNA polymers, and who provides the binding energies of the different DNA basepairs, including nearest-neighbor contributions.

The second energy, called E_ext, is associated with the elasticity of the single-stranded and double-stranded parts of the molecular construction. The elasticity of the single-stranded parts (which increase in length as the opening progresses) is described in a freely jointed chain model with entropic and enthalpic contributions (Smith et al., 1996). The elasticity of double-stranded DNA is treated in a wormlike chain model (Smith et al., 1992; Odijk, 1995) with parameters taken from Wang et al. (1997). Our description of the elasticity of the single- and double-stranded parts of the molecular construction is compatible with data from the literature (Allemand et al., 1998; Smith et al., 1996; Wang et al., 1997) and with measurements that we have performed pulling single DNA molecules from opposite ends.

The third energy, called E_trap, is the potential energy of the bead in the trap; E_trap = k_trapx²/2. The force-induced displacement of the bead is given by x = x_0  1  2, the difference between the sample displacement x₀, the length ₁ of the DNA strand attached to the microscope slide, and the length ₂ of the DNA strand attached to the bead.

The fourth energy is the thermal energy kT. It leads to the Brownian motion and directly enters the statistical weight in our calculation of the thermal averages in the canonical ensemble (Eq. 12). In this article we use T = 300 K and do not consider temperature variations. The thermal average of an observable A is calculated according to

(12)

with the total energy given by

(13)

The theoretical description has been published in closer detail elsewhere (Bockelmann et al., 1998). In our earlier work we compared calculations to unzipping measurements performed with soft glass microneedles, and we neglected the elasticity of double-stranded DNA and nearest-neighbor contributions to the binding energy of the basepairs.

No velocity dependence is included in our theoretical description. For the displacement velocities of the measurements considered in this article, Stokes' formula gives a frictional force on the bead well below 1 pN. When we unzip DNA with the optical tweezer setup at velocities above 1 µm/s, the opening force increases with velocity. This velocity dependence increases with the length of the DNA to be opened, suggesting that it arises from viscous friction of the rotating DNA tail (Ph. Thomen et al, manuscript in preparation). Without considering rotational friction, Cocco et al. (2001) have theoretically studied force and kinetic barriers to unzipping DNA and predict an increase in the opening force starting at loading rates above 10^5.5 pN/s. However, with the stiffness range of optical tweezers (typically 50-300 pN/µm) and displacement velocities below 1 µm/s, we are more than three orders of magnitude below such loading rates.

	RESULTS

TOP ABSTRACT INTRODUCTION THE TRAPPING INTERFEROMETER SAMPLE PREPARATION THEORETICAL DESCRIPTION RESULTS CONCLUSIONS REFERENCES

Relation between stiffness and basepair sensitivity

In Fig. 11 we present a force versus displacement curve measured upon opening the molecular construction. The force F starts to rise sizably when the displacement approaches the total crystallographic length of the double-stranded linker arms. Around 10-15 pN the force ceases to increase and the regime of increasing elastic DNA deformation changes into a regime of mechanical unzipping, where the two strands of the DNA molecule separate and the opening fork progresses through the genomic sequence with ~1000 bp per µm of additional displacement. In the unzipping regime, which for the -DNA extends over a displacement range of almost 50 µm, the force exhibits a very rapid variation with a typical amplitude of ~10%. This force variation is due to the differences in the pairing and stacking energies among the different basepairs, and thus reflects the known sequence of the -DNA in the molecular construction. As expected from the relative stability of the Watson-Crick basepairs, a G/C-rich region corresponds to higher unzipping force than an A/T-rich one (Bockelmann et al., 1997, 1998; Essevaz-Roulet et al., 1997). This result was confirmed by Rief et al. (1999), who derived an unzipping force of 10 pN from AFM measurements on a poly-A/T oligonucleotide duplex and a value of 20 pN from measurements on a poly-G/C duplex.

View larger version (0K): [in this window] [in a new window]   FIGURE 11   Force versus displacement curves corresponding to mechanical unzipping of a single -phage DNA molecule. The trace is recorded during opening of the double helix with a constant displacement velocity of 1 µm/s (sampling rate of 400 Hz, aliasing filter cutoff frequency of 176 Hz).

Fig. 12 provides a zoom into the force signal of a 100 nm/s measurement for three different displacement intervals. The experimental data (bottom curves) are compared to a calculation (upper curves, upshifted by 4 pN for presentation and based on the assumption of thermal equilibrium as described in the preceding section). The force signal is composed of a succession of sawtooth-like structures. For a given additional displacement the opening progresses a little during the gradual rise in force, and a lot during the drop in force. This behavior can be understood in the frame of equilibrium statistical mechanics as an interplay of the complex energy landscape caused by the DNA base sequence, the Brownian motion, and the compliances of the molecule and the optical tweezer (Bockelmann et al., 1997, 1998).

