Correlation of Capture Efficiency with the Geometry, Transport, and

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Correlation of capture efficiency with the geometry, transport and reaction parameters in heterogeneous immunosensors Dharitri Rath, and Siddhartha Panda Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b00041 • Publication Date (Web): 14 Jan 2016 Downloaded from http://pubs.acs.org on January 19, 2016

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Correlation of capture efficiency with the geometry, transport and reaction parameters in heterogeneous immunosensors Dharitri Rath1, 2 and Siddhartha Panda1, 2, 3,* 1

2

Department of Chemical Engineering,

Centre for Environmental Sciences and Engineering, 3

Samtel Centre for Display Technologies, Indian Institute of Technology Kanpur, Kanpur - 208 016, Uttar Pradesh, India

* Corresponding author Tel: +91-512-259-6146 Fax: +91-512-259-0104 E-mail: [email protected]

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Abstract: Higher capture efficiency of biomarkers in heterogeneous immunosensors would enable early detection of diseases. Several strategies are used to improve the capture efficiency of these immunosensors including the geometry of the system along with the transport and reaction parameters. Having a prior knowledge of the behavior of the above parameters would facilitate the design of an efficient immunosensor. While the contribution of the transport and reaction parameters toward understanding of the mechanism involved in capture have been well studied in literature, their effect in combination with the geometry of the sensors has not been explored till now. In this work, we have experimentally demonstrated that the capture efficiency of the antigen-antibody systems is inversely related to the size of the sensor patch. The experimental system was simulated in order to get an in-depth understanding of the mechanism behind the experimental observation. Further, the extent of heterogeneity in the system was analyzed using the Sips isotherm to obtain the heterogeneity index ( α ) and the reaction rate constant ( K D ) as fitted parameters for a sensor patch of 1.5 mm radius. The experimental kinetic data obtained for the same sensor patch matched reasonably with the simulation results by considering K D as the global affinity constant, which indicated that our system can be considered to be homogeneous. Our simulation results associated with the size dependency of the capture efficiency were in agreement with the trends obtained in our experimental observations where an inverse relation was observed owing to the fact that the mass-transfer limitation decreases with the decrease in the size of the sensor patch. The possible underlying mechanism associated with size dependency of capture efficiency was discussed based on the time-dependent radial variation of captured antigens obtained from our simulation results. A study on the parametric variation was further 2


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conducted for the non-mixed and mixed systems on the transport ( Deff ), reaction ( K D ) and geometric parameters ( R ). Two different correlations were established for the non-mixed and t mixed systems between the capture efficiency ( f ) and a nondimensional number  D   tR 

consisting of the above mentioned parameters. Such unified relations will be useful in designing heterogeneous immunosensors, and can be extended to microfluidic immunosensors.

Keywords: Capture efficiency, sensor patch, correlation, transport, reaction.

1. Introduction: Heterogeneous immunosensors are used for the diagnosis of diseases through the detection of signature proteins or biomarkers in body fluids. In heterogeneous immunosensors, the antibodies are immobilized on the sensor surface, and the antigens are transported (through diffusion and electromigration, with added convection for flow systems) from the bulk to the surfaces where the binding reactions occur. Most of these processes are mass-transfer limited due to faster reaction kinetics.1 Along with the transport and reaction, the geometry of the sensor surface is important for the design of these sensors. An important performance parameter of these sensors is the capture efficiency which is defined as the ratio of the amount of antigens captured to the maximum available antibodies on the sensor surface. Higher capture efficiency will enable lower detection limits and thus enable early disease detection. Capture efficiency depends on the interplay between the transport and reaction parameters,2 and also the geometry of the systems. 3


