JOURNAL OF APPLIED PHYSICS 100, 074315 共2006兲

Dynamic-force spectroscopy measurement with precise force control
using atomic-force microscopy probe
Osamu Takeuchi,a兲 Takaaki Miyakoshi, Atsushi Taninaka, Katsunori Tanaka, Daichi Cho,
Machiko Fujita, Satoshi Yasuda, Suzanne P. Jarvis,b兲 and Hidemi Shigekawac兲
Institute of Applied Physics, 21st Century COE, CREST-JST, University of Tsukuba,
Ibaraki 305-8573, Japan

共Received 6 January 2006; accepted 11 July 2006; published online 11 October 2006兲
The accuracy of dynamic-force spectroscopy 共DFS兲, a promising technique of analyzing the energy
landscape of noncovalent molecular bonds, was reconsidered in order to justify the use of an
atomic-force microscopy 共AFM兲 cantilever as a DFS force probe. The advantages and disadvantages
caused, for example, by the force-probe hardness were clarified, revealing the pivotal role of themolecular linkage between the force probe and the molecular bonds. It was shown that the feedback
control of the loading rate of tensile force enables us a precise DFS measurement using an AFM
cantilever as the force probe. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2355432兴
I. INTRODUCTION

The properties of noncovalent bonding between organic
molecules have been extensively studied not only for understanding the biochemical reactions in living bodies but also
for practical applications, realizing efficient biochemical materials with the ability of molecular identification or producing complex nanodevices by using self-assembly techniques.
In this context, Evans and co-workers developed dynamicforce spectroscopy 共DFS兲 in 1997,1,2 which had distinct advantages over other thermodynamic experimental methods.
DFS measurement provides not only an energy barrier height
for the unbinding reaction but also a length scale to which
the interaction between two molecules extends. Such a microscopic analysis of intermolecular interactions is of great
importance in understanding the origin of the function of amolecule from its complex three-dimensional structure.
In DFS measurement, rupture force, the force required torupture the molecular bond, is measured under a gradually
increasing tensile force applied to a bonded molecular pair.
To date, the biomembrane force probe 共BFP兲 has been
mainly used as the force probe in DFS measurements insteadof the atomic-force microscopy 共AFM兲 probe, although thelatter is much more popular as an ultrasensitive force probe
in other applications. This is because of the belief that asofter force probe is more accurate for DFS measurement. In
this study, we reconsider this issue by revealing the role ofthe molecular linkage between the force probe and the molecular bonds. The advantages of using an AFM probe for
DFS measurement are demonstrated.
II. DYNAMIC-FORCE SPECTROSCOPY

When a molecular pair is ruptured by a tensile force in a
DFS measurement, the obtained rupture forces have a linear
relationship to the logarithmic loading rate of the tensilea兲Electronic mail: osamu@big.or.jp
Permanent address: SFI Nanoscience Laboratory, Lincoln Place Gate, Trinity College, Dublin 2, Ireland.
c兲
Website: http://dora.ims.tsukuba.ac.jp/b兲

0021-8979/2006/100共7兲/074315/6/$23.00

force.1–4 This is a consequence of the negligible rebinding
rate of once-ruptured molecules, differing from the case of
the unzipping model which has competitive unbinding andrebinding rates.5–7 From the obtained linear relationship, we
can obtain information about the energy landscape of themolecular interaction, i.e., energy barrier position xb andnatural lifetime of the molecular linkage toff.
When the rupture force of a molecular pair is repeatedly
measured, the rupture force is distributed over a finite range.
The shape of its probability density function P兵f rup其 depends
on xb, toff, loading rate of tensile force r0, and thermal energy
kBT as given by Evans.4 We note that the equation given canbe further transformed into
P兵f rup其 = C exp兵共f rup − f *兲/f ␤其exp关1 − exp兵共f rup − f *兲/f ␤其兴,
共1兲with the normalizing constant C, the thermal energy of thesystem in the force dimension f ␤ = kBT / x␤, the distance between the energy barrier and the most stable point measured
along the reaction path x␤ = xb cos ␪, and the most probablerupture force f * that gives the maximum value of P兵f rup其,where
f * = f ␤ ln共toffr0/f ␤兲 = f ␤兵ln共r0兲 − ln共f ␤/toff兲其.

