Available online at www.sciencedirect.com

Review Article

Mathematical modeling of nonlinear reaction–diffusion

processes in enzymatic biofuel cells

L. Rajendran

1 , M. Kirthiga

1 and E. Laborda

2

Enzymatic biofuel cells convert the chemical energy of biofuels into electrical energy by employing oxidoreductase enzymes as catalysts. They have attracted considerable attention due to

their potential use as a promising alternative to traditional power sources. Also, enzyme-modified electrodes have important applications in electrochemical biosensors, bioreactors, implantable medical devices as well as in biochemistry as a source of information on the action of enzymes. Among the different electrochemical techniques available, enzymatic biofuel cells and electrodes are generally studied via cyclic voltammetry, chronoamperometry, polarization curves and impedance measurements to gain better insight into the enzymatic system. The main aim is to

assess the influence of the mediator species and its diffusivity, the loading of biocatalysts, the amount of substrates, mediators and inhibitors, the strategy of enzyme immobilization, etc. as well as to study the enzymatic kinetic mechanism. Within this context, mathematical models must be used to understand, predict and optimize the performance of enzymatic biofuel cells and electrodes as a function of the chief experimental parameters above mentioned. In this review

article, major recent research activity concerning the mathematical modeling of enzymatic electrodes and fuel cells is discussed, highlighting the main contributions as well as current problems and challenges.

Addresses 1 Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti 626115, Tamil Nadu, India 2 Department of Physical Chemistry, Regional Campus of International Excellence “Campus Mare Nostrum”, University of Murcia, 30100 Murcia, Spain

Corresponding author : Rajendran, L. ( raj_sms@rediffmail.com )

Current Opinion in Electrochemistry 2017, 1 :121–132

This review comes from a themed issue on Fundamental and Theo- retical Electrochemistry 2017

Edited by Angela Molina

For a complete overview see the Issue and the Editorial

Available online 11th January 2017

http:// dx.doi.org/ 10.1016/ j.coelec.2016.11.003

2451-9103/© 2016 Elsevier B.V. All rights reserved.

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Introduction

A fuel cell is basically an energy device, like a battery,that converts chemical energy into electricity providingalmost constant output power as long as the reactantsare continuously supplied and the products are contin-uously removed [1] . Enzymatic fuel cells are a type offuel cells that employ enzymes (biocatalysts) instead ofnoble metal catalysts. The working principle is the sameas in conventional fuel cells ( Figure 1 ) [2] , that is: the fuel(sugar, alcohol, hydrogen,...) is oxidized at the anode andthe electrons released by the oxidation reaction are driventhrough an outer electrical circuit to the cathode wherethey combine with the oxidant (typically oxygen) and pro-tons to yield the by-products (water).

In the last decade, the most significant advances in the im-provement of the performance of enzymatic biofuel cells[ 3

•] are related to device miniaturization and to the ’de-sign’ of more active enzymatic systems via enzyme engi-neering [4] , new methods of immobilization, stabilizationand preservation of enzymes at the electrode [5] and ofenhancing direct bioelectrocatalysis [6] as well as of bio-electrodes that can operate in air-breathing mode [7] . Asa result, a wide range of applications of enzymatic bio-fuel cells have arisen over years, some of the most im-pressive and recent ones being the development of self-powered [8] and implantable [9] devices. The former con-cept was introduced by Katz, Bückmann and Willner in2001 [8] when they employed a biofuel cell as a biosensorfor the fuel. More recently, Katz et al . have developed theuse of biological logic gates for self-powered biosensing[10,11] . Regarding implantable devices in living organ-isms, enzymatic biofuel cells have been implanted in rats[12] , cockroaches [13] , snails [14] and clams [15] . Also, ithas been considered the fabrication of enzymatic biofuelcells on contact lens [16,17] or patches [18,19] for power-ing portable devices [20] , bio batteries [21–23] , microflu-idic prototypes [24–32] , and paper-based cells [33–35] .

The overall performance of enzymatic biofuel cells(and also of electrochemical biosensors) are the conse-quence of the interplay between several physical and bio-electrochemical phenomena, mainly: species transport,enzymatic reactions and heterogeneous electron transferprocesses between the electrode and the enzyme (directelectron transfer, DET) or, more frequently, a mediator(mediated electron transfer, MET). This review will coverimportant contributions to the modeling and simulation

Current Opinion in Electrochemistry 2017, 1 :121–132

122 Fundamental and Theoretical Electrochemistry 2017

Figure 1

Schematic of an enzymatic biofuel cell. In general the enzyme and the mediator are immobilized on the electrode surface through polymeric/gel films and nanomaterials.

oa

T

cd

c

T

i

FbTa

S

T

ee

(

E

E

a

E

E

M

w

ac

mP

a(M

p(

I

eo

f

t

s

cd

b

tpe

[

C

f the above processes, which enables us to understand

nd optimize the performance of enzymatic electrodes.he focus will be on the mathematical resolution of the

orresponding nonlinear reaction–diffusion problems that escribe the response of this kind of electrodes with spe-ial emphasis on the contributions in the last 3 years (seeable 1 ); earlier theoretical works in the field can be found

n [ 36

•-39 ].

