A Simple Analytical Expression of a Non-Linear Boundary Value Problem for an Immobilized Oxidase Enz

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32

A Simple Analytical Expression of a Non-Linear Boundary Value Problem for an Immobilized Oxidase Enzyme Electrode Using the New Homotopy Perturbation Method *V.Ananthaswamy1, C. Chowmiya2, M. Subha3

1Department of Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India 2M.Phil.,Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India 3 Department of Mathematics, MSNPM Women’s College, Poovanthi, Sivagangai Dt., Tamil Nadu, India

1 [email protected]*; [email protected]; 3subhaamsc@rediffmailcom

Abstract

In this article a mathematical model of an immobilized oxidase enzyme electrode is presented. The model is based on the three reaction-diffusion equations containing a non-linear reaction term under the steady state conditions. A simple analytical expressions pertaining to concentrations of the immobilization of three enzyme substrates are obtained by using the new Homotopy perturbation method (HPM). A simple analytical expression of the concentrations of substrate, oxygen and oxidized mediator and current was obtained in terms of the thiele moduli and the small values of the normalized surface concentration of substrate sB , thenormalized surface concentration of oxygen oB and the normalized surface concentration of oxidized mediator mB . These analytical solutions are compared with the numerical simulation. A good agreement between analytical expressions and numerical results is noted.

Keywords

Enzyme Electrodes; Non-linear Boundary Value Problem; Reaction-diffusion Equations; Biosensor; New Homotopy Perturbation Method; Numerical Simulation

Introduction

The basic concepts of enzyme electrodeswereintrodu-ced by Clark and Lyons [Clark et. al (1962), Lyons et.al (1962)]. Many kinds of biosensors based on electrochemical enzyme electrodes have been studied and a majority of this device operates in an amperometric mode [Scheller et.al (1992), schulbert et. al (1992)]. Biosensors are usually classified into various groups either by type of transducer employed (electrochemical, optical, piezoelectric, and thermal) or by the kind of bio-recognition element utilized (antibody, enzymes, nucleic acids, and whole cells). Both components of the biosensor, namely, the bio

recognition element (referred as a receptor) and transduction platform (referred as a transducer) play an important role in the construction of a sensitive and specific device for the analyte of interest.

Schulmeister has described models for multilayerand multienzyme electrodes; these models assumed operationof the electrode under diffusion control, such that the enzymekinetics are linear with substrate [Schulmeister et. al (1990) and Pfeiffer et. al (1993)]. The kinetics is described by a parabolic differential equation with linear inhomogeneities. Thefact that relatively few enzyme electrodes have been commercialized may be due to the technical problemsassociated with either the enzyme reaction or the underlying product/substrate-sensitiveelectrode, such as the availability of appropriate enzymes, their successful immobilization, andprobably most importantly the achievement of an acceptablecatalytic lifetime of the enzyme.

Leypoldt and Gough have described a model for a two substrate enzyme electrode with the non-linear enzyme reaction[Leypoldt et. al (1984) , Gough et.al (1984)].This model to describe the behaviorof a glucose oxidase electrode. Bergel and Comtat describe the transient response of a mediated amperometric enzyme electrode by using an implicit finite difference method [Bergal et. al (1984), Comtat et. al (1984)]. A two-substrate enzyme electrode was described by Gooding and Hall [Gooding et. al (1996) , Hall et. al (2012)].The development of models for enzyme electrodesprovides a better understanding of the individual processesinfluencing the response of the device, and this informationmay be used as a guide for directions for improvement of thesensor design.

mailto:[email protected]

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33

The importance of enzyme electrode and the various approaches to enzyme electrochemistry has been discussed. The current response has been also observed experimentally for mediated enzyme electrodes employing glucose oxidase [Martens et. al (1995), Hindle et. al (1995), Hall et. al (1995) and Pallachi et. al (1990), Turner et.al (1990)].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 [Trevan et. al (1981)]. Homogeneous membranes are used as carries for immobilization of enzymes [Simon et. al (2002), Hailweil et. al (2002), Cass et. al (2002)] and also used in biomaterials, bio-separators and biosensors [Liu et.al (1996) ,Zhang et. al (1996) , Yu et.al , Deng et. al (1996) ].

