Applied Mathematics, 2013, 4, 1603-1608
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412217
Open Access AM
Analytical Solution of Substrate Concentration in the
Biosensor Response
Seyed Ali Madani Tonekaboni1, Ali Shahbazi Mastan Abad2, Amin Afshari2, Ali Khalilzadeh2,
Shahab Karimi2, Mitra Shabanisamghabady2
1School of Mechanical Engineering, University of Waterloo, Waterloo, Canada
2School of Mechanical Engineering, University of Tehran, Tehran, Iran
[e-mail], [e-mail], [e-mail]
Received September 8, 2013; revised October 8, 2013; accepted October 15, 2013
Copyright © 2013 Seyed Ali Madani Tonekaboni et al. This is an open access article distributed under the Creative Commons Attri-
bution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
ABSTRACT
Homotopy analysis method (HAM) is employed to investigate amperometric biosensor at mixed enzyme kinetics and
diffusion limitatio n. Math ematical modeling of th e problem is d ev elop ed utilizing non -Michaelis-Menten k inetics of the
enzymatic reaction. Different results of the problem are obtained for different values of the dimensionless parameters.
Accuracy of the obtain ed results is verified by comparing them with the available actual and simulated ones. It is con-
cluded that the obtained solution can be considered as a promising one to investigate different aspects of the phenom-
ena.
Keywords: Homotopy Analysis Method; Amperometric Biosensor; Mathematical Modeling; Non-Michaelis-Menten
Kinetics
1. Introduction
Biosensors play important roles as components of the
transduction mechanisms [1] and can be employed as
measurement devices to gauge biologically relevant in-
formation such as neural interfaces and oxygen elec-
trodes [2]. Furthermore, they can be used as transducers
which translate the biomolecular responses into electrical
signals [3]. Biosensors produce signals harmonized to the
concentrations of the measured analytes. These devices
are used in so many applications like detection of patho-
gens [4], toxic metabolites (such as mycotoxins [5]), de-
tection of pesticides and river water contaminants such as
heavy metal ions [6], etc. The mentioned examples show
the importance of biosensors and their applications in
different branches of Science and Engineering which
reveals the requirement of analysis of these highly de-
manded instruments. One of the popular and perspective
trends of biosensorics is amperometric biosensor [7].
Since they were first introduced by Clark and Lyons in
1962 [8], several studies have investigated different as-
pects of amperometric biosensors. In principle, they
measure the changes of the current of indicator electrode
by direct electrochemical oxidation or reduction of the
products of the biochemical reaction [9-11]. They are
widely used today because of the reliability and high
sensitivity for environment, clinical and industrial appli-
cations.
Design of biosensors is based on understanding the
kinetic characteristics of these devices. Generally, meas-
uring the concentration of substrate inside enzyme mem-
branes is not possible. Hence, various mathematical
models of amperometric biosensors have been presented
and used as an important tool in order to obtain analytical
characteristics of actual biosensors [12,13], such as in-
vestigative monolayer membrane model used to study the
biochemical treatment of biosensors [14,15]. Their ma-
thematical models are based on reaction-diffusion equa-
tions including non-linear term that relate to non-Micha-
elis-Mentenkinetics of the enzymatic reaction [16,17].
Hence, high accurate analytical and numerical methods
should be employed to investigate this important nonlin-
ear chemical equation.
Most scientific problems in engineering are inherently
nonlinear. Except for a few of them, the majority of
nonlinear problems does not have analytical solutions.

