FIGURE 12.4 Hybridization (binding) of 0.5 /¿M complementary DNA fragment in solution to DNA tethered to a gold surface (Peterlinz et al., 1997).

that as the fractal dimension increases by about 9.21%—from Dfi = 2.747 to Df = 3.0—the binding rate coefficient increases by a factor of 1.29—from ki = 2.141 to k2 = 2.768. (Recall that the highest value that the fractal dimension can have is 3.0.) Also, the changes in the fractal dimension and in the binding rate coefficient are in the same direction. An increase in the degree of heterogeneity on the SPR surface leads to an increase in the binding rate coefficient. This also has been observed for analyte-receptor binding reactions occurring on other biosensor surfaces (Sadana and Sutaria, 1997; Sadana, 1998).

Berger et al. (1998) employed SPR multisensing to monitor four separate immunoreactions simultaneously by using a multichannel SPR instrument. These authors utilized a plasmon-carrying gold layer onto which a four-channnel cell was pressed. The gold layer was imaged at a fixed angle of incidence, which permitted the monitoring of changes in reflectance. Antibodies were coated to the surface, and antigens in solution were applied to the surface. Three different monoclonals of the oe-hCG (human chorionic gonadotrophin) (1C, 7B, and 3A) were utilized. Human chorionic gonadotrophs and luteinizing hormone (LH) were used as antigens.

Figure 12.5a shows the binding curves obtained using a single-fractal analysis [Eq. (12.1a)] and a dual-fractal analysis [Eq. (12.1b)] for 2 x

4 6 8 Time, arbitrary units

4 6 Time, arbitrary units

4 6 8 Time, arbitrary units

4 6 Time, arbitrary units

Time, arbitrary units

Time, arbitrary units

Time, arbitrary units

Time, arbitrary units

FIGURE 12.5 Binding of analyte in solution to receptor immobilized on a multichannel SPR surface (Berger et al., 1998). (a) 2 x 10~7M human chorionic gonadotrophin (hCG) in solution/a-hCG [1C] immobilized on surface (—, single-fractal analysis; —, dual-fractal analysis; this applies throughout the figures); (b) 10~8M luteinizing hormone (LH) in solution/a-hCG [1C] immobilized on surface; (c) 10~8M luteinizing hormone (LH) in solution/a-hCG [7B] immobilized on surface; (d) 2 x 10~7M hCG in solution/a-hCG [3A] immobilized on surface; (e) 2 x 10~7M hCG in solution/a-hCG [3A] immobilized on surface; (f) 10~8M LH in solution/a-hCG [3A] immobilized on surface.

10 ~ 7M hCG in solution to the a-hCG [1C] immobilized to a multisensing SPR surface. In this case, a single-fractal analysis does not provide an adequate fit, and thus a dual-fractal analysis was used. Table 12.2b shows the values of k and Df for a single-fractal analysis and kx, k2, Dr. and Dj2 for a dual-fractal analysis. Once again, the dual-fractal anaysis clearly provides a better fit. Also for the dual-fractal analysis, note once again that as the fractal dimension increases by a factor of 3.35—from Dj1 = 0.8172 to D/2 = 2.746—the binding rate coefficient increases by a factor of 5.31—from k1 = 0.9986 to k2 = 5.304. Thus, the binding rate coefficient is, once again, sensitive to the degree of heterogeneity that exists on the surface.

Figure 12.5b shows the binding of 10~8M luteinizing hormone (LH) in solution to a-hCG [1C] immobilized to a multisensing SPR surface. Again, a single-fractal analysis does not provide an adequate fit so a dual-fractal analysis was utilized. Table 12.2b shows the values of k and Df for a single-fractal analysis and ki, k2, Df,, and Dj2 for a dual-fractal analysis. For the dual-fractal analysis, as the fractal dimension increases by a factor of 3.23—from Dji = 0.9282 to Df2 = 3.0—the binding rate coefficient increases by a factor of 14.1—from ki = 2.467 to k2 = 37.37. In this case, the binding rate coefficient is very sensitive to the degree of heterogeneity that exists on the surface. Once again, the changes in the fractal dimension and in the binding rate coefficient are in the same direction.

