Dynamical Response of Nanomechanical Resonators to Biomolecular
Interactions
Kilho Eom1,*, Tae Yun Kwon1,2, Dae Sung Yoon3, Hong Lim Lee2, and Tae Song Kim1
1
Nano-Bio Research Center, Korea Institute of Science and Technology (KIST), Seoul
136-791, Republic of Korea
2
School of Advanced Materials Science and Engineering, Yonsei University, Seoul 120749, Republic of Korea
3
Department of Biomedical Engineering, Yonsei University, Kangwon-do 220-710,
Republic of Korea
We studied the dynamical response of a nanomechanical resonator to biomolecular (e.g.
DNA) adsorptions on a resonator’s surface by using a theoretical model that considers
the Hamiltonian H composed of elastic bending energy of a resonator and the potential
energy for biomolecular interactions. It was shown that biomolecular interactions (e.g.
DNA-DNA interaction) govern the dynamic behavior of a nanomechanical resonator,
suggesting that nanomechanical resonators may enable one to quantify biomolecules.
PACS: 81.07.-b, 45.10.-b, 68.43.-h, 82.20.Wt
Nanomechanical resonators have recently allowed one to not only gain insight into
fundamentals of quantum mechanics [1-3] but also detect the molecules even in
extremely low concentrations [4, 5]. For instance, Yang, et al. reported the ultrahigh
*
To whom the correspondence should be addressed. E-mail: eomkh@kist.re.kr
1
sensitive mass sensing of molecules even in a zepto-gram resolution by using a
nanomechanical resonator [6]. Moreover, it was recently reported that resonating
cantilevers have enabled the sensitive in-vitro biomolecular detection [7-10]. The high
sensitivity in detecting molecules is attributed to scaling down that leads to highfrequency dynamical range of a resonator. Accordingly, nanomechanical resonators
have been a strong candidate for ultrahigh sensitive in-vitro biomolecular detection.
The detection principle is the direct transduction of biomolecular adsorption on
a resonator’s surface into the resonant frequency shift. It was well known that the mass
of adsorbed molecules makes contribution to resonant frequency shift [11], as long as
molecular interactions between adsorbed molecules do not play a critical role on the
elastic bending behavior of a resonator. In recent studies [12, 13], it was found that
resonant frequency shift for in-vitro biomolecular detection is ascribed to molecular
interactions, e.g. electrostatic repulsion, hydration, etc., between adsorbed biomolecules.
Specifically, it was reported that the surface stress induced by biomolecular interactions
dominates the resonant frequency shift for in-vitro biomolecular detection [14, 15].
However, a continnum model with a constant surface stress in recent studies [14, 15]
may be debatable, since it was provided that, in classical elasticity, the constant surface
stress may not induce any resonant frequency shift [16, 17]. Moreover, it is hard to
quantitatively relate the surface stress to biomolecular interactions. Thus, it is demanded
to develop the model based on molecular model of biomolecular interactions for gaining
insight into quantitative descriptions on relationship between biomolecular interactions
and resonant frequency shift.
In this Letter, we developed a model, which allows one to quantitatively
describe the relationship between biomolecular interactions and resonant frequency shift,
2
on the basis of the molecular model for biomolecular interactions. Specifically, a model
considers the Hamiltonian H for the adsorption of double-stranded DNA (dsDNA) on
the surface of nanomechanical resonator such that Hamiltonian H includes the elastic
bending energy of a resonator and the potential energy for intermolecular interactions
between dsDNAs. It was shown that ionic strength, which is responsible for
intermolecular interactions for dsDNAs, plays a vital role on the resonant frequency
shift. Furthermore, the resonant frequency shift is also related to the mechanical
characteristics of a resonator (e.g. geometry, mechanical properties). The results allow
one to gain insight into not only the relationship between molecular interactions and
resonant frequency shift but also how to design the nanomechanical resonator for highly
sensitive in-vitro biomolecular detection.
