Approximate Analytical Expressions of Non-linear Boundary Value Problem in an Amperometric Biosensor...

Copyright © 2014 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2014, 10(3): 201-219

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN: 2166-286X

Florida, USA

Article

Approximate Analytical Expressions of Non-linear Boundary

Value Problem in an Amperometric Biosensor Using the New

Homotopy Perturbation Method

D. Shanthi2 , V. Ananthaswamy1 and L. Rajendran1, *

1Department of Mathematics, The Madura College, Madurai-625011, Tamil Nadu, India

2PG Assistant, Government Higher Secondary School, Podhumbu, Madurai, Tamil Nadu, India.

*Author to whom correspondence should be addressed1: [email protected]; Ph.No. 0452-

4208051, Mob. No. +91-9442228951.

Article history: Received 2 February 2014, Received in revised form 20 May 2014, Accepted 30 May

2014, Published 6 June 2014.

Abstract: A theoretical model of modified enzyme-membrane electrode forsteady-state

condition is discussed. This model contains a non-linear term related to enzyme kinetics

reactions. The closed and simple approximate analytical expressionsofthe concentrations of

the species and current are obtained by new approach of Homotopy perturbation method.

These expressions are derived for all possible values of parameters 2 (Thiele modulus), 0B

(normalized surface concentration of oxidized mediator) and SB (normalized surface

concentration of substrate). The theoretical results thus obtained were then verified by the

numerical results. A good agreement between theoretical predictions and numerical results

is observed.

Keywords Enzyme-membrane; Permeable electrode; Immobilised enzyme layer; Biosensor;

Numerical Simulation; New Homotopy perturbation method.

Mathematics Subject Classifications (2000): Mathematical Modeling and non-linear

differential equation.

mailto:[email protected]

Int. J. Modern Math. Sci. 2014, 10(3): 201-219


202

1. Introduction

Polymer membranes have been utilized in biomaterials, bio-separators and biosensors [1-2]. The

membranes provide an ideal support for the immobilization of the biocatalyst. Substrate partition at the

membrane/fluid inter-phase can be used to improve the selectivity of the catalytic reaction towards the

desired products [3]. Recently a new method for enzyme immobilization [4] has been successfully used

for the building of enzymatic bio-sensors and also of a chemically active membrane [5]. In the recent

three decades, much effort has been devoted to the development of various biosensors involving

biologically sensitive component and transformers - devices with many fields of applications [6-8].

Atwo-substrate non-linear enzyme reaction model has been developedexperimentally [9-10]. The

behaviour of a glucose oxidize (GOx) electrode [11-12] is discussed in this model. It has been found that

the mediators could not totally replace the natural co-substrate when both were present in the assay

solution. So that here, a three-substrate model would be required. In these cases, various complex

calibration curve of the enzyme electrode was observed [13-14].

Inspite of extensive experimental investigations for the design of bio-sensor, only a few studies

concerned the modeling or theoretical design of such system. Loghambal and Rajendran [15] have

described the mathematical model in an amperometric oxidase enzyme-membrane electrode. Simulation

results for amperometric enzyme electrode [16-18] reported using Runge-Kutta method [16]and

shooting method [17-18]. But in this paper, the same system is modeled analytically. However, to the

best of our knowledge, till date no general simple analytical results for the concentration of oxidized

mediator, substrate and reduced mediator for all values of the parameters have been reported [17-19].

The purpose of this communication is to derive the closed-form of analytical expressions of

concentrations of mediator, substrate and reduced mediator by solving the system of non-linear reaction-

diffusion equations using the New Homotopy perturbation method [34-35]. The theoretical models of

enzyme electrodes give information about the mechanism and kinetics operating in the biosensor. This

theoretical results gained from this modeling can be useful in sensor design, optimization and prediction

of the electrode-membrane’s response.

