Flexoelectric origin of nanomechanic deflection in DNA-microcantilever system

Fei Liu a,*, Yong Zhang a, Zhong-can Ou-Yang a,b

a Institute of Theoretical Physics, The Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, People’s Republic of Chinab Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China

Received 11 May 2002; received in revised form 27 November 2002; accepted 9 January 2003

Abstract

The membrane theory is used to study the recently observed nanomechanical bending of cantilevers, which have processed

biomolecular adsorption or biochemical reactions. To be different from entropy-controlling bending mechanism discussed before,

we propose that the flexoelectric effect induces cantilever bending. With the introduction of flexoelectric spontaneous curvature, the

relation between the bending and biopolymer character is constructed by a simple analytical formula. The cantilever motion induced

by adsorption of single-strand DNA and DNA hybridization reaction is quantified analytically and our results show good

agreement with experiments.

# 2003 Elsevier Science B.V. All rights reserved.

Keywords: Cantilever-based biosensor; Cantilever-DNA system; Nanomechanical deflection; Theoretical model; Flexoelectric origin

1. Introduction

Recently, the development of microcantilever-based

biosensors for detection of biochemical species adsorp-

tion and specific biochemical reactions has attracted

increasing attention (Baller et al., 2000; Raiteri et al.,

1999; Fritz et al., 2000; Moulin et al., 1999, 2000; Wu et

al., 2001a,b; Hansen et al., 2001). The inexpensive

sensors not only show fast response, high sensitivity,

and suitable for mass production (Baller et al., 2000),

but also provide common platforms for label-free

analysis of different biochemical reactions such as

protein�/protein binding, DNA�/protein interactions,

and drug discovery (Wu et al., 2001b).

Compared to the results in research of cantilever-

based biosensors in protein case, more detailed quanti-

tative results have been achieved in research of DNA

adsorption and hybridization by Wu et al. (2001a,b).

They found that end-thiolated single-strand DNA

(probe ssDNA) molecules adsorbed on one surface

always create a compressive stress, that is the cantilever

bends away from adsorbed surface. The steady-state

deflection increases as the number of nucleotides of

probe ssDNA increases. While injecting complementary

ssDNA (target ssDNA) molecules, hybridization reac-

tion with this end-grafted probe ssDNA relieves the

compressive stress created during immobilization of

probe ssDNA. The compressive stress decreases as

length of target ssDNA increases (Figs. 2 and 3 in Wu

et al., 2001a). All steady-state deflections are sensitive to

different salt concentration. Remarkably, further ex-

perimental data demonstrate that deflection can discern

the number and location of base pair mismatches

happened in hybridization (Figs. 2 and 3 in Hansen et

al., 2001). In protein case, more complex phenomena

have been observed (Baller et al., 2000; Moulin et al.,

1999, 2000; Wu et al., 2001a), e.g. the adsorption of

immunoglobulin G (IgG) and albumin (BSA) induce

compressive stress and tensile stress, respectively (Mou-

lin et al., 1999, 2000). These quantitative differences can

be used in detecting single-nucleotide polymorphisms

(SNPs) and disease-related proteins (Hansen et al., 2001;

Wu et al., 2001a).

The cantilever-based biosensors show promising pro-

spects in application, but it is difficult to give a physical

* Corresponding author. Tel.: �/86-10-6254-1808; fax: �/86-10-

6256-2587.

E-mail address: liufei@itp.ac.cn (F. Liu).

