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Molecular & Cellular Proteomics 5:1681-1696, 2006.
© 2006 by The American Society for Biochemistry and Molecular Biology, Inc.
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From the Divisions of Functional Genome Analysis and ¶ Biophysics of Macromolecules, Deutsches Krebsforschungszentrum, Im Neuenheimer Feld 580, D-69120 Heidelberg, Germany, || Advalytix AG, Eugen-Sänger-Ring 4, Gewerbegebiet Brunnthal Nord, D-85649 Brunnthal, Germany, ** ConvaTec, Deeside Industrial Park, Deeside, Flintshire CH5 2NU, United Kingdom, and Institute of Physics, University of Augsburg, Universitätsstr. 1, D-86135 Augsburg, Germany
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ABSTRACT |
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 TOP ABSTRACT EXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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380 well characterized antibodies (11) and Zeptosens, protein microarrays enabling detection to 10 pg/ml analyte in serum (12). However, this situation is inconsistent with the performance predicted by ambient analyte theory (13, 14). The ready availability of standard scanner resolutions of 0.1 and less Cy molecules/µm2, antibodies with a picomolar affinity (10–9–10–11 m), and typical density of binding sites in the range of 105/µm2 should ideally allow achievement of attomolar sensitivity even with simple detection approaches. Nevertheless to the best of our knowledge such a high sensitivity has not been demonstrated thus far without using expensive amplification techniques.
In addition to some known reasons such as low stability of antibody molecules, strong background signal due to inevitable protein adsorption on all kinds of surfaces, or even insufficient sensitivity of the detection approach itself, we have demonstrated recently a strong domination of the mass transport constraints in the reaction kinetics on examples of a typical antibody microspot assay (15–17). Due to a small binding area, the microspot kinetics depends intrinsically on the analyte concentration. Therefore, even if ideal mass transport-independent incubation conditions could be achieved, the reaction on a microspot may still require many hours, or even tens of hours, to reach the thermodynamic equilibrium at relatively low analyte concentrations (L0), i.e. at L0 << Kd where Kd is the binding affinity constant in m. In contrast to this, the reaction may be accomplished relatively fast at L0 >> Kd (few minutes or seconds). To enable analysis of the mass transport dependence of binding reactions on microspots, a mathematical tool was developed in our previous study (15). This tool is based on the so-called two-compartment model (TCM), which is widely used for interaction studies using the Biacore instruments (18–20). TCM dissects the mass transport-dependent binding into two steps: (i) transport of the analyte from the bulk compartment to the reaction area (reaction compartment) and (ii) the subsequent binding process. Using this model, the overall reaction rate in the antibody microspots was found to be strongly impaired by the mass transport constraints, being capable of prolonging the times required for a solution reaction by many orders of magnitude. As a consequence of this, the saturation of signal intensity on a highly affine antibody spot can be achieved only after unrealistically long incubation of tens, hundreds, or even thousands of hours (15, 16).
In contrast to other kinetically relevant effects, e.g. related to antibody immobilization (partial denaturation, heterogeneous affinity, steric hindrances for binding, etc.), the strong mass transport limitation of the reaction kinetics seems to be the primary physicochemical shortcoming of the current microarray technology. Therefore, the kinetically relevant optimization of microarray parameters aimed to reduce mass transport limitations has to be one of the first and most important considerations in protein microspot assay design. We carried out a kinetic analysis experimentally and using a modified TCM theory to assess some basic design characteristics of a microarray immunoassay such as incubation geometry, spot size, stirring, and binding site density on a spot. Additionally we investigated possible reaction mechanisms underlying the signal development on the antibody microspot and tried to understand the physicochemical nature of several experimental microarray characteristics. Finally by optimizing a conventional assay format with a simple signal generation and detection approach, we achieved am and low to mid-fM sensitivities for several labeled antigens. Moreover we were able to detect a group of low abundance cytokines in the blood plasma with very high signal-to-noise ratios.
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EXPERIMENTAL PROCEDURES |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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(IFNG), monoclonal anti-human interferon- antibody (anti-IFNG), keyhole limpet hemocyanin (KLH), and affinity-isolated anti-hemocyanin antibody (anti-KLH) were obtained from Sigma. Thyroglobulin (TG) and monoclonal anti-thyroglobulin antibody (anti-TG) were obtained from HyTest Ltd. (Turku, Finland). For profiling experiments, anti-IL1A, anti-IL1B, anti-IL2, anti-IL4, anti-IL6, anti-IL8, anti-IL10, anti-IL12B, IL15, anti-interferon-
, anti-IFNG, anti-tumor necrosis factor , anti-transforming growth factor ß1, anti-granulocyte-macrophage colony-stimulating factor 2, anti-vascular endothelial growth factor, and anti-fibroblast growth factor 2 were obtained from Acris Antibodies (Hiddenhausen, Germany).