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Hence, there is a need for an efficient design which considers the effect of all these parameters together namely transport, reaction and geometry of the sensors. A prior knowledge of the dependencies of the capture efficiency on these parameters would facilitate the design of an efficient immunosensor. Several strategies to improve the sensitivity of the heterogeneous immunosensors, including modification of the geometry of the systems, have been reported in the literature. Abolfath-Beygi et al. have shown that in a microfluidic surface based sensor operating in the mass-transfer limited regime, the overall capture obtained for multiple patches of smaller areas was higher than a single patch of the equivalent area,3 which signifies the importance of the miniaturization of the sensing surface as a promising design knob. Lynn et al. have shown that the capture efficiency is inversely proportional to the height of the sensing chamber of the microfluidic flow cells.4 Subsequently, they have studied the utility of various passive mixing structures to enhance the rate of transport in the microfluidic channels resulting in enhanced sensor performnace5, and further used the staggered herringbone mixer-sensors for the enhanced performance of the biosensors.6 Further, Sadeghi et al. conducted studies to show the effect of the aspect ratio of a rectangular microchannel on the concentration of the surface bound analytes, and the dependencies of capture efficiency on the Damkohler number ( Da ).7 Nair and Alam have shown the dependence of the capture with the dimensionality of the nanostructures, and suggested a scaling relationship based on a reaction-diffusion model.8 Leary et al. conducted parametric studies to obtain dependency of the surface concentration of the target on the nondimensional numbers associated with transport and reaction parameters such as Peclet number ( Pe ), and Da , and thus obtained the regimes of higher capture of analytes.9 Moreover, role of the size of the sensor surfaces was specifically studied for microarray-based techniques, 4


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showing the significance of miniaturization to improve the capture of analytes.10,

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Ekins and

Chu performed simulations to show that the efficiency increased and the saturation time decreased with decrease in the size of the micro-spot.12 Peterson et al. performed experiments on the DNA hybridization efficiency, and concluded that both the capture efficiency and the kinetics of capture of targets scale as inverse of the surface density of the probe molecules.13 Dandy et al. investigated the effect of the feature size of the sensor on the DNA hybridization efficiency, and showed that the radial distributions of capture molecules have a crucial role which needs to be considered to explain the inverse dependence of capture efficiency on the spot size.14 However to the best of our knowledge, a detailed exploration of the influence of a combined effect of the geometrical parameter (size of sensor patch) and the transport and reaction parameters for the enhancement in the capture of antigens has not been investigated. In the current study, the effect of the size of the sensor patch (defined as the surface of the sensor where the reaction occurs) on the capture efficiency in case of a non-flow heterogeneous immunosensor was studied for the detection of PSA molecules. The amount of capture of PSA was quantified experimentally for the circular sensor patches of different radii. The mechanism of the capture was investigated through simulating the system to obtain the capture efficiencies. Further, the parametric variations were obtained for the transport parameter (diffusivity - Deff ), the reaction parameter (dissociation rate constant - K D ) and the geometry (radius of the sensor patch - R ) in case of both non-mixed and mixed systems. Further, the scaling analysis was performed to obtain the dependency of the capture efficiency on the transport, reaction and geometric parameters of the system using correlations for theses antigen-antibody systems. This

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study can be extended to the design of flow systems in microfluidic immunosensors which is a future scope of this work.

2. Experimental Section: 2.1.

Chemicals:

The chemicals used for the silicon processing such as 3-aminopropyltriethoxysilane (APTES), glutaraldehyde, phosphate buffer saline (PBS), protein A, milk protein casein and Tween 20 were purchased from different sources as mentioned in our previous works.15, 16 Prostate specific antigen (PSA) was obtained from Fitzgerald Industries International, USA, anti-PSA IgG, and HRP-tagged anti-PSA antibodies from Genetech Laboratory, Biotech. Park, Lucknow, India. DAB kit, and FITC-tagged anti-PSA were purchased from GeNie™, Bangalore, India. 2.2.

Experimental methodology:

Silicon wafers were cut into circles of six different radii- 0.5, 1, 1.5, 2, 2.5 and 3 mm using a laser system (Nd: YAG Laser, Laser Lab India Pvt. Ltd.). Silicon processing, and the preparation of the different stacks upto the antibody immobilization followed by immobilization of antigens were performed following our previous methodology detailed elsewhere.15 FITC-tagged antiPSA molecules were used as the capture antibodies, and the fluorescence images were taken using a Fluorescence Microscope (Leica Microsystems, Germany). The quantification of the number of antigens immobilized was performed using an HRP enzyme tagged colorimetric method: in this experiment, enzyme (HRP) tagged anti-PSA IgG was used as the standard to obtain a linear fit of the calibration curves having R2 ≥ 0.99 with known concentrations. The 6


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antigens were allowed to react with the detection anti-PSA antibodies in a non-mixed system which were conjugated with HRP. The amount of captured PSA was quantified using the enzyme- substrate reaction (with DAB as the substrate), and measured in an enzyme-linked immunosorbent assay (ELISA) reader (Thermo Scientific Multiskan® EX, Vantaa, Finland). This gave the total amount of PSA captured by each of the sensor surfaces which was used to calculate the capture efficiencies. Experiments were conducted in triplicate and the error bars represent one standard deviation about the mean. 2.3.

