A single- and a dual-fractal analysis of antigen–antibody binding kinetics for different biosensor applications

Biosensors & Bioelectronics 14 (1999) 515 – 531 www.elsevier.com/locate/bios

A single- and a dual-fractal analysis of antigen–antibody binding kinetics for different biosensor applications Ajit Sadana Chemical Engineering Department, Uni6ersity of Mississippi, MS 38677 -9740, USA

Abstract The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The data is adequately described by a single- or a dual-fractal analysis. Initially, the data was modelled by a single-fractal analysis. If an inadequate fit was obtained then a dual-fractal analysis was utilized. The regression analysis provided by Sigmaplot, 1993 (Scientific Graphing Software: User’s Manual. Jandel Scientific, San Rafael, CA) was utilized to determine if a single-fractal analysis is sufficient, or a dual-fractal analysis is required. In general, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction (except for a single example) for the antigen–antibody systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the antigen–antibody binding systems indicate a high sensitivity of the binding rate coefficient on the fractal dimension when both a single -as well as a dual-fractal analysis is used. For example, for a single-fractal analysis and for the binding of human endothelin-1 (ET-1) antibody in solution to ET-115 – 21·BSA (bovine serum albumin) immobilised on a surface plasmon resonance surface, the order of dependence of the binding rate coefficient, k on the fractal dimension, Df is 7.0945. Similarly, for a dual-fractal analysis and for the binding of parasite L. dono6ani diluted pooled sera in solution to fluorescein isothiocyanate-labeled anti-human immunoglobulin IgG immobilized on an optical fibre, the order of dependence of k1 and k2 on Df1 and Df2 were 6.8018 and −4.393, respectively. Binding rate coefficient expressions are also developed as a function of the analyte (antigen or antibody) concentration in solution. The binding rate coefficient 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 emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Antigen – antibody binding; Biosensor; Dual-fractal analysis; Kinetics

1. Introduction Sensitive detection systems (or sensors) are required to detect a wide range of substances. Sensors find application in the areas of biotechnology, physics, chemistry, medicine, aviation, oceanography, detection of explosives (e.g. buried land mines and in aircraft luggage), and in environmental control. The importance of providing a better understanding of the mode of operation of biosensors to improve their sensitivity, stability, and specificity has been emphasized (Scheller et al., 1991). Whetton (1997) refers to biosensors as electronic noses, and emphasizes the reduction in time required in the detection process. The solid-phase immunoassay technique represents a convenient method for the separation and/or detection of reactants (e.g. antigen) in solution. The binding of antigen to an

antibody-coated surface (or vice versa) is sensed directly and rapidly. Also, due to its ease of use and relatively low costs, immunoassays such as enzymelinked immunosorbent assay (ELISA) are used extensively for biomedical analysis (Loomans et al., 1997). Besides, interesting extensions of the basic immunosensing method to say the capacitive method (wherein the capacitance of the receptor layer is measured) are available. Thus, there is a need to characterize the antigen– antibody reactions that are occurring on different surfaces used for different types of biosensor applications. The details of the association of antibodies (or antigens) with antigen-coated (or antibody-coated) surface is of tremendous significance for the development and characterization of immunodiagnostic devices as well as for biosensors (Pisarchick et al., 1992). The receptor

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A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

surface may be far from optimum due to poor receptor binding/adsorption, random orientation of the receptor molecule, alteration of conformation of the receptor molecule on interaction with the surface, steric hindrance, and altered kinetics (e.g. due to surface roughness or diffusional limitations). External diffusional limitations play a role in the analysis of such assays (Stenberg and Nygren, 1982; Nygren and Stenberg, 1985; Stenberg et al., 1986; Bluestein et al., 1987; Sadana and Sii, 1992a,b; Sadana and Madagula, 1994; Sadana and Beelaram, 1995). An analysis is available for the influence of partial mass transport limitation on protein–ligand binding for the BIAcore biosensor (Christensen, 1997). The analysis is available for both the heterogeneities of the analyte (in solution) and the ligand (on the surface). Kopelman (1988) indicates that surface-diffusioncontrolled 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 (e.g. binding) coefficients. Fractals are disordered systems and the disorder is described by non-integral dimensions (Pfeifer and Obert, 1989). These authors 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 (Pajkossy and Nyikos, 1989). Details about fractal applications in the biological and other sciences are available in previous publications (Sadana, 1998a,b,c). Thus, they are not repeated here to avoid repetition. Considering that both the chemistry as well as the environment plays a significant role on the reactions occurring at interfaces, the application of fractals to biosensor development is an attempt to try to understand the dynamic morphology of the reaction interface, and how it may be utilized to enhance the stability, sensitivity, reaction time, reliability, and other biosensor performance parameters of importance. A fractal analysis would provide novel physical insights into the diffusion-controlled reactions occurring at the surface. Fractal aggregate scaling relationships have been developed for both diffusion-limited and diffusion-limited cluster aggregation (DCLA) processes in spatial dimensions 2, 3, 4 and 5 (Sorenson and Roberts, 1997). These authors noted that the ‘prefactor’ displays uniform trends with the fractal dimension, Df. Fractal dimension values for the kinetics of antigen – antibody binding (Milum and Sadana, 1997; Sadana, 1997) and for analyte–receptor (Sadana and Sutaria, 1997a) binding are available. In these studies the influence of the experimental parameters such as analyte concentration on the fractal dimension and on the binding rate coeffi-

cient (the ‘prefactor’ in this case) were analysed. A direct attempt was then made to relate the influence of the degree of surface roughness, such as the fractal dimension on the binding rate coefficient for analyte– receptor reactions (Sadana, 1998a,b,c). One would like to further delineate the role of surface roughness on antigen–antibody reactions occurring on different types of biosensor arrangements, such as the optical fibre, surface plasmon resonance (SPR), capacitance biosensor, grating coupler chip, etc. We do this both for a single-fractal analysis and also for a dual-fractal analysis. The dual-fractal analysis is utilized only if the single-fractal analysis does not provide an adequate fit, and the dual-fractal analysis leads to a significant improvement in the fit. Thereafter, we present quantitative relationships for the binding rate coefficient(s) as a function of (a) the systems parameters (such as analyte concentration) and (b) of the fractal dimensions obtained for antigen–antibody reactions occurring on different types of biosensors. The non-integer orders of dependence obtained for (a) the binding rate coefficient, k, and the fractal dimension, Df for a single-fractal analysis, and for (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 for a dualfractal analysis further reinforces the multi-fractal nature of the antigen–antibody binding systems occurring on different types of biosensors.