Biosensors and Bioelectronics 18 (2003) 655�/660

www.elsevier.com/locate/bios

0956-5663/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0956-5663(03)00047-2

interpretation of how biomolecules adsorption and

biochemical reactions change surface stress. So far,

only the Stoney’s equation (Stoney, 1908),

Dz�4Df (1 � n)l2

0

Et2;

where Df is the difference between two surface stress, Dz

is change in deflection, E , n , t , and l0 are Young’s

modulus, Poisson’s ratio, thickness, and length of

cantilever material, respectively, which has been applied

in experiments (Raiteri et al., 1999; Moulin et al., 1999,

2000; Wu et al., 2001a,b; Hansen et al., 2001; Berge etal., 1996), is just to describe the relation between

cantilever free-end deflection and changes of surface

stress. Obviously, it cannot explain why different

biomolecular structures or specific biochemical reaction

create different deflections. Wu et al. (2001a) and

Chakraborty and Golumbfskie (2001) have argued

that system’s free energy, which is composed of canti-

lever strain energy, biomolecules conformational en-tropy, and intermolecular interaction, determines final

cantilever deflection. Especially, they emphasized that

molecular conformational entropy directly determines

the deflection directions, e.g. the relieving of the

compressive stress during hybridization was contributed

to the conformation entropy losing during hybridiza-

tion. Their experiment seems to demonstrate their

argument. Moulin et al. (1999, 2000) have given similarresults about protein adsorption. Because the adsorbates

are molecularly complex and charged, e.g. partially

hybridized short ssDNA and protein�/ligand complex,

a quantitative analysis is very difficult.

Although the viewpoint of Wu et al. may be true, in

this paper we try to give another mechanism to explain

this nanomechanical bending generated by biomolecular

adsorption and biochemical reaction. Considering thatcharged biomolecules distribute uniformly on cantilever

surface, and cantilever minute deflection is sensitive to

pH or salt concentration (Wu et al., 2001a; Moulin et

al., 1999), we propose that the deflection is induced by

curvature electricity effect (Meyer, 1969). To be different

from free-energy viewpoint, our model suggests that

change of the deflection is due to change of the electric

potential difference across molecular layer, that ispotential instead of forces such as entropy or repulsive

forces (Wu et al., 2001a; Chakraborty and Golumbfskie,

2001) controlling the cantilever bending. In the follow-

ing sections, we first investigate the surface-adsorbed

cantilever as an asymmetric membrane. With 1D open

asymmetric membrane shape equation, we calculate the

deflection as function of spontaneous curvature c0

(Helfrich, 1973). Then, based upon the modified poly-electrolytes theory and the relation between c0 and

electric potential Dc across molecule layer, we study

the DNA-cantilever system (Wu et al., 2001a,b) and find

an apparent semi-microscopic relation for cantilever

deflection, probe ssDNA length, and salt concentration.

Finally, the model is extended to analyze hybridization

reaction using smearing trick. These results show goodagreement in quantitative with experimental observa-

tions mentioned above. Though we limit ourselves to the

DNA-cantilever system (Wu et al., 2001a,b), we believe

that our model partially satisfies protein-cantilever case

(Moulin et al., 1999, 2000).

2. Theoretical model

As shown in Fig. 1, we can see the discussed cantilever

as an asymmetric membrane with 1D deformation, i.e.

no bending happens along x -direction. According tocurvature free energy of asymmetric membrane (Hel-

frich, 1973), we derive corresponding 1D shape equation

by variational calculus (Elsgolc, 1961):

lc1(y)�12k(c2

0�c21(y))c1(y)�k92c1(y)�0; (1)

where 1D Laplace�/Beltrami operator 92 and principle

curvature c1(y) are written as

92�1

(1 � z?2)1=2

d

dy

�1

(1 � z?2)1=2

d

dy

�;

c1(y)�zƒ

(1 � z?2)3=2:

(2)

z (y) is the deflection of the cantilever at y , z ?�/dz /dy ,

and k and l are bending rigidity and effective tensile

stress, respectively. The spontaneous curvature c0 takes

care of the asymmetry on the two sides of cantilever.

Eq. (3) is a higher order nonlinear differential

equation (Zhang and Ou-Yang, 1996). However, be-

Fig. 1. Schematic diagram illustrating the cantilever downward

bending induced by ssDNA immobilization.

