Analytical expressions of steady-state concentrations of species in potentiometric and amperometric biosensor

A mathematical model of potentiometric and amperometric enzyme electrodes is discussed. The model is based on the system of non-linear steady-state coupled reaction diffusion equations for Michaelis-Menten formalism that describe the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate, product and corresponding flux response have been derived for all values of parameters using Homotopy analysis method. The obtained solution allow a full characterization of the response curves for only two kinetic parameters (The Michaelis constant and the ratio of overall reaction and the diffusion rates). A simple relation between the concentration of substrate and products for all values of parameter is also reported. All the analytical results are compared with simulation results (Scilab/Matlab program). The simulated results are agreed with the appropriate theories. The obtained theoretical results are valid for the whole solution domain.


INTRODUCTION
Theoretical models of enzyme electrodes give information about the mechanism and kinetics of the biosensor. The information gained from mathematical modeling can be useful in sensor design, optimization and predict tion of the electrode's response. Enzyme-based systems, i.e. the combination of enzyme-reactor sensor and enzyme electrodes have gained considerably more importance for chemical analysis and also for clinical diagnostics [1,2]. Blaedel et al. [3] have derived explicit solution of electrodes for steady-state conditions which can apply to very high substrate concentration or to sufficiently low substrate concentration. Guilbault and Nagy [4] proposed an approximation which is used for sensors with very thin enzyme membranes in contact with diluted substrate samples. Hameka and Rechnitz [5] have used Laplace transforms to treat potentiometric enzyme electrodes. Tranh-Minh and Broun [6] also solved the differential equations for reaction and diffusion processes by using a numerical method. Morf [7,8] presented a relatively simple approach for the electrode response that applies to the whole range of substrate concentrations by obtaining an explicit result. Glab et al. [9][10][11] extended this theory to account for additional influence arising from the presence of pH-buffering systems. Mell and Maloy [12,13] have obtained the current response as a function of time and substrate concentration using numerical simulation. Olesson et al. [14] treated the behavior of these sensors in flow-injection analysis applications. Albery and Bartlett [15] studied the steady-state current response of modified enzyme-based sensors where electron transfer from the enzyme to the electrode occurs directly by a redox mediator. Later these authors [16] have derived 11 equations to differentiate between 7 limiting cases by offering a complete theoretical treatment of the basic reaction and diffusion processes in homogeneous solution. Mackey et al. [17] recently studied the current efficiency of sensors with a monolayer of two enzymes.
Several groups [18][19][20] introduced numerical method for the study of amperometric and cyclic-voltammetric enzyme electrodes based on Michaelis-Menten kinetics. The available theoretical descriptions of enzyme electrodes seem to be quite complex. Reaction and diffusion processes in the enzyme layer are modeled by a finitedifference method [21][22][23][24][25][26][27][28][29][30][31][32]. Morf et al. [33] presented the theoretical model for an amperometric glucose sensor for low substrate concentration and high benzoquinone concentration compared with respective Michaelis constants. To my knowledge no rigorous analytical expression of concentration of substrate and mediator of ampereometric glucose sensor under steady-state conditions for all values of parameters have been reported. The purpose of this communication is to derive approximate analytical expressions for the steady-state concentrations and flux for all values of parameters α, β and γ using Homotopy analysis method. This dimensionless parameter are defined in the Eq.6.

MATHEMATICAL FORMULATION OF THE PROBLEM
In this paper we consider the analytical systems based on an enzyme-containing bulk membrane of thickness d that is contacted on one side with an aqueous solution of the substrate S (refer Figure 1).
The substrate molecules entering the membrane phase react according to a Michaelis-Menten type enzymecatalyzed reaction [31,32] to yield an electroactive product P. The corresponding governing equations in Cartesian coordinates for planar diffusion and reaction in a membrane may be written as follows [33,34]: where   em S and   em P are the concentrations of the species in the enzyme membrane, D S and D P are the corresponding diffusion coefficients,   tot is the total concentration of free enzymes and enzyme-substrate complexes that is assumed to be constant within the membrane including the surface zones, v is the number of product species obtained per substrate molecule, k 3 is the rate constant for the irreversible step of product formation, and K M the Michaelis constant [31,32]. The Eqs.1 and 2 are solved for the following boundary conditions by assuming the zero fluxes at x = 0 and of equilibrium distribution at x = d [33,34].
For enzyme reactors, the outward flux of product species at x = d is described by: where J P is the outward flux of the product species and D P is the diffusion coefficient of the product.

