Analyte–Receptor Binding Kinetics for Biosensor Applications: A Single-Fractal and a Dual-Fractal Analysis of the Influence of the Fractal Dimension on the Binding Rate Coefficient

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

208, 455– 467 (1998)

CS985832

Analyte–Receptor Binding Kinetics for Biosensor Applications: A Single-Fractal and a Dual-Fractal Analysis of the Influence of the Fractal Dimension on the Binding Rate Coefficient Anand Ramakrishnan and Ajit Sadana1 Chemical Engineering Department, University of Mississippi, University, Mississippi 38677-9740 Received May 12, 1998; accepted August 25, 1998

The diffusion-limited binding kinetics of antigen (analyte) in solution to antibody (receptor) immobilized on a biosensor surface is analyzed within a fractal framework. Most of the data presented are adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (“Scientific Graphing Procedure, User’s Manual,” Jandel Scientific, San Rafael, CA, 1993). A couple of examples of a dual-fractal analysis are also presented. It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes in the same direction for the analyte–receptor systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the analyte–receptor binding systems indicate the high sensitivity of the binding rate coefficient on the fractal dimension when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis and for the binding of cell surface proteins from Helicobacter pylori strain in solution to sialyl-(a-2,3)-lactose-conjugated (20 mol%) polyacrylamide immobilized on a resonant mirror biosensor (S. Hirmo et al., Anal. Biochem. 257, 63, 1998), the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, was 14.15. The fractional order of dependence of the binding rate coefficient(s) on the fractal dimension(s) further reinforces the fractal nature of the system. The binding rate coefficient(s) expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface and further emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance. © 1998 Academic Press

INTRODUCTION

Sensitive detection systems (or sensors) are required to distinguish a wide range of substances. Sensors find application in the areas of biotechnology, physics, chemistry, medicine, aviation, oceanography, the food industry, detection of explosives (for example, buried land mines and explosives in aircraft luggage), and environmental control. The importance of providing a better understanding of the mode of operation of 1

To whom correspondence should be addressed.

biosensors to improve their sensitivity, stability, and specificity has been emphasized (1). Whetton (2) refers to biosensors as electronic noses and emphasizes the reduction in time required in the detection process. For example, the Robinson test requires 48 h to determine if chocolate has adsorbed any flavor from its packaging. A prototype biosensor takes 5 min for this test. The solid-phase immunoassay technique represents a convenient method for the separation and/or detection of reactants (for example, antigen) in a solution. The binding of antigen to an antibody-coated surface (or vice versa) is sensed directly and rapidly. Recently, Kleinjung et al. (3) emphasize the further utilization of biosensors with the increasing understanding of the basic nature of nucleic acids. For example, they have utilized a high-affinity RNA as a molecular recognition element in a biosensor. There is a need to characterize the reactions occurring at the biosensor surface. The details of the association of antibodies (or antigens) with antigen-coated (or antibody-coated) surface are of tremendous significance for the development and characterization of immunodiagnostic devices and biosensors (4). Furthermore, external diffusional limitations play a role in the analysis of such assays (5–13). The particle-enhanced sensitivity of the surfaceplasmon-resonance biosensor has been analyzed (14). These authors indicate that the fractal dimension is a measure of the clustering of particles. For example, for colloidal gold there is clustering of particles that leads to significant signal enhancement. Kopelman (15) indicates that surface-diffusion-controlled reactions that occur on clusters or islands are expected to exhibit anomalous and fractal-like kinetics. These fractal kinetics exhibit anomalous reaction orders and time-dependent (for example, binding) coefficients. Fractals are disordered systems and the disorder is described by nonintegral dimensions (16). These authors further indicate that as long as surface irregularities show scale invariance that is dilatational symmetry they can be characterized by a single number, the fractal dimension. The fractal dimension is a global property and is insensitive to structural or morphological details (17). In a recent communication Avnir et al. (18) emphasize that the

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power law utilized in describing the fractal nature of systems very appropriately condenses the complex nature of the system being analyzed. Furthermore, it provides a simple picture of the correlation between the system structure and the dynamics of its formation. This type of information is particularly relevant in the study of analyte–receptor binding reactions occurring on surfaces. The optical amplification of ligand–receptor binding using liquid crystals has been analyzed (19). Gupta et al. schematically show the change in the surface heterogeneity (or the fractal dimension, D f) as avidin or IgG molecules in solution bind to ligands attached to self-assembled monolayers of molecules supported on a gold film. Their schematic indicates that the surface roughness increases on the binding of the analyte (Av or IgG) in solution to the ligands on the surface. The widespread nature of fractals has been indicated (20). Fractals are present in sols and gels, in soot and smoke, on rough surfaces, in disordered layers on surfaces, and in porous objects. Lee and Lee (21) indicate that the fractal approach has been applied to surface science, for example, adsorption and reaction processes. These authors emphasize that the fractal approach provides a convenient means to quantitatively represent the different structures and morphologies at the reaction interface. They analyzed simulations of Eley–Rideal diffusionlimited reactions on different objects. The primary advantage is that this permits the development of a predictive approach in the field of catalysis. Lee and Lee emphasize using the fractal approach to develop optimal structures. This is of importance because there are indications that today’s sensors tend to be costly, cumbersome, and specialized (22). This author indicates that it would be helpful to develop new sensors that are based on dirt-cheap starting materials. They could then be effectively used as low-cost detectors for medical diagnostics and industrial monitoring and in environmental testing. Antibodies are heterogeneous and their immobilization on a fiber-optic surface, for example, would exhibit some degree of heterogeneity. This is a good example of a disordered system, and a fractal analysis is appropriate for such systems. Besides, the antibody–antigen reaction on the surface is a good example of a low-dimension system in which the distribution tends to be less random (15). A fractal analysis would provide novel physical insights into the diffusion-controlled reactions occurring at the surface. Fractal kinetics have been reported in other biochemical reactions such as the gating of ion channels (23, 24), enzyme reactions (25), and protein dynamics (26). It has been emphasized that the nonintegral dimensions of the Hill coefficient used to describe the allosteric effects of proteins and enzymes is a direct consequence of the fractal property of proteins (25). Goetze and Brinkmann (27) have analyzed the scaling properties of 53 protein surfaces. They emphasize that these surfaces show self-similarities, and this self-similarity is measured by the fractal dimension of the surface. Allometric scaling laws, including the metabolic reactions, have been analyzed (28). These authors indicate that these laws

