Theoretical and Numerical Analysis of Nonlinear Processes in Amperometric Enzyme Electrodes with Cyclic Substrate Conversion

A theoretical model of amperometric enzyme electrodes has been developed in which chemical amplification occurs in a single enzyme membrane via cyclic substrate conversion. The system is based on non-stationary diffusion equations with a nonlinear factor related to the Michaelis– Menten kinetics of the enzymatic reaction. By solving the nonlinear equations using the AGM technique, simple analytical expressions of concentration substrate, product, and amperometric current response are derived. Further, biosensor sensitivity, resistance, and gain are obtained from the current. MATLAB programming was used to carry out the digital simulation. The analytical results are validated with the numerical results. The effect of substrate concentration, maximum enzymatic rate, and membrane thickness on biosensor response was evaluated.


Introduction
Biosensors are two-component devices that consist of a biological entity, typically a redox enzyme that recognises the target analyte, and a transducer that converts the bio-recognition event between the biocatalyst and reactant/substrate (typically a binding/adduct formation process between the active site and substrate) into a measurable signal. Electrochemical or amperometric biosensors measure variations in a current at a working indicator electrode arising from the direct electrochemical oxidation or reduction of the biochemical reaction products [1][2][3][4][5][6][7][8][9]. Amperometric biosensors typically operate in a batch mode under conditions of constant applied electrode potential with concomitant monitoring of the steady-state current response arising from an electron transfer reaction at the sensor/electrode interface. This current is directly related to the concentration of the analyte. Amperometric biosensors are safe, inexpensive, and environmentally friendly, and, consequently, have found significant clinical and industrial applications.
It is very important to be able to quantitatively describe reaction and diffusion within bounded surface regions of the type encountered in amperometric biosensor systems. Typically, such systems may be very complex, and the challenge is to develop the simplest mathematical model that enables the description of the steady-state or transient current as a function of the substrate concentration. Typically, the mathematical model involves proposing a geometry-dependent reaction/diffusion equation subject to well-defined initial and boundary conditions for the substrate concentration within the bounded biosensing layer, which reflect the experimental conditions relevant to the sensing process. Typically, it is assumed that substrate transport is described by Fick's law of diffusion and the reaction kinetics quantified in the first instance by the Michaelis-Menten rate equation. This is usually a good choice since Michaelis-Menten kinetics describe a binding event or adduct

Mathematical Formulation of the Problem
The biosensor may be considered as an enzyme electrode consisting of a thin film membrane of thickness d containing an immobilised enzyme applied onto the surface of the electrochemical transducer. Consider the very simple scheme of a substrate S electrochemically converted to a product P at the surface of the support electrode. This product is then enzymatically transformed back to substrate S using an enzyme catalyst E. This is perhaps the most simple scheme for substrate cyclic conversion.
In the case of phenol-sensitive biosensors, for example, 1,2-benzoquinone (S) is reduced electrochemically and the product pyrocatechol (P) formed is then subsequently oxidised in the membrane using immobilised laccase as the enzyme catalyst (E). Coupling of the enzyme-catalysed reaction in the enzyme layer with planar diffusion, described by Fick's law, leads to the following steady-state reaction/diffusion equations [26]: For 0 < x < d, where x stand for the distance coordinate and S(x) and P(x) denote the concentration functions of the substrate and reaction product, respectively. Note that V max is the maximal enzymatic rate attainable with that amount of enzyme when the enzyme is fully saturated with substrate. This is given by V max = k cat e Σ , where k cat denotes the catalytic rate constant and e Σ denotes the total enzyme concentration in the layer. Note that K M denotes the Michaelis constant, d the thickness of the enzyme layer, and D S and D P the diffusion coefficients of the substrate and product, respectively. Let x = 0 represent the electrode surface, while x = d represents the bulk solution/membrane interface. The operation of the biosensor begins when some substrate appears over the surface of the enzyme layer. The electrode potential is chosen to ensure complete transformation of substrate to product at the electrode so that the substrate concentration at the electrode surface is zero. During electrochemical conversion, the product is produced. The rate of the product generation at the electrode is proportional to the rate of conversion of the substrate. When the solution is well-stirred outside the membrane, the diffusion layer (0 < x < d) remains at a constant thickness. Consequently, the concentration of substrate, as well as product, over the enzyme surface (bulk solution/membrane interface) remains constant while the biosensor contacts the substrate solution. This is reflected in the boundary conditions given by: where S 0 is the concentration of substrate in the bulk solution. Note that we have neglected concentration polarization in the solution phase adjacent to the immobilized enzyme layer. This can, of course, be readily accounted for in the analysis. The current is measured at the support electrode/enzyme layer interface and defines the response of the biosensor device to the chemical activity in the enzyme layer. The current depends upon the flux of the substrate S at the electrode surface, i.e., at x = 0. Consequently, the current density i can be obtained explicitly from Faraday's law and Fick's law via: where n e is a number of electrons involved in a charge transfer at the electrode surface and F is Faraday constant F ≈ 9.65 × 10 4 C/mol.

