Investigation of a Mathematical Model and Non-Linear Effects Based on Parallel-Substrates Biochemical Conversion

: According to the broad applicability and advancements made in addressing complex non-linear problems, Researchers have been actively involved in addressing the accurate approaches to solve non-linear problems in fields such as chemical science, material science, and image processing. In this paper, the Amperometric response of catalase-peroxidase (parallel-substrates) conversion has been discussed. The Mathematical model relies on a set of non-linear differential equations that describe the reaction and diffusion of the system. The analytical methods were extended to derive the approximate solution of the non-linear reaction-diffusion equation. The straightforward and concise analytical expressions for the concentrations and current of the Biosensor are developed. This study includes the computational resolution of the problem by utilizing a MATLAB program. A comparison between analytical outcomes and those derived numerically has been conducted. The analytical findings presented are dependable and offer an effective comprehension of the behaviour of this system.


Introduction
Biosensors are sensing instruments that convert a biological interaction into an electrical output.An oxygen-sensitive electrode is a type of Biosensor designed to detect and measure oxygen concentration in a given solution.The oxygen electrode typically consists of a sensitive element that reacts with oxygen, generating a measurable signal as a current.Amperometric Biosensors detect the output current due to the electrochemical reaction [1][2][3].While the Biosensor operates, the substrate being examined undergoes a biochemical transformation to produce a specific product.Typically, the Biosensor's response exhibits a direct relationship with the concentration of the reaction product [4].Amperometric Biosensor find applications across various fields.Some of their notable applications include Biochemical applications, Environmental monitoring, the food and beverage industry, Biotechnology and Pharmaceuticals, Industrial applications, and wearable health devices [5,6].The development of mathematical models for enzyme Biosensors began with modelling the Amperometric enzyme electrode [7][8][9][10].
The unique characteristics of biosensors are essential for designing and enhancing them.Employing Mathematical modelling replaces physical experiments to improve the configuration for Biosensors.Effectively detailing the quantitative explanation of how reactions and diffusion occur within confined surface areas, as observed in Amperometric Biosensor systems, holds great significance.Baronas et al. developed a computational model that accurately simulates an amperometric biosensor using catalase-peroxidase-catalyzed two-substrate conversion through this approach.Additionally, it analyzes how various physical and kinetic parameters impact the Biosensor's response [11].The exploration of Mathematical models for two-enzyme biosensors began with a focus on creating models for an Amperometric monolayer enzyme electrode containing two co-immobilized enzymes.Subsequent advancements led to the creation of non-linear mathematical models tailored for Amperometric two-enzyme biosensors utilizing varied enzyme combinations.
This study aims to create an analytical solution to the boundary value problem that defines how the Biosensor operates.This solution will depict the steady or non-steady concentration rates of the substances involved in the reaction, travelling through the catalytic layer based on distance and time.Analytical solutions for different kinds of biosensors have been documented in recent years.A straightforward Analytical expression is not available to determine the steady-state concentrations and current in an amperometric-based biosensor that uses parallel-substrate conversion within a biochemical reaction.Analytical methods such as Akbari-Ganji's Method (AGM) and Differential Transform Method (DTM) are applied to solve this model.The numerical results are gained using the Computational Software MATLAB.This paper introduces a straightforward and closed version of the Analytical expressions for determining substrate and product concentrations.These expressions are derived using a novel approach called the Akbari-Ganji Method (AGM).

Mathematical formulation of the problem
The catalase-peroxidase (parallel-substrate conversion) process can be used to describe the chemical reaction in the following manner.
Two simultaneous conversions of two substrates (S 1 and S 2 ), to get resulting products (P 1 and P 2 ) utilizing two enzymes (E1 and E2) to accelerate the reaction.This Biosensor works in two parallel faces.Figure 1 illustrates the schematic depiction of the system.Combining the reactions catalyzed by catalase and peroxidase within the enzyme layer, as explained by Fick's law, results in formulating the reaction-diffusion equations that depict how the Biosensor operates within the enzyme layer as follows: Where x stands for space.S e1 , S e2 are the concentration of substrates S 1 and S 2 and P e1 , P e2 are the concentration of products P 1 and P 2 .D S e1 , D S e2 , D P e1 , D P e2 are the corresponding diffusion coefficients and k 1 , k 21 and k 22 are the reaction rate that occurs in the system.This Kinetic model Amperometric Biosensor has been developed under the following assumptions [12,13].
✓ During the first stage of the Biosensor's function, the solution exclusively comprises hydrogen peroxide (S 1 ).When the Biosensor response stabilizes at the end of this initial phase.The second substrate(S 2 ) is introduced into the solution, initiating the second phase of the Biosensor's operation.
✓ The movements of substrates and product through diffusion happens beyond the enzyme layer, specifically in a region referred to as the Nernst diffusion layer.
✓ The boundary conditions of the system are given as follows.
where d refers to the thickness of the enzyme layers.
With the boundary conditions, ✓ With the dimensionless Parameters ✓ These equations were developed by assuming the model parameters [13] as, ✓ The dimensionless current response of the system is given by,

Approximate analytical expressions for the concentration of substrates and products using AGM
In general, non-linear systems do not possess precise analytical solutions.Researchers typically prefer analytical solutions as they offer a more lucid insight into how model parameters impact performance [14][15][16][17].AGM has proven effective in addressing non-linear systems.AGM is an innovative approach of solving differential equations with algebraic expressions [18][19][20][21][22].Many complex non-linear differential models encountered in science and engineering have been effectively addressed by utilizing the AGM.This method has proven its ability to provide highly accurate and approximate analytical solutions for these models.The detailed computation and analytical solution are given below, To solve the differential equations, it is assumed that the answers to the non-linear differential equations Eqn ( 9), ( 10) and ( 12) are as follows: , n, m and q are obtained by applying the boundary conditions given in Eqns ( 13) and ( 14).The following are the approximate analytical solutions for the concentration of reactant species using AGM, The Normalized current response of the system is derived as,