The binding curve in Fig. 12.5b shows a maximum, and one might argue that a monotonically increasing function of time cannot explain the section of the curve where the signal decreases. It is for this reason that a dual-fractal analysis is required. k, and Df are obtained from the increasing section of the curve, and k2 and Df. are obtained from the decreasing section. For the decreasing section of the curve, the slope, b, of the signal versus time curve is negative. The fractal dimension, Dj2, in this case is evaluated from (3 — Dj)/2 = b. Since b is negative, the value of Df2 is set at the highest possible value—3.0. Note that the value of Dj2 reported in the table is Df2 = 3.0 — 0.093. We did not use the + sign since Df, cannot be higher than 3.0. This is just one possible explanation for the maximum exhibited in the binding curve. Another might be to resort to a model including saturation, simple first-order adsorption with normal diffusion limitation but with limited amounts of sites and/or reversibility. But these types of models seem to have a serious deficiency in that they do not incorporate the heterogeneity that exists on the surface. It is this heterogeneity on the surface under diffusion-limited conditions that we are trying to characterize using a single-or a dual-fractal analysis.

Of course, there is room to provide a better explanation for the decreasing section of the curve in a model that includes fractals, diffusion, and heterogeneity on the surface. Another possible explanation could be that desorption may lead to a decrease in the signal. (In the examples analyzed, no desorption is assumed.) Structural changes in the sensing layer may also lead to a decrease in the signal, but this is less likely. Ramsden et al. (1994) have also utilized random sequential adsorption to explain some of the deviations of protein adsorption kinetics from simple Langmuir first-order kinetics (and from pure diffusional limitations). Recognize that this represents "receptor-less" adsorption, wherein the protein adsorbs directly to a surface.

Figure 12.5c shows the binding of 10~8M luteinizing hormone (LH) in solution to a-hCG [7B] immobilized to a SPR multisensing surface. In this case, in contrast to the curves in Fig. 12.5b for this multisensing system, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 12.2b shows the values of k and Dj. The fact that a single-fractal analysis is adequate to describe the binding kinetics, in contrast to the previous two cases, implies that there is change in the binding mechanism between the binding of LH in solution to a-hCG [7B] immobilized to an SPR surface and the binding of both 10~8M LH in solution to a-hCG [1C] immobilized to an SPR surface and 2 x 10" 7M hCG in solution to a-hCG [1C] immobilized to an SPR surface.

Figure 12.5d shows the binding of 2 x 10~7M hCG in solution to a-hCG [3A] immobilized to a multisensing SPR surface. A single-fractal analysis does not provide an adequate fit, and a dual-fractal analysis is required. Table 12.2b shows the values of k and Dj for a single-fractal analysis and k,, k2, Dji, and Dj2 for a dual-fractal analysis. For the dual-fractal analysis, as the fractal dimension increases by a factor of about 1.41—from Dj = 2.115 to Dj2 = 2.980—the binding rate coefficient increases by a factor of about 2.13—from kx = 3.177 to k2 = 6.779. As observed previously, the changes in the fractal dimension and the binding rate coefficient are in the same direction. It is interesting to compare the results obtained for the binding of the hCG/a-hCG [1C] and hCG/a-hCG [3A] systems. They are the same system, except for the monoclonal antibodies (a-hCG), which are type 1C and 3A. Note that the fractal dimensions (Dji and Dj2) and the binding rate coefficients C^i and k2) are all higher for type 3A than for type 1C. Apparently, for type 3A there is a higher degree of heterogeneity (Dji and Dj2) on the surface when compared to type 1C, and this leads to higher values of the binding rate coefficients (ki and k2).

Figure 12.5e shows the binding of 2 x 10 "7 M hCG in solution to a-hCG [3A] immobilized to a multisensing SPR surface. Since this is exactly the same system as plotted in Fig. 12.5d, one may consider this as a repeat, or reproducibility, run. Note, however, that the values of the corresponding binding rate coefficients and fractal dimensions obtained for the single-fractal as well as the dual-fractal analysis are quite different from each other. This indicates that the results are not quite reproducible, at least for this case. Differences between similar systems with regard to signal size and kinetics, however, may depend on how many binding sites (antibodies) one has bound to the coupling matrix. Once again, the binding curve in Fig. 12.5e exhibits a maximum as in Fig. 12.5b. The same explanation applies here, so it is not repeated.

Figure 12.5f shows the binding of 10 ~8 M luteinizing hormone (LH) in solution to a-hCG [3A] immobilized to a multisensing SPR surface. Once again, a single-fractal analysis does not provide an adequate fit, and a dual-fractal analysis is required. Table 12.2b shows the values of k and Dj for a single-fractal analysis and klt k2, Dfi and Dj2 for a dual-fractal analysis. For the dual-fractal analysis, as the fractal dimension increases by a factor of about 2.56—from Dfi = 1.024 to Dj2 = 2.625—the binding rate coefficient increases by a factor of about 3.25—from ki = 1.726 to k2 = 5.622.