are characteristic of all organisms. For example, they were able to describe the 3/4 power law for metabolic reactions using a model of transport of essential materials through space-filling fractal networks of branching tubes. A characteristic feature of fractals is the self-similarity at different levels of scale. When the heart rate (beats per minute) of a healthy individual is recorded for 3, 30, and 300 min, the quick erratic fluctuations seem to vary in a manner similar to that of the slower fluctuations (29). This indicates a self-similarity. Note that selfsimilarity does not imply identical. Self-similarity implies that the features of a structure or process look alike at different scales of length or time. Fractal aggregate scaling relationships have been determined for both diffusion-limited and diffusion-limited cluster aggregation processes in spatial dimensions 2, 3, 4, and 5 (30). These authors noted that the prefactor displays uniform trends with the fractal dimension, D f. Fractal dimension values for the kinetics of antigen–antibody binding (31, 32) and for analyte– receptor (33) binding for fiber-optic biosensor systems are available. In these studies the influence of experimental parameters such as analyte concentration on the fractal dimension and on the binding rate coefficient (the prefactor in this case) was analyzed. An initial attempt was made to relate the influence of the degree of surface roughness, such as the fractal dimension, on the binding rate coefficient for antigen–antibody binding for biosensor applications (34). One would like to further delineate the role of surface roughness on the speed of response, specificity, stability, and sensitivity of fiber-optic and other biosensors. Here we extend these studies to analyte–receptor binding reactions (the general case) occurring on biosensor surfaces. In a more general sense, the analysis is also applicable to analyte–receptor reactions for nonbiosensor applications, such as membrane reactions. Finally, the noninteger orders of dependence obtained for the binding rate coefficient, k, on the fractal dimension, D f, further reinforces the fractal nature of these analyte–receptor binding systems. THEORY

An analysis of the binding kinetics of antigen in solution to antibody immobilized on the biosensor surface is available (31). The influence of lateral interactions on the surface and variable rate coefficients is also available (35). Here we initially present a method of estimating actual fractal dimension values for analyte–receptor binding systems utilized in fiberoptic biosensors. Variable Binding Rate Coefficient Kopelman (15) has recently indicated that classical reaction kinetics is sometimes unsatisfactory when the reactants are spatially constrained on the microscopic level by walls, phase

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boundaries, or force fields. Such heterogeneous reactions, for example, bioenzymatic reactions, that occur at interfaces of different phases exhibit fractal orders for elementary reactions and rate coefficients with temporal memories. In such reactions, the rate coefficient exhibits a form given by k 1 5 k9t 2b.

[1a]

In general, k 1 depends on time, whereas k9 5 k 1 (t 5 1) does not. Kopelman (15) indicates that in three dimensions (homogeneous space) b equals zero. This is in agreement with the results obtained in classical kinetics. Also, with vigorous stirring, the system is made homogeneous and b again equals zero. However, for diffusion-limited reactions occurring in fractal spaces, b . 0; this yields a time-dependent rate coefficient. The random fluctuations in a two-state process in ligand binding kinetics have been analyzed (36). The stochastic approach can be used as a means to explain the variable binding rate coefficient. The simplest way to model these fluctuations is to assume that the binding rate coefficient, k 1 (t), is the sum of its deterministic value (invariant) and the fluctuation ( z(t)) (36). This z(t) is a random function with a zero mean. The decreasing and increasing binding rate coefficients can be assumed to exhibit an exponential form (35, 37): k 1 5 k 1,0exp~2b t! k 1 5 k 1,0exp~ b t!.

Single-fractal analysis. Havlin (43) indicates that the diffusion of a particle (antibody (Ab)) from a homogeneous solution to a solid surface (antigen (Ag)-coated surface) where it reacts to form a product (antibody–antigen complex; Ab z Ag) is given by

[1b]

Here, b and k 1,0 are constants. The influence of a decreasing and an increasing binding rate coefficient on the antigen concentration near the surface has been analyzed when the antibody is immobilized on the surface (35). These authors noted that for an increasing binding rate coefficient after a brief time interval, as time increases, the concentration of the antigen near the surface decreases, as expected for the cases when lateral interactions are present or absent. The diffusion-limited binding kinetics of antigen (or antibody or substrate) in solution to antibody (or antigen or enzyme) immobilized on a biosensor surface has been analyzed within a fractal framework (31, 32). Furthermore, experimental data presented for the binding of HIV virus (antigen) to the antibody anti-HIV immobilized on a surface display a characteristic ordered disorder (38). This indicates the possibility of a fractal-like surface. It is obvious that the above biosensor system (wherein either the antigen or the antibody is attached to the surface) along with its different complexities that include heterogeneities on the surface and in solution, diffusion-coupled reaction, time-varying adsorption or binding rate coefficients, etc., can be characterized as a fractal system. The diffusion of reactants toward fractal surfaces has been analyzed (39 – 42). Havlin (43) has briefly reviewed and discussed these results.