F. Liu et al. / Biosensors and Bioelectronics 18 (2003) 655�/660656

cause one end of the cantilever is free, and the other is

fixed, the equation has a simple solution, and deflection

of the free end of the cantilever is given by

Dz� 12c0l2

0 : (3)

Generally, the spontaneous curvature c0 is hard to

calculate from microscopic viewpoint (Ou-Yang et al.,1999). While for charged membrane, using the concept

of curvature electricity in the field of liquid crystal

(Meyer, 1969), a simple phenomenological relation

between c0 and electric potential difference across the

membrane Dc has been obtained as (Ou-Yang et al.,

1991, 1992)

c0�e11Dc

k; (4)

where e11 is flexoelectric coefficient. In red blood cells,

Dc can be measured by experiment, however, in present

case of biopolymer-cantilever structure, we need to

estimate Dc afresh by analysis. In what follows, weonly discuss the DNA-cantilever system.

2.1. ssDNA immobilization

We first consider ssDNA immobilization case (Wu et

al., 2001a). The electric potential of polyelectrolyte

brush with long chains has been extensively studied

using analytical (Zhulina et al., 1992) and numerical

self-consistence field methods (Miklavic and Marcelja,

1988; Misra et al., 1989; Fleer et al., 1993). But simpler

solution is possible if brush structure is known pre-viously. Our model for calculating electric potential of

ssDNA brush is schematically shown in Fig. 2. Accord-

ing to the polymer theory (Alexander, 1977; de Gennes,

1979), when the surface coverage s is larger than an

overlap critical surface coverage sc, the average mono-

mer concentration f(z ) can be seen as uniform through

brush and we have

fh�Ns; (5)

where h is the equilibrium height of the ssDNA brush,

and N is the fragment length of the ssDNA. In terms of

linearized Poisson�/Boltzmann (PB) equation, the elec-

tric potential c(z ) is described by the following equa-

tions:

d2cI

dz2�

�4pe2b

P2

i�1 niz2i

o

�cI�

�4pzpe

o

�f; 05z5h;

d2cII

dz2�

�4pe2b

P2

i�1 niz2i

o

�cII; z�h;

(6)

where zi and ni are the valence and concentration of the

i th ion of equilibrium bulk electrolyte, respectively, zp

the valence of ssDNA segments (each nucleotide carries

a net negative charge, i.e. zp:/�/1), e the electron

charge, o the dielectric constant taken to be the same

inside and outside the ssDNA region (Miklavic and

Marcelja, 1988), and b denotes 1/kBT , where kB and T

are Boltzmann constant and temperature, respectively.Considering that the cantilever is coated by gold film at

z�/0 (Wu et al., 2001a; Berge et al., 1996), and electric

field must be continuous at z�/h , we have

cI(0)�0;c?II(h)��kcI(h);

�(7)

where k�(4pe2ba2i�1niz

2i =o)

1=2 is inverse Debye screen-

ing length and c?II�dcII=dz: The solution outside the

brush region decays by an exponential term cII(z )8/

exp(�/kz). Solving Eq. (6) subjected to boundary

conditions (7) gives the potential difference across the

membrane. Substituting the potential difference into Eq.

(4), Eq. (3) yields

Dz�2pl20e11

k

zpef

ok2exp(�kh)(1�cosh(kh)): (8)

How about height h or concentration f? Because

ssDNA persistence length is about 2 nt (Baumann et

al., 1997) at the ionic strengths 0.05�/1.0 M (Wu et al.,

2001a; Hansen et al., 2001), we use the scaling method

(Dan and Tirrell, 1993) to predict ssDNA brush

structure. In our model, the solvent is supposed to be

poor rather than be good (Dan and Tirrell, 1993) by

taking into account the realistic images of ssDNA brushon surface (see Figs. 2�/4 in Rekesh et al., 1996). First,

the ratio of persistence of ssDNA with nucleotide length

a (about 0.38 nm) is inversely proportional to the square

of the Debye length k , which is scaled as �/1/g8 , where

8 is salt concentration, and the parameter g proposed

here accounts for asymmetric or multivalent electrolyte.