DIMENSIONLESS FORM OF THE PROBLEM
To compare the analytical results with simulation results, we make the above non-linear partial differential equations Eqs.1 and 2 in dimensionless form by defining The Eqs.1 and 2 reduces to the following dimensionless form: Recently, Homotopy analysis method (HAM) has been developed to solve different types of non-linear problems. Many problems of engineering and allied disciplines have been successfully solved [35,36]. The major advantages of this method are that it does not require the presence of small/large parameter unlike the perturbation method or delta decomposition method and can be applied to any type of nonlinearity. Further, working of HAM reduces to those of other methods for a certain choice of auxiliary quantities defined later [36,37], and it can easily be implemented in various symbolic soft computing tools e.g. MATHEMATICA, MAPLE etc. A details of this method can be found in the original work of Liao [36]. The Homotopy analysis method [38,39] was first proposed by Liao in 1992. The basic concept of Homotopy analysis method is given in [40]. By solving the above Eq.7 using HAM [38][39][40][41][42][43][44], we can obtain the concentration of substrate in dimensionless form as follows: Detailed derivation of the above Eq.11 is described in Appendix A. Adding the Eqs.7 and 8 we get Integrate the above Eq.12 twice and using the boundary conditions (9) and (10), we can obtain the concentration of product as follows: The normalized flux becomes, The above equation represent the dimensional flux for all values of parameter for steady state conditions.

Limiting Case: 1-Unsaturated (First-Order) Catalytic Kinetics
We consider the situation when the substrate concentration is sufficiently low, that is for   M em , the catalytic reaction follows first-order kinetics. This situation will pertain when the product Hence the non-linear reaction/diffusion equation Eqs.7 and 8 has been reduced to the linear equation. Hence the concentration profiles of substrate and product are given by: In this case the dimensionless flux becomes tanh P j v The Eqs.17 and 18 are same as Eqs.7 and 8 in [33].

Limiting Case: 2-Saturated (Zero-Order) Catalytic Kinetics
For very high substrate concentrations,   M em , the enzyme-catalyzed reaction becomes zero-order kinetics. In this case S K  1 u   and Eqs.7 and 8 reduces to Now the concentration of substrate and product becomes as follows: The normalized flux is given by Recently Morf et al. [33] have derived the above (Eqs.22-24) same results (Eqs.14 and 15) in Ref. [33].

NUMERICAL SIMULATION
The two non-linear reaction/diffusion equations (Eqs.7 and 8) corresponding to the boundary conditions (Eqs.9 and 10) were solved by numerical methods. We have used pdex4 to solve these equations (Pdex4 in MATLAB is a function to solve the initial-boundary value problems of differential equations). The numerical solution is compared with our analytical results and is shown in  are the principal determining factor. The concentration gradient near the electrode, and hence the flux of substrate to the electrode, decreases with an increase in γ. For a given layer, and hence value of γ, the concentration profile is also altered according to the bulk concentration of the substrate. Thus an increased substrate concentration causes a decrease in the flux to the electrode. Eqs.11 and 13 are the new and simple analytical expressions of concentrations of substrate and product respectively. The analytical expression of concentration of substrate, product and flux are summarized in Tables 5 and 6.
In Figures 2 and 3, the profiles of substrate as well as product concentration in the enzyme layer are presented. The normalized concentration of substrate is represented in Figures 2(a) and (b). From this Figure,      when the reaction diffusion parameter γ increases for all values of α. When γ is very small , the concentration of substrate is uniform and the curve becomes straight line.
In Figures 3(a)-(c) represent the normalized concentration of product P for various values of γ. From this Figure it is obvious that the value of the product concentration P increases when the reaction dif-fusion parameter γ increases for various values of α. Figures 4(a) and (b) represent the normalized concentration of the substrate S and the product P for various values of γ. From this Figure it is clearly that the value of substrate concentration S decreases and the value of the product concentration P increases. From the Figures 2-4, it is also confirmed that when the thickness of the mem Copyright © 2012 SciRes.
OPEN ACCESS Table 5. Analytical expression of concentration of substrate, product and flux for limiting cases. Table 6. Analytical expression of concentration of substrate, product and flux.

Flux
The parameter of greatest interest in an amperometric biosensor is the current which is related to the flux of electroactive material to the electrode surface. For a particular membrane, the variation in γ can be achieved practically, by varying the thickness of the membrane, d, or the loading of the enzyme,   tot . As can be seen in Eq.14, and confirmed above, the thickness appears as a squared term and thus small changes will have a much more pronounced effect on the response of the enzyme electrode than the enzyme loading.

CONCLUSION
We have theoretically analyzed the behavior of potentiometric and amperometric enzyme electrodes, which is previously described in Ref. [33]. The coupled time independent steady-state non-linear reaction/diffusion equations have been solved analytically and numerically. In this work, we have presented analytical expression corresponding to the substrate and product concentration within the polymer film in terms of parameters using Homotopy analysis method. Moreover, we have also obtained an analytical expression for the steady state current. A good agreement with numerical simulation data is noticed. The analytical results presented here will be used to determine the kinetic characteristic of the biosensor. The theoretical model described here can be used for optimization of the design of the biosensor. Further, based on the outcome of this work it is possible to extend the procedure to find the approximate amounts of substrate and product concentrations and current for reciprocal competitive inhibition process. brane decrease, the concentration of substrate increase and the product decrease.

Convergence Parameter h
The auxiliary convergent parameter h controls the convergence of the analytical solution series [36]. The Eq.11 and Eq.13 contain the parameter h, which gives the convergence region for the Homotophy analysis method. Multiple h curves are plotted to define the h region such that the analytical solution series is independent of h. The common region among and its derivative is known as the convergence region for the corresponding function u. The h curves in