Mathematical modeling of amperometric biosensor with cyclic reaction is discussed. Analytical expressions pertaining to the concentration of substrate, cosubstrate, reducing agent and medial product and current for hybrid enzyme biosensor are obtained in terms of Thiele module and saturation parameters. In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to solve the system of nonlinear differential equations. Furthermore, in this work the numerical simulation of the problem is also reported using Scilab/Matlab program. Our analytical results are compared with simulation results. A good agreement between analytical and numerical results is noted.
Biosensor (Figure
Basic scheme of a biosensor [
The biosensor was first described by Clark and Lyons in 1962, when the term enzyme electrode was adopted [
Amperometric electrodes have been used in the design of biosensors for glucose, aminoacids, and other molecules [
Recently, Rajendran and coworkers [
In enzymebased catechol biosensor, the cyclic reaction scheme for the substrate, cosubstrate, reducing agent and medial product can be represented as follows [
A cyclic reaction between catechol and 1,2benzoquinone takes place by combining the tyrosinase reaction and the chemical reduction of 1,2 benzoquinone to catechol by
Here
By introducing the following set of nondimensional variables,
The homotopy analysis method (HAM) [
Using HAM method (Appendix
In order to investigate the accuracy of this analytical method with a finite number of terms, the system of differential equations ((
Comparison between the analytical normalized substrate concentration






Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  
0.0  0.0000  0.0000  0.00  0.5000  0.5000  0.00  1.0000  1.0000  0.00 
0.2  0.0974  0.0976  0.21  0.4870  0.4869  0.02  0.9835  0.9833  0.02 
0.4  0.0951  0.0954  0.32  0.4781  0.4779  0.04  0.9725  0.9723  0.02 
0.6  0.0935  0.0938  0.32  0.4730  0.4729  0.02  0.9665  0.9664  0.01 
0.8  0.0929  0.0932  0.32  0.4712  0.4710  0.04  0.9644  0.9642  0.02 
1.0  0.0929  0.0931  0.22  0.4711  0.4707  0.08  0.9643  0.9639  0.04 
 
Average deviation  0.23  Average deviation  0.03  Average deviation  0.01 
Comparison between the analytical normalized medial product concentration






Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  
0.0  0.1000  0.1000  0.00  0.5000  0.5000  0.00  1.0000  1.0000  0.00 
0.2  0.0963  0.0963  0.00  0.4798  0.4798  0.00  0.9591  0.9592  0.01 
0.4  0.0934  0.0935  0.11  0.4640  0.4643  0.06  0.9272  0.9278  0.06 
0.6  0.0913  0.0915  0.22  0.4527  0.4533  0.13  0.9044  0.9056  0.13 
0.8  0.0901  0.0902  0.11  0.4459  0.4467  0.18  0.8907  0.8924  0.19 
1.0  0.0896  0.0898  0.22  0.4436  0.4446  0.23  0.8861  0.8880  0.21 
 
Average deviation  0.11  Average deviation  0.10  Average deviation  0.10 
Comparison between the analytical normalized reducing agent concentration






Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  
0.0  0.1000  0.1000  0.00  0.5000  0.5000  0.00  1.0000  1.0000  0.00 
0.2  0.0934  0.0934  0.00  0.4671  0.4671  0.00  0.9343  0.9341  0.02 
0.4  0.0884  0.0884  0.00  0.4421  0.4420  0.02  0.8843  0.8840  0.03 
0.6  0.0849  0.0849  0.00  0.4246  0.4244  0.05  0.8491  0.8488  0.03 
0.8  0.0828  0.0828  0.00  0.4141  0.4139  0.05  0.8283  0.8279  0.05 
1.0  0.0821  0.0821  0.00  0.4107  0.4105  0.05  0.8214  0.8210  0.05 
 
Average deviation  0.00  Average deviation  0.03  Average deviation  0.03 
Comparison between the analytical normalized cosubstrate concentration






Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  Our work ( 
Numerical  % Error  
0.0  0.1000  0.1000  0.00  0.5000  0.5000  0.00  1.0000  1.0000  0.00 
0.2  0.0800  0.0800  0.00  0.4000  0.4000  0.00  0.8000  0.7999  0.01 
0.4  0.0600  0.0600  0.00  0.3000  0.2999  0.03  0.6000  0.5999  0.02 
0.6  0.0400  0.0400  0.00  0.2000  0.1999  0.05  0.4000  0.3999  0.03 
0.8  0.0200  0.0200  0.00  0.1000  0.1000  0.00  0.2000  0.2000  0.00 
1.0  0.0000  0.0000  0.00  0.0000  0.0000  0.00  0.0000  0.0000  0.00 
 
