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Sensors and Actuators B 166– 167 (2012) 535– 543

Contents lists available at SciVerse ScienceDirect

Sensors and Actuators B: Chemical

journa l h o mepage: www.elsev ier .com/ locate /snb

luctuations of the number of adsorbed molecules in biosensors due to stochasticdsorption–desorption processes coupled with mass transfer

vana Jokic a,∗, Zoran Djuric b, Milos Frantlovic a, Katarina Radulovic a, Predrag Krstajic a, Zorana Jokic c

Institute of Microelectronic Technologies and Single Crystals (IHTM–MTM), University of Belgrade, Njegoseva 12, 11000 Belgrade, SerbiaInstitute of Technical Sciences of SASA, Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Belgrade, SerbiaFaculty of Biology, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia

r t i c l e i n f o

rticle history:eceived 26 December 2011eceived in revised form 25 February 2012ccepted 1 March 2012vailable online 10 March 2012

eywords:iosensordsorption–desorption noiseass transfer

luctuations

a b s t r a c t

We derived a simple theory of fluctuations of the equilibrium number of adsorbed molecules in biosen-sors, caused by the stochastic nature of adsorption–desorption (AD) processes coupled with mass transfer.The two-compartment model is used for approximation of the spatial dependence of analyte concentra-tion in the reaction chamber, which is justified when a thin layer depleted of the analyte exists closeto the surface on which the binding reaction occurs. By using the obtained analytical expression for thepower spectral density of fluctuations we perform for the first time the quantitative analysis of the influ-ence of the mass transfer on the fluctuations spectrum. For realistic parameter values, the influence ofmass transfer proved to be significant, causing the increase in the fluctuations level of up to two orders ofmagnitude compared to the rapid mixing case. The dependences of the mass transfer influenced fluctu-ations spectrum on various parameters of the analyte–receptor binding process are also systematicallyinvestigated. The presented theoretical model of fluctuations enables good estimation of the AD noise,

which affects the total noise and the minimal detectable signal of biosensors. It provides the guidelinesfor improvement of the limits of detection, and for optimization of detection methods. The theory is alsoproposed as a basis for development of highly sensitive methods for analyte detection and characteriza-tion of biomolecular binding processes, based on the measured fluctuations spectrum. It is applicable forvarious types of sensors whose operation principle relies on the adsorption process of analyte molecules.

. Introduction

Detection and identification of biomolecules in solutions areery important tasks in medicine, environmental protection, agri-ulture, and homeland security. Investigation of biomolecularnteractions, being the basis of various vital biochemical processess also of great significance, especially for the fundamental under-tanding of biology, as well as for pharmaceutical research. Theres a variety of different platforms for detection of biological entitiesnd characterization of biomolecular interactions. However, theres a need to explore novel techniques which improve the efficiencynd sensitivity of detection of analyte molecules in liquids and pro-ide additional information about biomolecular processes and theirinetics.

The principle of operation of one group of biological sen-ors relies on the process of affinity-based binding of analyteolecules from the solution to the specific probe molecules used for

∗ Corresponding author. Tel.: +381 11 2638188; fax: +381 11 2182995.E-mail address: ijokic@nanosys.ihtm.bg.ac.rs (I. Jokic).

925-4005/$ – see front matter © 2012 Elsevier B.V. All rights reserved.oi:10.1016/j.snb.2012.03.004

© 2012 Elsevier B.V. All rights reserved.

functionalization of the surface of a transducer. Binding of analytemolecules to the probe molecules results in a measurable changeof one or more of the transducer’s physical (mechanical, electri-cal, and optical) parameters [1–6]. In the steady state, establishedwhen binding and unbinding processes of interacting moleculesbecome balanced under given conditions, the coverage of the sur-face by bound molecules depends on the concentration of analytemolecules in the solution. The analyte concentration and the rateconstants of the interaction processes of biomolecules are usu-ally obtained from the steady state data and from the change ofthe measured parameter in time during the period of the steadystate establisment [4–8]. Mass transfer (transport of the analytemolecules existing in a solution to and from the sensing surface)is an important effect that has to be considered when biomolecu-lar binding-unbinding processes are characterized and also whendecisions are made during sensor design [9–12].

Fluctuations of the number of adsorbed molecules in the

steady state also contain information about the analyte, theadsorption–desorption (AD) process and its kinetics [13](the terms binding–unbinding, adsorption–desorption andassociation–dissociation will be used as synonyms in this paper).

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36 I. Jokic et al. / Sensors and Ac

his implies that methods based on frequency domain analysis ofeasured fluctuations can be developed for detection of analytes

nd for investigation of biomolecular interactions, as a comple-entary approach to conventional time domain measurements

14].Fluctuations of the measured parameter, induced by the

tochastic nature of AD processes, have traditionally been regardeds noise. Fluctuations of the number of adsorbed molecules, causedy the stochastic nature of AD processes, result in fluctuations ofhe sensor’s output signal, which have traditionally been regardeds noise. They are fundamental by their origin and thereforenavoidable. In devices that contain micro/nanosized function-

ng parts, fluctuations generated by fundamental mechanismsut a hard limit on their performance [15]. As the dimensionsf the structures decrease, the influence of AD induced fluctu-tions increases and there is a possibility for the AD noise totart to dominate over other fundamental noises [13,16,17]. Com-ared to other micro/nanodevices, scaled down biological sensorsre especially affected by fluctuations caused by AD processesf analyte particles (large polymer molecules, proteins, nucleiccids, cells, etc.). Therefore, with the current trend towards small-cale micro/nanofabricated biosensors, the analysis of this kindf fluctuations becomes increasingly important. Comprehensivenderstanding and theoretical modeling of AD induced fluctua-ions in biological micro/nanosensors is essential for determinationf their limiting performance and for improvement of detectionethods.In general, the phenomenon of fluctuations in biosensors

esponse belongs to the category of processes occurring in systemsn which physical quantities are both space and time dependent.luctuations arise due to both the stochastic transition eventsbinding or unbinding of the interacting molecules) and the massransfer. Therefore, the combined influence of these effects has toe considered.

