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Sciknow Publications Ltd.
Research Journal of Modeling and Simulation
©Attribution 3.0 Unported (CC BY 3.0)
RJMS 2015, 2(2):48-54
DOI: 10.12966/rjms.05.03.2015
Simple Analytical Expression of Steady State Substrate
Concentration in the Biosensor Response
Vembu Ananthaswamy1,*, and Elangovan Sreejee2
1
Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India
M. Phil., Mathematics, The Madura College, Madurai, Tamil Nadu, India
2
*Corresponding author (Email: [email protected])
Abstract - In this paper, it’s easy to use analytic tool for non-linear differential equations in general, namely the New Homotopy
perturbation method, is further improved and systematically described through a typical non-linear differential equations, it has
been used 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. An explicit analytical solution is
given for the first time, with recursive formulas for coefficients.
Keywords - Amperometric biosensor; Non-Michaelis-Menten kinetics; Mathematical modeling; Non-linear differential
equation; New Homotopy perturbation method
1. Introduction
Biosensor is a device which measures biologically pertinent information such as oxygen electrodes, neutral interfaces, etc.
(Sheller et al., 1988). It is also utilized as a component of the transduction mechanisms (Sheller et al., 1988). Furthermore, they
can be used as transducers which translate the biomolecular responses into electrical signals (Wollenberger et al., 1997).
Biosensors use specific biochemical reactions catalyzed by enzymes powerless on electrodes. Many enzymes are repressed by
their own substrates, leading to velocity curves that rise to a maximum and then descend as the substrate concentration increases.
In the literature, mathematical models have been widely used as a vital tool to study and optimize the analytical characteristics of
actual biosensors (Schulmeister et al., 1993). 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 (Wollenberger et al., 1997; Chaubey et al.,
2002; Guilbault et al., 1973 ).Mathematical modeling is widely used as an important tool to inspect and optimize the analytical
characteristics of biosensors (Aris, 1975). Exploratory monolayer membrane contained in the model biosensors are used to study
the biochemical treatment of biosensors (Baeumner et al., 2004; Schulmeister et al., 1990). The mathematical model developed is
based on reaction-diffusion equations including none-linear terms that relate to non-Michaelis-Menten kinetics of the enzymatic
reaction(Aris, 1975; Bieniasz et al., 2004).In addition to several numerical methods for solving linear and nonlinear differential
equations, there exists some analytical methods such as perturbation method (Cole, 1978) and Homotopy perturbation method
(HPM) (He, 1998 & 1999). All of the above mentioned methods including the numerical methods have certain restrictions, such
as necessity for existence of small parameters, incapability of determining convergence regions, etc.
Design of biosensors is based on understanding the kinetic characteristics of these devices. Generally, measuring the
concentration of substrate inside enzyme membranes 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
( He, 1998 & 1999; Mell et al. (1975)., such as investigative monolayer membrane model used to study the biochemical
treatment of biosensors (Schulmeister, 1990; Aris, 1975). Their ma- thematical models are based on reaction-diffusion equations
including non-linear term that relate to non-Michaelis-Menten kinetics of the enzymatic reaction (Bieniasz et al., 2004; He,
1998). Hence, high accurate analytical and numerical methods should be employed to investigate this important nonlinear
chemical equation. Most scientific problems in engineering are inherently nonlinear. Except for a few of them, the majority of
nonlinear problems do not have analytical solutions.
Using the linear property of homotopy, one can transform a nonlinear problem into an infinite number of linear sub problems
regardless of the existence of small parameters in the original non-linear problem. HPM is a powerful mathematical technique
and has already been applied to several nonlinear problems (Hoffman, 1992; He, 1998 & 2004; Ananthaswamy et al., 2014). The
Research Journal of Modeling and Simulation （2015）48-54
49
majority of non-linear problems do not have analytical solutions. Therefore, the constitutive laws of these problems should be
solved using other schemes such as numerical or perturbation methods. In the numerical method, stability and convergence of the
solution should be considered so as to avoid divergence or inappropriate results (Nayfeh, 1993)In the perturbation method, the
small parameter is inserted in the equation; thus, finding the small parameter and exerting it into the equation is one of the
deficiencies of this method (Ozis et al., 2007).
The main focus of this paper is on amperometric bio-sensor at mixed enzyme kinetics and diffusion limitation by utilizing
NHPM as a powerful method. Non-Michaelis-Menten kinetics of the enzymatic reaction is used to obtain the constitutive
equation of the problem. Several non-dimensional parameters are defined to the dimensionless equation. Finally, the obtained
solution is analyzed to investigate the effects of varying each dimensionless parameter in the procured equation of the problem
(Li et al., 2006).
2. Mathematical Formulation of the Problem
Spatial dependency of enzyme kinetics on biochemical systems has recently attracted much attention by considering the effect of
diffusion in these processes (Aris, 1975; Bieniasz et al., 2004). The simplest scheme of non-Michaelis-Menten kinetics may for
instance be described by adding to the michaelis-menten scheme eqn.(1) the relationship of the interaction of the enzyme
substrate complex (ES ) with another substrate molecule (S ) eqn. (2) followed by the generation of non-active complex ( ES2 )
as
E  S  ES  E  P
(1)
ES  S  ES 2
(2)
The reaction is said to display michaelis-menten kinetis in which the relationship between the rate of an enzyme catalyzed
reacton and the substrate concentration takes the form
V [S ]
(3)
v  max
K M  [S ]
Where v and V max are called initial reaction velocity and maximum velocity respectively. K M is known as michaelis
constant for S . K M and Vmax are constants at a given temperature and enzyne concentration.
The non-michaelis-menten hypothesis,
kc [ E ]0[ S ]
v