~Ab z Ag! ,

H

t ~32Df!/ 2 5 t p t , t c t 1/ 2

t . tc

.

[2a]

Here D f is the fractal dimension of the surface. Equation [2a] indicates that the concentration of the product Ab z Ag (t) in a reaction Ab 1 Ag 3 Ab z Ag on a solid fractal surface scales at short and intermediate scales as Ab z Ag ;t p with the coefficient p 5 (3 2 D f)/2 at short time scales and p 5 1/ 2 at intermediate time scales. This equation is associated with the short-term diffusional properties of a random walk on a fractal surface. Note that in a perfectly stirred kinetics on a regular (nonfractal) structure (or surface) k 1 is a constant; that is, it is independent of time. In other words, the limit of regular structures (or surfaces) and the absence of diffusion-limited kinetics leads to k 1 being independent of time. In all other situations one would expect a scaling behavior given by k 1 ; k9t 2b , with 2b 5 p , 0. Also, the appearance of the coefficient p different from p 5 0 is the consequence of two different phenomena, that is, the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited) condition. Havlin (43) indicates that the crossover value may be determined by r 2c ; t c. Above the characteristic length, r c, the self-similarity is lost. Above t c, the surface may be considered homogeneous, since the self-similarity property disappears and regular diffusion is now present. For the present analysis, t c is arbitrarily chosen. One may consider the analysis to be presented as an intermediate heuristic approach in that in the future one may also be able to develop an autonomous (and not time-dependent) model of diffusion-controlled kinetics. Dual-fractal analysis. The single-fractal analysis presented above is extended to include two fractal dimensions. At present, the time (t 5 t 1 ) at which the first fractal dimension changes to the second fractal dimension is arbitrary and empirical. For the most part it is dictated by the data analyzed and experience gained by handling a single-fractal analysis. A smoother curve is obtained in the transition region if care is taken to select the correct number of points for the two regions. In this case, the product (Ab z Ag) concentration on the biosensor surface is given by

~Ab z Ag! ,

5

t ~32Df1!/ 2 5 t p1 t , t 1 t ~32Df2!/ 2 5 t p2 t 1 , t , t 2 5 t c . t

1/ 2

t . tc

[2b]

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FIG. 1. Binding of hemagglutinating and poorly hemagglutinating H. pylori cells to sialyl-(a-2,3)-lactose-conjugated (20 mol%) polyacrylamide (39SL-PAA) immobilized on a resonant mirror biosensor (44): (a) hemagglutinating, (b) poorly hemagglutinating.

RESULTS

Hirmo et al. (44) have recently characterized Helicobacter pylori strains using their sialic acid binding to a resonant mirror biosensor. These authors indicate that H. pylori is a gastric pathogen that causes type B gastritis and duodenal ulcer disease and possesses a variety of cell surface proteins (45). The binding of the cell surface proteins (receptor–ligand interaction) using sialic acid binding specific for a-2,3-sialyl lactose was characterized using this new optical biosensor technique. These authors indicate that as the molecules bind to the sensing surface there is a change in the refractive index. This results in a shift in the resonant angle (46). Hirmo et al. (44) emphasize that the advantages of their technique are that it is label-free and that real-time monitoring of biomolecular events is possible. Figure 1 shows the curves obtained using Eq. [2a] for the binding of hemagglutinating and poorly hemagglutinating H. pylori cells to sialyl-(a-2,3)-lactose-conjugated (20 mol%) polyacrylamide (39SL-PAA, MW ; 30 kDa) immobilized on a resonant mirror biosensor (RMB) using bacterial cell suspen-

sions. In these two cases a single-fractal analysis is adequate to describe the binding kinetics. Table 1 shows the values of the binding rate coefficient, k, and the fractal dimension, D f. The values of the binding rate coefficient, k, presented in Table 1 were obtained from a regression analysis using Sigmaplot (47) to model the experimental data using Eq. [2a], wherein (Ab z Ag) 5 kt p . The k and D f values presented in Table 1 are within 95% confidence limit. For example, for the binding of hemagglutinating H. pylori cells in solution to silayl-(a-2,3)-lactose immobilized on the RMB, the value of k reported is 150.022 6 1.432. The 95% confidence limit indicates that 95% of the k values will lie between 148.590 and 151.454. This indicates that the values are precise and significant. The curves presented in the figures are theoretical curves. Note that as one goes from the poorly hemagglutinating to the hemagglutinating cells there is (a) about a 7.8% increase in the fractal dimension, D f, value from 2.4298 to 2.620, and (b) an increase in the binding rate coefficient value by a factor of 21.3, from 7.043 to 150.0. Note that the change in the fractal dimension and the change in the binding rate coefficient, k, are in the same direction. These two results indicate that the binding rate coefficient, k, is rather sensitive to the fractal dimension or the degree of heterogeneity that exists on the surface. Figures 2a and 2b show the binding curves obtained using Eq. [2a] for the binding of 250 mg/ml cell surface proteins extracted from H. pylori strains 52 and 33 in solution, respectively, to immobilized 39SL-PAA immobilized on an RMB. Once again, a single-fractal analysis is adequate to describe the binding kinetics. Table 1 shows the values of the binding rate coefficient, k, and the fractal dimension, D f. Once again, note that the values of the fractal dimension and the binding rate coefficient exhibited by strain 52 are higher than those exhibited by strain 33. A 27.2% increase in the fractal dimension from a D f value of 2.24 to 2.85 leads to an increase in the binding rate coefficient k value by a factor of 7.19, from 7.446 to 51.27. Figures 3a through 3c show the curves obtained using Eq. [2a] for the binding of cell surface proteins from H. pylori strain 52 in solution in the absence (Fig. 3a) and in the presence of 5 and 10 mM free sialyl-(a-2,3)-lactose (Figs. 3b and 3c) to 39SL-PAA immobilized on a RMB. Once again a single-fractal analysis is adequate to describe the binding kinetics. Table 1 shows the values of the binding rate coefficient, k, and the fractal dimension, D f. In the free sialyl concentration range of 0 to 10 mM, the binding rate coefficient, k, is given by k 5 ~161.34 6 3.579!@sialyl lactose# 0.072760.0024.