Average deviation  0.00  Average deviation  0.01  Average deviation  0.01 
Normalized concentration profiles of reactants
Normalized concentration profiles of substrate
Normalized concentration profiles of medial product
Normalized concentration profiles of reducing agent
Normalized concentration profiles of cosubstrate
The concentration of substrate, cosubstrate, reducing agent and medial product depends upon Thiele module and saturation parameters. The Thiele module
The concentration profiles for the four reactants for some fixed values of parameters are shown in Figure
The active membrane thickness
The normalized current
Diagrammatic representation of the normalized current
The theoretical model of hybrid amperometric enzyme biosensor with cyclic reaction and biochemical amplification for steadystate condition is discussed. The system of three nonlinear differential equations for pingpong enzyme kinetics has been solved analytically. Influence of Thiele module and active membrane thickness is investigated. The obtained results have a good agreement with those obtained using numerical method. This analytical result will be useful in sensor design, optimization, and prediction of the electrode response. Using this result, the action of biosensor is analyzed at critical concentration of substrate and enzyme activities. Theoretical results obtained in this paper can also be used to analyze the effect of different parameters such as active membrane thickness and saturation parameters.
Consider the following differential equation [
From (
Adding (
function pdex4
m = 0;
x = linspace(0,1);
t=linspace(0,100000);
sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);
u1 = sol(:,:,1);
u2 = sol(:,:,2);
u3 = sol(:,:,3);
u4 = sol(:,:,4);
figure
plot(x,u1(end,:))
title(‘u1(x,t)’)
xlabel(‘Distance x’)
ylabel(‘u1(x,2)’)
figure
plot(x,u2(end,:))
title(‘u2(x,t)’)
xlabel(‘Distance x’)
ylabel(‘u2(x,2)’)
figure
plot(x,u3(end,:))
title(‘u3(x,t)’)
xlabel(‘Distance x’)
ylabel(‘u3(x,2)’)
figure
plot(x,u4(end,:))
title(‘u4(x,t)’)
xlabel(‘Distance x’)
ylabel(‘u4(x,2)’)
function [c,f,s] = pdex4pde(x,t,u,DuDx)
c = [1; 1; 1; 1];
f = [1; 1; 1; 1].* DuDx;
Q=9;
p1=0.92;
p=1.2;
m1=0.2;
m2=0.05;
m3=0.001;
n1=0.8;
n2=0.1;
n3=0.6;
F=Q*p1*((1/(1+1/(u(1))+1/(u(4)))m1*(u(2))* (u(3))));
F1=Q*n2*(m2*(u(2))1/(1+1/(u(1))+1/(u(4))));
F2=m3*n3*Q*(u(3));
F3=Q*n1*p*(1/(1+1/(u(1))+1/(u(4))));
s=[F; F1; F2; F3];
function u0 = pdex4ic(x);
u0 = [1; 1; 1; 1];
function [pl,ql,pr,qr]=pdex4bc(xl,ul,xr,ur,t)
pl = [ul(1)0.09;ul(2)−0;ul(3)1;ul(4)0.54];
ql = [0; 0; 0; 0];
pr = [0; 0; 0; ur(4)];
qr = [1; 1; 1; 0];
The analytical solution represented by (
The
Measured substrate concentration of catechol (mM)
Medial product concentration of 1,2 benzoquinone (mM)
Reducing agent concentration of Lascorbic acid (mM)
Cosubstrate concentration of oxygen (mM)
Maximal rate (mmol/(l.s))
Diffusion coefficient for substrate (
Diffusion coefficient for medial product (
Diffusion coefficient for reducing agent (
Diffusion coefficient for cosubstrate (
Reaction rate constants (mmol/(l.s))
Reaction rate constants (mM)
Distance coordinate (
Active membrane thickness (
Convergence control parameter
Normalized measured substrate concentration (dimensionless)
Normalized medial product concentration (dimensionless)
Normalized reducing agent concentration (dimensionless)
Normalized cosubstrate concentration (dimensionless)
Normalized distance coordinate (dimensionless)
Saturation parameters (dimensionless)
Linear enzyme kinetic coefficient (dimensionless)
Ratio of diffusion coefficients (dimensionless)
Ratio of reaction rate constants (dimensionless)
Thiele module (dimensionless)
Normalized current (dimensionless).
This work was supported by the University Grants Commission (F. no. 39–58/2010(SR)), New Delhi, India. The authors are thankful to Dr. R. Murali, The Principal, The Madura College, Madurai, and Mr. M. S. Meenakshisundaram, The Secretary, Madura College Board, Madurai, for their encouragement. The author K. Indira is very thankful to the Manonmaniam Sundaranar University, Tirunelveli, for allowing to do the research work.