2. Mathematical Formulation of the Problems

Gooding and Hall [17] presented a concise discussion and derivation of the dimensionless non-

linear mass transport equation for this model, which is summarized briefly for completeness. In this

model the substrate and co-substrate penetrate through a permeable electrode to the enzyme layer and

then reduces to the form of co-substrate which diffuses back to the electrode. The general reaction

scheme for an immobilized oxidase in the presence of two oxidants is given as follows [17]:



203

where mk is the rate constant for the forward direction of the mth reaction and 1k is the rate constant for

the backward direction. If ]E[ T is the total enzyme concentration in the matrix then at all times,

][EES][]E[][E redOXT (4)

where ]E[ OX , ]ES[ and ][Ered are the oxidized mediator, enzyme-substrate complex and reduced

mediator enzyme concentrations respectively. At steady state, the diffusion of a substrate into the

enzyme layer is equal to the reaction rate of the substrate within the matrix. We examine a planar matrix

of thickness y = d, where diffusion is considered in the y- direction only (edge effects are neglected)

(Fig.1).

Fig. 1.Schematic representation of typical enzyme-membrane electrode geometry [17].

The system of non-linear differential equations for this scheme is given as follows [17]:

1

OX

OST22

OX2

M 1][Med[S]

][Edy

]Med[d

kD (5)



204

1

OX

OST22

2

S 1][Med[S]

][Edy

]S[d

kD (6)

1

OX

OST22

red2

M 1][Med[S]

][E dy

]Med[d

kD (7)

where MD is the diffusion coefficient of the oxidized and reduced forms of the mediator (assumed to be

equal) and SD is the diffusion coefficient of substrate within the enzyme layer. ]Med[ OX , ]Med[ red and

S][ are the concentration of oxidized mediator, reduced mediator and substrate at any position in the

enzyme layer. S and O are the dimensionless rate constants

( 121S /)( kkk and 42O / kk ). The Eqns. (5)-(7) are solved for the following boundary

conditions:

At the far wall, y = 0

0dy]Med[ddy/]S[ddy]Med[d redOX (8)

at the electrode, y = d

0]Med[ ,][S]S[]S[ ,]Med[]Med[]Med[ redSbOXObOXOX KK (9)

bOX ]Med[ and b]S[ are the concentration of oxidized mediator and substrate at the enzyme layer|

electrode boundary, and ]Med[ OX and ][S are the bulk solution concentrations. OK and SK are the

equilibrium partition coefficients for oxidized mediator and the substrate respectively. We make the

non-linear differential eqns. (5)-(7) to dimensionless form by defining the following dimensionless

variables,

][S][Med and ]Med[]E[

,]S[ ,]Med[ , y

,]Med[]Med[ ,]S[]S[ ,]Med[]Med[

bSbOXMSbOXMT222

SbS0bOX0

bredredRbSbOXOX0

DDDkd

BBdx

FFF

(10)

where OF , SF and RF are the normalized surface concentrations of oxidized mediator, substrate and

reduced mediator and x is the normalized distance. OB and SB are the normalized surface concentration

of oxidized mediator and substrate. 2 is the Thiele modulus for the oxidized mediator which governs

reaction/diffusion. The dimensionless form of the oxidized mediator, substrate and reduced mediator are

as follows:

S0S0SS00

S0S02

2

02

FFBBFBFB

FFBB

dx

Fd (11)

S0S0SS00

S0S02S2

S2

FFBBFBFB

FFBB

dx

Fd (12)



205

S0S0SS00

S0S02

2

R2

FFBBFBFB

FFBB

dx

Fd (13)

The consumption of oxidized mediator, substrate and reduced mediator, are all related processes. So

there is only one independent variable for which to solve

S0S0SS00

S0S02

2

R2

2

S2

S2

02

1

FFBBFBFB

FFBB

dx

Fd

dx

Fd

dx

Fd

(14)

The normalized boundary conditions are given by:

0)0('0 F 0)0('S F 0)0('