The theory of fluctuations of the number of adsorbed parti-les for the cases where mass transfer is of negligible influence iserived in [13,18–20]. Such case is, for example, adsorption from aaseous environment, where mass transfer is much faster then iniquids. A complex analysis of the stochastic processes in biosensorsperating in liquids can be found in [21–23]. A stochastic analysishich includes mass transfer effects is considered in literature from

he field of chromatographic separation-based assays [24]. How-ver, to the best of our knowledge, a quantitative analysis of thenfluence of mass transfer on fluctuations spectrum of the numberf adsorbed particles in the affinity-based sensors is still missing inhe literature.

In this paper we analyze the fluctuations of the number ofolecules adsorbed on a biosensor surface from the liquid phase.

he objective of our work is the development of a simple theoryhich is useful for the quantitative estimation of the effects ofass transfer on the equilibrium fluctuations of the measured sig-

al and also for investigation of biosensor performance limits. Thedditional motivation for the development of this theory is its appli-ation in designing of new methods for detection of biomoleculesn solutions, and also for characterization of biomolecular bind-ng processes coupled with mass transfer, using the analysis of the

easured spectrum of fluctuations.In the following sections, we derive a simple mathemati-

al model of stochastic adsorption of analyte particles, takingnto account the mass transfer process. We introduce the two-ompartment model approximations, in order to obtain thequation describing the change of the number of adsorbed

articles in time due to the combined effect of the intrinsicdsorption–desorption process and the mass transfer. Then, bypplying the approach used for generation–recombination (GR)uctuations of the number of charge carriers in semiconductors,

s B 166– 167 (2012) 535– 543

which are described by an analogous stochastic equation, we obtainthe analytical expression for the power spectral density of fluc-tuations of the number of adsorbed molecules. The numericalcalculations are performed using the parameter values correspond-ing to realistic experimental and practical conditions in biosensors.Finally, the analysis of the results demonstrates the significanceof taking into account the effects of mass transfer in biosensorsoperating in liquids, when fluctuations of their output signal areconsidered.

It is important to realize that, due to the complexity of the phe-nomena on the sensor surface and in the reaction chamber, it wouldbe very difficult to establish a complete mathematical model thatincorporates all the relevant physical processes. Like all approxi-mate approaches, the theory we present in this paper is devised byintroducing assumptions which limit the scope of its applicability.First, the reaction is assumed to be a simple bimolecular reversiblebinding reaction according to the Langmuir adsorption model. It isassumed that the functionalization of the surface is homogeneous,that there is no non-specific binding, as well as that neither analytenor receptor particles undergo changes that can affect statisticalproperties of the binding process in time or over the ensemble.Also, the two-compartment model is valid for the cases of masstransfer influenced binding which leads to formation of a thin layerdepleted of the analyte adjacent to the surface where adsorptionprocess occurs.

We would like to emphasize that the fluctuations of the numberof adsorbed molecules, determined by the presented theory, can beeasily converted into fluctuations of a quantity (e.g. mass, resonantfrequency, refraction index, etc.) that characterizes a given sensor(quartz crystal microbalance (QCM), thin film bulk acoustic res-onator (FBAR), plasmonic, nanoplasmonic, etc.) whose operatingprinciple relies on the adsorption process of analyte molecules.

2. Theoretical analysis

In the surface based biosensing techniques one type of reactingmolecules (called probe molecules, capturing molecules or recep-tors) is immobilized on a surface of a sensing element, where theyserve as the sites for binding of molecules of an another reactant(i.e. the analyte). A sensing element functionalized in this way islocated in a reaction chamber. When a solution containing ana-lyte molecules is introduced in a reaction chamber, the bindingreaction may occur when the analyte molecules come in the vicin-ity of the binding sites. We assume a simple one-to-one reactionmodel: the association process, whereby one analyte molecule andone receptor molecule bind to each other and create the immobi-lized complex, and the dissociation process, whereby this complexdissociates. The sensor response corresponds to the total numberof formed analyte–receptor complexes. We also assume that theanalyte particles do not interact with each other, and that all thebinding sites on the sensor surface are equivalent. These assump-tions are in accordance with the Langmuir adsorption model [10].The equation describing the time evolution of the reversible bind-ing process between analyte and receptor molecules is then

d�b(t, x, y)dt

= kf Cs(t, x, y)(�m − �b(t, x, y)) − kr�b(t, x, y) (1)

where �b(t,x,y) is the surface density of bound analyte molecules,�m is the surface density of receptor sites (constant in time andassumed to be uniform on the surface), Cs(t,x,y) is the free ana-lyte concentration in the immediate vicinity of the functionalizedsensor surface, kf is the association rate constant, and kr is the dis-

sociation rate constant (these are the intrinsic rate constants for theanalyte–receptor binding–unbinding reaction). All the variables inEq. (1) depend on an exact position on the functionalized sensorsurface (i.e. on coordinates in that plane, x and y).

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Besides the association–dissociation reaction described by Eq.1), transport processes of diffusion and flow also occur in theensor’s reaction chamber. They bring analyte molecules to, andake them away from the binding sites. While the direction of thenalyte flow is along the reaction chamber (x axis direction), i.e. its parallel with the functionalized sensor surface, the direction ofhe diffusion depends on the spatial distribution of concentrationf the free analyte molecules. The continuity equation describeshe rate of change of the spatially dependent analyte concentration(t,x,y,z), influenced by transport processes as