Here, we consider the dynamic behavior of a nanomechanical resonator in
response to biomolecular adsorption on its surface. Let us denote the packing density, θ,
of adsorbed biomolecules on a surface as θ = N/L, where N is the number of adsorbed
biomolecules on a surface and L is a resonator’s length. As shown in Fig. 1, once
biomolecules are adsorbed on the surface, the intermolecular interaction (e.g. DNADNA interaction) induces the additional bending of a resonator. In Fig. 1, the
interspacing distance, d, between biomolecules (e.g. DNA) is given by
d ( s ) = d 0 ⎡⎣1 + κ c (1 + s / c ) ⎤⎦
(1)
where d0 = 1/θ, κ is a curvature defined as κ = ∂2w(x, t)/∂x2 with given deflection w(x, t),
2c is a thickness of a resonator, and s is a distance from a resonator’s surface. Since the
length scale of biomolecule (i.e. chain length), Lc, is smaller than a resonator’s thickness
2c, i.e. Lc/c < 1, an interspacing distance between biomolecules can be approximated as
d = d0(1 + κc).
3
With the prescribed potential energy, U(d), for intermolecular interactions
between adsorbed biomolecules, the effective potential energy, V, for a nanomechanical
resonator upon biomolecular adsorption on its surface consists of elastic bending energy,
Eb, of a resonator and potential energy, Eint, for intermolecular interactions between
adsorbed biomolecules.
L
V = Eb + Eint =
L
1
ξκ 2 dx + ∫ θUdx
∫
20
0
(2)
where ξ is a bending modulus for a resonator. By using Taylor series expansion of U
with respect to curvature κ at κ = 0, the total potential energy V is in the form of
L
V = ∫ ⎡⎣ v0 + ϕκ + (1/ 2 )( ξ + ψ ) κ 2 + O (κ 3 ) ⎤⎦ dx
(3)
0
Here, the coefficients v0, φ, and ψ are defined as follows: v0 = θU|κ=0, φ = ∂(θU)/∂κ|κ=0,
and ψ = ∂2(θU)/∂κ2|κ=0. The kinetic energy, T, of a nanomechanical resonator is given by
L
T=
1
2
( μ + θ m )( ∂w / ∂t ) dx
∫
20
(4)
where μ is a resonator’s mass per unit length, and m is the mass of a biomolecule (e.g.
DNA chain). The oscillating deflection motion of a resonator can be represented in the
form of w(x, t) = u(x)exp[iωt], where u(x) is a deflection eigenmode and ω is a resonant
frequency. The mean value of Hamiltonian, <H>, per oscillation cycle is
H = T + V
=−
ω2
2
L
∫ ( μ + θ m) u
0
L
2
2
dx + ∫ ⎡v0 + ϕ u′′ + (1/ 2 )( ξ + ψ )( u ′′ ) ⎤ dx
⎣
⎦
(5)
0
where angle bracket < > indicates the mean value per oscillation cycle, and prime
represents the differentiation with respect to coordinate x. The variational method with a
4
Hamiltonian <H> provides the weak form of equation of motion [18].
L
δ H = ∫ ⎡⎣ −ω 2 ( μ + θ m ) u + ( ξ +ψ ) ( d 4u / dx 4 ) ⎤⎦ δ u
0
(6)
+ ⎡⎣ϕ + ( ξ + ψ ) u ′′ ⎤⎦ δ u ′ − ( ξ + ψ ) u ′′′δ u 0 = 0
L
L
0
Here, a symbol δ indicates the variation, and one may regard δu as a virtual deflection
eigenmode that satisfies the essential boundary condition. In Eq. 6, the first term
represents the equation of motion for a resonator with biomolecular adsorptions on its
surface, whereas the other terms provide the boundary conditions. Thus, from Eq. 6, the
equation of motion for an oscillating resonator upon biomolecular adsorption on its
surface is given by (ξ + ψ)(d4u/dx4) – ω2(μ + θm)u = 0. Consequently, the resonant
frequency, ω, of a nanomechanical resonator upon biomolecular adsorptions on its
surface is
1 + (ψ / ξ )
ω
=
1 + (θ m / μ )
ω0
(7)
where ω0 is a reference resonance, which is a resonance without any biomolecular
adsorption, given by ω0 = (λ/L)2(ξ/μ)1/2. As shown in Eq. 7, the resonant frequency shift
due to biomolecular adsorption is attributed to not only the mass of adsorbed
biomolecules but also the bending stiffness change induced by the intermolecular
interactions between adsorbed biomolecules. Specifically, the bending stiffness change
induced by biomolecular interactions is dictated by the harmonic (second-order) term ψ
in the potential energy for intermolecular interactions.