共2兲

As is obvious from Eqs. 共1兲 and 共2兲, the peak width ofthe density function is determined only by x␤ and kBT,
whereas the peak position also depends on toff and r0. Thus,
when a molecular system is measured with different loading
rates, the distribution of the rupture force is only shiftedalong the force axis without changing its peak shape. Consequently, although previous DFS measurement was only
concerned with the dependence of the peak position on theloading rate of the tensile force, if an ideal experiment could
be performed, the peak width and peak position of the rupture force distribution for a single loading rate would provide
us with both x␤ and toff.

100, 074315-1© 2006 American Institute of Physics

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FIG. 1. 共Color online兲 Soft molecular linkage acting as molecular rubber.

III. ACCURACY OF DFS MEASUREMENT

To date, it is believed that a soft force probe is better
than a hard force probe for DFS measurement. As a result,the BFP has mainly been adopted for the force probe, which
is a few orders of magnitude softer than the most commonlyused ultrasensitive force probe, AFM cantilevers. Here, we
review the issues that affect the accuracy of DFS measurement and reconsider why the harder force probe has not been
preferred. We demonstrate that understanding the hidden roleof the flexible molecular chain, which connects the force
probe and the molecular pair, fundamentally changes theconcept of DFS measurement that has been accepted untilnow.
A. Molecular linkages

In DFS measurement, a bonded molecular pair is fixed
onto the apex of a force probe and a substrate surface via
molecular chains of ⬃30 nm, as shown in Fig. 1. Such molecular chains act kinetically as molecular rubber, whose
force-distortion curves can be described by freely jointed
chain 共FJC兲 model or wormlike chain 共WLC兲 model7,8 andare highly nonlinear, particularly for small forces. The nonlinearity of the force-distortion curves often makes it difficult
to control the loading rate of tensile force, as discussed below, particularly when the molecular pair is more flexible
than the force probe. However, a flexible molecular chain,i.e., a long molecular chain, is favorable from other aspects.
The flexibility of a molecular chain provides the molecularpair with freedom of motion, which enables the rupture process to occur along the most energetically favorable reaction
path. In addition, long molecular linkages separate the apexof the force probe and the substrate, reducing the effect of
the long-range interaction between the probe and the substrate.
In general, the force probe has a much larger mass thanthe molecular system and has large damping in a liquid environment. Thus, its motion is many orders of magnitude
slower than that of the molecular system, whose trial frequency is usually ⬎109 Hz. In such a situation, the molecular chain, rather than the force probe, effectively exerts tensile force on the molecular pair. However, the force probe
merely measures the average force that the molecular chain
applies to the molecular pair. One must keep this point inmind when discussing the accuracy of the measurement.
B. Thermal fluctuation of force probe

In the situation shown in Fig. 1, the thermal fluctuation
amplitude of the force-probe deformation can be expressedas ⌬xfp = 冑kBT / 共kfp + kmol兲, with the spring constant of theforce probe kfp and the stiffness of the molecular linkage
kmol. In the force measurement, this fluctuation corresponds
to the uncertainty of the measured force value ⌬f fp= kfp⌬xrms = 冑kfpkBT / 共1 + kmol / kfp兲. This equation suggeststhat a softer force probe, namely, one with low kfp, gives asmaller uncertainty; thus, this indicates an advantage of using a softer probe. However, this is true only if the measurement bandwidth covers the whole spectral distribution of thethermal fluctuation. When the force probe has a high resonant frequency and a high quality factor, the frequency distribution of its thermal fluctuation is almost fully out of themeasurement bandwidth. In such cases, a very accurate measurement can be realized.9 Therefore, since a harder force
probe gives a higher resonant frequency and a higher quality
factor in a liquid environment, its thermal noise level can be
as low as that of softer probes.
One might be afraid that accurate measurement of tensile
force with a limited bandwidth is not sufficient but precise
control of the tensile force along with the loading rate isrequired. If the tensile force contains a large fluctuation, an
instantaneous large force can break the molecular bond even
if the average force is small. Thus, the fluctuation of tensile
force must be suppressed, regardless of the measurement
bandwidth. Then, does a hard probe cause larger fluctuationon the molecular bond?
The answer is “no.” According to statistical mechanics,
the thermal fluctuation that the molecular pair receives isalways kBT / 2. Namely, the amplitude of thermal fluctuation
is independent of the hardness of the force probe. For better
understanding, imagine that the force probe is thermally fluctuating around an averaged position and momentarily bent
towards the sample by the fluctuation. Then, the rupture
probability becomes smaller because the length of the molecular chain is decreased, although the apparent tensile force
is increased by the bending. The deformation of the force
probe is only induced by thermal fluctuation and the molecular pair does not experience the exact force of the instantaneously measured value. Consequently, one can use hard
force probes in DFS measurements, as long as the measurement precision of tensile force is sufficiently high within arequired bandwidth.
C. Rebinding probability