∂C(t, x, y, z)∂t

= −div j(t, x, y, z) (2)

here j(t,x,y,z) is the flux vector (z-axis is perpendicular to theunctionalized sensor surface). The components of j depend on thepatial distribution of the free analyte concentration, the spatial dis-ribution of flow velocity of the solution in the reaction chamber,nd the analyte diffusion coefficient. Therefore, Eq. (2) is a partialifferential equation, which with appropriate initial and boundaryonditions describes the change of the free analyte concentrationn the reaction chamber, due to transport and AD processes. Eqs.1) and (2) can be simultaneously solved by using numerical finite-lement methods. However, since we are interested in an analyticalolution for Cs(t,x,y) = C(t,x,y,0) which we can use in further analysisf the fluctuations of the number of adsorbed molecules, a simplerathematical model is required.There are many possible sets of transport and binding conditions

n the reaction chamber, which differ depending on the geometrynd dimensions of the system (reaction chamber, sensing surface),he solution flow rate, the receptor surface density, and also onhe diffusivity and affinity-based binding properties of the ana-yte molecules. Therefore, a general analytic solution of Eqs. (1) and2) does not exist. Instead, analytical solutions can be found withertain approximations only for some of the physically relevantases. In microfluidic systems, two transport regimes are consid-red depending on the ratio of the time scale required for diffusionf analyte particles across the chamber height and the time scale foronvection in the axial direction [9]. When the diffusion time scales shorter than the convection time scale [9,10], molecules enteringhe chamber at an arbitrary height z are able to reach by diffusionhe surface on which the binding reaction occurs. Such transportegime (which we will call type 1 transport regime) is common, forxample, in shallow microfluidic channels. In other cases (the diffu-ion time scale greater than the convection time scale), the analyteolecules in the flow chamber that are not able to reach the reac-

ive surface exit the sensing region, carried by the flow. Thus, onlyhe molecules from the layer adjacent to the sensing surface canarticipate in the binding reaction. Higher flow rates, thicker chan-els and slower-diffusing analytes are in favor of such transportegime. This situation (type 2 transport regime) is typically encoun-ered in SPR systems. It is also common in flow-through devices,uch as QCM and some other surface-based sensors [9–12,25]. Theollowing consideration and the analysis that we perform in thisaper refer to type 2 transport regime.

When introducing the approximations which significantly sim-lify the equations to be solved, we first consider the case of theo-called rapid mixing [11]. In that case, the transport of analytearticles is assumed to be so fast compared to the binding reaction,hat it leads to a rapid establishment of the uniform analyte con-entration Co in the entire reaction chamber, which is also constantn time (Co is the analyte concentration in the solution entering thehamber). Then the approximation CS(t,x,y) = Co can be used in Eq.

1). It acquires the same form as the equation used for descriptionf reversible adsorption of particles from the gas phase [13],hich can be solved analytically for �b(t), yielding the number of

ound molecules that approaches equilibrium exponentially. This

Fig. 1. Schematic representation of the processes in a biosensor reaction chamber,assuming that the analyte concentration distribution is according to the two-compartment model.

corresponds to the case of reaction-limited kinetics. In other cases,in systems with type 2 transport regime, the analyte concentrationnear the sensing surface is spatially dependent. When the diffusionflux of analyte molecules on the reactive surface is smaller thanthe adsorption flux (diffusion-limited kinetics), the zone in thevicinity of the functionalized surface is depleted of the analytedue to the binding reaction, i.e. a thin mass transfer boundarylayer is formed. In that case, the simplified system of equationsdescribing the processes of analyte binding and transport can beobtained by introducing the assumptions of the two-compartmentmodel regarding the analyte concentration dependence on spatialcoordinates in the sensor’s reaction chamber.

2.1. Two-compartment model equations

The two-compartment model provides a good approximation ofthe mass transfer influenced binding kinetics (i.e. of the change ofthe number of adsorbed molecules on the sensor surface in time) incases when a thin zone depleted of the analyte is formed close to thesurface on which the binding reaction occurs. Its accuracy is exper-imentally confirmed [11,12,25,26]. This model uses the simplifieddependence of the analyte concentration on the spatial coordinates.The reaction chamber is divided into two compartments: the innercompartment, in the immediate contact with the sensor surface,and the outer compartment, which occupies the remaining part ofthe chamber (Fig. 1).

The model assumes that the analyte concentration in the outercompartment is uniform in space and constant in time, and isequal to the concentration in the bulk solution, Co (the short timeinterval at the beginning of analyte injection, when the concen-tration increases from 0 to Co, is neglected). The concentrationof analyte molecules in the inner compartment, Ci, is also uni-form within the compartment (thus Cs = Ci), but it changes in timedue to both the transport of analyte molecules between compart-ments and the association–dissociation process taking place onthe sensor’s surface. After a short period of time from the begin-ning of the AD process, this change becomes slow, so that thequasi-steady state approximation becomes valid (i.e. dCi/dt ≈ 0)[11], but the time dependence of the concentration in the innercompartment cannot be neglected. The important benefit of thetwo compartment model is the elimination of the dependence ofthe analyte concentration on spatial coordinates in each of thetwo compartments, so all the variables depend only on time andtheir values used in the equations are averaged over the sensor’s

surface.

Based on the assumptions of the two-compartment modeland the quasi-steady state approximation, the rate of change ofthe number of adsorbed molecules per unit area equals the net

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umber of molecules that enter the inner compartment in unitime per unit boundary area between the compartments, so theollowing equation is valid

d�b(t)dt

= km(Co − Ci(t)) (3)

ere km is the mass transfer coefficient, which characterizeshe transport between the inner and the outer compartmentphysically, it is the volumetric flux). The model assumes that km

s constant.Spatial uniformity of the variable �b(t) over the entire function-

lized sensor surface of the area A implies that the total number ofound analyte molecules on the sensor surface, Nb(t), is obtaineds A�b(t). The total number of immobilized receptor sites, Nm,quals A�m. Accordingly, the system of equations describing theime change of the number of molecules adsorbed on the sensorurface is simply obtained from Eqs. (1) and (3) as

m(Co − Ci(t))A = kf Ci(t)(Nm − Nb(t)) − krNb(t) (4)

dNb(t)dt

= kf Ci(t)(Nm − Nb(t)) − krNb(t) (5)

qs. (4) and (5) constitute the system of equations of the two-ompartment model. It enables us to turn to the analysis of theondimensional quantity Nb(t) which does not refer to a particularosition on the sensing surface, but rather to the whole surface.