In this work, we consider the case where dsDNA molecules are adsorbed on the
surface of resonator. The intermolecular interactions, U(d), between dsDNAs on the
surface was provided by Strey, et al. [19, 20].
5
U (d )
exp ( − d / λH )
exp ( − d / λD )
=α
+β
+ Econf ( d )
Lc
d / λH
d / λD
(8)
Here, intermolecular interaction U consists of hydration repulsion with amplitude α and
screening length scale λH, electrostatic repulsion with amplitude β and Debye length λD,
and configurational entropic effect Econf(d) that enhances the hydration and electrostatic
repulsions. It should be noted that hydration and electrostatic repulsions are governed
by the ionic strength, [I], of a solvent in such a way that the screening lengths and
repulsion amplitudes depend on the ionic strength, e.g. λD ≈ 3.08 / [ I ] Å and λH ≈ 2.88
Å for monovalent salt [20]. From Eqs. 1, 3 and 8, the induced bending rigidity, i.e.
bending stiffness change, ψ for a resonator due to DNA-DNA interactions is computed
as ψ = Lcc2f(θ, [I]). This indicates that the induced bending rigidity ψ depends on
geometry parameters for both dsDNA and resonator, i.e. dsDNA chain length Lc,
resonator’s thickness 2c. Moreover, in Fig. 2, it is shown that induced bending rigidity ψ
due to DNA-DNA interactions is dependent on ionic strength of monovalent salt of a
solvent, [I], which governs the hydration and electrostatic repulsions, as well as dsDNA
packing density. A high packing density and 1M NaCl concentration of a solvent
induces the larger repulsive forces between dsDNAs than a low packing density and 0.1
M NaCl concentration of a solvent, resulting in a larger elastic bending motion of a
resonator for a high packing density and 1M NaCl concentrations of a solvent. This is
consistent with a recent study [21], which reported that the nanomechanical bending
motion of a cantilever is originated from intermolecular interactions between adsorbed
biomolecules.
As stated earlier in Eq. 7, with an assumption of θm << μ, a resonant frequency
shift, Δω, induced by DNA-DNA interactions on the resonator’s surface is given by
6
Δω/ω0 = (ω – ω0)/ω0 ≈ (1 + ψ/ξ)1/2 – 1. It suggests that the resonant frequency shift
depends on not only the amount of intermolecular interactions between dsDNAs but
also the mechanical characteristics for a nanomechanical resonator, i.e. resonator’s
geometry and mechanical property. Let us take into account the nanomechanical
cantilevers with a dimension of b × 2c × L (width × thickness × length), in order to
understand the resonant frequency shift induced by dsDNA adsorption on a cantilever
surface with respect to geometry of a cantilever, i.e. cantilever’s thickness. In Fig. 3, it
is shown that a normalized resonant frequency shift increases as a cantilever’s thickness
decreases. As a cantilever’s thickness decreases in such a way that a thickness is
comparable to a size of biomolecules, e.g. DNA chain length, a normalized resonant
frequency shift, Δω/ω0, induced by DNA-DNA interactions becomes enormously larger.