undamentals of the modeling of enzymatic

iofuel cells

he overall reaction scheme of an enzymatic half-cell re- ction can be represented as follows:

enzyme −−−−→ P

+ n H

+ + n

e

− (1)

he mechanism of this reaction depends on the type oflectron transfer mechanism between the enzyme and the

lectrode. In the case of DET, the mechanism of reaction 1 ) can be written as [39] :

ox + S

K ES ←→ ES

k cat −→ E red + P

(2)

red → E ox + n H

+ + n

e

− (3)

nd in the case of MET as [39–41] :

ox + S

K ES ←→ ES

k cat −→ E red + P

(4)

red + M ox K EM ←→ EM

k m −→ E ox + M red (5)

urrent Opinion in Electrochemistry 2017, 1 :121–132

red → M ox + n H

+ + n

e

− (6)

here E ox , E red , ES, M ox , M red and EM are the oxidizednd reduced forms of the enzyme, the enzyme–substrate

omplex, the oxidized and reduced forms of the mediatorolecule and the enzyme–mediator complex, while S and

are the substrate (reactant) and the product of the over-ll enzymatic process, respectively. The DET mechanism

Equations (2) and ( 3 )) has the form of pure Michaelis–enten kinetics, while the MET mechanism is an exam-

le of the so-called two-substrate ping-pong mechanism

Equations (4) –( 6 )).

n the case of enzymatic electrodes, one deals with a het-rogeneous process which means that the concentration

f the species that are taking part in reactions ( 2 )–( 6 ) is aunction of both time and space. In general for the specieshat can freely diffuse in the biocatalyst layer (e.g., sub-trate and mediator molecules), the change of their con-entration profile with time considering transport only by

iffusion can be described by Fick’s second law modifiedy the corresponding enzymatic kinetic terms. Thus, forhe frequent ’ping-pong’ mechanism, the following cou- led system of reaction–diffusion nonlinear partial differ- ntial equations is to be solved within the enzymatic layer42,43] :

∂ c m

∂t = D m

∂ 2 c m

∂ x 2 − k cat K

−1 m

c e c m

c s c m

K

−1 ( c s + K s ) + c s (7)

m

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Mathem

atical m

od

eling o

f no

nlinear reactio

n–diffusio

n p

rocesses

in enzym

atic b

iofuel

cells R

ajendran,

Kirthiga

and Lab

orda

123

Table 1

Recent contributions to the theoretical modeling of the signal of enzymatic electrodes/biosensors ( ∗∗).

Publication

Experimental techniques Enzymatic scheme

Modeling method

(HPM, HAM, ADM, etc.) Other aspects

Author(s) Reference

Analytical solutions

L. Rajendran et al. Electrochimica Acta 147 (2014) 678–687

Chronoamperometry, Normal pulse voltammetry, Steady state voltammetry and

limiting current

S + E ox ↔ ES → P + E red

M ox + E red → M red + E ox

M red ↔ M ox + n e −

New approach to

homotopy perturbation

method (NHPM)

• with and without redox polymer/hydrogel film

M. Rasi et al. Journal of The Electrochemical Society 162 (2015) H671–H680

Chronoamperometry, Normal pulse voltammetry, Steady state voltammetry and

limiting current

S + E red ↔ ES → P + E ox

M red + E ox ↔ EM → M ox + E red

M ox + n e − ↔ M red

Homotopy perturbation

method (HPM) • Gas-diffusion and

flow-through electrodes (under steady state)

M. Rasi et al. Sensors and

Actuators B 208 (2015) 128–136

Chronoamperometry, Normal pulse voltammetry, Steady state voltammetry and

limiting current

S + E red ↔ ES → P + E ox

M red + E ox ↔ EM → M ox + E red

M ox + e − ↔ M red

New approach to

homotopy perturbation

method (NHPM)

• no redox polymer/hydrogel film

K. Saravanakumar et al. Chemical Physics Letters 621 (2015) 117–123

Steady state voltammetry and

limiting current

S + E red ↔ ES → P + E ox

M red + E ox ↔ EM → M ox + E red

M ox + e − ↔ M red

Homotopy perturbation method (HPM)

R. Malini Devi et al. Applied

Mathematics 6 (2015) 1148–1160

Limiting current chronoamperometry, Steady state limiting current

S + E ↔ ES → P + E P ± e − → P ′

Homotopy perturbation method (HPM)

O. M. Kirthiga et al. Journal of Electroanalytical Chemistry 751 (2015) 119–127

Homotopy analysis method (HAM)

K. Saravanakumar et al. Fuel Cells 15 (2015) 523–536

Chronoamperometry, Normal pulse voltammetry, Steady state voltammetry and

limiting current

S + E ox ↔ ES → P + E red

2 M ox + E red → 2 M red + E ox

M red ↔ M ox + e −

New approach to homotopy perturbation method

(NHPM)

( continued on next page )

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Fundam

ental and

Theo

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Table 1 ( continued )

Publication Experimental techniques

Enzymatic scheme Modeling method

(HPM, HAM, ADM, etc.) Other aspects

Author(s) Reference

SP. Ganesan et al . International Journal of Computational Mathematics (2014) http: //dx.doi.org/10. 1155/2014/694037

Steady state limiting current

S + E 1 ↔ E 1 S → P 1 + P 2 + E 1 I + E 1 → E 1 I P 1 + E 2 , ox ↔ E 2 P 1 → P 3 + E 2 , red

