Viscoelastic Measurements of Single Molecules on a Millisecond Time

the power spectrum to be obtained rapidly and analyzed with a suitable model. .... Single Molecule TechniquesAtomic Force Microscopy and Optical T...
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Langmuir 2005, 21, 4765-4772

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Viscoelastic Measurements of Single Molecules on a Millisecond Time Scale by Magnetically Driven Oscillation of an Atomic Force Microscope Cantilever Masaru Kawakami,† Katherine Byrne,† Bhavin S. Khatri,† Tom C. B. Mcleish,† Sheena E. Radford,‡ and D. Alastair Smith*,† Institute of Molecular Biophysics, Astbury Centre for Structural Molecular Biology, School of Physics and Astronomy and School of Biochemistry and Microbiology, University of Leeds, Leeds LS2 9JT, United Kingdom Received December 10, 2004. In Final Form: March 4, 2005 The dynamical nature of biomolecular systems means that knowledge of their viscoelastic behavior is important in fully understanding function. The linear viscoelastic response can be derived from an analysis of Brownian motion. However, this is a slow measurement and technically demanding for many molecular systems of interest. To address this issue, we have developed a simple method for measuring the full linear viscoelastic response of single molecules based on magnetically driven oscillations of an atomic force microscope cantilever. The cantilever oscillation frequency is periodically swept through the system resonance in less than 200 ms allowing the power spectrum to be obtained rapidly and analyzed with a suitable model. The technique has been evaluated using dextran, a polysaccharide commonly used as a test system for single molecule mechanical manipulation experiments. The monomer stiffness and friction constants were compared with those derived from other methods. Excellent agreement is obtained indicating that the new method accurately and, most importantly, rapidly provides the viscoelastic response of a single molecule between the tip and substrate. The method will be a useful tool for studying systems that change their structure and dynamic response on a time scale of 100-200 ms, such as protein folding and unfolding under applied force.

* Corresponding author: tel, 00 44 113 343 3875; fax, 00 44 113 343 1881; e-mail, [email protected]. † School of Physics and Astronomy. ‡ School of Biochemistry and Microbiology.

To date various techniques have been developed that are capable of measuring the viscoelastic response of biomolecules, for example, by oscillating the AFM cantilever holder8 or the sample stage using an external piezo actuator between the scanner and sample.9,10 In these experiments, the mechanical response of the molecule is measured via the amplitude and phase of the cantilever oscillations and both the viscous friction and elasticity can be determined. However, the frequency of the oscillation is limited by the response of the piezo element to a narrow bandwidth, usually below a few kilohertz. Under these conditions only the stiffness of a molecule can be derived with any degree of accuracy8 and therefore, while a few qualitative discussions have been made using lowfrequency oscillations, these have not explored quantitative viscoelastic parameters.9,10,12 Recently, Humphris et al. reported a magnetically driven cantilever oscillation technique, which they used to determine the viscoelastic response of a single molecule of the polysaccharide dextran for the first time.13 One consideration in the case of actively driven oscillations is that the system can be near or far from equilibrium, depending on the amplitude and frequency of oscillation. In general, if the conformational transitions of the molecule are rapid compared with the frequency of oscillation, then the system can be regarded as being close to equilibrium (quasi-equilibrium). By

(1) Binning, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930-933. (2) Smith, S. B.; Finzi, L.; Bustamante, C. Science 1992, 258, 11221126. (3) Rief, M.; Clausen-Schaumann, H.; Gaub, H. E. Nat. Struct. Mol. Biol. 1999, 6, 346-349. (4) Marszalek, P. E.; Oberhauser, A. F.; Pang, Y. P.; Fernandez, J. M. Nature 1998, 396, 661-664. (5) Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H. E. Science 1997, 275, 1295-1297. (6) Oberhauser, A. F.; Marszalek, H. P.; Erickson, H. P.; Fernandez, J. M. Nature 1998, 393, 181-185. (7) Florin, E. L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415-417.

(8) Chtcheglova, L. A.; Shubeita, G. T.; Sekatskii, S. K.; Dietler, G. Biophys. J. 2004, 86, 1177-1184. (9) Mitsui, K.; Nakajima, H.; Arakawa, H.; Hara, M.; Ikai, A. Biochem. Biophys. Res. Commun. 2000, 272, 55-63. (10) Okajima, T.; Arakawa, H.; Alam, M. T.; Sekiguchi, H.; Ikai, A. Biophys. Chem. 2004, 107, 51-61. (11) Florin, E. L.; Radmacher, M.; Fleck, B.; Gaub, H. E. Rev. Sci. Instrum. 1994, 65, 639-643. (12) Sakai, Y.; Ikehara, T.; Nishi, T.; Nakajima, K.; Hara, M. Appl. Phys. Lett. 2002, 81, 724-726. (13) Humphris, A. D. L.; Tamayo, J.; Miles, M. J. Langmuir 2000, 16, 7891-7894.