This is consistent with a recent work [22], which reported the role of a cantilever’s
thickness on a resonant frequency shift driven by biomolecular adsorptions. The
normalized resonant frequency shift, Δω/ω0, can be analytically obtained as a function
of a cantilever’s thickness. Since a bending modulus of a cantilever for a rectangular
cross-section is ξ = (2/3)Ebc3, where E is the Young’s modulus of a cantilever, and the
induced bending rigidity due to DNA-DNA interaction is given by ψ = Lcc2f(θ, [I]), the
normalized resonant frequency shift is obtained as Δω/ω0 = (1 + 3Lcf(θ, [I])/2Ebc)1/2 – 1.
By asymptotic analysis, we have Δω/ω0 = (3/4)(Lcf(θ, [I])/Ebc) + O(1/c2), showing that
a normalized resonant frequency shift induced by DNA-DNA interaction is inversely
proportional to a cantilever’s thickness. In a similar manner, it can be easily found that a
normalized resonant frequency shift is also inversely proportional to the Young’s
modulus for a cantilever.
In our model, it should be reminded in that the mass of adsorbed biomolecules
7
is neglected for a resonant frequency shift. If the total mass of adsorbed biomolecules
(e.g. macromolecules) is comparable to the mass of a nanomechanical resonator, then a
resonant frequency shift induced by biomolecular adsorptions may be determined by
both biomolecular mass and biomolecular interactions. Moreover, if the dimension of a
resonator is a nano-scale (c, b, L < 1 μm), the dynamic behavior of a resonator may be
also affected by the thermal fluctuations renowned as Brownian motion. However, our
model is theoretically valid, because our model employs the small biomolecules (i.e.
short DNA chain) such that adsorbed biomolecular mass is much smaller than a
resonator’s mass, i.e. θm/μ << 1. Further, our model adopts the resonator with a micron
length scale (e.g. L = 2 μm), so that continuum model for a resonator is sufficient to
represent the dynamic behavior of resonators, that is, the Brownian motion is
insignificant on the dynamic motion of micron-length-scale resonators.
In conclusion, our model showed the significant role of biomolecular
interactions on the dynamical response of a nanomechanical resonator to biomolecular
adsorption on its surface. Specifically, intermolecular interactions for adsorbed
biomolecules generate the change of bending rigidity of a resonator, resulting in a
resonant frequency shift. It was shown that a resonant frequency shift induced by DNA
adsorption depends on both ionic strength of a solvent and DNA packing density, which
govern the intermolecular interactions for DNAs, and that a resonant frequency shift is
also dependent on the geometry (i.e. thickness) of a resonator. This implies that smallscale resonators, whose thickness is comparable to the size of biomolecules, are suitable
for sensitive detection of biomolecular interactions. It is proposed that, based on our
model, the nanomechanical resonators may enable one to quantify the adsorbed
biomolecules on the surface.
8
This work was supported by Intelligent Microsystem Center sponsored by the
Korea Ministry of Science and Technology as a part of the 21st Century’s Frontier R&D
projects (Grant No. MS-01-133-01) and the National Core Research Center for
Nanomedical Technology sponsored by KOSEF (Grant No. R15-2004-024-00000-0).
9
References
[1]
V. B. Braginsky and F. Ya Khalili, Quantum Measurements (Cambridge
University Press, 1992).
[2]
K. C. Schwab and M. L. Roukes, Phys. Today 58, 36 (2005).
[3]
D. Kleckner and D. Bouwmeester, Nature 444, 75 (2006).
[4]
B. Ilic, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, J. Appl.
Phys. 95, 3694 (2004).
[5]
M. Li, H. X. Tang, and M. L. Roukes, Nat. Nano. 2, 114 (2007).
[6]
Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, Nano Lett.
6, 583 (2006).
[7]
A. Gupta, D. Akin, and R. Bashir, Appl. Phys. Lett. 84, 1976 (2004).
[8]
J. H. Lee, K. H. Yoon, K. S. Hwang, J. Park, S. Ahn, and T. S. Kim, Biosens.
Bioelectron. 20, 269 (2004).