S. A. M. TONEKABONI ET AL.
1604
Therefore, the constitutiv e laws of these problems should
be solved using other schemes such as numerical or per-
turbation methods. In the numerical method, stability and
convergence of the solution should be considered so as to
avoid divergence or inappropriate results [18]. In the
perturbation method, the small parameter is inserted in
the equation; thus, finding the small parameter and ex-
erting it into the equation is one of the deficiencies of this
method [19]. One of the semi-exact methods for solving
nonlinear equation which does not need small/large pa-
rameters is Homotopy Analysis Method (HAM), first
proposed by Liao [20,21]. Homotopy Analysis Method is
now widely used to solve different types of nonlinear
problems. Various papers on nonlinear physical and en-
gineering problems [22,23] have proved the validity of
HAM. Moreover, recently the application of HAM on
medical and chemistry problems has gotten so much at-
tention among researchers. Counting reaction network
equilibria [24], reaction-diffusion Brusselator model [25],
predicting the lowest energy conformations of proteins
[26] and many other examples can be considered as ap-
plications of HAM in Chemistry and Medicine. Several
auxiliary parameters and functions available in the pro-
cedure of HAM need to be chosen so properly for the
convergence of the solution. Through practice of auxil-
iary parameter h, convergence region of the solution is
readily adjustable to a wide range of variables.
This paper presents the analytical solution for an am-
perometric biosensor at mixed enzyme kinetics and dif-
fusion limitation by utilizing HAM as a strong method.
Non-Michaelis-Menten kinetics of the enzymatic reac-
tion is used to obtain the constitutive equation of the
problem. Different non-dimensional parameters are de-
fined so as to non-dimensionalize the equation. The ob-
tained non-dimensional equation is used to procure mth-
order deformation equation as an important step of the
procedure of the solution. The h-curves are obtained for
several cases illustrated in the paper to clarify the con-
vergence region of the solution. In addition, results are
obtained to investigate the effects of the variations of
each dimensionless parameter of the procured equation.
Finally, some of the results are compared with the actual
and simulated results available in the literature [27] to
verify the accuracy of the method.
2. Mathematical Modeling
Spatial dependency of enzyme kinetics on biochemical
systems has recently attracted much attention by consid-
ering the effect of diffusion in these processes [16,17].
The simplest scheme of non Michaelis-Menten kinetics
may for instance be described by adding to the Micha-
elis-Menten scheme (2.1) the relations hip of the interact-
tion of the enzyme substrate complex 
with an-
other substrate molecule

ES


S
ES
(2.2) followed by the gen-
eration of non-active complex as

2
ES
EP

ES


S

2
ES
(2.1)
ES



(2.2)
The reaction is sometimes said to display Michaelis-
Menten kinetics in which the relationship between the
rate of an enzyme catalyzed reaction and the substrate
concentration takes the form


max
M
VS
K
S (2.3)
 
where

and max V are the so-called “initial reaction
velocity” and maximum velocity respectively.
In addition,
M
K
is known as Michaelis constant for
.
S
M
K
and max V are constants at a given temperature
and a given enzyme concentration.
The reactions exhibit non-Michaelis-Menten kinetics,
in which the kinetic behavior does not obey the Equation
(2.3). The velocity function

for the simple reaction
process without competitive inhibition is given by Pao
[28] and Baronas et al. [27], which is based on the non-
Michaelis-Menten hypothesis,








max
022
c
M
iM i
kE SVS
K
SSK SK
 
 K S

(2.4)
where the constants

max 0 c
Vk , E
M
K
and i
K
are
Michaelis-Menten and inhibition constants respectively.
The Equation (2.4) conforms to Equation (2.3) for large
values of i
K
with respect to
M
K
. On the basis of
Equation (2.4), the rate is maximized by increasing the
concentration. It is then said to be inhibited by the sub-
strate. In addition, the constant i
K
(which has the di-
mension of a concentration) is called the substrate inhibi-
tion constant. For obtaining the rate of change of sub-
strate concentration


t,SS

 at time t and position


 throughout the domain, the following equation
given by Pa o [ 28] is used.

,

S
SDS t
t




 (2.5)
S is the substrate diffusion coefficient and D S

is
the gradient operation. On the basis of non-Michaelis-
Menten kinetics, Equation (2. 5) bec omes
2
22
1
S
M
iM
SS S
D
tSK KK




K
S (2.6)
in which 0cM
K
KE K

. In this article, steady state
condition is considered which results in changing Equa-
tion (2.6) to the following equation
Open Access AM

S. A. M. TONEKABONI ET AL. 1605
2
22
0
1
S
MiM
SKS
DSKS KK


  (2.7)
Equation (2.7) is changed to the non-dimensional form
(Equation (2.8)) [27] using the following non-dimen-
sional para meter s
2
22
22
0, 0 1
1
,,, ,
SM
uKu u
xuu
SkLks
uxK
LDK KK ks

 





iM
ks
(2.8)
Equation (2.8) must be solved to satisfy the following
boundary conditions which are based on the location of
electrodes and diffusion layer in the boundaries of the
membarne
1 at 1
0 at 0
ux
ux
x



(2.9)
3. HAM Solution
In this section the solution procedure of this problem
using HAM is discussed. The appropriate form of non-
linear differential Equation (2.8) for the procedure of
HAM is presented as follows:
  

  
2
2
2
,1 ,,
,,0
Nxqxq xq
xqK xq
x


 