Introduction The atomic force microscope (AFM) has been widely adopted as a powerful tool for imaging samples in air with the potential for atomic resolution.1 However, a particular strength of the AFM is its ability to operate in fluids, thus providing a high-resolution imaging technique for biological samples capable of functioning under physiological conditions. In addition to its role as a microscope, the AFM has been developed as a powerful complementary tool to optical tweezers2 in force spectroscopy, especially of biomolecules such as DNA/RNA,3 polysaccharides,4,5 proteins,6 and receptor-ligand pairs.7 Conventional AFM constant loading rate and force clamp experiments provide only a measurement of mechanical resistance or stiffness of the system under study. However, under physiological conditions biomolecules are dynamic in both their local and global conformations, and these fluctuations play an important role in biological function. The dynamical nature of these systems implies that an understanding of the viscoelastic behavior may aid in developing a full understanding of the function of these biological molecules.

10.1021/la0469699 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/12/2005

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contrast, large amplitude oscillations may probe a nonlinear or nonharmonic regime of response. Some molecules have conformational transitions which are accompanied by large changes in viscoelasticity, and they will also respond nonlinearly to the forced oscillation when it is large. Recently we developed a new technique in which the thermal noise of the cantilever/single molecule system under constant force conditions (force-clamp) is recorded, and the power spectral density (PSD) of these fluctuations is calculated and analyzed using a viscoelastic model. This approach is unique in that the measurements are made very close to equilibrium.14 However, the drawback of this approach is that it requires thermal noise to be collected for the order of seconds to tens of seconds at each force clamp set point in order to provide a PSD with good enough signal-to-noise ratio for analysis. This means that the molecule has to be held between the cantilever tip and substrate for many tens of seconds, which is difficult to achieve. Chemical (covalent) attachment of a molecule to the substrate and cantilever15 is one approach to prolonging the period of time for which a molecule can be held and therefore extending the time during which thermal noise data can be captured, providing thermal drift of the cantilever and scanner do not affect the results. However, a more serious impediment to such experiments is the fact that biomolecules such as proteins spontaneously unfold after a short time under force (for example, a typical survival time for the folded state of ubiquitin with 100 pN applied is ∼1 s).16,17 Therefore, a faster method of acquiring the PSD of the cantilever/molecule system is required. Here we report a new approach based on the use of a magnetically driven cantilever incorporating a fast sweeping of the magnetic field drive frequency that enables us to gain a very sensitive cantilever response and hereby to perform a quick and highly reproducible acquisition of the PSD of the system. A high signal-to-noise ratio is achieved within a hundred milliseconds opening up the possibility of studying the viscoelastic response of conformational transitions of proteins and other biomolecules. In this paper, we validate the new technique using the polysaccharide dextran (R-1-6-glucan), since this polymer has been well characterized by AFM using both mechanical unfolding experiments and by theoretical approaches.4,5,14,18 Material and Methods Single Molecule Force Spectroscopy. The experiment has been designed so that the AFM cantilever is driven by an ac magnetic field, the frequency of which is rapidly swept through the system resonance permitting the PSD to be obtained in approximately 200 ms. A conventional AFM (PicoForce with a Nanoscope IIIa controller; Digital Instruments, Santa Barbara, CA) was modified for these experiments as follows. Two further PCs were introduced to the system to allow maximum flexibility of the control and data collection functions. Custom programs running under Igor Pro (WaveMetrics, Portland, OR) were developed to record the deflection signal at a high sampling rate (200 kHz) using a data acquisition board (NI-6014, from National Instruments, Austin, TX) and to sample the cantilever deflection at ∼8 (14) Kawakami, M.; Byrne, K.; Khatri, B.; Mcleish, T. C. B.; Radford, S. E.; Smith, D. A. Langmuir 2004, 20, 9299-9303. (15) Mitsui, K.; Hara, M.; Ikai, A. FEBS lett. 1996, 385, 29-33. (16) Oberhauser, A. F.; Hansma, P. K.; Carrion-Vazquez, M.; Fernandez, J. M. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 468-472. (17) Schlief, M.; Li, H.; Fernandez, J. M. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 7299-7304. (18) O’Donoghue, P.; Luthey-Schulten, Z. A. J. Phys. Chem. B 2000, 104, 10398-10405.