[9]
J. H. Lee, K. S. Hwang, J. Park, K. H. Yoon, D. S. Yoon, and T. S. Kim, Biosens.
Bioelectron. 20, 2157 (2005).
[10]
B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead,
Nano Lett. 5, 925 (2005).
[11]
T. Braun, V. Barwich, M. K. Ghatkesar, A. H. Bredekamp, C. Gerber, M. Hegner,
and H. P. Lang, Phys. Rev. E. 72, 031907 (2005).
[12]
J. H. Lee, T. S. Kim, and K. H. Yoon, Appl. Phys. Lett. 84, 3187 (2004).
[13]
S. Cherian and T. Thundat, Appl. Phys. Lett. 80, 2219 (2002).
[14]
J. Dorignac, A. Kalinowski, S. Erramilli, and P. Mohanty, Phys. Rev. Lett. 96,
186105 (2006).
[15]
K. S. Hwang, K. Eom, J. H. Lee, D. W. Chun, B. H. Cha, D. S. Yoon, T. S. Kim,
10
and J. H. Park, Appl. Phys. Lett. 89, 173905 (2006).
[16]
M. E. Gurtin, X. Markenscoff, and R. N. Thurston, Appl. Phys. Lett. 29, 529
(1976).
[17]
P. Lu, H. P. Lee, C. Lu, and S. J. O'Shea, Phys. Rev. B. 72, 085405 (2005).
[18]
T. Mura and T. Koya, Variational Methods in Mechanics (Oxford University
Press, 1992).
[19]
H. H. Strey, V. A. Parsegian, and R. Podgornik, Phys. Rev. Lett. 78, 895 (1997).
[20]
H. H. Strey, V. A. Parsegian, and R. Podgornik, Phys. Rev. E. 59, 999 (1999).
[21]
G. H. Wu, H. F. Ji, K. Hansen, T. Thundat, R. Datar, R. Cote, M. F. Hagan, A. K.
Chakraborty, and A. Majumdar, Proc. Natl. Acad. Sci. USA. 98, 1560 (2001).
[22]
A. K. Gupta, P. R. Nair, D. Akin, M. R. Ladisch, S. Broyles, M. A. Alam, and R.
Bashir, Proc. Natl. Acad. Sci. USA. 103, 13362 (2006).
11
Figure Captions
Fig. 1. Schematic for a bending of a resonator (e.g. cantilever) induced by
intermolecular interactions between adsorbed biomolecules (e.g. DNA)
Fig. 2. Induced bending of a resonator, ψ/Lcc2 ≡ f(θ, [I]), was computed as a function of
packing density, θ, and ionic strength, [I], of a solvent. The high packing density
(e.g. θ ≈ 1010) and high ionic strength (e.g. [I] = 1 M NaCl) induce the larger
repulsive intermolecular forces, leading to the larger elastic bending motion of a
cantilever, than low packing density (e.g. θ < 109) and/or low ionic strength (e.g.
[I] = 0.1 M NaCl).
Fig. 3. Normalized resonant frequency shifts induced by DNA-DNA interactions were
numerically computed as a function of a resonator’s thickness. Normalized
resonant frequency shift increases as a thickness decreases in such a way that a
resonant frequency shift is significantly amplified when a resonator’s thickness
becomes comparable to the size of biomolecules (i.e. t ≡ 2c ≈ 100 nm). Here, we
used [I] = 1M NaCl for ionic strength of a solvent, Lc = 100 nm for a DNA chain
length, and the characteristic parameters for a cantilever as follows: b = 10 nm,
L = 2 μm, E = 190 GPa.
12
Figures
Fig. 1.
Eom, K., et al.
13
Fig. 2.
Eom, K., et al.
14
Fig. 3.
Eom, K., et al.
15