2. Theory An analysis of the binding kinetics in solution to antibody immobilised on the biosensor surface is available (Milum and Sadana, 1997). The influence of lateral interactions on the surface and variable rate coefficients is also available (Sadana and Madagula, 1993). Here we initially present a method of estimating actual fractal dimension values for antigen–antibody binding systems utilized in different types of biosensors.

2.1. Variable binding rate coefficient Kopelman (1988) indicates that classical reaction kinetics are sometimes unsatisfactory when the reactions are spatially constrained on the microscopic level by either walls, phase 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: k1 = k%t − b

(1a)

In general, k1 depends on time, whereas k%= k1 (t= 1) does not. Kopelman (1988) indicates that in three dimensions (homogeneous space), b=0. This is in

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

agreement with the results obtained in classical kinetics. Also, with vigorous stirring, the system is made homogeneous and b=0 again. 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 has been analysed (Di Cera, 1991). 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, k1 is the sum of its deterministic value (invariant) and the fluctuation [z(t)] (Di Cera, 1991). This z(t) is a random function with a zero mean. The decreasing and increasing binding rate coefficients can both be assumed to possess an exponential form (Cuypers et al., 1987; Sadana and Madagula 1993): k1 = k1,0 exp(bt)

(1b)

Here, b and k1,0 are constants, b \0 and bB0 stands for increasing and decreasing binding rate, respectively. The influence of a decreasing and an increasing binding rate coefficient on the antigen concentration near the surface has been analysed when the antibody is immobilized on the surface (Sadana and Madagula, 1993). 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) immobilised on a biosensor surface has been analysed within a fractal framework (Milum and Sadana, 1997; Sadana, 1997). Furthermore, experimental data presented for the binding of HIV virus (antigen) to the antibody anti-HIV immobilized on a surface displays a characteristic ordered ‘disorder’ (Anderson, 1993). 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 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 towards fractal surfaces has been analysed (Sadana and Beelaram, 1994, 1995; Sadana, 1997; Sadana and Sutaria, 1997b). Havlin (1989) has briefly reviewed and discussed these results.

2.2. Single-fractal analysis Havlin (1989) 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

517

form a product [antibody–antigen complex; Ab·Ag] is given by: (3 − Df )

(Ab·Ag) t

2

=tp

(Ab·Ag) t 1/2

tB tc (2a)

t\ tc

Here Df is the fractal dimension of the surface. The fractal dimension in our case reflects the extent of heterogeneity that exists on the biosensor surface. Eq. (2a) indicates that the concentration of the product Ab·Ag(t) in a reaction Ab + Ag“ Ab·Ag on a solid fractal surface scales at short and intermediate scales as Ab·Ag t p with the coefficient p= (3− Df)/2 at short time scales, and p=1/2 at intermediate time scales. This equation is associated to 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), k1 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 k1 being independent of time. In all other situations one would expect a scaling behavior given by k1  k%t − b with − b=pB0. Also, the appearance of the coefficient, p, different from p=0 is the consequence of two different phenomena, that is the heterogeneity (fractality) of the surface, and the imperfect mixing (diffusion-limited) condition. Havlin (1989) indicates that the crossover value may be determined by r 2c  tc. Above the characteristic length, rc, the self-similarity is lost. Above tc, the surface may be considered homogeneous, since the selfsimilarity property disappears, and ‘regular’ diffusion is now present. For the present analysis, tc, 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.

2.3. Dual-fractal analysis The single-fractal analysis presented above is extended to include two fractal dimensions. At present, the time (t= t1) 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 analysed and the 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·Ag) concentration on the biosensor surface is given by: (3 − Df1) 2

(Ab·Ag) t

= t p1

tB t1

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

518 (3 − Df2)

(Ab·Ag)t

2

= t p2

(Ab·Ag)t 1/2

t \tc

t1 Bt Bt2 =tc (2b)

3. Results At the outset it is appropriate to indicate that the mathematical approach is straightforward. It is assumed that the fractal approach applies which may be a limitation. No physical basis or independent observation of the model is presented. The analysis is performed on experimental material available in the literature. The experimental points presented in the figures are the data available in the literature. An HP ScanJet 4C (Hewlett Packard, Boise, ID, USA) may be used to digitize plots of kinetic data available in the literature. The software Un-Scan-it (Un-Scan-it, 1995) may then be used to obtain numerical values for concentration versus time for each of the analyte – receptor reactions presented. The modelling of this experimental data by Eqs. (2a) and (2b) is the contribution made in this manuscript. In the tables the experimental conditions are those available in the literature. The values of the binding rate coefficients and fractal dimensions presented are the contributions made in this manuscript. Laricchia-Robbio et al. (1997) have analyzed the binding of protein-A-purified rabbit anti-ET-1 antibody incubated with the peptide ET-116 – 21 in solution to ET-1·BSA (bovine serum albumin) coupled to an extended carboxymethylated hydrogel matrix in a BIAcore biosensor system. Fig. 1a shows the curves obtained using Eq. (2a) for a single-fractal analysis. Table 1 shows the values of the binding rate coefficient, k and the fractal dimension, Df obtained using Sigmaplot (1993) to fit the data. The equation: [Ab·Ag]=kt