[3a]

Figure 4a shows that this predictive equation fits the values of the binding rate coefficient, k, presented in Table 1c reasonably well. Note that only three data points are available, and one data point is available in the absence of silayl lactose

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TABLE 1 Influence of Different Parameters on Fractal Dimensions and Binding Rate Coefficients for the Binding of Helicobacter pylori to Sialylglycoconjugates Using a Resonant Mirror Biosensor (44)

Analyte concentration in solution/receptor on surface

Binding rate coefficient, k

Fractal dimension, Df

150.022 6 1.432 7.0433 6 0.3579

2.620 6 0.00712 2.4298 6 0.03720

7.4460 6 0.0425

2.2424 6 0.00695

(a) Hemagglutinating H. pylori cells/immobilized sialyl-(a-2,3)-lactose-conjugated (20 mol%) polyacrylamide (39SL-PAA, MW ;30 kDa) on a resonant mirror biosensor (RMB) Poorly hemagglutinating H. pylori cells/immobilized 39SL-PAA on RMB (b) 250 mg/ml cell surface proteins extracted from H. pylori strain 52/immobilized 39SL-PAA on RMB 250 mg/ml cell surface proteins extracted from H. pylori strain 33/immobilized 39SL-PAA on RMB

51.274 6 1.016

2.8501 6 0.0152

82.995 6 2.699

2.4836 6 0.0204

(c) 250 mg/ml cell surface proteins from H. pylori strain 52 (in the absence of free sialyl-(a2,3)-lactose)/immobilized 39SL-PAA on RMB 250 mg/ml cell surface proteins from H. pylori strain 52 in the presence of 5 mM free sialyl-(a-2,3)-lactose/immobilized 39SL-PAA on RMB

178.36

6 6.31

2.6356 6 0.0202

(d) 250 mg/ml cell surface proteins from H. pylori strain 52 in the presence of 10 mM free sialyl-(a-2,3)-lactose/immobilized 39SL-PAA on RMB Two times diluted H. pylori cells of strain 52/immobilized 39SL-PAA on RMB Nondiluted H. pylori cells of strain 52/immobilized 39SL-PAA on RMB Two times concentrated H. pylori cells of strain 52/immobilized 39SL-PAA on RMB Five times concentrated H. pylori cells of strain 52/immobilized 39SL-PAA on RMB

concentration. Due to the lack of experimental data points, this point was also used in the modeling. For this case a very small (equal to 0.0001 mM) silayl concentration was used. Using this value a predicted k value of 82.91 was obtained. This compares very favorably with the experimental value of k equal to 82.995 obtained in the absence of free sialyl concentration in solution. Also, more data points would have provided for a better fit. Note the very low value of the exponent. In the free sialyl lactose concentration range from 0 to 10 mM utilized in solution, the fractal dimension, D f, is given by D f 5 ~2.5997 6 0.0207!@sialyl lactose# 0.0049260.0087.

[3b]

Figure 4b shows that this predictive equation fits the values of the fractal dimension presented in Table 1c reasonably well. Once again, and as indicated above for the lack of data points, the data point for the absence of free sialyl lactose concentration in solution was used. Here too, a very low concentration of 0.0001 mM of sialyl lactose was used. In this case too, a predicted value of 2.4846 was obtained for 0.0001 mM of sialyl lactose. This compares very favorably with the estimated D f value of 2.4836 obtained in the absence of free sialyl lactose

193.438 6 4.655 0.1774 6 0.0109 2.2641 6 0.1078 8.0477 6 0.6249 163.486 6 2.822

2.6152 6 0.0152 1.2240 6 0.0430 1.7980 6 0.0226 1.9884 6 0.0352 2.6484 6 0.0080

concentration. Once again, note the very low value of the exponent. This indicates that the fractal dimension is not very sensitive to the free sialyl lactose concentration in solution. Figure 4c shows that the binding rate coefficient, k, increases as the fractal dimension, D f, increases. This is in accord with the prefactor analysis of fractal aggregates (30), and with the analyte–receptor binding kinetics for biosensor applications (34). For the data presented in Table 1c, the binding rate coefficient, k, is given by . k 5 ~0.000216 6 0.000032! D 14.14962.980 f

[3c]

This predictive equation fits the values of the binding rate coefficient k presented in Table 1c reasonably well. The very high exponent dependence indicates that the binding rate coefficient is very sensitive to the degree of heterogeneity that exists on the surface. Some of the deviation may be attributed to the depletion of the analyte in the vicinity of the surface (imperfect mixing). No correction is presented, at present, to account for the imperfect mixing. More data points are required to more firmly establish a quantitative relationship in

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k 5 ~1.512 6 2.421! 3 @normalized cell concentration# 2.866660.2239. [4a] The normalized cell concentration is the cell concentration divided by the nondiluted cell concentration. Figure 6a

FIG. 2. Binding of 250 mg/ml cell surface proteins extracted from H. pylori strain 52 (a) and strain 33 (b) in solution to immobilized 39SL-PAA immobilized on a resonant mirror biosensor (44).

this case. Perhaps one may require a functional form for k that involves more parameters as a function of the fractal dimension. One possible form could be k 5 aD bf 1 cD df .

