Modeling a nanocantilever based biosensor using a stochastically perturbed harmonic oscillator

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by  Patrick Snyder

Modeling a nanocantilever based biosensor using a stochastically perturbed harmonic
oscillator
Patrick Snyder and Amitabh Joshi∗
Department of Physics, Eastern Illinois University, Charleston, Illinois 61920
Juan D. Serna†
School of Mathematical and Natural Sciences, University of Arkansas at Monticello, Monticello, Arkansas 71656
(Dated: January 22, 2013)
A nanoscale biosensor most suitable for clinical purposes could be based on the use of nanocan-
3
tilevers. These microscopic ‘diving boards’ are coated with binding probes that have an affinity to
1
a specific amino acid, enzyme or protein in the body. The binding probes on the silicon cantilever
0
attract target particles, and as target biomolecules bind to the cantilever, it changes the vibrating
2
frequency of the cantilever. The process of target binding is a random process and hence creates

n fluctuations in the frequency and damping mechanism of the cantilever. The effect of such fluc-
a tuations are given in our model calculations to provide a broader quantitative understanding of
J nanocantilever biosensors.

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1 PACSnumbers: 05.40.-a,05.40.Ca,87.85.gp
Keywords: biosensors,stochasticharmonicoscillator,nanocantilever

]
h
p I. INTRODUCTION ofsuchnanodevicesisalsolargeandthatfurtheraddsto
- their sensitivity of detection.2,3
o
i Biosensorsare composite devices consisting of biologi- Recently,aslightlydifferentkindofnanocantileverhas
b
cal sensing elements and transducer systems. The work- emerged. These nanocantilever biosenesors contain mi-
.
s ing principle of these devices involves the binding of the crofluidic channels that are used to detect small mass
c
i desired analyte to the biorecognition element fixed on a species, e.g., cells, proteins etc. These microfluidic chan-
s
suitable support matrix connected to a transducer. The nels allow one to operate under a low pressure back-
y
h bindingofanalytescausechangesinphysicalorchemical ground which enhances the quality factor of detector.4
p properties of bioreceptive elements together with a sup- In orderto measure the changein frequency ofnanocan-
[ portmatrix,whichcanthenbesensedbyatransducerto tilevers it is required that system parameters should re-

generateanelectricalsignal. Thisgeneratedsignalquan- main unaltered. Ideally speaking, the analyte once at-
1
tifies the amount of analyte deposited. Classification of tached to nanocantilever should not detach or move.5
v
3 biosensors can be based on either their biorecognition Suchattachment-detachmentoradsorption-desorptionof
3 mechanisms or on the methodology of signal transduc- analyteparticles,6 their momentumtransfertonanocan-
5 tion.1 tilevers,7 and the random motion over detector,5 causes
4 Nanobiosensors utilizing nanocantilevers can provide fluctuations in resonant frequency along with shifts in
.
1 extreme sensitivity in the detection of biomolecules (an- frequency. These fluctuations, according to fluctuation-
0 alytes) down to a single-molecule level. Detecting par- dissipation theorem broaden the spectrum of nanocan-
3 ticular biomolecules can help biomedicalresearchersrec- tilever oscillatory motion.
