Hindawi Publishing CorporationISRN Physical ChemistryVolume 2013 Article ID 202781 12 pageshttpdxdoiorg1011552013202781

Research ArticleAn Approximate Analytical Method for the Evaluation ofthe Concentrations and Current for Hybrid Enzyme Biosensor

Indira Krishnaperumal and Rajendran Lakshmanan

Department of Mathematics The Madura College Madurai 625011 Tamil Nadu India

Correspondence should be addressed to Rajendran Lakshmanan raj smsrediffmailcom

Received 22 November 2012 Accepted 13 December 2012

Academic Editors I Anusiewicz H Reis and H Saint-Martin

Copyright copy 2013 I Krishnaperumal and R Lakshmanan This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Mathematical modeling of amperometric biosensor with cyclic reaction is discussed Analytical expressions pertaining to theconcentration of substrate cosubstrate reducing agent andmedial product and current for hybrid enzyme biosensor are obtained interms 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 ofthe problem is also reported using ScilabMatlab program Our analytical results are compared with simulation results A goodagreement between analytical and numerical results is noted

1 Introduction

Biosensor (Figure 1) is a device that uses specific biochemicalreactions mediated by isolated enzymes immunosystemstissues organelles or whole cells to detect chemical com-pounds usually by electrical thermal or optical signals[1] They involve a biological (recognition) element and atransduction element The biological or recognition elementmay be an antibody an enzyme DNA RNA a whole cell ora whole organ or system The transduction element whereinthe biological event or signal is converted to a measurablesignal may include anyone of the following forms chemicalelectrical magnetic mechanical optical or thermal

The biosensor was first described by Clark and Lyonsin 1962 when the term enzyme electrode was adopted [2]The term ldquobiosensorrdquo was introduced by Cammann in 1977[3] Since then research communities from various fieldssuch as physics chemistry and material science have cometogether to develop more sophisticated reliable and maturebiosensing devices for applications in the fields of medicineagriculture biotechnology as well as in the military forbioterrorism detection and prevention [4] Biosensors offerthe prospects of simplified virtually nondestructive analysisof turbid biological fluids Also biosensors for medical care

have demanded the greatest attention for technical develop-ment [5]

Amperometric electrodes have been used in the design ofbiosensors for glucose aminoacids and other molecules [6ndash9] In cases of amperometric enzyme biosensors the potentialat the electrode is held constant while the current flowis measured Amperometric biosensors are quite sensitiveand more suited for mass production than the potentio-metric ones [10 11] Electropolymerized films offer wideimmobilization capabilities and extremely large diversity inthe development of biosensors [12] The development ofamperometric biosensors is an area of growing interest inmany branches of science [13ndash17] Hybrid biosensors arebiosensors withmore than one biosensitivematerial-enzymetissue microorganism [18] or other agents [19] Hybridbiosensors are less selective than enzymatic ones but thechoice of the type of biosensor should be made according toseveral additional factors like availability of biocatalysts theirstability and sensitivity required [20]

Recently Rajendran and coworkers [21ndash23] obtainedthe analytical expressions of substrate and product foramperometric biosensors Eswari and Rajendran [24 25]obtained the steady-state current at microdisk biosensor andmicrocylinder biosensor Recently Rangelova [26] found the

2 ISRN Physical Chemistry

Bioelement

Signal

Analyte

Biosensor

Transducer

Figure 1 Basic scheme of a biosensor [1]

concentration profiles of substrate cosubstrate reducingagent medial product using finite difference technique andMatlab programTo our knowledge no general simple analyt-ical expressions for the concentrations and current for hybridbiosensors have been reported for all values of the Thielemodule and saturation parameters However in generalanalytical solution of nonlinear differential equations is moreinteresting and useful than purely numerical solutions asthey are amenable to various kinds of manipulation and dataanalysis Analytical expressions are usually derived from thebasic physical principles and free from numerical dispersionsand other truncation errors that often occurred in numericalsimulations For this reason in this paper we have derivedan analytical expression for the concentrations and currentfor hybrid enzyme biosensor for all values of the normalizedparameters using homotopy analysis method (HAM)

2 Mathematical Formulation and Analysis ofthe Problem

21 Mathematical Formulation In enzyme-based catecholbiosensor the cyclic reaction scheme for the substrate co-substrate reducing agent and medial product can be repre-sented as follows [26]

119878 + 1198621198701

997888997888rarr 119871 + 1198751

119871 + 1198771198702

997888997888rarr 119878 + 1198752

(1)

where 1198701and 119870

2are reaction rate constants 119875

1is the first

product and1198752is the second product (dehydroascorbic acid)

119878 is the measured substrate (catechol) 119862 is the cosubstrate(oxygen) 119877 is the reducing agent (119871-ascorbic acid) and 119871 isthe medial product (1 2 benzoquinone)

A cyclic reaction between catechol and 12-benzoquinonetakes place by combining the tyrosinase reaction and thechemical reduction of 12 benzoquinone to catechol by L-ascorbic acidThis acid is well known as an effective reducingagent and 12 benzoquinone would be reduced to catecholand may drive the reaction in the opposite direction to thatof enzymatic oxidation of catechol The oxygen consumedin the enzymatic reaction is not compensated for by thechemical reduction Therefore if L-ascorbic acid does notaffect the enzyme activity of tyrosinase a cyclic reactionshould take place and the consumption of the dissolvedoxygen will continue until its concentration becomes zero

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

Nor

mal

ized

con

cen

trat

ion

s

Reducing agent Rh = minus044

Cosubstate C

Substrate S

Medial product L

Normalized current coordinate x

Figure 2 Normalized concentration profiles of reactants119878 119862 119877 and 119871 for various values of the normalized parametersare plotted The key to the graph stacked line represents (12)ndash(15) and dotted line represents the numerical simulation Thefollowing values have been taken for Figure 2 1206012 = 9 120588 =

12 1205881= 092 120572 = 009 120573 = 054 120574 = 1 120582 = 0 119898

1= 02

1198982= 005 119898

3= 0001 120583

1= 08 120583

2= 01 120583

3= 06

[27]Thedifferential equations for this ping-pongmechanismat the steady-state condition are as follows [26]

119863119878

1198892[119878]

1198891205752=

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

minus1198702 [119871] [119877]

1198702119898

119863119871

1198892[119871]

1198891205752=1198703 [119871]

119870119898

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

119863119877

1198892[119877]

1198891205752=1198704 [119877]

119870119898

119863119862

1198892[119862]

1198891205752=

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

(2)

Here [119878] [119862] [119877] and [119871] (mM) denote the concentra-tion of substrate cosubstrate reducing agent and medialproduct respectively 119863

119878 119863119862 119863119877 and 119863

119871(m2s) are diffu-

sion coefficients 1198702 1198703 and 119870

4(mmol(ls)) are the reac-

tion rate constants 120575(120583m) is the distance coordinate119870119878 119870119862 and 119870

119898(mM) are the reaction rate constants and

119881119898(mmol(ls)) is the maximal rate The boundary condi-

tions are given by [26]

[119878] = 119878119874 [119871] = 119871

119874 [119877] = 119877

119874

[119862] = 119862119874

when 120575 = 0

119889 [119878]

119889120575= 0

119889 [119871]

119889120575= 0

119889 [119877]

119889120575= 0

[119862] = 0 when 120575 = 119889

(3)

where 119889 is the active membrane thickness The diffusionlimiting current can be expressed as follows

119868 = 119899FAD119862(119889 [119862]

119889120575)

120575=119889

(4)

ISRN Physical Chemistry 3

00

01

005

015

02

025

03

035

04

045

05

01 02 03 04 05 06 07 08 09 1

m1 = 1m3 = 0001μ3 = 06h = minus076

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μmφ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Normalized current coordinate x

α = 09

α = 07

α = 05

α = 03

α = 009

φ2 = 1μ3 = 1h = minus092

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(b)

Figure 3 Normalized concentration profiles of substrate 119878 for various values of theThiele module 1206012 active membrane thickness 119889 and thenormalized parameters are plotted using (12) The following values have been taken for Figure 3 (a) 120574 = 1 120582 = 01 120572 = 009 120573 = 0541205881= 092 (b) 1206012 = 1 120574 = 1 120582 = 01 120573 = 054119898

1= 1119898

3= 7 120588

1= 092 The key to the graph stacked line represents (12) and dotted line

represents the numerical simulation

where 119899 is the number of electrons taking part in theelectrochemical reaction on the cathode 119865 is the Faradayrsquosconstant and 119860 is the surface of cathode

22 Normalized Form By introducing the following set ofnondimensional variables

119909 =120575

119889 119878 =

[119878]

119870119878

119862 =[119862]

119870119862

119877 =[119877]

119870119898

119871 =[119871]

119870119898

1206012=

1198892119881119898

119863119878119870119898

120588 =119870119898

119870119862

1205881=119870119898

119870119878

120572 =119878119874

119870119878

120573 =119862119874

119870119862

120574 =119877119874

119870119898

120582 =119871119874

119870119898

1198981=1198702

119881119898

1198982=1198703

119881119898

1198983=1198704

119881119898

1205831=119863119878

119863119862

1205832=119863119878

119863119871

1205833=119863119878

119863119877

(5)

the nonlinear reactiondiffusion equation (2) take the follow-ing normalized form

1198892119878

1198891199092= 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877] (6)