[3d]

Here a, b, c, and d are the coefficients to be determined by regression. This functional form may describe the data better, but one would need, as indicated above, more points to justify the use of a four-parameter equation. Figures 5a–5d show the binding curves obtained using Eq. [2a] for the binding of H. pylori cells of strain 52 in solution to 39SL-PAA immobilized on a RMB. In this case different cell concentrations were used. Once again a single-fractal analysis is adequate to describe the binding kinetics. Table 1d shows the values of the binding rate coefficient, k, and the fractal dimension, D f. Diluted and concentrated cell concentrations were used. For the diluted and concentrated cell concentrations used, the binding rate coefficient, k, is given by

FIG. 3. Binding of cell surface proteins from H. pylori strain 52 in solution in the absence of (a) and in the presence of (b) 5 mM and (c) 10 mM free sialyl-(a-2,3)-lactose) to immobilized 39SL-PAA on a resonant mirror biosensor (44).

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461

coefficient is quite sensitive to the normalized cell concentration as indicated by the value of the exponent. Figure 6b indicates that the fractal dimension, D f, increases as the normalized cell concentration increases. In the normalized cell concentration analyzed, the fractal dimension, D f, is given by D f 5 ~1.6246 6 0.1474! 3 @normalized cell concentration# 0.316160.0509. [4b] Figure 6b shows that this predictive equation fits the values of the fractal dimension, D f, presented in Table 1d reasonably well. Note the low value of the exponent. This indicates that the fractal dimension is not very sensitive to the normalized cell concentration. An increase in the fractal dimension, D f, leads to an increase in the binding rate coefficient, k. For this cell concentration range, the binding rate coefficient, k, is given by k 5 ~0.0218 6 0.0145! D 8.779560.9171 . f

FIG. 4. Influence of (a) sialyl-(a-2,3)-lactose concentration (in mM) on the binding rate coefficient, k, and (b) sialyl-(a-2,3)-lactose concentration (in mM) on the fractal dimension, D f.

shows that the predictive equation fits the values of the binding rate, k, presented in Table 1d reasonably well. This is despite the high error in the estimate of the coefficient in the predictive equation. There is some scatter in the data at the higher normalized cell concentration. More data points would more firmly establish this relation. The binding rate

[4c]

There is some scatter in the data at the highest value of the fractal dimension, D f, utilized. Nevertheless, the predictive equation fits the values of the binding rate coefficient, k, presented in Table 1d reasonably well (Fig. 6c). More data points, especially at the higher fractal dimension, D f values, would more firmly establish this equation. Once again, some of this deviation may be attributed to the depletion of the analyte in the vicinity of the surface (imperfect mixing). No correction is presented, at present, to account for imperfect mixing. Furthermore, and as indicated earlier, one may require a functional form for k that involves more parameters as a function of the fractal dimension. The possible form suggested is given in Eq. [3d]. The coefficients a, b, c, and d given in Eq. [3d] are determined by regression. Once again, as observed previously, the high exponent dependence indicates that the binding rate coefficient, k, is very sensitive to the degree of heterogeneity that exists on the surface. Wink et al. (48) have recently utilized surface plasmon resonance (SPR) to analyze liposome-mediated enhancement of immunoassay sensitivity of proteins and peptides. They developed a sandwich immunoassay for interferon-g (IFN-g). A 16-kDa cytokine was used as the capture monoclonal antibody that was physically adsorbed on a polystyrene surface. These authors were careful to point out the advantage and the disadvantage of this technique. The advantage is that no chemical labeling is required. The disadvantage is that SPR detects an aspecific parameter, that is, refractive index. Figure 7a shows the binding of 20 ng/ml interferon-g in solution to cytokine physically adsorbed on a polystyrene surface. Once again, a single fractal analysis is adequate to describe the binding kinetics. Table 2 shows

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FIG. 5. Binding of different cell concentrations of H. pylori strain 52 in solution to 39SL-PAA immobilized on a resonant mirror biosensor (44): (a) two times diluted, (b) nondiluted, (c) two times concentrated, (d) five times concentrated.

the values of the binding rate coefficient, k, and the fractal dimension, Df. A high-affinity RNA has recently been utilized as a recognition element in a biosensor (3). These authors have taken advantage of the recent developments in the understanding of the basic nature of nucleic acids. They indicate that the inherent nature of some nucleic acids to combine the geneotype (nucleotide sequence) and a phenotype (ligand binding or catalytic activity) permits one to identify molecular targets. These authors immobilized an L-adenosine-specific RNA via a biotin–avidin bridge to an optical fiber core. They measured binding using total internal reflection fluorescence of L-adenosine conjugated to fluorescein isothiocyanate (FITC). Figure 7b shows the curve obtained using Eq. [2a] for the binding of FITC L-adenosine to high-affinity RNA attached to a fiber-optic biosensor via a avidin– biotin bridge. Once again, a singlefractal analysis is sufficient to adequately describe the binding kinetics. Table 2 shows the values of the binding rate coefficient, k, and the fractal dimension, D f. A new biosensor platform for analyzing antibody–antigen and streptavidin– biotin interactions has recently been developed (49). They analyzed the binding of 0.02 ng/ml antibody (anti-biotin) to biotin immobilized to a transparent indium–tin oxide (ITO) working electrode. These authors utilized total internal reflection fluorescence (TIRF) to monitor biospecific interactions. Electrochemical polarization was used to control the interactions between biotin and anti-biotin. Figure 7c shows the curve obtained for the