R F (15)

1)1(0 F 1)1(S F 0)1(R F (16)

From the eqn. (14) we get,

2

02

2

S2

S

1

dx

Fd

dx

Fd

(17)

and

2

R2

2

S2

dx

Fd

dx

FdS (18)

Integrating the eqns. (17) and (18) twice and applying the appropriate boundary conditions eqns. (15)

and (16) we get,

)1)(()/1(1)( S0 xFxF S (19)

)(1)/1()(R xFxF SS (20)

The corresponding normalized current response is given by

1

R

xdx

dFI (21)

3. Analytical Expression of the Normalized Surface Concentrations Using New

Homotopy Perturbation Method

Linear and non-linear phenomena are of fundamental importance in various fields of science and

engineering. Most models of real life problems are still very difficult to solve. Therefore, approximate

analytical solutions such as Homotopyperturbation method (HPM) [20-33] were introduced. This

method is the most effective and convenient ones for both linear and non-linear equations. Perturbation

method is based on assuming a small parameter. The majority of non-linear problems, especially those

having strong non-linearity, have no small parameters at all and the approximate solutions obtained by

the perturbation methods, in most cases, are valid only for small values of the small parameter.

Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter is small.



206

However, we cannot rely fully on the approximations, because there is no criterion on which the small

parameter should exists. Thus, it is essential to check the validity of the approximations numerically

and/or experimentally. To overcome these difficulties, HPM have been proposed recently.

Recently, many authors have applied the Homotopy perturbation method (HPM) to solve the

non-linear boundary value problem in physics and engineering sciences [20-23]. Recently this method

is also used to solve some of the non-linear problem in physical sciences [24-26]. This method is a

combination of Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the

Lighthill equation [24], the Diffusion equation [25] and the Blasius equation [26-27]. The HPM is unique

in its applicability, accuracy and efficiency. The HPM [18-33] and New HPM [34-35] uses the

imbedding parameter p as a small parameter, and only a few iterations are needed to search for an

asymptotic solution.The analytical expression of concentration (see Appendix B) of the substrate is as

follows:

A

AxxF

cosh

cosh)(S (22)

Using the eqn. (22), we can obtain the concentrations of oxidized mediator 0F and reduced mediator

RF from the eqns. (19) and (20).

1cosh

cosh11)(0

A

AxxF

S (23)

cosh

cosh1

1)(R

A

AxxF

S (24)

From the eqns. (21) and (24) we get the dimensionless current is as follows:

hAAIS

tan 1

(25)

whereSS

Ss

BBBB

BBA

00

00

(26)

4. Numerical Simulation

The non-linear diffusion equations (eqns. (11)-(13)) for the boundary conditions (eqns. (15) and

(16)) are also solved numerically. We have used the function pdex4 in Scilab/Matlab numerical software

to solve numerically, the initial-boundary value problems for parabolic-elliptic partial differential

equations. This numerical solution is compared with our analytical solutions in Figs. 2 -5.



207

Fig. 2. Normalized concentrations of (a) Oxidized mediator 0F (eqn. (23)) (b) Substrate SF (eqn.(22))

and (c) Reduced mediator RF (eqn. (24)) computed for some fixed values of the dimensionless

parameters 05.0 ,0052.0 ,1.0 SSO BB and various values of thiele modulus 2 .



208

Fig. 3. Dimensionless concentrations of (a) Oxidized mediator OF (eqn. (23)) (b) Substrate SF (eqn.

(22)) and (c) Reduced mediator RF (eqn.(24)) for some fixed values of parameters

25 and 5.0 ,1.0 2SO B and various values of normalized surface concentration of substrate SB

, when (i) 001.0S B (ii) 005.0S B (iii) 01.0S B (iv) 05.0S B (v) 1.0S B and (vi) 1S B .



209

Fig. 4. Dimensionless concentrations of (a) Oxidized mediator OF (eqn. (23)) (b) Substrate SF (eqn.