Simulation technique:

The heterogeneous immunoassay for the detection of PSA has been depicted schematically in Figure 1. The process of capture of PSA is governed by a system of partial and ordinary differential equations (PDEs/ODEs).

Figure 1: Schematic of the transport and reaction process for the capture of PSA. The coordinate system is provided in the figure.

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The transport of PSA from the bulk solution to the sensor patch placed at the bottom of the beaker can be described by equation 1.

∂C = Deff ∂t

 1 ∂  ∂C  ∂ 2C  vθ ∂C  r ∂r  r ∂r  + ∂z 2  − r ∂θ    

(1)

where C is the concentration of analytes at time t and Deff is the effective diffusivity of antigens in solution, vθ = ω × r , is the θ component of velocity. ω = 0 for all non-mixed cases (discussed in sections 3.1.2, 3.2.1 and 3.2.2) and ω = 500 rpm for the mixed cases (discussed in sections 3.2.3 and 3.2.4). The initial condition (IC) for equation 1 is given by equation 2.

C ( z, r , t = 0) = Cb

(2)

where Cb is the initial bulk concentration of antigens. The boundary conditions (BCs) for equation 1 are given by equations 3, 4, 5 and 6.

∂C ( z = 0, r , t )

= K onC ( z = 0, r , t )(θ max − θ t ) − K off θt

(3)

=0

(4)

Deff

∂C ( z , r = R, t ) =0 ∂r

(5)

Deff

∂C ( z , r = 0, t ) =0 ∂r

(6)

Deff

Deff

∂z ∂C ( z = h, r , t ) ∂z

where h is the height of the reaction chamber, R is the radius of the sensor patch, and K on and K off are the forward and reverse rate constants respectively. After reaching the sensor patch, the

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PSA molecules react with anti-PSA molecules following a 1st order reversible kinetics given in equation 7, dθt = K onC ( z = 0, r , t )(θ max − θt ) − K off θt dt

(7)

where θ t is the surface density of antigen-antibody bound complex at time t , and θ max is the maximum surface density of the immobilized antibodies. The IC for equation 7 is given by equation 8.

θt (r , t = 0) = 0

(8)

The above equations were solved in COMSOL multiphysics software using the finite element method, utilizing two application modules: “transport of diluted species” module and “surface reaction” module, using the “PARDISO” solver in the minimal residual iterative mode.

3. Results and Discussions: 3.1.

Experimental results: 3.1.1. Distribution of anti-PSA antibodies:

Figure 2 shows a fluorescence micrograph of the distribution of anti-PSA antibodies (capture antibodies) on a sensor surface of 1 mm radius. Such images were taken to verify the process of immobilization of the FITC tagged anti-PSA antibodies on the surface of the sensor patch. We could observe the fluorescence signals obtained on the sensor surface which verifies the immobilization of the capture antibodies and the distribution can be considered to be uniform here. Similar observations were obtained with patches of smaller and higher sizes (data not 9


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shown). This observation is similar to that reported in our previous work 15. This is attributed to the method of immobilization, and the chemistry of ligands used to attach the capture antibodies.

Figure 2: Fluorescence micrograph of the immobilized antibodies on the sensor patch of 1 mm radius.