1
ognizepathogensanddiseasesduringclinicalmonitoring. In order to develop a theoretical model related to de-
:
v Likemanyotherdetectors,nanoscalebiosensorsarechar- tection process for these nanocantilevers it is incredibly
Xi acterized by a quantity called the dynamic range. The important to keep all macroscopic and microscopic fac-
dynamic range is characterized by two extreme points tors involved in the detection process in mind. For such
r
a starting at the minimum detection limit for detecting a tiny system having dimensions on several tens to hun-
mass concentration (intrinsic to the detector) to the up- dred nanometers, it is indeed possible to analyze some
per limit when saturation occurs in the detector.2 variables macroscopically, but was critical to recognize
Nanocantilever based biosensors are based on the me- the elements that were less obvious and would be negli-
chanicalmotionofcantilevers. Acantileverisanoscillat- gible if one was not working on the nanoscale. Because
ingmechanicaldevicewhosemasscontinuallychangesas amino acids are one of the main bioreceptors used in
biological analytes attach to it. The attachment of ana- nanocantilever-based biosensors, the affinity and speci-
lytesleadstoachangeinthedevice’sresonantfrequency. ficity of amino acid sequences was a very important fac-
The amount of mass deposited on such detectors can be tor in our analyses. These amino acids are critically de-
estimatedby measuringthe shift infrequency of the res- pendent on ligands to which they are attached. Ligands
onator. These detectors possess high sensitivity because play a major role in the determination of geometric di-
their intrinsic mass is small but the sticking of analyte mensions. This change in geometry could cause receptor
molecules causes significant mass change and hence pro- amino acid sequences to become inactive or even initi-
ducesaverylargechangeinfrequency. Thequalityfactor ate other events like nonlinear expansion or stochastic
2
motion of the cantilever. These chaotic events need to ofanalytemoleculesislow,wecansaythatF ispropor-
d
be noted when mathematically modeling nanocantilever tional to the number of analyte particles N striking the
a
biosensors. nanocantileverpersecond. However,ifthedraggingforce
This paper is organized as follows: In section II, we is random, it can be described by the term ξ(t)(dx/dt).
giveabriefdescriptionofhowmassdepositioncausesfre- The term brings random damping to the nanocantilever
quencyshiftsintheharmonicoscillator. InsectionIII,we oscillator.7 Another possible type of dragging is called
model nanocantilevers mathematically using a stochas- inertial dragging,and it is proportionalto the molecular
tically perturbed harmonic oscillator. We explain how acceleration. We will not discuss this case here.
noise introduces measurable random frequency fluctua- Some other type of phenomenon taking place is the
tions in the oscillating nanocantilever that are propor- adsorption-desorption of the analyte molecules on the
tionaltothemassdepositedonitssurface. Thisfeatureis nanocantilever surface.6,7 Such thermally driven effect
used by biosensors as the detecting mechanism. Finally, produces random frequency fluctuations in the oscilla-
some concluding remarks are presented in section IV. tor.6–8 Theprocessofadsorption-desorptioncanbemod-
eledbyamolecularflux-dependentadsorptionrateanda
thermallyactivatedrate ofdesorption.6,7 Another mech-
anism that produces frequency fluctuations is the one
II. MASS DEPOSITION AND FREQUENCY
occurringwithnanocantilevershavingmicrofluidicchan-
SHIFT
nels. In this case, the fluctuations are produced by the
diffusion of an adsorbed particle along the microfluidic
Nanocantilevers can be simply modeled as damped,
channelinsidethe vibratingnanocantilever.5 Allofthese
driven harmonic oscillators. The oscillating mechanism
fluctuations or random stochastic processes mentioned
we consider in this work consists of a mass-spring sys-
above can be incorporated as delta-correlated or expo-
tem, and its equation of motion will be used to describe
nentially correlatednoise processes (Ornstein-Uhlenbeck
the classical behavior of nanocantilevers. As it is well
processes) in the modeling equations.
known, this equation includes a damping and a driving
terms. Random damping in the oscillator changes its
frequency. If these frequency changes can be measured,
III. STOCHASTICALLY PERTURBED
thentheoscillators(nanocantilevers)canbeusetodetect HARMONIC OSCILLATOR
simple physicalphenomenasuchasmassdeposition. For
a nanocantilever,it is commonto measure changesinits
Letus considerthe simple caseofa nanocantileverde-
frequency of resonance. This frequency shift is directly
scribed by a damped harmonic oscillator of mass m and
proportionaltothe massdepositedonitaswewillseein
spring constant k, and driven by a sinusoidal external
the following.
force. The differential equation describing the motion is
Foramass-springsystemwithmassmandspringcon-
stant k, the characteristic angular frequency under sim- d2x dx
ple harmonic motion is given by ω0 = k/m. A small mdt2 +bdt +kx=F0sin(ωt), (2)
change in the mass of the system due to mass deposi-
p
tion modifies the frequency of oscillation of the system. where F0 and ω are the amplitude and frequency of the
Taking finite differences of the angular frequency with external driving force, respectively, and b is a positive
respect to mass and solving for ∆m, we may get damping coefficient.
