Fractal Dimension and the Binding Rate Coefficient

Wink et al. (1998) analyzed interferon-)' (IFN-y) in solution using a liposome sandwich immunoassay developed specifically for SPR analysis. A 16-kDa cytokine (a capture monoclonal antibody, MD-2) was directly adsorbed onto a polystyrene layer. This polystyrene layer covers a gold surface attached to microtiter plates. These authors indicate that after the addition of the IFN-y, a hiotinylated detecting antibody is added. Avidin is the bridging molecule between the biotinylated antibody and the biotinylated liposomes. Figure 12.1 shows the curves obtained using Eq. (12.1a) for the binding of 20 ng/ml IFN-y in solution to 16-kDa cytokine (capture antibody, MD-2) adsorbed on a polystyrene surface. Note that without the addition of the liposomes there was hardly any noticeable shift in the resonance angle.

A single-fractal analysis is adequate to describe the binding kinetics. The entire binding curve is utilized to obtain the fractal dimension and the binding rate coefficient. Table 12.1a shows the values of the binding rate coefficient, k, and the fractal dimension, Dj. The values presented in the table were obtained from a regression analysis using Sigmaplot (1993) to model the experimental data using Eq. (12.1a), wherein (analyte • receptor) = kf. The k and Df values presented in the table are within 95% confidence limits. For example, for the binding of 20 ng/ml IFN-y in solution to 16-kDa cytokine (capture monoclonal antibody, MD-2) adsorbed on a polystyrene surface, the k value is 263.76 ± 3.915. The 95% confidence limit indicates that 95% of the k values will lie between 259.845 and 267.675. This indicates that the values are precise and significant. The curves presented in the figures are theoretical

Interferon

FIGURE 12.1 Binding of 20 ng/ml interferon-y in solution to 16-kDa cytokine (capture monoclonal antibody, MD-2) adsorbed on a polystyrene surface (Wink et ah, 1998).

Time, sec

FIGURE 12.1 Binding of 20 ng/ml interferon-y in solution to 16-kDa cytokine (capture monoclonal antibody, MD-2) adsorbed on a polystyrene surface (Wink et ah, 1998).

TABLE 12.1 Influence of Different Parameters on Fractal Dimensions and Binding Rate Coefficients for Different Analyte-Receptor Binding Reactions Utilizing Surface Plasmon Resonance: Single-Fractal Analysis

Analyte in solution/ receptor on surface

Binding rate coefficient, k Fractal dimension, Dj

Reference

16-kDa cytokine (capture monoclonal antibody, MD-2) absorbed on polystyrene surface

(b Protein-A-purified rabbit 3.3927 ± 0.1401

anti-ET-1 antibody + ET-116_21/ ET-l15_2i-BSA immobilized on extended carboxymethylated hydrogel matrix Polyclonal rat anti-l15.21/ 62.781 ± 0.915

ET-l15_2i-BSA immobilized on extended carboxymethylated hydrogel matrix Polyclonal rabbit anti-ET-115.21/ 39.143 ±0.816

ET-115.21-BSA immobilized on extended carboxymethylated hydrogel matrix Rabbit anti-ET-1 antibody/ 113.63 ± 4.428

immobilized ET-1

1.3658 ±0.0544 (Laricchia-Robbio et al., 1997)

2.112±0.0122 (Laricchia-Robbio et al., 1997)

2.0622 ± 0.0278 (Laricchia-Robbio et al, 1997)

2.3492 ± 0.0474 (Laricchia-Robbio et al, 1997)

curves. All of the examples to be presented involve the presence of SPR. In general, the trends to be presented are similar to the ones observed when either the SPR or another type of biosensor is utilized.

Laricchia-Robbio et al (1997) used a surface plasmon resonance biosensor for the detection and epitope mapping of endothelin-1. Endothelin-1 (ET-1) is a vasoconstrictor peptide that consists of 21 amino acids (Yanagisawa et al, 1988). Yanagisawa et al. originally isolated this peptide from porcine endothelial cells. Three isoforms of human ET (ET-1, ET-2, ET-3) have been identified (Inoue et al, 1989). Larrichia-Robbio et al. indicate that since ET-1 and its isoform exhibit high activity in hypertension and vasospasm, diagnostic tests are being developed for these compounds. Thus it is important to understand the structure-function relationships of these compounds and analyze their interactions with other receptors. Thus, these authors developed antibodies not only against ET-1, but also against the C-terminal eptapeptide, ET-l15_2i, due to its importance for receptor binding in ET-1.