Fabrication of Antibody Arrays—
Homemade epoxysilanized slides were manufactured according to the following protocol. Untreated slides were washed with ethanol, then etched overnight by immersion in 10% NaOH, cleaned by sonicating for 15 min, rinsed four times in water, washed twice in ethanol, and derivatized in a 100% GPTS solution at room temperature for 3 h. After silanization, GPTS-treated slides were washed thoroughly with dichloroethane and dried with N2. The reason for usage of the homemade slides was their lower variability in comparison with the different batches of commercial epoxysilane-coated slides. PBS buffer supplemented with 0.5% trehalose was used as a spotting buffer (21). The antibodies were spotted using an SDDC-2 Micro-Arrayer from Engineering Services Inc. (Toronto, Canada) and SMP15, SMP10, SMP3, and SMP2 pins (TeleChem). The slides were spotted with antibody concentrations of 2 mg/ml in spotting solution if not stated otherwise. It was initially found that different pins produce spots with similar antibody density at this antibody concentration (variation less then 15%; data not shown). The slides for all kinetics experiments were produced so that only one spot could centrally be positioned in every incubation chamber. After spotting the slides were incubated at 4 °C overnight and subsequently blocked for 3 h at room temperature in PBST (PBS with 0.005% Tween 20) supplemented with 4% milk powder. The slides for profiling experiments were spotted with 2 mg/ml antibody in spotting solution. So-called FullArea chips were made by pipetting 30 µl of 150 µg/ml anti-IFNG in spotting buffer on the bottom of a Flexiperm well (Sigma). The Flexiperms were of 3.3-mm well radius and 10-mm height and fixed on the slide surface using double adhesive tape. FullArea chips, where the bottom of the incubation chamber is completely coated by antibodies, were designed to be comparable to the classical ELISA incubation geometry. Incubation and blocking was done as described above.
Antigen Labeling and Incubation—
Antigen solution of 1 mg/ml was labeled with the monofunctional NHS ester of Cy3 dye (Amersham Biosciences) as recommended by the manufacturer. Unreacted dye was blocked from further reaction by adding hydroxylamine to a final concentration of 1 m and separated from the labeled proteins by PD-10 columns (SephadexTM G-25, Amersham Biosciences).
The slides before incubation were first rinsed for a few minutes in PBS and dried by centrifugation to remove an aqueous film on the slide surface that may potentially prolong and modify the reaction kinetics due to the additional diffusion time through this film. Incubation of the microarrays with antigens always occurred in Flexiperms and with the incubation volume of 100 µl (PBS, 4% skim milk, and 0.01% sodium azide) unless stated otherwise. For stirring we usually used the SlideBooster (Advalytix, Brunnthal, Germany), which allows for the simultaneous incubation of up to 12 slides with high agitation efficiency so that e.g. up to 48 reactions in Flexiperm format could be processed in parallel. The surface acoustic wave mixing technology of the instrument has no dead volume and allows for the agitation of various reaction geometries (cover glass, microtiter plate well, or open drop) making it well suited for our investigations (22). The SlideBooster agitation parameters were optimized as indicated previously (15). To test the suitability of the incubation geometry, the following incubation chambers were also used: "slidebox" (chamber size, 15 x 26 x 80 mm), EasySeal with dimensions 16 x 28 x 0.5 mm (ThermoHybaid Ltd., Ashford, UK), and Flexiperm as mentioned above. In contrast to EasySeals and Flexiperms, slideboxes were incubated on a shaker. To avoid potential photobleaching effects especially at long incubation times, all incubation chambers were isolated from light. Incubations were performed under stirring and non-stirring conditions. After the incubation, the slides were rinsed several times with PBST.
For profiling experiments, 1 mg of serum was labeled with 100 µg of biotin-PEG4-NHS (Quanta-Biodesign), and the unreacted reagent was separated from the proteins using Microcon centrifugal units (Millipore, Schwalbach, Germany). After 14 h of incubation with 300 µl/well, the slides were washed two times for 5 min each with PBST and incubated for 30 min with 100 nm ExtrAvidin (Sigma) labeled with Dy647-NHS (Dyomics) in the SlideBooster. Finally the slides were washed five times (5 min each) with PBST.
Scanning and Data Analysis—
Fluorescence signals were recorded using a ScanArray5000 unit (Instrument Co.) and analyzed with the GenePix software package (Axon Instruments). The results were stored and managed in an appropriate Microsoft Access database. All data points in this work represent an average of three to six individual measurements obtained.
