#### 3.1. The Model

The following reaction sequence can be proposed for the case of mediated electrocatalytic reduction of H_{2}O_{2} at HRP electrode [38,39]:

where HRP is horseradish peroxidase. Fe^{3+} is the HRP cofactor. Compound (I) (oxidation state +5) and Compound(II) (oxidation state +4) are oxidized intermediates. QH_{2} and Q represent hydroquinone and its oxidized form (benzoquinone), respectively. In these reactions, a two-electron oxidation of the ferriheme moiety of HRP enzyme is caused by H_{2}O_{2}. The resulting intermediate Compound (I) consists of oxyferryl iron and a porphyrin π cation radical. Compound (I) is reduced to Compound (II) and then it was further reduced to the original form of HRP (Fe^{3+}) by the mediator QH_{2}. Q is finally reduced back to QH_{2} by a rapid reaction involving the acceptance of two electrons from the electrode.

The inhibition of phenylhydrazine to HRP is an anticompetitive inhibition [39]. When the inhibitor (phenylhydrazine) is added into the electrolyte solution, it can combine with Compound (I) causing the decrease of the current i. The possible mechanism of inhibition could be represented as follows:

where compound(I) - I represents complex of compound(I) with inhibitor.After adding substrate or inhibitor dropwise, the current rapidly returns toward equilibrium (steady state) with increasing time. For the pre-steady-state part of enzymatic reaction, the current obeys exponential decay equation, i.e.,

where i is the faradaic current at time t. i_{∞} is the steady-state current. τ is relaxation time, being equal to the time constant of an exponential return of a faradaic current to equilibrium after adding substrate or inhibitor, which reflects the rate at which excited current return to steady-state current.

Suppose that a steady-state current (i_{∞}) is the current when the biosensor system approaches to equilibrium as t → ∞:

where i(t) is the current at time t. In practice, the current at the response time t_{R} is assumed as i_{∞}. t_{R} is defined as the time when the absolute current slope falls below a given small value (ε < 0.0001) [27], i.e.,

A reduction current (i_{∞}) observed on electrode is a combination of the mass transport-limited current (i_{∞,L}) and the reaction-rate-limited current (i_{∞,K}) [40], accords with the Koutecky-Levich equation, i.e.,

For a rotating disk electrode, the mass-transport-limited current (i_{∞,L}) depends on the angular rotation velocity (ω) of the electrode and the bulk concentration ([H_{2}O_{2}]) of H_{2}O_{2}, i.e., i_{∞,L} is determined by the Levich equation [22,41]:

where n is the number of electrons transferred to the enzyme in one catalytic cycle (n = 2). F is the Faraday constant. A is the area of the electrode surface. D is the diffusion coefficient of H_{2}O_{2}. ν is the kinematic viscosity of water. For a given biosensor that is rotating at a constant angular velocity, i_{∞,L} is proportional to [H_{2}O_{2}], Equation (7) reduces to:

where a_{1} is constant (a_{1} = 0.62nFAD^{2/3}ν^{−1/6}ω^{1/2}).

When the transport of H_{2}O_{2} is high enough to keep its concentration at the electrode surface equal to that in the bulk solution, the reaction-rate-limited current i_{∞,K} is governed by the ping-pong kinetic scheme, being expressed by [24,39]:

Where Γ is the surface concentration of HRP enzyme. [QH_{2}] is the concentration of hydroquinone. [I] is the concentration of phenylhydrazine inhibitor. Assuming that a_{2} = 1/nFAΓk_{1}, b = 1/nFAΓk_{I}, and c_{1} = (k_{2} + k_{3})/(nFAΓk_{2}k_{3}[QH_{2}])+1/nFAΓ, Equation (9) can be also reduced to:

Substituting Equations (8) and (10) into Equation (6), the Koutecky-Levich equation can be reduced to:

where a_{3} = a_{2} + 1/a_{1}.

Combining b[I] term with constant c_{1} in Equation (11) in a constant phenylhydrazine concentration, we obtain c_{2} = c_{1} + b[I]. The dependence of the steady-state current on the substrate concentration is written as:

Combining a_{3}/[H_{2}O_{2}] term with constant c_{1} in Equation (11) in a constant H_{2}O_{2} concentration, we obtain c_{3} = c_{1} + a_{3}/[H_{2}O_{2}]. The dependence of the steady-state current on the inhibitor concentration is written as:

#### 3.3. Electrochemical Enzyme Assay

From Figure 2(a), the relation between i_{∞} and [H_{2}O_{2}] is not linear. The calibration curve is valid only under low concentration of substrate conditions. More specifically, when [H_{2}O_{2}] < 0.4 mM, the relation is very close to a straight line (calibration curve): i_{∞} = −7.86(±0.074) × [H_{2}O_{2}] − 0.025(±0.018) with r = −0.999; p < 0.0001. This linear correlation is a prerequisite for applying the electrochemical biosensors. Thus electrocatalysis reduction to H_{2}O_{2} with HRP electrode can detect trace levels of H_{2}O_{2}. In fact, our result shows the steady-state current follows the Michaelis-Menten kinetic model within a broader concentration range. Kinetic constants are determined by fitting the initial rate data to Lineweaver-Burk plot Figure 2(b). Thus, the current is regarded as a relative enzyme reaction rate. This methodology could monitor in real-time subtle changes in enzyme activity.

The pre-steady-state time course of the i → i_{∞} process is fitted to Equation (3) for a single exponential decay, in which τ is the apparent relaxation time for output current (i.e., the reciprocal of the apparent first-order rate constant). It is found, however, that τ is not a simple function of the substrate concentration. As shown in Figure 3, τ is low in both the low- and high-concentration regions; a relatively larger τ occurs in at a moderate concentration ([H_{2}O_{2}] = 0.8–1.4 mM). This suggests that, within this concentration range, the rate at which the equilibrium is approached is slowed down.

If the kinetic control becomes operative, there is a possible explanation: a “shoulder” appears on the curve of t_{0.5}versus[H_{2}O_{2}] in the two-dimensional-in-space biosensor model, as has been proposed in reference [27]. Where t_{0.5} is the half time (t_{0.5} = 0.693τ). Baronas et al.[27] showed that the shoulder occurs at [H_{2}O_{2}] near to the Michaelis constant K_{m}. At [H_{2}O_{2}] ≪ K_{m} (τ = 1/k_{1}K_{m}) the reaction kinetics for [H_{2}O_{2}] is a zero order throughout the biosensor, whereas for [H_{2}O_{2}] ≫ K_{m} (τ = 1/k_{1}[H_{2}O_{2}]) the kinetics is a first order throughout. At intermediate values of [H_{2}O_{2}] the kinetics undergoes a transition from zero order to first order. This phenomenon is also manifested in inhibition assays, so it is not obvious whether agreement between our data and the shoulder of t_{0.5} is not a coincidence. Inhibition kinetics also experiences a transition from zero order to first order, but it is not proof.