In this paper, we describe a model for thisthree-substrate enzyme electrode and employ the model toinvestigate the influence of the oxygen on the current responseof a mediated oxidase electrode. To our knowledge, no general analytical expressions that describe concentration of substrate, oxygen and oxidized mediator for various of the thiele modulus 2

sϕ , 2oϕ and 2

mϕ and the normalized parameters ,sB oB and

mB have been reported. However, in general, analytical solutions of non-linear differential equations are more important manipulation and analysis. For this reason, we have derived that analytical expressions corresponding to the concentrations of substrate, oxygen and oxidized mediator in an oxidase enzyme electrode using new Homotopy perturbation method. This method is reliable and highly accurate in handling non-linear problems.

Mathematical Formulation of the Boundary Value Problem

The details of the model adopted have been fully described in Martens and Hall [Martens et. al (1994), Hall et.al (1994)]. Figure.1 represents the general kinetic reaction scheme of enzyme-membrane electrode geometry [Gooding et. al (1996), Hall et.al (1996)].The general reaction scheme for an immobilized oxidase inthe presence of two oxidants can be written as follows:

31

2

kkOX red

k

E S ES E P+ → → +

← (1)

42 2 2

kred OXE O E H O+ → + (2)

5kred OX OX redE Med E Med+ → + (3)

We assume that the concentrations of all reactants and enzyme intermediates remain constant for all time. Also the concentration of total active enzyme [ ]tE and the reactants in the bulk electrode remain constant. We can consider that the diffusion of the reactants can be described by Fick’s second law and the enzymes are assumed to be uniformly dispersed throughout the matrix. The enzyme activity is not a function of position. The reaction/diffusion equations correspond-ing to the concentrations of substrate, oxygen and oxidized mediator within a matrixcan be expressed as[14]

[ ] [ ] [ ][ ]

1122

32 1 OX ss t

o m

Medd S OD k ESdyβ

β β

−− = + + +

(4)

[ ] [ ] [ ][ ]

[ ] [ ] [ ][ ]

22

2

112

3

/ //

1

OX o OX mm

OX m

OX st

o m

d Med O MedD

Meddy

MedOk E

S

β ββ

ββ β

−−

+

= + + +

(5)

[ ] [ ] [ ][ ]

[ ] [ ] [ ][ ]

222

22

112

3

/ //

1

o OX mo

o

OX st

o m

O Medd OD

Ody

MedOk E

S

β ββ

ββ β

−−

+

= + + +

(6)

where [ ]S , [ ]OXMed and [ ]2O are the concentrations of substrate, oxidized mediator and oxygen and y is distance from the electrode. sD , mD and oD are the diffusion coefficients of substrate, oxidized mediator and oxygen. [ ]tE Denotes the concentration of the total active enzyme in the matrix and ( )2 3 1/s k k kβ = + ,

3 4/o k kβ = & 3 5/m k kβ = represent reaction constants of substrate, oxygen and oxidized mediator respectively. The boundary conditions are given by

When y d= ,

[ ] [ ] [ ]2 2 2obO O K O∞

= = , [ ] [ ] [ ]sb bS S K S= = ,

[ ] [ ] [ ]OX OX m OXbMed Med K Med∞

= = (7)

when 0,y =

[ ] [ ]OX OX bMed Med= [ ] [ ]2 / / 0d O dy d S dy= = (8)

Here [ ]bS , [ ]2 bO [ ]OX bMed and [ ]S∞ , [ ]2O∞

, [ ]OXMed∞

are the bulk concentrations of substrate, oxygen and oxidized mediator species. oK , sK and mK are the


34

partition coefficients for oxygen, substrate and oxidized mediator respectively. Equation (4) can be represented in the normalized form using the following normalized parameters.

[ ][ ]s

b

SF

S= ; [ ]

[ ]2

2o

b

OF

O= ; [ ]

[ ]OX

mOX b

MedF

Med= ; yx

d= ; (9)

[ ]bs

s

SB

β=

[ ]2 bo

o

OB

β= ;

[ ]OX bm

m

MedB

β= ; (10)

[ ][ ]

232 t

ss b

d k ED S

ϕ = ; [ ][ ]