Kawakami et al.

kHz and time averaged for 10 ms which was used to control tip-sample separation. The Picoforce closed-loop control was turned on during all the experiments. The software from Digital Instruments (Nanoscope v6.1) was used only for automatic approach of the cantilever to the surface. For driving the cantilever, a commercial MAD (magnetically actuated driving) system from Digital Instruments was used. This system consists of a cantilever holder within which is an electromagnetic coil and a MAD drive unit which produces a constant amplitude of ac magnetic field in the coil independent of drive frequency. A function generator (TG1304; Thurlby Thandar, Cambridge, U.K.) was used to control the MAD drive output amplitude. Frequency sweep of driving signal was achieved via a commercial signal access module (Digital Instruments). Usually with active driving of the cantilever a lock-in amplifier is used and the amplitude and phase are measured simultaneously. However, in the case when the drive frequency is changed rapidly, a lock-in amplifier may not be able to respond fast enough and large uncertainties occur in the phase and amplitude values. Our approach is to record the deflection signal and calculate the PSD of the system as a postprocessing step. All AFM experiments were carried out in pure water at room temperature. Prior to each measurement the system was left for at least 1 h to thermally equilibrate. In the free cantilever analysis, the PSD of thermally driven fluctuations and magnetically driven oscillations at various drive amplitudes were measured in triplicate. The analysis of thermal noise and magnetic response will be described in detail later. Force spectroscopy of dextran with magnetic oscillation was performed as follows. The tip was initially approached to the sample at a speed of 100 nm/s and pressed into the polysaccharide monolayer (see below) with a force of several tens of nanonewtons for half a second. Then the tip was retracted at a constant speed of 15 nm/s while the cantilever was driven by the ac magnetic field whose frequency was repeatedly swept from 1 to 40 kHz in 200 ms while the deflection signal was sampled at 200 kHz and recorded. After the experiment, the deflection response to each frequency sweep was selected and analyzed as described below. Magnetization of the Cantilever Using a Samarium-Cobalt Particle. One of the keys to the success of this technique lies in the method of endowing the cantilever with a response to an external magnetic driving field. Cantilevers with evaporated magnetic layers (Chrome coated, MADOTR4 from DI) and nickel/polystyrene spheres glued to the back (10-30 µm diameter from Duke Science/ Magnetized Polystyrene Bead from Spherotex) were tested, but their magnetization was too small to obtain good response to the external magnetic field and their magnetization degraded with time even during experiments. A permanent magnetic particle (ferrite) glued to the back of the cantilever has been used,11 but we found that this material’s magnetization was not large enough and it is vulnerable to oxidation and thus not suited for use in aqueous solutions. Instead, we used a samarium-cobalt (SmCo) magnet, whose magnetization is very large and almost permanent. A SmCo magnet was first smashed with a hammer and a small piece (∼1 mm diameter) was then ground into a fine powder using a pestle and mortar. The resulting small particles clump together very strongly making it difficult to obtain a single particle to attach to the AFM cantilever. Separation of a single SmCo particle from the clump of particles produced by grinding was achieved as follows. A small amount of the powder was manually separated from the clump of particles and placed on the center of a glass slide. A small

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amount of methanol was dropped onto the powder and another glass slide was placed on top of the first. The glass slides were washed with methanol and dried with nitrogen gas prior to the procedure. The two glass slides were then moved with respect to each other spreading the suspension of SmCo particles around. The glass slides were then separated gently and the methanol was allowed to evaporate leaving scattered and isolated SmCo particles on the two slides. Under a light microscope, a single SmCo particle (approximately spherical in appearance and ∼10 µm in diameter) was picked up using a finely sharpened (∼3 µm end diameter) glass needle and glued onto the backside of a cantilever with epoxy-based resin which sets in hours. We found that the adhesion of molecules (e.g., polysaccharides or proteins) to the tip is small when the tip is gold-coated (data not shown). Therefore we used a cantilever purchased from MikroMasch (CSC38/Si3N4/ Al/BS, type c, nominal spring constant 80 pN/nm (when measured the spring constant was in fact closer to 300 pN/nm)), which is Si3N4 and aluminum coated on the backside. This rectangular cantilever gives a distinct resonance at around 8 kHz in fluid, the shape of which is hardly affected by the distance between the cantilever and surface (see Supporting Information). Usually, in the case of triangular cantilevers, the resonance peak is heavily broadened when it is approached closer than few hundreds of nanometers from the surface in fluid. This means that there is a strong tip-sample distance dependence of the viscoelastic parameters of the cantilever which can be difficult to subtract from the response of the cantilever/molecule system. Additionally, when a SmCo particle and magnetic field is used to drive a triangular cantilever, the resonance is distorted and sometimes sidebands appear. We suggest that this is due to complex hydrodynamic flow around the triangular-shaped cantilever or the excitation of harmonics. In the case of rectangular cantilevers, however, these phenomena were not observed and excellent cantilever resonances were always obtained. Analysis of Thermally and Magnetically Driven Cantilever Deflection Signals. Analysis of the thermal noise of the free cantilever was performed with the simple harmonic oscillator (SHO) model as described elsewhere.14 Thermal noise data were converted from the photo detector signal (V) to cantilever displacement (nm) using the deflection sensitivity (V/nm) and Fourier transformed into the frequency-domain with Hahn-type windowing with segment lengths of 8192 data points. Fitting the PSD was performed using eq 119-22

〈|z(ν)|2〉 )