3.1.2. Dependency of capture efficiency on size: The capture efficiencies ( f ) of the circular sensor patches for the detection of PSA molecules are defined as: r

∫ 2πrθ (r )dr t

f =

0

(9)

θ maxπr 2

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Experimentally, the capture efficiency was obtained by taking the ratio of the total moles of PSA captured to the total moles of immobilized antibodies on the reaction surface, i.e. ( θ max πr 2 ) for a non-mixed system. Figure 3 shows the capture efficiency as a function of size of the sensor patch (ranging from 0.5 to 3 mm). It is observed that the capture efficiency decreases with increasing size of the reaction patch. A maximum efficiency of ~ 40% was obtained for the 0.5 mm patch whereas a minimum efficiency of ~11% was obtained for the 3 mm patch. A similar observation of enhanced f with decreased size was reported earlier for DNA hybridization used in the microarray technology.14 Thus the above result validates our work in case of immunosensors to detect PSA. Such an enhancement in capture efficiency due to miniaturization is desirable, and warrants further understanding of the mechanism in detail. Hence we investigated the effect of various transport and reaction parameters along with the size of the sensor patch in order to understand the role of each of the parameters towards the overall capture process. In particular, we have simulated the transport and surface reaction of the PSA molecules using the species balance and surface reaction equations, described in detail in the following section.

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Figure 3: Capture efficiency obtained experimentally as a function of the size of the reaction patch for a non-mixed system.

3.2.

Heterogeneity analysis:

In order to investigate the heterogeneities associated with the surface reaction on the sensor patches, which may arise due to various possible orientations of the linker molecules or the capture antibodies, we have used the Sips isotherm17, 18 and conducted experiments to quantify the capture efficiency with varying antigen concentrations for a particular case of the sensor patch having a radius of 1.5 mm. Figure 4a shows the values of f as a function of the bulk concentration of antigens. We fitted the experimental data according the equation 1019 α

f

( Cb / K D ) = α 1 + ( Cb / K D )

(10)

where the exponent α , termed the “heterogeneity index”20 which represents the extent of heterogeneity in the system was calculated to be 0.85. Another important parameter in the equation is K D , which is the average reaction constant for the distribution of various types of affinities possible for the antigen-antibody interactions. We obtained the value of K D = 1.27 x 10-7 M with R 2 = 0.99 from fitting with the Sips model. In literature, there are two different approaches of simulating the affinity-based surface reactions- one where a homogeneous system with a single value of the reaction constant is used for global fitting of the data, and the other which accounts for the heterogeneity of the receptor molecules wherein a distribution of the reaction constants is taken into consideration. The first

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method has been used by several authors without mentioning the heterogeneity of the system, and the Langmuir model is used to describe the surface reaction in their systems21, 22. The second approach has been used by Vijayendran et al.19, 20, who determined the extent of heterogeneity in various types of immobilization schemes used for the capture molecules (with values of α in the range 0.07-0.99), and Schuck group who considered the surface reaction on the sensor surface having unknown distribution of the surface sites along with the mass transport limitation, but did not mention the heterogeneity index for their systems

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. For our experimental system, we

assumed the sensor surfaces to be homogeneous, and used the fitted K D value representing an average value for the distribution of the surface sites. To validate this assumption, we conducted experiments to obtain the kinetics of the antigen-antibody interaction. Figure 4b shows the plot for the experimental kinetic data of the capture ( θt ) of antigens for a patch of 1.5 mm radius, and the simulated curve obtained by solving the species balance and the surface reaction equations (1-8), where the K D value was 1.27 x 10-7 M (fitted parameter from Figure 4a). It can be observed from the figure that our experimental and simulation results are reasonably in good agreement ( R 2 = 0.97 ), and this assumption can be utilized for estimating the capture efficiency for these systems and has been used for rest of simulation studies and scaling analysis discussed in section 3.3. However, using the complete distribution of reaction constants is beyond the scope of the present work.

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Figure 4: (a) Equilibrium binding isotherm for the interaction of PSA and anti-PSA, with a least square fitting to the Sips model (solid line) for the estimation of α and K D (b) Kinetic data for the same interaction taking the average value of K D , for the sensor patches of 1.5 mm radius, matched with the simulation results.

3.3.

Simulation results:

In this section, we simulated the species balance and the surface reaction equations (1-8) in order to explain our experimental findings associated with the enhancement in the overall capture efficiency with decrease in the patch size. We started with the radial variation of the amount of PSA captured for patches of different sizes, and used this to calculate the corresponding overall capture efficiencies for a non-mixed system, as discussed in sections 3.2.1 and 3.2.2. To get more physical insight about the relative effects of transport, reaction and geometric factors, we conducted studies on parametric variations in section 3.2.3 for both non-mixed and mixed 14


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systems. Further, we performed scaling analysis in order to get correlations between the capture efficiency and a dimensionless number for both non-mixed and mixed systems, which incorporate all the parameters associated with the above mentioned factors; and this is described in section 3.2.4.

