World Journal of Engineering and Technology, 2013, 1, 27-32
Published Online November 2013 (http://www.scirp.org/journal/wjet)
http://dx.doi.org/10.4236/wjet.2013.13005
Open Access WJET
27
New Solution of Substrate Concentration in the Biosensor
Response by Discrete Homotopy Analysis Method
Seyyed Ali Madani Tonekaboni1, Ali Shahbazi Mastan Abad2, Shahab Karimi2, Mitra Shabani2
1School of Mechanic, University of Waterloo, Ontario, Canada; 2School of Mechanics, University of Tehran, Tehran, Iran.
Received June 22nd, 2013; revised July 28th, 2013; accepted August 26th, 2013
Copyright © 2013 Seyyed Ali Madani Tonekaboni et al. This is an open access article distributed under the Creative Commons At-
tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-
erly cited.
ABSTRACT
In this article, Discrete Homotopy Analysis Method (DHAM), as a new numerical method, is employed to investigate
amperometric biosensor at mixed enzyme kinetics and diffusion limitation. Mathematical modeling of the problem is
developed utilizing non-Michaelis-Menten kinetics of the enzymatic reaction. Different results are obtained for differ-
ent values of the dimensionless parameters described in the paper. The presented solution is then compared with the
available actual and simulated results.
Keywords: Discrete Homotopy Analysis Method; Amperometric Biosensor; Mathematical Modeling;
Non-Michaelis-Menten Kinetics
1. Introduction
Biosensor is a device which measures biologically rele-
vant information such as oxygen electrodes, neutral in-
terfaces, etc. [1]. It is also utilized as a component of the
transduction mechanisms [1]. Furthermore, it has been
applied as a transducer, mapping the change in bio-
molecules into electrical signals [2]. Biosensors produce
a signal indicative of the concentration of the measured
analyte. As such, they are used in many industrial, envi-
ronmental, food safety [3], and medical applications.
Examples of such use are detection of pathogens [4],
toxic metabolites such as mycotoxins [5], and pesticides
and water contaminants such as heavy metal ions [6].
These applications showcase the wide usage and studies
of biosensors and highlight the requirement of low detec-
tion limits and quicker analysis with high specificity for
biosensors [2]. Mathematical modeling is widely used as
an important tool to investigate and op timize the analyti-
cal characteristics of biosensors [9]. Investigative mono-
layer membrane contained in the model biosensors are
used to study the biochemical treatment of biosensors
[7,8]. The mathematical model developed is based on
reaction-diffusion equations including none-linear terms
that relate to non-Michaelis-Mentenkinetics of the enzy-
matic reaction [9,10].
In addition to several numerical methods employed for
solving linear and nonlinear differential equations, there
exists some analytical methods such as perturbation
method [11], δ-expansion method [12], Adomian de-
composition method (ADM) [13,14], and Homotopy
perturbation method (HPM) [15,16]. All of the above
mentioned methods including the numerical methods
have certain restrictions, such as necessity for existence
of small parameters, incapability of determining conver-
gence regions, etc. One of the analytical methods pro-
posed in the last couple of decades is homotopy analysis
method (HAM) in which many of these restrictions have
been omitted. In 1992, Liao introduced homotopy analy-
sis method (HAM) for solving strongly nonlinear differ-
ential equations [17]. Using the linear property of ho mo-
topy, one can transform a nonlinear problem into an infi-
nite number of linear sub-problems regardless of the ex-
istence of small parameterss in the original non-linear
problem. HAM is a powerful mathematical technique
and has already been applied to several nonlinear prob-
lems [16-22].
Since HAM has many advantages in comparison to
other analytical methods, it is employed to solve con-
tinuous problems. Hence, after the discrete ADM method
[23], discrete Homotopy analysis method (DHAM) was
introduced in 2010 by Zhu et al. [24]. This method can
be applied to complex problems containing discontinuity

New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analy sis Method 28
in fluid characteristics and the geometry of the problem.
In addition, it needs little computational cost as a nu-
merical method in comparison to HAM as an analytical
approach. DHAM has similar advantages to continuous
HAM. For instance, by means of introducing an auxiliary
parameter one can adjust and control the convergence
region of the solution series. This method should be em-
ployed for solving various differential equations to high-
light its high capabilities in comparison with other nu-
merical methods.
The main focus of this paper is on amperometric bio-
sensor at mixed enzyme kinetics and diffusion limitation
by utilizing DHAM as a powerfull method. Non-Micha-
elis-Menten kinetics of the enzymatic reaction is used to
obtain the constitutive equation of the problem. Several
non-dimensional parameters are defined to the dimen-
sionless equation. The obtained non-dimensional equa-
tion is used to procure the mth-order deformation equa-
tion as an important step towards obtaining the solution.
The h-curves obtained for several cases are illustrated in
this paper to clarify the convergence region of the solu-
tion. Finally, the obtained solution is analyzed to inves-
tigate the effects of varying each dimensionless parame-
ter in the procured equation of the problem. In addition,
some of the results are compared with the actual and
simulated results available in the literature [25].
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 [9,10].
The simplest scheme of non-Michaelis-Menten kinetics
may for instance be described by adding to the Micha-
elis-Menten scheme (2.1) the relationship of the interac-
tion of the enzyme substrate complex 
with an-
other substrate molecule (2.2) followed by the gen-
eration of non-active complex

ES

S


2
ES

as
ESES EP (2.1)
2
ES SES (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
[26] and Baronas et al. [27], which is based on the non-
Michaelis-Menten hypothesis,








max
022
c
M
iM
kE SVS
i
K
SSKK SSK
 
 (2.4)
where the constants


max E0 c
Vk ,
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


,SS t

 at time t and position


 throughout the domain, the following equation
given by Pa o [ 2 6] is us ed.