2kBTζ [k - m(2πν)2]2 + ζ2(2πν)2

(1)

where kB is Boltzmann’s constant, ν is the frequency in hertz, T is the absolute temperature, m is the mass of the system, z is the cantilever deflection, ζ is the friction constant, and k is the spring constant of the system. The fit to the PSD was performed between 4 and 20 kHz to avoid several significant spikes that typically occur in the PSD around 1-3 kHz, arising from mechanical vibrations in the scanner, and a broad signal from the photodetector (19) Sader, J. E. J. Appl. Phys. 1998, 84, 64-76. (20) Chon, J. W. M.; Mulvaney, P.; Sader, J. E. J. Appl. Phys. 2000, 87, 3978-3988. (21) Schaffer, T. E.; Cleveland, J. P.; Ohnesorge, F.; Walters, D. A.; Hansma, P. K. J. Appl. Phys. 1996, 80, 3622-3627. (22) Roters, A.; Johannsmann, D. J. Phys.: Condens. Matter 1996, 8, 7561-7577.

seen at 50 kHz. An additional, small, and constant PSD amplitude was used to take into account the static response of the spatial second harmonic of the cantilever. By consideration of frequencies well below the Nyquist frequency of 100 kHz, no problems caused by aliasing were encountered. The oscillatory motion of the cantilever in fluid can be described as forced damped simple harmonic motion, and the amplitude z at driving frequency ν can be expressed as23

z(ν) )

F

x(k - m(2πν)2)2 + ζ2(2πν)2

(2)

where F is the magnetic driving force. The deflection signal with magnetic oscillation data was converted from the photo detector signal (V) to cantilever displacement (nm) using the deflection sensitivity (V/nm) and Fourier transformed into the frequency-domain with Hahn-type windowing with segment lengths of 512 data points. A short segment length ensures a small spread in frequency in each window, resulting in an effectively flat input force spectrum (see Supporting Information). The PSD of the cantilever deflection from forced vibration is given as (see Supporting Information)

〈|z(ν)|2〉 )

F2/2RT (k - m(2πν)2)2 + ζ2(2πν)2

(3)

where R is the frequency slew rate (200 kHz/s) and T is the duration of each frequency chirp (0.2 s). At least one of the parameters (F, ζ, k, or m) needs to be determined by another independent method and fixed in the fitting to determine absolute values of other parameters. In the fitting of the free cantilever data, the spring constant was fixed to the value that was obtained from a measurement of the thermal noise. Therefore the fitting to the PSD of magnetic oscillation provides the force, effective friction, and mass of the system. This value of force gives the amplitude of oscillations for a given voltage applied to the magnetic coil, and once it is determined, we allow the other parameters of the forced PSD to be unconstrained. The PSDs due to forced oscillations are then collected on approach to the surface, when the cantilever is free, and also when retracting from the surface with a dextran polymer attached. The effective stiffness, friction, and mass of the cantilever determined on the approach phase are then subtracted from those determined on the retract phase to give the stiffness and friction of the dextran polymer. The analysis described above is not valid if the sweeping of the frequency is not performed linearly as a function of time. However, we made a close inspection of the sweep signal from the function generator and confirmed that the frequency versus time response is linear for the range from 3 to 40 kHz (data not shown). The PSD spectrum may be affected if the Q-factor of the cantilever is high, because of the long settling time of cantilever oscillations. However, from the thermal noise spectrum analysis, we find that the resonance is considerably broadened in fluid and the half-width of the PSD is about 5 kHz (Q ≈ 1), which gives ∼30 µs as the settling time. This is much faster than the frequency sweeping speed in this study. The sweeping speed was 2 × 105 Hz/s ≈ 6 Hz/30 µs. Normalization of Viscoelastic Values. The slowly varying force-distance data (sampled at 8 kHz and (23) Landau L. D.; Lifshitz E. M. Mechanics, 3rd ed.; ButterworthHeinemann: Oxford, 1976.

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In initial experiments the free cantilever was approached to within 300 nm from the dextran monolayer surface in fluid and the thermal noise was collected for 10 s. A typical thermal noise PSD is shown in Figure 1b (bottom trace). The PSD has a distinct peak at around 7 kHz (slightly shifted to lower frequency in comparison with the unmodified cantilever due to the addition of the small piece of SmCo). Using eq 1 thermal noise PSDs were fitted and the effective spring constant, friction, and mass of the free cantilever were estimated to be 0.274 ( 0.003 N/m, 2.621 ( 0.016 × 10-6 kg/s and 1.231 ( 0.018 × 10-10 kg, respectively. Next the free cantilever was oscillated by the magnetic drive, the frequency of which was swept from 1 to 40 kHz in ∼200 ms as a linear function of time with a driving amplitude of 200 mV corresponding to ∼500 pN of driving force. An example of the deflection signal response of the cantilever to the linear chirped signal is shown in Figure 1a, with the PSD of this response shown in Figure 1b. The changes in the PSD of the cantilever response, as a result of changing the driving amplitude