(3 − Df )/2

(2c)

was used to obtain the value of k and Df for a singlefractal analysis. The values of the parameters presented in Table 1 are within 95% confidence limits. For example, for the binding of protein-A-purified rabbit antiET-1 antibody+ET-116 – 21 in solution to ET-1·BSA immobilised on extended carboxymethylated hydrogel matrix in a BIAcore biosensor the value of k reported is 3.392790.1401. The 95% confidence limits indicates that 95% of the k values will lie between 3.2526 and 3.5328. Note that an important implication derived from the estimated value of Df is that the relevant process or analyte– receptor binding in this case must occur in an interval of (Df +1)-dimensional space. This also applies to all of the results presented below. Fig. 1b shows the binding of polyclonal rat anti-ET115 – 21 in solution to ET-115 – 21·BSA immobilised on

extended carboxymethylated hydrogel matrix. In this case too, 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, Df. Fig. 1c shows the binding of polyclonal rabbit anti-ET115 – 21 in solution to ET-115 – 21·BSA immobilised on extended carboxymethylated hydrogel matrix. 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, Df. It is of interest to note that as one goes from the polyclonal rabbit anti-ET-115 – 21 to the polyclonal rat anti-ET-115 – 21 the fractal dimension increases by about 2.4% from Df = 2.0622 to Df = 2.1120 and the binding rate coefficient, k increases by about 37.4% from k= 39.143 to k= 62.781. Note that increases in the fractal dimension and in the binding rate coefficient are in the same direction. An increase in the degree of heterogeneity on the biosensor surface (increase in Df) leads to an increase in the binding rate coefficient. Laricchia-Robbio et al. (1997) performed experiments of epitope mapping to determine whether the binding of the first antibody to the immobilized antigen would affect the binding of a second antibody, or vice versa. This also provided them with information on the number of distinct active epitopes on the surface of ET-1. Fig. 1d shows the binding of rat anti-ET-115 – 21 monoclonal (MAb 24/24) in solution to immobilized ET-1. The analyte was injected to saturate all ET-115 – 21 binding sites. Table 1 shows the values of the binding rate coefficient, k and the fractal dimension, Df. A single-fractal analysis is adequate to describe the binding kinetics. Fig. 1e shows the binding of rabbit antiET-1 monoclonal (MAb 24/24) in solution to immobilized ET-1. Table 1 shows the values of the binding rate coefficient, k and the fractal dimension, Df. A single-fractal analysis is again adequate to describe the binding kinetics. In this case, it is of interest to note that as one goes from the rabbit anti-ET-1 to the rat ET-1 antibody in solution, a slight decrease in the fractal dimension by about 4.7% from Df = 2.3492 to Df = 2.2388 leads to slightly more than a doubling in the binding rate coefficient, k from a value of 113.63 to 228.79. In this case, a slight decrease in the degree of heterogeneity (Df) on the surface leads to an increase in the binding rate coefficient, k. This is in contrast to the result presented above where the opposite trend was obtained. No explanation is offered, at present, to explain this change in behavior. It would be of interest to make comparisons and conclusions of the influence of the surface on the binding rate coefficients. However, this assumes that data is available from a single study wherein sufficient experiments have been performed under different conditions and are made available. When such data is available attempts have been made to make comparisons and conclusions. Sometimes very few data is available in a single study. In that case

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

guarded or cautionary comments may be made when comparing experimental data from very different systems and conditions. Fig. 2 shows that the binding rate coefficient, k increases as the fractal dimension, Df increases. This is in accord with the prefactor analysis of fractal aggregates (Sorenson and Roberts, 1997), and for the analyte –receptor binding kinetics for biosensor applications (Sadana, 1998a). For the data presented in Table 1, the binding rate coefficient, k is given by: 9 1.2299 k =(0.3510 90.2473)D 7.0945 f

(3a)

519

This predictive equation fits the values of the binding rate coefficient, k presented in Table 1 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. There is some scatter in the data at the higher fractal dimension values. 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 this case, at least at the higher

Fig. 1. Binding and epitope mapping of human endothelin-1 (ET-1) by surface plasmon resonance (Laricchia-Robbio et al., 1997). (a) Protein-A-purified rabbit anti-ET-1 antibody + ET-116 – 21 in solution to ET-115 – 21·bovine serum albumin (BSA) immobilised on extended carboxymethylated hydrogel matrix; (b) polyclonal rat anti-ET-115 – 21 in solution to ET-115 – 21·BSA immobilised on extended carboxymethylated hydrogel matrix; (c) polyclonal rabbit anti-ET-115 – 21 in solution to ET-115 – 21·BSA immobilised on extended carboxymethylated hydrogel matrix; (d) rat MAb 24/24 antibody anti-ET-115 – 21 in solution to immobilised ET-1; (e) rabbit anti-ET-1 antibody in solution to immobilised ET-1.

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520

Table 1 Influence of different parameters on the binding rate coefficients and fractal dimensions for the binding of human endothelin-1 (ET-1) using a surface plasmon resonance-based biosensor (BIAcore Technology): single-fractal analysis (Laricchia-Robbio et al., 1997)a Analyte in solution/receptor on surface

Binding rate co- Fractal dimension, Df efficient, k

Protein-A-purified rabbit anti-ET-1 antibody+ET-116-21/ET-115–21·BSA immobilized on extended carboxy-methylated hydrogel matrix Polyclonal rat anti-ET-115–21/ET-115–21·BSA immobilized on extended carboxy-methylated hydrogel matrix Polyclonal rabbit anti-ET-115–21/ET-115–21·BSA immobilized on extended carboxy-methylated hydrogel matrix Rat monoclonal Mab24/24 antibody anti-ET-115–21/immobilized ET-1 Rabbit anti-ET-1 antibody/immobilized ET-1

3.3927 90.1401

1.3658 90.0544

62.781 90.915

2.112 9 0.0122

39.143 9 0.816

2.0622 90.0278

228.79 9 10.123 113.63 94.248

2.2388 9 0.0390 2.3492 90.0474

a

BSA, bovine serum albumin.

fractal dimension values. 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 = aD bf +cD df

(3b)

