Mathematical Modeling of Multienzyme Biosensor System

A mathematical model of hybrid inhibitor biosensor system is discussed. This model consists of five nonlinear partial differential equations for bisubstrate sensitive amperometric system. Simple and closed form of analytical expressions for concentration of glucose-6-phosphate (substrate), potassium dihydrogen phosphate (inhibitor), oxygen (co-substrate), glucose (product 1), and hydrogen peroxide (product 3) is obtained in terms of rate constant using modified Adomian decomposition method (MADM). In this study, behavior of biokinetic parameters is analyzed using this theoretical result.Theobtained analytical results (concentrations) are compared with the numerical results and are found to be in satisfactory agreement.


Introduction
A biosensor is an analytical device, used for the detection of an analyte or substrate that combines a biological component with a physicochemical detector [1].Biosensors are potentially useful as they detect components quickly and have been active in the research area for certain years.The development of these biosensing systems depends on a quantitative measurement of the enzyme activity before and after exposure to a target analyte.Inhibition-based biosensors have been the topic of several recent reviews [2,3].Biosensors based on the principle of enzyme inhibition have been applied for a wide range of significant analytes.Some recent approaches and developments in enzyme inhibition-based biosensors for food safety and environmental monitoring have been discussed by Amine et al. [4].The application of whole-cell-based biosensors to different environmental media, such as water, soil, and atmospheric monitoring has been reviewed by Gu et al. [5].An algal biosensor for toxicity assessment of estuarine waters was tested at different concentrations [6].On biosensor technology, initial studies for the construction of a diffusive sampling device was described for monitoring personal exposure to formic acid [7].The determination of benzene in workplace air samples using a bacterial biosensor based on flow injection analysis (FIA) has been developed by Lanyon et al. [8].Kok and Hasirci [9] reported that the total anticholinesterase activity of binary pesticide mixtures was lower than the sum of the individual inhibition values.Mazzei et al. [10] immobilized acid phosphatase and glucoseoxidase on a separate dialytic membrane.The bienzyme modified membrane was placed on platinum electrode using an "O" ring.On enzymatic biosensor, the steady state analytical expression of concentration of substrate is derived using variational iteration method [11].A monolayer potentiometric biosensor response for a polymer-modified electrode system has been discussed by Rahamathunissa et al. [12].The biosensors can be used for biomedical applications [13] and electric circuits [14].Some nonlinear problems in different types of biosensors are solved using various numerical methods [15] and analytical methods [16].
The multienzyme biosensor system was used for the detection of phosphorous and fluoride ions.Rangelova et al. [17] investigated the influence of concentration and reaction rate constant of inhibitor with two enzymes in the dynamic mode.In this work, we study the steady state performance of multienzyme inhibitor system performing glucose-6phosphate hydrolysis at different conditions.The analytical expression for concentrations and current are derived using modified Adomian decomposition method.The effect of internal mass transfer parameters and kinetic parameters of inhibitor is discussed.

Mathematical Formulation of the Problem
In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products.Almost all chemical reactions in a biological cell need enzymes in order to occur at rates sufficient for life.The working mechanism of hybrid biosensor for phosphate is based on the following sequential biochemical reactions [8]: The biochemical reaction, which is biocatalyzed by acid phosphatase (AP), involves the hydrolysis of glucose-6-phosphate (C 6 H 13 O 9 P) with the liberation of glucose (C 6 H 12 O 6 ) and inorganic phosphate (HPO 2− 4 ).In the second reaction with glucose oxidase (GOD), the oxygen present oxidizes the glucose into glucono delta-lactone (C 6 H 10 O 6 ) and hydrogen peroxide (H 2 O 2 ).The amount of hydrogen peroxide being produced is measured electrochemically.On the basis of the biochemical reactions, the following successive enzyme reactions with competitive inhibition have been proposed: where , , ,  1 , and  3 are the concentrations of the glucose-6-phosphate (substrate), potassium dihydrogen phosphate (inhibitor), oxygen (co-substrate), glucose (product 1), and hydrogen peroxide (product 3) respectively. 1 (acid phosphatase) is the first enzyme and  2 (glucose oxidase) is the second enzyme. 2 and  4 represent the inorganic phosphate and glucono delta-lactone.Figure 1 presents the overall reaction scheme for a multienzyme inhibitor system.We assume that diffusion is one-dimensional in space and is described with second Fick's law.The concentration profiles of five reagents of the system are described by the following nonlinear equations: where   ,   ,   1 , and   3 are diffusion coefficients for substrate, co-substrate, product 1, and product 3.   denote the reaction constant for concentration profiles ( = , , ,  1 ), respectively. 1 is the reaction velocity of acid phosphatase and  2 is the reaction velocity of glucose oxidase.The initial conditions are  = 0,  (, 0) =  0 ; (, 0) =  0 ;  (, 0) =  0 ;  1 (, 0) = 0;  3 (, 0) = 0. ( Equations ( 4) are subjected to the following boundary conditions: The output current is defined as where  is the number of electrons,  is the Faraday's constant, and  is the area of the electrode surface.
International Journal of Computational Mathematics 3