If a system is subject to both random and periodic
2m forces, the well-known phenomenon of stochastic reso-
∆m= ∆ω . (1)
−ω 0 nance (SR) may emerge.9,10 In the following,we will de-
0
scribe the effects of including random damping and fre-
Forthisequationtomakephysicalsense,weassumethat quencyasstochasticallyperturbationsinthe equationof
theelasticandgeometricpropertiesofthenanocantilever evolution of the nanocantilever. They will enter Eq. (2)
remain unchanged after the small mass deposition. as multiplicative noise ξ(t), such as the new equations of
When analyte molecules are deposited on a nanocan- motion can be written as
tilever,thereisamomentumexchangebetweentheoscil-
d2x dx
lator and the particles colliding with it. The process can +2β[1+ξ(t)] +ω2x=Asin(ωt) (3)
becalledaprocessofrandomcollisions.7 Thesecollisions dt2 dt 0
bring a random dragging force that affects the nanocan-
for the random damping, and
tilever motion. If the concentration of analyte molecules
is low, then this dragging force can be attributed to a d2x dx
+2β +ω2[1+ξ(t)]x=Asin(ωt) (4)
molecular drag proportional to the molecular velocity dt2 dt 0
and contributing to the dissipative part of the equation
of motion of the oscillator. This dragging force can be for random frequency. Here β b/2m is the damping
≡
writtenasF =b(dx/dt)andcorrespondstothedamping parameter, ω = k/m is the characteristicangular fre-
d 0
termoftheoscillator’sequation. Whentheconcentration quency in the absence of damping, and A= F /m. The
0
p
3
randomvariableξ(t)willbeconsideredasbothGaussian The presence of noise in the system brings fluctua-
white noise with the correlator tions in the nanocantilever frequency, resulting in spec-
tral broadening. Hence it becomes necessary to deter-
ξ(t)ξ(t′) =Dδ(t t′), (5) mine the minimum measurable frequency shift that can
h i −
beobservedinthisnoisyenvironment. Thespreadinthe
and color noise with exponential correlator
frequencyδω canbeobtainedbyintegratingthespectral
0
ξ(t)ξ(t′) =α2e−λ|t−t′|. (6) density of frequency fluctuations S(ω)
h i
1/2
The parameters D and α represent the white and color ω0+∆ω0
noise strengths, respectively, and λ the correlational de- δω0 S(ω)dω . (13)
cay rate. For color noise, we are only interested in sym- ≈"Zω0−∆ω0 #
metricaldichotomousnoise(randomtelegraphsignal)for
Here, we assume that a measurement of the nanocan-
whichξ(t)cantakeoneofthevaluesξ = α,andtheav-
eragewaiting time for eachofthese state±s is λ−1. Under tileverfrequencywasdonewithasquare-shapedtransfer
function over the bandwidth 2∆ω , and centered at ω .
the following limiting conditions: α2 and λ , 0 0
Eq. (6) reduces to (5), provided α2/λ→=D∞(see Re→f. 1∞0). Equation (13) is an estimate for any real system. The
spectral density S(ω) is determined by the nature of the
noise present in the system, and can be determined by
taking the Fourier transform of the white noise correla-
A. Nanocantilever with random damping
tor7
∞
Inthe absenceofanexternaldrivingforce(A=0)the
S(ω)= ξ(t)ξ(0) e−iωtdt. (14)
equation of motion (3) takes the form h i
Z−∞
Oˆ x = 2βξ(t)dx, (7) Atresonance,thesecondmomentofthenanocantilever
D{ } − dt displacement, x2 , satisfies the relation
h i
where the operator OˆD is defined by the following ex- 1mω2 x2 = 1κT, (15)
pression 2 0h i 2
Oˆ d2 +2β d +ω2. (8) whereκ istheBoltzmannconstantandT isthe absolute
D ≡ dt2 dt 0 temperature at resonance. By using (14), we can find
the spectraldensitycorrespondingtothese displacement
If the random damping is produced by the delta-
fluctuations
correlated white noise (5), then (3) reduces to
const S(ω)
ddt22hxi+2β(1−2βD)ddthxi+ω02hxi=0. (9) Shx2i = [(ω2−ω02)2+4β2×ω2(1−2βD)2]1/2. (16)
Here, we have use the fact that averages split in the There are two damping mechanisms present in Eq. (16):
form11 The thermo-mechanical fluctuations governed by the
damping parameter β, and the momentum exchanged
dx(t′) dx(t′)
ξ(t)ξ(t′) = ξ(t)ξ(t′) . (10) with white noise, whose spectral density is proportional
dt h i dt to βD. In particular, that momentum exchange is re-
(cid:28) (cid:29) (cid:28) (cid:29)
sponsiblefortakingthesystemintoresonantstates. The
Equation(9)modelsthecollisional dampingproducedby
spectraldensityforthefluctuationsinthenanocantilever
analytes getting stuck on the nanocantilever. If 2βD <
displacements turns out to be a resonance frequency
1, the presence of white noise will lead to a damping
curve in the parameter D (the noise strength) which
decrease (weak noise). On the other hand, if 2βD >
shows a maximum at D =(4β)−1. This behavior resem-
1, the effective damping turns negative increasing the
bles the stochastic resonance phenomenon (see Fig. 1).