Here a, b, c and d are the coefficients to be determined by regression. The 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. Loomans et al. (1997) have monitored peptide antibody binding using reflectometry. They were able to obtain the association and dissociation constants of the binding reaction between the antibody in solution and the immobilized antigen. The curve obtained using Eq. (2a) for the binding between 100 mg/ml mouse anti-human chorionic gonadotrophin (hCG) monoclonal antibody OT-3A in solution to a derivative of peptide 3A (K-7) peptide physically adsorbed to the surface is shown in Fig. 3a. A single-fractal analysis is adequate to describe the binding kinetics. The values of the binding rate coefficient, k and the fractal dimension, Df are given in Table 2. Similarly, Fig. 3b shows the binding curve obtained between 100 mg/ml mouse antihCG monoclonal antibody OT-3A in solution to a derivative of peptide 3A (H-peptide) physically adsorbed to a surface. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient, k and the fractal dimension, Df are given in Table 2. It is of interest to note that in both of these cases the fractal dimension is almost the same. Df is equal to 2.5512 and 2.5974 for the K-7 peptide and the H-peptide, respectively. However, there is a significant difference in the binding rate coefficient for these two cases. The binding rate coefficient is larger by a factor of 2.28 for the K-7 peptide (k =1.6233) when compared with the corresponding coefficient for the H-peptide (k= 0.7103). This indicates that the OT-3A:K-7 peptide

binding reaction is more sensitive to the degree of heterogeneity (Df) on the reflectometer surface than the OT-3A:H-peptide binding reaction. Also, since these two binding reactions are adequately described by a single-fractal analysis, one can say, with caution, that the binding mechanisms for both of these cases are similar. It is also of interest (and as mentioned above) to note that the binding reaction of the antibody OT3A in solution to these two peptides physically adsorbed to a surface leads to values of the fractal dimension (or the degree of heterogeneity) that are within 1.81% of each other. Fig. 3c shows the binding curves obtained using a single-fractal analysis [Eq. (2a)] and a dual-fractal analysis [Eq. (2b)] for 100 mg/ml OT-3A in solution to Ata-peptide (a derivative of peptide 3A) physically adsorbed on a reflectometer surface. In this case, a singlefractal analysis does not provide an adequate fit, and thus a dual-fractal analysis is used. Table 2 shows the values of (a) the binding rate coefficient, k and the fractal dimension, Df for a single-fractal analysis, and (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 for a dual-fractal analysis. Clearly, the dual-fractal analysis provides a much better fit than that obtained from a single-fractal analy-

Fig. 2. Influence of the fractal dimension, Df on the binding rate coefficient, k.

Analyte in solution/receptor on surface

k

Df

OT-3A (100 mg/ml) in PBS/physically adsorbed derivative of peptide 3A (K-7 peptide) OT-3A (100 mg/ml) in PBS/physically adsorbed derivative of peptide 3A (H-peptide) OT-3A (100 mg/ml) i25n PBS/physically adsorbed derivative of peptide 3A (Ata-peptide) HRP (100% MeCn medium)/anti-HRP HRP (aqueous medium)/anti-HRP 16.5 mg/ml HSA /polyclonal anti-HSA immobilized on capacitance immunosensor 16.5 mg/ml HSA/monoclonal anti-HSA immobilized on capacitance immunosensor

1.6233 9 0.123

2.5512 90.028

NA

NA

NA

NA

0.7103 9 0.032

2.5974 9 0.0234

NA

NA

NA

NA

0.4869 90.065

2.3138 9 0.076

0.3790 90.036

1.6524 90.018

2.07829 0.092

2.92649 0.023

0.0740 90.002 0.1671 90.035 0.1488 90.112

2.4060 90.031 2.024 9 0.129 2.1034 90.037

NA 0.1050 90.032 NA

NA 0.3363 90.013 NA

NA 1.6302 90.364 NA

NA 2.368 9 0.081 NA

0.2300 90.0465

2.1602 90.0528

0.1912 90.0091

0.5857 9 0.0068

1.76309 0.0528

2.7364 9 0.0174

a

PBS, phosphate-buffered saline; HRP, horseradish peroxidase; HSA, human serum albumin.

k1

k2

Df1

Reference

Df2

Loomans et al., 1997 Loomans et al., 1997 Loomans et al., 1997 Lu et al., 1997 Lu et al., 1997 Mirsky et al., 1997 Mirsky et al., 1997

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

Table 2 Influence of different parameters on fractal dimensions and binding rate coefficients for different analyte–receptor binding reactions: a single- and a dual-fractal analysisa

521

522

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Lu et al. (1997) have recently analyzed the influence of the water-miscible organic solvent acetonitrile on the enzymatic activity of horseradish peroxidase (HRP) and on the HRP–anti-HRP binding interaction. These authors noted that an increase in the concentration of this organic solvent led to increases in the enzymatic activity as well as in the binding reaction. These authors indicated that HRP was selected since it may be easily monitored, and also it is reasonably stable both in aqueous and in organic solvents. These authors immobilised anti-HRP to amine binding microwells. They indicated that a polystyrene microwell plate coated with a thin layer of dextran polymer containing N-oxysuccinimide groups can be readily utilized for the immobilisation of the antibody through the available amine moieties. Fig. 4a shows the binding curve obtained using a single-fractal analysis [Eq. (2a)] of HRP in 100% acetonitrile (20 ml of 100 mg/ml HRP solution mixed with 10 ml of absolute acetonitrile) to immobilised antiHRP. Table 2 shows the values of the binding rate coefficient, k and the fractal dimension, Df. Once again, a single-fractal analysis is adequate to describe the binding kinetics. Fig. 4b shows the binding curves obtained using a single-fractal analysis [Eq. (2a)] and a

Fig. 3. Binding of OT-3A antibody in solution to different physically adsorbed derivatives of peptide 3A by reflectometry (Loomans et al., 1997). (a) K-7 peptide; (b) H-peptide; (c) Ata-peptide (- - - -, singlefractal analysis; — , dual-fractal analysis).

sis. For the dual-fractal analysis it is of interest to note that as the fractal dimension increases by about 48% from Df1 =2.0782 to Df2 =2.9264 the binding rate coefficient increases by a factor of about 4.36 from k1 = 0.3790 to k2 =1.6524. Thus, the binding rate coefficient is quite sensitive to the degree of heterogeneity that exists on the reflectometer surface. Also, the changes in the fractal dimension and in the binding rate coefficient observed in this case are in the same direction. It would be of interest to provide some structural basis for this change in the binding mechanism of OT-3A to either the K-7 or the H-peptide (single-fractal mechanism applicable) and to the Ata-peptide (dual-fractal mechanism applicable). However, no such structural basis or other possible explanation is offered at present.