(22)) and (c) Reduced mediator RF (eqn.(24)) for some fixed values of parameters

100 ,05.0 ,0052.0 2

SS B and various values of normalized surface concentration of oxidised

mediator 0B when, (i) 001.00 B (ii) 005.00 B (iii) 01.00 B (iv) 05.00 B (v) 1.00 B and (vi)

10 B .



210

Fig. 5. Dimensionless concentrations of the Oxidized mediator OF (eqn. (23)), Substrate SF (eqn. (22))

and the Reduced mediator RF (eqn.(24)) versus the normalized distance x when

05.0 ,0052.0 ,1.0 SSO BB and 4002

Fig. 6. Variation of normalized current Iwith (a) normalized surface concentration of oxidized mediator

OB ,(b)& (c)normalized surface concentration of the substrate SB (d) normalized parameter S for

various values of thiele modulus 2 using eqn. (25), when (i) 502 (ii) 1002 (iii) 2002 (iv)

3002 (v) .4002



211

5. Results and Discussions

Fig. 1 is the Schematic representation of typical enzyme-membrane electrode geometry [17].

Fig. 2 is the normalized concentrations of (a) oxidized mediator OF , (b)substrate SF and (c) reduced

mediator RF versus the dimensionless distance x. From Fig. 2 (a) and (b), it is clear that when the thiele

modulus 2 increases, the corresponding normalized concentrations of oxidized mediator 0F and the

substrate SF decreases for some fixed values of ,0 SBB and .S From Fig. 2 (c), it is noted that when

the thiele modulus 2 increases, the normalized concentrations of the reduced mediator RF also

increases for some fixed values of the surface concentrations of oxidized mediator 0B and the substrate

SB and the dimensionless parameter .S

Fig. 3 is the normalized concentrations of (a) oxidized mediator 0F , (b)substrate SF and (c)

reduced mediator RF versus the dimensionless distance x. From Fig. 3 (a) and (b), we infer that when the

dimensionless parameter 0B increases, the corresponding the corresponding surface concentrations of

oxidized mediator 0F and the substrate SF decreases for some fixed values of the dimensionless

parameters , 2S B and .S From Fig. 3(c), it is noted that when the dimensionless parameter 0B

increases, the normalized surface concentrations of the reduced mediator RF also increases for some

fixed values of the dimensionless parameters , 2S B and .S

Fig. 4 is the normalized concentrations of (a) oxidized mediator OF , (b)substrate SF and (c)

reduced mediator RF versus the dimensionless distance x. From Fig. 4 (a) and (b), we infer that when the

dimensionless parameter SB increases, the corresponding the corresponding concentrations of oxidized

mediator 0F and the substrate SF decreases for some fixed values of the dimensionless parameters

, 20 B and .S From Fig. 4(c), it is noted that when the dimensionless parameter SB increases, the

normalized concentrations of the reduced mediator RF also increases for some fixed values of the

dimensionless parameters , 20 B and .S

Fig. 5 is the normalized concentrations of the oxidized mediator 0F , substrate SF and the reduced

mediator RF versus the dimensionless distance x . From this figure, we note that the oxidized mediator

and the substrate increases and the reduced mediator decreases for some fixed values of the

dimensionless parameters , , 20 SBB and .S



212

The normalized current I can be calculated using the eqn. (25). Fig. 6 is the normalized current I

versus (a) the normalized surface concentrations of the oxidized mediator 0F , (b) &(c) the normalized

surface concentrations of the substrate SF and (d) the normalized parameter S . From these figures, it

is clear that the normalized current increases for various values of the dimensionless parameters 2 .

6. Conclusions

The non-linear reaction diffusion equationsin an amperometric biosensor was solved analytically.