Dimensionless Form of Problem
To facilitate further analysis, it is often useful to cast the reaction/diffusion problem into dimensionless form. We can make the reasonable assumption that D S = D P = D. The dimensionless steady-state model of the system under consideration can be obtained from Equations (2) and (3) and is given as follows: where the dimensionless variables are given by The boundary conditions are transformed as follows: The current density in dimensionless form is [17] The steady-state current I of the biosensor is expressed as follows: We note that the reaction/diffusion expressions presented in Equations (9) and (10) are nonlinear. Previously, such nonlinear equations were solved in the limit of low and high substrate concentrations. In this work we will proceed to use a recently established protocol to derive a useful analytical solution to these expressions which is valid over a wide region of substrate concentration.
The approximate analytical expression for the dimensionless stationary concentration of substrate u and product v is obtained as follows (Appendix A): where we note that: We can also show that the normalised current becomes: Electrochem 2022, 3

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The steady-state current I is given by This general expression for the steady-state current response is valid over a wide range of substrate concentrations. We now examine some interesting limiting cases corresponding to the situation where the concentration of substrate is much less than the Michaelis constant and when it is much greater.

Limiting Case: Unsaturated (First-Order) Catalytic Kinetics
We first consider the situation where the substrate concentration level in the enzyme layer is less than the Michaelis constant K M , (S 0 << K M ). This situation will pertain when av 1. In such circumstances, Equations (8) and (9) reduce to the linear reaction-diffusion equations: By solving Equations (21) and (22) with their corresponding boundary conditions (Equations (12) and (13)), the exact expression of the concentrations of substrate u and product v and current has been obtained as follows: and The steady-state current I is This expression is valid for all values of the parameter σ.

Limiting Case: Saturated (Zero-Order) Catalytic Kinetics
We now examine the situation where the substrate concentration level in the enzyme layer is greater than the Michaelis constant K M , (S 0 >> K M ). This situation will pertain when av 1. Here, Equations (9) and (10) reduce to the linear reaction-diffusion equations: By solving Equations (26) and (27) with their corresponding boundary conditions (Equations (12) and (13)), the exact expression of the concentrations of substrate u and product v and current has been obtained as follows: and v(X) Electrochem 2022, 3

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The steady-state current I is We note that the latter expression contains two terms. The first corresponds to the diffusion of substrate through the matrix. The second corresponds to the decomposition of the product/enzyme complex to regenerate the substrate. The factor of 2 arises from the fact that the product generated at the electrode may also be lost by diffusion through the enzyme layer, as well as reacting at the enzyme to generate substrate S. Here, the product concentration is much larger than the Michaelis constant of the enzyme (p/K M >> 1).