Rebinding of the ruptured molecules was investigated by
Evans as related to the disadvantage of using a hard force
probe.4 Since hard probes do not deform much for a small
rupture force, the two molecules are not pulled far apart after
the rupture. For example, a force probe with a 100 pN/ nm
spring constant pulls the molecules apart only by 1 Å when
the rupture force is 10 pN. In this case, these two molecules
may have a high probability of rebinding, which causes a
critical discrepancy from the ideal DFS theory. However, in areal system, the coupled stiffness of the force probe and molecular chains is given by kfpkmol / 共kfp + kmol兲. Thus, evenwhen the force probe is very hard, the contraction of the soft
molecular chain separates the two molecules if the molecular
chain is sufficiently flexible. Moreover, the motion of the two

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FIG. 2. DFS measurement system.

molecules has three-dimensional freedom after the rupture.
Therefore, the rebinding probability is negligible when weuse sufficiently long molecular chains.

D. Constant loading rate

When a force probe ideally follows Hooke’s law, theretraction of the base of the force probe at a constant velocity
gives a constant loading rate of the tensile force in rupture
force measurement. However, the actual force probe is combined with a molecular chain that exhibits a nonlinear forcedistance relationship. As a result, when the stiffness of theforce probe is greater than that of the molecular chain, retraction of the base at a constant velocity does not give a
constant loading rate. This problem was previously pointed
out and has been a difficulty in DFS measurement.10
As will be discussed, if the measurement bandwidth of
the force probe is sufficiently wide, we can solve this problem by performing feedback control of the retraction velocity
to maintain a constant loading rate.

E. Stability of measurement system

The instability of the measurement system due to thermal drift and the creep/hysteresis of the piezoelectric device
also affect the measurement accuracy. Such instability causesa change in the probe-substrate distance. When the loading
rate is determined by the constant velocity of the force probe,
this instability affects the DFS measurement. Since this effect is proportional to the stiffness of the force probe, a softer
probe seems to be superior. However, the effective stiffnessof the force probe is determined by the combination of theprobe stiffness and that of the molecular linkage. Thus, if themolecular chain is sufficiently flexible, the high stability ofthe measurement system can be maintained. Moreover, if one
uses the feedback control of the loading rate proposed above,
this type of instability is automatically compensated.
The instability of the detection of probe deformation
should be independently discussed. The response of a hard
force probe is affected more by such instability than that of asoft force probe. Thus, the detection signal often drifts
gradually during a long-term measurement. This difficulty
should be solved by calibrating the natural position of theforce probe every time before starting each rupture force
measurement, particularly when a hard force probe is used.

F. Advantage of using AFM probe in DFS measurement

Since its invention two decades ago, the measurement
accuracy of AFM has undergone many improvements. Now,
many good AFM systems are commercially available. Suchsystems generally have a higher resolution and a wider bandwidth for the detection of probe deformation than those of
BFPs and laser optical tweezers 共LOTs兲.11–13 Therefore,
AFM systems, with a relatively hard probe, possess a highaccuracy of force detection. In addition, the faster responseof the measurement system enables feedback control of theloading rate of tensile force, as demonstrated in the next
section.
IV. EXPERIMENT
A. DFS measurement system