View larger version (4K): [in this window] [in a new window]   FIGURE 12   Evolution of the unzipping force signal with increasing displacement. The measured curve (bottom) corresponds to an opening performed with a displacement velocity of 100 nm/s (sampling rate, 400 Hz; anti-aliasing filter cutoff frequency, 176 Hz). The upper curve is the result of a calculation assuming thermal equilibrium, as described in the text. Three different displacement intervals are shown. They roughly correspond to opening from basepair 1 to 1800 (left), 3600 to 5800 (middle), and 19,600 to 24,100 (right). The measured and the calculated curve exhibit the same scaling but are vertically displaced for convenience. The feature around 8500 nm is a nonreproducible experimental event. Such events occur exceptionally and we do not know the origin.

One way to quantify the sensitivity of the force signal to the base sequence is to consider the calculated variance in the number j of opened basepairs (amplitude of the thermal breathing of the opening fork). This variance var j exhibits a rapid variation as a function of the local sequence (see Bockelmann et al., 1998; Fig. 5). During opening of the first 2000 basepairs (left graph of Fig. 12), the stick phases correspond to an amplitude of 5-10 bp, while during the slip events the averaging involves 20-50 bp. Assuming a constant sequence, we analytically find that var j is proportional to k. The total local stiffness of the system is given by

(14)

with k_ss, k_ds, and k_trap being the local stiffness at the opening force of the single-stranded DNA, the double-stranded DNA, and the measurement device, respectively. In the present case, k_tot is limited by the molecule and not by the measurement device. The stiffness k_tot decreases with increasing displacement because the length of the ss-DNA increases. This leads to the systematic decrease in the slope of the rising part of the sawteeth in Fig. 12 (notice the different horizontal scale of the rightmost graph). Concomitant with that, the average size of the sawteeth increases, the density of features decreases, and the average noise amplitude increases.

In Fig. 13 we directly compare the unzipping signal of our earlier microneedle experiment with the one of the trapping interferometer. Opening the same sequence of basepairs, the optical trap data give a significantly higher density of structures in the force versus displacement curve. The trapping interferometer is about two orders of magnitude stiffer than the glass microneedle used in the earlier work (250 versus 1.7 pN/µm) and the two ds-linker arms of the new molecular construction are about three times stiffer than the original -DNA linker (total length of 14,560 bp versus 48,502 bp). As shown in Fig. 13, our calculations indeed predict that the resulting increase in total stiffness leads to the experimentally observed gain in basepair sensitivity. In the microneedle case, the amplitude of the molecular stick-slip motion is stronger. This is exemplified by the presence of pronounced steps and plateaus in the curve describing the progression of the opening fork with increasing displacement (upper part of Fig. 13).

View larger version (27K): [in this window] [in a new window]   FIGURE 13   Comparison of an unzipping signal measured with a soft microneedle (middle) and a signal measured with the trapping interferometer (bottom). Each experimental curve is compared to a calculation based on the assumption of thermal equilibrium. The microneedle data have been recorded with a displacement velocity of 20 nm/s, the optical trap data with a velocity of 200 nm/s. Both measurements correspond to opening the first 3200 basepairs of the -DNA. For both cases the average number of opened basepairs j has been calculated as a function of displacement and is presented in the upper part of the figure (position in the DNA sequence).

A different way to investigate basepair sensitivity is to study the effect of mutations in the DNA sequence on the calculated force signal. In Fig. 14 we present force curves calculated for opening a -DNA sequence with different single-basepair mutations. We have chosen the positions of the mutations to be located on small features of the force signal, because in the presence of experimental drift, the modification of a small feature would probably be easier to detect than a change of same amplitude in a smoother part of the force curve. In the calculations a different base sequence always gives a different force curve, and any single-basepair mutation is detectable. Even the interchange of the bases of a given pair (see example 189 C  G) induces a modification because the binding energy depends not only on the local basepair, but also on the neighboring bases (SantaLucia, 1998). The twin peak at 5100 nm displacement can be suppressed in the calculation by changing basepair 393 from A/T to C/G. Similarly, the small peak occurring at 4800 nm displacement disappears upon a mutation C/G to A/T on basepair 189. The small peak at 5700 nm is either reinforced or replaced by a deep valley by single-basepair mutations at position 1079 (T/A to G/C) and 1080 (C/G to A/T). As illustrated for basepair 189, a mutation from a C/G to an A/T pair (or vice versa) typically induces a stronger change in the force signal than a basepair reversal (C/G to G/C). These theoretical considerations already indicate that the different mutations that induce biological phenotypes (replacements, deletions, insertions, mispairings) will be more or less difficult to observe in the unzipping configuration. The difficulty will decrease with increasing number of modified basepairs and will also depend on the type(s) of the mutation(s) and its (their) local sequence environment.