1198892119871

1198891199092= 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

] (7)

1198892119877

1198891199092= 119898312058331206012119877 (8)

1198892119862

1198891199092= 12060121205831120588(1 +

1

119878+1

119862)

minus1

(9)

The transformed boundary conditions are

119878 = 120572 119871 = 120582 119877 = 120574 119862 = 120573 when 119909 = 0

119889119878

119889119909= 0

119889119871

119889119909= 0

119889119877

119889119909= 0 119862 = 0 when 119909 = 1

(10)

The dimensionless form of the current is given by

120595 =119868119889

119899FAD119862119870119862

= (119889119862

119889119909)

119909=1

(11)

3 Approximate Analytical Expression ofConcentrations of Four Reactants UsingHomotopy Analysis Method

31 Homotopy Analysis Method The homotopy analysismethod (HAM) [28ndash36] is a general analytic method toget series solutions of various types of nonlinear equationsincluding ordinary differential equations partial differentialequations and coupled nonlinear equations Unlike pertur-bation methods the HAM is independent of smalllargephysical parameters More importantly different from allperturbation and traditional nonperturbation methods theHAM provides us with a simple way to ensure the conver-gence of solution series Besides different from all pertur-bation and previous nonperturbation methods the HAMprovides us with great freedom to choose proper basefunctions to approximate a nonlinear problem [29 34]Liaorsquos book [29] for the homotopy analysis method was firstpublished in 2003 Now more and more researchers havebeen successfully applying this method to various nonlinearproblems in science and engineering In this paper we employHAM to solve the nonlinear differential equations (6) (7)and (9) The basic concept of Homotopy analysis method isgiven in Appendix A

4 ISRN Physical Chemistry

32 Solution of Boundary Value Problem Using the Homo-topy Analysis Method Using HAM method (Appendix B)we obtain the analytical expression corresponding to theconcentrations of substrate co-substrate reducing agent andmedial product as follows

119878 (119909) = 120572 +ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(12)

119871 (119909) = 120582 + ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(13)

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (14)

119862 (119909) = 120573 (1 minus 119909) +ℎ12060121205881205831120572

(1 + 120572)

times [119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(15)

where 119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 = log(120572(120573(1 + 120572) + 120572)) and h is the convergence controlparameter Equation (12)ndash(15) represent the new analyticalexpression of the dimensionless reactant concentrationsUsing (11) and (15) we can obtain the current as follows

120595 = minus120573 minusℎ12060121205881205831120572

(1 + 120572)[1

2+

120572

(1 + 120572) 120573(1 + 119873 +

120572119873

(1 + 120572) 120573)]

(16)

4 Result and Discussion

41 Numerical Simulation In order to investigate the accu-racy of this analytical method with a finite number of termsthe system of differential equations ((6)ndash(9)) also solvedby numerical methods The function pdepe (finite elementmethod) in ScilabMatlab software which is a function ofsolving PDE is used to solve these nonlinear equations [37]The ScilabMatlab program is also given in Appendix CTo validate the results the convergence studies are carried

out The convergence region of auxiliary parameter h isgiven in the Appendix D To show the efficiency of thepresent analytical method our results are compared with thenumerical solution (ScilabMatlab program) in Tables 1 2 3and 4 and Figures 2ndash6 The average relative errors betweenour analytical results and numerical results are 023 011003 and 001 for the concentrations of substrate co-substrate reducing agent and medial product respectively

42 Effect of the Thiele Module The concentration of sub-strate co-substrate reducing agent and medial productdepends upon Thiele module and saturation parametersThe Thiele module 1206012(= 119889

2119881119898119863119878119870119898) essentially compares

the rate of enzyme reaction (119881119898119870119898) and diffusion in the

enzyme layer (1198892119863119878) We observe the rise and downfall of

concentration profiles in two cases (i) If Thiele module issmall (1206012 lt 1) then enzyme kinetics predominates in thebiosensor response The overall kinetics is governed by thetotal amount of active enzyme (ii) The response is underdiffusion control if theThielemodule is large (1206012 gt 1) whichis observed at high catalytic activity and active membranethickness or at low reaction rate constant (119870

119898) or diffusion

coefficient values (119863119878)

The concentration profiles for the four reactants forsome fixed values of parameters are shown in Figure 2 Theanalytical results are compared with the numerical resultgraphically Upon comparison it is evident that both resultsgive satisfactory agreement The concentration of substrate 119878increases and attains its maximum whenThiele module andsaturation parameter 120572 increase (refer to Figure 3) Thereforethe profile deviates more from the linearity The concen-tration profile representing medial product 119871 increases asThiele module and saturation parameter 120582 increase (refer toFigure 4) From Figure 5 it is inferred that the increasingvalue of 119909 coordinate decreases the concentration of reducingagent119877 Figure 6 represents the concentration of co-substrate119862 versus the normalized distance coordinate It reaches theminimum value zero at electrode interface

43 Influence of ActiveMembraneThickness Theactivemem-brane thickness 119889 is one of the important technical param-eters and it has a considerable effect on the concentrationprofiles for the four reactants Also the Thiele module isdirectly proportional to the thickness of the membrane Theconcentration profiles depend significantly on themembranethickness 119889 If the active membrane thickness is large (119889 gt

50 120583m or thick membrane) the concentration profiles ofsubstrate and medial product increase whereas the concen-tration of reducing agent 119877 and co-substrate 119862 decreases Ifthe membrane thickness is small (119889 lt 50 120583m or thinnermembrane) the concentration profiles of substrate medialproduct and reducing agent have uniform values Theseuniform values are equal to the concentration of the abovereactants at 119909 = 0 The concentration profile of co-substrate119862 increases when thickness of the membrane decreases Theincreasing value is not significant Also the concentrationof co-substrate 119862 increases when the saturation parameter 120573increases (refer to Figures 3ndash6)

ISRN Physical Chemistry 5

01

005

015

02

025

03

035

04

045

05

055

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 005μ2 = 3h = minus076λ = 01

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

(a)

0

01

005

015

02

025

03

035

04

045

05

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 5μ2 = 05h = minus05

λ = 05

λ = 04

λ = 03

λ = 02

λ = 01

Normalized current coordinate x

φ2 = 1

(b)

Figure 4 Normalized concentration profiles of medial product 119871 for various values of the Thiele module 1206012 active membrane thickness 119889and the normalized parameters are plotted using (13) The following values have been taken for Figure 4 (a) 120572 = 009 120573 = 054 (b) 1206012 =1 120572 = 009 120573 = 054 The key to the graph stacked line represents (13) and dotted line represents the numerical simulation

075

08

085

09

095

1

105

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 01

μ3 = 06

γ = 1

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 7 d = 13229 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

(a)

04

05

06

07

08

09

1

11

0 02 04 06 08 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 0001

m3 = 03

m3 = 1

m3 = 3

m3 = 5μ3 = 06

γ = 1

φ2 = 1

(b)

Figure 5 Normalized concentration profiles of reducing agent119877 for various values of theThielemodule 1206012 activemembrane thickness 119889 andthe normalized parameters are plotted using (14) The key to the graph stacked line represents (14) and dotted line represents the numericalsimulation

0

06

05

04

03

02

01

minus010 02 04 06 08 1

Normalized current coordinate x

φ2 = 9 d = 150 μmφ2 = 4 d = 100 μm

φ2 = 1 d = 50 μmφ2 = 01 d = 158 μmφ2 = 001 d = 5 μm

μ1 = 08h = minus06

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(a)

0

02

04

06

08

1

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

μ1 = 08

h = minus1

φ2 = 01β = 1

β = 07

β = 054

β = 03

β = 01

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(b)

Figure 6 Normalized concentration profiles of co-substrate 119862 for various values of theThiele module 1206012 active membrane thickness 119889 andthe normalized parameters are plotted using (15) The following values have been taken for Figure 6 (a) 120572 = 009 120573 = 054 120588 = 12 (b)1206012= 01 120572 = 009 120588 = 12 The key to the graph stacked line represents (15) and dotted line represents the numerical simulation

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

2 ISRN Physical Chemistry

Bioelement

Signal

Analyte

Biosensor

Transducer

Figure 1 Basic scheme of a biosensor [1]

concentration profiles of substrate cosubstrate reducingagent medial product using finite difference technique andMatlab programTo our knowledge no general simple analyt-ical expressions for the concentrations and current for hybridbiosensors have been reported for all values of the Thielemodule and saturation parameters However in generalanalytical solution of nonlinear differential equations is moreinteresting and useful than purely numerical solutions asthey are amenable to various kinds of manipulation and dataanalysis Analytical expressions are usually derived from thebasic physical principles and free from numerical dispersionsand other truncation errors that often occurred in numericalsimulations For this reason in this paper we have derivedan analytical expression for the concentrations and currentfor hybrid enzyme biosensor for all values of the normalizedparameters using homotopy analysis method (HAM)