binding of 0.02 mg/ml antibody anti-biotin–FITC in solution to biotinylated ITO. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 2 shows the values of the binding rate coefficient, k, and the fractal dimension, Df. Kyono et al. (50) have recently utilized the scintillation proximity assay to detect hepatitis C virus helicase activity. These authors indicate that hepatitis C virus (HCV) is a major etiologic agent of non-A and non-B viral hepatitis (51, 52). Kyono et al. (50) indicate that the C-terminal two-thirds of the nonstructural protein 3 (NS3) of hepatitis C virus possesses RNA helicase activity. This enzyme is expected to be one of the target molecules of anti-HCV drugs. These authors have utilized the scintillation proximity assay system to detect the helicase activity of NS3 protein purified by an immunoaffinity column. A polyclonal antibody to HCV was adsorbed on the immunoaffinity column. Figure 7d shows the curves obtained using Eqs. [2a] and [2b] for the time course of helicase activity by purified HCV NS3 protein. Note that in this case a dual-fractal analysis is required to provide a reasonable fit. Table 1d shows the values of (a) the binding rate coefficient, k, and the fractal dimension, Df, obtained using a single-fractal analysis, and (b) the binding rate coefficients, k1 and k2, and the fractal dimensions, Df1 and Df2, using a dual-fractal analysis. Though only single examples of the four different systems are presented in Table 2, they do provide estimates of the binding rate coefficient(s) and the fractal dimen-

ANALYTE–RECEPTOR BINDING KINETICS

463

[2b] for the binding of the analyte in solution to the receptor immobilized on the ITO surface. Figure 8a shows the binding of 0.03 mg/ml IgG–FITC in solution and in the presence of 0.01 mg/ml BSA to the biotinylated ITO surface. Note that a single-fractal analysis does not provide an adequate fit, and a dual-fractal analysis is required to provide a reasonable fit. Table 3 shows the values of (a) the binding rate coefficient, k, and the fractal dimension, D f, obtained using a single-fractal analysis, and (b) the binding rate coefficients, k 1 and k 2 , and the fractal dimensions, D f1 and D f2, using a dual-fractal analysis. Figure 8b shows the binding of 0.03 mg/ml IgG–FITC in solution in the absence of BSA to the biotinylated ITO surface. In this case, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 3 shows the values of (a) the binding rate coefficient, k, and the fractal dimension, Df, using a single-fractal analysis. On comparing this with the previous result (Fig. 8a), one notes that the presence of BSA in solution leads to complexities in the binding. Figures 8c and 8d show the binding of 0.03 mg/ml g-globulin–FITC in solution in the absence and in the presence of 0.01 mg/ml BSA to a biotinyated ITO surface. In both of these cases a single-fractal analysis is adequate to describe the binding kinetics. Table 3 and Fig. 9 show that for the cases where a singlefractal analysis is involved (Figs. 8b– 8d), an increase in the fractal dimension leads to an increase in the binding coefficient irrespective of the analyte (IgG or g-globulin) in solution to the biotinylated ITO surface. For the values of the fractal dimension, D f, and the binding rate coefficient, k, presented in Table 3, the binding rate coefficient, k, is given by k 5 ~0.00447 6 0.00129! D 2.544760.8626 . f

FIG. 6. Influence of the normalized cell concentration of H. pylori strain 52 in solution on (a) the binding rate coefficient, k, and (b) the fractal dimension, D f. (c) Influence of the fractal dimension, D f, on the binding rate coefficient, k.

sion(s) obtained for the different systems analyzed. More such systems need to be analyzed even if only a single example is available in each case. Asanov et al. (49) have analyzed the nonspecific and biospecific adsorption of IgG–FITC and g-IgG–FITC in solution to a biotinylated ITO surface. They analyzed this in the absence and in the presence of bovine serum albumin (BSA). Figures 8a– 8d show the curves obtained using Eqs. [2a] and

[5]

The above predictive equation fits the values of the binding rate coefficient, k, presented in Table 3 and in Fig. 9 reasonably well. Only three data points were available. More data points would more firmly establish this equation. Also, as indicated earlier, some of the deviation may be attributed to the depletion of the analyte in the vicinity of the surface (imperfect mixing). No correction is presented, at present, to account for the imperfect mixing. Besides, one may also use a functional form for k that involves more parameters as a function of the fractal dimension (Eq. [3d]). However, one would need more points, as indicated earlier, to justify the use of a four-parameter equation. The binding rate coefficient is quite sensitive to the degree of heterogeneity that exists on the surface as indicated by the value of the exponent. Table 4 summarizes the predictive binding rate coefficient expressions obtained as a function of the fractal dimension for both a single- and a dual-fractal analysis. For the single-

464

RAMAKRISHNAN AND SADANA

FIG. 7. (a) Binding of 20 ng/ml interferon (IFN)-g in solution to 16-kDa cytokine (monoclonal antibody) adsorbed on a polystyrene surface (with liposome amplification) (48). (b) Binding of L-adenosineFITC in solution to high-affinity RNA attached to a fiber-optic biosensor via an avidin– biotin bridge (3). (c) Binding of 0.02 mg/ml anti-biotin in solution to biotinylated indium–tin oxide electrode (49). (d) Binding of 20 ng/ml per well helicase in solution to immobilized nonstructural protein 3 of hepatitis C virus (50) (broken line, single-fractal analysis; solid line, dual-fractal analysis).

fractal analysis binding systems presented, the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, ranges from 2.5447 (analyte, for example,