M ox + E 2 , red ↔ E 2 M → M red + E 2 , ox

M red ↔ M ox + e −

Adomian decomposition

method (ADM) • multienzyme inhibitor system

K. Saravanakumar et al . Applied

Mathematical Modelling 39 (2015) 7351–7363

Steady state limiting current

S + E ox ↔ ES → P + E red

M ox + E red → M red + E ox

M red polymer −−−→ M ox + e −

M red electrode −−−−→ M ox +e

Homotopy analysis method (HAM)

• conducting polymer film

R. Muthuramalingam

et al . Russian Journal of Electrochemistry 52 (2016) 143–153

Steady state voltammetry and

limiting current

S + E ox ↔ ES → P + E red

2 M ox + E red → 2 M red + E ox

M red ↔ M ox + e −

Adomian decomposition

method (ADM) • porous electrode

J. Saranya et al. Chemical Physics Letter (2016, in

press)

Steady state limiting current

New approach to homotopy perturbation method

(NHPM)

Numerical methods Multilayer electrodes

D. Simelevicius et al . Sensors 14 (2014) 2578–2594

Limiting current chronoamperometry, Steady state limiting current

S + E ox → P + E red

M ox + E red → M red + E ox

M red + O 2 → M ox + P ′

M red → M ox + n e −

Implicit finite difference method

• multilayer enzymatic electrode • O 2 influence on biosensor operation

D. Simelevicius et al . SIMUL 2014: The Sixth International Conference on

Advances in System

Simulation

Chronoamperometry, Normal pulse voltammetry, Steady state voltammetry and

limiting current

S + E ox → P + E red

M ox + E red → M red + E ox

M red ↔ M ox + n e −

Implicit finite difference method

• multilayer enzymatic electrode •heterogeneous mediator ET of any reversibility (Butler–Volmer)

( continued on next page )

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n p

rocesses

in enzym

atic b

iofuel

cells R

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Kirthiga

and Lab

orda

125

Table 1 ( continued )

V. Aseris et al . Electrochimica Acta 146 (2014) 752–758

Limiting current chronoamperometry, Steady state limiting current

S + E ox → P + E red

2 M ox + E red → 2 M red + E ox

2 M

′ ox + E red → 2 M

′ red + E ox

M ox + M

′ red ↔ M red + M

′ ox

M red → M ox + n e −M

′ red → M

′ ox + n ′ e −

Explicit finite difference method

• bilayer enzymatic electrode • synergistic conversion scheme

F. Achi et al . Sensors and

Actuators B 207 (2015) 413–423

Limiting current chronoamperometry, Steady state limiting current

S + E 1 ↔ E 1 S → S ′ + E 1 I + E 1 → E 1 I S ′ + E 2 ↔ E 2 S ′ → P + M red + E 2 M red → M ox + n e −

Implicit finite difference method

• bilayer enzymatic electrode • multienzyme inhibitor system

F. Garay Sensors and

Actuators B 207 (2015) 581–587

Limiting current chronoamperometry, Steady state limiting current

M ox + E red ↔ E ox + M red

E ox ↔ E int + P ′ S + E ox ↔ ES ↔ P + E ox

S + E int ↔ ES ↔ P + P ′ + E red

P + 2 e − → S

Explicit finite difference method

• sandwich-type biosensor

R. A. Croce Jr. et al . International Journal of High

Speed Electronics and Systems 24 (2015) 1550012

Limiting current chronoamperometry, Steady state limiting current

S + E ox ↔ ES → P + E red

M ox + E red ↔ EM → M red + E ox

M red → M ox + 2 e −

Explicit finite difference method

• multilayer enzymatic electrode

V. Ašeris et al . Computational and

Applied

Mathematics 35 (2016) 405–421

Limiting current chronoamperometry, Steady state limiting current

S + E ox ↔ ES → P + E red

M ox + E red → M red + E ox

M red → M ox + e −

Implicit finite difference method

• bilayer enzymatic electrode

( continued on next page )

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ental and

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retical E

lectrochem

istry 2017

Table 1 ( continued )

Publication Experimental techniques

Enzymatic scheme Modeling method

(HPM, HAM, ADM, etc.) Other aspects

Author(s) Reference

Porous electrodes

T.Q.N. Do et al . Electrochimica Acta 137 (2014) 616–626

Steady state voltammetry and

limiting current

S + E red ↔ ES → P + E ox

E ox + e − + H

+ ↔ E int

E int + e − + H

+ ↔ E red

Numerical • porous electrode •direct electron transfer following a Butler–Volmer kinetics

T.Q.N. Do et al . Bioelectrochemistry 106 (2015) 3–13

Steady state voltammetry and

limiting current

S + E ox → P + E red

2 M ox + E red → 2 M red + 2 H

+ + E ox

M red ↔ M ox + e −

Numerical • porous electrode •heterogeneous mediator ET of any reversibility (Butler-Volmer)

2D model

D. Britz et al . Electrochimica Acta 152 (2015) 302–307

Limiting current chronoamperometry, Steady state limiting current

S + E ↔ ES → P + E P ± e − → P ′ Implicit finite difference

method Finite element method (COMSOL Multiphysics ®)

• disk microelectrode (2D

model)

( ∗∗) Unless otherwise indicated, a single film entrapping a single enzyme and a single mediator are considered and the heterogeneous electron transfer at the electrode is assumed to be reversible or taking place under limiting current conditions. S = substrate; P = product; E ox = enzyme in the oxidized form; E red = enzyme in the reduced form; E int = enzyme in a partly reduced form; M ox = mediator in the oxidized form; M red = mediator in the reduced form; I = enzyme inhibitor.