2 Mathematical Formulation and Analysis ofthe Problem

21 Mathematical Formulation In enzyme-based catecholbiosensor the cyclic reaction scheme for the substrate co-substrate reducing agent and medial product can be repre-sented as follows [26]

119878 + 1198621198701

997888997888rarr 119871 + 1198751

119871 + 1198771198702

997888997888rarr 119878 + 1198752

(1)

where 1198701and 119870

2are reaction rate constants 119875

1is the first

product and1198752is the second product (dehydroascorbic acid)

119878 is the measured substrate (catechol) 119862 is the cosubstrate(oxygen) 119877 is the reducing agent (119871-ascorbic acid) and 119871 isthe medial product (1 2 benzoquinone)

A cyclic reaction between catechol and 12-benzoquinonetakes place by combining the tyrosinase reaction and thechemical reduction of 12 benzoquinone to catechol by L-ascorbic acidThis acid is well known as an effective reducingagent and 12 benzoquinone would be reduced to catecholand may drive the reaction in the opposite direction to thatof enzymatic oxidation of catechol The oxygen consumedin the enzymatic reaction is not compensated for by thechemical reduction Therefore if L-ascorbic acid does notaffect the enzyme activity of tyrosinase a cyclic reactionshould take place and the consumption of the dissolvedoxygen will continue until its concentration becomes zero

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

Nor

mal

ized

con

cen

trat

ion

s

Reducing agent Rh = minus044

Cosubstate C

Substrate S

Medial product L

Normalized current coordinate x

Figure 2 Normalized concentration profiles of reactants119878 119862 119877 and 119871 for various values of the normalized parametersare plotted The key to the graph stacked line represents (12)ndash(15) and dotted line represents the numerical simulation Thefollowing values have been taken for Figure 2 1206012 = 9 120588 =

12 1205881= 092 120572 = 009 120573 = 054 120574 = 1 120582 = 0 119898

1= 02

1198982= 005 119898

3= 0001 120583

1= 08 120583

2= 01 120583

3= 06

[27]Thedifferential equations for this ping-pongmechanismat the steady-state condition are as follows [26]

119863119878

1198892[119878]

1198891205752=

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

minus1198702 [119871] [119877]

1198702119898

119863119871

1198892[119871]

1198891205752=1198703 [119871]

119870119898

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

119863119877

1198892[119877]

1198891205752=1198704 [119877]

119870119898

119863119862

1198892[119862]

1198891205752=

119881119898

1 + (119870119878 [119878]) + (119870119862 [119862])

(2)

Here [119878] [119862] [119877] and [119871] (mM) denote the concentra-tion of substrate cosubstrate reducing agent and medialproduct respectively 119863

119878 119863119862 119863119877 and 119863

119871(m2s) are diffu-

sion coefficients 1198702 1198703 and 119870

4(mmol(ls)) are the reac-

tion rate constants 120575(120583m) is the distance coordinate119870119878 119870119862 and 119870

119898(mM) are the reaction rate constants and

119881119898(mmol(ls)) is the maximal rate The boundary condi-

tions are given by [26]

[119878] = 119878119874 [119871] = 119871

119874 [119877] = 119877

119874

[119862] = 119862119874

when 120575 = 0

119889 [119878]

119889120575= 0

119889 [119871]

119889120575= 0

119889 [119877]

119889120575= 0

[119862] = 0 when 120575 = 119889

(3)

where 119889 is the active membrane thickness The diffusionlimiting current can be expressed as follows

119868 = 119899FAD119862(119889 [119862]

119889120575)

120575=119889

(4)

ISRN Physical Chemistry 3

00

01

005

015

02

025

03

035

04

045

05

01 02 03 04 05 06 07 08 09 1

m1 = 1m3 = 0001μ3 = 06h = minus076

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μmφ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Normalized current coordinate x

α = 09

α = 07

α = 05

α = 03

α = 009

φ2 = 1μ3 = 1h = minus092

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(b)

Figure 3 Normalized concentration profiles of substrate 119878 for various values of theThiele module 1206012 active membrane thickness 119889 and thenormalized parameters are plotted using (12) The following values have been taken for Figure 3 (a) 120574 = 1 120582 = 01 120572 = 009 120573 = 0541205881= 092 (b) 1206012 = 1 120574 = 1 120582 = 01 120573 = 054119898

1= 1119898

3= 7 120588

1= 092 The key to the graph stacked line represents (12) and dotted line

represents the numerical simulation

where 119899 is the number of electrons taking part in theelectrochemical reaction on the cathode 119865 is the Faradayrsquosconstant and 119860 is the surface of cathode

22 Normalized Form By introducing the following set ofnondimensional variables

119909 =120575

119889 119878 =

[119878]

119870119878

119862 =[119862]

119870119862

119877 =[119877]

119870119898

119871 =[119871]

119870119898

1206012=

1198892119881119898

119863119878119870119898

120588 =119870119898

119870119862

1205881=119870119898

119870119878

120572 =119878119874

119870119878

120573 =119862119874

119870119862

120574 =119877119874

119870119898

120582 =119871119874

119870119898

1198981=1198702

119881119898

1198982=1198703

119881119898

1198983=1198704

119881119898

1205831=119863119878

119863119862

1205832=119863119878

119863119871

1205833=119863119878

119863119877

(5)

the nonlinear reactiondiffusion equation (2) take the follow-ing normalized form

1198892119878

1198891199092= 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877] (6)

1198892119871

1198891199092= 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

] (7)

1198892119877

1198891199092= 119898312058331206012119877 (8)

1198892119862

1198891199092= 12060121205831120588(1 +

1

119878+1

119862)

minus1

(9)

The transformed boundary conditions are

119878 = 120572 119871 = 120582 119877 = 120574 119862 = 120573 when 119909 = 0

119889119878

119889119909= 0

119889119871

119889119909= 0

119889119877

119889119909= 0 119862 = 0 when 119909 = 1

(10)

The dimensionless form of the current is given by

120595 =119868119889

119899FAD119862119870119862

= (119889119862

119889119909)

119909=1

(11)

3 Approximate Analytical Expression ofConcentrations of Four Reactants UsingHomotopy Analysis Method

31 Homotopy Analysis Method The homotopy analysismethod (HAM) [28ndash36] is a general analytic method toget series solutions of various types of nonlinear equationsincluding ordinary differential equations partial differentialequations and coupled nonlinear equations Unlike pertur-bation methods the HAM is independent of smalllargephysical parameters More importantly different from allperturbation and traditional nonperturbation methods theHAM provides us with a simple way to ensure the conver-gence of solution series Besides different from all pertur-bation and previous nonperturbation methods the HAMprovides us with great freedom to choose proper basefunctions to approximate a nonlinear problem [29 34]Liaorsquos book [29] for the homotopy analysis method was firstpublished in 2003 Now more and more researchers havebeen successfully applying this method to various nonlinearproblems in science and engineering In this paper we employHAM to solve the nonlinear differential equations (6) (7)and (9) The basic concept of Homotopy analysis method isgiven in Appendix A

4 ISRN Physical Chemistry

32 Solution of Boundary Value Problem Using the Homo-topy Analysis Method Using HAM method (Appendix B)we obtain the analytical expression corresponding to theconcentrations of substrate co-substrate reducing agent andmedial product as follows

119878 (119909) = 120572 +ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(12)

119871 (119909) = 120582 + ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(13)

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (14)

119862 (119909) = 120573 (1 minus 119909) +ℎ12060121205881205831120572

(1 + 120572)

times [119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(15)

where 119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 = log(120572(120573(1 + 120572) + 120572)) and h is the convergence controlparameter Equation (12)ndash(15) represent the new analyticalexpression of the dimensionless reactant concentrationsUsing (11) and (15) we can obtain the current as follows

120595 = minus120573 minusℎ12060121205881205831120572

(1 + 120572)[1

2+

120572

(1 + 120572) 120573(1 + 119873 +

120572119873

(1 + 120572) 120573)]

(16)

4 Result and Discussion

41 Numerical Simulation In order to investigate the accu-racy of this analytical method with a finite number of termsthe system of differential equations ((6)ndash(9)) also solvedby numerical methods The function pdepe (finite elementmethod) in ScilabMatlab software which is a function ofsolving PDE is used to solve these nonlinear equations [37]The ScilabMatlab program is also given in Appendix CTo validate the results the convergence studies are carried

out The convergence region of auxiliary parameter h isgiven in the Appendix D To show the efficiency of thepresent analytical method our results are compared with thenumerical solution (ScilabMatlab program) in Tables 1 2 3and 4 and Figures 2ndash6 The average relative errors betweenour analytical results and numerical results are 023 011003 and 001 for the concentrations of substrate co-substrate reducing agent and medial product respectively

42 Effect of the Thiele Module The concentration of sub-strate co-substrate reducing agent and medial productdepends upon Thiele module and saturation parametersThe Thiele module 1206012(= 119889