Fig. 4. Binding of horseradish peroxidase (HRP) in solution to anti-HRP immobilised to amine binding microwells (Lu et al., 1997). (a) 100% MeCN (acetonitrile) medium; (b) aqueous medium (- - - -, single-fractal analysis; — , dual-fractal analysis).

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dual-fractal analysis [Eq. (2b)] for HRP in aqueous solution to immobilized anti-HRP. Table 2 shows (a) the values of the binding rate coefficient, k and the fractal dimension, Df for a single-fractal analysis, and (b) the values of the binding rate coefficients, k1 and k2, and the fractal dimensions, Df1 and Df2 for a dual-fractal analysis. Clearly, the dual-fractal analysis provides a better fit than that obtained from a single-fractal analysis. For the dual-fractal analysis it is of interest to note that as the fractal dimension increases by about 44.7% from a value of Df1 = 1.6302 to Df2 =2.368 the binding rate coefficient increases by a factor of about 3.17 from k1 = 0.1050 to k2 =0.3363. Thus, once again, the binding rate coefficient is quite sensitive to the degree of heterogeneity that exists on the surface. Also, once again, the changes in the fractal dimension and in the binding rate coefficient are in the same direction. It is of interest to note that the binding of HRP in the organic solvent and in aqueous solution may be adequately described by a single- and a dual-fractal analysis, respectively. This indicates a change in the binding mechanism for these two cases. Apparently, there is some complexity in the HRP – anti-HRP interaction in aqueous solution as compared to the case when this interaction occurs in the organic solvent. No explanation is offered at present to help explain this change in the binding mechanism. Mirsky et al. (1997) have recently utilized an alternative method to detect antigen – antibody interactions. They measured the electrical capacitance of the receptor layer. This method has been used to analyze the adsorption of different species on metal electrodes. These authors indicate that the binding of human serum albumin (HSA) to the antibody layer leads to an increase in the effective dielectric thickness of the layer. This leads to a decrease in the electrical capacity. Fig. 5a shows the binding of the HSA in solution to the polyclonal antibody anti-HSA immobilised to a gold surface. A single-fractal analysis is adequate to describe the binding kinetics. Table 2 shows the values of the binding rate coefficient, k and the fractal dimension, Df. Fig. 5b shows the binding of HSA in solution to monoclonal antibody anti-HSA immobilised to a gold surface. Table 2 shows the values of (a) the binding rate coefficient, k and the fractal dimension, Df for a singlefractal analysis, and (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1, and Df2 for a dual-fractal analysis. Clearly, the dual-fractal analysis provides a better fit than that obtained from a singlefractal analysis. For the dual-fractal analysis it is of interest to note that as the fractal dimension increases by a factor of about 1.55 from a value of Df1 =1.7630 to Df2 =2.7364, the binding rate coefficient increases by a factor of 3.06 from k1 =0.1912 to k2 =0.5857. There is apparently a change in the binding mechanism in the

523

Fig. 5. Binding of human serum albumin (HSA) in solution (Mirsky et al., 1997). (a) Polyclonal anti-HSA immobilised on a capacitance immunosensor; (b) monoclonal anti-HSA immobilised on a capacitance immunosensor (- - - -, single-fractal analysis; — , dual-fractal analysis).

two cases since for the monoclonal antibody case a dual-fractal analysis is required, whereas a single-fractal analysis is adequate for the polyclonal antibody case. Mirsky et al. (1997) suggest that the lower response of the polyclonal antibodies is due to the formation of a less dense layer with corresponding lower electrical conductivity when compared with the monoclonal antibodies. If we may extend our analysis then one can conclude that, at least for this case, and as observed for the monoclonal antibodies when compared with the polyclonal antibodies, a higher densely packed layer would yield a higher electrical conductivity, and subsequently result in a more complex binding interaction (dual-fractal analysis). No structural basis or any other explanation, is offered at present to indicate in some cases that a single-fractal analysis was adequate, whereas in others it failed, and a dual-fractal analysis was required. Brynda et al. (1998) have analyzed the binding of HRP from phosphate-buffered physiological saline (PBS) to anti-HRP immobilised on a grating coupler chip. They examined the binding of HRP to both an anti-HRP monolayer, as well as to two cross-linked anti-HRP layers. They indicate that the increase in

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activity due to multi-layers may be due to (a) an increase in the number of receptors, and (b) also due to a better activity of the anti-HRP molecules that are not in direct contact with the solid support. Fig. 6a shows the binding curves obtained using a single-fractal analysis [Eq. (2a)] and a dual-fractal analysis [Eq. (2b)] for the binding of HRP in PBS in solution to an anti-HRP monolayer immobilised to a grating coupler chip. In this case, a single-fractal analysis does not provide an adequate fit, and thus a dual-fractal analysis was used. Table 3 shows the values of (a) the binding rate coefficient, k and the fractal dimension, Df for a single-fractal analysis, and (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 for a dualfractal analysis. Clearly, the dual-fractal analysis provides a better fit than that obtained from a single-fractal analysis. It is of interest to note that as the fractal dimension increases by a factor of about 28.4% from Df1 =2.3362 to Df2 =3.0 (the maximum value), the binding rate coefficient increases by a factor of about 7.65 from k1 =2.4852 to k2 =19.031. As observed earlier, the changes in the fractal dimension and in the binding rate coefficient are in the same direction.