,
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 KS
D
tSKS KK



  (2.6)
in which 0cM
K
KE K

.
In this paper, the steady state condition is accounted
for and hence, Equation (2.6) is changed to the non-di-
mensional form [25] using the following non-dimen-
sional paramet e rs
2
22
1
S
M
iM
SKS
DSKS KK


  (2.7)
22
, x=, K=, =, =
SM
SkLks
uLDK KK
ks
 



iM
ks
This results in the following n on-dimensional differen -
tial equation
2
22
0, 0<1
1
uKu u
xuu 



  (2.8)
Equation (2.8) must be solved such that it satisfies the
following boundary conditions
1 at 1
0 at 0
ux
ux
x






(2.9)
3. Analytical and Numerical Solutions
DHAM Solution
The discrete form of the nonlinear differential equation
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New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analy sis Method 29
(Equation (2.8)) is obtained as the first step of DHAM’s
procedure of the solution
  

 
2
11
2
1
2
iii
iii
i
Nqq q
qqq
K
q
x



 

 

(3.1)
where is the node number, is the nonlinear op-
erator, and the function is defined as
i N

iq

 

0,
0
1
lim ,
lim
ii
q
ii
q
quX
qu






where is the unknown field variable at node number
i, is the embedding parameter, and 0,i is the
initial guess which is employed to meet the requirements
of the boundary conditions. Here, 0,i is valued at “1”
satisfying all the boundary conditions stated in Equation
(2.9).
i
u

0, 1


qu
U
Through the generalizing concept of DHAM, the so-
called zero-order deformation equation can be written as
 
0,
1iiii qLq uqhHNq 

 
 
where is the non-zero auxiliary parameter, i
0h
H
is the auxiliary function, and is the auxiliary linear
operator which is chosen here as
L

1
2
2
iii
i
1
f
ff
fx


  (3.2)
Expanding in Taylor series with respect to the
embedding parameter , one obtains

iq

q


0, ,
1
,
0
1
!
m
iim
m
m
i
mi m
q
qu uq
q
umq








i
With due attention to the procedure of DHAM [27],
,mi should be chosen so as the following equation is sa-
tisfied
u

,0, ,
31 4
2
mmimi
m
uuu
u
x 
 0
mi
i
1
(3.3)
If the series converges at , then the se-
ries solution is

iq
 1q

0, ,
1
1
ii
m
uu 




where ,mi could be obtained by the so-called high-
order deformation equation. For obtaining the mth-order
deformation equation, the following vector is defined as
u

0, 1,,
,,,
niin
uu uu
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

,1,,mim miimim
Lu uhHR 




u
where



1
,1 1
0
0, 1
1, otherwise
1
1!
m
m
i
mi mm
q
m
Nq
Rmq











 u
Therefore, the following relation is obtained

1
,1 11
0
1
11
00
m
mimmj mj
j
j
m
mj kjkm
jk
Ruuu
uuuKu



 


 

 

 


u
We are free to choose the auxiliary parameter , the
auxiliary function i
h
H
, the initial guess 0,i, and the
auxiliary linear operator so that the validity and
flexibility of the DHAM solution to control the conver-
gence region is proven. Due to the rule of solution ex-
pression [27], the auxiliary function is chosen as follows
u
L
1
i
H
According to the DHAM, the valid region of the aux-
iliary parameter h for convergence of the solution series
is the flat regions of h-curves. To see the proper values of
h, the h-curves are plotted for different values of dimen-
sionless parameters , and
K

 in Figure 1 for ob-
taining the valid results for the considered conditions.
4. Results and Discussion
The procedure for solving the non-dimensional equation
of enzyme reaction (Equation (2.8)) which is based on
the non-Michaelis-Menten kinetics theory utilizing
DHAM is described in the Section 3. It is mentioned
there that the mth-order deformation equation should be
employed to solve the problem. As the first step towards
the solution, the diagrams for variation of non-dimen-
sional parameter

 uX versus auxiliary parameter h for
different investigated cases are illustrated in Figure 1.
Then, the flat region of h-curves in each case is obtained
from these diagrams.
On the basis of the chosen values of auxiliary parame-
ter h in the flat regions of h-curves (Figure 1), some use-
ful diagrams including variations of versus

uX
X
(Figure 2) are procured to clarify the dependency of
these variations on different non-dimensional parameters
defined in Equation (2.8). It is shown in Figure 2 that the
effect of variation of non-dimensional parameter K on
the profiles of

 uX is substantial. Values of

 uX at
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New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analy sis Method 30
(a)
(b)
(c)
(d)
Figure 1. Variations of u (X) versus non-dimensional pa-
rameter X for (a) α = 1.0, β = 0.1, (b) α = 0.1, β = 1.0, (c) α =
10.0, β = 0.1 and (d) α = 10.0, β = 1.0.
(a)
(b)
(c)
(d)
Figure 2. Variations of u (X) versus auxiliary parameter h
for (a) α = 1.0, β = 0.1, (b) α = 0.1, β = 1.0, (c) α = 10.0, β =
0.1 and (d) α = 10.0, β = 1.0.
Open Access WJET