amplitude of oscillation x and leading to instabilities
in the dynamics of the nahnoicantilever (strong noise).10
In this problem, however,the phenomenon of stochas-
Whenthenanocantileverisdrivenbyanexternalforce,
tic resonance is somehow counterintuitive. The reason
a solution to (9) can be written as
is that the “unwanted” noise coherently adds up to the
external force signal, increasing the signal to noise ra-
x =Bsin(ωt+ϕ). (11)
h i tio instead of reducing it.9 From Fig. 1, it is clear that,
Uponsubstituting (11)into(9),wecansolveforthe am- for these nanocantilever based biosensors, the resonance
plitude B and get the following expression peak is not sharp but has a finite width. The peak
width is determined by the intrinsic damping (β) of the
A
nanocantilever,andthestochasticfluctuationscausedby
B = . (12)
[(ω2 ω2)2+4β2ω2(1 2βD)2]1/2 analyte molecules sticking on it. To measure the precise
− 0 −
4
shift in the resonance frequency of the nanocantilever, With random frequency, the mathematical expression
the sharpest possible peak from the figure should be the used to find the spectral densities of noise fluctuations
appropriate choice. Therefore, knowing precise values of becomes a bit more complicated than that used in ran-
β andD parameters,andestimatingthemassdeposition dom damping (see Eq. (14)). Hence, the analytical solu-
correctlybecomes criticalin designing suitable nanocan- tion(11)forthefirstmomentinrandomdampingcannot
tilever based biosensors. be usedto obtainasolutionforrandomfrequency. How-
ever, for dichotomous color noise, the amplitude associ-
atedwiththefirstmomentshowsastochasticresonance-
B. Nanocantilever with random frequency like feature in α and λ similar to that observed in (13).
Therefore, a similar analysis can be applied to get the
In the absence of an external driving force (A = 0) resonanceprofile andfind the changein frequency of the
equation (4) takes the form nanocantilever due to analyte deposition.
The adsorption-desorption of noise by the nanocan-
d2x dx tilever gives rise to fluctuations in its frequency. This
+2β +ω2[1+ξ(t)]x=0. (17)
dt2 dt 0 is caused mostly by the constant bombardment of ana-
lyte molecules on its surface. This noise mechanism can
Here, the multiplicative noise ξ(t) introduces random
be understood from the following perspective. The an-
fluctuations in the nanocantilever frequency. It can be
alyte molecules are adsorbed due to their affinity to the
shown that, if the noise is white with correlator (5), the
nanocantilever substrate and are desorbed because of a
firstmoment of the oscillatoris not affected by the noise finite temperature change. This creates some fractional
and x(t) = x(t). On the other hand, for the exponen-
frequency noise. It is interesting to note that this pro-
h i
tially correlated noise (6), the first moment x satisfies
cess of adsorption-desorption of analyte molecules does
h i
thefollowingequationinaccordancewiththecumulative
not produce a damping mechanism per se. The ran-
expansion of van Kampen12
domness in the sticking and releasing particles on the
cantilever does not contribute to the average change in
d2 d
x +(2β ω2q ) x +ω2[1 ω q ] x =0, (18) the energy,but it changes the frequency of the nanocan-
dt2h i − 0 1 dth i 0 − 0 2 h i tilever in a nondeterministic manner.6,7 Thus, this pro-
where the parameters q and q are given by8 cessintroducesadifferentparametricnoisethatdoesnot
1 2
culminate into dissipation.