Compare the results obtained in Figs. 12.5c and 12.5f. They are both the same systems, LH/a-hCG, except that for Fig. 12.5c we have type 7B for a-hCG and for Fig. 12.5f we have type 3A for oc-hCG. Note that in Fig. 12.5c a single-fractal analysis is sufficient to adequately describe the binding kinetics, whereas for Fig. 12.5f a dual-fractal analysis is required. This indicates that there is a difference in the binding mechanisms when these two different types of a-hCG are immobilized to the multisensing surface, everything else being the same.

It would be of interest to note the influence of the fractal dimension (or the degree of heterogeneity that exists on the surface) on the binding rate coefficient. However, not enough data is available for a particular condition. In lieu of that, we will use all the data presented in Table 12.2b for the dual-fractal analysis of the binding of analyte-receptor systems using multichannel SPR. Figure 12.6 plots the data for these binding rate coefficients as a function of the fractal dimension. Note that data for both a-hCG and LH are plotted together. Because of this, the result we present should be viewed with caution.

The binding rate coefficient, ki, is given by h = (1.7647+ 0.6660)Dji'1774±0'3105. (12.3)

Figure 12.6a shows that this predictive equation is very reasonable, considering we have plotted data for two different sets of systems and that in one system there are three different types of monoclonal antibodies of a-hCG (1C, 3A, and 7B) immobilized to the SPR multichannel surface. Equation (12.3) indicates that the binding rate coefficient, klt is only marginally sensitive to the surface roughness or the degree of heterogeneity that exists on the surface, as seen by the low exponent dependence of the binding rate coefficient on the fractal dimension (slightly more than 1).

We also made an initial attempt to analyze the influence of the fractal dimension, D/2 on the binding rate coefficient, k2. For the SPR multichannel

Fractal dimension, Dfl

FIGURE 12.6 (a) Influence of the fractal dimension, Dy- , on the binding rate coefficient, fei. (b) Influence of the fractal dimension, Dj7 on the binding rate coefficient, k2 (— ), and of the fractal dimension, Dji and (Djl and Df2), on the binding rate coefficient, k2 (—).

Fractal dimension, Dfl

FIGURE 12.6 (a) Influence of the fractal dimension, Dy- , on the binding rate coefficient, fei. (b) Influence of the fractal dimension, Dj7 on the binding rate coefficient, k2 (— ), and of the fractal dimension, Dji and (Djl and Df2), on the binding rate coefficient, k2 (—).

data given in Table 12.2b, the binding rate coefficient, k2 is given by k2 = (0.1489 + 0.0404)Dj2644° ± 19434. (12.4a)

Figure 12.6b shows that there is quite a bit of scatter (dotted line), due to the noticeable bend in the curve. This is indicated by the error in the exponent. A better fit could be obtained if more parameters were used. Initially, there is a degree of heterogeneity on the surface (DjJ, which leads to another (higher value) degree of heterogeneity on the surface (Dj2 ) as the reaction proceeds. Let us presume that the binding rate coefficient, k2, at any time depends on not only the present degree of heterogeneity on the surface (Dj ), but also on any previous one (Dj ). In that case, the binding rate coefficient is given by k2 = (1.0752±0.1136)D/1i'6598±L1037

This predictive equation fits the data presented in Fig. 12.6b (solid line) slightly better than does Eq. (12.4a). Recall that the highest value that the fractal dimension can have is 3, and near that value the binding rate coefficient rises rather sharply, which contributes to the high value of the exponent for Dj2 . More data points are required to more firmly establish the predictive equations around the fractal dimension value of 3. Note that we have used only the two lowest values of k2 when the SPR multichannel surface exhibited a Dj2 value of 3. Apparently, in the region close to Dji =3, the k2 versus Dj2 curve would exhibit asymptotic characteristics.