By elimination of the variable Ci(t) from Eqs. (4) and (5), thequation is derived which can be written in the form

dNb(t)dt

= g(Nb(t)) − r(Nb(t)) (6)

here the functions g(Nb(t)) and r(Nb(t)) are given as analyticalxpressions depending only on time. In this way, the two-ompartment model enables us to describe the change in theumber of adsorbed molecules by one equation which takes intoccount the mass transfer effects and is independent of spatialoordinates, instead of the complex system of Eqs. (1) and (2).q. (6) has the same form as the one used to express the rate ofhange of the number of charge carriers in semiconductors due toeneration–recombination processes [13,27]. The following equiv-lence of the quantities can be established: the rate of increase ofhe number of adsorbed particles (g) is equivalent to the carriereneration rate, and the rate of decrease of the adsorbed particlesumber (r) is equivalent to the rate of carrier recombination. Inquilibrium, fluctuations of the number of adsorbed particles orig-nate from the difference between the instantaneous values of thetochastic rates corresponding to g and r, as will be shown in theext section.

.2. Fluctuations of the number of adsorbed analyte molecules inquilibrium

Eq. (6) describes the deterministic behavior of the number ofdsorbed particles in time. It enables us to obtain the numberf bound molecules in the steady state (Nbe), established whenll the transient processes are finished. The condition dNb/dt = 0pplied to Eq. (6), i.e. g(Nbe) = r(Nbe), yields the equilibrium numberf adsorbed molecules

be = kf Co

kr + kf Co· Nm (7)

This expression corresponds to the Langmuir isotherm equation.

Upon reaching the equilibrium, the number of adsorbed parti-

les still fluctuates due to the stochastic nature of the AD processoupled with mass transfer. The instantaneous value of the numberf adsorbed particles in equilibrium is Nb(t) = Nbe + �Nb(t), where

s B 166– 167 (2012) 535– 543

�Nb(t) denotes small fluctuation around the equilibrium value atthe moment t (�Nb(t) � Nbe).

In order to perform the analysis of fluctuation phenomenait is generally possible to classify the fluctuation processes intwo categories. The first category includes the processes thatare independent of spatial coordinates. This is the case in spa-tially homogeneous systems whose total number of constituentsfluctuates. Carrier fluctuations in semiconductors, arising fromthe generation and recombination processes, belong to this class[27,28]. The resulting noise is called generation–recombinationnoise.

Fluctuations in spatially continuous systems, in which bothspatial and time dependences of variables exist, belong to the sec-ond category. Here, fluctuations arise due to combined effects ofthe stochastic transitions between states (e.g. free/bound) and thetransport of the system constituents.

The phenomenon of fluctuations in chemical and biological sen-sors generally belongs to the second category. However, in the caseof rapid mixing, where there is no spatial dependence of variablesdue to the fast mass transfer, the analogy of AD and GR processescan be directly applied to model the fluctuations, as it is done forAD processes in gases [13,20]. In other cases belonging to type 2transport regime, the two-compartment model leads to simplifica-tions that permit the fluctuations of the total number of adsorbedparticles to be also analyzed by applying the approach used for anal-ysis of GR fluctuations. This is possible since the two-compartmentmodel provides the analytical expressions for the effective ratesof transition of the analyte molecules from the free state to theadsorbed state, g(Nb), and from the adsorbed to the free state, r(Nb)(equivalent to generation and recombination rates, respectively).These effective rates incorporate the influences of both the AD pro-cess and the mass transfer. It is important to note that the effectiverates we obtained are spatially independent, explicitly independentof time, but dependent on Nb(t), which is the situation analogousto GR processes. Therefore, for the analysis of the random processNb(t) (the fluctuating number of bound particles on the surface) weuse the theory [29] that yields the probability distribution, mean,variance, correlations and other statistical properties of GR pro-cesses characterized by generation and recombination rates whichare functions of the number of charge carriers n. These processes areknown in the probability theory as the birth-death processes, a classof the discrete-state, continuous time Markov processes. Ergodic-ity of the process Nb(t) is assumed (there is no disturbing factorsin the system that would compromise stationarity and ergodicity).Starting from the master equation (in which the probabilities oftransition from the state Nb to Nb + 1 and from Nb to Nb − 1 duringthe time interval �t → 0 are related to the transition rates g(Nb)and r(Nb), respectively), it can be shown that the random variableNb has the steady state probability distribution [27,29]

P(Nb) = P(0)

∏Nb−1�=0 g(�)∏Nb�=1r(�)

(8)

where P(0) is obtained from the condition that the sum of proba-bilities over all possible values of Nb equals 1. The variance of therandom variable Nb is [27–29]

⟨(�Nb)2⟩ = g(Nbe)

(dr/dNb − dg/dNb)

∣∣∣Nb=Nbe

(9)

After substitution of Nb(t) = Nbe + �Nb(t) in Eq. (6), followed byexpansion of the expression on the right side in a Taylor series in

the vicinity of Nbe, the following equation is obtained

d�Nb

dt=

(dg

dNb− dr

dNb

)∣∣∣Nb=Nbe

· �Nb = −1�

· �Nb (10)

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t defines the exponential decay in time of the excess number ofound molecules �Nb(t) towards the equilibrium value Nbe.