2119881119898119863119878119870119898) essentially compares

the rate of enzyme reaction (119881119898119870119898) and diffusion in the

enzyme layer (1198892119863119878) We observe the rise and downfall of

concentration profiles in two cases (i) If Thiele module issmall (1206012 lt 1) then enzyme kinetics predominates in thebiosensor response The overall kinetics is governed by thetotal amount of active enzyme (ii) The response is underdiffusion control if theThielemodule is large (1206012 gt 1) whichis observed at high catalytic activity and active membranethickness or at low reaction rate constant (119870

119898) or diffusion

coefficient values (119863119878)

The concentration profiles for the four reactants forsome fixed values of parameters are shown in Figure 2 Theanalytical results are compared with the numerical resultgraphically Upon comparison it is evident that both resultsgive satisfactory agreement The concentration of substrate 119878increases and attains its maximum whenThiele module andsaturation parameter 120572 increase (refer to Figure 3) Thereforethe profile deviates more from the linearity The concen-tration profile representing medial product 119871 increases asThiele module and saturation parameter 120582 increase (refer toFigure 4) From Figure 5 it is inferred that the increasingvalue of 119909 coordinate decreases the concentration of reducingagent119877 Figure 6 represents the concentration of co-substrate119862 versus the normalized distance coordinate It reaches theminimum value zero at electrode interface

43 Influence of ActiveMembraneThickness Theactivemem-brane thickness 119889 is one of the important technical param-eters and it has a considerable effect on the concentrationprofiles for the four reactants Also the Thiele module isdirectly proportional to the thickness of the membrane Theconcentration profiles depend significantly on themembranethickness 119889 If the active membrane thickness is large (119889 gt

50 120583m or thick membrane) the concentration profiles ofsubstrate and medial product increase whereas the concen-tration of reducing agent 119877 and co-substrate 119862 decreases Ifthe membrane thickness is small (119889 lt 50 120583m or thinnermembrane) the concentration profiles of substrate medialproduct and reducing agent have uniform values Theseuniform values are equal to the concentration of the abovereactants at 119909 = 0 The concentration profile of co-substrate119862 increases when thickness of the membrane decreases Theincreasing value is not significant Also the concentrationof co-substrate 119862 increases when the saturation parameter 120573increases (refer to Figures 3ndash6)

ISRN Physical Chemistry 5

01

005

015

02

025

03

035

04

045

05

055

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 005μ2 = 3h = minus076λ = 01

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

(a)

0

01

005

015

02

025

03

035

04

045

05

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 5μ2 = 05h = minus05

λ = 05

λ = 04

λ = 03

λ = 02

λ = 01

Normalized current coordinate x

φ2 = 1

(b)

Figure 4 Normalized concentration profiles of medial product 119871 for various values of the Thiele module 1206012 active membrane thickness 119889and the normalized parameters are plotted using (13) The following values have been taken for Figure 4 (a) 120572 = 009 120573 = 054 (b) 1206012 =1 120572 = 009 120573 = 054 The key to the graph stacked line represents (13) and dotted line represents the numerical simulation

075

08

085

09

095

1

105

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 01

μ3 = 06

γ = 1

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 7 d = 13229 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

(a)

04

05

06

07

08

09

1

11

0 02 04 06 08 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 0001

m3 = 03

m3 = 1

m3 = 3

m3 = 5μ3 = 06

γ = 1

φ2 = 1

(b)

Figure 5 Normalized concentration profiles of reducing agent119877 for various values of theThielemodule 1206012 activemembrane thickness 119889 andthe normalized parameters are plotted using (14) The key to the graph stacked line represents (14) and dotted line represents the numericalsimulation

0

06

05

04

03

02

01

minus010 02 04 06 08 1

Normalized current coordinate x

φ2 = 9 d = 150 μmφ2 = 4 d = 100 μm

φ2 = 1 d = 50 μmφ2 = 01 d = 158 μmφ2 = 001 d = 5 μm

μ1 = 08h = minus06

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(a)

0

02

04

06

08

1

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

μ1 = 08

h = minus1

φ2 = 01β = 1

β = 07

β = 054

β = 03

β = 01

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(b)

Figure 6 Normalized concentration profiles of co-substrate 119862 for various values of theThiele module 1206012 active membrane thickness 119889 andthe normalized parameters are plotted using (15) The following values have been taken for Figure 6 (a) 120572 = 009 120573 = 054 120588 = 12 (b)1206012= 01 120572 = 009 120588 = 12 The key to the graph stacked line represents (15) and dotted line represents the numerical simulation

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

ISRN Physical Chemistry 3

00

01

005

015

02

025

03

035

04

045

05

01 02 03 04 05 06 07 08 09 1

m1 = 1m3 = 0001μ3 = 06h = minus076

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μmφ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(a)

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

Normalized current coordinate x

α = 09

α = 07

α = 05

α = 03

α = 009

φ2 = 1μ3 = 1h = minus092

Nor

mal

ized

su

bstr

ate

con

cen

trat

ionS

(b)

Figure 3 Normalized concentration profiles of substrate 119878 for various values of theThiele module 1206012 active membrane thickness 119889 and thenormalized parameters are plotted using (12) The following values have been taken for Figure 3 (a) 120574 = 1 120582 = 01 120572 = 009 120573 = 0541205881= 092 (b) 1206012 = 1 120574 = 1 120582 = 01 120573 = 054119898

1= 1119898

3= 7 120588

1= 092 The key to the graph stacked line represents (12) and dotted line

represents the numerical simulation

where 119899 is the number of electrons taking part in theelectrochemical reaction on the cathode 119865 is the Faradayrsquosconstant and 119860 is the surface of cathode

22 Normalized Form By introducing the following set ofnondimensional variables

119909 =120575

119889 119878 =

[119878]

119870119878

119862 =[119862]

119870119862

119877 =[119877]

119870119898

119871 =[119871]

119870119898

1206012=

1198892119881119898

119863119878119870119898

120588 =119870119898

119870119862

1205881=119870119898

119870119878

120572 =119878119874

119870119878

120573 =119862119874

119870119862

120574 =119877119874

119870119898

120582 =119871119874

119870119898

1198981=1198702

119881119898

1198982=1198703

119881119898

1198983=1198704

119881119898

1205831=119863119878

119863119862

1205832=119863119878

119863119871

1205833=119863119878

119863119877

(5)

the nonlinear reactiondiffusion equation (2) take the follow-ing normalized form

1198892119878

1198891199092= 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877] (6)

1198892119871

1198891199092= 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

] (7)

1198892119877

1198891199092= 119898312058331206012119877 (8)

1198892119862

1198891199092= 12060121205831120588(1 +

1

119878+1

119862)

minus1

(9)

The transformed boundary conditions are

119878 = 120572 119871 = 120582 119877 = 120574 119862 = 120573 when 119909 = 0

119889119878

119889119909= 0

119889119871

119889119909= 0

119889119877

119889119909= 0 119862 = 0 when 119909 = 1

(10)

The dimensionless form of the current is given by

120595 =119868119889

119899FAD119862119870119862

= (119889119862

119889119909)

119909=1

(11)

3 Approximate Analytical Expression ofConcentrations of Four Reactants UsingHomotopy Analysis Method

31 Homotopy Analysis Method The homotopy analysismethod (HAM) [28ndash36] is a general analytic method toget series solutions of various types of nonlinear equationsincluding ordinary differential equations partial differentialequations and coupled nonlinear equations Unlike pertur-bation methods the HAM is independent of smalllargephysical parameters More importantly different from allperturbation and traditional nonperturbation methods theHAM provides us with a simple way to ensure the conver-gence of solution series Besides different from all pertur-bation and previous nonperturbation methods the HAMprovides us with great freedom to choose proper basefunctions to approximate a nonlinear problem [29 34]Liaorsquos book [29] for the homotopy analysis method was firstpublished in 2003 Now more and more researchers havebeen successfully applying this method to various nonlinearproblems in science and engineering In this paper we employHAM to solve the nonlinear differential equations (6) (7)and (9) The basic concept of Homotopy analysis method isgiven in Appendix A

4 ISRN Physical Chemistry

32 Solution of Boundary Value Problem Using the Homo-topy Analysis Method Using HAM method (Appendix B)we obtain the analytical expression corresponding to theconcentrations of substrate co-substrate reducing agent andmedial product as follows

119878 (119909) = 120572 +ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(12)

119871 (119909) = 120582 + ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(13)

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (14)

119862 (119909) = 120573 (1 minus 119909) +ℎ12060121205881205831120572

(1 + 120572)

times [119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(15)

where 119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 = log(120572(120573(1 + 120572) + 120572)) and h is the convergence controlparameter Equation (12)ndash(15) represent the new analyticalexpression of the dimensionless reactant concentrationsUsing (11) and (15) we can obtain the current as follows

120595 = minus120573 minusℎ12060121205881205831120572

(1 + 120572)[1

2+

120572

(1 + 120572) 120573(1 + 119873 +

120572119873

(1 + 120572) 120573)]

(16)