It is of interest to analyze not only analyte-receptor binding kinetics using the SPR biosensor but also analyte-receptorless binding kinetics, for example, the binding of proteins in solution directly (receptorless) to an SPR surface. Figures 12.7a-12.7c show the binding of 10 to 10~4MbSA in solution directly to an SPR multichannel surface (Berger et al., 1998). In each case, a single-fractal analysis does not provide an adequate fit, and a dual-fractal analysis is required. Table 12.2c shows the values of k and Dy for a single-fractal analysis and ki, k2, Dji, and Dj2 for a dual-fractal analysis. Note for the dual-fractal analysis that as the bSA concentration in solution increases from 10"6M to 10 4M both binding rate coefficients (ki and k2) exhibit increases, the fractal dimension, Dj , decreases, and the fractal dimension, Dj2, increases. The variation in the signal versus time is very small in Figs. 12.7b and 12.7c where one calculates the binding rate coefficient, k2, and the fractal dimension, Dj2 . It is this very small variation that leads to very small values of b (the slope of the curve). The fractal dimension is evaluated from (3 - Dj)/2 = b. It is this very small value of b that leads to Dj values close to 3 (the maximum value). In other words, the degree of heterogeneity on the surface is now close to or at its maximum value.

In the bSA concentration range (10 ~ 6 to 10 ~4 M) in solution analyzed, the binding rate coefficient is given by

Figure 12.8a shows that only three data points are available. Nevertheless, the fit is quite reasonable. More data points would more firmly establish this equation. The low exponent dependence of fei on the bSA concentration indicates that the binding rate coefficient, kly is only mildly sensitive to the analyte concentration in solution.

Similarly, in the bSA concentration range (10 ~6 to 10~4M) in solution analyzed, the binding rate coefficient, k2, is given by fei = (284.775 ±33.095)[bSA]

Once again, Fig. 12.8b shows that only three data points are available.

Time, min

FIGURE 12.7. Binding of different concentrations of bSA in solution directly to the SPR biosensor surface (Berger et al, 1998). (a) 10"6 M; (b) 10"5 M; (c) 10~4 M.

Time, min

FIGURE 12.7. Binding of different concentrations of bSA in solution directly to the SPR biosensor surface (Berger et al, 1998). (a) 10"6 M; (b) 10"5 M; (c) 10~4 M.

bSA concentration, M

FIGURE 12.8 Influence of the bSA concentration (in M) in solution on (a) the binding rate coefficient, kL, and (b) the binding rate coefficient, k2.

bSA concentration, M

FIGURE 12.8 Influence of the bSA concentration (in M) in solution on (a) the binding rate coefficient, kL, and (b) the binding rate coefficient, k2.

Nevertheless, the fit is quite reasonable. More data points would more firmly establish this equation. Once again, the low exponent dependence of k2 on the bSA concentration indicates that k2 is only mildly sensitive to the analyte concentration in solution. The exponent dependence exhibited by both k1 and k2 on the bSA concentration in solution are quite close to each other (0.3676 and 0.4630, respectively), with k2 being slightly more sensitive than k}.

Figure 12.9a shows that Dji decreases as the bSA concentration in solution increases. In the bSA concentration range (10 ~6 to 10~4M) in solution analyzed, the fractal dimension is given by

Dfi = (0.1897 + 0.0241) [bSA] 1429 ± 0 0366 _ (12.6a)

This predictive equation fits the values of Dj presented in Table 12.2c reasonably well. Since only three points are available, more data points are required to more firmly establish this equation. The fractal dimension, Dj, is only mildly sensitive to the bSA concentration in solution, as noted by the low value (magnitude) of the exponent.

0 2E-05 4E-05 6E-05 8E-05 0.0001 bSA concentration, M

FIGURE 12.9 Influence of the bSA concentration (in M) in solution on (a) the fractal dimension, Dfi, and (b) the fractal dimension, Dj.

0 2E-05 4E-05 6E-05 8E-05 0.0001 bSA concentration, M

FIGURE 12.9 Influence of the bSA concentration (in M) in solution on (a) the fractal dimension, Dfi, and (b) the fractal dimension, Dj.

In the bSA concentration range (10 6 to 10 4 M) in solution analyzed, the fractal dimension Dj2 is given by

Figure 12.9b shows that Dj2 increases as the bSA concentration in solution increases. Once again, this predictive equation fits the values of D/2 presented in Table 12.2c reasonably well. Once again, only three data points are available, so more data points are required to more firmly establish this equation. The fractal dimension, Dj2 , is only very slightly sensitive to the bSA concentration in solution, as noted by the very low value of the exponent. Interestingly, neither Dft nor Dj. are sensitive to the bSA concentration in solution. In other words, the degree of heterogeneity on the surface is, for all practical purposes, independent of the bSA concentration in solution.

Figure 12.10a shows the decrease in ki with an increase in the Dj . In the 10" 6 to 10 MbSA concentration range in solution analyzed, the binding

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Fractal dimension, Df

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