The corresponding stochastic equation used for the fluctuationsnalysis has the form

d�Nb

dt= −1

�· �Nb + �g(t) − �r(t) (11)

i.e. the same as the equation for GR fluctuations. The terms �g(t)nd �r(t) are the stochastic components of the effective transitionates, which account for the randomness of all the involved pro-esses. These stochastic terms give the previous equation the formf the Langevin equation, where �(t) = �g(t) − �r(t) is the Langevintochastic source function. �g(t) and �r(t) are statistically inde-endent processes. The Langevin equation yields the relationshipetween the power spectral density (PSD) of the random variableb(t) and the uniform spectrum of �(t) [28]

2Nb

(f ) =S2

��2

1 + (2�f )2�2(12)

By integrating the expression (12) over the frequencies f from 0o ∞, and by equating the result with Eq. (9) which equals g(Nbe)�based on Eq. (10)), the spectrum of the Langevin stochastic sources obtained as S2

�= S2

�g+ S2

�r= 4g(Nbe). Thus, the PSD of the fluc-

uations of the number of adsorbed analyte molecules is obtainedn the form

2Nb

(f ) = �N2b

(f ) = 4g(Nbe)�2

1 + 4�2f 2�2(13)

he quantity SNb(f) is the rms value of fluctuations of the num-er of adsorbed molecules per square root Hertz, and it is alsoalled the spectral density of fluctuations, or simply, the spectrumf fluctuations.

Eq. (10) implies that the time constant of the fluctuation processs

= 1kr + kf Co

+ krkf �m

(kr + kf Co)2km

(14)

The first term on the right side of Eq. (14) equals the timeonstant in the case when the rapid-mixing model of the analyteoncentration establishment is valid. It will be denoted by �R. Theecond term, which we will denote by �m, shows that the time con-tant is influenced by the value of the mass transfer coefficient. It isbvious that �R is the minimal possible value of �. It depends onlyn the time constants of adsorption (�A = 1/kfCo) and desorption�D = 1/kr). As the mass transfer flux decreases, � increases due tohe increase of �m. It is interesting to note that the time constant ofstablishment of equilibrium of the binding process in the case ofxponential binding kinetics is also given by Eq. (14) [12]. Then theatio �/�R is a measure of the influence of mass transfer on the bind-ng process kinetics, because it shows how much the mass transferecelerates the establishment of the equilibrium, compared to thease of rapid-mixing. In the cases when kinetics deviates from thexponential behavior the same expression (Eq. (14)) is valid for theime parameter which influences the rate of increase of the numberf bound molecules [12].

By introducing the expressions for both � and g(Nbe), Eq. (13)an be written in the explicit form

2Nb(f ) = 4kf krCo�mA(kr + kf Co + (kf kr/km)�m)2(kr + kf Co)−1

(kr + kf Co)4 + (2�f )2(kr + kf Co + (kf kr/km)�m)2

This expression enables the analysis of the spectra of fluctua-

ions when the influence of mass transfer is taken into account. Inhe case when the mass transfer effects are neglected the fluctu-tions power spectrum has the same form as in the case of rapidixing, which can be obtained from Eq. (13) by replacing � with �R.

s B 166– 167 (2012) 535– 543 539

By using Eq. (13) it is easy to determine the power spectrumof fluctuations of the coverage of the receptor sites by analytemolecules (the coverage is defined as (t) = Nb(t)/Nm)

S2 (f ) = �2(f ) = �N2

b(f )

N2m

= S2Nb

(f )

N2m

Since S2(f ) is proportional to 1/A, a smaller sensing area leads to

increased fluctuations of the surface coverage. Eq. (13) also enablesobtaining of the PSD of fluctuations of a physical quantity by meansof which the detection of an analyte is performed. For example, sucha quantity can be the adsorbed mass

S2m(f ) = �m2(f ) = M2

a �Nb(f ) = M2a S2

Nb(f )

where Ma is the mass of an analyte particle. However, this can alsobe some other quantity (resonant frequency, refraction index, etc.),depending on the type of the sensor.

3. Numerical calculations: results and discussion

In this section we report the results of the analysis of thespectrum of fluctuations of the number of adsorbed particles inbiosensors, based on numerical calculations performed using thederived expressions. The parameter values (kf, kr, km, Co, �m) usedin the analysis are within the range that corresponds to the realisticexperimental and practical conditions [7,11,12,25]. They are cho-sen so to include the cases when the mass transfer influences thespectrum of fluctuations, as well as the cases when the spectrum isnot affected by the transport of analyte molecules.

Fig. 2a and b show the spectrum of fluctuations of the numberof adsorbed molecules, as a function of both the frequency f and theanalyte bulk concentration Co. The parameters kf, kr and �m equal8 × 107 1/(Ms), 0.08 1/s and 5 × 10−12 Mm, respectively, and theyare common for both diagrams. A sensing area A = 100 × 10 �m2

is assumed. For the case shown in Fig. 2a, km = 2 × 10−5 m/s, whilekm = 2 × 10−2 m/s for the diagram presented in Fig. 2b. Fig. 2c showsthe ratio RFS of the fluctuations spectra which are given in Fig. 2aand b. In Fig. 2a–c crossed vertical planes are used to make thedependences of the shown quantity on f (for Co = const) or on Co

(for f = const) more obvious.Fig. 3 shows the same quantity as Fig. 2a and b, but only as a func-

tion of frequency, with the mass transfer coefficient as a parameter(Co = 1 nM, the order of magnitude of km changes from 10−6 m/s to10−2 m/s, while the other parameters are the same as above).

The spectrum of fluctuations is of Lorentzian type, so the func-tion SNb(f) for a constant Co has a flat part in the low frequency range(ω � 1/�), which is called plateau. At higher frequencies the valueof the same function monotonically decreases (Figs. 2a, b and 3).The position of the characteristic “knee” is determined by the fre-quency ω = 1/� at which SNb(f) has 21/2 times lower value comparedto the value at the plateau.

By comparing the diagrams in Fig. 2a and b it becomes obvi-ous that the fluctuation spectrum significantly differs for differentvalues of km. The mass transfer affects both the magnitude of theplateau of the spectrum and the position on the frequency axis ofthe knee of the curve SNb(f) for Co = const. In the case of lower masstransfer coefficient, shown in Fig. 2a, the level of the spectral den-sity of fluctuations is higher compared to the case of higher masstransfer flux (Fig. 2b), for the same analyte bulk concentration. Theincrease of the level of fluctuations due to the influence of masstransfer can be clearly seen in Fig. 2c. For the parameter valuesgiven above, the level of fluctuations spectrum in the region of the

plateau is about 20 times higher in the case of the lower transportflux. As km decreases, the increase of the level of fluctuations spec-trum in the plateau zone becomes more prominent (Fig. 3). Thisis logical, because in this frequency range the magnitude of the

540 I. Jokic et al. / Sensors and Actuators B 166– 167 (2012) 535– 543

Fig. 2. The spectrum of fluctuations of the number of adsorbed molecules, as a func-tion of both the frequency and the analyte bulk concentration, for the case of: (a)km = 2 × 10−5 m/s, (b) km = 2 × 10−2 m/s. (c) The ratio of the fluctuations in the caseof slow mass transfer (shown in a) and in the case of fast mass transfer (shown inb).