The block diagram of our measurement system is shownin Fig. 2. An AFM system with a liquid cell 共Molecular
Imaging, Pico SPM兲 is connected to a commercial AFM
electronics system 共Digital Instruments Nanoscope III兲 thatis modified to extract the cantilever deflection signal and tomix additional signals into Z-piezoelectric voltage. After thecantilever is placed to gently touch the substrate, the feedback circuit of the AFM controller is turned off. Then, the
DFS measurement is fully controlled by a digital signal processor 共DSP兲 board 共Toro, Innovative Integration兲 with a
150 MHz TMS320C6711 共Texas Instruments兲 installed intoa PCI slot of another personal computer.
The algorithm of the DFS force curve measurement is asfollows. 共1兲 The cantilever is retracted 150– 200 nm from thesubstrate, and the deflection signal is recorded to calibrate
the natural position of the cantilever with zero probe-sample
interaction. 共2兲 The cantilever is moved towards the substrate
at a constant velocity, and when the deflection signal detects
100– 200 pN of a specified repulsive force from the surface,
the feedback control of Z-piezoelectric displacement isturned on to maintain the deflection signal at the same value.
共3兲 The cantilever and the substrate are kept gently touchingfor 10 ms– 1 s to form a molecular bond between them. 共4兲
The reference value for Z-piezoelectric feedback is changedin accordance with the desired loading rate. As a result, thecantilever is retracted from the substrate at a constant loading
rate until bond rupture. During this process, the deflection
signal and Z-piezoelectric displacement are recorded at a
100 kHz sampling rate onto a hard disk drive. 共5兲 Once thecantilever is 150– 200 nm from the substrate, the control
jumps back to step 共1兲 and the next force curve measurement
is started.

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J. Appl. Phys. 100, 074315 共2006兲

With this procedure, we can calibrate the natural positionof the force probe every time before starting the force curvemeasurement, to maintain an accurate contact pressure andcontact time of the probe apex with the substrate. In addition,
since Z-piezoelectric feedback automatically increases retracting velocity after bond rupture, this procedure can decrease the total acquisition time, which is efficient particularly for small loading rates.
B. Preparation of molecules on probe apex
and substrate

A gold-coated cantilever 共Bio-Lever, Olympus,
30 pN/ nm, rectangular shape兲 was immersed into a solutionof 8-amino-1-octanethiol, hydrochloride 共Sigma-Aldrich,
1 mM in ethanol兲 for 48 h to form a close-packed
self-assembled monolayer 共SAM兲 with an amino group at
the surface. After rinsing with ethanol, the cantilever was
immersed into a solution of biotin-PEG3400-COO-NHS关Shearwater
Polymers,
0.1 mMinethanol:DMF
共dimethylformamide兲 = 99: 1兴 for 20 h to fix biotin onto theprobe apex. Finally, the cantilever was rinsed with ethanol.
A gold thin film 共100 nm兲 was evaporated onto a freshly
cleaved mica surface in high vacuum at 400 ° C and wasflame annealed using a hydrogen gas burner for ⬃30 s.
Then, the gold substrate was immersed into a solution of
1,8-octanedithiol 共Sigma-Aldrich, 1 mM in ethanol兲 for 48 h
to form a close-packed SAM with a thiol group at the surface. After rinsing with ethanol, the substrate was immersed
into a solution of streptavidin-maleimide 共Sigma-Aldrich,
10 mg/ l in phosphate buffered saline 共PBS兲, pH 7.4: DMF= 99: 1兲 for 5 h to fix streptavidin molecules to the substrate.
Finally the substrate was rinsed with PBS.
The formation of closely packed SAMs on the probe
apex and the substrate was necessary to decrease the strong
interaction between the gold layers on both surfaces. DFS
measurements were performed at room temperature in PBS,
in which a small amount of free biotin molecules was dissolved. The free biotin molecules occupy the reaction site of
the streptavidin molecules on the substrate surface and reduce the probability of biotin-streptavidin bonding down to
10% for each measurement. This procedure is necessary tomeasure the rupture force at the single-molecule level, by
avoiding the formation of multiple molecular bonds.
V. RESULTS AND DISCUSSION
A. Rupture force measurement with feedback control

A typical force curve of streptavidin-biotin obtained using our system is shown in Fig. 3. The force signal during
retraction is plotted against 共a兲 time and 共b兲 cantilever displacement. At first, in region A, the probe apex and the substrate are in contact. Then, in region B, tensile force is applied to the molecular pair through the molecular linkage
关polyethylene glycol 共PEG兲兴. At the end of region B, themolecular pair is ruptured, and the cantilever returns to its
natural position. Although the force-distance curve is nonlinear in region B, as shown in Fig. 3共b兲, the loading rate is kept
almost constant by the feedback circuit, as shown in Fig.
3共a兲. The low-pass signal with a cutoff frequency of 200 Hz

FIG. 3. Typical force curve of rupture force measurement. 共a兲 Force andcantilever displacement plotted against time. 共b兲 Force plotted against cantilever displacement. The gray and black curves in the force plot are raw andlow-pass filtered signals with 50 kHz and 200 Hz bandwidths, respectively.