View larger version (0K): [in this window] [in a new window]   FIGURE 14   Calculated effect of different single basepair mutations on the force versus displacement curve. The superposition of three curves presented in the inset gives an enlarged view on the effect of a (C/G to A/T) mutation and of a (C/G to G/C) inversion with respect to the nonmutated sequence.

Experimentally, the basepair sensitivity also depends on the quality of the force measurement, in particular on the signal-to-noise ratio and the proper control of the relevant experimental parameters. The detection of very local sequence events becomes increasingly difficult with increasing amplitude of the thermal motion of the opening fork, and hence with the length of the DNA to be opened. However, the precision of the force measurement can be improved by decreasing the displacement velocity that allows low-pass-filtering the Brownian motion with increasing efficiency. Therefore, the noise on the experimental force curves can be (and in our measurements sometimes is) smaller than the calculated thermal variance of the force. Brownian motion therefore does not represent a fundamental limitation, but may nevertheless induce a very serious experimental difficulty. For instance, residual drifts, which are often due to changes in thermal expansion of the setup, have to be controlled on increasingly long time scales if increasing time-averaging of the signal is needed to low-pass-filter the Brownian motion. In this sense, the experimental performances determine to which precision the thermal average of the force is measured and what sensitivity can be achieved in terms of the genomic sequence.

Our measurement configuration allows us to perform several measurements on the same molecule. The double helix reforms spontaneously when the direction of the stage motion is reversed and, after this closing, a new opening measurement can be engaged. We observe no systematic difference in the degree of reproducibility between two consecutive opening measurements on the same molecule with respect to the degree of reproducibility between two recordings performed on two different molecules. This shows that under the present conditions the DNA double helix is able to renaturate to its native B-form under an external force in the 10-15 pN regime. In Fig. 15 we compare three measurements (E1-E3) and a calculation (Th), corresponding to the opening of the first 2500 bp of the -DNA. A comparison of Figs. 14 and 15 suggests that our measurements provide rather local information on the base sequence. The twin peak at 5100-nm displacement is weakly resolved in measurements E1 and E3. This structure can be suppressed in the calculation by changing basepair 393 from A/T to C/G. The small peak in a valley occurring at 4800 nm displacement is present in measurements E1 and E2, and disappears upon a mutation C/G to A/T on basepair 189. The small peak at 5700 nm may be guessed in the noise of the experimental traces E1 and E2. In the calculation this feature is either reinforced or replaced by a deep valley by single-basepair mutations at position 1079 (T/A to G/C) and 1080 (C/G to A/T).

View larger version (1K): [in this window] [in a new window]   FIGURE 15   Variation of the opening force during the first 2500 nm of the opening. Measurements on two different molecules are presented. One has been opened with a velocity of 100 nm/s (E1), the other one with 50 nm/s (E2), and 200 nm/s (E3). For all three measurements we used a sampling rate of 400 Hz and a cutoff frequency of 176 Hz. The calculated curve (Th) is based on the assumption of thermal equilibrium and therefore corresponds to the theoretical limit of zero displacement velocity.

The differences between the measurements presented in Fig. 15 are only partly due to the different displacement velocity. Our measurements show a certain degree of nonreproducibility. On the one hand, this can have several experimental reasons, in particular thermal drifts of the setup, mechanical, acoustic, and electrical noise, spatial inhomogeneity of the sample transmission, etc. A major remaining limitation of our setup arises from residual drifts between the trapping beam (position controlled with respect to the commercial microscope stand) and the part of the (custom-built) coarse displacement stage that holds the piezo stage. This probably induces the small shifts between the experimental curves. On the other hand, there is the possibility of an intrinsic statistical variation between different force measurements, even in the regime of small displacement velocity. In the following section, a possible mechanism for a statistical variation of the force signals is discussed.

Force fluctuations: bistability of the opening fork

In this section we consider the fluctuations in the force signal occurring at low displacement velocity. In the measurement presented in Fig. 16 a DNA molecule is opened with a stage velocity of 20 nm/s, the displacement is stopped for a short time, and we let the molecule close itself by moving back the stage (again with a velocity of 20 nm/s). A rough mirror symmetry appears between the left and right halves of the figure. In particular, we note that the orientation of the sawtooth structures is reversed in time during the closing of the molecule, and that there is no significant reduction of the force during closing as compared to the opening. Of course, our theoretical description predicts a perfect symmetry between opening and closing because it assumes that thermal equilibrium is reached at each instant t. The presented experimental data show a significant deviation from this prediction. In Fig. 17 we have directly superposed the opening and the closing data. There are regions where the two curves are similar, and there are regions where they differ. In general, a more important difference is observed between an opening and a closing measurement than between two opening measurements. We also find that the degree of reproducibility between two closing measurements is typically smaller than the one between two opening measurements. This indicates that equilibrium is not systematically reached, in particular during closing (even at the small velocity of 20 nm/s, corresponding to zipping 20 basepairs per second on average). In Fig. 16 the molecule starts to open at a force between 10 and 15 pN, while during closing important sawtooth structures persist down to 5 pN. Clearly, such differences are not explained by our current theoretical description. To account for them theoretically, we would have to go beyond the assumption of thermal equilibrium.