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Mathematical modeling of nonlinear reaction–diffusion processes in enzymatic biofuel cells Rajendran, Kirthiga and Laborda 127

∂ c s ∂t

= D s ∂ 2 c s ∂ x 2

− k cat K

−1 m

c e c m

c s c m

K

−1 m

( c s + K s ) + c s (8)

where D i is the diffusion coefficient of the free diffusivespecies i , K m

and K s are apparent Michaelis constants, andc e , c m

and c s are the concentration of enzyme, oxidizedmediator and substrate, respectively.

Redox polymer-mediated systems have also been mod-eled from the above differential equations but taking intoaccount that D m

in Equation (7) corresponds to an ap-parent electron diffusion coefficient that reflects chargepropagation by electron-hopping across the redox poly-mer layer. Andrieux and Savéant [44–46] and more re-cently Costentin and Savéant [ 47

•–49 ] have thoroughlyinvestigated these systems and related limiting processesto the concentration profiles internal and external tothe film. Bartlett et al . extended the analysis by consid-ering the Michaelis–Menten enzyme kinetics, present-ing a one-dimensional diffusion-only model of the cat-alytic film with steady state material balances of the sub-strate and mediator that has been extensively used forinterpreting the kinetics of immobilized and diffusiveenzyme–mediator systems [39,42] . Using some of the lim-iting cases from Bartlett and Pratt [42] , Calvo and cowork-ers obtained the kinetic parameters for several layer-by-layer self-assembled ultrathin films of osmium andferrocene-mediated glucose oxidase (GOx) electrodes[50–53] .

Complete enzymatic biofuel cells

Only few records can be found in the literature regardingthe modeling of complete enzymatic fuel cell systems. The-oretical works on specific bioelectrodes are more abun-dant (in Enzymatic electrodes section), either from theperspective of representing a main performance-limitingfactor of the cell [54

•] or from their use as enzymatic elec-trochemical biosensors [39, 40] .

Different theoretical approaches to enzymatic fuel cellmodeling have been followed, which are based on ei-ther conventional mass and charge balances combinedwith enzymatic reactions and mass transport [55–59] ,metabolic control analysis [60

•] or statistical analysis [61] .Polarization and power curves (and fuel utilization [62] )are predicted under different operation conditions asinfluenced by the rate of proton generation and con-sumption [55] , the morphology of the electrodes [56] ,mass transport limitations of the mediator [57

•] and oxy-gen [59] as well as the electrolyte composition [61] .An interesting approach was presented by Glykys et al .[60

•] where a common enzymatic fuel cell based on anosmium-mediated glucose oxidase anode and a laccasecathode was investigated. Saranya and Banta [68] havevery recently developed a mathematical model of an en-zymatic glucose membraneless fuel cell with DET. Thetheoretical results can be used in the optimization of such

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fuel cell to attain the highest power value and also in theestimation of kinetic parameters.

Enzymatic electrodes

Analytical and numerical mathematical methods (seeTable 1 ) have been employed to solve the reaction–diffusion problems associated with enzymatic electrodes[ 36

•–39 ] and to predict their electrochemical response,typically in amperometric mode. Whereas the formermethods lead to the deduction of analytical expressionsfor more direct, general and rapid study of the system’sperformance, the latter have enabled researchers to covermore complex situations in terms of the kinetic schemeand/or the electrode structure and morphology.

Analytical solutions

Rasi and coauthors [63

•] developed a theoretical analyti-cal model to describe the transient response of the elec-troreduction of oxygen to water catalyzed by enzyme lac-case via a ping-pong kinetic scheme. The coupled time-dependent nonlinear partial differential equations weresolved analytically using a modified homotopy perturba-tion method (HPM) in conjunction with the complex in-version formula in the Laplace plane. The time requiredto reach the steady state current was explicitly reported asa useful parameter to estimate the power density of bio-fuel cells. Devil et al . [64] followed the same approach toderive analytical expressions for the response of an en-zymatic electrode following a Michaelis–Menten type ki-netics.

The HPM method was also employed by Saravanaku-mar and coworkers [65] to derive analytical expressionsfor (i) the steady state concentrations of the redox medi-ator and oxygen substrate (as a function of the film thick-ness) and (ii) for the steady state current density (in termsof the electrode potential and the enzyme characteristics)corresponding to the oxygen reduction in biofuel cellsvia a ping-pong kinetic scheme. Furthermore, two differ-ent graphical procedures were suggested for estimatingthe Michaelis–Menten constants and global rate constantsfrom the experimental current–potential response.

Saravanakumar et al . [66] derived analytical expressionsfor the concentration of the glucose substrate and oxi-dized mediator and of the current density. The expres-sions were obtained using a new approach to HPM andcomplex inversion formula. The influence of several pa-rameters of the system were considered: mediator diffu-sivity, enzyme film thickness, enzyme kinetics and elec-trode potential. Moreover, a graphical procedure was pro-posed for the quantitative estimation of the Michaelis–Menten constants and the turnover rate.