Approximate analytical expressions for the concentration of substrates and products using DTM
In the year 1986, Zhou introduced DTM for solving differential equations.It is the extension of Taylor's series method.DTM approach is based on finding coefficients of Taylor's series.DTM is an existing method that provides a series expression of non-linear boundary value problems [23].The detailed computation using DTM is given below, Considering the differential equation given in Eqn (9) with the initial conditions The differential transformed form of Eqn (9) concerning the conditions given in Eqn (30) is given by,

Contemporary Mathematics
Now, Assume that, Letting k = 0 in eqn (31), we get Now, taking the differential inverse transform s 1 (k) is defined as, and letting y 0 = 0, we obtain the series expression solution as follows, In a similar way we can find approximate analytical solutions for the concentration of reactant species using DTM, as follows: The Normalized current response of the system is derives as The assumed constants a, b, c and f are obtained using the boundary condition given in Eqn (14).At y = 1 and for particular values of reaction-diffusion parameters γ 1 = 0.1, γ 2 = 0.1 and µ = 5.

Comparison of analytical results with numerical simulation
In this Section, Numerical simulations are carried to evaluate how accurate and dependable the proposed method is.Recently, Mathematical modelling has been investigated with Numerical simulations [24][25][26].The Mathematical expressions derived in this paper have been cross-referenced with a highly precise numerical solution acquired through MATLAB (pdex4) software.The code relevant to this is provided in Appendix A. Furthermore, the analysis involves comparing the numerical solutions with the AGM solutions Eqns ( 21) & ( 29) and the results are presented in the following tabular form.

Results and discussion
The amount of the specific substance present in it influences the Biosensor's reaction rate.Figure 2 shows that the AGM results are very close to the Numerical solution, and the DTM results have minor variation from the Numerical results.Therefore, Eqns ( 21), ( 22), ( 23) and ( 29) are the new simple and closed analytical expressions for the Normalized concentrations of Substrates (S E1 , S E2 ) and Normalized concentrations of Products (P E1 , P E2 ). Figure 3   The concentration profiles of the substrate and product within the enzyme layer, obtained through both analytical solution and computational simulation, are compared and displayed in Figure 3.The continuous line illustrates the analytical solution, while the dashed line represents the numerical solution.These particular curves were calculated using the subsequent standard parameter values, µ = 0.1, γ 1 = 0.1, γ 2 = 22 for S E1 , µ = 5, γ 1 = 0.1, γ 2 = 20 for S E2 , µ = 100, γ 1 = 100, γ 2 = 500 for P E1 , µ = 100, γ 1 = 100, γ 2 = 100 for P E2 .Both the calculated analytical and numerical solutions were represented on a shared graph, encompassing a broad spectrum of potential values for the highlighted parameters within the problem.Table 1 and 2 evidently shows an exceptional agreement between the analytical solution depicting reactant and product concentration profiles for the reaction-diffusion problem and the corresponding values obtained through numerical evaluation using MATLAB.On average, the variation between numerical and analytical outcomes is typically 0.39% for substrate (S 1 ) and 0.92% for product (P 1 ), indicating a highly accurate match.(21) with Numerical (Num) result for the fixed value of parameter µ = 5   Observing the figure, it's evident that the analytical result we obtained aligns closely with the outcomes from numerical experiments.We have also analyzed the unique characteristics of the Biosensor activity by employing our analytical findings across different parameter values.As the rate of the reaction goes up, there is a corresponding rise in the concentration of the Product.The concentration of the substrate declines as the reaction rate increases.The Biosensor's present reaction is reliant on the concentration of the product (P 1 ).The analytical derivation provides the equation for the dimensionless current as expressed in Eqn (24).The current generated by the Biosensor is independent on the reaction rate.Instead, it steadily rises when reaction rate increases and decreases consistently as reaction rate γ 1 increases.It is displayed in Figure 6, keeping the parameters as constant, adjust the reaction parameter γ 1 and γ 2 across different values.This work may extend for predicting biosensor response with multi-substrate conversion and biosensor with substrate inhibition.And also, the Mathematical formulation can be developed for the enzyme electrode with different shapes.

Conclusions
A Mathematical representation illustrating the steady-state reaction behaviour of an amperometric biosensor that operates using catalase-peroxidase biochemical reactions (parallel-substrate conversion) is presented.A coupled nonlinear Differential Equations has been effectively addressed by applying a novel technique known as the AGM.The obtained analytical expressions offer a simple, straightforward and more effective means to grasp and anticipate how the system will behave.The results obtained from our analytical approach, which solves for the substrates and resulting products, have been verified against a numerical method, showing reliable and matching outcomes.The Biosensor current response concerning the kinetic parameters is also discussed.

Figure 1 .
Figure 1.Schematic represeantation of the system depicts all the concentration curves verses dimensionless space.

Figure 2 .
Figure 2. Comparison of AGM and DTM results with Numerical results (NUM)

Figure 4 .
Figure 4. Plots between dimensionless concentration of First Product (S E1 ) and dimension space y

Figure 5 .Figure 6 .
Figure 5. Plots between dimensionless concentration of First Product (P E1 ) and dimension space y

Table 1 .
Comparison of the molar concentration of Substrate S 1 (S E1 ) given in Eqn

Table 2 .
1Comparison of the molar concentration of Substrate P 1 (P E1 ) given in Eqn (29) with Numerical (Num) result for the fixed value of parameter µ = 5 1