Theory of the Transient Current Response for the Homogeneous Mediated Enzyme Catalytic Mechanism at the Rotating Disc Electrode

This paper presents the approximate analytical expression for transient and steady-state concentration profiles of enzymes, mediator, substrate and current. The transport and kinetics of the reaction in the diffusion layer with a rotating disc electrode are described using closed-form solutions of homogeneous systems. These new approximate analytical expressions are valid for all values of parameters. Furthermore, in this work, the numerical simulation is also presented using the Matlab program. The analytical results are compared with simulation results, and satisfactory agreement is noted.

Nonlinear equation occurs in the homogeneous mediated enzyme reaction mechanism. Albery and coworkers [38] presented a comprehensive theoretical treatment for an amperometric enzyme electrode that uses a mediator interacting in a homogeneous solution to transfer the electrons. Bartlett where = ∞ /( + ∞ ) and ∞ is the substrate concentration in the bulk solution.
Substrate present to regenerate reduced form of the enzyme, E'. Building upon earlier work for these mechanisms, Albery et al. [38] and Bartlett et al. [39] presented a concise discussion and derivation of the mass transport equation for these mechanisms for non-steady-state condition, which is summarized briefly for completeness. The following schematic diagram depicts the reaction system. is a bimolecular rate constant and a pseudo-first-order rate constant for the reaction between E and S. When the substrate concentration is high enough, the enzyme becomes saturated and equals . Nonlinear reaction diffusion-reaction equations for the four species [38,39] can be written as follows: In the above analysis, the only diffusion of enzyme and mediator is examined. We assume that substrate is present in excess, so the reaction-diffusion of S may be neglected. If not, the further reactiondiffusion equation must be considered, and the equation of the substrate is as follows: respectively. S D is the diffusion coefficients of the substrate. The concentration of total enzyme  in the solution is considered to be uniform. This means that the oxidized and reduced forms of the enzyme Here D Z is the Levich diffusion layer thickness given by where  is the kinematic viscosity and W (Hz) is the rotation speed. Now the Eq. (4), Eq. (6) and Eq. (8) are reduced in dimensionless form as follows: The corresponding initial and boundary conditions for the above two equations are given by, (20) In dimensionless terms the current becomes [39]

ANALYTICAL EXPRESSION OF CONCENTRATION OF THE SPECIES USING NEW APPROACH OF HOMOTOPY PERTURBATION METHOD (HPM)
Solving systems of nonlinear equations, which is one of the most fundamental problems in mathematics, can be used to solve several applied problems. In the physical and chemical sciences, novel methods have recently been used to solve nonlinear problems [23]. HPM is a common method used to solve a differential equation.
The HPM was proposed by He in 1999 [24]. This approach has recently been used in nanotechnology to solve nonlinear oscillator problems [25][26]. This method is also applied to solve coupled nonlinear differential equations in the microelectromechanical system [27], and axial vibration system [28], etc. Using the new approach of HPM, the approximate analytical expressions for the concentration of the mediator, enzyme and substrate are obtained (Appendix-A) as follows: The dimensionless current is given as follows:

LIMITING CASES
Here we have derived the concentration of the mediator, enzyme and corresponding expression of current for various special cases.

Limiting case 1: Enzyme-mediator kinetics and high reaction rate
We begin by considering the case where 1   , 1   (enzyme-mediator reaction is rate limiting) and 1  E  (the most E will escape from the diffusion layer before regenerated to E'). The rate of reaction is sufficiently high ( 1  M  and M will be more likely to react than escape). Since both u and v are less than unity, the term are neglected. In this situation the Eqs. (15) and (16) becomes as follows: Using the initial and boundary conditions (18)(19)(20), we can solve Eqs. (27) and (28) to obtain exact analytical expressions for mediator and enzyme concentrations (Appendix B).
The steady-state current

Limiting case 2: Enzyme-substrate kinetics and high rate of reaction
When the rate of reaction is sufficiently high ( 1  M  and M will be more likely to react than escape) and 1  E  (the most E will escape from the diffusion layer before regenerated to E' ), and 1   (enzyme-substrate reaction is the rate limiting step), 1   (equal diffusion coefficients of enzyme and mediator), the nonlinear reaction diffusion equations (15)(16) becomes as follows: The approximate expressions of mediator and enzyme concentrations are obtained by solving the above two equations with the boundary conditions (18)(19)(20).
When M  is very large and the maximum value of

Limiting case 3: Enzyme-mediator kinetics and low reaction rate
When the rate of reaction is sufficiently low ( , Using the initial and boundary conditions (18)(19)(20), we get exact solution as follows: . This result is also confirmed in Bartlett et al. [39]. Approximate analytical expressions of dimensionless concentrations and current for the above different limiting cases is also given Table. 1.

RESULT AND DISCUSSION
Equations (22), (23) and (24) are the new, general and simple analytical expressions of concentration profiles for the mediator (u), enzyme (v) and substrate (S) for transient conditions. Albery and co-workers [38], Bartlett and Pratt [39] derived the different approximate solutions for various limiting cases for steady-state only. Logambal and Rajendran [42] applied He's variational iteration method to find an approximate analytical solution of steady-state nonlinear differential equations describing the transport and kinetics of the enzyme and of the mediator in the diffusion layer of the electrode. But in this method, it is very difficult to find the unknown parameter in the concentration.
We have also derived analytical expressions of concentration of mediator, enzyme and current for transient conditions for the three main limiting cases such as (i) Enzyme-mediator kinetics and high reaction rate, (ii) Enzyme-substrate kinetics and high rate of reaction, (iii) Enzyme-mediator kinetics and low reaction rate. Approximate analytical expressions of concentration of mediator and current are also validated with the numerical results and limiting case result in Tables 2-3 and Fig. 2. A satisfactory agreement is noted. Also, from the Table, it is observed that when the distance from the electrode surface increases, the concentration of mediator decreases.   for all values of other parameters. The three-dimension plot (Fig.4(a-c)) of current versus other parameters also confirm this results.
For solving the above equations, we need to take Laplace transformation. Therefore, Eqs. (A5) and (A6) becomes, where s is the Laplace variable and an over bar indicates a Laplace-transformed quantity, Solving the Eq. (A8), and using the boundary conditions and (A9) and (A10) we can find the following results Now, we indicate how Eq. (A11) can be inverted using the complex inversion formula. If The real number c is chosen such that c s  lies to the right of all the singularities, but is otherwise assumed to be arbitrary. In practice, the integral is evaluated by considering the contour integral presented on the right-hand side of Eq. (A12), which is then evaluated using the so-called Bromwich contour. The contour integral is then evaluated using the residue theorem which states for any analytic function where the residues are computed at the poles of the function Hence, in order to invert Eq. (A11), we need to evaluate The poles are obtained from   0 . Hence there is a simple pole at and there are infinitely many poles given by the solution of the equation and so ) ( 4 Applying L-Hospital rule in the above equation, Applying the complex inversion formula for the above equation similar to Eq. (A11) we get,