For the solution of sodium phosphate buffer (Na3PO4)

(Wu et al., 2001a) g is about 6. Then, following similarcalculation method developed by Dan and Tirrell, we

have h �/aNd�/a3Ns /(g8 )2, sc�/N�2/3(g8 )4/3/a2,

where d�/sa2/(g8 )2 is effective height coefficient. Com-

Fig. 2. Schematic diagram illustrating ionic distribution and polymer

brush structure, where local part of bent cantilever is approximately

viewed as planar one; circles with plus and minus signs stand for Na�

and PO43�, respectively.

F. Liu et al. / Biosensors and Bioelectronics 18 (2003) 655�/660 657

bining Eq. (8) with these relations, an apparent analy-

tical formula connecting deflection with polymer struc-

ture and salt concentration is obtained as

Dz(N)�2pl20e11

k

zpe

ok2

s

adexp(�kaNd)

� (1�cosh(kaNd)): (9)

Expanding Eq. (9) up to lowest order in N , we find that

Dz (N )8/N2, and Dz tends to be constant when N

increases. Only one parameter, the flexoelectric coeffi-

cient e11 for ssDNA-cantilever system, is still lacking.

We simply make use of e11 in LC and set e11:/1.0�/

10�5 dyn1/2 (Helfrich, 1971). Based on our calculation,

we find s�/sc when N � /[20,50], where s is about 6�/

1012 chains/cm2 and 8�/0.1 M (Wu et al., 2001a), so Eq.(9) is reasonable assumption. A comparison between

experiment and present prediction Eq. (9) is shown in

Fig. 3; obviously, the agreement is good.

2.2. ssDNA hybridization

Now we would like to see whether our simple model

could be extended to explore the deflection changes

induced by DNA hybridization (Wu et al., 2001a;

Hansen et al., 2001). Nonuniform hybridization (Wuet al., 2001a) makes us to give up traditional diblock

copolymer theories (Hariharan et al., 1998; Biver et al.,

1997). But we try to smear inhomogeneous distribution

in the hybridized ssDNA layer by slightly revising Eq.

(9) as

fh� (N�jn)s; (10)

where j is the percentage of the probe ssDNA that is

hybridized with the n length complementary target

strands (60�/80% in Wu et al., 2001a). To be same

with ssDNA immobilization case, we need the scaling

relation between persistence of partial hybridized

ssDNA and nucleotide length. We suppose that theratio is scaled as �/v (j ,n)/g8 , where the undefined

function v (j ,n ) accounts for the effect of hybridization.

In present paper, we only discuss the hybridization

reaction happened at the same salt concentration.

Considering that the hybridization model should be

identical to Eq. (9) when either the hybridization

percentage j vanishes or the length of target ssDNA

tends to zero, we let v (j ,n)�/1, as j or n vanishes.Then, following the same process, we have h �/aN-

d (j ,n ), where function d (j ,n )�/v2(j ,n)sa2/(g8 )2. And

Eq. (9) becomes

Dz(N;j; n)�2pl20e11

k

zpe

ok2

s

ad(j; n)

� (1�jn=N) exp(�kaNd(j; n))

� (1�cosh(kaNd(j; n))): (11)

Here, we suppose that the flexoelectric coefficient e11 is

same with immobilization model. Considering that

parameters j and target ssDNA length n are indepen-

dent, and the persistence of hybridized ssDNA increases

with length n (Wu et al., 2001a), we expand v2(j ,n ) as

v2(j; n):1�r1(j)n�r2(j)n2; (12)

where ri (0)�/0 (i�/1,2). The first nonlinear term left is

to consider great difference of the persistence lengthbetween ssDNA and dsDNA. Now the hybridization

model has two undefined functions ri(j). In the present

model, we get their values by fitting experiment data.

Fig. 4 shows the numerical result by fitting Eq. (11)

with experiment (Wu et al., 2001a), and reveals an

unexpected downward motion of cantilever induced by

hybridization when the length of target ssDNA is

shorter than critical value of 5. This effect has notbeen observed (Wu et al., 2001a).

