1 1.5 2 mole percent

1 1.5 2 mole percent

FIGURE 5.13 Influence of different doping densities (in mol%) on a sensor tip on (a) the binding rate coefficients, ki and fe2; (b) the fractal dimension, Df, ; (c) the fractal dimension, Df2.

FIGURE 5.13 Influence of different doping densities (in mol%) on a sensor tip on (a) the binding rate coefficients, ki and fe2; (b) the fractal dimension, Df, ; (c) the fractal dimension, Df2.

the best fit curve, which indicates that both ki and k2 increase as the mole percentage of biotin lipid surface density increases. More data points are required to more firmly establish the trend presented. Nevertheless, the trend is of interest since it shows one possible way of changing the binding rate coefficients, kx and k2. For example, as the biotin lipid surface density increases by a factor of 9.64—from 0.28 to 2.7 mol%, the binding rate coefficient, k1, increases by a factor of 5.15—from a value of 0.010 to 0.0515—and the binding rate coefficient, k2, increases by a factor of 4.92— from a value of 0.0475 to 0.234.

In the concentration range analyzed, the binding rate coefficient, fej, can be described by h = (0.0177+ 0.0026)[biotin surface density, mol%]056610539, (5.14)

and the binding rate coefficient, k2, can be described by k2 = (0.129 ±0.0441)[biotin surface density]0'757*0175. (5.15)

These predictive equations fit the Table 5.5 values of kj and k2 reasonably well (Fig. 5.13a).

Similarly, Figs. 5.13b and 5.13c show that both fractal dimensions, D^ and Df2, exhibit increases as the biotin lipid surface density increases in the range from 0.28 to 2.70 mol%. There is, however, considerable scatter in the data for both Df. and Df2. An increase in biotin surface lipid density from 0.28 to 2.70 mol% leads to a small increase in the fractal dimension, Df2, by about 7.5%—from Df2 — 2.53 (at the lowest surface biotin surface density of 0.28 mol%) to D{2 — 2.72 (at the highest biotin surface density of 2.70 mol%).

In the concentration range analyzed, the fractal dimension, D{1, is given by

Dfl = (1.461 ±0.237) [biotin surface density, mol%]0 0143± 0 0890, (5.16) and the fractal dimension, Df2, is given by

Df2 = (2.594±0.201)[biotin surface density, mol%]00529± 0 0441. (5.17)

The fractional exponent dependence on biotin surface density exhibited by the binding rate coefficients and by the fractal dimensions D{1 and Df2 provide further support for the fractal nature of the system.

As mentioned, there is scatter in the data. A better fit may be obtained if an expression such as

Df. or Df2 = a[biotin surface density, mol%]b

is used. Here a, b, c, and d are coefficients to be determined by regression. But, at present, this just introduces more variables, and this leads to a better fit. The fractal dimension, Df1, exhibits a complex dependence on biotin surface density. Apparently, the Df, versus biotin surface density curve exhibits a maximum. More data points are required to more firmly describe the trend exhibited.

Ikariyama et al (1997) developed and analyzed a biosensor to detect environmental pollutants. These authors indicate that some microorganisms can assimilate benzene-related and other compounds since they possess a series of enzymes that can digest these chemicals (Koga et al, 1985; Yen and Gunsalus, 1982). Ikariyama et al indicate that the genetic information is encoded in a series of degradation plasmids and that the TOL plasmid in Pseudomonas putida mt-2 contains a series of genes that can degrade xylene and toluene. The authors utilized a fiber-optic biosensor to monitor benzene derivatives by recombinant E. coli that contained the luciferase gene. They constructed a fusion gene between TOL plasmid and the luciferase gene. Recombinant E. coli bearing this fusion gene was then immobilized on the fiber-optic end.

Figure 5.14a shows the curve obtained using Eq. (5.1) for the binding of m-xylene-saturated STE buffer solution to the microorganism immobilized to the fiber-optic tip and covered with a polycarbonate membrane. In this case, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 5.5 shows the values of the binding rate coefficient and the fractal dimension. Ikariyama et al indicate that there is a fluctuating relationship between m-xylene and the luminescence, which is reflected in the "error" observed for estimating the binding rate coefficient, k. Note that this fluctuating relationship does not significantly affect the error in the estimated value of the fractal dimension or the degree of heterogeneity that exists on the biosensor surface.