Validation of Analytical Results
In this section, we present numerical simulations to test the accuracy and reliability of the proposed method. The analytical expressions obtained in this paper have been compared with the highly accurate numerical solution obtained by the MATLAB routine bvp4c, which is a finite-difference code that implements the three-stage Lobatto IIIA formula. The analytically derived (Equations (16) and (17)) and computationally simulated concentration profiles of substrate and product within the enzyme layer are compared in Figure 1. These curves were computed for the following typical parameter values: The analytical and numerical solutions were plotted on the same coordinates for a wide range of possible values of the underlined problem parameters. It is clear from Table 1 that the fit between the analytical solution to the reaction/diffusion problem in the form of reactant and product concentration profiles, and the corresponding quantity numerically evaluated using MATLAB, is excellent, with average deviation between numerical and analytical results being typically 0.36% for substrate and 0.02% for product.

Influence of Parameter on Concentration of Substrate and Product
The dimensionless concentration profiles of substrate u and product v involved the cyclic substrate conversion following Michaelis-Menten kinetics are depicted in   (16) and (17)) and the dotted line represents the numerical result.

Influence of Transport and Reaction Parameters on the Current Response
In Figure 2a,b the dependence of the maximal biosensor current on the substra concentration for various values of diffusion coefficient and the thickness of th enzyme layer is outlined. We note that the steady-state current response is a monot nously decreasing function of for all values of and . From Figure 3, it is note that the biosensor current rises as increases and decreases. Further, it can b  (16) and (17)) and the dotted line represents the numerical result.

Results and Discussion
We note that the solution of the nonlinear reaction/diffusion equations as expressed in Equations (16) and (17) represents new and simple analytical expressions for the concentration substrate and product, respectively. Furthermore, the amperometric current response is described by Equation (20) for all parameter values.

Influence of Parameter on Concentration of Substrate and Product
The dimensionless concentration profiles of substrate u and product v involved in the cyclic substrate conversion following Michaelis-Menten kinetics are depicted in Figure 1.
The experimental values of the parameters used in this figure are K M = 10 −7 mol cm −3 , S 0 = 10 −7 mol cm −3 , D = 3 × 10 −6 cm 2 s− 1 , d = 0.02 cm, and V max = 10 −7 mol cm −2 s −1 [26]. From the figure, it can be noted that our obtained analytical result matches quite well with the numerical experiments. Peculiarities of the biosensor activity have been examined using our analytical result for various values of the parameters. The nonlinear distribution of concentration of both compounds throughout the membrane can be observed, and it is noted that the substrate concentration u decreases and the product concentration v increases significantly with decrease of dimensionless distance X from the outer edge of the membrane to the inner electrode/film interface.

Influence of Transport and Reaction Parameters on the Current Response
In Figure 2a,b the dependence of the maximal biosensor current ψ on the substrate concentration S 0 for various values of diffusion coefficient D and the thickness of the enzyme layer d is outlined. We note that the steady-state current response is a monotonously decreasing function of S 0 for all values of D and d. From Figure 3, it is noted that the biosensor current rises as V max increases and K M decreases. Further, it can be deduced that S 0 > 1 mol cm −3 and the biosensor current is independent of the bulk substrate concentration S 0 . This is to be expected since, under these conditions, the enzyme sites in the matrix are saturated with substrate. Electrochem 2022, 3, FOR PEER REVIEW 8 Figure 2. Dependence of current in Equation (19) on substrate concentration for the varying values of (a) and (b) , and (c) the 3D representation of the variation of normalised current density with KM and Vmax.