3.3.1. Temporal and spatial dependence of the captured PSA: Figure 5 shows the radial variation of the amount of PSA molecules captured on a sensor patch of 0.5 mm by the immobilized anti-PSA antibodies at six different reaction times for a nonmixed system. It can be observed that the concentration of the captured antigens increases monotonically from the center to the periphery of the circular reaction patch for all cases. The common feature with all curves is that the radial concentration gradient or radial mass

 ∂C  flux,   , is sharper at the periphery, and decreases in the radial inward direction and becomes  ∂r  negligible near the center (as indicated by the flat concentration profile of the captured antigens). Furthermore, the region having lesser gradient extends upto a larger length as time progresses. Both observations suggest that as time progresses, more and more reaction sites are occupied, and thus the molecules reaching further to these occupied sites remain unreacted and get accumulated over the sensor patch. Hence gradually the radial concentration profile becomes less and less shaper as time progresses.

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Figure 5: Radial variation of the amount of antigen molecules captured on a sensor patch of 0.5 mm radius at t = 100s, 300s, 500s, 700s, 900s and 1000s (from bottom to top) for a non-mixed system. The radial variation of the captured antigens at different times shown in Figure 5, is attributed to the mass transfer limitation, and a proposed explanation for this has been depicted by the schematic in Figure 6. The circular patch shows the bottom of the reaction chamber where the sensor patch has been placed. The inner blue colored patch is the reaction patch placed over the bottom of the non-reacting surface of the reaction chamber. Initially, the antigen molecules are present all over the reaction chamber, which are transported onto the sensor patch where the capture antibodies are immobilized. The transport of antigens can occur in both the normal ( nˆ ) and the radial ( rˆ ) directions; thus we have two fluxes - in the normal and the radial directions. It is noticeable that the radial concentration profile shown is attributed to the fact that the reaction sites are located at different positions radially; hence molecules have to travel different radial distances in order to reach the various reaction sites, which leads to radial dependency of the radial diffusional flux. As the concentration at any place affects the concentration gradient in all 16


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directions, this leads to radial dependence of the normal mass flux. To understand the above mentioned fact associated with normal mass flux, let us consider a case wherein diffusion occurs only in the normal direction, and radial diffusion is completely absent. In this case all the traveling molecules see identical environment throughout the surface, hence the normal flux should not show radial variation, and it will only show variation in the normal direction. Moreover, since the reaction sites at the periphery of the patch are at the closest approach to molecules travelling radially inward, the concentration gradient is the sharpest at the periphery, which becomes weaker towards the center. As a result, we obtain a radial variation of the amount of captured PSA molecules.

Figure 6: Schematic showing the transport of PSA molecules onto the sensor patch.

3.3.2. Dependency of capture efficiency on size and θ max : The effect of the size of the sensor patch on the capture of PSA molecules was quantified from the radial profile of θ t obtained from the simulation results, and these data were used to calculate

f using equation 9 for a non-mixed system. Figure 7 shows the capture efficiency as a function of size of the sensor patch with radii ranging from 0.5 mm to 3 mm, keeping other parameters constant. The plot shows that the capture efficiency decreases with increase in size of the 17


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reaction patch similar to the experimental results reported in Figure 3. We obtained the maximum efficiency (~ 32%) for the smallest patch, and the lowest efficiency (~ 17%) for the largest patch. This observation is in agreement with that of the previous reports for the amount of DNA hybridization used in the microarray technology.14 However, the system used in the previous study was DNA-DNA hybridization for microarray technology, and here we have demonstrated similar concept for the variation of the size of the sensor patch on the capture efficiency of the heterogeneous immunoassay used for the detection of PSA molecules. Further, we have obtained the dependency of the capture efficiencies with the size, transport and reaction parameters which has not been reported elsewhere. To the best of our knowledge, this is first demonstration for the detection of PSA, which validates the previous results obtained with the microarray sensors.

Figure 7: Capture efficiency as a function of the size of the sensor patch obtained from simulation results for a non-mixed system.