Fig. 6. Binding of horseradish peroxidase (HRP) from phosphatebuffered saline (PBS) in solution to anti-HRP immobilised on a grating coupler chip (Brynda et al., 1998). (a) Anti-HRP monolayer; (b) two cross-linked anti-HRP layers (- - - -, single-fractal analysis; — , dual-fractal analysis).

Fig. 6b shows the binding curves obtained using a single-fractal analysis [Eq. (2a)] and a dual-fractal analysis [Eq. (2b)] for the binding of HRP in PBS in solution to two cross-linked anti-HRP layers immobilised on a grating coupler chip. Once again in this case a single-fractal analysis does not provide an adequate fit, and thus a dual-fractal analysis was used. Table 3 shows the values of (a) the binding rate coefficient, k and the fractal dimension, Df for a single-fractal analysis, and (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 for a dualfractal analysis. Clearly, the dual-fractal analysis provides a better fit than that obtained from a single-fractal analysis. Once again, it is of interest to note that as the fractal dimension increases by about 34.7% from Df1 = 2.1174 to Df2 = 2.8560, the binding rate coefficient increases by a factor of about 8.76 from k1 = 2.1435 to k2 = 19.031. Note as observed above, the changes in the fractal dimension and in the binding rate coefficient are in the same direction. It is of interest to note that the values of the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 are all slightly lower for the two crosslinked anti-HRP layer case when compared to the monolayer anti-HRP case. This is in spite of the fact that the final saturation value is higher, as expected, for the two cross-linked anti-HRP layer case when compared to the monolayer case. Apparently, the degree of heterogeneity on the surface or the fractal dimension has a significant influence on the binding rate coefficient(s) value(s). Further careful studies are required to delineate this aspect better. Nath et al. (1997) have utilized a fibre optic evanescent sensor to detect L dono6ani antibodies in sera of kala azar patients. Cell surface protein of L. dono6ani was immobilised on a fibre optic sensor. In the first step the antigen on the fiber reacts with the L. dono6ani infected sample. In the second step this reacts with fluorescein isothiocyanate (FITC)-labeled anti-human IgG to generate the signal. Fig. 7a–e show the binding of parasite L. dono6ani diluted pooled sera (1:1600 to l:256 000) to FITC-labeled anti-human IgG immobilized on an optical fiber. The diluted pooled sera was diluted by a factor of two starting from 1:1600. In each of these cases a singlefractal analysis does not provide an adequate fit, and thus a dual-fractal analysis was used. Table 3 shows the values of (a) the binding rate coefficient, k and the fractal dimension, Df for a single-fractal analysis, and (b) the binding rate coefficients, k1 and k2 and the fractal dimensions, Df1 and Df2 for a dual-fractal analysis. In each of these cases it is of interest to note that, and as indicated above, an increase in the fractal dimension from Df1 to Df2 leads to an increase in the binding rate coefficient from k1 to k2. Note that, and as indicated above, the changes in the fractal dimension

Analyte in solution/receptor on surface

k

Df

k1

k2

Df1

Df2

Reference

HRP–PBS /anti-HRP monolayer immobilized on grating coupler chip HRP–PBS/two cross-linked anti-HRP layers immobilized on grating coupler chip Parasite L. dono6ani diluted pooled sera (1:1600)/FITC-labeled anti-human IgG immobilized on optical fiber Parasite L dono6ani diluted pooled sera (1:3200)/FITC-labeled anti-human IgG immobilized on optical fiber Parasite L. dono6ani diluted pooled sera (1:6400)/FITC-labeled anti-human IgG immobilized on optical fiber Parasite L dono6ani diluted pooled sera (1:128 000)/FITC-labeled anti-human IgG immobilized on optical fiber Parasite L. dono6ani diluted pooled sera (1:256 000)/FITC-labeled anti-human IgG immobilized on optical fiber

4.4467 9 0.424

2.5698 90.048

2.4852 90.1033

19.031 9 0.351

2.336290.042

3.0−0.0007

4.9370 90.517

2.4528 9 0.051

2.1435 90.075

18.791 9 0.535

2.117490.042

2.85609 0.047

2.4453 90.144

1.9032 9 0.041

2.4021 90.028

3.1270 9 0.093

1.756090.015

2.20509 0.072

Brynda et al., 1998 Brynda et al., 1998 Nath et al., 1997

1.7039 9 0.149

1.8732 90.061

1.6628 9 0.062

2.2714 9 0.060

1.62769 0.049

2.237290.064

Nath et al., 1997

1.1052 9 0.127

1.7866 90.079

1.0708 90.066

1.5406 9 0.0408

1.46949 0.081

2.21449 0.064

Nath et al., 1997

0.6188 90.079

1.8160 9 0.088

0.5979 90.050

0.9221 9 0.061

1.528490.108

2.31709 0.158

Nath et al., 1997

0.4523 90.075

2.0150 90.011

0.4315 90.025

0.8247 9 0.035

1.569890.077

2.76449 0.102

Nath et al., 1997

a

HRP, horseradish peroxidase; PBS, phosphate-buffered saline; FITC, fluorescein isothiocyanate; Ig, immunoglobulin.