View larger version (1K): [in this window] [in a new window]   FIGURE 16   Opening and closing a DNA molecule with a displacement velocity of 20 nm/s. The force signal of the trapping interferometer (upper curve) and the displacement of the translation stage (lower curve, measured with a capacitive position sensor) are shown as a function of time. The data are recorded with a sampling rate of 100 Hz and an anti-aliasing filter cutoff frequency of 44 Hz.

View larger version (30K): [in this window] [in a new window]   FIGURE 17   Comparison of the force signals recorded during the opening and during the closing phase of the measurement of Fig. 16. The lower horizontal time scale corresponds to the opening (upper dark curve for the force, lower dark curve for the displacement). The closing signals (red curves) are presented with a reversed time axis (time increase from right to left and is given by the upper horizontal scale), such that the displacement curves of the opening and closing (lower lines) coincide.

In Fig. 18 a detail of the force curve is presented where the opening and closing signals show a rather close agreement. The corresponding theoretical results are shown in Fig. 19. We notice that, at this scale, the experimental force curves exhibit important differences compared to the calculated curve. The decreasing part of the sawteeth is smoother in the calculated force curve (Fig. 19, upper line) than in the measurement. In addition, force flips appear in the measured signal that are not borne out by the calculation. In Fig. 18, pronounced flips appear in the opening data in the time interval 60-64 s and for closing in the interval 133-137 s. The corresponding displacement interval (4.95-5.03 µm) is given by the displacement curves (straight lines, right scale). Such bistabilities are observed only at elevated stiffness (with the trapping interferometer but not with the soft microneedle), at low displacement velocity and in the vicinity of the decreasing part of a sawtooth. Under these conditions, sudden flips between force values are frequently observed and occur also without sample displacement (data not shown).

View larger version (31K): [in this window] [in a new window]   FIGURE 18   Detail of the force and displacement signals as a function of time. For the opening data (dark lines) time increases from left to right and is given by the lower horizontal scale. For the closing data (red lines) time increases from right to left and is given by the upper horizontal scale. For convenience, we have vertically upshifted (by 2 pN) the force signal recorded upon closing and have positioned opening and closing data horizontally, such that the displacement curves slightly separate and the features of the force curves closely coincide.

View larger version (0K): [in this window] [in a new window]   FIGURE 19   Calculation of the force F (upper curve) and the statistical variance in the force var F (lower curve) for the displacement interval of Fig. 18. The force variance is defined by var F =  , where ... indicates the ensemble average (Eq. 12) of our theoretical description.

The amplitude of the force fluctuations on the measured curves of Fig. 18 exhibits a sizable variation with displacement. Qualitative agreement is obtained between the displacement dependence of the calculated force variance (lower curve of Fig. 19) and the amplitude of the measured fluctuations. The weak shoulder at t = 59 s (displacement of 4.94 µm) corresponds to an enhanced force fluctuation in experiment and theory. Maximum fluctuation occurs around the drop of the big sawtooth (around 61.5 s or 4.97 µm), again both in experiment and theory. The increased force fluctuations measured around t = 63 s are also predicted theoretically. The amplitude of the noise on the measured signal should not, however, be equated with the calculated value of var F, because the former is reduced by low-pass filtering and includes instrumental noise.

Fig. 20 illustrates our interpretation of the flips between different force levels. For a sample displacement corresponding to a position in the vicinity of the force drop of a sawtooth (left part, corresponding to ~61-62 s in Fig. 18, where there is a bistability in the signal), the total energy, presented as a function of the number j of opened basepairs, exhibits at least two local minima, which are close in energy and are separated by a potential barrier. One minimum corresponds to higher force and lower j, the other one to lower force and higher j, leading to a bistability of the opening fork. At slow displacement (or at fixed stage) and provided that the height of the potential barrier does not exceed a few k_BT, the system fluctuates thermally between the different minima, and therefore the Brownian motion resembles telegraph noise. Thermal equilibrium theory is appropriate if the system disposes enough time to exploit the whole relevant parameter space to establish the average value. If the imposed displacement velocity is too high, we expect differences between the force signals recorded upon opening and closing. Hysteresis may occur because during opening the system statistically remains longer close to the minimum of small j, while during closing it remains longer close to the minimum of higher j.

View larger version (2K): [in this window] [in a new window]   FIGURE 20   Bottom: Calculated energy landscape for a displacement of 4970 nm (left, corresponding to the rapidly decreasing part of a sawtooth in th