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How does the molecular linker in dynamic force spectroscopy affect probing molecularinteractions at the single-molecule level?

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2016 Jpn. J. Appl. Phys. 55 08NB01(http://iopscience.iop.org/1347-4065/55/8S1/08NB01)
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REGULAR PAPER

Japanese Journal of Applied Physics 55, 08NB01 (2016)
http://doi.org/10.7567/JJAP.55.08NB01

How does the molecular linker in dynamic force spectroscopy affect probing
molecular interactions at the single-molecule level?
Atsushi Taninaka, Kota Aizawa, Tatsuya Hanyu, Yuuichi Hirano, Osamu Takeuchi, and Hidemi Shigekawa
Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Received October 6, 2015; accepted December 12, 2015; published online March 1, 2016
Dynamic force spectroscopy (DFS) based on atomic force microscopy, which enables us to obtain information on the interaction potential between
molecules such as antigen–antibody complexes at the single-molecule level, is a key technique for advancing molecular science and technology.
However, to ensure the reliability of DFS measurement, its basic mechanism must be well understood. We examined the effect of the molecular
linker used to ﬁx the target molecule to the atomic force microscope cantilever, i.e., the force direction during measurement, for the ﬁrst time, which
has not been discussed until now despite its importance. The effect on the lifetime and barrier position, which can be obtained by DFS, was foundto be >10 and >50%, respectively, conﬁrming the high potential of DFS. © 2016 The Japan Society of Applied Physics

1.

Introduction

The understanding of molecular interactions in the formationof speciﬁc bonds based on reciprocal recognition is a key
factor for understanding and utilizing the functions ofmolecules such as DNA and antigen–antibody complexes.1–3)
The probing of the interaction potential between two
molecules is the basis for describing molecular reactions.1–5)
However, since functional molecules generally have complex
structures, and local characteristics and reaction processes
determine the observed molecular interactions, it is diﬃcult
to understand their functions from the results of conventional
macroscopic analyses, which provide averaged information.
Dynamic force spectroscopy (DFS) is a technique based on
atomic force microscopy (AFM), which provides informationon the interaction potential landscape between two molecules
at the single-molecule level.3,5–10) For example, sincemolecules such as streptavidin and avidin have multiple
bonding sites, it is diﬃcult to separately probe them in detailusing conventional macroscopic methods which gives
averaged value. In DFS, using the variation in the crosslinkers, selective analysis of the bonding sites has beenrealized, and direct and bridging interactions at each reaction
site in a ligand–receptor system have been distinguished andindividually analyzed.8–11)
In DFS based on the Bell–Evans model,4,12,13) the
unbinding force applied to a pair of molecules is increased
at a constant rate, and the force required to rupture the bond ismeasured. However, although the analysis is based on theassumption that the force is applied to a sample placed
immediately below the probe tip [Fig. 1(a)], the molecular
linker used to ﬁx the target molecule to the cantilever may
change the direction of the force [Fig. 1(b)]. Despite its
importance, its eﬀect on measurement has not been closely
examined until now. To ensure the reliability of DFS
measurement, it must be thoroughly understood and should
be taken into account.
Here, we report the ﬁrst results of simulations carried outto examine the eﬀect of the direction of the force applied to atarget molecule through a molecular linker.
2.

Experimental procedure and theory

Figure 1 shows schematic illustrations of DFS measurement.
As shown in Fig. 1(a), after forming a ligand–receptor
structure, it is ruptured with the force applied by an atomic

Fig. 1. (Color online) Schematic illustrations of DFS measurement for(a) d = 0 and (b) d ≠ 0. (c) Relationship between modal rupture force f + andlogarithm of loading rate r. The position of the barrier xb and the lifetime of
the bond can be obtained from the slope and the intercept of the straight line,respectively. (d) Potential landscape of molecular interaction. xb indicates theposition of the barrier. (e) Typical system with biotin (ligand)–streptavidin(receptor) used in DFS measurement. PEG: poly(ethylene glycol).

force microscope. In the DFS method, as described above,
the unbinding force applied to a molecular bond is increased
at a constant rate (loading rate r), and the force required
to rupture the molecular bond is measured.8,9) The modalrupture force f + is obtained as the most frequent rupture forceof the histogram in the rupture force [right side in Fig. 1(a)].