of the function generator from 100 to 2000 mV, are shown in Figure 1b (upper traces). The PSDs were fitted with eq 3 with the effective stiffness fixed to the value obtained from the thermal noise measurements (0.274 N/m) to yield the effective damping, mass, and force applied to the cantilever. The results of fitting are shown in parts c and d of Figure 1. Remarkably the effective friction and effective mass do not show any dependence on the driving amplitude within the range in this study (100-2000 mV, in Figure 1c,d). The effective mass would not be expected to change; however, the insensitivity of effective friction to drive amplitude is more surprising and must imply that over this range of oscillation amplitudes the dissipative friction against solvent is constant. The mean force shows excellent linearity with the driving amplitude (Figure 1e). A comparison of the values of effective damping and mass derived from magnetically driven and thermal oscillations shows that there is close agreement between the two techniques to within a few percent (6% and 3% for the effective damping and mass, respectively). A conventional constant pulling speed experiment using dextran was performed including the rapid and repetitive sweeping of the magnetic drive from 1 to 40 kHz with amplitude of 200 mV (∼500 pN) to the cantilever. Figure 2a shows the force distance curve with the approach phase shown offset above the retract phase. The constant sweeping of the magnetic field drive through the cantilever resonance results in the “fishbone” structure observed in the cantilever displacement. Note that the fluctuation of force shown here (maximum amplitude is ∼(1000 pN) is not the force applied to the molecule of interest but the force applied to the cantilever. Since the cantilever and molecule are in parallel, they experience the same extension not the same applied force. The amplitude of force fluctuation given to the molecule must be calculated using the stiffness of the molecule being stretched. Despite the repetitive sweeping drive frequency and relatively large maximum amplitude ((3 nm), the interaction between the Si3N4 tip and the dextran is sufficient to hold the molecule up to forces of ∼2.0 nN. The time trace of deflection signal shown in Figure 2a was divided into 163 segments, each of which comprises one frequency sweep event (from 1 to 40 kHz), as seen in Figure 1a. The data in each segment were Fourier transformed, and a series of PSDs were produced. In Figure 3a, example PSDs of the cantilever-molecule system at four different tensile forces and a PSD of the free cantilever at ∼130 nm from the surface (obtained in the approach phase) are shown. In all cases, each PSD has a distinct resonance peak between 7 and 10 kHz, which is clearly influenced by the force applied to the dextran molecule. Figure 3b shows a closeup of the force-extension curve of the dextran molecule in Figure 2 and the corresponding signal collected at 8 kHz and averaged over 10 ms. The force-extension trace shows the typical response of dextran which has been discussed in detail elsewhere.4,5,25 Before stretching, almost all the pyranose rings (∼98%) are in the chair conformation. The tensile force rises gently as the molecule is stretched (100-120 nm in Figure 3b), which is attributed to the entropic elasticity of polymer in the chair form. The corresponding PSD is slightly reduced in amplitude, broadened, and shifted to higher frequency (squares in Figure 3a) with respect to the free cantilever PSD (circles in Figure 3a). As the extension is increased up to 130 nm (∼0.6 nN), the enthalpic contribution to the elasticity starts to dominate,

(24) Smith, S. B.; Cui, Y. J.; Bustamante, C. Science 1996, 271, 795799.

(25) Marszalek, P. E.; Li, H. B.; Oberhauser, A. F.; Fernandez, J. M. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 4278-4283.

averaged over 10 ms) were fitted in the case of dextran extension using an extended freely-jointed-chain (FJC+) model.24 In the FJC model the chain is characterized by a number of Kuhn units Nk, which have a length and elasticity represented by b and κ. Here we take account of the chair-to-boat transition of the monomers, by assuming that the total length of the chain is given by a multiplication of the number of Kuhn units Nk by the effective monomer length averaged over a Boltzmann distribution between the chair and boat states. We define the monomer lengths in the chair and boat states using two different Kuhn segment lengths bCH, bBT and two elasticity values κCH and κBT. Thus the length of the dextran molecule between tip and substrate is given by the following equation

〈∆R(F)〉 )

Nk 1+e

(

(

)

F + κCHbCH F e-∆G(F) (4) bBTL(FbBT) 1 + κBTbBT

-∆G(F)

bCHL(FbCH) 1 +

(

)

)

where L represents the Langevin function and ∆G(F) ) ∆G0 - F∆x, with ∆x meaning the distance between the chair and boat “ground” states along the reaction coordinate. (Note that in the usual kinetic study with AFM the term “∆x” is used to represent the distance between the “ground and transition” states.) The viscoelastic properties of a single polysaccharide chain determined by fitting the experimental data (friction and stiffness) were then normalized to those of the individual monomer (pyranose ring) by multiplying by the number of Kuhn segments Nk. This permits the direct comparison of results of experiments in which different lengths of dextran chain are held between the tip and substrate. Polysaccharide Samples. Dextran powder (D-5251; lot no, 69H1267, average MW ) 473000) was purchased from Sigma (Deisenhofen, Germany) and dissolved in pure water at a concentration of 10 wt %. An aliquot of the polysaccharide solution was dropped on a glass coverslip (10 mm diameter, 0.2 mm thick; Agar Scientific, Essex, U.K.) and left to dry overnight. The glass coverslip was rinsed extensively with water leaving a monolayer of the polysaccharide. Results and Discussion