(3.1)
where is the node number, is the nonlinear op-
erator, and the function
iN


iq
 is defined as
 

0
0
1
lim ,,
lim ,
q
q
x
qux
x
qux





 (3.2)
where is the unknown field variable,

ux


0, 1q
is the embedding parameter, and is the initial
guess which is employed to meet the requirements of the
boundary conditions. In this paper, the

0
ux


0
ux 1

has
been chosen which correctly satisfies all the boundary
conditions stated in Equation (2.9).
So through the generalizing concept of HAM the so-
called zero-order deformation equation can be written as:
  
0
1,qLxquxqhHxNxq   

,  

(3.3)
where is the non-zero auxiliary parameter,
0h


H
x is the auxiliary function, and is the auxiliary
linear operator which is chosen here as L
 

2
2
d
d
f
x
fx x 
 (3.4)
with the following property:


12 12
0 when 0CC CCxx

 (3.5)
Expanding


,
x
q  in Taylor series with respect to
the embedding parameter
q
, one obtains

 


01
0
,
,
1
!
m
m
m
m
mm
q
x
qux uxq
xq
ux mq









(3.6)
With due attention to the procedure of HAM [21],


m
ux should be chosen so as the following equation is
satisfied

0
d10
d
m
m
X
uu
X

 (3.7)
If the series


,
x
q  converges at , then the se-
ries solution is 1q

 
01
,1 m
m
x
uxu x 



 (3.8)
where


m
ux could be obtained by the so-called high-
order deformation equation. For obtaining the mth-order
deformatio n equation, the following vector is defined as:






01
,,,
n
uxux uxun (3.9)
Differentiating both sides of the zero-order equation m
times with respect to and then setting , the so-
called mth-order deformation equation can be obtained as
q0q



 
11
,
mmm mm
LuxuxhHxRx 

 

u

(3.10)
where
0, 1
1, otherwise
m
m




 (3.11)



1
11
0
,
1
,1!
m
mm m
q
Nxq
Rx
mq






 u (3.12)
Therefore, the following relation is obtained

1
11
0
1
11
00
,m
mm mimi
i
mi
mi kikm
ik
Rxu uu
uuuKu



1







 



 


u
(3.13)
We are free to choose the auxiliary parameter , the
auxiliary function h


H
x, the initial guess


x
0, and
the auxiliary linear operator so that the validity and
flexibility of the HAM solution to control the conver-
gence region is proven. Due to the rule of solution ex-
pression [21] the auxiliary function is chosen as follows
u
L


1Hx (3.14)
According to the HAM, the valid region of the auxil-
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S. A. M. TONEKABONI ET AL.
1606
iary parameter h for convergen ce of the solutio n series is
the flat regions of h-curves.
4. Results and Discussion
To see the proper values of h, the h-curves are plotted for
different values of dimensionless parameters
, and
K

 in Figure 1 to obtain the valid results of
the considered conditio ns.
Figure 1. Variations of


ux
, 0β
versus non-dimensional pa-
ramete for (a) ; (b) , ; (c)
; and (d) .
x
, 
1.0 .1α
10.0α
0.1 1.0αβ
1.0β10.0 0.1αβ, 
The procedure of solving the non-dimensional equa-
tion of enzyme reaction (Equation (2.8)) which is based
on the non-Michaelis-Menten kinetics theory utilizing
HAM is described in Section 3. It is mentioned that
mth-order deformation equation should be employed to
solve the problem. As the first step of the solution, the
diagrams of variation of non-dimensional parameter


ux versus auxiliary parameter h for different investi-
gated cases are illustrated (Figure 1). Then, flat regions
of h-curves are obtained employing these diagrams.
On the basis of the chosen values of auxiliary parame-
ter h in the flat regions of h-curves (Figure 1) the varia-
tions of


ux versus
x
were examined (Figure 2) to
clarify the dependency of these variations on different
non-dimensional parameters defined in Equation (2.8).
Figure 2 clearly demonstrates that the effect of variation
of non-dimensional parameter K on the profiles of


ux
is so important which causes large differences between
values of


ux for different values of K. Values of


ux at different locations are presented in Table 1 and
Table 2 for better clarifying the effects of K as well as
other no n-dimensi onal param e t e r s