Governing Equations in Dimensionless Form
Equations ( 4)-( 7) can be reduced to the corresponding dimensionless forms by introducing the following dimensionless parameters: Consequently, ( 4)-( 7) can be written in dimensionless form for steady state as follows: The boundary conditions are Here we can assume that  is very small ( = 0) at  = 0.The dimensionless current is as follows:

Analytical Expressions of the Concentrations and Current Using Modified Adomian Decomposition Method
In order to solve the boundary value problem, ( 9)-( 13), we used the modified Adomian decomposition method [18,19].
The basic idea of this method is described in Appendix A [20].Detailed derivations of the dimensionless concentrations , V, , , and  of glucose-6-phosphate, potassium dihydrogen phosphate, oxygen, glucose, and hydrogen peroxide are described in Appendix B. As a result, we have obtained International Journal of Computational Mathematics where Equations ( 16)-( 20) represent the analytical expression of the concentration profiles of five reagents for all values of the dimensionless parameters , , , ,  and .Also, these expressions satisfy the boundary conditions (14).The current response is given by

Numerical Simulation
The nonlinear differential equations ( 9)-( 13) with boundary conditions (14) have been solved numerically using SCILAB/MATLAB software.A respective script pdex4 is provided in Algorithm 1.The numerical solution is compared with our analytical results in Figures 2-6.The comparison reveals that the analytical dimensionless concentrations and numerical results are in satisfactory agreement.The dimensionless concentration of substrate and product 3 obtained using ( 16) and ( 20) are reported in Tables 1 and 2. In these tables, our result is compared with simulation results.The average relative error between our analytical result for the concentration of substrate and simulation result are 1.487% when  = 25 and 0.00% when  = 50, 100 and 250 (Table 1) whereas the average relative error in the case for concentration of product 3 is 0.00% for all values of   (Table 2).

Results and Discussion
The normalized nonlinear differential equations were solved using modified Adomian decomposition method.Equations ( 16)-( 20) represent the analytical expressions for the dimensionless concentrations of glucose-6-phosphate (), potassium dihydrogen phosphate V(), oxygen (), glucose (), and hydrogen peroxide ().Figure 2 demonstrates quantitatively the effects of dimensionless substrate inhibition modulus, dimensionless substrate Thiele modulus, and dimensionless inhibitor inhibition modulus on the dimensionless concentration profile of substrate () as a function of dimensionless distance  using (16).It shows that the value of () approaches unity when  = 0 for all values of ,   , and .It is evident that the substrate concentration () increases as a result of increasing dimensionless inhibition modulus  and dimensionless substrate modulus .Further, the substrate concentration () decreases when dimensionless Thiele modulus   or thickness of the membrane increases.
Figure 3 represents the dimensionless concentration profile of the inhibitor V() versus dimensionless distance  using (17).It is clear that for the increasing values of dimensionless inhibitor Thiele modulus   and dimensionless substrate inhibition modulus , the concentration profile V() deviates from the linear form.Also, the inhibitor concentration V() increases when dimensionless inhibitor inhibition modulus  increases (see Figure 3(f)).The dimensionless concentration profile of co-substrate () against dimensionless distance  is plotted in Figure 4. From these Figures, we conclude that the value of () decreases when the dimensionless co-substrate Thiele modulus   increases.Furthermore, the co-substrate profile () increases when dimensionless Michaelis-Menten constant of glucose  and oxygen  increases (see Figures 4(e) and 4(f)).

Conclusions
The system of nonlinear, steady-state reaction diffusion equations in multienzyme biosensor has been solved analytically using modified Adomian decomposition method.The accuracy of the approximate analytical solutions has been verified by comparison with numerical solutions.Satisfactory agreement is noted.This method is very effective, simple, and time saving.It also gives the better results.This method can be easily extensive to all kinds of nonlinear differential equations.Moreover, the behavior of inhibitor starting concentration over biosensor can be analyzed through the current expression.

Figure 1 :
Figure 1: Reaction scheme of the enzyme with competitive inhibition process.

Figures 7 (
a)-7(b) show the effect of dimensionless Michaelis-Menten constant  on the dimensionless current as a function of product 3 dimensionless Thiele modulus   with .The dimensionless current  approaches zero and it is independent of  at low dimensionless Thiele modulus.Also, the dimensionless current increases with decreasing  at high dimensionless Thiele modulus.Figures7(c)-7(d) represent the effect of Thiele modulus   on the dimensionless current as a function of product 3 dimensionless Michaelis-Menten constant  with .The dimensionless current  decreases with decreasing dimensionless Thiele modulus   .From the Figures 7(e)-7(f), it is inferred that the dimensionless current  increases when dimensionless Michaelis-Menten constant of glucose  decreases for some fixed values of   .

Figure 2 :Figure 3 :Figure 4 : 2 𝛽 2 𝛼Figure 5 :Figure 6 :Figure 7 :
Figure 2: Plot of dimensionless substrate concentration  versus dimensionless distance  for various values of , , and   .The key to the graph (solid line) represents our approximate solution presented in this study (see(16)) and (dotted line) represents the numerical solution.

Table 1 :
Comparison of dimensionless concentration of glucose-6-phosphate  with numerical simulation for various values of dimensionless distance .(seeFigure2(a)).