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Figure 1. Response of a free cantilever to the frequency sweep magnetic drive in fluid 300 nm above the sample surface. (a) Typical cantilever deflection as a function of magnetic drive frequency (top axis). Lower axis shows the time scale of the frequency sweep (1-40 kHz in approximately 200 ms). (b) Power spectral density (PSD) of the free cantilever obtained by analyzing the thermal noise (lowest trace) and magnetically driven oscillation with varying driving amplitude using the simple harmonic oscillator (SHO) model of eq 1 (solid fit lines). Spikes seen at 3 and 4 kHz in the thermal noise PSD (indicated with asterisks) are due to mechanical vibrations from scanner. (c, d, and e) Results of fitting the magnetically driven PSDs with the SHO model of eq 3. In this case the free cantilever spring constant (determined from the thermal noise measurements) was kept fixed and the fit procedure yields the effective damping, mass, and force on cantilever. All PSD measurements were carried out in triplicate, and their standard errors on the mean are shown as error bars. The error bars of force are too small to be shown in (e). Line in (e) is linear function fit with no intercept value.

Figure 2. Stretching of a single dextran molecule in fluid with frequency sweep magnetically driven oscillation. The forceextension curve of the experiment is shown. The approach phase is shown offset above the retract phase for clarity. The “fishbone” appearance is due to the repeated sweeping of the magnetic drive frequency through the resonance of the cantilever. The difference in the appearance of the approach and retract phase is due to the different speeds of scanner motion in the two phases. (The fishbone structure is tilted in comparison with Figure 1 because the cantilever displacement is plotted here as a function of extension which is given by the z scanner position corrected for the cantilever deflection. In Figure 1a the cantilever displacement is plotted as a function of time and therefore no tilt is observed.)

resulting in a steeper rise of the force extension curve. Up to this point, the extended freely jointed chain model is

usually used to describe the force extension curve.24 The influence of the applied force on the PSD spectrum becomes

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Figure 3. Viscoelastic analysis of magnetically forced vibration of the cantilever/molecule system in fluid. (a) Examples of PSD spectra of the free cantilever 130 nm from surface (circles) and the cantilever-dextran system with applied force of 0.18, 0.57, 1.01, and 1.93 nN. Lines are fits to the forced damped SHO model of eq 3. (b) The force-extension curve of a single dextran molecule. The low pass-filtered force-extension data and its fit using eq 4 are shown offset for clarity (upper thin curve and thick line). Dependences of the molecular friction (c), stiffness (d), and mass (e) of the dextran. The gradient (dF/dx) of lowpass-filtered force-distance curve in (b) fitted with eq 4 is overlayed as solid line in (d).

more significant at this point (triangles in Figure 3a). Upon further extension, the pyranose rings in the dextran chain undergo the conformational transition from chair to boat, which elongates the length of the displacement between two consecutive glycosidic oxygen atoms by ∼18% (from 0.44 to 0.52 nm).4,5 This conformational change gives a plateau region in the force-extension curve seen between 135 and 150 nm in Figure 3b. Recent MD simulations have revealed that a rotation around the C5-C6 bond which occurs simultaneously with the chair-boat transition also affects the elasticity of the dextran chain.5,26 During the transition, the amplitude of the PSD starts to rise again, is sharpened, and shifted to lower frequency,