,

 .
Table 1. Values of non-dimensional variable


ux
1.0β
at dif-
ferent locations for , ,1.00.1, 0.1αβα

 
K
and for
different values of non-dimensional parameter.
xα = 1.0, β = 0.1 α = 0.1, β = 1.0
K = 0. 1K = 1.0K = 2. 0K = 5.0 K = 0.1 K = 1.0 K = 2.0K = 5.0
00.97640.78310.60950.3012 0.9762 0.7675 0.56970.2517
0.20.97730.79160.62640.3245 0.9771 0.7767 0.58620.2517
0.40.98020.81720.66950.3971 0.9800 0.8044 0.63620.3515
0.60.98490.86010.74580.5261 0.9848 0.8507 0.72090.4885
0.80.99150.92090.85510.7225 0.9914 0.9159 0.84170.7002
1.01.00001.00001.00001.0000 1.0000 1.0000 1.00001.0000
Table 2. Values of non-dimensional variable


ux at
diferent locations for , 10.00.1 αβ

,
and for different values of non-dimensi onal parameter .
, 1αβ
K
10.0 .0
x α = 10.0, β = 0. 1 α =10.0, β = 1. 0
K = 0.1K = 1.0K = 2.0K = 5.0 K = 0.1 K = 1. 0 K = 2.0K = 5.0
00.99550.95510.91050.7795 0.9958 0.9583 0.91670.7927
0.20.99570.95690.91410.7882 0.9960 0.9600 0.92000.8010
0.40.99620.96230.92480.8146 0.9965 0.9650 0.93000.8258
0.60.99710.97120.94270.8586 0.9973 0.9733 0.94670.8672
0.80.99840.98380.96780.9204 0.9985 0.9850 0.97000.9253
1.01.00001.00001.00001.0000 1.0000 1.0000 1.00001.0000
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S. A. M. TONEKABONI ET AL.
Open Access AM
1607
5. Conclusion
Verification of the Solution
In order to verify the accuracy of the obtained solution,
results are compared with available simulation and exact
solutions available in the literature [28] for the special
case of 0, 0

  (Table 3). It is shown that the
values of the results of HAM and exact solution in the
considered special case are identical to each other at the
considered numerical precision. It should be noted that
exact solutions are available only for this special case.
The excellent agreements between HAM and exact solu-
tions in Table 3 suggest that HAM can yield highly ac-
curate solutions not only for the special cases but also for
the general cases for which exact solu tion does not exist.
Hence, the results presented in this paper can be utilized
as promising data for investigating the behavior of the
general enzyme reaction.
Analytical solution of the amperometric biosensor at
mixed enzyme kinetics and diffusion limitation is pre-
sented utilizing HAM. Dimensionless equation of the
problem is obtained using the mathematical modeling
presented in the paper which is based on non-Micha-
elis-Menten kinetics of the enzymatic reaction. Solution
procedure of the non-dimensional equation of enzyme
reaction is described and mth-order deformation equation
is obtained on the basis of the non-dimensional enzyme
reaction equation presented in this article. Several
h-curves are presented to show the convergence region of
the solution. Results of the solution are presented for
different quantities of the dimensionless parameters used
to non-dimensionalized the enzyme reaction equation. It
is clarified that the most effective parameter in the reac-
Figure 2. Variations of


ux versus auxiliary parameter x for (a) , 1.0 0.1αβ

; (b) ; (c)
; and (d) .
, 0.1 1.0αβ
, 10.00.1 αβ , 1.0β10.0α
Table 3. Comparison of the results of the HAM with simulation and actual results of the problem at different locations and
for different values of non-dimensi onal parameter


, 00Kαβ

.
K = 0.1 K = 1.0 K = 5.0
x Simulation HAM Exact SimulationHAM Actual Simulation HAM Actual
0 0.9500 0.9520 0.9520 0.6500 0.6481 0.6481 0.2100 0.2113 0.2113
0.25 0.9529 0.9550 0.9550 0.6666 0.6684 0.6684 0.2502 0.2452 0.2452
0.50 0.9613 0.9639 0.9639 0.7293 0.7303 0.7303 0.3585 0.3578 0.3578
0.75 0.9767 0.9789 0.9789 0.8366 0.8390 0.8390 0.5893 0.5851 0.5851
1.0 0.9976 1.0000 1.0000 0.9940 1.0000 1.0000 0.9970 1.0000 1.0000

S. A. M. TONEKABONI ET AL.
1608
tion and local dependency of the dependent variable of
the problem


ux is K. Conclusively, some available
results in the literature are used to prove the high accu-
racy of the presented solution.
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