Vmax S 
(4)
K M  [S ]  [S ] / Ki K M  [S ]  [ S ]2 / Ki
Where the constants Vmax  k c [E ]0 , K M and K i are Michaelis-Menten and inhibition constants. On the basis of the
eqn.(4), rate is maximised by increasing the concentration.It is said to be inhibited by the substrate.The constant K i is called the
substrate inhibition constant.The rate of change of substrate concentration S  S ( , t ) at time t and position    throughout
the domain,
S
 Ds .(S )  v( , t )
(5)
t
Ds is the substrate diffusion coefficient and S is the gradient operation.On the basis of non-michealis-menten kinetics ,
Equation (5) becomes
2
S
2S
KS
 Ds

2
t

1  S / K M  S 2 / Ki K M
(6)
In which K  K c E0 / K M .
Ds
2S

2

KS
1  S / K M  S 2 / Ki K M
0
(7)
Using the following dimensionless variables
S

kL2
ks
ks
, x ,K 
2 ,  
,
ks
L
Ds
KM
Ki K M
Now, the dimensionless form of the eqn.(7) and the corresponding boudary conditions are as follows:
u
(8)
Research Journal of Modeling and Simulation （2015）48-54
50
d 2u

Ku
dx
1  u   u 2
u 1
at x  1
2
u
0
x
at
 0,
0  u 1
x0
(9)
(10)
3. Solution of the Non-linear Boundary Value Problem Using the New Homotopy
Perturbation Method (NHPM)
Recently, many authors have applied the Homotopy perturbation method (HPM) to solve the non-linear problem in physics and
engineering sciences (He, 1999 & 2003; Ariel, 2010; Ananthaswamy et al., 2012 & 2013; Shanthi et al., 2013). This method is
also used to solve some of the non-linear problem in physical sciences (Shanthi et al., 2014; Ananthaswamy et al., 2014). This
method is a combination of Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the Lighthill
equation, the Diffuing equation and the Blasius equation (He, 1999 & 2003; Ariel, 2010; Ananthaswamy et al., 2013). The HPM
is unique in its applicability, accuracy and efficiency. The HPM uses the imbedding parameter p as a small parameter, and only
a few iterations are needed to search for an asymptotic solution. Using this method (Shanthi et al., 2014; Ananthaswamy et al.,
2014), we can obtain solution of the eqn. (9) and (10) is as follows:
cosh A( x)
u ( x) 
(11)
cosh( A)
Where
A  k /(1     )
(12)
4. Results and Discussion
Figure (1) & (2) represents the dimensionless concentration u(x) versus dimensionless distance x. From Fig.1 it is clear that
when the effect of variation of dimensionless parameter K increases , the coresponding dimentionless concentration u(x)
increases in some fixed vales of other dimensionless parameters  and  . From Fig.2 it is observe that the effect of variation of
dimensionless parameter K increaces , the coresponding dimensionless concentration u(x) increases at some fixed values of
other dimensionless parameters  and  .
Fig.1:Dimensionless concentration u(x) versus the dimensionless concentration x for various values of the dimensionless
parameters  ,  and K , when (a)  = 1.0,  = 0.1 and K = 0.1, 1,2,5; (b)  = 0.1,  = 1.0 and K = 0.1,1,2,5 ; (c)  = 10,
Research Journal of Modeling and Simulation （2015）48-54
51
 =0. 1 and K = 0.1,1,2,5 ;(d)  =10,  =1.0 and K =0.1,1,2,5.
Fig.2:Dimensionless concentration u(x) versus the dimensionless distance x for various values of the dimensionless parameters
 ,  and K , when (a)  = .5,  = 0.1 and K = 0.1, 1,2,5; (b)  = 0.5,  = 1.0 and K = 0.1,1,2,5 ; (c)  = 5,  =0. 1 and
K = 0.1,1,2,5 ;(d)  =5,  =1.0 and K =0.1,1,2,5.
5. Conclusion
The non-linear reaction diffusion equation in an amperometric biosensor was solved analytically. A simple analytical solution of
the steady state concentration of the amperometric biosensor at mixed enzyme kinetics is derived by using the New Homotopy
perturbation method. The primary result of this work is simple and approximate expressions of the concentrations for all values
of the dimensionless parameters K ,  and  . This analitical solution is clarified that the most effective parameter in the
reaction and local dependency of the dimensionless concentration u(x) is K . This method can be easily extended to solve other
strongly non-linear boundary value problems in chemical and biological sciences.
Acknowledgement
The authors are thankful to Shri. S. Natanagopal, Secretary, The Madura College Board, Dr. R. Murali, The Principal and Mr. S.
Muthukumar, Head of the Department of Mathematics, The Madura College (Autonomous), Madurai, Tamil Nadu, India for
their constant encouragement.
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Mell, L. D., & Maloy, J. T. (1975). A Model for the Am- perometric Enzyme Electrode Obtained through Digital Simulation and Applied to the Glucose Oxidase
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Nayfeh, A. H. (1993). Problems in Perturbation, 2nd Edition, Wiley, New York.
Ozis, T. & Yildirim, V., (2007). A Comparative study of He’s Homotopy perturbation method for determining frequency-amplitude relation of a nonlinear
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Appendix A
Basic concept of the Homotopy perturbation method (HPM) (He 1998, 1999, 2003 & 2004)
To explain this method, let us consider the following function:
(A.1)
Do (u)  f (r )  0, r  
with the boundary conditions of
u
(A.2)
Bo (u, )  0,
r
n
where Do is a general differential operator, Bo is a boundary operator, f (r ) is a known analytical function and  is the
boundary of the domain  . In general, the operator Do can be divided into a linear part L and a non-linear part N . The eqn.
(A.1) can therefore be written as
(A.3)
L(u)  N (u)  f (r )  0
By the Homotopy technique, we construct a Homotopy v(r, p) :   [0,1]   that satisfies
H (v, p)  (1  p)[ L(v)  L(u0 )]  p[ Do (v)  f (r )]  0.
(A.4)
H (v, p)  L(v)  L(u0 )  pL(u0 )  p[ N (v)  f (r )]  0.
(A.5)
where p  [0, 1] is an embedding parameter, and
u0 is an initial approximation of eqn. (A.1) that satisfies the boundary
conditions. From the eqns. (A.4) and (A.5), we have
H (v,0)  L(v)  L(u0 )  0
(A.6)
H (v,1)  Do (v)  f (r )  0
(A.7)
When p=0, the eqns. (A.4) and (A.5) become linear equations. When p =1, they become non-linear equations. The process of
changing p from zero to unity is that of L(v)  L(u0 )  0 to Do (v)  f (r )  0 . We first use the embedding parameter p as a
“small parameter” and assume that the solutions of eqns. (A.4) and (A.5) can be written as a power series in p :
v  v0  pv1  p 2v2  ...
Setting p  1 results in the approximate solution of the eqn. (A.1):
(A.8)
Research Journal of Modeling and Simulation （2015）48-54
u  lim v  v0  v1  v2  ...
53
(A.9)
p1
This is the basic idea of the HPM.
Appendix B
Solution of the boundary value problem eqn. (8) using the Homotopy perturbation method
(Ananthaswamy et al., 2014, Shanthi et al., 2013 & 2014)
In this Appendix, we indicate how the eqn. (11) in this paper is derived. To find the solution of eqns.(9) and (10) we construct the
new Homotopy as follows :
  2u

