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Vol.3, No.1, 1-8 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.31001
Copyright © 2011 SciRes. OPEN ACCESS
Analytical solution of the concentration and current
in the electoenzymatic processes involved in a
PPO-rotating-disk-bioelectrode
Govindhan Varadharajan, Lakshmanan Rajendran*
Department of Mathematics, The Madura college, Madurai, India; *Corresponding Author: [email protected]
Received 16 October 2010; revised 20 November 2010; accepted 25 November 2010.
ABSTRACT
A mathematical model for electroenzymatic
process of a rotating-disk-bioelectrode in which
polyphenol oxidase occurs for all values of
concentration of catechol substrate is pre-
sented. The model is based on system of reac-
tion-diffusion equations containing a non-linear
term related to Michaelis-Menten kinetics of the
enzymatic reaction. Approximate analytical me-
thod (He’s Homotopy perturbation method) is
used to solve the non-linear differential equa-
tions that describe the diffusion coupled with a
Michaelis-Menten kinetics law. Closed analytical
expressions for substrate concentration, prod-
uct concentration and corresponding current
response have been derived for all values of
parameter 2
1
()( )==
E
 
and
Ls
using pertur-
bation method. These results are compared with
simulation results and are found to be in good
agreement. The obtained results are valid for
the whole solution domain.
Keywords: Mathematical Model; Polyphenol
Oxidase; Steady-State; Homotopy Perturbation
Method; Simulation
1. INTRODUCTION
Enzymes electrodes are powerful tools for under-
standing the mechanism and kinetics of fast reactions.
Owing to their specificity and sensitivity, enzyme elec-
trodes including various amplification, schemes have
been developed for many applications such as electro-
chemical immunoassays, [1-2] water pollutant detection,
[3-7] and monitoring of biological metabolities [8-11].
The sensitivity of enzyme electrodes is very often in-
creased by incorporation of a substrate-recycling scheme
and several strategies including chemical, enzymatic, or
electrochemical recycling have been developed. In the
view of numerous application of such bio-sensor with
amplified response, we are interested in investigating the
concentration s and p in order to improve the metrologi-
cal characteristics further. In addition, this theoretical
approach is of practical interest since this kind of bio-
sensor can be used for the determination of phenolic
compounds and catecholamine neurotransmitters in the
field of environmental control and clinical analysis [12-
22]. Such a theoretical and kinetic analysis is a powerful
approach to rationalize functions of biosensors. Desprez
and Labbe [23] obtained the analytical expression con-
centration and current for the limiting cases only. The
purpose of this communication is to derive a simple ac-
curate polynomial expressions of concentrations gener-
ated at a enzyme electrode using Homotopy perturbation
method.
2. MATHEMATICAL FORMULATION OF
THE PROBLEM
Figure 1 is a schematic representation of the rotating
disk PPO-electrode working with catechol S substrate.
The different assumption which lead to the electrode
Electrode Enzyme-Layer Diffusion
Convection
Layer
Bulk Solution
SS
P
P
PPO
2e
-
0L
L
+ δ X
Figure 1. Schematic representation for the electrocatalytic
process that occur at PPO-rotating-disk-bio electrode. s and p
respectively denote catechol substrate and product of PPO. L
and δ are respectively the thickness of the enzyme-layer and
the diffusion-convection layer.
G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
2
response are as given below: 1) In our model we have
assumed that, because of their structural similarity, s and
p have the same diffusion coefficient D’ in the bulk so-
lution. 2) Diffusion of s and p within the enzymes layer
of thickness L with the same diffusion coefficient D and
a partition coefficient equal to unity. 3) Enzymatic reac-
tion between substrate s and enzymes E within the en-
zyme layer. The rate of the enzymatic step is
cat S
T
kESK S
, Where S
K
is the Michaelis
constant for substrate s and cat
k the rate of the enzy-
matic rate-limiting step in the Michaelis-Menten formal-
ism.
T
E is the total concentration of enzyme within
the enzyme layer.(iv) Depending on the applied elec-
trode potential, electroreduction of enzymatically pro-
duced orthoquinone p or electro-oxidation of non-en-
zymatically oxidized substrates occurs on the electrode
surface. Electrode potential values are assumed to be
sufficiently Cathodic or anodic so that this electro-
chemical is not rate limiting. We consider the equations
of Barlett and Whitaker [24], Desprez and Labbe [23],
describing the concentrations of s and p at steady state as
follows:
2
20
1
dS S
S
dx