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

Table 3 Influence of different parameters on fractal dimensions and binding rate coefficients for different analyte–receptor reactions: a dual-fractal analysisa

525

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Fig. 7. Binding of parasite L. dono6ani diluted pooled sera in solution to fluorescein isothiocyanate (FITC)-labeled anti-human IgG immobilised on an optical fiber (Nath et al., 1997). (a) 1:1600; (b) 1:3200; (c) 1:6400; (d) 1:128 000; (e) 1:256 000 (- - - -, single-fractal analysis; — , dual-fractal analysis).

and in the binding rate coefficient are in the same direction. It is of interest to note that the binding rate coefficients, k1 and k2 both increase as the dilution factor (defined as the reciprocal of the dilution of the pooled positive serum) increases (see Fig. 8), e.g. the dilution factor for the 1:1600 case is 0.000625. In the dilution factor range analyzed, the binding rate coefficient, k1 is given by: k1 = (284.981920.363)[dilution factor]0.6426 9 0.0314 (4a) This predictive equation fits the values of k1 presented in Table 3 and in Fig. 8a reasonably well. The low exponent dependence of the binding rate coefficient, k1 on the dilution factor indicates that the binding rate

coefficient exhibits a rather low dependence on the dilution factor in this range analyzed. The fractional exponent dependence exhibited by k1 on the dilution factor further reinforces the fractal nature of the system. Similarly, in the dilution factor range analyzed, the binding rate coefficient, k2 is given by: k2 = (138.734915.344)[dilution factor]0.5143 9 0.0478 (4b) This predictive equation fits the values of k2 presented in Table 3 and in Fig. 8b reasonably well. The low exponent dependence of the binding rate coefficient, k2 on the dilution factor indicates, once again, that the binding rate coefficient exhibits a rather low dependence on the dilution factor in the range analyzed.

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

527

Once again, the fractional exponent dependence exhibited by k2 on the dilution factor further reinforces the fractal nature of the system. In the dilution factor range analyzed, the fractal dimension, Df1 is given by: Df1 =(2.281890.1363) [dilution factor]0.0414 9 0.0264 (5a) There is considerable scatter in the data (see Fig. 9a). A better fit could be obtained using the following equation: Df1 =a[dilution factor]b +c[dilution factor]d

(5b)

Such an equation would represent the points better, but this would be at the cost of additional parameters. No theoretical justification is available at present; thus this form is not used. It would, however, help model the ‘minima’ exhibited a lot better. The availability of more data points would also lead to a better fit and model development. Nevertheless, the predictive equation is of interest since it does indicate the influence of the dilution factor on the fractal dimension, Df or the degree of heterogeneity on the biosensor surface. In this case, due to the very low value of the exponent, the Df exhibits a negligible dependence on the dilution factor. Similarly, in the dilution factor range analyzed, the fractal dimension, Df2 is given by:

Fig. 9. Influence of the dilution factor on (a) the fractal dimension, Df1; (b) the fractal dimension, Df2.

Df2 = (1.26349 0.0851)[dilution factor] − 0.0702 9 0.0297 (5c) Once again, there is considerable scatter in the data (see Fig. 9b). A better fit could be obtained using Eq. (5b). Also, as indicated above, the availability of more data points would lead to a better fit and model development. In this case, however, in contrast to Eq. (5a) above, the fractal dimension, Df2 decreases with an increase in the dilution factor. No explanation is offered at present to explain this difference in behavior between Df1 and Df2 on the dilution factor. In any case, the fractal dimension, Df2 does exhibit a low dependence on the dilution factor as noted by the low value of the exponent (− 0.0702). In the dilution factor range analyzed, the binding rate coefficient, k1 is given by: 9 4.5751 k1 = (0.04409 0.0378)D 6.8081 f1

Fig. 8. Influence of the dilution factor on (a) the binding rate coefficient, k1; (b) the binding rate coefficient, k2.

(6a)

Fig. 10a shows the fit of this predictive equation. There is scatter in the data. Note the very high dependence of the binding rate coefficient, k1 on the fractal dimension. As observed previously [see Eq. (3a)], the binding rate coefficient is sensitive to the surface roughness or the degree of heterogeneity that exists on the surface. In this case, a correction to the data to account for the depletion of the analyte in the vicinity of the surface (imperfect mixing) would be of assistance. How-

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

528

binding rate coefficient, k2 is given by: 4.393 9 2.3720 k2 = (61.4956935.2369)D − f2

Fig. 10. Influence of (a) the fractal dimension, Df1 on the binding rate coefficient, k1; (b) the fractal dimension, Df2 on the binding rate coefficient, k2.

ever, at present, no such correction is presented. The availability of more data points would also be of assistance. A better fit could be obtained using a form of the equation given by Eq. (3b), with Df1 and Df2 replacing Df since now we are talking of a dual-fractal analysis. Similarly, in the dilution factor range analyzed, the

(6b)

Fig. 10b shows that k2 decreases with an increase in the fractal dimension, Df2. There is scatter in the data. Once again, a correction to the data to account for the depletion of the analyte in the vicinity of the surface (imperfect mixing) would be of assistance. However, at present, no such correction is presented. The availability of more data points would be assistance. Also, as indicated above, a better fit could be obtained using a form of the equation given by Eq. (3b). As observed previously [Eq. (3a)] and for k1 the binding rate coefficient is rather sensitive to the surface roughness or the degree of heterogeneity that exists on the surface. Table 4 summarizes the expressions for (a) the binding rate coefficient obtained as a function of the fractal dimension and (b) the fractal dimension obtained as a function of a system variable, such as reactant concentration (dilution factor in our case). Only single examples are presented for the binding rate coefficient for a single-fractal analysis (Laricchia-Robbio et al., 1997), and for a dual-fractal analysis (Nath et al., 1997). Nevertheless, for both of these cases, the binding rate coefficient is sensitive to the degree of heterogeneity that exists on the surface. The exponent dependence of k on Df is 7.0945 for a single-fractal analysis (LaricchiaRobbio et al., 1997). Similarly, the exponent dependence of k1 on Df1 and of k2 on Df2 are 6.8018 and −4.393, respectively, for a dual-fractal analysis (Nath et al., 1997). Even though only a couple of examples are presented for the dependence of the binding rate coefficient on the fractal dimension, the table does provide some perspective of the dependence of the binding rate coefficient on the fractal dimension. Finally, expressions for the fractal dimension as a function of a

Table 4 Binding rate coefficient expressions as a function of the reaction parameter (dilution factor) and the fractal dimension, and fractal dimension expressions as a function of the dilution factora Antigen–antibody system