  2u

Ku(1)
Ku
(1  p) 

 p


0
2
2
2
2
1  u (1)  u (1) 
1  u  u 
 x
 x
  2u

  2u

K
Ku
(1  p) 

 p


0
 x 2 1     
 x 2 1  u  u 2 
(B.1)
(B.2)
The analytical solution of (B.1) is
u  u0  pu1  p 2u2  ...
(B.3)
Substituting Equation (B.3) in (B.2) w get,
  2 (u 0  pu1  p 2 u 2  ...)
Ku(1) 
(1  p) 


1     
x 2

  2 (u 0  pu1 p 2 u 2  ...)



x 2


 p
0
2
K (u 0  pu1  p u 2  ...)


 1   (u  pu  p 2 u  ...)   (u  pu  p 2 u  ...) 2 
0
1
2
0
1
2


(B.4)
Comparing the coefficients of like powers of p in the eqn.(B.4) we get
p0 :
 2u
x 2

K
1  
0
(B.5)
The initial approximations is as follows
u 0 (1)  1 ;
u ' 0 (0)  0
u i (0)  0 ;
u ' i (0)  0
(B.6)
i  1,2,3...
Solving the eqns. (B.4) and using the boundary condition (B.6), we obtain the following results:
cosh( Ax)
u 0 ( x)  u 0 
cosh( A)
According to HPM, we conclude that
u  lim u ( x)  u 0
(B.8)
p1
After putting the eqn. (B.7) into an eqn. (B.8), we obtain the solutions in the text eqns.(10) and (11).
Appendix C
Nomenclature
Symbol
[ES]
[S]
(B.7)
Meaning
Enzyme concentration of the substrate
Enzyme substrate complex
54
Research Journal of Modeling and Simulation （2015）48-54
V
Vmax
Initial reaction velocity
Maximum velocity
KM
Michaelis constant
Ki
Substrate inhibition constant
Ds
S
E
 ,
S
T

Substrate diffusion constrate
ES2 
Gradient operation
Binds to enzyme
Diffusion parameters
Substrate
Time
Position
Generation of non-active complex