By making use of a modified Adomian decompositionmethod (ADM), Ganesan et al . [67] obtained analytical ex-pressions for the steady state concentration profiles and

Current Opinion in Electrochemistry 2017, 1 :121–132

128 Fundamental and Theoretical Electrochemistry 2017

lipt

NGtbd

T

t

s

tlb

t

p

t

epcca

di

b

ttt

E(wteh

T

dcwc

bemc

taa

PM

ai

em

SM

rl

P

.

ps

i

s

f

t

e

t

I

o

om

oape

o

u

t

ALD(PIaFMCC

AareAT

pt

i

m

i

m

e

i

T

r

fe

a

C

imiting current of a complex enzymatic scheme involv- ng two different enzymes and a competitive inhibition

rocess. The basic concepts of the homotopy perturba- ion and ADM are given in the Appendix.

umerical simulations

alaway et al . used a one-dimensional numerical model o obtain kinetic information of O 2 reduction on laccase- ased enzymatic electrodes modified by different Os re- ox hydrogels from cyclic voltammetry experiments [43] .hey found out that the rate constant for the reaction be-

ween laccase and oxygen was slightly lower than in freeolution. It was also found out that the rate constant forhe reaction between the mediator and laccase depends inearly on the mediator potential in the potential range

etween ca. 0.5 and 0.8 V, while in the potential range be-ween ca. 0.1 and 0.5 V it was independent of the mediatorotential. Tamaki et al . [41] modeled the glucose oxida-ion on high surface area carbon black electrode in pres-nce of glucose oxidase (GOD) encapsulated in a redox

olymer film. They assumed different thicknesses of the

atalyst layer and, by changing the diffusion coefficient of harge through the redox polymer, they obtained that the

pparent electron diffusion in the polymer is not the rateetermining step of the overall electrode kinetics. Accord-

ng to their study, current densities up to 0.1 A cm

−2 coulde obtained by using high surface area carbon black elec-rodes modified with a redox polymer and by increasing

he enzyme loading in the catalyst layer and the enzyme

urnover rate [41] .

nzymatic electrodes with multilayered coatings sandwich-type bioelectrodes) have been designed

ith different purposes, such as better preservation of he enzyme loading and production of more stable bio- lectrodes. The theoretical modeling of these systems as been mainly tackled by finite difference methods.hus, Ašeris et al . [69

•] studied the response of a glucoseehydrogenase-based biosensor following a synergistic onversion scheme including an enzyme layer together ith a dialysis membrane. A similar bilayer structure was

onsidered by Achi et al . [70

•] to simulate an inhibition-ased biosensor for mercury ions involving two different nzymes: invertase in solution and glucose oxidase im- obilized on the electrode surface. The influence of the

onfiguration of the biosensor in terms of the thickness ofhe enzymatic membrane and the diffusion layer on the

mperometric response was investigated theoretically

nd validated with experimental data.

orous enzymatic electrodes following DET [71

•] and

ET [72] mechanisms have also been simulated by Do etl . The theoretical results, along with experimental stud- es, point out the key influence of the morphology of thelectrode, which is frequently disregarded in theoretical

odels. s

urrent Opinion in Electrochemistry 2017, 1 :121–132

ummary

ost mathematical models of biofuel cells are based oneaction–diffusion differential equations containing non- inear terms related to the kinetics of the enzyme reaction.owerful and accurate analytical (HPM, HAM, ADM,

..) and numerical mathematical methods have been em-loyed for their resolution under steady and non-steady

tate conditions. The results provide very useful insightnto the effects on the performance of the thickness andtructure of the enzymatic film, the loading of the dif-erent species, the diffusivity of the mediator, etc. Also,he theoretical modeling and simulation of these systemsnable us to characterize the enzymatic reactions (i.e.,urnover rate and Michaelis–Menten constant).

n spite of the above benefits, there are only limited the-retical studies addressing enzymatic fuel cells and mostf them include a number of simplifying assumptions ainly related to the mass and charge transport inside and

utside the biocatalyst film, the enzymatic kinetic scheme

nd the electrode morphology. Experimental validation of roposed models is even more seldom. Therefore, more

ffort in the future research is needed in this direction inrder to develop more detailed models and accurate sim-lations that can assist the rational development and op-imization of biofuel cells.

cknowledgments

R and MK gratefully acknowledge the financial support from the epartment of Science and Technology , New Delhi, Government of India

SB/SI/PC-50/2012 ). In addition, we wish to thank the College Chairman, rincipal and Head of the department, Department of Mathematics, Sethu nstitute of Technology, Pulloor, Kariapatti-626115 for their encouragement nd support. EL greatly appreciates the financial support provided by the undación Séneca - Agencia de Ciencia y Tecnología de la Región de urcia (Project 18968/JLI/13 ) as well as by the Ministerio de Economía y ompetitividad of the Spanish Government (fellowship “Juan de la ierva-Incorporación 2015 ”).