Figure 5.14b shows the curve obtained using Eq. (5.1) for the binding of m-xylene-saturated STE buffer solution to the immobilized microorganism immobilized to the fiber-optic tip and covered with a dialysis membrane. In this case too, a single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 5.5 shows the values of the binding rate coefficient and the fractal dimension. In this case, the fluctuating relationship observed when the polycarbonate membrane was used was not present. Ikariyama et al indicate that the less-hydrophilic property of the polycarbonate membrane is the reason for the fluctuations. Note that there is an increase in the fractal dimension, Df, and a corresponding increase in the binding rate coefficient, k, as one goes from the dialysis membrane to the polycarbonate membrane. An 89.6% increase in Df, from 0.9664 (dialysis membrane) to 1.8532 (polycarbonate membrane), leads to an increase in k by a factor of about

FIGURE 5.14 Binding of m-xylene-saturated STE buffer solution to the immobilized microorganism immobilized on the fiber-optic tip (Ikariyama et al., 1997), and covered with (a) polycarbonate membrane and (b) dialysis membrane. Influence of the absence (c) and presence (d) of 1 mM methylbenzyl alcohol on the binding of m-xylene in solution to cell suspension immobilized on the fiber-optic tip with a polycarbonate membrane. Luciferin added after 2 h of luciferase induction. For (d): — single-fractal analysis; —, dual-fractal analysis.

FIGURE 5.14 Binding of m-xylene-saturated STE buffer solution to the immobilized microorganism immobilized on the fiber-optic tip (Ikariyama et al., 1997), and covered with (a) polycarbonate membrane and (b) dialysis membrane. Influence of the absence (c) and presence (d) of 1 mM methylbenzyl alcohol on the binding of m-xylene in solution to cell suspension immobilized on the fiber-optic tip with a polycarbonate membrane. Luciferin added after 2 h of luciferase induction. For (d): — single-fractal analysis; —, dual-fractal analysis.

8—from a value of 119.82 to 959.58. This indicates that the binding rate coefficient is rather sensitive to the fractal dimension or the degree of heterogeneity that exists on the biosensor surface.

Figures 5.14c and 5.14d show the influence of the absence and presence of 1 mM methylbenzyl alcohol on the binding kinetics using a polycarbonate membrane. Ikariyama et al. wanted to analyze the influence of induction time on luminescence intensity. Two hours after luciferase induction, luciferin was added. The authors noted that few ppm of methylbenzyl alcohol could be detected in an hour. Table 5.5 shows the values of the binding rate coefficients and the fractal dimensions in the absence and in the presence of 1 mM methylbenzyl alcohol. In the absence of methylbenzyl alcohol, there was no detectable luminescence for about 300 min. After that time period, the binding kinetics could be described by a single-fractal analysis. The values of the binding rate coefficient, k, and the fractal dimension, Df, are presented in Table 5.5. In the presence of 1 mM methylbenzylalcohol (Fig. 5.14d) a dual-

fractal analysis clearly provides a better fit. The parameter values for both analyses are presented in Table 5.5.

It would be of interest to determine 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 set of conditions. In lieu of that, Figure 5.15 plots values of the binding rate coefficient as a function of the fractal dimension for two different sets of conditions. Two points are taken when a single-fractal analysis was applicable. One point is taken when a dual-fractal analysis was applicable. For this case, the first set of parameter values (ki and Df.) are plotted as k and Df, respectively. Because of this, the result that follows should be viewed with caution. Nevertheless, the binding rate coefficient is given by k = (114.04 + 43.58)Df'314± 0,692. (5.19)

Figure 5.15 indicates that this predictive equation is very reasonable, considering that data were plotted from two different sets of conditions and that the final analysis also includes both a single- and a dual-fractal analysis. However, the predictive equation does indicate that the binding rate coefficient is very sensitive to the surface roughness or the degree of heterogeneity that exists on the biosensor surface. This is because of the high exponent dependence of the binding rate coefficient on the fractal dimension.

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