(1)
2
20
1
dP S
S
dx

(2)
where

1
2
2
11
,, S
Scat
T
DK
KkE






(3)
Using the following boundary conditions:
,0 when SS PPxL

  (4)
0,0 when 0,SP x  (5)
We introduce the following set of dimensionless
variables:
2
1
,,, ,
E
SPx
UV XLS
SP L


  (6)
The governing non-linear reaction/diffusion Eqs.1-5
is expressed in the following non- dimensional form as :
2
2
1
0
1
EU
dU
U
dX

(7)
2
2
1
0
1
EU
dV
U
dX

(8)
with the boundary conditions:
0,1, when 1VUX L
 (9)
0,1, when 0VU X
 (10)
The dimensionless current is given by

0
2
aX
IFADS dUdX
 (11)
3. ANALYTICAL SOLUTION OF STEADY
STATE CONCENTRATION USING
HPM
Recently, many authors have applied the HPM to
various problems and demonstrated the efficiency of the
HPM for handling non-linear structures and solving
various physics and engineering problems [25-28]. This
method is a combination in topology and classic pertur-
bation techniques. Ji Huan He used the HPM to solve the
Lighthill equation [29], the Duffing equation [30] and
the Blasius equation [31]. The idea has been used to
solve non-linear boundary value problems, integral
equations and many other problems [32-33]. 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 (see Ap-
pendix A), we can obtain the following solution to Eqs.7
and 8 (Appendix-A)

3
2
1
22 324
1
1
11121121
24
12121
EEE
EEE
UX
X
LLL
XXX
L


 

 
 

 

 





(12)

3
2
1
224 23
1
112 11211
24
12121
EEE
EEE
VX
X
LLL
XX X
L


 

 


 

 





(13)
The current response is given by
 

1
3
2
2
0.510.51
1
a
EE
E
IFADS
LL
L
 

 

(14)
4. LIMITING CASE RESULTS
The kinetic response of amperometric biosensor de-
G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
33
pends on concentrations of U and V. The concentrations
of the species depend upon concentration of the substrate
U. But substrate U depends on two factors
E
and 1
.
S
K
is the Michaelis-Menten constant, an intrinsic char-
acter of an enzyme.
4.1. Unsaturated (First Order) Catalytic
Kinetics
When 11U
, the reaction rate can be simplified to
cat T
vk E as a zero reaction. In zero order reaction
1
U
is small. Now the Eq.7 becomes
2
2
1
0
E
dU
dX

The solution of the above equation becomes
2
11
11
22 L
EE
X
UX



 


(15)
4.2. Unsaturated (Zero Order) Catalytic
Kinetics
If 11U
, the rate will be
cat S
T
vk E SK as
a first order reaction. Now the Eq.7 becomes
2
20
E
dU U
dX

From the above equation we can obtain the concentra-
tion of U as follows:
E
E
E
1
E
E
1
L-
E
2cosh1e 1
e
2cosh1
e 1
e
2cosh1
E
L
X
X
L
U
L
L









 


















(16)
5. NUMERICAL SIMULATION
The non-linear differential Eq.7 is also solved using
numerical methods. The functionbvp4c in Scilab soft-
ware which is a function of solving two-point boundary
value problems (BVPs) for ordinary differential equa-
tions is used to solve this equation. It’s numerical solu-
tion is compared with Homotopy perturbation method
and it gives satisfactory result. The Scilab program is
also given in Appendix (B).
6. RESULTS AND DISCUSSION
In other cases the order is between zero and one. For
an enzyme electrode to be analytically useful, its re-
sponse must be quantitatively related to the substrate
concentration. Based on this principle, 11U
is not
the proper case for an enzyme electrode, because in the
zero order reaction product concentration is independent
of the substrate concentration. An order between zero
and one is favorable. Eq. 12 represents the most general
approximate new analytical expression for the substrate
concentration profiles for all values of
E
and 1
.
Recently Labbe et al. [23] obtained the solution of this
model for the limiting cases. A comparison of numerical
simulation results with our Eq.12 is shown in Figures
2(a-c). The agreement between simulation results and
Eq.12 is quite good. Figure 3(a-c) show the dimen-
sionless steady-state concentration V using Eq.13 for
various values of
E
and 1
. From these figures, we
can see that the value of the concentration V matches
well with the simulation results. The concentration U
attains minimum or V attains maximum when