4 Result and Discussion

41 Numerical Simulation In order to investigate the accu-racy of this analytical method with a finite number of termsthe system of differential equations ((6)ndash(9)) also solvedby numerical methods The function pdepe (finite elementmethod) in ScilabMatlab software which is a function ofsolving PDE is used to solve these nonlinear equations [37]The ScilabMatlab program is also given in Appendix CTo validate the results the convergence studies are carried

out The convergence region of auxiliary parameter h isgiven in the Appendix D To show the efficiency of thepresent analytical method our results are compared with thenumerical solution (ScilabMatlab program) in Tables 1 2 3and 4 and Figures 2ndash6 The average relative errors betweenour analytical results and numerical results are 023 011003 and 001 for the concentrations of substrate co-substrate reducing agent and medial product respectively

42 Effect of the Thiele Module The concentration of sub-strate co-substrate reducing agent and medial productdepends upon Thiele module and saturation parametersThe Thiele module 1206012(= 119889

2119881119898119863119878119870119898) essentially compares

the rate of enzyme reaction (119881119898119870119898) and diffusion in the

enzyme layer (1198892119863119878) We observe the rise and downfall of

concentration profiles in two cases (i) If Thiele module issmall (1206012 lt 1) then enzyme kinetics predominates in thebiosensor response The overall kinetics is governed by thetotal amount of active enzyme (ii) The response is underdiffusion control if theThielemodule is large (1206012 gt 1) whichis observed at high catalytic activity and active membranethickness or at low reaction rate constant (119870

119898) or diffusion

coefficient values (119863119878)

The concentration profiles for the four reactants forsome fixed values of parameters are shown in Figure 2 Theanalytical results are compared with the numerical resultgraphically Upon comparison it is evident that both resultsgive satisfactory agreement The concentration of substrate 119878increases and attains its maximum whenThiele module andsaturation parameter 120572 increase (refer to Figure 3) Thereforethe profile deviates more from the linearity The concen-tration profile representing medial product 119871 increases asThiele module and saturation parameter 120582 increase (refer toFigure 4) From Figure 5 it is inferred that the increasingvalue of 119909 coordinate decreases the concentration of reducingagent119877 Figure 6 represents the concentration of co-substrate119862 versus the normalized distance coordinate It reaches theminimum value zero at electrode interface

43 Influence of ActiveMembraneThickness Theactivemem-brane thickness 119889 is one of the important technical param-eters and it has a considerable effect on the concentrationprofiles for the four reactants Also the Thiele module isdirectly proportional to the thickness of the membrane Theconcentration profiles depend significantly on themembranethickness 119889 If the active membrane thickness is large (119889 gt

50 120583m or thick membrane) the concentration profiles ofsubstrate and medial product increase whereas the concen-tration of reducing agent 119877 and co-substrate 119862 decreases Ifthe membrane thickness is small (119889 lt 50 120583m or thinnermembrane) the concentration profiles of substrate medialproduct and reducing agent have uniform values Theseuniform values are equal to the concentration of the abovereactants at 119909 = 0 The concentration profile of co-substrate119862 increases when thickness of the membrane decreases Theincreasing value is not significant Also the concentrationof co-substrate 119862 increases when the saturation parameter 120573increases (refer to Figures 3ndash6)

ISRN Physical Chemistry 5

01

005

015

02

025

03

035

04

045

05

055

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 005μ2 = 3h = minus076λ = 01

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

(a)

0

01

005

015

02

025

03

035

04

045

05

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 5μ2 = 05h = minus05

λ = 05

λ = 04

λ = 03

λ = 02

λ = 01

Normalized current coordinate x

φ2 = 1

(b)

Figure 4 Normalized concentration profiles of medial product 119871 for various values of the Thiele module 1206012 active membrane thickness 119889and the normalized parameters are plotted using (13) The following values have been taken for Figure 4 (a) 120572 = 009 120573 = 054 (b) 1206012 =1 120572 = 009 120573 = 054 The key to the graph stacked line represents (13) and dotted line represents the numerical simulation

075

08

085

09

095

1

105

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 01

μ3 = 06

γ = 1

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 7 d = 13229 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

(a)

04

05

06

07

08

09

1

11

0 02 04 06 08 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 0001

m3 = 03

m3 = 1

m3 = 3

m3 = 5μ3 = 06

γ = 1

φ2 = 1

(b)

Figure 5 Normalized concentration profiles of reducing agent119877 for various values of theThielemodule 1206012 activemembrane thickness 119889 andthe normalized parameters are plotted using (14) The key to the graph stacked line represents (14) and dotted line represents the numericalsimulation

0

06

05

04

03

02

01

minus010 02 04 06 08 1

Normalized current coordinate x

φ2 = 9 d = 150 μmφ2 = 4 d = 100 μm

φ2 = 1 d = 50 μmφ2 = 01 d = 158 μmφ2 = 001 d = 5 μm

μ1 = 08h = minus06

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(a)

0

02

04

06

08

1

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

μ1 = 08

h = minus1

φ2 = 01β = 1

β = 07

β = 054

β = 03

β = 01

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(b)

Figure 6 Normalized concentration profiles of co-substrate 119862 for various values of theThiele module 1206012 active membrane thickness 119889 andthe normalized parameters are plotted using (15) The following values have been taken for Figure 6 (a) 120572 = 009 120573 = 054 120588 = 12 (b)1206012= 01 120572 = 009 120588 = 12 The key to the graph stacked line represents (15) and dotted line represents the numerical simulation

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

4 ISRN Physical Chemistry

32 Solution of Boundary Value Problem Using the Homo-topy Analysis Method Using HAM method (Appendix B)we obtain the analytical expression corresponding to theconcentrations of substrate co-substrate reducing agent andmedial product as follows

119878 (119909) = 120572 +ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(12)

119871 (119909) = 120582 + ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(13)

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (14)

119862 (119909) = 120573 (1 minus 119909) +ℎ12060121205881205831120572

(1 + 120572)

times [119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(15)

where 119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 = log(120572(120573(1 + 120572) + 120572)) and h is the convergence controlparameter Equation (12)ndash(15) represent the new analyticalexpression of the dimensionless reactant concentrationsUsing (11) and (15) we can obtain the current as follows

120595 = minus120573 minusℎ12060121205881205831120572

(1 + 120572)[1

2+

120572

(1 + 120572) 120573(1 + 119873 +

120572119873

(1 + 120572) 120573)]

(16)

4 Result and Discussion

41 Numerical Simulation In order to investigate the accu-racy of this analytical method with a finite number of termsthe system of differential equations ((6)ndash(9)) also solvedby numerical methods The function pdepe (finite elementmethod) in ScilabMatlab software which is a function ofsolving PDE is used to solve these nonlinear equations [37]The ScilabMatlab program is also given in Appendix CTo validate the results the convergence studies are carried

out The convergence region of auxiliary parameter h isgiven in the Appendix D To show the efficiency of thepresent analytical method our results are compared with thenumerical solution (ScilabMatlab program) in Tables 1 2 3and 4 and Figures 2ndash6 The average relative errors betweenour analytical results and numerical results are 023 011003 and 001 for the concentrations of substrate co-substrate reducing agent and medial product respectively

42 Effect of the Thiele Module The concentration of sub-strate co-substrate reducing agent and medial productdepends upon Thiele module and saturation parametersThe Thiele module 1206012(= 119889

2119881119898119863119878119870119898) essentially compares

the rate of enzyme reaction (119881119898119870119898) and diffusion in the

enzyme layer (1198892119863119878) We observe the rise and downfall of

concentration profiles in two cases (i) If Thiele module issmall (1206012 lt 1) then enzyme kinetics predominates in thebiosensor response The overall kinetics is governed by thetotal amount of active enzyme (ii) The response is underdiffusion control if theThielemodule is large (1206012 gt 1) whichis observed at high catalytic activity and active membranethickness or at low reaction rate constant (119870

119898) or diffusion

coefficient values (119863119878)

The concentration profiles for the four reactants forsome fixed values of parameters are shown in Figure 2 Theanalytical results are compared with the numerical resultgraphically Upon comparison it is evident that both resultsgive satisfactory agreement The concentration of substrate 119878increases and attains its maximum whenThiele module andsaturation parameter 120572 increase (refer to Figure 3) Thereforethe profile deviates more from the linearity The concen-tration profile representing medial product 119871 increases asThiele module and saturation parameter 120582 increase (refer toFigure 4) From Figure 5 it is inferred that the increasingvalue of 119909 coordinate decreases the concentration of reducingagent119877 Figure 6 represents the concentration of co-substrate119862 versus the normalized distance coordinate It reaches theminimum value zero at electrode interface

43 Influence of ActiveMembraneThickness Theactivemem-brane thickness 119889 is one of the important technical param-eters and it has a considerable effect on the concentrationprofiles for the four reactants Also the Thiele module isdirectly proportional to the thickness of the membrane Theconcentration profiles depend significantly on themembranethickness 119889 If the active membrane thickness is large (119889 gt

50 120583m or thick membrane) the concentration profiles ofsubstrate and medial product increase whereas the concen-tration of reducing agent 119877 and co-substrate 119862 decreases Ifthe membrane thickness is small (119889 lt 50 120583m or thinnermembrane) the concentration profiles of substrate medialproduct and reducing agent have uniform values Theseuniform values are equal to the concentration of the abovereactants at 119909 = 0 The concentration profile of co-substrate119862 increases when thickness of the membrane decreases Theincreasing value is not significant Also the concentrationof co-substrate 119862 increases when the saturation parameter 120573increases (refer to Figures 3ndash6)