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resulting in an appearance very close to the initial PSD (see crosses and squares in Figure 3a). When all the pyranose rings are in the boat conformation (at an extension of 150-155 nm), the tensile force starts to rise more rapidly due to the deformation of the pyranose rings and the bending of the covalent bonds in the linkage between the pyranose rings. At this stage the peak in the PSD spectrum reduces rapidly in amplitude, is heavily broadened, and shifted to higher frequency (diamonds in Figure 3a). Finally the dextran molecule detaches from the tip at ∼2.0 nN (155 nm extension). The behavior of the PSD as a function of applied force shown in Figure 3a is exactly the same as observed when the Brownian motion of the dextran-cantilever system is monitored in a force clamp experiment.14 Fitting of the PSDs was performed using eq 3 with constant amplitude of magnetic driving force, and the effective damping, stiffness, and mass of the free cantilever (approach phase) and of the cantileverdextran system (retract phase) were obtained. When a polymer is stretched by a cantilever and the fluctuations of the cantilever-molecule system observed, the cantilever and the polymer can be considered to be in parallel since the changes in extension of the polymer and cantilever are the same at their point of contact. Thus, the stiffness and friction of the cantilever and polymer are summed to give the total system stiffness and friction. In other words, the effective stiffness and friction of the molecule can be given by subtracting the values of free cantilever from those of the cantilever-molecule system at the corresponding tip-sample distance.11,13,14 Implicit in this procedure is that we model the biopolymer as an overdamped spring and dashpot in parallel, which is justified when conformational internal friction is dominant. The results are shown in parts c, d, and e of Figure 3, respectively. As the dextran molecule is stretched by the cantilever, the molecular friction and stiffness show a sigmoidal change depending on the extension (130-150 nm), in agreement with thermal noise experiments14 showing that these parameters are affected by the conformational transition of the pyranose rings. As expected, the molecular mass is estimated to be almost zero because the mass of a single dextran molecule is ∼10-21 kg, which is negligibly small compared to that of the free cantilever (∼10-10 kg) and is independent of the extension. This suggests that it would be possible to fix the value of effective mass in the fitting of the PSDs, but this did not in fact give a better fit (data not shown). The molecular stiffness can be obtained from the gradient of the force-extension curve in Figure 3b since it has been corrected for cantilever deflection. If the signalto-noise ratio of the force-extension curve is poor, then estimation of the molecular elasticity directly from its gradient may be inaccurate.8 Obviously the gradient of the force-extension curve shown in Figure 3b cannot be obtained accurately due to the intermittent large oscillations. Therefore, the low pass-filtered force curve (Figure 3b, upper trace) was fitted using eq 4 and the gradient (dF/dx) was obtained by differentiating the fit line. In Figure 3d, the gradient obtained in this way is superimposed over the elastic constant obtained from the PSD fitting (filled circles). Close agreement is obtained between these two methods for obtaining the elastic constant of the molecule indicating that the analysis of PSDs with the forced damped simple harmonic oscillator model is valid. Seven further independent single molecule extension experiments were successfully achieved giving in total (26) Lee, G.; Nowak, W.; Jaroniec, J.; Zhang, Q.; Marszalek, P. E. Biophys. J. 2004, 87, 1456-1465.

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the viscoelastic force dependence of eight different single molecules. It should be noted that the length of the dextran molecule between tip and substrate in these experiments is not controlled since the “pick-up” is a random process. Therefore normalization of the molecular viscoelastic parameters to the monomer values is necessary in order to compare the results of different experiments. Since the glycopyranose rings are connected in series, the monomeric stiffness can be found by multiplying the molecular stiffness of the molecule by the number of pyranose rings between tip and substrate (Nk), which can be determined by fitting the low pass-filtered force-extension curve in Figure 3b using eq 4. We see an excellent convergence of the monomeric stiffness as can be seen in Figure 4a, highlighting the high reproducibility of the experiments. This approach to normalization is only applicable for the case where the chain friction has a negligible dependence on solvent friction. If solvent friction is significant compared with conformational internal friction, then the normalized monomeric friction constant would scale with the chain length (Nk). Conversely, if internal friction dominates, the friction of the monomer should scale as 1/Nk.27 We find that normalization of the friction by multiplication by Nk leads to good convergence of the effective friction constant of each monomer in Figure 4b (open circles). The small deviation means not only the good reproducibility of the measurement but also the high validity of the assumption that the dissipation is domi-

nated by the internal friction of the chain and the contribution from the solvent friction is negligibly small. A further comparison of the results of this magnetic drive sweeping method with those of the analysis of Brownian fluctuations of dextran reported previously14 may now be made. The viscoelastic parameters of each monomer unit of dextran determined for 15 molecules by the Brownian noise method are also shown in Figure 4 (crosses). There is good agreement between the values of the viscoelastic properties of each monomer in dextran obtained for 23 different molecules by these two different techniques. It is noteworthy that the two different methods are executed using completely different types of cantilever. They have different shapes and stiffness (thermal noise experiments were carried out with a triangular cantilever of nominal stiffness 0.35 N/m and the magnetically driven experiments used rectangular cantilevers of stiffness 0.27 N/m). Despite this, the viscoelastic values of each monomer derived from two methods show excellent agreement indicating that they are independent of the cantilever characteristics. It is also surprising that even with the relatively large amplitude of magnetically driven oscillations ((3 nm maximum amplitude), the viscoelastic values determined by this method are very close to those derived from the thermal noise method over the force range used in this study (0-1.5 nN). The shortest dextran molecule measured in these experiments was ∼74 nm (Nk ) 165); thus the fluctuation in length induced is approximately (4% of the chain length. However, in the case of the thermal driving experiment, the fluctuation of the cantilever is less than (0.5 nm. This result indicates that the dextran molecules respond linearly to forced oscillation over a large dynamic range even after the chairto-boat transition. The enthalpic elasticity of the pyranose ring of each monomer in the chair conformation is estimated to be 15 ( 1.8 N/m from thermal noise experiments14 and 13.5 ( 2.5 N/m from the magnetic driving experiments. Both of these values are in good agreement with values determined previously by other groups (14.6 ( 2.7,4 14.6 ( 3.7,25 8 ( 1 N/m,5,28) lending further confidence to both of the methodologies considered here. During the transition from chair to boat, the stiffness of each monomer shows a sudden drop, meaning that the chair-boat transition gives considerable extra elasticity to the ring. After the transition, the stiffness of the pyranose ring in the boat conformation shows drastic rise with force. It has been reported that in the boat conformation the stiffness of the pyranose ring of dextran has a maximum value at high force (40 ( 10,5,28 37.6 ( 6.5 N/m25). The largest value of stiffness estimated from the data in Figure 4a is 45 ( 5 N/m. The friction constants of each monomer yielded by both methods also show a clear sigmoidal dependence on applied force similar to the monomeric stiffness (Figure 4b). A broad peak occurs around 0.6 nN, a minimum around 0.9 nN, and sharp rise at forces over 1.0 nN. As discussed above, the close resemblance of the behavior of the friction to that of the elasticity indicates that the friction of the pyranose monomer is strongly influenced by the chair-to-boat transition. Before the transition, i.e., below 0.6 nN of tensile force, the friction of the pyranose monomer gives a gradual rise with force which is due to the friction provided by the bending motions or forced rotation of dihedral angles in the backbone of the polymer. The fall in the friction constant accompanies the chair-