3
1
1
121 1121
24 1
E
LL L
X


 
 
 
 
Dimensionless distance X
Dimensionless concentration U(X)
α
1
= 1, γ
E
= 0
0
L
δ
α
1
= 0.5, γ
E
= 0.5
α
1
= 0.1, γ
E
= 1
α
1
= 0.01, γ
E
= 1
0 0.1 0.2 0.3
0.4 0.5
0.6
0.7 0.8 0.9 1
1.02
1
0.98
0.96
0.94
0.92
0.9
0.88
(a)
Dimensionless distance X
Dimensionless concentration U(X)
α
1
= 0.1, γ
E
= 1
1
L
δ
α
1
= 0.5, γ
E
= 0.5
α
1
= 0.01, γ
E
= 0.01
α
1
= 0.01, γ
E
= 1
0 0.2 0.4 0.6
0.8 1 1. 2 1.4 1.6 1.8 2
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
(b)
G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
4
Dimensionless distance X
Dimensionless concentration U(X)
α
1
= 0.1, γ
E
= 0.5
2
L
δ
α
1
= 0.5, γ
E
= 0.1
α
1
= 0.1, γ
E
= 0.1
α
1
= 0.01, γ
E
= 0.01
0
0.5
1
1.5
2 2.5
3
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
(c)
Figure 2. Normalised concentration profile U(X) as a function
of dimensionless parameter X = x/L. The concentrations were
computed using Eq.12 for various values of the reaction/
diffusion parameter1E
,
and for the values (a) 0L
(b)
1L
(c) 2L
. The line denotes Eq.12 and the dot
denotes the numerical simulation.
Dimensionless distance X
Dimensionless concentration V(X)
α
1
= 0.1, γ
E
= 1
0
L
δ
α
1
= 0.5, γ
E
= 0.5
α
1
= 1, γ
E
= 0.1
α
1
= 0.01, γ
E
= 1
0
0.1 0.2
0.3
0.4
0.5 0.6
0.7 0.8 0.9
1
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
(a)
Dimensionless distance X
Dimensionless concentration V(X)
α
1
= 0.01, γ
E
= 1
1
L
δ
α
1
= 0.1, γ
E
= 1
α
1
= 0.5, γ
E
= 0.5
α
1
= 0.01, γ
E
= 0.01
0
0.2 0.4
0.6 0.8
1
1.2
1.4 1.6 1.8
2
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
(b)
Dimensionless distance X
Dimensionless concentration V(X)
α
1
= 0.01, γ
E
= 0.01
2
L
δ
α
1
= 0.1, γ
E
= 0.5
α
1
= 0.5, γ
E
= 0.1
α
1
= 0.1, γ
E
= 0.1
0
0.5
1
1.5
2
2.5
3
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
(c)
Figure 3. Normalised concentration profile V(X) as a function
of dimensionless parameter X = x/L. The concentrations were
computed using Eq.13 for various values of the reaction/
diffusion parameter 1E
,
and for the values (a) 0L
(b) 1L
(c) 2L
The line denotes Eq.12 and the dot
denotes the numerical simulation.
0
L
δ
γ
E
α
1
= 0.1
0
1 2
3
4
5 6
7 8 9
10
3
2.5
2
1.5
1
0.5
0
α
1
= 1
α
1
= 10
α
1
= 20
Current response ψ = I
a
/ 2FAD
s
∞
Figure 4. Dimensional current
versus E
for various
values of 1
and for the fixed value 0L
The line de-
notes Eq.14 and the dot denotes the numerical simulation.
Figure 4 represents the dimensionless current
versus
E
for various values of 1
. From this figure, it
is observed that the value of the current increases slowly
when
E
increases and 1
decreases. Our analytical
expression of current is also compared with simulation
result in Figure 4. It gives satisfactory agreement.
7. CONCLUSIONS
The time independent non-linear reaction-diffusion
equation has been formulated and solved analytically
and numerically. Analytical expressions for the concen-
trations and current are derived by using the HPM. The
G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
55
primary result of this work is simple approximate calcu-
lations of concentrations and current for all values of
dimensionless parameters 1
,
E
and L
. The HPM
is an extremely simple method and it is also a promising
method to solve other non-linear equations. This method
can be easily extended to find the solution of all other
non-linear equations.
8. ACKNOWLEDGEMENTS
The authors are very grateful to the referees for their valuable sug-
gestions. This work was supported by the Department of Science and
Technology (DST), New Delhi, Government of India. The authors are
also thankful Dr. T. V. Krishnamoorthy, The Principal and Mr. M.S
Meenakshisundaram, The Secretary, The Madura College, Madurai for
their encouragement.
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G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
77
APPENDIX A
Solution of the Eqs.7 and 8 using Homotopy pertur-
bation method. In this Appendix, we indicate how
Eqs.12 and 13 in this paper are derived. Furthermore, a
Homotopy was constructed to determine the solution of
Eqs.7 and 8.