232

2

to

o b

d k ED O

ϕ = ; [ ][ ]2

32 tm

m OX b

d k ED Med

ϕ = (11)

where sF , oF and mF represent the normalized concentrations of substrate, oxygen and oxidized mediator and sB , oB and mB are the corresponding normalized surface concentrations. The surface concentration is the ratio of the bulk concentration and the reaction constants. 2

sϕ , 2oϕ and 2

mϕ denote the thiele moduli of substrate, oxygen and oxidized mediator, respectively. Thiele modulus 2ϕ represents the ratio of the characteristic time of the enzymatic reaction to that of substrate diffusion and dis the thickness of the enzyme layer. The systems of three non-linear reaction/diffusion equations in normalized form are

22

21

1 11

ss

o o m m s s

d Fdx

F B F B F B

ϕ

= + + +

(12)

22

21

1 11

o o oo

o o m m

o o m m s s

d F F BF B F Bdx

F B F B F B

φ

= + + + +

(13)

22

21

1 11

m m mm

o o m m

o o m m s s

d F F BF B F Bdx

F B F B F B

φ

= + + + +

(14)

The boundary conditions become

1 1s o mF F F at x= = = = (15)

1mF = , 0 0o sdF dFat x

dx dx= = =

(16)

The normalized current OXJ is given by,

0

mOX

x

dFJ

dx =

= −

(17)

Analytical Expression of the Normalized Surface Concentrations Using the 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 Homotopy perturbation method (HPM) [Gohri et. al (2007), Ozis et. al (2007), Li et. al (2006), Mousa et. al (2008), He (1999 and 2003), Ariel (2010), Ananthaswamy et. al (2012 and 2013)] 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. 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 [Ghori et. al (2007), Ozis et. al (2007), Mousa et. al (2008)]. Recently this method is also used to solve some of the non-linear problem in physical sciences [He 1999 and 2003 ]. This method is a combination of Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the Lighthill equation [Mousa et. al (2008)], the Diffusion equation [He 1999] and the Blasius equation [He 2003]. The HPM is unique in its applicability, accuracy and efficiency. The HPM uses the imbedding parameter p as a small parameter, and only a few iterations are needed to search for an asymptotic solution. The simple analytical expressions of the concentrations of substrate, oxygen and oxidized mediator are as follows:

( ) ( )2

21 1 12s

kF x x= − +

(18)

( ) ( )2

22 1 12o

kF x x= − +

(19)


35

( ) ( )2

23 12m

kF x x x= − +

(20)

where,

2 21

( )( ) ( )

o m ss

o m s s o m

B B Bk

B B B B B Bϕ

+= + + + +

(21)

2 22 ( ) ( )

o so

o m s s o m

B Bk

B B B B B Bϕ

= + + + +

(22)

2 23 ( ) ( )

m sm

o m s s o m

B Bk

B B B B B Bϕ

= + + + +

(23)

The corresponding normalized current using eqn. (20): 23

2OXk

J = (24)

where 23k is defined by the eqn. (23).

Numerical Simulation

The non-linear reaction-diffusion equations eqns. (12)-(14) 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 simple analytical equations in Figs. (2) - (6). A satisfactory agreement is noted for various values of the thiele moduli 2

sϕ , 2oϕ and 2

mφ and possible small values of the dimensionless parameters sB , oB and mB .

Results and Discussions

Figure.1 is the schematic model of an enzyme-membrane electrode. Figure2 is the normalized concentration of (a)substrate sF , normalized concentration of (b)oxygen oF and normalized concentration of (c) mediator mF versus the normalized distance x. From Fig.2 (a), (b) and (c) it is clear that when the thiele modulus 2 2 2, ands o mϕ ϕ ϕ increases, the corresponding normalized concentrations of substrate ,sF oxygen oF , mediator

mF decreases for some fixed values of the normalized surface concentrations of the substrate sB , normalized surface concentration of oxygen oB and normalized surface concentration of the mediator mB respectively.

Figure3 is the normalized concentrations of (a)substrate sF , (b)oxygen oF and (c) mediator mF versus the normalized distance x. From Fig. 3 ((a)-(c)) we infer that when the normalized parameter sB

increases, the corresponding normalized concentration of substrate ,sF oxygen oF and mediator mF decreases

for some fixed values of oB , mB when 2 1,sφ = 2 1oϕ =

and 2 1mϕ = .

Figure4 is the normalized concentrations of (a)substrate sF , (b)oxygen oF and (c) mediator mF versus the normalized distance x. From Fig. 4 (a)-(b) it is clear that when the normalized parameter oB increases, the corresponding normalized concentrations of substrate ,sF oxygen oF decreases for some fixed values of sB , mB when 2 1,sφ = 2 1oϕ = . Figure. 4 (c) it is clear that when the normalized parameter sB increases, the corresponding normalized concentrations of mediator mF increases for some fixed values of sB ,

mB and 2 1mϕ = .