The development of the microspot intensities over time was analyzed using the analytical solution of two-compartment theory (18–20) as described previously (15),
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where S
= SmaxL0/(L0 + Kd), S and Smax are the maximum steady state, and current and maximally attainable signal intensities, respectively, all three parameters expressed in signal units (SU); Kd = k–/k+ where k+ and k– are the association and dissociation rate constants (m–1 s–1 and s–1, respectively); L0 is the initial analyte concentration (m); and t is the time in seconds. W(x) in Equation 1 is the Lambert special function defined as the solution of the equation W(x)exp(W(x)) = x (23), c = aexp(a) where a = k+L0Smax/(k–Smax + Km(Kd + L0)) is a dimensionless parameter that measures the deviation of the kinetics in Equation 1 from an ideal single exponential kinetics, and 
= (k– + k+L0)/(1 + k–Smax/Km(Kd + L0)) is the rate of approaching the steady state where Km is the phenomenological mass transport constant (SU x m–1 s–1). Alternatively the experimental data can be fitted to the following equation.
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Equation 1 can be obtained by inversion of Equation 2 and vice versa. To describe the initial development of signal intensity (S(t) << S), one can derive from Equation 2 a simple linear expression,
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where v0 is the initial binding reaction velocity which is given by 1/v0 = 1/videal + 1/vm where videal = k+L0 is the ideal initial binding velocity and vm = KmL0/Smax is the mass transport contribution. In the case of mass transport-limited binding (vm << videal), the initial binding velocity equals approximately the mass transport velocity, v0 
vm.
Under non-stirring conditions, the corresponding binding velocity and mass transport rate can be found from the following approximation. Let us assume a fully absorbing disc (videal 
) of radius R homogenously covered by antibodies with the density of binding sites 
. All of the remaining surface is assumed to be totally reflecting. In the stationary Smoluchowski limit, the rate of absorption by such a disc is known to be KS = 4DRL0 where D is the bulk diffusion coefficient of the analyte molecules (in cm2/s). The number of antibodies with antigen bound increases with time as N(t) = KSt, and the signal increases accordingly as S(t) = SmaxN(t)/ Nmax where Nmax = 
R2 is the total number of antibodies available. On the other hand, in the considered limit we have S(t) = Smaxvmt with vm = KmL0/Smax. From this it follows that KmL0R2 = SmaxKS and thus
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Furthermore the overall reaction becomes obviously mass transport-limited when the experimentally estimated Damkoehler number,
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is large, Da-exp >> 1. For this particular model with Km in Equation 4,
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where Da-theo is the theoretically determined Damkoehler number. It grows linearly both with the spot radius R and with the antibody density , implying that smaller spots with lower antibody density are preferable to minimize mass transport limitations under the model assumptions.
When the stirring is applied, the situation becomes more complicated. In this case, the mathematical response of the rate of mass transport or the Damkoehler number to stirring intensity, diffusion coefficient, spot radius, and density of binding sites is currently not known. Nevertheless the two-compartment model remains a powerful and insightful tool for kinetic analysis if the rate constant of mass transport is experimentally determined. On the phenomenological grounds, an effective diffusion coefficient can be introduced for all the remaining parameters. The effective diffusion coefficient might then be determined using Equation 4 used merely for its definition. However, to have an experimental definition of the effective diffusion coefficient that does not depend on the two-compartment modeling, we determined it experimentally from the FullArea chip measurements (from non-stationary initial binding kinetics) as described below.
To analyze the data obtained from FullArea chips, an equation from the reaction-diffusion theory by Stenberg and Nygren (24–26) was used.
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Previously determined affinity parameters for anti-IFNG (15) (k+ = 527,000 m–1 s–1, k– = 0.000322 s–1, and Kd = 611.01 pm as measured by Biacore) as well as binding site density ( = 10–11 mol/cm2) were used for this mathematical analysis. Affinity of anti-TG was also measured using Biacore (k+ = 45,520 m–1 s–1, k– = 0.000998 s–1, and Kd = 21.93 nm). We observed that in the presence of stirring the initial FullArea chip kinetics was described by a square root dependence (as in Equation 7). For this reason, the experimental definition of the effective diffusion coefficient in the presence of stirring was obtained by applying Equation 7 for the corresponding data analysis.
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RESULTS |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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All geometries were also incubated with 100 times higher (20 nm) and lower antigen concentrations (2 pm). Predictably at 20 nm, the time required for saturation was much shorter, and the differences between the signal intensities developing under stirring and non-stirring conditions were smaller (Fig. 1ii, B). When incubating the slides at 2 pm antigen, we first detected signal in the most systems after about 10–20 h of incubation (data not shown). Development of signal intensities was also measured with a few other antibodies against IFNG, TG, and KLH, under different incubation conditions. The reaction times described above were often even more prolonged for these antibody-antigen pairs and were less reproducible (see supplemental Fig. S1 for anti-KLH).