The binding of protein-A-purified rabbit anti-ET-1 antibody incubated with the peptide ET-116_2i in solution to ET-1i5_2i-BSA coupled to an extended carboxymethylated hydrogel matrix in a BIACORE biosensor was analyzed by Laricchia-Robbio et al (1997). Figure 12.2a shows the curves obtained using Eq. (12.2a) for a single-fractal analysis. Table 12.1b shows the values of k and Dj. Figure 12.2b shows the binding of polyclonal rat anti-ET-115_2i in solution to ET-115_2i-BSA immobilized on an extended carboxymethylated hydrogel matrix. Here too, a single-fractal analysis is sufficient to describe the binding kinetics and Table 12.1b shows the values of k and Dj.

Figure 12.2c shows the binding of polyclonal rabbit anti-ET-li5_2i in solution to ET-115_21-BSA immobilized on an extended carboxymethylated hydrogel matrix. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 12.1b shows the values of k and Dj. Note that as one goes from the polyclonal rabbit anti-ET-1 i5_2i to the polyclonal rat anti-ET-115_21, Dj increases by about 2.4%—from 2.0622 to 2.1120—and k increases by about 37.4% from 39.143 to 62.781. Note also 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 Dj) leads to an increase in the binding rate coefficient. At present, no explanation is offered for this increase in the binding rate coefficient exhibited by a more heterogeneous surface, except that this is a phenomenological conclusion from the fractal analysis.

Larrichia-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. Their work also provided them with information on the number of distinct epitopes

FIGURE 12.2 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 ™ solution to ET-115_2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (b) polyclonal rat anti-ET-115_2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (c) polyclonal rabbit anti-ET-l15_2i in solution to ET-115„2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (d) rabbit anti-ET-l antibody in solution to immobilized ET-1.

FIGURE 12.2 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 ™ solution to ET-115_2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (b) polyclonal rat anti-ET-115_2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (c) polyclonal rabbit anti-ET-l15_2i in solution to ET-115„2i-BSA immobilized on extended carboxymethylated hydrogel matrix; (d) rabbit anti-ET-l antibody in solution to immobilized ET-1.

on the surface of ET-1. Figure 12.2d shows the binding of rabbit anti-ET-1 in solution to immobilized ET-1. Table 12.1b shows the values of k and Dj. Once again, a single-fractal analysis is adequate to describe the binding kinetics.

Figure 12.3 shows that the binding rate coefficient increases as the fractal dimension increases. This is in accord with the prefactor analysis of fractal aggregates (Sorenson and Roberts, 1997) and of the analyte-receptor binding kinetics observed for biosensor applications (Sadana and Sutaria, 1997; Sadana, 1998). For the data presented in Table 12.1, the binding rate coefficient, k, is given by k = (0.4467 ±0.0799)D/6-4405± 03980. (12.2)

This predictive equation fits the values of k presented in Table 12.1 reasonably well. The very high exponent dependence indicates that the binding rate coefficient is very sensitive to the degree of heterogeneity that

FIGURE 12.3 Influence of the fractal dimension, Df, on the binding rate coefficient, k.

Fractal dimension, Df

FIGURE 12.3 Influence of the fractal dimension, Df, on the binding rate coefficient, k.

exists on the surface. More data points are required to more firmly establish this equation.

Peterlinz et al. (1997) analyzed the hybridization of thiol-tethered DNA on a passivated gold surface using two-color surface plasmon resonance spectroscopy. The novel two-color SPR method (Peterlinz and Georgiadis, 1996) allowed them to make quantitative the number of ss-DNA molecules per unit area for tethered DNA films. Peterlinz et al. tethered an immobilized oligonucleotide array that contains ss-DNA molecules with known sequence (probe) to a surface. When this surface was exposed to the target molecule (an ss-DNA molecule of unknown sequence), only those molecules whose sequence is complementary to the tethered molecule will bind (hybridize) to the probe on the surface.

Figure 12.4 shows the hybridization of 0.5 /¿M solution of complementary DNA fragment in 1.0 M NaCl to a DNA mercaptohexanol film. Peterlinz et al. indicate that in the film, the 25-base oligomer is tethered to the gold surface via an alkanethiol covalently linked to the 5' position of the ss-DNA. The mercaptohexanol was added to minimize nonspecific adsorption. In this case, a single-fractal analysis does not provide an adequate fit, and thus a dual-fractal analysis was used. Table 12.2a shows the values of k and Dj for a single-fractal analysis and kl, k2, Dj, and Df , for a dual-fractal analysis. Clearly, the dual-fractal analysis provides a better fit. For the dual-fractal analysis, note

TABLE 12.2 Influence of Different Parameters on Fractal Dimensions and Binding Rate Coefficients for Different Anaiyte-Receptor Binding Reactions Utilizing Surface Plasmon Resonance: Single and Dual-Fractal Analysis

Analyte in solution/receptor on surface

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