Figure.5 is the normalized concentrations of (a)substrate sF , (b)oxygen oF and (c) mediator mF versus the normalized distance x. From Figs. 5 (a) and (c) it is clear that when the normalized parameter mB increases, the corresponding normalized concentrations of substrate sF and mediator mF decreases for some fixed values of sB , oB when 2 1sφ = and 2 1.mϕ = Figure. 5 (b) it is clear that when the normalized parameter mB increases, the corresponding normalized concentrations of oxygen oF increases for some fixed values of sB , oB and 2 1oϕ = .

Figure.6 is the normalized concentrations of the substrate s ,F oxygen oF and the mediator mF versus the dimensionless distance x . From this figure, we note that the substrate, oxygen and mediator increasesfor some fixed values of the parameters , ,s o mB B B and

2 2 2 1s o mϕ ϕ ϕ= = = Figure.7 is the variation of normalized current OXJ with (a) normalized thiele modulus 2

mϕ for the mediator (b) normalized surface concentration of the substrate sB (c) normalized surface concentration of oxygen oB and (d) normalized surface concentration of mediator mB . From these Figs. 7 (a), (c) and (d) the normalized current decreases and (b) increases for some fixed values.

FIGURE 1: GEOMETRY OF THE PROBLEM


36

(a)

(b)

(c)

FIGURE 2: NORMALIZED CONCENTRATIONS OF (A)

SUBSTRATE sF (EQN. (18)) (B) OXYGEN oF (EQN.(19)) AND

(C)MEDIATOR mF (EQN. (20)) COMPUTED FOR SOME FIXED

VALUES OF THE PARAMETERS 0.5Bs = , 0.05oB = , 0.1rB =

AND VARIOUS VALUES OF THIELE MODULUS 2sφ , 2

oϕ AND 2mϕ .

(a)

(b)

(c)




VALUES OF THE PARAMETERS 0.05oB = , 0.1rB = AND 2 1sφ = AND VARIOUS VALUES OF NORMALIZED SURFACE

CONCENTRATION OF SUBSTRATE SB , WHEN (I) S 0.001B = (II)

S 0.005B = (III) S 0.01B = (IV) S 0.05B = (V) S 0.1B = AND

(V) 0.5Bs =


37

(a)

(b)

(c)




VALUES OF THE PARAMETERS 0.5Bs = , AND 2 1oϕ = AND

VARIOUS VALUES OF NORMALIZED SURFACE CONCENTRATION OF OXYGEN WHEN, (I) 0.001oB = (II)

0.05oB = (III) 0.1oB = (IV) 0.5oB = (V) 1oB = AND (VI) 2oB = .

(a)

(b)

(c)




VALUES OF THE PARAMETERS 0.5Bs = , 0.05oB = AND 2 1mϕ =

AND VARIOUS VALUES OF NORMALIZED SURFACE CONCENTRATION OF MEDIATOR(II) 0.005mB = (III) 0.01mB =

(IV) 0.05mB = (V) 0.1mB = (VI) 0.2mB = AND (VI) 0.5mB = .


38

FIGURE6: NORMALIZED CONCENTRATIONS OF SUBSTRATE

sF (EQN. (18)) VERSUS THE NORMALIZED DISTANCE x ,

OXYGEN oF (EQN. (19))VERSUS THE NORMALIZED DISTANCE

x WHEN 0.5Bs = , 0.05oB = , 0.1mB = AND 2 1oϕ = AND

MEDIATOR mF (EQN.(20)) VERSUS THE NORMALIZED

DISTANCE x WHEN 0.5Bs = , 0.05oB = , 0.1mB = AND 2 1mϕ = .

(a)

(b)

(c)

(d)

FIGURE7: VARIATION OF NORMALIZED CURRENT OXJ WITH (A) NORMALIZED THIELE MODULUS FOR THE MEDIATOR (B)

NORMALIZED SURFACE CONCENTRATION OF THE SUBSTRATE sB (C) NORMALIZED SURFACE CONCENTRATION

OF OXYGEN oB AND (D) NORMALIZED SURFACE

CONCENTRATION OF MEDIATOR mB .