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We have also conducted a study on the effect of the maximum available capture molecules (antibodies in this case) on the sensor surface by varying θ max . This study has relevance in case of the use of mixture of the receptor molecules on the sensor surfaces pertinence in multianalyte detection systems

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, for example the detection of multianalytes in a

checkerboard pattern27. The use more than one type of receptor molecule for multi-analyte detection has been incorporated in our simulations in terms of the decrease in θ max for the given antibody. The rationale for assumption is that the pairs of antigen and antibody used in the immunosensors are highly specific to each other and rarely has cross-reactivity. Hence we have obtained the capture efficiencies for the cases of 25%, 50% and 75% of θ max to compare the efficiencies with that of the 100% case, and have plotted the values in Figure 8 for these four cases. It can be observed that the value of f increases with increasing θ max which is attributed to the higher number of the capture antibodies available for the reaction with the incoming antigens.

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Figure 8: Capture efficiency as a function of the % of θ max on the sensor patch obtained from simulation results for a non-mixed system.

3.3.3. Parametric variations: In order to understand the mechanism responsible for the trends obtained for the capture efficiencies with the radial size, we further conducted parametric studies. The most relevant parameters to further characterize the dependence of f on size of the sensor patch are the transport and reaction parameters – dissociation rate constant ( K D ) and effective diffusivity ( Deff ) respectively, which are discussed in detail in this section for both non-mixed and mixed systems.

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Effect of KD: Figure 9 shows the capture efficiencies as a function of radius of the sensor patch for different values of K D at a fixed values of Deff (= 3.2x 10-10 m2/s), for non-mixed case (figure 9a), and for a mixed system with a constant mixing speed (figure 9b) of 500 rpm. The choice of the values of the parameters used was based on our previous experimental study as well as values reported in literature for antigen-antibody systems16, 28. A mixing speed of 500 rpm was chosen based on our previous experiments where the capture of antigens could be enhanced by using different mixing speeds2, 16. Other mixing speeds could also have been used, but, we have considered only one speed as an exemplary case. It can be observed from the Figure 9a that for a given radial distance, the capture efficiencies decrease with increase in K D . Moreover, a values of f obtained for a lower value of R and a higher value of K D can also be obtained at higher R and lower K D (within the range of various parameters explored in this study). For example, we can obtain the same f = 0.479 for two combinations of R = 1 mm, K D = 8 x 10-6 M, and R = 1.5 mm, K D = 1 x 10-7 M which is indicated by a horizontal line in Figure 9a. The above trend is attributed to the fact that higher K D implies a lesser rate of reaction, and a lesser rate of reaction gets compensated by enhanced mass transport for the smaller size of the sensor patch thereby resulting in higher amount of capture. The above observation signifies that we can get a combination of geometry and reaction parameters to obtain a particular value of f by knowing the dependencies prior to performing the actual experiments. As mentioned, we have also obtained the capture efficiencies for the sensor patches for four values of K D where we introduced a uniform mixing in the system, depicted in Figure 9b. It is observed from the figure

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that the trends of variation of capture efficiencies with the size of the sensor patch are same in both the non-mixed and the mixed systems with enhanced capture for the mixed system.

Figure 9: Capture efficiency as a function of size of the sensor patch for different values of KD (M) (where Deff =3.2 x 10-10 m2/s) for (a) non-mixed and (b) mixed ( ω = 500 rpm) system. The horizontal line in (a) indicates that the same capture efficiency can be obtained for various combination of R and K D . Moreover, it can also be observed that the effect of K D is more pronounced for smaller values of radius as compared to the higher values. In order to explain this observation, we consider two time scales: the reactive time scale ( t R ) and the diffusive time scale ( t D ), which are associated with the two competing phenomena - reaction and transport respectively. The reactive time scale is obtained assuming a quasi-steady state and is given by22: t R = (k on C + k off ) . The −1

2 diffusive time scale is given by: t D = R

D

. As there is an experimental observation of radial

dependency of the captured antigens (shown in figure 5), the patch radius is considered as the 22


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characteristic length. The values of tR vary from ~ 5 s to ~ 45 s for the values of K D used. For any given curve in Figure 9, tR is constant (as K D is constant), while t D varies from ~ 830 s for the 0.5 mm patch to ~ 30,000 s for the 3 mm patch. As t D is lesser for smaller patches, the limitations due to transport of analytes are relatively less dominant and the analytes can reach the reaction sites faster. Hence the reaction rate is enhanced resulting in higher capture of PSA molecules, and a pronounced difference in the values of the capture efficiencies is observed. Effect of the reaction constants have been reported in several works earlier29, 30, however the combined effects of reaction constant and the size of sensor patch was not explored till now.