(27) de Gennes, P. G. Scaling Concepts in Polymer Physics, 2nd ed.; Cornell Univeristy Press: Ithaca, NY, 1985.

(28) Rief, M.; Fernandez, J. M.; Gaub, H. E. Phys. Rev. Lett. 1998, 81, 4764-4767.

Figure 4. Force spectra of the (a) stiffness and (b) friction of the data normalized by the number of monomeric units Nk, i.e., these data represent the values for a single glycopyanose ring. These parameters were determined from 8 magnetically driven oscillation experiments (open circles) and from 15 thermal noise experiments14 (crosses). The number of monomers in each molecule was determined by fitting the force-extension curve using eq 4. The values of Nk in these experiments were 452, 336, 165, 290, 181, 253, 421, and 271 for the magnetic oscillation experiments and 403, 387, 401, 295, 500, 414, 86, 199, 189, 357, 566, 160, 195, 74, and 631 for the thermal noise measurements.

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to-boat transition, and surprisingly its value decreases almost to zero value despite the high tensile force in this region (1.0 nN). Recent work suggests that this phenomenon is microscopic in origin and related to the fact that force changes the heights of transition barriers and hence the time scale of conformational transitions, which itself is intimately linked to internal friction. Theoretical/ computational studies such as molecular dynamics simulations,26 ab initio calculations,18 or more sophisticated microscopic models of internal friction are needed to elucidate the processes which govern the complex viscoelastic response. In conclusion, we have demonstrated a new technique that is capable of rapidly (on a time scale of 100-200 ms) acquiring the viscoelastic response of single molecules based on a magnetically driven AFM cantilever and rapid sweeping of the drive frequency. We have reported the application of this technique to the polysaccharide dextran. In the case of this polymer, the viscoelastic response measured by magnetically driven oscillations of the cantilever can be compared directly with recently published data in which the thermally driven Brownian motion of dextran was analyzed to yield the viscoelastic parameters of interest.14 The disadvantage of the latter technique is the relatively long period of time required to collect sufficient thermal oscillations to achieve good signal-to-noise ratio in the measured PSD (of the order of seconds at every force clamp set point). The two methods

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give values of effective friction and elasticity of each pyranose ring in the polymer that agree closely despite the use of completely different cantilever shapes and stiffness and an order of magnitude greater amplitude of oscillation when using the magnetic drive. The next step is to combine this new approach with a force clamp14,29,30 technique to permit kinetic measurement of viscoelastic response in biopolymer such as DNA/RNA or in the refolding/unfolding transitions of proteins. Acknowledgment. We thank G. S. Beddard, P. D. Olmsted, D. J. Brockwell, and members of the Institute of Molecular Biophysics for valuable discussions. This work was supported in part by the EPSRC. M.K. was a JSPS Visiting Research Fellow and is now supported by the Institute of Molecular Biophysics, University of Leeds, and S.E.R. is a BBSRC Professorial Fellow. Supporting Information Available: Figures showing typical PSD cantilever spectra and distance dependence of effective damping of various cantilevers and discussion of the power spectral density of sampled chirped signal. This material is available free of charge via the Internet at http://pubs.acs.org. LA0469699 (29) Oberhauser, A. F.; Hansma, P. K.; Carrion-Vazquez, M.; Fernandez, J. M. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 468-72. (30) Fernandez, J. M.; Li, H. Science 2004, 303, 1674-1678.