Conclusions

The system of non-linear reaction diffusion equations of the model has been solved analytically. The approximate analytical expressions for the concentrations of substrate, oxygen, mediator and the current at the enzyme-membrane electrode geometry are obtained using the new Homotopy perturbation method. This analytical result is useful for improvingthe sensor design. The extension of the procedure to the model with reduced mediator in the bulk solution seems possible. These analytical expressions can be used for the optimization of the


39

thickness of the enzyme layer or Thiele modulus which produces significant change in both the magnitude of the current and the general behavior of the system.

ACKNOWLEDGEMENT

The authors are thankful to the Secretary, the Principal and the Head of the Department of Mathematics, The Madura College, Madurai, India for their constant encouragement.

Appendix A: Basic Concept of the Homotopy Perturbation Method

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

( ) ( ) 0, r oD u f r− = ∈Ω (A.1)

with the boundary conditions of

( , ) 0, rouB un∂

= ∈Γ∂

(A.2)

where oD is a general differential operator, oB is a boundary operator, ( )f r is a known analytical function and Γ is the boundary of the domain Ω . In general, the operator oD can be divided into a linear part L and a non-linear part N . The eqn.(A.1) can therefore be written as

( ) ( ) ( ) 0 L u N u f r+ − = (A.3)

By the Homotopy technique, we construct a Homotopy ( , ) : [0,1]v r p Ω× →ℜ that satisfies

0( , ) (1 )[ ( ) ( )] [ ( ) ( )] 0oH v p p L v L u p D v f r= − − + − = (A.4)

0 0( , ) ( ) ( ) ( ) [ ( ) ( )] 0H v p L v L u pL u p N v f r= − + + − = (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 H v L v L u= − = (A.6)

( ,1) ( ) ( ) 0oH v D v f r= − = (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( ) ( ) 0L v L u− = to ( ) ( ) 0oD v f r− = . 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 :

20 1 2 ... v v pv p v= + + + (A.8)

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

0 1 21lim ...p

u v v v v→

= = + + + (A.9)

This is the basic idea of the HPM.

Appendix B: Analytical Solution of the System of Non-linear Equations (12)-(16) Using New Homotopy Perturbation Method [16-27]

In this Appendix, we indicate how the eqns. (18)-(20) in this paper is derived. To find the solution of the eqns.(12) - (14), we construct the new Homotopy as follows:

( )( )( )

( )( )( )

22

2

22

2

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

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

(1) (1)

0

s o o m m s ss

o o m m s s s s

o o m m

o o m m s sss

o o m m s s s s

o o m m

F B F B F Bd Fp

F B F B F B F BdxF B F B

F B F B F Bd Fp


ϕ

ϕ

+ − − + +

+ +

+ + − = + +

+ +

(B.1)

( )( )( )

2 2

2

2

2

2

( )(1 )

( ) ( )

0

s s o m s

o m s s o m

s

o o m m s ss

o o m m s s s s

o o m m

d F B B Bp

B B B B B Bdx

d Fdx

pF B F B F B

F B F B F B F B

F B F B

ϕ

ϕ

+− − + + + +

+ = + − + + + +

(B.2)

( )( )

( )( )

2

2

2

22

2

(1 )(1) (1)

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

(1) (1)

0

o

o o o s s

o o m m s s s s

o o m m

o o o s so

o o m m s s s s

o o m m

d Fdx

pF B F B

F B F B F B F B

F B F B

d F F B F Bp


ϕ

ϕ

− − + + + +

+ − = + + + +

(B.3)


40

( )( )

2 2

2

2

2

2

(1 )( ) ( )

0

o o o s

o m s s o m

o

o o s so

o o m m s s s s

o o m m

d F B Bp

B B B B B Bdx

d Fdx

pF B F B

F B F B F B F B

F B F B

ϕ

ϕ

− − + + + +

+ = − + + + +

(B.4)

( )( )

( )( )

2

2

2

2

2

2

(1 )(1) (1)

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

(1) (1)

0

m

m m m s s

o o m m s s s s

o o m m

m

m m s sm

o o m m s s s s

o o m m

d Fdx

pF B F B

F B F B F B F B

F B F B

d Fdx

pF B F B

F B F B F B F B

F B F B

ϕ

ϕ

− − + + + + + = − + + + +

(B.5)

( )( )

2 2

2

2

2

2

(1 )( ) ( )