08NB01-1© 2016 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 55, 08NB01 (2016)

A. Taninaka et al.

Figure 1(c) shows the relationship between f + and thelogarithm of r. From the reciprocal value of the slope ofthe straight line and its intercept, the position of the potential
barrier landscape xb [Fig. 1(d)] and the lifetime of the bond τ
can be obtained.4,7,9,11,12)When the molecular linker used to ﬁx the molecule to theatomic force microscope tip is long, the coupling may belocated at a distance d below the atomic force probe as shownin Fig. 1(b). In this case, when the cantilever is pulled up,
the force f is applied obliquely to the target molecule, which
results in a discrepancy in the rupture force measured byAFM as described above.
Figure 1(e) shows an illustration of a typical system with abiotin ligand and streptavidin receptor, which is widely usedin this ﬁeld4–12,14,15) and was adopted as a sample for theanalysis in this work. Poly(ethylene glycol) (PEG) was used
as a cross-linker molecule to reduce the eﬀect of the probe on
the measurement, i.e., steric barrier to hamper the naturalbonding. Streptavidin was ﬁxed using maleimide on a Au
substrate covered with a ﬁlm of 1-octanethiol and 1,10decanedithiol (100 : 1) formed by self-assembly.
To analyze the behavior of a molecular linker, thewormlike chain (WLC) model has generally been adopted,
which is particularly suitable for describing stiﬀer polymers
with successive segments exhibiting cooperativity.16–19) In
the WLC model, the energy of a molecular chain used as across-linker is described in terms of bending, twisting, andstretching. When the contribution of stretching is neglected
and the molecular chain is suﬃciently stiﬀ to neglect a smallextension, the energy of the molecular chain can be expressedin terms of the bending and extension energies that aregenerated when a force is applied to the end of the molecularchain.16–19)
Assuming the WLC model, the contractile force of PEG fis written askB T
11 L
f¼ þ;
ð1ÞA4ð1  L=L0 Þ2 4 L0
where A is the persistencelength,ﬃ20) L0 is the contour length
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ(molecular length), L ¼ z2 þ d2 is the extension length ofthe molecule, kB is the Boltzmann constant, and T is thetemperature, and z is the retraction distance of the probe-tip
apex shown in Fig. 1. The persistence length is the scale used
to describe the stiﬀness of polymers. At room temperature,kBT is 4.14 × 10−21 J. For A, 0.38 nm was used.20)
Figure 2(a) shows a plot of f against z for the cases ofd = 0, 5, 10, and 15 nm. Since the cantilever measures theforce in the z direction, fz, the relationship between fz and f
in the direction along the molecular linker iszfz ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f:z2 þ d2

ð2ÞFigure 2(b) shows the relationship between f and fz in thecases of d = 0, 5, 10, and 15 nm. When d is nonzero, f has anonzero value even when fz = 0. This is due to that fact thatthe molecular linker is elongated from the start of measurement when d is nonzero. Even the case of d = 5 nm, thediscrepancy between fz, and f was ∼3 pN.
According to the Bell–Evans model,4,12,13) the rupture ratek (rupture probability per unit time) under force f can be
written as

Fig. 2. (Color online) Force along (a) molecular linker f and(b) z-direction fz, which are indicated in Fig. 1(b).

xb f:k ¼ k0 exp
kB T

ð3Þ

Thus, the diﬀerential equation for S, the probability that twomolecules coupled at t = 0 remain coupled at t = t, and its
solution are written in terms of k asdSðtÞ
¼ kðtÞSðtÞ;
dt
 Zt

00
SðtÞ ¼ Sð0Þ exp kðt Þ dt

ð4Þ

0

 Zt

¼ exp kðt0 Þ dt0 ;0ð5Þ

where S(0) = 1 was used. Since the loading rate is constant(r = rc) and feedback is applied to the measured force fz, thenfz can be written as,zfz ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f ¼ rc t:
2z þ d2

ð6Þ

Therefore, the distribution of the rupture force fz can bewritten askSðtÞ ¼ kSð fz Þ:ð7Þ

Then, the modal rupture force f z that gives the maximum
value of kS(t) is expressed as d kB TkB Txb off
fz ¼ln rc þln;
ð8Þ
xb
xb
kB T
dis the lifetime under the condition of fz = 0 ( f ≠ 0where off
if d is nonzero) and is given by
 
1d
off
¼
:
ð9Þ
k fz ¼008NB01-2© 2016 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 55, 08NB01 (2016)

A. Taninaka et al.

Fig. 4. (Color online) f + as a function of rc for L = 30 nm obtained bysimulation.

Table I.

Fig. 3. (Color online) Rupture rate k(t) and bonding probability S(t) as afunction of time, and rupture force distribution and normalized f + as afunction of d for (a) rc = 104 pN=s and (b) rc = 102 pN=s.