The approximate analytical expressions for the steady state concentrations of oxidised mediator,

substrate and reduced mediator for all values of parameters 2 , OB and SB at the enzyme-membrane

electrode geometry are obtained using the new Homotopy perturbation method. A satisfactory agreement

with the numerical simulation result is noted. These analytical expressions can be used to analyze the

effect of different parameters such as membrane thickness, type of buffer in the external solution and

enzyme loading in the membrane. This theoretical result is also useful for the optimize the sensitivity of

the bio-sensor.

Acknowledgements

This work was supported by the Council of Scientific and Industrial Research (CSIR No.:

01(2442)/10/EMR-II), Government of India. The authors are also thanks to the Secretary Shri. S.

Natanagopal, Madura College Board, Madurai, and Dr. R. Murali, The Principal, The Madura College,

Madurai, Tamilnadu, India for their constant encouragement.

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Appendix A

Basic concept of the Homotopy perturbation method[20-35]

To explain this method, let us consider the following function:

r ,0)()( rfuDo (A.1)

with the boundary conditions of

r ,0) ,(

n

uuBo (A.2)

where oD is a general differential operator, oB is a boundary operator, )r(f is a known analytical

function and is the boundary of the domain . In general, the operator oD can be divided into a

linear part L and a non-linear part N . Equation (A.1) can therefore be written as

0)()()( rfuNuL (A.3)

By the Homotopy technique, we construct a Homotopy ]1,0[:),( prv that satisfies

0)]()([)]()()[1(),( 0 rfvDpuLvLppvH o (A.4)

0)]()([)()()(),( 00 rfvNpupLuLvLpvH (A.5)

where p[0, 1] is an embedding parameter, and 0u is an initial approximation of the eqn.(A.1) that

satisfies the boundary conditions. From the eqns. (A.4) and (A.5), we have

0)()()0,( 0 uLvLvH (A.6)

0)()()1,( rfvDvH o (A.7)

When p=0, the eqns. (A.4) and (A.5) become linear equations. When p =1, they become non-linear

equations. The process of changing p from zero to unity is that of 0)()( 0 uLvL to 0)()( rfvDo .

We first use the embedding parameter p as a small parameter and assume that the solutions of the eqns.

(A.4) and (A.5) can be written as a power series in p :

...22

10 vppvvv (A.8)



216

Setting 1p results in the approximate solution of eqn. (A.1):

...lim 2101

vvvvup

(A.9)

This is the basic idea of the HPM.

Appendix B

Analytical solution the normalized concentration of the substrate (eqn.(12)) using New Homotopy

perturbation method

In this Appendix, we indicate how the eqn. (22) in this paper is derived. To find the solution of eqns.(11)

- (13) we construct the new Homotopy as follows [24-25]:

0

)1()1()1()1(

)1()1(

S0S0SS00

S0S02S2

S2

0000

002

2

2

FFBBFBFB

FFBB

dx

Fdp

FFBBFBFB

FFBB

dx

Fdp

SSSS

SSSS

(B.1)

0)1(S0S0

S0S02S2

S2

00

02

2

2

BBBB

FFBB

dx

Fdp

BBBB

FBB

dx

Fdp

SS

SSSS

(B.2)

The analytical solution of the eqn.(B.2) is

..........2

2

10 SSSS FppFFF (B.3)

Similarly the analytical solutions of eqns. (11) and (13) be

..........20

2

1000 0 FppFFF

(B.4)

..........2

2

10 RRRR FppFFF

(B.5)

Substituting (B.3) -(B.5) in (B.2) we get

0

....)...)((

...)(....)(

.....)....)((

.....)(

....)(...)()1(

202

1002

2

1S0

2

2

1S202

1000

2

2

1202

100S02S

2

2

2

1

2

00

2

2

102

2

2

2

1

2

00

00

00

0

00

FppFFFppFFBB

FppFFBFppFFB

FppFFFppFFBB

dx

FppFFd

p

BBBB

FppFFBB

dx

FppFFdp

SSS

SSS

SSS

SSS

SS

SSSSSSSS

(B.6)