0

m m m s

o m s s o m

m

m m s sm

o o m m s s s s

o o m m

d F B Bp

B B B B B Bdx

d Fdx

pF B F B

F B F B F B F B

F B F B

ϕ

ϕ

− − + + + +

+ = − + + + +

(B.6)

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

0 1 2

2 ..........s s s sF F pF p F= + + + (B.7)

Similarly the analytical solutions of the eqns. (B.4) and (B.6) are

10

22 ..........

oo o oF F pF p F= + + + (B.8)

0

21 2 ..........m m m mF F pF p F= + + + (B.9)

Substituting the eqn.(B.7)-(B.9)into an eqn.(B.2) we get

0 1 2

2 2

2

2

( ...)

(1 )( )

( ) ( )

s s s

s o m s

o m s s o m

d F pF p F

dxpB B B

B B B B B Bϕ

+ + + − + −

+ + + +

0 1 2

1 2 3

0 1 2

0 1 2

1 2 3

0 1 2

0 1 2

0 1 2

2 2

2

2

2

22

2

2

2

2

( ...)

( ...)

( ...)

( ...)

( ...)

( ...)

( ...)

( ..

s s s

o o o o

m m m m

s s s ss

o o o o

m m m m

s s s s

s s s

d F pF p F

dx

F pF p F B

F pF p F Bp

F pF p F B

F pF p F B

F pF p F B

F pF p F B

F pF p F

ϕ

+ + +

+ + + + + + +

++ + +

− + + + + + + +

+ + +

+ + + +

1 2 3

0 1 2

2

2

0

.)

( ...)

( ...)

s

o o o o

m m m m

B

F pF p F B

F pF p F B

= + + + + + + + +

(B.10)

Substituting the eqns. (B.7)-(B.9) into an eqn.(B.4) we get

0 1 2

0 1 2

0 1 2

0 1 2

1 2 3

0 1 2

2 2

2

2

2 2

2

2

22

2

2

( ...)

(1 )

( ) ( )

( ...)

( ...)

( ...)

( ...)

( ...)

o o o

o o s

o m s s o m

o o o

o o o o

s s s so

o o o o

m m m m

d F pF p F

dxpB B

B B B B B B

d F pF p F

dx

F pF p F Bp

F pF p F B

F pF p F B

F pF p F B

ϕ

ϕ

+ + + − −

+ + + +

+ + +

+ + ++

+ + +−

+ + + + + + +

0 1 2

0 1 2

1 2 3

0 1 2

2

2

2

2

( ...)

( ...)

( ...)

( ...)

s s s s

s s s s

o o o o

m m m m

F pF p F B

F pF p F B

F pF p F B

F pF p F B

+ + + + + + + + + + + + + + +

0=

(B.11)

Substituting the eqns.(B.7)-(B.9) into an eqn.(B.6) we get

0 1 2

2 2

2

2

( ...)

(1 )

( ) ( )

m m m

m m s

o m s s o m

d F pF p F

dxpB B

B B B B B Bϕ

+ + + − −

+ + + +


41

0 1 2

0 1 2

0 1 2

1 2 3

0 1 2

0 1 2

0 1 2

1 2 3

2 2

2

2

22

2

2

2

2

2

( ...)

( ...)

( ...)

( ...)

( ...)

( ...)

( ...)

( ...)

(

m m m

m m m m

s s s sm

o o o o

m m m m

s s s s

s s s s

o o o o

m

d F pF p F

dx

F pF p F Bp

F pF p F B

F pF p F B

F pF p F B

F pF p F B

F pF p F B

F pF p F B

F

ϕ

+ + +

+ + ++

+ + +−

+ + + + + + +

+ + +

+ + + +

+ + ++

+0 1 2

2

0

...)m m mpF p F B

= + + + (B.12)

Comparing the coefficients of like powers of p in (B.10) we get

0

20 2

2( )

: 0( ) ( )

s o m ss

o m s s o m

d F B B Bp

B B B B B Bdxϕ

+− = + + + +

(B.13)

The initial approximations are as follows

0 1(1) 1; ' (0) 0s sF F= = (B.14)

(1) 0; ' (0) 0 , 1,2,3,...i is sF F i= = = (B.15)

Solving the eqn. (B.13) and using the boundary conditions eqns.(B.14)-(B.15), we obtain the following result:

0

221( ) ( 1) 1

2s skF x F x= = − + (B.16)

Where 21k is defined in the text eqn. (21).