New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analy sis Method
Open Access WJET
31
different locations are presented in Tables 1 and 2 for
better clarifying the effects of K, as well as other non-
dimensional parameter


,

 . It is clearly shown that
the values of variable for lower value of

are lower
than he higher ones. In addition, the spatial variation of
variables which is also shown in Figure 2 is clarified.
Verification of the Solution
The results of the problem obtained by employing DHAM
and the results procured by simulation and actual results
[25] are compared in Table 3 to show the accu- racy of
the presented solution. As such, the presented result in
this paper can be utilized as promising data for investi-
gating the behavior of the enzyme reaction in the consi-
dered conditions.
5. Conclusions
Solution to the amperometric biosensor at mixed en-
zyme kinetics and diffusion limitation is presented util-
izing DHAM as a new numerical method. Dimensionless
equation of the problem is obtained using the mathe-
matical modeling presented in this paper, which is based
on non- Michaelis-Menten kinetics of the enzymatic re-
action. Solution procedure of the non-dimensional equa-
tion of enzyme reaction is described and mth-order de-
formation equation is obtained on the basis of the
non-dimensional enzyme reaction equation presented in
this paper. Several h-curves are dipicted to show the
convergence region of the solution. Results of the solu-
tion are presented for different quantities of the dimen
sionless parameters used to non-dimensionalize the en-
zyme reaction equation. It is shown that the most effect-
tive parameter in the reaction and local dependency of
the dependent variable of the problem

 uX is K.
Available results in the literature are used conclusively to
prove the high accuracy of the presented solution.
On the basis of the presented solution for the consid-
ered problem in the area of enzyme kinetics, it can be
concluded that DHAM can be employed to solve differ-
Table 1. Values of non-dimensional variable u (X) at different locations for α = 1.0, β = 0.1 and α = 0.1, β = 1.0 for different
values of non-dimensional parameter K.
α = 1.0, β = 0.1 α = 0.1, β = 1.0
x K = 0.1 K = 1.0 K = 2.0 K = 5.0 K = 0.1 K = 1.0 K = 2.0 K = 5.0
0 0.9764 0.7831 0.6097 0.3012 0.9762 0.7675 0.5694 0.2532
0.2 0.9773 0.7916 0.6246 0.3244 0.9771 0.7767 0.5860 0.2770
0.4 0.9802 0.8172 0.6695 0.3967 0.9800 0.8044 0.6360 0.3521
0.6 0.9849 0.8601 0.7457 0.5255 0.9848 0.8507 0.7207 0.4884
0.8 0.9915 0.9209 0.8551 0.7221 0.9914 0.9159 0.8416 0.7000
1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 2. Values of non-dimensional variable u (X) at different locations for α = 10.0, β = 0.1 an d α = 10.0, β = 1.0 for different
values of non-dimensional parameter K.
α = 10.0, β = 0.1 α = 10.0, β = 1.0
x K = 0.1 K = 1.0 K = 2.0 K = 5.0 K = 0.1 K = 1.0 K = 2.0 K = 5.0
0 0.9955 0.9551 0.9105 0.7790 0.9958 0.9583 0.9167 0.7924
0.2 0.9957 0.9569 0.9141 0.7878 0.9960 0.9600 0.9200 0.8007
0.4 0.9962 0.9623 0.9248 0.8142 0.9965 0.9650 0.9300 0.8255
0.6 0.9971 0.9712 0.9427 0.8583 0.9973 0.9733 0.9467 0.8670
0.8 0.9984 0.9838 0.9678 0.9202 0.9985 0.9850 0.9700 0.9252
1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 3. Comparison of results of the DHAM with simulation and actual results of the problem at different location and for
different values of non-dimensional parameter K.
K = 0.1 K = 0.1 K = 5.0
x Simulation DHAM Actual SimulationDHAM Actual Simulation DHAM 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 0.9618 0.9639 0.9639 0.7295 0.7308 0.7308 0.3585 0.3578 0.3578
0.75 0.9767 0.9789 0.9789 0.8386 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

New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analy sis Method 32
ent nonlinear ordinary differential equations used to
model different problems in Engineering and Science.
The accuracy is clearly shown and the ablility of the
aproach to control the convergence of the solution is ob-
viously shown. Therefo re, the employed method not only
can be used to solve different complicated nonlinear
problems but also can be considered as a promising nu-
merical technique.
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