Sensitivity of Biosensor
The term sensitivity refers to the change of the biosensor response with regard to a change in the substrate concentration. Since both the current and the substrate concentration vary in order of magnitude, dimensionless sensitivity is another valuable parameter to be considered. The mathematical representation of the non-dimensional biosensor sensitivity for substrate concentration can be estimated from the following [42]: where is biosensor sensitivity and is steady-state current density measured at the substrate concentration . Therefore, from Equation (20), we can readily show that:

Sensitivity of Biosensor
The term sensitivity refers to the change of the biosensor response with regard to a change in the substrate concentration. Since both the current and the substrate concentration vary in order of magnitude, dimensionless sensitivity is another valuable parameter to be considered. The mathematical representation of the non-dimensional biosensor sensitivity for substrate concentration S 0 can be estimated from the following [42]: where B s is biosensor sensitivity and I(S 0 ) is steady-state current density measured at the substrate concentration S 0 . Therefore, from Equation (20), we can readily show that: where α, γ are defined in Equations (11) and (18). As the maximal enzymatic rate drops down, i.e., V max → 0, it can be observed that B s also tends to zero. Quick diffusivity contributes to relatively low sensitivity. Figure 3a supports this inference. When the membrane thickness and enzymatic rate vary by two orders of magnitude, a similar effect is again observed. This shows that thickness of the membrane plays a crucial role in determining the biosensor sensitivity and Figure 3b confirms the same. It is possible to observe from the smooth curves of Figure 3a,b that 0 ≤ B s ≤ 0.5 for all values of the parameters. This assertion receives further confirmation in the 3D plot as outlined in Figure 3c.

Resistance of Biosensor
The normalised resistance B R is represented as the gradient of steady-state biosensor current concerning the thickness of membrane d [42]: where I is the steady-state current density measured at membrane thickness d. From Equation (32) we obtain From the Equations (32) and (33), the new relationship between sensitivity and resistance can be shown as B s (S 0 ) = α 2(α+1) B R (d). Hence, there is almost zero resistance for negligibly small values of V max . This can also be observed in the three-dimensional representation of B R as a function of transport and enzyme kinetics, as presented in Figure 4c. Moreover, from the latter curve we note that the biosensor resistance increases as the diffusion coefficient decreases. The range of biosensor resistance as depicted by the nonlinear curves of Figure 4a,b lies within the bounds from 0 to 1 for all values of the parameters. Both K M and S 0 have a similar effect in the sense that, for their lower values, the device resistance increases.

Gain of the Biosensor
Sensitivity gain is described as the ratio of the trigger biosensor's steady-state current (I CEC ) to the corresponding CE biosensor's steady-state current (I CE ). This is used to compare the amplified and non-amplified biosensor response. Now, if the enzyme is absent and substrate recycling is absent, then the applied potential dictates that the device is operating under mass transport control, a steady-state limiting current is observed, and the normalised current density is given by ψ = ψ L = 1. Then, the amplification, or gain factor G, can be defined by: We note that the gain factor ψ/ψ L = √ γcoth √ γ depends on the Michaelis-Menten parameters associated with the enzyme catalysis P → S reaction. When √ γ << 1, we note that coth √ γ ∼ = 1 √ γ 1 + γ 3 , and so the gain factor is given by: In contrast, when √ γ 1, then coth √ γ ∼ = 1 and the gain factor is given by: These expressions are valid for all values of the saturation parameter α. Figure 5 displays the dependence of the gain G of the steady-state current on the substrate concentration S 0 for thinner biosensor film thicknesses d. The gain increases up to 27 times approximately at S 0 > 1 mol cm −3 and V max = 10 −3 mol cm −3 s −1 . For V max ≤ 10 −6 mol cm −3 s −1 , the gain is less than 4 and is linear, whereas for K M ≤ 10 −3 mol cm −3 , the gain is the linear function of S 0 . Electrochem 2022, 3, FOR PEER REVIEW 9 where , are defined in Equations (11) and (18). As the maximal enzymatic rate drops down, i.e., → 0, it can be observed that also tends to zero. Quick diffusivity contributes to relatively low sensitivity. Figure 3a supports this inference. When the membrane thickness and enzymatic rate vary by two orders of magnitude, a similar effect is again observed. This shows that thickness of the membrane plays a crucial role in determining the biosensor sensitivity and Figure 3b confirms the same. It is possible to observe from the smooth curves of Figure 3a,b that 0 0.5 for all values of the parameters.
This assertion receives further confirmation in the 3D plot as outlined in Figure 3c.  negligibly small values of . This can also be observed in the three-dimensional representation of as a function of transport and enzyme kinetics, as presented in Figure 4c. Moreover, from the latter curve we note that the biosensor resistance increases as the diffusion coefficient decreases. The range of biosensor resistance as depicted by the nonlinear curves of Figure 4a,b lies within the bounds from 0 to 1 for all values of the parameters. Both and have a similar effect in the sense that, for their lower values, the device resistance increases.