After putting the eqn. (B.16) into an eqn. (B.7), we obtain the solution in the text eqn.(18).

Comparing the coefficients of like powers of p intoan eqn.(B.11) we get

0

20

2

2

:

0( ) ( )

o

o so

o m s s o m

d Fp

dxB B

B B B B B Bϕ

− = + + + +

(B.17)

The initial approximations are as follows

0 0(1) 1; ' (0) 0o oF F= = (B.18)

(1) 0; ' (0) 0 , 1,2,3,...i io oF F i= = = (B.19)

Solving the eqn. (B.17) and using the boundary conditions eqns. (B.18)-(B.19), we obtain the following result:

0

222( ) ( 1) 1

2o okF x F x= = − + (B.20)

Where 22k is defined in the text eqn.(22).

After putting the eqn.(B.20) into an eqn.(B.8), we obtain the solution in the text eqn.(19).

Comparing the coefficients of like powers of p into an eqn.(B.12) we get

0

20

2

2

:

0( ) ( )

m

m sm

o m s s o m

d Fp

dxB B

B B B B B Bϕ

− = + + + +

(B.21)

The initial approximations are as follows:

0 0(1) 1; ' (0) 1m mF F= =

(B.22)

(1) 0; ' (0) 0i im mF F= = (B.23)

Solving the eqn.(B.21) and using the boundary conditions eqns.(B.22)-(B.23), we obtain the following result:

0

223( ) ( ) 1

2m mk

F x F x x= = − +

(B.24)

Where 23k is defined in the text eqn. (23).

After putting the eqn.(B.24) into an eqn.(B.9), we obtain the solution in the text eqn.(20).

Appendix C: Scilab/Matlabprogram for the Numerical Solution of the Systems of Non-linear Differential Eqns. (12)-(16)

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); %plot(x,u1(end,:)) %xlabel('Distance x') %ylabel('u1(x,2)') %figure


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%plot(x,u2(end,:)) %xlabel('Distance z') %ylabel('u2(x,2)') %figure %plot(x,u3(end,:)) %xlabel('Distance x') %ylabel('u3(x,2)') % -------------------------------------------------------------- function [c,f,s]=pdex4pde(x,t,u,DuDx) c = [1;1; 1]; f = [1; 1; 1] .* DuDx; Bs=0.5; B0=0.05; Br=0.1; a1=sqrt(0.001); a2=sqrt(0.001); a3=sqrt(0.001); F1 =-a1^2/(1+(1/(u(2)*B0+u(3)*Br))+(1/(u(1)*Bs)));

F2 =-(a2^2*(u(2)*B0)/(u(2)*B0+u(3)*Br))*(1/(1+(1/(u(2)*B0+u(3)*Br))+(1/(u(1)*Bs)))); F3 =-(a3^2*(u(3)*Br)/(u(2)*B0+u(3)*Br))*(1/(1+(1/(u(2)*B0+u(3)*Br))+(1/(u(1)*Bs)))); s = [F1; F2; F3]; % -------------------------------------------------------------- function u0 = pdex4ic(x) u0 = [1; 1;1]; % -------------------------------------------------------------- function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [0; 0; ul(3)-1]; ql = [1; 1; 0]; pr = [ur(1)-1; ur(2)-1; ur(3)-1]; qr = [0; 0; 0];

Greek Symbols 2sϕ Thiele modulus for the substrate (Normalized) 2oϕ Thiele modulus for the oxygen (Normalized) 2mϕ Thiele modulus for the mediator (Normalized)

Subscripts

o Oxygen s Substrate m Mediator

OX Oxidized species red Reduced species t Total ∞ Bulk solution

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Appendix D: Nomenclature

Symbol Meaning

[ ]tE Total active enzyme concentration in the matrix (mmol/L)

[ ]OXE Enzyme concentration of the oxidized mediator (mmol/L)

[ ]ES Enzyme concentration of the substrate (mmol/L)

[ ]redE Reduced enzyme concentration (mmol/L)

[ ]2O Concentration of oxygen at any position in the enzyme layer (mmol/L)

[ ]2 bO Oxygen concentration in the bulk electrolyte (mmol/L)

[ ]2O∞

Oxygen concentration in the bulk solution (mmol/L)

[ ]S Concentration of substrate at any position in the enzyme layer (mmol/L)