Mass Transport Dependence on Differently Sized Spots—
The observed progression curves (Fig. 1ii) indicate a strong dominance of mass transport in the reaction. As follows from several theoretical considerations (13, 25, 27, 28), the velocity of microspot reactions when limited by diffusion (not mass flux due to the stirring) depends on the radius of the spot, increasing as the spot radius decreases, in an inversely proportional relationship (Equation 4). It is important to note that these theories assume a non-adsorptive, completely reflecting surface as well as a steady-state reaction (or Km = const, constant mass flux over the whole reaction time). In contrast to this, the reaction velocity would be independent of the spot size if the non-steady-state mass flux is dominant. To analyze the character of this reaction, differently sized anti-IFNG spots were incubated under stirring and non-stirring conditions with 5 nm analyte concentration (see Fig. 2 and Table I). The data obtained were fitted to Equation 1 and Equation 3 where Smax was set to 1 for all spots to simplify the analysis and interpretation. In these units, the signal intensities obtained experimentally corresponded to the fractional occupancy of binding sites on the spot, and derived Km values also related to the fractional occupancy.
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The same experiments were also carried out with the TG antigen-antibody pair, which has a substantially larger analyte (molecular mass of 670 versus 17 kDa for IFNG) (Fig. 3) and consequently a 3–4-fold lower diffusion coefficient. The diffusion coefficients can be estimated by the relation 3
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= D2/D1 where m is the molecular mass in kDa. The TG antigen/antibody interaction may therefore demonstrate an alternative reaction behavior and is interesting as an example of a microspot reaction with very large molecules. In this case, the signal velocity was independent of the spot radius under non-stirring conditions. Moreover the signal intensity for the large pins developed even faster than for the small pins under stirring conditions. Even under strong stirring conditions, the kinetics for both antigen-antibody pairs still remained substantially mass transport-dependent (Da-exp >> 1, see also Tables I–Table III). Moreover the reactions on spots of different sizes demonstrated relatively different susceptibilities to stirring and seem to have an irregular dependence on spot radius for both antigen-antibody pairs (Figs. 2 and 3).
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, the microspot reaction remained strongly mass flux-dependent. Also the Km/Smax and/or S-exp values obtained at 50 µg/ml differed strongly between both tested concentrations of IFNG, whereas, especially in the case of SMP2 spots, S-exp obtained for 4 nm was nearly 3-fold lower in comparison with 20 nm (both concentrations are well above Kd of antibody), and the progression curves seem to remain in a stationary state, not developing further. The same irregularities were also observed for the case of TG antigen-antibody pairs where they were even more imprinted (Fig. 5, and Table III). Namely based on the affinity parameter of anti-TG as measured by Biacore, the difference between S values for 20 and 4 nm has to be about 3-fold. Although the expected differences were still obtained at 1 mg/ml on SMP15 and SMP2 spots, spots printed with 50 µg/ml demonstrated much lower, seemingly saturated signal intensities (e.g. about 50-fold lower S-exp at 4 nm as compared with 20 nm on SMP2 spots). Also as in the case of anti-IFNG, Km constants differed strongly at low , whereas the progression curves still seemed to keep their exponential form and steady-state regime. It should be emphasized that these effects appear especially strong on the smallest SMP2 spots for both observed antibodies. In some preliminary experiments, we were unable to obtain signal intensities analyzing 50 µg/ml printed SMP2 spots with subnanomolar (800 and 160 pm) IFNG concentrations (data not shown). To prove whether this quasistationary state can be overcome by a much longer incubation time, the slides printed with SPM3 and different antibody concentrations from 1000 to 125 µg/ml were incubated over 30 h with the dilutions of the corresponding antigens from 100 nm to 2 pm (see supplemental Fig. S2). The signal intensities obtained were normalized against the signal obtained at the 100 nm concentration. Even at 125 µg/ml, both antibodies, anti-IFNG and anti-TG, still demonstrated a strong, 2–4-fold decrease of the relative signal intensities at relatively low L0 so that the dose-response curves deviated from the results expected from the corresponding Kd values.
Analysis of the Mass Transport Dependence of Classical Well Assays—
With the dual goals of estimating the diffusion coefficient of the IFNG molecules in our medium and comparing the kinetics of classical well assays with microspots, the FullArea chips were incubated with 100 µl of 4 nm IFNG for an incubation time of up to 56 h. In the context of our study, the FullArea chip can also be considered as a spot with an infinite radius representing non-steady-state kinetic limit.