Gain of the Biosensor
Sensitivity gain is described as the ratio of the trigger biosensor's steady-state current ( ) to the corresponding CE biosensor's steady-state current ( ). This is used to compare the amplified and non-amplified biosensor response. Now, if the enzyme is absent and substrate recycling is absent, then the applied potential dictates that the device is operating under mass transport control, a steady-state limiting current is observed, and the normalised current density is given by 1. Then, the amplification, or gain factor G, can be defined by: We note that the gain factor / √ coth √ depends on the Michaelis-Menten parameters associated with the enzyme catalysis P  S reaction. When 1   , we note that coth √ ≅ √ 1 , and so the gain factor is given by: In contrast, when √ ≫ 1, then coth √ ≅ 1 and the gain factor is given by: These expressions are valid for all values of the saturation parameter . Figure 5 displays the dependence of the gain of the steady-state current on the substrate concentration for thinner biosensor film thicknesses . The gain increases We now mention a subtle but important point [43]. We need to evaluate the relationship between the observed flux, which is manifested as a current flow arising from the heterogeneous electrochemical reaction of the substrate S at the electrode surface, and the reaction flux corresponding to the formation of substrate S formed via reaction of the product species P at the enzyme surface. As shown in Appendix B, the observed flux giving rise to the experimentally measured current is related to the substrate regeneration flux f S at the enzyme in the diffusion layer via Here, we have introduced an efficiency factor η which determines how much substrate regenerated by the enzymatic reaction of P is lost to the solution and which cannot take place in amplification. We note that the efficiency factor takes the following form: We now mention a subtle but important point [43]. We need to evaluate the relationship between the observed flux, which is manifested as a current flow arising from the heterogeneous electrochemical reaction of the substrate S at the electrode surface, and the reaction flux corresponding to the formation of substrate S formed via reaction of the product species P at the enzyme surface. As shown in Appendix B, the observed flux giving rise to the experimentally measured current is related to the substrate regeneration flux fS at the enzyme in the diffusion layer via ⁄ (38) Here, we have introduced an efficiency factor  which determines how much substrate regenerated by the enzymatic reaction of P is lost to the solution and which cannot take place in amplification. We note that the efficiency factor takes the following form: This function is presented in Figure A1 in Appendix B. When  = 1, then  = 0.75.
Hence, under these circumstances, 25% of the substrate regenerated via the reaction of the product P with the enzyme E is lost via diffusion to the bulk solution and cannot participate in the electrochemical reaction at the electrode surface, and hence in signal amplification.

Analysis of 'Mixed Case' Moving Boundary Scenario Corresponding to Varying Levels of Saturation across the Enzyme Membrane.
We have examined the limiting situations for totally saturated kinetics when ≫ 1 and unsaturated kinetics when ≪ 1. We can also envisage the more complex situation where the level of saturation varies across the thickness of the enzyme film [44]. The simplest strategy involves the assumption that the inner part of the film is unsaturated, while the outer region of the film remains saturated. A moving normalized distance parameter * located between the two regions under consideration can be introduced. As the substrate diffuses into the film, a reaction front will have been established at * .
When * 1, the entire region is unsaturated, while at * 0, the entire region is saturated. Now, at * , . Hence, Equation (17) becomes This function is presented in Figure A1 in Appendix B. When Φ = 1, then η = 0.75. Hence, under these circumstances, 25% of the substrate regenerated via the reaction of the product P with the enzyme E is lost via diffusion to the bulk solution and cannot participate in the electrochemical reaction at the electrode surface, and hence in signal amplification.

