Zero-Flux-Boundary Sensors

In spite of its importance and popularity, the fine details of the P-D-glucose oxidase mechanism are not completely known. The proposed model (Fig. 2.14) includes both the catalase cascade and the protonation equilibria (Caras et al., 1985b). The pH-dependent reaction term corresponding to this model is quite complex. 

The verification of the model is again performed by fitting the experimental calibration (Fig. 2.15) and time response (Fig. 2.16) curves. 

The fits in this case are not as good as the ones obtained for the penicillin case (Figs. 2.10 and 2.11). This is due to the fact that the glucose oxidation mechanism is not yet completely understood and the kinetic equations are only approximate. Nevertheless, it is again possible to plot the profiles of the most important species in the gel layer and from this fit to estimate the optimum thickness of the gel layer (Fig. 2.17). For the glucose sensor, the optimum thickness appears to be 150pm, 

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With these assumptions in mind, we now complete the outline of the solution of the diffusion-reaction problem as it applies to the most difficult case, the pH-based enzymatic sensors (potentiometric or optical). We assume only that there is no depletion layer at the gel/solution boundary (7), and that there is no fixed buffer capacity (4). The objective of this exercise is to find out the optimum thickness of the gel layer that is critically important for all zero-flux-boundary sensors, as follows from (2.26). 

The actual solution for both transient and steady-state response of any zero-flux-boundary sensor can be obtained by solving (2.26) through (2.33) for the appropriate boundary and initial conditions. Fitting of the experimental calibration curves (Fig. 2.10) and of the time response curves (Fig. 2.11) to the calculated ones, validates the proposed model. 

Enzymatic reactions coupled to optical detection of the product of the enzymatic reaction have been developed and successfully used as reversible optical biosensors. By definition, these are again steady-state sensors in which the information about the concentration of the analyte is derived from the measurement of the steady-state value of a product or a substrate involved in highly selective enzymatic reaction. Unlike the amperometric counterpart, the sensor itself does not consume or produce any of the species involved in the enzymatic reaction it is a zero-flux boundary sensor. In other words, it operates as, and suffers from, the same problems as the potentiometric enzyme sensor (Section 6.2.1) or the enzyme thermistor (Section 3.1). It is governed by the same diffusion-reaction mechanism (Chapter 2) and suffers from similar limitations. 

Next, we have to define the boundary and the initial conditions. For the zero flux sensors (Fig. 2.9), the first space derivatives (i.e., fluxes) of all variables at the transducer/gel boundary (point x = 0) are zero ... 

In Fig. 2.10, the boundary between the enzyme-containing layer and the transducer has been considered as having either a zero or a finite flux of chemical species. In this respect, amperometric enzyme sensors, which have a finite flux boundary, stand apart from other types of chemical enzymatic sensors. Although the enzyme kinetics are described by the same Michaelis-Menten scheme and by the same set of partial differential equations, the boundary and the initial conditions are different if one or more of the participating species can cross the enzyme layer/transducer boundary. Otherwise, the general diffusion-reaction equations apply to every species in the same manner as discussed in Section 2.3.1. Many amperometric enzyme sensors in the past have been built by adding an enzyme layer to a macroelectrode. However, the microelectrode geometry is preferable because such biosensors reach steady-state operation. 

Besides the differential equations the complete formulation of the model requires a set of initial and boundary conditions. These must reflect the situation at the interface between measuring solution and enzyme electrode membrane and between membrane and sensor. For the models considered, it is assumed that the measuring solution is perfectly mixed and contains a large amount of substrate as compared to the substrate converted in the enzyme membrane. It has been shown experimentally (Carr and Bowers, 1980) that in measuring solutions diffusion is much more rapid than in membranes. A boundary layer effect is not considered. On the sensor side all electrode-inactive substances fulfill zero flux conditions. If the model contains more than one layer the transfer between the layers may be modeled by using relations of mass conservation. The respective equations will be given in the following sections. 

Next we have to define the boundary and the initial conditions. For so called zero flux sensors there is no transport of any of the participating species across the sensor/enzyme layer boundary. Such condition would apply to, e.g., optical, thermal or potentiometric enzyme sensors. In that case the first space derivatives of all variables at point x are zero. On the other hand amperometric sensors would fall into the category of non-zero-flux sensors by this definition and the flux of at least one of the species (product or substrate) would be given by the current through the electrode. 

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