Half of the maximum signal intensity was detected in this system after about 30 min of incubation with stirring and after 7 h of incubation without stirring (Fig. 6). The initial part of the progression curve was fitted using Equation 7, which gave the IFNG diffusion coefficient of 1.05·10–7 cm2/s without stirring and the effective diffusion coefficient with stirring of 1.1·10–6 cm2/s. Measuring the diffusion coefficient using fluorescence correlation spectroscopy (Refs. 29 and 30; see also supplemental material) under non-stirring conditions, values in the same range were found ((3.1 ± 0.89)·10–7 cm2/s for IFNG as well as (1.09 ± 0.24)·10–7 for TG). This difference in the diffusion coefficient determinations may be attributed to the impact of the high milk protein concentration in the incubation solution. The maximum signal intensity reached on the FullArea chips was about 20 times lower than that for the SMP2 (45 µm) microspot chips incubated in parallel. Incubation of the FullArea chips with 200- and 50-µl volumes resulted in a nearly 2-fold increase and decrease, respectively, in signal intensity obtained upon saturation. Correspondingly a similar slope of progression curve at initial time points was observed for all three volumes (data not shown). This indicates that the FullArea chip detects the whole amount of analyte in the well (not concentration) and that its kinetic behavior corresponds to a classical well assay.
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To prove the viability of our approach for blood plasma protein profiling, a microarray containing 16 anti-cytokine antibodies was constructed. Whenever possible, the antibodies were chosen in accordance with their binding affinity so that most had Kd values between 1 nm and a few dozen pm. After overnight incubation (14 h) with stirring, we could detect signals from all the spotted antibodies with the signal-to-noise ratios ranging from about 30 up to several hundreds (see Fig. 7B). Background was almost undetectable with this scanner adjustment. Because normal concentrations for the investigated cytokines are in the range of a few pg/ml (supplemental Table S1), a low-to-middle femtomolar sensitivity is in fact achieved in our plasma profiling experiments.
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DISCUSSION |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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The magnitude of the mass flux effects on assay kinetics varies enormously. At L0 >> Kd, the reaction duration depends on L0 because the time constants for intrinsic ideal reaction and mass transport component are 
ideal 1/(k+L0) and m 1/(KmL0), respectively. It may therefore be extremely short: a few seconds or minutes (15, 16). On the contrary, more than 4 h of incubation were required at L0 << Kd to obtain the signal saturation for the TG antigen-antibody pair at the highest spotting concentration shown in Table III and Fig. 3A ( = ideal + m 1/k– + Smax/KmKd). To estimate the maximal duration of the reaction in the limit of mass transport at L0 << Kd, the reaction course can be approximated by the following equation (cf. ideal 1/k– (15, 16)).
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According to Equations 8 and 9, to reach one-half of the maximal signal intensity for the IFNG antigen-antibody pair, about 16 h are required at the best conditions demonstrated in Table I, whereas saturation would only be attained after more than 60 h. In contrast to this, the same values for SMP15 and SMP10 under non-stirring conditions (Table I) are more than 200 and 900 h, respectively! The corresponding incubation times may have to be prolonged enormously for the assay formats that are not kinetically optimized, for analyte molecules with a low diffusion coefficient, or for antibodies with lower Kd (Equation 9). Because this situation cannot be completely overcome by stirring, the question of crucial importance for those of us developing microarray assays is this: what are the underlying physical principles of antibody microspot kinetics?;
In a first order approximation, our system can be modeled in the mass transport-limited environment as a completely absorbing disc placed on a reflecting planar surface. In this situation, one can distinguish between two spot size-dependent reaction regimes: an initial non-steady-state mass transport phase in which the signal develops proportionally to 
and a subsequent steady-state phase where the signal development is proportional to the incubation time t. To match this description with the two-compartment theory, we use a slightly modified equation derived from the diffusion-limited immunoassay theory of Stenberg and Nyngren (24–26). It describes the initial stage of the signal development and reads
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where Da is the corresponding Damkoehler number. Setting both terms in Equation 10 to be equal to one another, one can extract steady.
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This is the time after which the first linear term dominates the signal development and the reaction enters the steady-state flux regime where the mass transport to the binding area remains constant (Km = const). Via the time steady, the applicability of our two-compartment approach, which is restricted by the steady-state flux conditions, can be estimated. From the values of the diffusion coefficients and the density of binding sites (10–11 mol/cm2), which both are determined experimentally, it follows that steady for the IFNG reaction varies from a few seconds to about 15 min (see Table I) depending on the spot sizes and the incubation conditions. Consequently the analyzed reaction is substantially in the steady-state mass transport regime. Also theoretical Da values calculated from Equation 6 are in satisfactory agreement with the values derived from our experimental data using Equation 5 (see Table I).