Analysis of 'Mixed Case' Moving Boundary Scenario Corresponding to Varying Levels of Saturation across the Enzyme Membrane
We have examined the limiting situations for totally saturated kinetics when av 1 and unsaturated kinetics when av 1. We can also envisage the more complex situation where the level of saturation varies across the thickness of the enzyme film [44]. The simplest strategy involves the assumption that the inner part of the film is unsaturated, while the outer region of the film remains saturated. A moving normalized distance parameter X * located between the two regions under consideration can be introduced. As the substrate P diffuses into the film, a reaction front will have been established at X = X * . When X * = 1, the entire region is unsaturated, while at X * = 0, the entire region is saturated. Now, at = X * , v = 1 α . Hence, Equation (17) becomes On solving the above Equation (40), X * can be obtained as follows: This expression indicates the manner in which the location of the moving boundary varies with the kinetic parameter γ and the saturation parameter α.

Conclusions
In this paper, we have shown that a simple mathematical model of an amperometric biosensor operation can be successfully used to investigate and predict the dynamics of the response of the device when a cyclic substrate conversion operates. New and simple approximate analytical expressions of substrate and product concentrations, steady-state current, sensitivity, and biosensor resistance are reported. The effect of various physical parameters, such as membrane thickness, Michaelis constant, and maximum catalytic rate on the latter key performance indicators, is also discussed. This theoretical result can be applied in designing novel, highly sensitive biosensors, which minimise enzyme volume without losing sensitivity.
The corresponding boundary conditions are as follows: The approximate analytical solution for the Equation (A2) has been taken as follows: Using the boundary conditions v(0) = v 0 and v(1) = 0, the values of the parameters A 1 , B 1 have been obtained as follows: Substituting Equation (A8) in Equation (A1), we get: Now, substituting x = 0 in Equation (A9) and on solving the equation, we get: Adding Equations (A1) and (A2), we get: The boundary condition becomes: Hence, Equation (A11) becomes: with the boundary conditions: at x = 0, d f (x) dX = 1, and at x = 1, f (x) = 1 (A15) On solving Equations (A14) and (A15), we get: Using Equation (A8) in Equation (A17), we get: Using the boundary condition u(0) = 0 in Equation (A18), we get v 0 = 1.
where the kinetic parameter γ is given by We recall that σ 2 = k cat e Σ d K M (D/d) compares the rate of enzyme kinetics with that of the diffusion of P to the enzyme after the reaction via electrochemical reaction at the support electrode surface. In this enzymatic reaction, the substrate S is recycled again and can repeat the cycle, thus generating amplification.

Appendix B
The relationship between the observed flux f obs is measured as a steady-state amperometric current and the enzymatically catalyzed substrate regeneration flux f S .
In this appendix, we present a discussion which we initially developed some time ago. We consider an electrode surface coated with a thin membrane of thickness d containing an immobilized enzyme, which is in contact with an aqueous solution containing substrate or reactant species S at a bulk concentration S 0 . In this appendix, we derive a relationship between the observed flux f obs , which measures the electrochemical transformation of substrate S to product P, which can then react catalytically with the enzyme E to regenerate substrate S. These fluxes are related via the following expression: We propose that the substrate S produced by the enzyme/product reaction is formed at a uniform rate given by f S /d. Furthermore, some S will diffuse through the enzyme membrane towards the outer edge of the enzyme membrane layer and into the solution phase. This is described by the Fick diffusion equation. At the steady state, these fluxes balance and we write: This equation is solved subject to the following boundary conditions: Note that the boundary condition at x = 0 corresponds to the situation at the support electrode/enzyme layer interface. The first statement in Equation (A24) is that the potential of the electrode is set to a value such that the reactant species is set to a value such that it is immediately oxidized or reduced at the electrode surface, and so its surface concentration is zero. The second statement refers to the fact that the observed flux arises from the oxidation