Fig. 3. The spectrum of fluctuations of the number of adsorbed molecules with themass transfer coefficient as a parameter.

spectrum is proportional to �, which increases with the decrease ofkm.

It can be seen in Fig. 2a and b that the frequency correspond-ing to the knee of the curve SNb(f) for Co = const differs for twodifferent values of km, as it is lower in the case of slower mass trans-fer. The knee continuously moves towards higher frequencies withthe increase of the mass transfer coefficient (Fig. 3). The maximalvalue of the knee frequency is 1/�R. The measure of the influenceof mass transport on the decrease of the knee frequency is the ratio�/�R which equals 1 + kfkr�m/(km(kr + kfCo)) = 1 + kf�m(1 − e)/km.This ratio also equals the ratio of the plateau magnitude of thefluctuations spectrum for a given km and the same quantity inthe case of rapid mixing, for the same set of parameter values(Co, �m, kr and kf). Based on these facts a criterion whether ornot the influence of mass transfer on the fluctuations spectrumis significant can be established. When km is comparable to orlesser than (1 − e)kf�m (the intrinsic adsorption volumetric fluxin equilibrium) the influence of the mass transfer on the fluctua-tions spectrum of the number of adsorbed particles is significant(transport influenced fluctuations spectrum). When the conditionkm � (1 − e)kf�m is valid, the effects of mass transfer are practi-cally negligible, and the fluctuations spectrum is the same as in thecase of rapid mixing (reaction-determined fluctuations spectrum).In the example illustrated by Fig. 2, the mass transfer whose coef-ficient obeys km � 3.64 × 10−4 m/s is of negligible influence on thefluctuations spectrum in the whole shown range of analyte con-centration. Therefore, the spectrum shown in Fig. 2b is practicallythe same as the spectrum which would be obtained without takinginto account the mass transfer effects.

The ratio �/�R is also an indication of the discrepancy betweenthe characteristic parameters (the plateau magnitude and the kneefrequency) of the actual fluctuation spectrum and the one obtainedby neglecting the mass transfer effects in the cases when they actu-ally exist. Fig. 2c illustrates this discrepancy for km = 2 × 10−5 m/sand the values of other parameters used in the presented example.

Now we will examine how the influence of the mass transfer,characterized by a certain value of km, depends on the analyte con-centration in the solution, on the receptor surface density, and alsoon the intrinsic association and dissociation rate constants. Theanalysis given above has shown that these dependences can be

investigated by observing the ratio �/�R as a function of the chosenparameter (Co, �m, kf or kr). The greater the ratio �/�R at fixed km,the stronger the influence of the mass transfer on the fluctuations

I. Jokic et al. / Sensors and Actuators B 166– 167 (2012) 535– 543 541

Fig. 4. The ratio R = �/� as a measure of mass transfer influence on the fluctua-tc

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ions spectrum, depending on: (a) the analyte concentration and the mass transferoefficient, (b) the binding sites surface density and the coefficient of mass transfer.

pectrum (the greater is the deviation of both the plateau magni-ude and the knee frequency of the spectrum curve from the casef rapid mixing). Both qualitative and quantitative analysis of theass transfer influence as a function of each of the parameters will

e given in the following paragraphs.The spectrum of fluctuations differs for different values of Co

t a given km (see the curves SNb(f) for different fixed values of Co

n Fig. 2a). Fig. 2c shows that the ratio of the spectrum for a cer-ain mass transfer flux and the spectrum when the mass transfers neglected also depends on Co. How much the influence of massransfer is affected by Co can be seen more clearly in Fig. 4a whichhows the ratio R� = �/�R as a function of both the analyte concen-ration and the mass transfer coefficient, for kf = 5 × 107 1/(Ms),r = 0.08 1/s, and �m = 1 × 10−11 Mm. The purpose of the shownrossed vertical planes is to make the dependence of �/�R on eacharameter more obvious (valid for Figs. 4 and 5). By observing theependence R�(Co) for km = const (Fig. 4a), it can be noticed thathe influence of the mass transfer on the fluctuations spectrumecreases with the increase of Co. For example, for km = 10−5 m/snd Co = 10−10 M the plateau magnitude of the fluctuations spec-rum will be 50 times greater, and the knee frequency 50 times

ower compared to the case of rapid mixing. However, for theame mass transfer flux and Co = 10−8 M the deviation will be lowerabout 8 times). Such behavior can be explained by the decreasen the adsorption flux ((1 − e)kf�m) due to the increase of the

Fig. 5. The dependence of the ratio R� = �/�R on: (a) the intrinsic association rateconstant and the mass transfer coefficient, (b) the intrinsic dissociation rate constantand the mass transfer coefficient.

equilibrium surface coverage as Co increases. Therefore, at highervalues of Co the factor by which km exceeds (1 − e)kf�m is greater,so �/�R is lower. The shown example also indicates that it dependson Co whether or not the influence of mass transfer characterizedby a certain value of km is negligible. The mass transfer is of neg-ligible influence for km ≥ 4.7 × 10−3 m/s when Co = 10−10 M, whileat Co = 10−8 M the minimal value of km for which the mass transferinfluence is negligible is lower for an order of magnitude. At somehigher values of Co, the fluctuations spectrum can be unaffectedby mass transfer at the same value of km that made the fluctua-tions spectrum transport-influenced at lower Co. As Co decreases,the fraction of free binding sites (1 − e) is closer to 1, so �/�R

increases. At Co � kr/kf (in the shown example kr/kf = 1.6 × 10−9 M),the ratio �/�R at fixed km asymptotically approaches the maximalvalue 1 + kf�m/km, and the influence of mass transfer is strongestfor the given km. However, if km � kf�m (here kf�m = 5 × 10−4 m/s),km � (1 − e)kf�m will be valid for an arbitrary value of Co, so theinfluence of mass transfer will be negligible regardless of Co.