ISRN Physical Chemistry 5

01

005

015

02

025

03

035

04

045

05

055

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 005μ2 = 3h = minus076λ = 01

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

(a)

0

01

005

015

02

025

03

035

04

045

05

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 5μ2 = 05h = minus05

λ = 05

λ = 04

λ = 03

λ = 02

λ = 01

Normalized current coordinate x

φ2 = 1

(b)

Figure 4 Normalized concentration profiles of medial product 119871 for various values of the Thiele module 1206012 active membrane thickness 119889and the normalized parameters are plotted using (13) The following values have been taken for Figure 4 (a) 120572 = 009 120573 = 054 (b) 1206012 =1 120572 = 009 120573 = 054 The key to the graph stacked line represents (13) and dotted line represents the numerical simulation

075

08

085

09

095

1

105

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 01

μ3 = 06

γ = 1

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 7 d = 13229 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

(a)

04

05

06

07

08

09

1

11

0 02 04 06 08 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 0001

m3 = 03

m3 = 1

m3 = 3

m3 = 5μ3 = 06

γ = 1

φ2 = 1

(b)

Figure 5 Normalized concentration profiles of reducing agent119877 for various values of theThielemodule 1206012 activemembrane thickness 119889 andthe normalized parameters are plotted using (14) The key to the graph stacked line represents (14) and dotted line represents the numericalsimulation

0

06

05

04

03

02

01

minus010 02 04 06 08 1

Normalized current coordinate x

φ2 = 9 d = 150 μmφ2 = 4 d = 100 μm

φ2 = 1 d = 50 μmφ2 = 01 d = 158 μmφ2 = 001 d = 5 μm

μ1 = 08h = minus06

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(a)

0

02

04

06

08

1

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

μ1 = 08

h = minus1

φ2 = 01β = 1

β = 07

β = 054

β = 03

β = 01

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(b)

Figure 6 Normalized concentration profiles of co-substrate 119862 for various values of theThiele module 1206012 active membrane thickness 119889 andthe normalized parameters are plotted using (15) The following values have been taken for Figure 6 (a) 120572 = 009 120573 = 054 120588 = 12 (b)1206012= 01 120572 = 009 120588 = 12 The key to the graph stacked line represents (15) and dotted line represents the numerical simulation

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

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Carbohydrate Chemistry

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Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Analytical Methods in Chemistry

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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CatalystsJournal of

ISRN Physical Chemistry 5

01

005

015

02

025

03

035

04

045

05

055

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 005μ2 = 3h = minus076λ = 01

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

Normalized current coordinate x

(a)

0

01

005

015

02

025

03

035

04

045

05

0 02 04 06 08 1

Nor

mal

ized

med

ial p

rodu

ct c

once

ntr

atio

nL

m2 = 5μ2 = 05h = minus05

λ = 05

λ = 04

λ = 03

λ = 02

λ = 01

Normalized current coordinate x

φ2 = 1

(b)

Figure 4 Normalized concentration profiles of medial product 119871 for various values of the Thiele module 1206012 active membrane thickness 119889and the normalized parameters are plotted using (13) The following values have been taken for Figure 4 (a) 120572 = 009 120573 = 054 (b) 1206012 =1 120572 = 009 120573 = 054 The key to the graph stacked line represents (13) and dotted line represents the numerical simulation

075

08

085

09

095

1

105

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 01

μ3 = 06

γ = 1

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 7 d = 13229 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

(a)

04

05

06

07

08

09

1

11

0 02 04 06 08 1Normalized current coordinate x

Nor

mal

ized

red

uci

ng

agen

t co

nce

ntr

atio

nR

m3 = 0001

m3 = 03

m3 = 1

m3 = 3

m3 = 5μ3 = 06

γ = 1

φ2 = 1

(b)

Figure 5 Normalized concentration profiles of reducing agent119877 for various values of theThielemodule 1206012 activemembrane thickness 119889 andthe normalized parameters are plotted using (14) The key to the graph stacked line represents (14) and dotted line represents the numericalsimulation

0

06

05

04

03

02

01

minus010 02 04 06 08 1

Normalized current coordinate x

φ2 = 9 d = 150 μmφ2 = 4 d = 100 μm

φ2 = 1 d = 50 μmφ2 = 01 d = 158 μmφ2 = 001 d = 5 μm

μ1 = 08h = minus06

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(a)

0

02

04

06

08

1

0 01 02 03 04 05 06 07 08 09 1Normalized current coordinate x

μ1 = 08

h = minus1

φ2 = 01β = 1

β = 07

β = 054

β = 03

β = 01

Nor

mal

ized

cos

ubs

trat

e co

nce

ntr

atio

nC

(b)

Figure 6 Normalized concentration profiles of co-substrate 119862 for various values of theThiele module 1206012 active membrane thickness 119889 andthe normalized parameters are plotted using (15) The following values have been taken for Figure 6 (a) 120572 = 009 120573 = 054 120588 = 12 (b)1206012= 01 120572 = 009 120588 = 12 The key to the graph stacked line represents (15) and dotted line represents the numerical simulation

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

6 ISRN Physical Chemistry

1 15 2 25 3

0

002

004

006

008

01

012

014

016

018

02

φ2 = 9 d = 150 μm

φ2 = 5 d = 1118 μm

φ2 = 3 d = 866 μm

φ2 = 1 d = 50 μm

φ2 = 01 d = 158 μm

φ2 = 7 d = 13229 μm

α = 5β = 00001h = minus1

μ1

Nor

mal

ized

cu

rren

tΨ

(a)

15

005 01 015 02 025 03 035 04 045 05

05

0

1

α = 1

α = 08

α = 05

α = 03

α = 02

α = 01

α = 009

α = 007

β

h = minus1

φ2 = 9μ1 = 3

Nor

mal

ized

cu

rren

tΨ

(b)

Figure 7 Diagrammatic representation of the normalized current 120595 versus the normalized parameters

Table 1 Comparison between the analytical normalized substrate concentration S (12) and numerical simulation for various values of 120572 and1206012= 1 120573 = 054 120574 = 1 120582 = 01 119898

3= 7 120588 = 12 120583

3= 06 ℎ = minus092

119909120572 = 01 120572 = 05 120572 = 1

Our work (12) Numerical Error Our work (12) Numerical Error Our work (12) Numerical Error00 00000 00000 000 05000 05000 000 10000 10000 00002 00974 00976 021 04870 04869 002 09835 09833 00204 00951 00954 032 04781 04779 004 09725 09723 00206 00935 00938 032 04730 04729 002 09665 09664 00108 00929 00932 032 04712 04710 004 09644 09642 00210 00929 00931 022 04711 04707 008 09643 09639 004

Average deviation 023 Average deviation 003 Average deviation 001

Table 2 Comparison between the analytical normalized medial product concentration 119871 (13) and numerical simulation for various valuesof 120582 and 1206012 = 1 119898

2= 5 120588 = 12 120572 = 009 120573 = 054 120583

2= 05 ℎ = minus092

119909120582 = 01 120582 = 05 120582 = 1

Our work (13) Numerical Error Our work (13) Numerical Error Our work (13) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00963 00963 000 04798 04798 000 09591 09592 00104 00934 00935 011 04640 04643 006 09272 09278 00606 00913 00915 022 04527 04533 013 09044 09056 01308 00901 00902 011 04459 04467 018 08907 08924 01910 00896 00898 022 04436 04446 023 08861 08880 021

Average deviation 011 Average deviation 010 Average deviation 010

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

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Carbohydrate Chemistry

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Quantum Chemistry

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Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

ISRN Physical Chemistry 7

Table 3 Comparison between the analytical normalized reducing agent concentration 119877 (14) and numerical simulation for various values of120574 and 1206012 = 01 119898

3= 7 120583

3= 06

119909120574 = 01 120574 = 05 120574 = 1

Our work (14) Numerical Error Our work (14) Numerical Error Our work (14) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00934 00934 000 04671 04671 000 09343 09341 00204 00884 00884 000 04421 04420 002 08843 08840 00306 00849 00849 000 04246 04244 005 08491 08488 00308 00828 00828 000 04141 04139 005 08283 08279 00510 00821 00821 000 04107 04105 005 08214 08210 005

Average deviation 000 Average deviation 003 Average deviation 003

Table 4 Comparison between the analytical normalized co-substrate concentration 119862 (15) and numerical simulation for various values of120573 and 1206012 = 001 120588 = 12 120572 = 009 120583

1= 08 ℎ = minus092

119909120573 = 01 120573 = 05 120573 = 1

Our work (15) Numerical Error Our work (15) Numerical Error Our work (15) Numerical Error00 01000 01000 000 05000 05000 000 10000 10000 00002 00800 00800 000 04000 04000 000 08000 07999 00104 00600 00600 000 03000 02999 003 06000 05999 00206 00400 00400 000 02000 01999 005 04000 03999 00308 00200 00200 000 01000 01000 000 02000 02000 00010 00000 00000 000 00000 00000 000 00000 00000 000