[ ]bS Substrate concentration in the bulk electrolyte

(mmol/L)

[ ]OXMed Concentration of oxidised mediator at any position in the enzyme layer(mmol/L)

[ ]OX bMed Oxidised mediator concentration in the bulk

electrolyte (mmol/L)

[ ]OXMed∞

Oxidized mediator concentration in the bulk solution (mmol/L)

oD Diffusion coefficient of oxygen (cm2 s-1 )

sD Diffusion coefficient of substrate (cm2 s-1 )

mD Diffusion coefficient of mediator (cm2 s-1) d Enzyme layer thickness (cm) y Distance from the electrode (cm)

1 4 5, ,k k k Rate constants ( L mol-1 s-1 )

2 5,k k Rate constants ( s-1 )

oK Partitioning coefficient for oxygen (none)

sK Partitioning coefficient for substrate (none)

mK Partitioning coefficient for mediator (none)

oB Normalized surface concentration of oxygen

sB Normalized surface concentration of the substrate

mB Normalized surface concentration of mediator

sF Normalized surface concentration

oF Normalized oxygen concentration

mF Normalized mediator concentration


43

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Dr. V. Ananthaswamy received his M.Sc. Mathematics degree from The Madura College (Autonomous), Madurai-625011, Tamil Nadu, India during the year 2000. He has received his M.Phil degree in Mathematics from Madurai Kamaraj University, Madurai, Tamil Nadu, India during the year 2002. He has

received his Ph.D., degree (Under the guidance of Dr. L.


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Rajendran, Assistant Professor, Department of Mathematics, The Madura College, Tamil Nadu, India) from Madurai Kamaraj University, Madurai, Tamil Nadu, India, during the year October 2013. He has 14 years of teaching experiences for Engineering Colleges, Arts & Science Colleges and Deemed University. He has 3 years of research experiences. At present he is working as Assistant Professor in Mathematics, The Madura College (Autonomous), Madurai-625 011, Tamil Nadu, India from 2008 onwards. He has published more than 23 research articles in peer-reviewed National and International Journals and communicated 7 research articles in National and International Journals. Presently he has Reviewer/Editorial Board Member/Advisory Board Member in 43 reputed National and International Journals. Currently he has doing one ongoing minor research project sanctioned at UGC. His present research interest includes: Mathematical modeling based on differential equations and asymptotic approximations, Analysis of system of non-linear reaction diffusion equations in physical, chemical and biological sciences, Numerical Analysis, Mathematical Biology, Mathematical and Computational Modeling, Mathematical Modeling for Ecological systems. Also, he has participated and presented research papers in National and International Conferences.

Mrs. M. Subha received her M.Sc.,degree (2012) and M.Phil., degree (2013) in Mathematics from The Madura College and E.M.G. Yadhava Women’s College, Madurai, Tamilnadu, India. At present, She is working as Assistant Professor in the Department of Mathematics, Madurai

Sivakasi Nadars Pioneer Meenakshi Women’s College, Sivagangai District, Tamil Nadu, India. Also, she is doing her Ph.D entitled “Asymptotic Methods for Solving Initial and Boundary Value Problems” at Madurai Kamaraj University, Madurai under the guidance of Dr. L. Rajendran, Assistant Professor, Department of Mathematics, The Madura College, Madurai. Her present research interest include: Mathematical modelling, Analytical solution of system of nonlinear reaction diffusion processes in biosensor, Homotopy analysis method, Homotopy perturbation and numerical methods. She has published 4 papers in International Journals and communicated one research paper in National Journal. Also, She has participated and presented research papers in International and National Conferences.

Ms. C. Chowmiya received her M.Sc., degree in Mathematics from Sri Meenakshi Government Arts and Science College for Women (Autonomous), Madurai, Tamil Nadu, India during the year 2013. She has completed her M.Phil., (Mathematics) from The Madura College (Autonomous), Madurai, Tamil Nadu,

India during the year2014. At present, She is working as Assistant Professor in the Department of Mathematics, Syed Ammal Engineering College, Ramanathapuram District, Tamilnadu, India. She has presented a paper at National Conference. She has published one National and International Journals. Her research area includes; Mathematical Modeling for solving Biosensor problems, Solving Non-linear differential equations by Numerical and Asymptotic Methods.

A Simple Analytical Expression of a Non-Linear Boundary Value Problem for an Immobilized Oxidase Enz

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