On the other hand, substantially smaller diffusion coefficients of the analyte may result in non-steady-state reaction (similar to the FullArea chip). Note that for Da >> 1 in Equation 11, steady is proportional to R2/D (assuming that Equation 6 is valid), and it is independent of k+. In this regime, the second term dominates in Equation 10, and the kinetics can be independent of R, k+, and if Da >> 1. This is the case for the TG antigen-antibody pair where a dependence of the signal velocity on the spot size under non-stirring conditions was not observed (Fig. 3B). Moreover because S(t) 
in such a case, a much stronger mixing strength is required (Fig. 3) to obtain the same amplification factor of the signal velocity as for the steady-state mass transport regime (Fig. 2). Furthermore the TG reaction with stirring (Fig. 3A) is also in line with our theory because the reverse dependence of the v0 on the spot size may be held if Da is sufficiently low and Da/(1 + Da) < 1 according to Equation 10. Summarizing, the kinetic mechanism described by Stenberg and Nyngren (24–26) seems to dominate our experiments demonstrated in Figs. 2 and 3. Moreover the domination of this mechanism can also be assumed for any other antigen/antibody interaction in our system because IFNG and TG antigen-antibody pairs represent two extremes with relatively low and high molecular weight proteins, respectively.
Apart from this simplified picture of the underlying reaction mechanism, nonspecific protein adsorption may significantly influence the binding reactions on surfaces. Proteins that are bound only weakly to the surface (like relatively small IFNG molecules) stay mobile and diffuse on surface until they bind specifically or desorb again (31, 32). It is worth mentioning that an accompanying two-dimensional diffusion is usually considered to be one of the main mass flux mechanisms in the related DNA microarray technology (33). Accounting for the impact of surface diffusion on the microspot kinetics, we can differentiate between two different mass transport pathways: two-dimensional diffusion within the spot (intraspot surface diffusion) and diffusion outside the spot (interspot surface diffusion).
Upon reaching the area of the spot, the reversibly adsorbed molecules require time to search for the binding sites within the spot in two dimensions, which may result in an additional prolongation of the reaction course. The similar problem of a combined three- and two-dimensional binding reaction was theoretically investigated nearly 2 decades ago in a series of seminal studies (33–36). However, the common feature of these reaction models is that even when the targets occupy as little as 1% of the total surface area, the three-dimensional capture efficiency still dominates the reaction pathway. According to our estimation of antigen-binding site density (10–11 mol/cm2) and assuming their radius is in the range of 10–7 cm, the percentage of surface occupied by targets would be well over 10% at high antibody spotting concentration and close to 1% at the lowest spotting concentration (S-exp for 50 µg/ml; see Table II). Heterogeneous DNA/DNA interaction, which has been investigated much more thoroughly in this context, reaches maximal hybridization efficiency (e.g. in the case of 20-bp oligonucleotides) at a surface density of immobilized DNA molecules in the range of 10–13–10–16 mol/µm2 (37, 38). Consequently in comparison with the Smoluchowski mass transport limit (Equation 4), the influence of such intraspot two-dimensional diffusion on the reaction kinetics seems to be insignificant at least for sufficiently high antibody spotting concentrations.
Alternatively the binding sites may also be accessible for analyte molecules by interspot surface diffusion pathway. Assuming again that our spot is a completely absorbing disc, one can apply the Adam and Delbrück model (27, 39) of a two-stage diffusion capture to analyze this important issue. The average diffusion time to find the disc directly from the bulk (Smoluchowski average time) is s = V/4DR where V is the volume of the samples (17). On the other hand, to find the disc in a two-stage mechanism, a ligand in the bulk first has to reach the surface, and assuming that the walls of the incubation chamber are completely reflecting, the average time, 0 = H2/3D (17), would depend only on the height H of the incubation chamber. Furthermore for an incubation chamber with the radius Rc and if Rc >> R it takes average time
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where D' is the two-dimensional diffusion coefficient, to reach the target of radius R via the surface diffusion (27, 34). Because normally D' < D, the time 0 is typically neglected within the two-stage diffusion mechanism. The same is also valid for our experimental case where 0 is much smaller than ' (at H 0.3 cm and Rc 0.35 cm). For the absorption rate of the considered disc by a two-stage mechanism one can get the following estimation (27, 34),
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where d is the width of the boundary layer of surface diffusion that has a characteristic size of several nanometers (the effective radius of action of attractive molecular and electrostatic forces at surface) (40) and Eb is a characteristic energy of nonspecific binding. Lb is the actual bulk concentration of analyte, which is related to L0 and Ls, concentration of the analyte in the interface area, by the following expression: Lb = L0 – LsRc2d/V. Ligand concentration is greatly enhanced by the factor of exp(Eb/kT) in the boundary layer with respect to its bulk concentration so that the initial L0 may be depleted to a greater or lesser extent due to nonspecific adsorption in the limited incubation geometry. The two-stage diffusion mechanism will provide a dominant route when
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The energy of nonspecific binding to the surface (Eb) is typically many times larger than the thermal energy (kT 2.5 kJ/mol at room temperature) for protein molecules (40). For example, the energy for adsorption of a 13-residue peptide on colloidal silica was measured to be about 36 kJ/mol (41); binding of relatively small proteins such as lysozyme (14.3 kDa) or RNase A (13.7 kDa) to different chromatographic solid supports was indicated in the range of 13–34 kJ/mol (42–44), and the binding energy of a series of proteins to the aluminum salt ranges from about 30 to 40 kJ/mol (45). Assuming Eb = 20 kJ/mol and d corresponds to the molecular size of the IFNG molecules (about 4-nm diameter), the two-stage diffusion mechanism can only be significant in our incubation chamber with Rc 0.35 cm for R of only a few µm or even less (the dominating mechanism if R < (D'/D)10–4 cm). However, for Eb > 25 kJ/mol, this mechanism will be important or even dominant (>30 kJ/mol) for today’s typical spots of 50–300-µm radius. From another perspective, the two-stage transport mechanism would quickly gain in dominance (Equation 14) if the spot radius decreases. For example, SMP2 spots (45 µm) would be influenced about 3.5 times more strongly by this mechanism than SMP15 spots (272 µm) in our well geometry. To the best of our knowledge, although such an interspot diffusion mechanism is obvious in hindsight, it has not been considered in connection with the microarray kinetics.