IgG–FITC in the absence and in the presence of 0.01 mg/ml BSA in solution/biotinylated ITO surface (49)) to 14.149 (cell surface proteins extracted from H. pylori strain 52 in

TABLE 2 Influence of Different Parameters on Fractal Dimensions and Binding Rate Coefficients for Different Antibody–Antigen and Analyte–Receptor Systems Analyte concentration in solution/ receptor on surface 20 ng/ml interferon (IFN)-g/16-kDa cytokine (monoclonal antibody) adsorbed on polystyrene surface (with liposome amplification) FITC L-Adenosine /high affinity RNA attached to a fiber-optic biosensor via an avidin–biotin bridge 0.02 mg/ml antibody (anti-biotin)– FITC/biotinylated indium–tin oxide (ITO) electrode 20 ng per well helicase/nonstructural protein 3 of hepatitis virus using a scintillation proximity assay system

k

Df

k1

k2

D f1

D f2

Ref.

266.61 6 2.832

2.7216 6 0.007

na

na

na

na

(48)

12.707 6 1.096

2.5192 6 0.068

na

na

na

na

(3)

0.0350 6 0.005

2.1950 6 0.115

3.6058 6 1.344

1.5936 6 0.271

(49)

1.5671 6 0.197

18.710 6 2.001

0.8782 6 0.1736

2.4402 6 0.173

(50)

465

ANALYTE–RECEPTOR BINDING KINETICS

FIG. 8. Binding of 0.03 mg/ml IgG–FITC in solution (a) in the presence of 0.01 mg/ml BSA (broken line, single-fractal; solid line, dual-fractal analysis) and (b) in the absence of BSA to the biotinylated ITO surface (50). Binding of 0.03 mg/ml g-globulin in solution (c) in the absence and (d) in the presence of BSA to the biotinylated ITO surface (50).

the absence and in the presence of free sialyl-(a-2,3)-lactose in solution/immobilized 39SL-PAA on a resonant mirror biosensor (44)). The table also gives the predictive equations obtained for the binding rate coefficient, k, and the fractal dimension, Df, as functions of the analyte concentration. The table does provide an overall perspective of the

nature of the predictive equations for (a) the binding rate coefficient, k, as a function of the analyte concentration and the degree of heterogeneity, Df, that exists on the surface and (b) the fractal dimension, Df, as a function of the analyte concentration in solution. More data need to be analyzed to further delineate this dependence.

TABLE 3 Influence of Nonspecific and Specific Adsorption on the Fractal Dimensions for the Binding of Anti-Biotin Antibody in Solution to Biotin Immobilized on a Transparent Indium–Tin Oxide Electrode (49) Analyte concentration in solution/ receptor on surface 0.03 mg/ml IgG–FITC adsorption in the presence of 0.01 mg/ml BSA/biotinylated ITO surface 0.03 mg/ml IgG–FITC in the absence of BSA/biotinylated ITO surface 0.03 mg/ml g-globulin–FITC adsorption in the absence of BSA/biotinylated ITO surface 0.03 mg/ml anti-biotin antibodies–FITC in the presence of 0.01 mg/ml BSA/ biotinylated ITO surface

k

Df

k1

k2

D f1

D f2

0.00657 6 0.0012

1.7812 6 0.1132

0.00251 6 0.00023

0.03962 6 0.0011

1.2766 6 0.1463

2.4570 6 0.061

0.00844 6 0.0004

1.3688 6 0.00048

0.0140 6 0.0009

1.4514 6 0.042

0.02593 6 0.0018

2.1070 6 0.0458

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RAMAKRISHNAN AND SADANA

FIG. 9. Influence of the fractal dimension, D f, on the binding rate coefficient, k.

CONCLUSIONS

A fractal analysis of the binding of antigen (analyte) in solution to the antibody (receptor) immobilized on the biosensor surface provides a quantitative indication of the state of the disorder (fractal dimension, D f) and the binding rate coefficient, k, on the surface. The fractal dimension value provides a quantitative measure of the degree of heterogeneity that exists on the surface for antibody–antigen (or, in general, analyte–receptor) systems. In all but two of the examples presented, a single-fractal analysis provides an adequate fit. This was done by the regression analysis provided by Sigmaplot (47). A dual-fractal analysis is required to provide an adequate fit for two of the examples presented. In accord with the prefactor analysis for fractal aggregates (30), quantitative (predictive) expressions are developed for the binding rate coefficient, k, as a function of the fractal dimension, D f, for a single-fractal analysis. Predictive expressions are also developed for the binding rate coefficient, k, and for the fractal dimension, D f, as function of the analyte concentration in solution. The fractal dimension, D f, is not a classical independent