∞ The nanoscale cantilevers are quite sensitive to the
q =2 ξ(t)ξ(t t′) [1 cos(2ω t′)]dt′, (19a)
1 h − i − 0 adsorption-desorptionofnoise whencomparedwith con-
Z0 ventional scale cantilevers. The reason is the differ-
∞ ence between the surface to volume ratio for each type.
q2 =2 ξ(t)ξ(t t′) sin(2ω0t′)dt′. (19b) This explains why the number of adsorptionlocations in
h − i
Z0 nanocantilevers is bigger than those on the counterpart.
In the limit of white noise both parameters q and q Thefrequencyfluctuationscausedbynoisecanalsobe
1 2
vanish. In other words, white noise does not change the described using some basic considerations different from
width of the resonance profile of the frequency. So the those discussed in (18). We describe this method briefly
rateatwhichthefrequencyofthenanocantileverchanges for the sake of completeness, closely following Refs. 6
due to mass deposition remains simple to calculate. and 7. Let the adsorption rate be R , which is depen-
a
dentonthestickingcoefficientofthe nanocantileversur-
face,andR thetemperaturedependentdesorptionrate.
0.0025 d
The probability of molecular occupation in a particu-
A lar area is given by p = R /(R + R ), and the cor-
a a d
0.002
responding variance in the occupation probability σ =
v
√R R /(R +R ). Thecorrelationtimeτ ofabsorption-
a d a d c
0.0015 desorption noise is also given by τ =(R +R )−1. The
S2xX\ SHL spectral density of noise in this cacse canabe wrditten as
0.001 B
C S (ω)= 2πω02Naσv2τc ∆m 2, (20)
0.0005 D a [1+(ω ω )2τ2] m
− 0 c (cid:18) (cid:19)
E
where ∆m is the mass of the molecules attached on the
0 20 40 60 80 100
 cantilever surface. The shift in frequency follows equa-
tion (1). The noise variance is a maximum when the
FIG. 1. Plot of Shx2i/S(ω) as a function of the driving field probability of occupation is 1/2, that is the adsorption
frequency ω, with parameter conditions: ω0 = 50 and β = rate equals the desorption rate. On the other hand, the
0.35. CurvesA,B,C,D,andEarefor D=5, 10, 20, 30 and noise is a minimum when the occupation probability is
40, respectively. either 0 or 1. This noise will be superimposed on the
5
frequency change of the biosensor and hence becoming IV. SUMMARY
critical in estimating the analyte mass deposition as we
discuss in the following. In this work, we presented a realistic model for a
Integration over the spectral density S (ω) provides nanocantilever biosensor based on the description of
a
some change in the nanocantilever frequency a stochastically perturbed harmonic oscillator. These
biosensors work by mass sensing the analytes through
shifts in the characteristic frequency of the oscillator.
2
ω0+π∆f Whenanalytesaredepositedonthenanocantilever,they
∆ω S (ω)dω (21)
0 ≈"Zω0−π∆f a # bgirvinesgraisvearfliuetcytuoaftinoonisseisnodnatmhpisinvgibarsatwinegllsaysstfermeqwuehnicchy
ω σ ∆m
= 0 v [N tan−1(2π∆fτ )]1/2, (22) of the cantilever. The estimation of such fluctuations on
a c
− 2πm thespectralresponseofcantileverisimportantinfinding
outtheexactamountofanalytedepositionforclinicaldi-
where ∆f defines the width of passband. Hence, the agnostic purposes. In this work we considered frequency
change in mass on the nanocantilever is given by fluctuationsaccordingtodifferenttype noisemodelsand
their effect on the mass deposition on the cantilever.
δm ∆mσ [N tan−1(2π∆fτ )]1/2. (23)
v a c
≈
ACKNOWLEDGMENTS
Note that this expression for the frequency shift (due
to mass deposition) may be more realistic than that ob- Funding supports from RCSA and URSCA are grate-
tained in (1). fully acknowledged.
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