Effect of Deff: Figure 10 shows the plot of the capture efficiencies as a function of the size of the reaction patches for different values of diffusivity, and a constant value of K D (= 4 x 10-6 M), for nonmixed case (figure 10a), and for a mixed case with a constant mixing speed (figure 10b) of 500 rpm. The choice of the values of Deff was based on our previous experimental study for antigenantibody system.31 It can be observed from the figure that for the sensor patch of a given radius, the capture efficiency increases monotonically with Deff for all patches irrespective of their sizes. Moreover the decrease in the capture efficiency with increasing size of the sensor patch progressively becomes lesser as Deff decreases. This behavior is attributed to the fact that the decrease in Deff results in lesser rate of transport of antigens, leading to the reduction of the capture efficiency. Figure 10b shows the plot of f vs R for four values of K D , and it can be observed that the trend of variation of capture efficiencies is similar to that observed in case of the non-mixed systems. Higher values of capture efficiencies were obtained with mixing which 23


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can be attributed to facilitation of analyte transport. Though there are reports in literature describing the effect of diffusive transport on the capture of analytes,

9, 32, 33

to the best of our

knowledge, the combined effect of both geometric and transport parameters were not explored till now.

Figure 10: Capture efficiency as a function of the size of the sensor patch for different values of Deff (m2/s) (where K D = 4 x 10-6 M) for (a) non-mixed and (b) mixed ( ω = 500 rpm) system.

Moreover, it can also be observed that the effect of variation of Deff is more pronounced in case of the patches of lower radius (0.5 mm) as compared to the higher ones (3 mm). In order to explain it, we again calculated t D and t R . In the current plot, since K D is fixed for all curves,

tR ~ 40 s for all cases, whereas t D for sensor patches having a radius of 0.5 mm ranges from ~ 7.8 x 103 s ( Deff = 32 x 10-11 m2/s) to ~ 2.5 x 104 s ( Deff = 1 x 10-11 m2/s), and for a radius of 3 mm, it ranges from 2.8 x 105 s to 9 x 105 s. So, for the 3 mm patch, t D is 3 - 4 orders of magnitude higher than tR , and the extent of enhancement in capture efficiency is correspondingly lesser as compared to that of the 0.5 mm patch. The above values indicate that 24


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for the sensor patches of smaller size, effect of variation of rate of transport ( Deff ) becomes more pronounced, which is attributed to the relatively lesser mass transfer limitations as compared to the patches of larger radii whereas for the patches of larger sizes, values of t D become significantly high.

3.3.4. Scaling laws: The results obtained for the variation of the transport and the reaction parameters as seen in the previous section indicate that there is a possibility of expressing the capture efficiency as a function of Deff , K D , and the radius of the sensor patch. As expected from the expression of t D , the efficiency obtained with a larger patch for a higher value of Deff can also be obtained for a smaller patch for a lower value of Deff , as observed in Figure 10. This can be attributed to the mass transfer limitations due to lower diffusivity, and can be counterbalanced by taking sensors of smaller sizes. Similarly, the mass-transfer limitation obtained in larger patches can be mitigated by increasing the diffusivity. Similarly, when we considered the effect of K D and r on the capture efficiency for various sensor patches for a fixed Deff (Figure 9), the capture efficiency obtained at a lower value of R and a higher value of K D can also be obtained at higher R and lower K D . The above observation can be explained by considering the competition between reaction kinetics and rate of transport. Since a higher value of K D corresponds to slower reaction rate, hence lesser amount of capture due to a slower reaction can be compensated by decreasing the patch size, which will lead to a larger mass flux or lesser mass transfer resistance. 25