Binding rate coefficient or fractal dimension expression

Reference

Human ET-1/ET-115–21·BSA or ET-1 immobilized on surface

9 1.2299 k = (0.3510 90.2473)D 7.0945 f

Parasite L. dono6ani diluted pooled sera/FITC-labeled anti-human IgG immobilized on an optical fiber

9 4.5751 k1 =(0.0440 9 0.0378)D 6.8018 f

Laricchia-Robbio et al., 1997 Nath et al., 1997

9 2.3720 k2 =(61.4956 935.2369)D−4.393 f k1 =(284.981 920.363)[dilution factor]0.6426 9 0.0314 k2 =(138.734 915.344)[dilution factor]0.5143 9 0.0478 Df1 =(2.2818 9 0.1363)[dilution factor]0.0414 9 0.0264 Df2 =(1.2634 9 0.0851)[dilution factor]−0.0702 9 0.0297 a

ET-1, endothelin-1 antibody; BSA, bovine serum albumin; FITC, fluorescein isothiocyanate; Ig, immunoglobulin.

A. Sadana / Biosensors & Bioelectronics 14 (1999) 515–531

reaction parameter (dilution factor in our case) are also presented. At least for the example presented, the fractal dimension(s) exhibits a very low dependence on the dilution factor. More data needs to be analyzed, and subsequently such (or similar) predictive equations obtained to further delineate the above dependence obtained. 4. Conclusion A fractal analysis of the binding of antigen (or antibody) in solution to the antibody (or antigen) immobilised on the biosensor surface provides a quantitative indication of the state of disorder (fractal dimension, Df) 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 systems. Both types of examples are given wherein either a single-or a dual-fractal analysis is required to adequately describe the binding kinetics. The dual-fractal analysis was used only when the single-fractal analysis did not provide an adequate fit. This was done by the regression analysis provided by Sigmaplot (1993). In accord with the prefactor analysis for fractal aggregates (Sorenson and Roberts, 1997), quantitative (predictive) expressions are developed for (a) the binding rate coefficient, k as a function of the fractal dimension, Df for a single-fractal analysis, and for (b) the binding rate coefficients k1 and k2 as a function of the fractal dimensions, Df1 and Df2, respectively for a dual-fractal analysis. Predictive expressions are also developed for the binding rate coefficient and the fractal dimension as a function of the analyte (antigen or antibody) concentration in solution. The fractal dimension, Df is not a classical independent variable such as analyte (antigen or antibody) concentration. Nevertheless, the expressions obtained for the binding rate coefficient for a single- and a dual-fractal 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 the binding of human ET-1 antibody in solution to ET-115 – 21·BSA immobilised on a surface (Laricchia-Robbio et al., 1997) the order of dependence of k on Df was 7.0945 using the SPR technique. Similarly, in the case of a dual-fractal analysis and for the binding of parasite L. dono6ani diluted pooled sera in solution to FITC-labeled anti-human IgG immobilised on an optical fibre (Nath et al., 1997) the order of dependence of k1 and k2 on Df1 and Df2 were 6.8018 and −4.393, respectively. This emphasizes the importance of the extent of heterogeneity on the biosensor surface and its impact on the binding rate coefficient.

529

Note that 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 deviation in the data that may be minimized by (a) providing a correction for the depletion of the antigen (or antibody) in the vicinity of the surface (imperfect mixing), and by (b) using a four-parameter equation [e.g. Eq. (3b)]. In this case, more data points would be required to justify the use of Eq. (3b). Table 4 summarizes the binding rate coefficient expressions obtained as a function of the fractal dimension for two different systems (one example each of a single- and a dual-fractal analysis) utilized in biosensor applications. Other predictive expressions developed for the binding rate coefficient and for the fractal dimension as a function of the analyte (antigen or antibody) concentration in solution also given in Table 4 are of significance since they provide a means by which these parameters may be controlled. In general, and in one of the examples presented, the binding rate coefficient increases as the fractal dimension increases (Sadana, 1998a,b). This was noted for the single-fractal analysis presented. The analysis provides physical insights into the antigen–antibody (and in general, analyte–receptor) reactions occurring on biosensor surfaces. For the most part, the analysis is also extendable to analyte–receptor reactions occurring on non-biosensor (e.g. cell surface reactions on membranes) surfaces. For the dual-fractal analysis example presented, however, the binding rate coefficient, k2 decreases with an increase in the fractal dimension, Df2 even though k1 increases with an increase in the fractal dimension, Df1. Such decreases in the binding rate coefficient with an increase in the fractal dimension are rare, but they do exist. For example, during the direct binding of TRITC-labeled low-density lipoprotein (LDL) to an optical fibre-based biosensor, and for a single-fractal analysis, an increase in the fractal dimension did lead to a decrease in the binding rate coefficient (Sadana, 1998a). Note however, that in this case of protein binding (or adsorption), there was no receptor on the surface. This is a case of ‘receptorless’ binding. The quantitative expressions developed for the different antigen–antibody systems should assist in the better control of biosensor performance parameters such as stability, 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 when ever either a single- or a dual-fractal analysis applies. An increase in the binding rate coefficient value 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

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a selective (or multiple) reaction system, if an increase in the Df value leads to an increase in the binding rate coefficient (of interest), then this would enhance selectivity. Stability is a more complex issue, and one might intuitively anticipate that a distribution or heterogeneity of the receptor on the biosensor surface would lead to a more stable biosensor. This is especially true if the receptor has a tendency to inactivate or lose its binding capacity to the analyte in solution. 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 (Malhotra and Sadana, 1987). Whenever a distribution exists, then it should be precisely determined, especially if different distributions are known to exist (Malhotra and Sadana, 1990). This would help characterize the distribution present on the surface, and would influence the temporal nature of the binding rate coefficient on the surface. The present analysis only makes quantitative the extent of heterogeneity that exists on the surface, with no attempt at determining the qualitative nature of the distribution that exists on the surface. Much more detailed and precise data are required before any such attempt may be carried out. Finally, 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 the binding rate coefficient and the fractal dimension would affect biosensor robustness.

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