ppendix. Basic concepts of various

nalytical schemes for solving nonlinear eaction–diffusion equations for modeling

nzymatic biofuel cells

. Homotopy perturbation method (HPM) he HPM is one of the important methods to find the ap-roximate solutions for nonlinear partial differential equa- ions in mathematical physics. The HPM, which was orig-nally proposed by He [73–74] in 1999, has been proved by

any authors to be a powerful mathematical tool for solv-ng various kinds of linear and nonlinear problems. This

ethod introduces an efficient approach for a wide vari-ty of scientific and engineering applications. The HPMs unique in its applicability, accuracy, and efficacy [75] .he HPM uses the imbedding parameter p as a small pa-

ameter, and only a few iterations are needed to searchor an asymptotic solution [73] . Recently Filobello-Nino

t al. [76] applied the Laplace transform–HPM with vari-ble coefficients, in order to find analytical approximate

olutions for nonlinear differential equations with variable

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Mathematical modeling of nonlinear reaction–diffusion processes in enzymatic biofuel cells Rajendran, Kirthiga and Laborda 129

coefficients in oxygen diffusion problems. The HPM hasovercome the limitation of traditional perturbation meth-ods. It can take full advantage of the traditional perturba-tion, so a considerable amount of research has been con-ducted to apply the Homotopy technique to solve variousstrong nonlinear equations [36

•] .

To illustrate the basic ideas of this method, we considerthe following nonlinear differential equation

A (u ) − f (r ) = 0 , r ∈ δ. (A.1)

with the following boundary conditions

B

(u,

∂u

∂n

)= 0 , r ∈ τ. (A.2)

where A is a general differential operator, B a boundaryoperator, f ( r ) is a known analytical function and τ is theboundary of the domain δ. The operator A can be decom-posed into two operators L and N , where L is a linear, andN a nonlinear operator.

Equation (A.1) can be written as follows:

L (u ) + N (u ) − f (r ) = 0 (A.3)

By using the homotopy technique, we construct a homo-topy:

v (r, p) : δ × [0 , 1] → R, (A.4)

which satisfies:

H(v, p) = (1 − p) [ L (v ) − L ( u 0 ) ] + p [ A (v ) − f (r ) ] = 0 ,

p ∈ [ 0 , 1 ] , r ∈ δ (A.5)

or

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

(A.6)

where p ∈ [0, 1] is an embedding parameter and u 0

is an initial approximation for the solution of Equation(A.1) that satisfies the boundary conditions. Obviously,from Equations (A.5) and (A.6) it holds that:

H(v, 0) = L (v ) − L ( u 0 ) = 0 , (A.7)

H(v, 1) = A (v ) − f (r ) = 0 . (A.8)

The changing process of p from zero to unity is just thatof v ( r , p ) from u 0 ( r ) to u ( r ). In topology, this is called homo-topy. According to the HPM, we can first use the embed-ding parameter p as a small parameter, and assume that

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the solution of Equations (A.5) and (A.6) can be writtenas a power series in p :

v = v 0 + p v 1 + p

2 v 2 + ... (A.9)

Setting p = 1 gives the solution of Equation (A.1)

u = lim

p→ 1 v = v 0 + v 1 + v 2 + ... (A.10)

Series (A.10) is convergent for most cases, some conver-gence criteria being suggested in [74] .

B. Adomian decomposition method (ADM) In recent years, much attention has been devoted to theapplication of the Adomian decomposition method tothe solution of various scientific models [77] . It provideswithout linearization, perturbation, transformation or dis-cretization, the analyst with an easily computable, readilyverifiable and rapidly convergent sequence of analyticalapproximate functions for the solution. A key notion isthe Adomian polynomials, which are tailored to the par-ticular nonlinearity to solve nonlinear operator equations[78] . The principle of the ADM when applied to a generalnonlinear equation is given as follows.

Consider the nonlinear differential equation.

y ′′ + N (y ) = g(x ) (B.1)

with the following boundary conditions

y (0) = A, y (b) = B (B.2)

Where N ( y ) is a nonlinear function, g ( x ) is the given func-tion and A , B , b are constants. The following new differ-ential operator is proposed:

L =

d

2

d x 2 (B.3)

such that Equation (B.1) can be written as

L (y ) = g(x ) − N (y ) (B.4)

The inverse operator L

−1 is therefore considered as a two-fold integral operator as follows

L

−1 (. ) =

∫ x

0

∫ x

b (. ) d x d x (B.5)

Applying the inverse operator L

−1 on both sides ofEquation (B.4) it yields

y (x ) = L

−1 (g(x )) − L

−1 (N (y )) + y ′ (b)(x − 0) + y (0) (B.6)

Current Opinion in Electrochemistry 2017, 1 :121–132

130 Fundamental and Theoretical Electrochemistry 2017

t(

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T

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hat upon using the boundary conditions Equation

B.2) becomes

(x ) = L

−1 (g(x )) − L

−1 (N (y )) + Bx + A (B.7)

he ADM introduces the solution y ( x ) and the nonlinearunction N(y) by infinite series

(x ) =

∞ ∑

n =0

y n (x ) (B.8)

nd

(y ) =

∞ ∑

n =0

A n (B.9)

here the components y n ( x ) of the solution y ( x ) will beetermined recurrently and the Adomian polynomials A n f N(y) are evaluated using the formula

n (x ) =

1

n !

d

n

d λn N

( ∞ ∑

n =0

λn y n

)