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Similarly, loss in capture efficiency due to mass-transfer limitation in the case of a larger patch can be compensated by enhancing the reaction kinetics. These important observations lead us to believe that the capture efficiencies in these systems could be scaled using the ratio of the two time scales, and we attempted to obtained the capture efficiencies with variation of Deff , K D and the size of the sensors patches. Hence, we combined all the parameters, and plotted the capture efficiency obtained for various cases as a function of

tD

tR

, shown in Figure 11, which led to

collapse of all data points onto a single universal master curve, for the non-mixed (11a) and the mixed system (11b), using which the capture efficiency can be determined for any set of parameters. The solid line in Figure 11a shows a power law fit: f ∝  t D   tR  t shows f ∝  D   tR 

−0.085

−0.3035

and in Figure 11b

. Similar correlations can be developed for other values of ω by

performing simulations to obtain the dependencies of f with the transport, reaction and geometry of the system. The earlier reports in literature were based on the analytical solutions where some dependencies on either the geometry of the sensor chamber or the relevant nondimensional parameters have been shown. For example, the capture efficiency was shown to

(

be proportional to the height ( h ) of the sensing chamber as4 ∝ ( h )

−2

3

) , the ratio ( ϕ ) of the

height of passive mixing structure to that of the channel microarray sensors as5 (1 − ϕ )

−2

3

. The

effects of the relevant nondimensional numbers (such as Pe , Da ) on the mass transfer of the target analyte to the reaction surface have been exploited recently in detail, and the regimes of 26


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higher capture have been identified.9 However, to the best of our knowledge no relation is reported so far combining the transport and reaction parameters with the geometry. Here we have obtained a correlation unifying the effects of Deff , K D and size of the sensor patch. It is noted in particular that the above observations are obtained by solving the governing equations for sensor patches of sizes in the range of micrometers to millimeters, and thus this analysis may not be applicable for systems having nano-electrodes spaced apart

34

. Although these exponents are

system specific, such analysis can also be implemented for other immunosensors in order to establish the correlation between capture efficiency, and the various parameters associated with transport, reaction and the geometry of the reacting surface.

Figure 11: Capture efficiency as a function of a nondimensional number  

tD

 , obtained by t R 

combining the parameters- R , Deff and K D for (a) the non-mixed and (b) the mixed ( ω = 500 rpm) systems respectively.

27


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4. Conclusions: In the present work, we have quantified the capture efficiencies of PSA molecules as a function of the radial size of the sensor patch in the heterogeneous immunosensor. We experimentally obtained a monotonic increase in the capture efficiency with decreasing size. We obtained a heterogeneity index of 0.85, and the reaction constant value of 1.27x10-7 M for experiments conducted using a sensor patch with a radius of 1.5 mm. Though the α value indicated some heterogeneity in the system, we obtained a reasonable good match of the kinetic data obtained from experiments with that of the simulation results assuming a homogeneous distribution of the affinity constant with an average K D = 1.27 x 10-7 M for that particular sensor patch. We further investigated the mechanism of capture through numerical simulations assuming the average K D values, and obtained a radial variation in the capture efficiencies in all the different sizes of the sensor patches. Furthermore, the dependency of the capture efficiency on the transport ( Deff ) , reaction ( K D ) and geometric parameters

( R)

were measured. This dependency was used to

obtain unified correlations between the capture efficiency and the above mentioned parameters for both the non-mixed and the mixed systems, which are important in governing the mechanism for the antigen-antibody interactions. The correlation obtained for the non-mixed system is given by f ∝  t D   tR 

−0.3035

t , and that for the mixed system at 500 rpm is given by f ∝  D   tR 

−0.085

; and

similar correlations could be developed for other values of ω . These correlations are useful in the design of heterogeneous immunosensors. The findings can be incorporated to enhance capture efficiencies of the microfluidic immunosensors as a future scope of this study.

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Acknowledgement: The authors gratefully acknowledge the financial support of the DST Science and Engineering Research Board, India (Grant No. SB/S3/CE/055/2013).

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“Table of Contents Graphic” Title: Correlation of capture efficiency with the geometry, transport and reaction parameters in heterogeneous immunosensors

Author list: Dharitri Rath1, 2 and Siddhartha Panda1, 2, 3,* 1

2

Department of Chemical Engineering,

Centre for Environmental Sciences and Engineering, 3

Samtel Centre for Display Technologies, Indian Institute of Technology Kanpur, Kanpur - 208 016, Uttar Pradesh, India

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Correlation of Capture Efficiency with the Geometry, Transport, and

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