共dark gray兲 contains a thermal fluctuation of approximately
3 pN, whereas that of the raw data sampled at 100 kHz
共gray兲 is approximately 20 pN.
B. Histogram of rupture force

Thousands of force curve measurements were performed
using the method described above for each loading rate andapproximately 10% of them exhibited a rupture, as expected.
The rupture forces are summarized in histograms, as shownin Fig. 4. The extraction of the rupture force is performed byfitting the corresponding part of the force curves with a tilted
step function. With this fitting procedure, an accuracy of⬃1 pN was achieved for loading rates below 105 pN/ s. For
larger loading rates, reduced number of data points available
for fitting and insufficient feedback control of rupture force
decreased the measurement accuracy.
From the fitting of the histograms using Eq. 共1兲, we obtain the values of f *, x␤, and toff for each loading rate. As canbe seen, for example, in Fig. 4共c兲, whereas some of the plotsshow good agreement with the theoretical expectation, others
do not. More specifically, the plots tend to have a small but
nonzero probability in high-rupture-force areas and lessprobability near zero force. The former may be due to thesmall probability that multiple molecular bonds are formed
between the tip and the substrate, whereas the latter is due tothe limitation of the force-signal detection of distinguishing avery small rupture force in the thermal noise. These two
factors result in the peak positions f * of the fitting parameter
being higher than the maximum peak positions of the histograms 共indicated by the black arrowhead in the plots兲.
C. Rupture force dependence on loading rate

The most probable rupture forces f * obtained by the
curve fitting of the histograms using Eq. 共1兲 共filled circles兲Downloaded 12 Oct 2006 to 130.158.131.51. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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FIG. 5. 共Color online兲 Loading rate dependence of rupture force. The numbers next to the data points represent the barrier positions x␤ obtained by thewidth analysis of each histogram and the black line shows the linear fitting.

The deviation of the data points from the linear relationship is approximately the same order of magnitude as that ofthe previous experiment on the same system measured using
BFP, which has a spring constant more than one order ofmagnitude softer 共0.1– 3 pN/ nm兲 than our AFM cantilever.2
This result demonstrates the validity of using a hard cantilever for DFS measurement.
In comparison with the previous experiment by Merkel
et al.,2 x␤ obtained from the slope of the curves in Fig. 5 isalmost consistent with their result, whereas toff obtained hereis much smaller. Although the discrepancy in the values of
toff is not yet fully understood, the preliminary results of ourongoing experiments suggest that the different chemical attachment states of the streptavidin molecules onto the substrates affect the result.
VI. CONCLUSION

FIG. 4. Histograms of rupture force for loading rates of 共a兲 34.7 pN/ s, 共b兲
967 pN/ s, 共c兲 3320 pN/ s, and 共d兲 16 300 pN / s. The black lines are thefitting functions given by Eq. 共1兲.

are plotted against the logarithm of loading rate ln共r0兲, asshown in Fig. 5. There is a clear linear relationship of f *= 共4.67 pN兲兵ln共r0兲 − ln共6.04 pN/ s兲其. From the comparison of
the gradient and x intersection of the line with Eq. 共2兲, we
can obtain the parameters x␤ = 0.89 nm and toff = 1.3 s.
The x␤ values obtained by fitting each rupture force histogram using Eq. 共1兲 are also shown at each data point. Since
any experimental errors result in the peak widths of the rupture force distribution becoming wider, these x␤ values are
smaller than that obtained from the curve slope. However,the consistency of the values obtained by the two different
methods indicates the validity of our measurement.

In order to show why and how one can utilize widely
available AFM cantilever for precise DFS measurement, previously suggested disadvantages of using hard force probes
in DFS measurement were reconsidered. With understanding
the pivotal role of the molecular chain between the force
probe and the molecular complex, following consequences
are derived. Use of a hard force probe does not increasemeasurement uncertainty due to the thermal fluctuation nor
the rebinding rate of a ruptured molecular pair. A feedback
control of tensile force in rupture force measurement cansolve the difficulty in maintaining a constant loading rate andcompensate the instability of the measurement system. As aresult, the same-order accuracy as that by DFS measurement
using a biomembrane force probe was obtained by an AFMbased measurement system. These results pave the way for
DFS measurement to become more widely available.
ACKNOWLEDGMENT

This work was supported in part by a Grant-in Aid for
Scientific Research from the Ministry of Education, Culture,
Sports, Science and Technology, Japan.
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