Average deviation 000 Average deviation 001 Average deviation 001

1002

1003

1004

1005

1006

1007

1008

minus14 minus12 minus1 minus08 minus06 minus04 minus02

Nor

mal

ized

con

cen

trat

ion

C (

01)

h

φ2 = 001β = 1μ1 = 08

ρ = 12α = 09

Figure 8The h curve to indicate the convergence region for119862(01)

44 Current Response The normalized current 120595 versus thediffusion coefficient ratio 120583

1is calculated at different values

of the active membrane thickness 119889 The results obtained forvarious values of the normalized parameters are depicted inFigure 7(a) The current response increases when the activemembrane thickness (119889 gt 50 120583m) increases Also for thinnermembrane (119889 lt 50 120583m) the value of the current is zero InFigure 7(b) the current response increases as the saturationparameter 120572 increases

5 Conclusions

The theoretical model of hybrid amperometric enzymebiosensor with cyclic reaction and biochemical amplification

for steady-state condition is discussed The system of threenonlinear differential equations for ping-pong enzyme kinet-ics has been solved analytically Influence of Thiele moduleand active membrane thickness is investigated The obtainedresults have a good agreement with those obtained usingnumerical method This analytical result will be useful insensor design optimization and prediction of the electroderesponse Using this result the action of biosensor is analyzedat critical concentration of substrate and enzyme activitiesTheoretical results obtained in this paper can also be usedto analyze the effect of different parameters such as activemembrane thickness and saturation parameters

Appendices

A Basic Idea of Liaorsquos HomotopyAnalysis Method

Consider the following differential equation [38]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator t denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin a similar way By means of generalizing the conventionalHomotopy method Liao constructed the so-called zero-order deformation equation as

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

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Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

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Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 ISRN Physical Chemistry

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

of u(t) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution u(t)

Expanding 120593(119905 119901) in Taylor series with respect to p we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

where

119906119898 (119905) = [

1

119898

120597119898120593 (119905 119901)

120597119901119898]

119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter h and the auxiliary function are so properlychosen the series (A4) converges at 119901 = 1 then we have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119899= 1199060 1199061 119906119899 (A7)

Differentiating (A2) for m times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthemby119898 wewill have the so-called119898th-order deformationequation as

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)N119898 (119898minus1) (A8)

where

N119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898=

0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)N119898 (119898minus1)] (A10)

In this way it is easy to obtain 119906119898for119898 ge 1 at119898th order we

have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation of theoriginal equation (A1) For the convergence of the previousmethod we refer the reader to Liao [29] If (A1) admitsunique solution then this method will produce the uniquesolution If (A1) does not possess unique solution the HAMwill give a solution among many other (possible) solutions

B Approximate Analytical Expression ofConcentrations of Substrate Co-SubstrateReducing Agent and Medial Product

From (8) it is clear that the concentration of normalizedreducing agent 119877 is

119877 (119909) =120574 cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601) (B1)

In order to solve (6) (7) and (9) by means of the HAMwe first construct the zeroth-order deformation equation bytaking119867(119905) = 1

(1 minus 119901)1198892119878

1198891199092

= 119901ℎ1198892119878

1198891199092minus 12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B2)

(1 minus 119901)1198892119871

1198891199092

= 119901ℎ1198892119871

1198891199092minus 12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B3)

(1 minus 119901)1198892119862

1198891199092= 119901ℎ[

1198892119862

1198891199092minus 12060121205831120588(1 +

1

119878+1

119862)

minus1

] (B4)

The approximate solutions of (B2)ndash(B4) are as follows

119878 = 1198780+ 1199011198781+ 11990121198782+ sdot sdot sdot (B5)

119871 = 1198710+ 1199011198711+ 11990121198712+ sdot sdot sdot (B6)

119862 = 1198620+ 1199011198621+ 11990121198622+ sdot sdot sdot (B7)

Substituting (B5) in (B2) (B6) in (B3) and (B7) in (B4)and equating the like powers of p we get

119901011988921198780

1198891199092= 0 (B8)

119901111988921198781

1198891199092=11988921198780

1198891199092(ℎ + 1)

minus ℎ12060121205881[(1 +

1

119878+1

119862)

minus1

minus 1198981119871119877]

(B9)

119901011988921198710

1198891199092= 0 (B10)

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

ISRN Physical Chemistry 9

119901111988921198711

1198891199092=11988921198710

1198891199092(ℎ + 1)

minus ℎ12060121205832[1198982119871 minus (1 +

1

119878+1

119862)

minus1

]

(B11)

119901011988921198620

1198891199092= 0 (B12)

119901111988921198621

1198891199092=11988921198620

1198891199092(ℎ + 1) minus ℎ120601

21205831120588(1 +

1

119878+1

119862)

minus1

(B13)

The boundary conditions equation (10) become

1198780= 120572 119871

0= 120582 119862

0= 120573 when 119909 = 0 (B14)

1198891198780

119889119909= 0

1198891198710

119889119909= 0 119862

0= 0 when 119909 = 1 (B15)

119878119894= 0 119871

119894= 0 119862

119894= 0 when 119909 = 0 119894 = 1 2 3

(B16)

119889119878119894

119889119909= 0

119889119871119894

119889119909= 0 119862

119894= 0 when 119909 = 1 119894 = 1 2 3

(B17)

From (B8) (B10) and (B12) and from the boundary condi-tions (B14) and (B15) we get

1198780= 120572 (B18)

1198710= 120582 (B19)

1198620= 120573 (1 minus 119909) (B20)

Substituting the values of 1198780 1198710 and 119862

0in (B9) (B11)

and (B13) and solving the equations using the boundaryconditions (B16) and (B17) we obtain the following results

1198781=

ℎ12060121205881120572

(1 + 120572)2120573[(1 + 120572) 120573119909 (1 minus

119909

2)

+ 120572119872(1 +120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

+ℎ12058811198981120582120574

11989831205833

(cosh (radic11989831205833120601 (1 minus 119909))

cosh (radic11989831205833120601)minus 1)

(B21)

1198711= ℎ120601212058321198982120582119909(1 minus

119909

2) minus

ℎ12060121205832120572

(1 + 120572)2120573

times [(1 + 120572) 120573119909 (1 minus119909

2) + 120572119872(1 +

120572

(1 + 120572) 120573)

minus120572119909 (119872 minus119873 minus 1) ]

(B22)

1198621=

ℎ12060121205881205831120572

(1 + 120572)[119909

2(1 minus 119909) +

120572119872

(1 + 120572) 120573

times(1 minus 119909 +120572

(1 + 120572) 120573) minus

1205722119873119909

(1 + 120572)21205732]

(B23)

where119872 = log((120573(1 + 120572)(1 minus 119909) + 120572)(120573(1 + 120572) + 120572)) 119873 =

log(120572(120573(1 + 120572) + 120572))Adding (B18) and (B21) we get (12) in the text Similarly

we get (13) and (15) in the text

C ScilabMatlab Program to Find theNumerical Solution of Nonlinear Equations(6)ndash(9)

function pdex4m = 0x = linspace(01)t=linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)u3 = sol(3)u4 = sol(4)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)figureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)figureplot(xu3(end))title(lsquou3(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou3(x2)rsquo)

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

10 ISRN Physical Chemistry

figureplot(xu4(end))title(lsquou4(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou4(x2)rsquo)function [cfs] = pdex4pde(xtuDuDx)c = [1 1 1 1]f = [1 1 1 1]lowast DuDxQ=9p1=092p=12m1=02m2=005m3=0001n1=08n2=01n3=06F=-Qlowastp1lowast((1(1+1(u(1))+1(u(4)))-m1lowast(u(2))lowast(u(3))))F1=-Qlowastn2lowast(m2lowast(u(2))-1(1+1(u(1))+1(u(4))))F2=-m3lowastn3lowastQlowast(u(3))F3=-Qlowastn1lowastplowast(1(1+1(u(1))+1(u(4))))s=[F F1 F2 F3]function u0 = pdex4ic(x)u0 = [1 1 1 1]function [plqlprqr]=pdex4bc(xlulxrurt)pl = [ul(1)-009ul(2)minus0ul(3)-1ul(4)-054]ql = [0 0 0 0]pr = [0 0 0 ur(4)]qr = [1 1 1 0]

D Determining the Validity Region of ℎ

The analytical solution represented by (12) (13) and (15)contains the auxiliary parameter h which gives the con-vergence region and rate of approximation for homotopyanalysis method The analytical solution should converge Itshould be noted that the auxiliary parameter h controls theconvergence and accuracy of the solution series In order todefine region such that the solution series is independentof ℎ a multiple of ℎ curves are plotted The region wherethe distribution of 119878 119871 and 119862 versus h is a horizontal lineis known as the convergence region for the correspondingfunction The common region among 119878(119909) 119871(119909) and 119862(119909)

is known as the overall convergence region To study theinfluence of h on the convergence of solution the h curvesof 119862(01) are plotted in Figure 8 This figure clearly indicatesthat the valid region of h is about (minus15 tominus01) Similarly wecan find the value of the convergence-control parameter h fordifferent values of constant parameters