The dependence of the ratio �/�R on both the receptor surface

density and the mass transfer coefficient is shown in Fig. 4b, whichis obtained for the following values of parameters: kf = 8 × 107

1/(Ms), kr = 0.08 1/s, Co = 5 × 10−10 M. It can be seen from the dia-gram that the increase of the time constant � relative to �R, caused

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42 I. Jokic et al. / Sensors and Ac

y the influence of mass transfer, is higher at greater adsorp-ion sites density. Therefore, as �m increases, the influence of

ass transfer on AD fluctuations spectral characteristics becomestronger. In the given example (Fig. 4b) for high receptor surfaceensities, the level of the plateau of the fluctuations spectrum is upo two orders of magnitude higher compared to rapid mixing sys-ems, and the frequency of the knee of the spectral curve is loweror the same order of magnitude. The value of km at which the

ass transfer influence becomes negligible (i.e. � approaches �R)ecreases with the decrease of the receptor surface density.

The influence of mass transfer is also more prominent forigher values of kf, as seen in Fig. 5a which shows the dependence�(kf, km) for Co = 1 × 10−9 M, �m = 2.5 × 10−11 Mm, kr = 0.02 1/s. Form = 10−5 m/s, � is 50 times greater than �R at kf = 108 1/(Ms), whileor the same km the ratio �/�R is only slightly greater than 1 atf = 105 1/(Ms). This can be explained by the fact that the adsorp-ion flux increases with the increase of kf. Thus at higher values off the spectrum of fluctuations will be influenced by the analyteransport of such values of km that result in a negligible influencet lower values of kf (e.g. see the curve R�(kf) for km = 10−5 m/sn Fig. 5a). This means that at higher intrinsic adsorption rateonstants the mass transfer must be faster in order for its influ-nce to be negligible. Fig. 5a shows that with the increase of kf,or km = const, the ratio �/�R asymptotically approaches the maxi-

al value that equals 1 + kr�m/(Cokm). This strongest influence iseached when kf � kr/Co (kf � 2 × 107 1/(Ms) in the given case).

hen kf � kmkr/(kr�m-kmCo), the influence of the mass transfer isnsignificant. However, if km � kr�m/Co (km � 5 × 10−4 m/s in theonsidered example) the influence of mass transfer is negligibleegardles of the value of kf.

A similar conclusion arises from the analysis of the dependencef the ratio R� on both the intrinsic dissociation rate constant andhe mass transfer coefficient (Fig. 5b, obtained for Co = 5 × 10−10 M,m = 1 × 10−11 Mm, kf = 5 × 107 1/(Ms)): the greater kr, the strongerhe mass transfer influence on the spectrum of fluctuations of theumber of adsorbed particles. At the same mass transfer rate, theatio �/�R can vary for an order of magnitude, depending on thealue of kr, as shown by our calculations. The dependence of theatio �/�R on kr also has a maximum which equals 1 + kf�m/km

hen kr � kfCo = 0.025 s. The mass transfer is negligible for anyalue of kr that satisfies the expression kr � kmkfCo/(kf�m − km).n the other hand, if km � kf�m is valid, the influence of mass

ransfer is negligible for any value of kr. The diagrams in Fig. 5ave shown the combined effect of the intrinsic kinetics of thessociation–dissociation process and the mass transfer on theesulting fluctuations spectrum.

From the presented analysis we can summarize that the massransfer can have a significant influence on the spectrum of fluc-uations of the number of adsorbed molecules. This is especiallyvident in biosensors operating in liquids, where the transfer ofnalyte molecules is much slower than in gases. The analysisas also shown that the extent of the influence of mass trans-

er depends not only on the mass transfer coefficient km, but onther parameters as well (the intrinsic association and dissocia-ion rate constants, the surface density of the binding sites, thenalyte concentration in the solution). More strictly, the influenceepends on the relation between the volumetric mass transferux and the volumetric equilibrium intrinsic adsorption flux. Forhe parameter values used in the analysis, which correspond toealistic experimental values, the level of fluctuation spectruman be up to two orders of magnitude higher compared to thease of rapid mixing. Under experimental conditions different than

hose we used for numerical calculations, but also realistic (e.g.or lower km which corresponds to larger biomolecules, or forery viscous solutions, etc.), these values can be even substantiallyigher.

s B 166– 167 (2012) 535– 543

The strong dependence of the fluctuations spectrum on dif-ferent parameters of a binding–unbinding molecular process, aswe obtained, imply that new methods can be developed for ana-lyte detection and for characterization of mass transfer influencedbiomolecular interactions, based on the frequency domain analy-sis of the measured fluctuations in the sensor’s output signal. Thebinding kinetics and intrinsic rate constants can be determined byadjusting the parameters of the derived analytical expression tofit the experimentally obtained spectrum. Since AD intrinsic rateconstants are related to the particular analyte–receptor bindingpair, such methods could provide additional information about theadsorbed analyte, for example, those necessary for identificationof the analyte if several substances have the affinity for bindingto the same capturing probes. If the influence of mass transferactually exists, but is neglected during the analysis of the mea-sured fluctuations spectra, interpretation of the results will beerroneous, leading to wrong determination of analyte concentra-tion and binding kinetics and possibly to wrong identification ofthe analyte.