The classical consideration above (Equation 14) is still based on the assumption of an unlimited incubation volume or constant source of the analyte in the system. However, both organic and inorganic surfaces can adsorb a considerable amount of protein from bulk solution (usually µg–ng/cm2) (46–49). Therefore, nonspecific adsorption, even by a preblocked surface, may still partially deplete the initial analyte concentration in a limited volume and separate it into two phases: bulk and interface portions. The reduction of the initial analyte concentration can be described as follows.
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For our cylindrical reaction chamber with the height H = 0.3 cm (V = 0.1 ml), more than one-half of all the analyte would be depleted if Eb > kJ/mol according to Equation 15. The change in the adsorption energy has an exponential effect so that the right terms in Equations 14 and 15 double with every further 0.69 kJ/mol increase. It is noteworthy that the two-stage reaction "pathway" is classically considered as an additional mechanism, leading to the enhancement of the reaction rate due to reduction of dimensionality (27, 33, 34, 36, 45). However, due to the bulk analyte depletion, it may affect even adversely the reaction velocity if D' is relatively low (see Equations 12 and 13).
The kinetic, diffusional aspects are practically absent in the modern protein microarray literature. As a result, they do not receive sufficient attention in the experimental design and development of multiple protein microarray applications. Contrary to this, an optimization of even the simplest design and incubation parameters can have a strong impact on the performance and sensitivity of a microarray assay. In the following, we consider a series of kinetically relevant factors from the viewpoints of the reaction mechanism, as described above, in their importance for the microarray design.
Although intensive stirring cannot reduce the mass transport limitations to a negligible level, the understanding of stirring effects on the microspot scale is extremely important in practical context. Highly affine binding reactions measured by Biacore systems also suffer from the mass transport constraints despite a high velocity of the solution pumped through the chip. The reason is that the effective rate constant of the mass transport (to which one can formally relate an effective diffusion coefficient) increases as a cubic root of the pumping velocity only (18, 20, 50); i.e. a 1000-fold increase of this velocity leads only to a 10-fold increase of the effective diffusion coefficient as observed in our study (Fig. 6). Unlike the Biacore system, where the receptor molecules cover the whole surface of the chip, the microspot reactions are additionally influenced by the two-stage interspot diffusion mechanism (Equation 14). However, stirring, which improves the access of the analyte from the bulk to the surface, can only weakly affect surface diffusion. Therefore, even if a significant portion of the analyte accumulates near the surface, it can still represent a bottleneck for further increasing the overall reaction rate in microspot assays. Furthermore the spot size- and the binding site density-dependent balance of bulk and surface reaction "pathways" are susceptible to stirring to a different extent. In the case where the intraspot diffusion is irrelevant, the signal velocity on relatively large spots would be amplified proportionally with the increase of the effective diffusion coefficient (see Equation 14 and Table I), whereas with decreasing spot radius only a strongly reduced portion of the analyte diffusing from bulk contributes to the acceleration of the signal development (Equations 12 and 13). As a result, one can obtain under stirring conditions either irregular (Figs. 2 and 3 and Table I) or comparable (51) spot size dependence of the signal velocity.