Using Eqs. (3), (5), and (7) with the values of xb =0.68 nm, k0 = 1 s−1, T = 300 K, and the spring constant of thecantilever of 6 pN=nm, which were used for the experimentschematically shown in Fig. 1(e), k, S(t), and the distributionof the rupture force were calculated and the results are shownin Fig. 3 for (a) rc = 104 pN=s and (b) rc = 102 pN=s. Forrc = 104 pN=s, the values of f z were estimated to be 45.3,44.1, 40.7, and 34.1 pN for d = 0, 5, 10, and 15 nm, whilefor rc = 102 pN=s, f z was 17.3, 16.5, and 13.8 pN for d =0, 5, and 10 nm, respectively. The distribution function ford = 15 nm was distorted for rc = 102 pN=s, indicating thatrupture occurred from the start of the measurement, asdiscussed above for the relation between f and f z . In the caseof d = 20 nm (not shown), the distribution function wasdistorted even for rc = 103 pN=s. These results suggest thatthe analysis can be carried out provided d ≲ 15 nm. The
normalized f z is summarized as a function of d in the bottomof Fig. 3; f z decreases with increasing d.
3.

Results and discussion

Next, we examined how a molecular linker aﬀects theaccuracy of the obtained xb and τ. Figure 4 shows therelationship between f z and the logarithm of rc for L0 =30 nm, which is given by Eq. (8). First, a linear relationshipremains even when d is nonzero. Secondly, the slopes
observed experimentally were similar for diﬀerent values ofd. From the reciprocal value of the slope of the straight lineand its intercept, as explained using Fig. 1(d), xb and τoﬀ wereobtained, respectively, which are summarized in Table I. xb

Values obtained from Fig. 4.

d(nm)doff(s)xb(nm)

051.00.940.680.69

100.640.71

150.300.77

increases with increasing d, and the increase in xb from its
value when d = 0 was ∼5% for d = 10 nm and ∼10% for
d = 15 nm. These values are suﬃciently small to ensure thevalidity of DFS analysis.
On the other hand, τ decreased with increasing d, whoseeﬀect was larger than that on xb. The reduction in τ from its
value when d = 0 was ∼35% for d = 10 nm and ∼70% for
d = 15 nm. This is due to the change in the intercept caused by
the change in the slope having a greater eﬀect. Values of τ
have been scattered over four orders of magnitude in pastworks.7,11) Here, it has been clearly shown that the molecular
linker does not cause such large scattering, which is therefore
considered to be caused by factors such as the method ofﬁxing the sample molecule to the substrate, the pH of thesolvent,8,10,11) and the modiﬁcation of the energy landscape,which cannot be simply treated by the Bell–Evans model.21,22)
The lifetime, which is determined by the potential height, issensitive to changes in the potential landscape because therupture rate k has the exponential form shown in Eq. (3).
These factors should be investigated by DFS with detailed
theoretical analysis. As has been shown, when d is nonzero
and the atomic force microscope cantilever is retracted, a force
f is applied to the coupled molecule at the angle of θ =arctan(z=d). Therefore, two-dimensional mapping of theforce, which provides the relationship between d and f z ,and thus θ and f z , may give more detailed information on suchinteractions at local sites. This issue is left for future work.
Similar calculations can be carried out for diﬀerent L0
because Eq. (1) has the single parameter of L=L0 for themolecular linker. For example, for L0 = 90 nm, d = 0, 15,30, and 45 nm give the same values of f z as those obtainedfor L0 = 30 nm with d = 0, 5, 10, and 15 nm, respectively.
Therefore, L0 cannot be theoretically optimized. However,08NB01-3© 2016 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 55, 08NB01 (2016)

A. Taninaka et al.

when the molecular linker does not have a suﬃcient length,
since the eﬀects of the cantilever apex and the size of thereceptor cannot be neglected, the use of a relatively longmolecular linker is considered to produce a better measurement condition. On the other hand, for a long L0, a coupling
may also be formed at neighboring sites with diﬀerent valuesof d, depending on the density of molecules on the substratesurface. Therefore, L0 should be chosen in consideration of
the measurement conditions.
4.

Conclusion

We examined the eﬀect of the molecular linker used to ﬁx thetarget molecule to the atomic force microscope cantileverused in DFS, i.e., the force direction during measurement, forthe ﬁrst time, which has not been discussed until now despite
its importance. The eﬀect was found to be small, conﬁrming
the high potential of DFS. Further advances in the DFS
technique are expected to enable more detailed analysis of themolecular interactions at the single-molecule level.
Acknowledgment

H.S. acknowledges the support from the Japan Society for thePromotion of Science through Grant-in-Aid for ScientiﬁcResearch, 15H05734.

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08NB01-4© 2016 The Japan Society of Applied Physics