variable like analyte concentration. Nevertheless, the expressions obtained for the binding rate coefficient for a singlefractal analysis as a function of the fractal dimension indicate the high sensitivity of the binding rate coefficient on the fractal dimension. This is clearly brought out by the high order and fractional dependence of the binding rate coefficient on the fractal dimension. For example, in the case of the single-fractal analysis and for (a) the binding of 250 mg/ml cell surface proteins extracted from H. pylori strain 33 and in the presence and absence of free sialyl-(a-2,3)-lactose in solution to immobilized 39SL-PAA immobilized on a resonant mirror biosensor (44) and (b) the binding of the analyte (IgG, g-globulin, or anti-biotin) in the absence or presence of BSA in solution to a biotinylated ITO surface (49), the order of dependence of k and D f is 14.149 and 2.5447, respectively. This emphasizes the importance of the extent of heterogeneity on the biosensor surface and its impact on the binding rate coefficient, k. As indicated earlier, the data analysis in itself does not provide any evidence for surface roughness or heterogeneity, and the existence of surface roughness or heterogeneity assumed may not be correct. Furthermore, there is some deviation in data that may be minimized by (a) providing a correction for the deviation due to the depletion of the analyte in the vicinity of the surface (imperfect mixing) and (b) using a four-parameter equation (Eq. [3d]). In this case, more data points would be required to justify the use of Eq. [3d]. Table 4 summarizes the binding rate coefficient expressions obtained as a function of the fractal dimension for different systems utilized in biosensor applications. Other predictive expressions developed for the binding rate coefficient, k, and for the fractal dimension, D f, as functions of the analyte concentration in solution also given in Table 4 are of significance because they provide a means by which these parameters may be controlled. In general, and in the examples presented, the binding rate coefficient, k, increases as the fractal dimension, D f, or the degree of heterogeneity increases on the biosensor surface. This was noted for the single-fractal analysis presented. More

TABLE 4 Binding Rate Coefficient Expressions as a Function of the Fractal Dimension, Df, and the Analyte Concentration, and Fractal Dimension Expressions as a Function of the Analyte Concentration for Different Analyte–Receptor Systems

Analyte–Receptor System

Binding rate coefficient or fractal dimension expression

Cell surface proteins from H. pylori strain 52 in the absence and in the presence of 5 and 10 mM free sialyl-(a-2,3-lactose)/ 39SL-PAA immobilized on a RMB

k 5 (161.34 6 3.579) [sialyl lactose]0.072760.0024 D f 5 (2.5997 6 0.0207) [sialyl lactose]0.0049260.00087 k 5 (0.000216 6 0.000032) D 14.14962.980 f

(44)

Diluted and concentrated H. pylori cells of strain 52/39SL-PAA immobilized on a RMB

k 5 (1.512 6 2.421) [normalized cell]2.866660.2239 D f 5 (1.6246 6 0.1474)[normalized cell]0.316160.0509 k 5 (0.0218 6 0.0145) D 8.779560.9171 f

(44)

Adsorption of analyte (IgG, g-globulin)/biotinylated ITO surface

k 5 (0.00447 6 0.00129) D 2.544760.8626 f

(49)

Ref.

ANALYTE–RECEPTOR BINDING KINETICS

examples need to be analyzed to see if this is true for other systems (both antigen–antibody, and in general, analyte–receptor) utilized in biosensor applications. The analysis provides physical insights into the antigen–antibody reactions occurring on the biosensor surface. In general, the analysis is extendable to analyte–receptor reactions occurring on nonbiosensor (for example, cell surface reactions on membranes) surfaces. The quantitative expressions developed for the different antigen– antibody systems should assist in better control of the biosensor performance parameters such as stability, selectivity, sensitivity, and response time. More detailed and precise studies are required to determine the influence of the degree of heterogeneity that exists on the biosensor surface on the binding rate coefficient. An increase in the binding rate coefficient, k, should lead to enhanced sensitivity and to a decrease in the response time of the biosensor. Both of these aspects would be beneficial in biosensor construction. For a selective (or multiple) reaction system, if an increase in the D f value leads to an increase in the binding rate coefficient, k (of interest), then this would enhance selectivity. Stability is a more complex issue, and one might intuitively anticipate that a distribution or heterogeneity that exists on the biosensor surface would lead to a more stable sensor. Similar behavior has been observed for the deactivation of enzymes wherein a distribution of activation energies for deactivation (as compared to a single activation energy for deactivation) leads to a more stable enzyme (53). Another parameter that is not considered (or rarely) in the biosensor literature, but often in control theory, is robustness. This may be defined as insensitivity to measurement errors as far as biosensor performance is concerned. At this point, it is difficult to see how D f and k would affect biosensor robustness. REFERENCES 1. Scheller, F. W., Hintsche, R., Pfeifer, D., Schubert, D., Rediel, K., and Kindevater, R., Sens. Actuators 4, 197 (1991). 2. Whetton, C., In. Tech. August, 24 (1977). 3. Kleinjung, F., Klussmann, S., Erdmann, V. A., Scheller, F. W., Furste, J. P., and Bier, F. F., Anal. Chem. 70, 328 (1998). 4. Pisarchick, M. L., Gesty, D., and Thompson, N. L., Biophys. J. 63, 215 (1992). 5. Bluestein, B. I., Craig, M., Slovacek, R., Stundtner, L., Uricouli, C., Walziak, I., and Luderer, A., in “Biosensors with Fiberoptics” (D. Wise and L. B. Wingard, Jr., Eds.), pp. 181–223. Humana Press, Clifton, NJ, 1991. 6. Eddowes, M. J., Biosensors 3, 1 (1987/1988). 7. Stenberg, M., Stiblert, L., and Nygren, H. A., J. Theor. Biol. 120, 129 (1986). 8. Nygren, H. A., and Stenberg, M., J. Colloid Interface Sci. 107, 560 (1985). 9. Stenberg, M., and Nygren, H. A., Anal. Biochem. 127, 183 (1982). 10. Sadana, A., and Sii, D., J. Colloid Interface Sci. 151(1), 166 (1992). 11. Sadana, A., and Sii, D., Biosens. Bioelectron. 7, 559 (1992). 12. Sadana, A., and Madagula, A., Biosens. Bioelectron. 9, 45 (1994). 13. Sadana, A., and Beela Ram, A., Biosens. Bioelectron. 10, 301 (1995). 14. Leung, P. T., Pollardknight, D., Malan, G. P., and Finlan, M. F., Sensors Actuators B 22(3), 175 (1994).

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