Why does hybridization relieve the compressive stress

created during immobilization? From free energy argu-

ments discussed by Wu et al. (2001a) and Chakraborty

and Golumbfskie (2001), the persistence length of

hybridized ssDNA is very longer than that of probe

ssDNA (about 70 times), which leads to that the

conformation entropy gain by forming a curved inter-face is insignificant compared with that of probe

ssDNA; therefore, curvature of cantilever is decreased

correspondingly. In our model, however, the upward

Fig. 3. The steady-state deflection jDz j as a function of length of the

probe ssDNA. Experimental data is from Fig. 2a (inset) of Wu et al.,

2001b. Theoretical curve is calculated from Eq. (9) with the bending

rigidity k�/Et3/12(1�/n ), where E�/180 GN/m2, t�/0.5 mm, l0�/200

mm, n�/0.25, the temperature T�/298 K (Wu et al., 2001b), and the

dielectric constant o:/80. The lower curve is for e11�/1.0�/10�5

dyn1/2. The best fitting value for e11 is about 1.33�/10�5 dyn1/2; see

upper curve.

F. Liu et al. / Biosensors and Bioelectronics 18 (2003) 655�/660658

bending of the cantilever is caused by the decrease of the

potential difference. According to Eqs. (5) and (8),

hybridization of probe ssDNA indeed increases total

charges, which may cause more downward deflection,

but increase in height h of the molecular layer induced

by hybridization can decrease the average monomer

concentration f , which may cause upward deflection.The observed upward bending in experiments and the

downward bending predicted by our model just reflect

competition between total charges and average mono-

mer concentration completely. In the experimental side,

DNA hybridization is generally considered thermody-

namically unstable when complementary length is less

than 6 base pairs, however, Hansen et al. (2001) have

demonstrated the possibility for hybridization withshorter complementary length. We think that shorter

hybridization experiment may give a judgment for two

viewpoints.

3. Conclusions

In this paper we give a new physical interpretation for

the elegant cantilever-based biosensors. We propose that

nanomechanical bending of the cantilever is caused byflexoelectric effect, instead of conformational entropy

force suggested before. The cantilever-biomolecule sys-

tem is investigated as an asymmetric membrane induced

by curvature electricity effect. With 1D open membrane

shape equation, we calculate the deflection as function

of spontaneous curvature c0. Based upon polyelectro-

lytes theory and the relation between c0 and electricpotential, we find an apparent semi-microscopic relation

between cantilever deflection, ssDNA length, and salt

concentration. The nanometer deflection changes in

hybridization experiment are also analyzed using smear-

ing trick. These results show good agreement in

quantitative with experimental observations. Of course,

our model has defect, especially as we could not define

the correct expression for ri(j). We have to leave it tofuture work. In addition to finding correct expressions,

two directions have to be considered in future. One is

the importance of salt concentration. According to our

scaling argument, the persistence length of polymers

plays a major role in determining electric potential

across molecular layer, while ssDNA is a special

biological material whose persistence length is almost

invariant under the ion concentration 0.05�/1 M (Wu etal., 2001a; Baumann et al., 1997). It implicates that the

brush height or monomer density is changed little in this

ionic region. Deflection change induced by salt concen-

tration therefore is mainly dependent on Debye screen-

ing length k�1 according to Eq. (8). To test this

theoretical prediction, more delicate theoretical models

and experiments are needed. Another is to predict

different deflections created by different hybridizationplaces such as proximal terminal, distal terminal or

inner (Hansen et al., 2001). Because of using smearing

trick to calculate the hybridization height, the model

presented here cannot predict any deflection changes

observed by Hansen et al. (2001). Hence, more delicate

theoretical model is needed.

Acknowledgements

We are grateful to Prof. Majumdar, who generouslyprovided us relevant papers (Wu et al., 2001a,b; Hansen

et al., 2001). We would also like to thank Prof. H.-W.

Peng, Dr. J.-J. Zhou, and Z.-C. Tu for many helpful

discussions on the results.

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