λ=0

(B.10)

hich gives

0 = N ( y 0 ) , 1 = N

′ ( y 0 ) y 1 ,

2 = N

′ ( y 0 ) y 2 +

1

2

N

′′ ( y 0 ) y 2 1 ,

3 = N

′ ( y 0 ) y 3 + N

′′ ( y 0 ) y 1 y 2 +

1

3! N

′′′ ( y 0 ) y 3 1 ,

(B.11)

ubstitution of Equations (B.8) and (B.9) in Equation

B.7) gives

∞

n =0

y n = L

−1 (g(x )) − L

−1

( ∞ ∑

n =0

A n

)

+ Ax + B (B.12)

nd, by equating the terms in the linear system ofquation (B.11) , the following recurrent relationship is btained:

0 = L

−1 (g(x )) + Bx + A, y n +1 = L

−1 ( A n ) , n ≥ 0 (B.13)

hich leads to:

0 = L

−1 (g(x )) + Ax + B,

1 = −L

−1 ( A 0 ) ,

2 = −L

−1 ( A 1 ) ,

3 = −L

−1 ( A 2 ) ,

(B.14)

urrent Opinion in Electrochemistry 2017, 1 :121–132

rom Equations (B.11) and (B.14) , the components y n ( x )an be determined and hence the series solution of y n ( x )n Equation (B.7) can be immediately obtained [79] .

eferences and recommended reading

apers of particular interest, published within the annual period of eview, have been highlighted as:

• of special interest

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his is a comprehensive yet concise review of milestones in the evelopment of enzymatic fuel cells and the main current research lines nd challenges in the field.

. Yakovleva ME , Gonaus C , Schropp K , ÓConghaile P , Leech D , Peterbauer CK , Gorton L : Engineering of pyranose dehydrogenase for application to enzymatic anodes in biofuel cells . Phys Chem Chem Phys 2015, 17 :9074–9081 .

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1. Zhou M , Zhou N , Kuralay F , Windmiller JR , Parkhomovsky S , Valdés-Ramírez G , Katz E , Wang J : A self-powered “sense-act-treat” system that is based on a biofuel cell and controlled by Boolean logic . Angew Chem Int Ed 2012, 51 :2686–2689 .

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16. Falk M , Andoralov V , Silow M , Toscano MD , Shleev S : Miniature biofuel cell as a potential power source for glucose-sensing contact lenses . Anal Chem 2013, 85 :6342–6348 .

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32. Zebda A , Renaud L , Cretin M , Innocent C , Ferrigno R , Tingry S : Membraneless microchannel glucose biofuel cell with improved electrical performances . Sens Actuators B 2010, 149 :44–50 .

33. Narváez Villarrubia CW , Lau C , Ciniciato GPMK , Garcia SO , Sibbett SS , Petsev DN , Babanova S , Gupta G , Atanassov P : Practical electricity generation from a paper based biofuel cell powered by glucose in ubiquitous liquids . Electrochem

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36. •

Rajendran L : Chemical sensors: simulation and modeling . In: Korotcenkov G. Electrochemical Sensors, vol. 5. New York: Momentum Press; 2013:339–398 .

A revision of the fundamentals of the modelling of enzymatic biosensorsof different geometries is included together with an introduction to the most relevant analytical mathematical methods highlighted in this review, that is: Variational Iteration Method, Homotopy Perturbation the Method, Homotopy Analysis Method and Adomian Decomposition Method.

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42. Bartlett PN , Pratt KFE : Theoretical treatment of diffusion and kinetics in amperometric immobilized enzyme electrodes. Part 1: redox mediator entrapped within the film . J Electroanal Chem

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43. Gallaway JW , Barton SAC : Kinetics of redox polymer-mediated enzyme electrodes . J Am Chem Soc 2008, 130 :8527–8536 .

44. Andrieux CP , Dumasbouchiat JM , Savéant JM : Catalysis of electrochemical reactions at redox polymer electrodes: kinetic model for stationary voltammetric techniques . J Electroanal Chem 1982, 131 :1–35 .

45. Andrieux CP , Savéant JM : Kinetics of electrochemical reactions mediated by redox polymer films—irreversible cross-exchangereactions: formulation in terms of characteristic currents for stationary techniques . J Electroanal Chem 1982, 134 :163–166 .

46. Andrieux CP , Savéant JM : Catalysis at redox polymer coated electrodes . In: Murray RW. Molecular Design of Electrode Surfaces. New York: Wiley; 1992:207–270 .

47. •

Costentin C , Savéant JM : Cyclic voltammetry analysis of electrocatalytic films . J Phys Chem C 2015, 119 :12174–12182 .

A systematic and comprehensive analysis of the voltammetric behaviour of catalytic films is presented, establishing diagnosis criteria for different mechanistic scenarios and for understanding the performance of catalytic films.

48. Costentin C , Savéant JM : Cyclic voltammetry of electrocatalytic films: fast catalysis regimes . ChemElectroChem 2015, 2 :1774–1784 .

49. Costentin C , Savéant JM : Cyclic voltammetry of fast conductingelectrocatalytic films . Phys Chem Chem Phys 2015, 17 :19350–19359 .

50. Calvo EJ , Danilowicz CB , Wolosiuk A : Supramolecular multilayer structures of wired redox enzyme electrodes . Phys Chem ChemPhys 2005, 7 :1800–1806 .

51. Calvo EJ , Battaglini F , Danilowicz C , Wolosiuk A , Otero M : Layer-by-layer electrostatic deposition of biomolecules on surfaces for molecular recognition, redox mediation and signalgeneration . Farad Discuss 2000, 116 :47–65 .