Comparing the coefficients of like powers of p in the eqn.(B.6) we get

0:

S0S0

0SS02S2

0S2

0

BBBB

FBB

dx

Fdp (B.7)



217

The initial approximations is as follows

0)1(,1)0( ' ii SS FF (B.8)

0)1(,1)0( ' ii SS FF , ......3,2,1i (B.9)

Solving the eqns.(B.7) and using the boundary conditions (B.8)-(B.9), we obtain the following results:

)cosh(

)cosh()(

0SA

AxFxF S (B.10)

where A is defined in the text eqn. (26).

After putting the eqn. (B.10) into an eqn. (B.3), we obtain the solution in the text eqn.(22). Substituting

the eqns. (22) into an eqns. (19) and (20), we obtain the solutions in the text eqns. (23) and (24)

Appendix C

Scilab/Matlabprogram for the numerical solution of the systems of non-linear eqns. (11)-(13) and

(15)-(16).

function pdex 4

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);

figure

%plot(x,u1(end,:))

xlabel('Distance x')

ylabel('u1(x,2)')

figure

plot(x,u2(end,:))


ylabel('u2(x,2)')

figure

%plot(x,u3(end,:))


ylabel('u3(x,2)')

% --------------------------------------------------------------



218

function [c,f,s] = pdex4pde(x,t,u,DuDx)

Bs=.0052;

B0=01;

A=sqrt(225);

us=0.05;

up=1;

c = [1;1; 1];

f = [1; 1; 1] .* DuDx;

F1 =-A^2*((B0*Bs*u(1)*u(2)/(B0*u(1)+Bs*u(2)+B0*Bs*u(1)*u(2))));

F2 =-us*A^2*((B0*Bs*u(1)*u(2)/(B0*u(1)+Bs*u(2)+B0*Bs*u(1)*u(2))));

F3 =up*A^2*((B0*Bs*u(1)*u(2)/(B0*u(1)+Bs*u(2)+B0*Bs*u(1)*u(2))));

s = [F1; F2; F3];

% --------------------------------------------------------------

function u0 = pdex4ic(x);

u0 = [1; 0; 1];

% --------------------------------------------------------------

function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)

pl = [0; 0; 0];

ql = [1; 1; 1];

pr = [ur(1)-1; ur(2)-1; ur(3)];

qr = [0; 0; 0];

Appendix D

Nomenclature

Symbol Meaning

]E[ T Total enzyme concentration (mM )

]E[ OX Enzyme concentration of the oxygen (mM )

ES][ Enzyme concentration of the substrate (mM )

][E red Reduced enzyme concentration (mM )

]Med[ OX Concentration of oxidized mediator at any position in the enzyme layer (mM)

]Med[ red Concentration of reduced mediator at any position in the enzyme layer (mole cm 3 )

MD Diffusion coefficient of oxidized mediator (cm2s-1 )

SD Diffusion coefficient of substrate (cm2s-1 )

d Thickness of the planar matrix (cm)



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bOX ]Med[ Oxidized mediator concentration at the enzyme layer electrode boundary (mM )

]Med[ OX Oxidized mediator concentration in bulk solution (mM)

]S[ Concentration of substrate at any position in the enzyme layer (mM)

b]S[ Substrate concentration at the enzyme layer| electrode boundary (mM )

][S Substrate concentration in bulk solution (mM)

1k , 3k , 4k Rate constants ( M-1s-1 )

1k , 2k Rate constants ( s-1 )

SB

0B Normalized surface concentration of the oxidized mediator

Greek Symbols

2 Thiele modulus for the oxidized mediator (none)

S Dimensionless parameter

Subscripts

OX Oxidized

T Total

red Reduced

o Oxygen

S Substrate

P Product

R Reduced

b boundry

Bulk

Approximate Analytical Expressions of Non-linear Boundary Value Problem in an Amperometric Biosensor...

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