Given the strong impact of the mass transport and the surface diffusion on the assay kinetics, it is not surprising that the geometry of the incubation chamber used (Fig. 1) may have a crucial effect on the reaction mechanism that determines the development of signal velocity in time. Flat incubation geometries may strongly disturb the initial analyte concentration. For example, the binding energy for the classical coverslip (H = 0.01 cm) would be Eb > 25.3 for 50% analyte depletion as calculated from Equation 15, whereas nearly all of the analyte (>95%) would be adsorbed at Eb > 32.7. This would lead to a stronger analyte depletion than in the well geometry and would in turn change the overall reaction mode by moving it toward the two-dimensional interspot diffusion mechanism. Consequently the acceleration of signal velocity by stirring would be hindered in flat incubation chambers (Equation 15 and Fig. 1) because stirring can only slightly influence surface diffusion. Because the coverslip itself plays the same adsorptive role as the reaction surface, a kinetically improved incubation can be achieved by using a well assay in an open geometry, similar to the one presented here, and by increasing the ratio of the sample volume to contact adsorptive surface.
In terms of the assay "ambience," known to be the most sensitive characteristics of the microarray, the minimally permissible sample volume is defined as V [A]R2/Kdfmax for the case of L0 << Kd where [A] is the surface concentration of antibody in mol/cm2 and fmax is the maximal fraction of the analyte molecules bound to the antibodies from the bulk solution (at equilibrium and for L0 << Kd). In the case of our IFNG pair with R = 0.01 cm and fmax = 0.05, this volume is 100 µl. Clearly because saturation cannot be achieved with high affinity antibodies, this volume may be many times lower in accordance with the degree of saturation at L0 << Kd and for relatively short incubation times. However, considering the adsorptive forces in the same manner as in the case of the incubation geometry, the minimal sample volume has to be limited by the degree of nonspecific analyte depletion in the system and, consequently, by the ratio of the sample volume to the adsorptive surface. Otherwise a too low volume would lead automatically to lower reaction rates.
As expected from both the work by Stenberg and Nyngren (24–26) and the modified Adam and Delbrück (27, 39) mechanism described above, a decrease in the binding site density would not influence the initial signal velocity. This also means that the parameter Km/Smax in Table II should not change at various , whereas Da should decrease proportionally with decreasing (Equation 6). In such a situation, the signal intensities obtained upon saturation (S-exp) are also expected to proportionally reflect the differences in . However, our experimental observations indicate that strongly suboptimal antibody spotting concentrations can surprisingly result in a kind of stationary regime where a quasi-equilibrium is established on a level that is many times lower than one expected from the thermodynamic equilibrium under the ambient analyte conditions (Figs. 4 and 5 and supplemental Fig. S2). This causes an additional strong decrease of the maximally attainable signal intensity, a phenomenon that seems to intensify with the decreasing analyte concentration and the reduction of spot radius (Tables II and III). A potential indication of the character of this effect might be an increasing role of the intraspot surface diffusion, which might also be a limiting factor in the overall reaction rates at relatively low under stirring conditions. Interestingly enough, the three-dimensional surface chemistries, which represent an additional barrier for diffusion, suffer even more strongly from non-optimal spotting concentrations. In a previous study (16), we demonstrated for the same antigen-antibody pairs (IFNG and TG) that a spotting concentration of about 100 µg/ml results (under different incubation conditions) in 10–40-fold lower signal intensity on a Hydrogel surface from PerkinElmer Life Sciences as compared with our homemade epoxysilane slide. Moreover the signal intensity was at least 100 times lower than one obtained for 2 mg/ml on both surfaces. An additional mechanism of a three-dimensional search of sparsely distributed binding sites within the dense polymer matrix may be responsible for this effect.
The impact of antibody spotting concentration, which can influence the resulting signal intensity (and therefore also sensitivity) tens or even hundreds-fold (see S-exp in Tables II and III), seems to be largely overlooked in the literature. The antibody spotting concentrations are often not adjusted to the same concentration value. They are usually relatively low and vary strongly in complex antibody microarrays (typically between a few dozen and few hundred µg/ml). The experimental finding of optimal spotting concentrations is usually done under strongly mass transport-limited incubation conditions, and the results do not allow differentiation between different binding site densities at relatively high (for details, see Ref. 16). It is interesting to speculate that the answer to the often asked question, namely, why do the majority of antibodies not work in arrays (52), may have a quite simple answer: the spotting concentration is much too low.
Another factor, which is important for the reaction rates on spots, is the viscosity of the incubation buffer. The diffusion coefficient is inversely proportional to it in accordance with the Stokes-Einstein relation (27). Therefore, the viscosity impacts directly the mass transport dependence of the reaction kinetics (see Equation 6). Also the spottin
) conditions.
) and non-stirring (•) conditions. The progression curves were fitted to 
), and 50 µg/ml (
). Slides were incubated with 20 nm (A and C) and 4 nm (B and D) IFNG concentrations. The signal intensities obtained were normalized upon saturation against S