52. Calvo EJ , Wolosiuk A : Wiring enzymes in nanostructures built with electrostatically self-assembled thin films . ChemPhysChem 2005, 6 :43–47 .

53. Calvo EJ , Etchenique R , Danilowicz C , Diaz L : Electrical communication between electrodes and enzymes mediated by redox hydrogels . Anal Chem 1996, 68 :4186–4193 .

Current Opinion in Electrochemistry 2017, 1 :121–132

132 Fundamental and Theoretical Electrochemistry 2017

5•Apoc

5

5

5•

Tc

5

5

6•Mordp

6

6

6•

Toduatic

6

6

6

6

6

6•

Tbtm

7•

Ttd

7•

To

i

7

7

7

7

7

7

7

7

C

4. Barton SC : Oxygen transport in composite mediated biocathodes . Electrochim Acta 2005, 50 :2145–2153 .

numerical simulation of an enzyme-catalyzed oxygen cathode is resented and applied to the analysis of transport limitations in perating electrodes, with the goal of predicting the limits of obtainable athode current density.

5. Rubin VZ : Current-voltage modeling of the enzymatic glucose fuel cells . Adv Chem Eng Sci 2015, 5 :164–172 .

6. Song Y , Penmatsa V , Wang C : Modeling and simulation of enzymatic biofuel cells with three-dimensional microelectrodes . Energies 2014, 7 :4694–4709 .

7. Osman MH , Shah AA , Wills RGA , Walsh FC : Mathematical modelling of an enzymatic fuel cell with an air-breathing cathode . Electrochim Acta 2013, 112 :386–393 .

he complex dynamic modelling of a complete enzymatic fuel cell is arried out including experimental validation.

8. Osman MH , Shah AA , Wills RGA : Detailed mathematical model of an enzymatic fuel cell . J Electrochem Soc 2013, 160 :F806–F814 .

9. Bedekar AS , Feng JJ , Krishnamoorthy S , Lim KG , Palmore GTR , Sundaram S : Oxygen limitation in microfluidic biofuel cells . Chem Eng Commun 2008, 195 :256–266 .

0. Glykys DJ , Banta S : Metabolic control analysis of an enzymatic biofuel cell . Biotechnol Bioeng 2009, 102 :1624–1635 .

etabolic control analysis is used to investigate a common smium-mediated glucose oxidase/laccase enzymatic biofuel cell. The esults of the analysis show that the control of the electron flux strongly epends on the total mediator concentrations and the extent of olarization of the individual electrodes.

1. Jeon SW , Lee JY , Lee JH , Kang SW , Park CH , Kim SW : Optimization of cell conditions for enzymatic fuel cell using statistical analysis . J Ind Eng Chem 2008, 14 :338–343 .

2. Kjeang E , Sinton D , Harrington DA : Strategic enzyme patterning for microfluidic biofuel cells . J Power Sources 2006, 158 :1–12 .

3. Rasi M , Rajendran L , Sangaranarayanan MV : Enzyme-catalyzed oxygen reduction reaction in biofuel cells: analytical expressions for chronoamperometric current densities . J Electrochem Soc 2015, 162 :H671–H680 .

he transient current-potential response of the enzyme-catalyzed xygen reduction reaction in biofuel cells is presented. The one imensional nonlinear reaction-diffusion equation is solved analytically sing the homotopy method for deriving the substrate concentrations nd current densities pertaining to chronoamperometric response. The ime required to obtain the steady state has been obtained and the nfluence of film thickness, diffusion coefficients, and enzyme haracteristics has been deciphered.

4. Devi1 RM , Kirthiga OM , Rajendran L : Analytical expression for the concentration of substrate and product in immobilized enzyme system in biofuel/biosensor . App Math 2015, 6 :1148–1160 .

5. Saravanakumar K , Rajendran L , Sangaranarayanan MV : Current–potential response and concentration profiles of redox polymer-mediated enzyme catalysis in biofuel cells: estimation of Michaelis–Menten constants . Chem Phys Lett 2015, 621 :117–123 .

6. Saravanakumar K , Ganesan SP , Rajendran L : Theoretical analysis of reaction and diffusion processes in a biofuel cell electrode . Fuel Cells 2016, 15 :523–536 .

urrent Opinion in Electrochemistry 2017, 1 :121–132

7. Ganesan SP , Saravanakumar K , Rajendran L : Mathematical modeling of multienzyme biosensor system . Int J Comp Math , vol 2014 2014 694037 .

8. Saranya J , Rajendran L , Wang L , Fernandez C : A new

mathematical modelling using homotopy perturbation method to solve nonlinear equations in enzymatic glucose fuel cells . Chem Phys Lett 2016, 662 :317–326 .

9. Ašeris V , Gaidamauskait e E , Kulys J , Baronas R : Modelling glucose dehydrogenase-based amperometric biosensor utilizing synergistic substrates conversion . Electrochim Acta 2014, 146 :752–758 .

he amperometric response of a glucose dehydrogenase-based iosensor is successfully simulated (at least qualitatively) confirming hat the biosensor follows a synergistic substrate conversion echanism.

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he mathematical modelling of enzyme inhibition is tackled by studying he effect of mercury ions on the response of a biosensor for glucose etection using an implicit finite difference method.

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n species mass transport and enzyme utilization.

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