Because it causes decelleration of the binding kinetics andthe increase in signal fluctuations, mass transfer can be consid-ered as one of the physical limitations of biosensor performance.AD fluctuations, which are already considerable in small-scalemicro/nanofabricated sensors, become even greater when theinfluence of mass transfer exists, so the possibility for them todominate over the fluctuations generated by other fundamentalmechanisms is higher. Therefore, the limiting performance (thetotal noise and the minimal detectable signal) of biosensors areespecially affected by AD fluctuations when the binding of analyteto receptor molecules is influenced by mass transfer. By neglect-ing the mass transfer effect when it actually exists, the noise andthe minimal detectable signal would be underestimated. Obviously,the effects of mass transfer must be taken into account when thenoise and the minimal detectable signal of biosensors used in liq-uids has to be determined, and also when the analysis of measuredspectrum of fluctuations is performed in order to obtain additionalinformation about the analyte or about the biomolecular interac-tion process.

4. Conclusion

In this paper we presented a simple theory of fluctuations ofthe number of adsorbed particles in the affinity-based biologicalsensors, which takes into account the mass transfer effects. Weobtained the analytical expression for the power spectral densityof the fluctuations and formulated the criterion intended for deter-mination whether or not the influence of mass transfer of a givenflux on the fluctuations spectrum is significant.

The theory relies on assumptions that limit the scope of itsapplicability to the cases of simple bimolecular reversible bindingreactions and sensors in which analyte concentration can be welldescribed by the two-compartment model.

The derived mathematical model is a convenient tool for esti-mation of the quantitative influence of the mass transfer on thespectrum of fluctuations of the measured signal. In this paper wepresented for the first time the quantitative data about the dif-ference of the fluctuations spectrum influenced by mass transferand the spectrum not affected by it. Our numerical calculationsshowed that the mass transfer process can significantly influencethe fluctuations spectrum. Slower mass transfer results in both thehigher fluctuations magnitude and the lower knee frequency of

the spectrum curve. This implies that the performance of biosen-sors operating in liquids, where the transfer of analyte moleculesis much slower than in gases, is especially affected by the masstransfer process.

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The obtained analytical expression for the PSD of fluctuations isseful for quantitative estimation of the sensor’s AD noise in thease of analyte to receptor binding influenced by mass transfer.hus, the derived theory enables realistic estimation of the limit-ng performance of biosensors operating in liquids. It can also betilized for evaluation of the dependence of sensor performanceharacteristics (AD noise, minimal detectable signal) on differ-nt parameters, which is a prerequisite for minimization of massransfer adverse influence. Therefore, the significance of this the-ry is in its applicability for optimization of biomolecular bindingxperiments and detection methods. It provides the guidelines formprovement of the limits of detection, and it is useful for devel-pment of low-noise biosensors based on miniature transducertructures.

Both the theory and the analysis presented in this paper cane applied for development of novel highly sensitive methods forxtraction of information about the analyte and the biomolecu-ar binding process from the measured spectrum of fluctuations inhe sensor’s output signal. The presented mathematical model ofD fluctuations, which includes mass transfer effects, is a signifi-ant step towards a comprehensive theory of fluctuations, whichnables proper interpretation of the experimentally obtained fluc-uations spectra.

In our future work we will investigate the influence of non-pecific binding, surface heterogenity and surface diffusion ofolecules on the fluctuations spectrum of the measured signal in

iosensors.

cknowledgements

This work is partially funded by the Serbian Ministry of Educa-ion and Science within the project TR 32008 and the EU FP7 projectEGMINA (205533).

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Biographies

Ivana Jokic received her Dipl. Ing. degree in electrical engineering in 2000 from theUniversity of Belgrade, Serbia. She is currently a Ph.D. student of the same university.As a researcher at the IHTM-MTM institute, University of Belgrade, she participatesin the research and development of chemical and biochemical sensors based onmicro- and nanoelectromechanical systems. Her particular fields of interest are thenoise in MEMS/NEMS devices, adsorption–desorption processes, and their influenceon the ultimate performance of sensors.

Zoran Djuric received his Ph.D. in electrical engineering in 1972 from the Universityof Belgrade, Serbia, after obtaining his Dipl. Ing. degree (1964) and Mag. Sci. degree(1967), from the same university. Since 1967 he is in the field of microelectronicsand optoelectronics. His main fields of interest are microsystem and nanosystemtechnologies, including MEMS and NEMS based sensors, detectors and actuators.He is a corresponding member of the Serbian Academy of Sciences and Arts (SASA),a member of the Serbian National Academy of Engineering, and the director of theInstitute of technical sciences of SASA.

Milos Frantlovic received his Dipl. Ing. degree in electrical engineering in 2000and his Mag. Sci. degree in 2010 from the University of Belgrade, Serbia. He is cur-rently a Ph.D. student of the same university. At the IHTM-MTM institute, Universityof Belgrade, he is in charge of the research and development of intelligent elec-tronic instrumentation based on MEMS sensors. He is also involved in the researchof various types of chemical and biochemical MEMS sensors.

Katarina Radulovic received her Ph.D. in electrical engineering in 2005 from theUniversity of Belgrade, Serbia, after obtaining her Dipl. Ing. (1995) and Mag. Sci.degree (2000) from the same university. She is with the IHTM-MTM as a seniorresearch associate. She is engaged in the field of FEM based modeling, simulationand characterization of MEMS and NEMS sensors. She also participates in fabrica-tion of thin films by cathode sputtering and their characterization by infrared andphotothermal spectroscopy.

Predrag Krstajic received his Dipl. Ing. degree in electrical engineering (techni-cal physics department) from the University of Belgrade, Serbia, in 2000, and hisPh.D. in physics from the University of Antwerp, Belgium in 2006. He was a post-doctoral researcher at the University of Antwerp, Belgium, and at the ConcordiaUniversity, Canada. He is currently a researcher with the IHTM-MTM, University ofBelgrade, where he is engaged in the field of semiconductors and physics of quantumstructures.

Zorana Jokic is a M.Sc. student at the Department of Genetics, Faculty of Biol-ogy, University of Belgrade, Serbia. Her research fields of interest are biosensors,especially those based on micro/nanotechnologies, and applications of atomic forcemicroscopy in genetic research.