Nomenclature

Symbols

[S] Measured substrate concentration ofcatechol (mM)

[119871] Medial product concentration of 12benzoquinone (mM)

[119877] Reducing agent concentration ofL-ascorbic acid (mM)

[119862] Co-substrate concentration of oxygen(mM)

119881119898 Maximal rate (mmol(ls))

119863119904 Diffusion coefficient for substrate

(m2s)119863119871 Diffusion coefficient for medial

product (m2s)119863119877 Diffusion coefficient for reducing

agent (m2s)119863119862 Diffusion coefficient for co-substrate

(m2s)1198702 1198703 1198704 119870119898 Reaction rate constants (mmol(ls))

119870119904 119870119862 Reaction rate constants (mM)

120575 Distance coordinate (120583m)119889 Active membrane thickness (120583m)ℎ Convergence control parameter119878 Normalized measured substrate

concentration (dimensionless)119871 Normalized medial product

concentration (dimensionless)119877 Normalized reducing agent

concentration (dimensionless)119862 Normalized co-substrate

concentration (dimensionless)119909 Normalized distance coordinate

(dimensionless)120572 120573 120574 120582 Saturation parameters (dimensionless)1198981 1198982 1198983 Linear enzyme kinetic coefficient

(dimensionless)1205831 1205832 1205833 Ratio of diffusion coefficients

(dimensionless)120588 1205881 Ratio of reaction rate constants

(dimensionless)1206012 Thiele module (dimensionless)

120595 Normalized current (dimensionless)

Acknowledgments

This work was supported by the University Grants Com-mission (F no 39ndash582010(SR)) New Delhi India Theauthors are thankful to Dr R Murali The Principal TheMadura College Madurai and Mr M S Meenakshisun-daram The Secretary Madura College Board Madurai fortheir encouragement The author K Indira is very thankfulto theManonmaniam Sundaranar University Tirunelveli forallowing to do the research work

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

ISRN Physical Chemistry 11

References

[1] A D McNaught and A Wilkinson IUPAC Compendium ofChemical TerminologymdashThe Gold Book Blackwell ScientificOxford UK 2nd edition 1997

[2] L C Clark Jr and C Lyons ldquoElectrode systems for continuousmonitoring in cardiovascular surgeryrdquo Annals of the New YorkAcademy of Sciences vol 102 pp 29ndash45 1962

[3] K Cammann ldquoBio-sensors based on ion-selective electrodesrdquoFreseniusrsquo Zeitschrift fur Analytische Chemie vol 287 no 1 pp1ndash9 1977

[4] S P Mohanty and E Koucianos ldquoBiosensors a tutorial reviewrdquoIEEE Potentials vol 25 no 2 pp 35ndash40 2006

[5] A ChaubeyMGerard V S Singh and BDMalhotra ldquoImmo-bilization of lactate dehydrogenase on tetraethylorthosilicate-derived sol-gel films for application to lactate biosensorrdquoApplied Biochemistry and Biotechnology vol 96 no 1ndash3 pp303ndash311 2001

[6] A J Reviejo C Fernandez F Liu J M Pingarron and J WangldquoAdvances in amperometric enzyme electrodes in reversedmicellesrdquo Analytica Chimica Acta vol 315 no 1-2 pp 93ndash991995

[7] M Stoytcheva N Nankov and V Sharcova ldquoAnalytical char-acterisation and application of a p-benzoquinone mediatedamperometric graphite sensor with covalently linked glu-coseoxidaserdquo Analytica Chimica Acta vol 315 no 1-2 pp 101ndash107 1995

[8] G G Guilbault and F R Shu ldquoEnzyme electrodes based on theuse of a carbon dioxide sensor Urea and L-tyrosine electrodesrdquoAnalytical Chemistry vol 44 no 13 pp 2161ndash2166 1972

[9] L H Larsen N P Revsbech and S J Binnerup ldquoAmicrosensorfor nitrate based on immobilized denitrifying bacteriardquoAppliedand Environmental Microbiology vol 62 no 4 pp 1248ndash12511996

[10] A L Ghindilis P Atanasov M Wilkins and E WilkinsldquoImmunosensors electrochemical sensing and other engineer-ing approachesrdquo Biosensors and Bioelectronics vol 13 no 1 pp113ndash131 1998

[11] J Wang ldquoAmperometric biosensors for clinical and therapeuticdrug monitoring a reviewrdquo Journal of Pharmaceutical andBiomedical Analysis vol 19 no 1-2 pp 47ndash53 1999

[12] DM Zhou Y Q Dai andK K Shiu ldquoPoly(phenylenediamine)film for the construction of glucose biosensors based onplatinized glassy carbon electroderdquo Journal of Applied Electro-chemistry vol 40 no 11 pp 1997ndash2003 2010

[13] A P F Turner I Karube andG SWilson EdsBiosensors Fun-damentals and Applications Oxford University Press OxfordUK 1989

[14] A P F Turner Ed Advances in Biosensors vol 1 JAI PressLondon UK 1991

[15] J R Flores and E Lorenzo ldquoAmperometric biosensorsrdquo inAnalytical Voltammetry M R Smyth and J G Vos Eds vol27 ofWilson and Wilsonrsquos Comprehensive Analytical ChemistryElsevier Amsterdam The Netherlands 1992

[16] F Scheller and F Schubert Biosensors Elsevier AmsterdamThe Netherlands 1992

[17] M J Song SWHwang andDWhang ldquoAmperometric hydro-gen peroxide biosensor based on amodified gold electrode withsilver nanowiresrdquo Journal of Applied Electrochemistry vol 40no 12 pp 2099ndash2105 2010

[18] R S Dubey and S N Upadhyay ldquoMicroorganism basedbiosensor for monitoring of microbiologically influenced cor-rosion caused by fungal speciesrdquo Indian Journal of ChemicalTechnology vol 10 no 6 pp 607ndash610 2003

[19] T Yao and S Handa ldquoElectroanalytical properties of aldehydebiosensors with a hybrid-membrane composed of an enzymefilm and a redox Os-polymer filmrdquo Analytical Sciences vol 19no 5 pp 767ndash770 2003

[20] F Amarita C Rodriguez Fernandez and F Alkorta ldquoHybridbiosensors to estimate lactose in milkrdquo Analytica Chimica Actavol 349 no 1ndash3 pp 153ndash158 1997

[21] K Indira and L Rajendran ldquoAnalytical expression of theconcentration of substrates and product in phenolmdashpolyphenoloxidase system immobilized in laponite hydrogels MichaelismdashMenten formalism in homogeneous mediumrdquo ElectrochimicaActa vol 56 no 18 pp 6411ndash6419 2011

[22] S Loghambal and L Rajendran ldquoMathematical modeling inamperometric oxidase enzyme-membrane electrodesrdquo Journalof Membrane Science vol 373 no 1-2 pp 20ndash28 2011

[23] P Manimozhi A Subbiah and L Rajendran ldquoSolution ofsteady-state substrate concentration in the action of biosensorresponse at mixed enzyme kineticsrdquo Sensors and Actuators Bvol 147 no 1 pp 290ndash297 2010

[24] A Eswari and L Rajendran ldquoAnalytical solution of steady statecurrent at a microdisk biosensorrdquo Journal of ElectroanalyticalChemistry vol 641 no 1-2 pp 35ndash44 2010

[25] A Eswari and L Rajendran ldquoAnalytical solution of steady-statecurrent an enzyme-modifiedmicrocylinder electrodesrdquo Journalof Electroanalytical Chemistry vol 648 no 1 pp 36ndash46 2010

[26] V Rangelova ldquoModeling amperometric biosensor with cyclicreactionrdquo Journal of Engineering Annals of the Faculty of Engi-neering Huhedoara vol 5 no 1 pp 117ndash122 2007

[27] S Uchiyama Y Hasebe H Shimizu andH Ishihara ldquoEnzyme-based catechol sensor based on the cyclic reaction between cat-echol and 12-benzoquinone using L-ascorbate and tyrosinaserdquoAnalytica Chimica Acta vol 276 no 2 pp 341ndash345 1993

[28] S J Liao The proposed Homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[29] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman ampHallCRC Press Boca Raton FlaUSA 2003

[30] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parametersmdashII An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[31] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[32] S-J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash1128 1999

[33] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[34] S Liao and Y Tan ldquoa general approach to obtain seriessolutions of nonlinear differential equationsrdquo Studies in AppliedMathematics vol 119 no 4 pp 297ndash355 2007

[35] S J Liao ldquoBeyond perturbation a review on the basic ideas oftheHomotophy analysismethod and its applicationsrdquoAdvancedMechanics vol 38 no 1 pp 1ndash34 2008

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

12 ISRN Physical Chemistry

[36] S Liao Homotopy Analysis Method in Nonlinear DifferentialEquations Springer and Higher Education Press HeidelbergGermany 2012

[37] R D Skeel andM Berzins ldquoAmethod for the spatial discretiza-tion of parabolic equations in one space variablerdquo SIAM Journalon Scientific and Statistical Computing vol 11 no 1 32 pages1990

[38] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of