22 2
1
22 2
10
E
dU dUdU
ppUU
dX dXdX

 

 
 
(A1)

222
1
222
10
E
dVdU dV
ppUU
dXdX dX

 

 
 
(A2)
and the initial approximations are as follows:
0, 0, 1XV U 
(A3)
1,0, 1XVU
L
 (A4)
The approximate solutions of (A1) and (A2) are
23
01 2 3
UUpU pUpU  (A5)
and
23
01 2 3
VVpV pVpV  (A6)
Substituting Eqs .A5 and A6 into Eqs.A1 and A2 and
comparing the coefficients of like powers of p
2
00
2
: 0
dU
pdX (A7)
2
2
10
1
10 0
22
:0
E
dU
dU
pUU
dX dX

 (A8)
2
22
20
21
10 111
222
:0
E
dU
dU dU
pUUU
dXdX dX

 (A9)
and
2
00
2
: 0
dV
pdX (A10)
2
2
10
1
10 0
22
: 0
E
dV
dV
pUU
dX dX

 (A11)
2
22
20
21
10 111
222
: 0
E
dV
dV dV
pUUV
dXdX dX


(A12)
Solving the Eqs.A7 to A12 , and using the boundary
conditions (A3) and (A4), we can find the following
results
01UX
(A13)

2
12
E
UXX 1X
L




(A14)

2
2
21
24 232
1
1 1121
24 L
1
2112
24
EE
EE E
UX X
L
X
XX
L


 

 
 

 
 






(A15)
and
00VX
(A16)

12
2
E
VX 1X- X
L




(A17)

3
2
21
224 23
1
1 1121
24 L
1
1221
24
EE
EE E
VX XZ
L
X
XX
L


 

 
 

 
 



 


(A18)
According to the HPM, we can conclude that
 
012
1
lim
p
UXUXUU U
 (A19)
01 2
1
lim
p
VXVXV VV
  (A20)
Using Eqs.A13, A14 and A15 in Eq.A19 and
Eqs.A16, A17 and A18 in Eq.A2 0, we obtain the final
results as described in Eqs.12 and 13.
G. Varadharajan et al. / Natural Science 3 (2011) 1-8
Copyright © 2011 SciRes. OPEN ACCESS
8
APPENDIX B
Scilab program to find the solutions of the Eqs.7 and 8
function pdex4
m = 0;
x = linspace(0,1);
t=linspace(0,100000);
sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);
u1 = sol(:,:,1);
u2 = sol(:,:,2);
figure
plot(x,u1(end,:))
title('u1(x,t)')
xlabel('Distance x')
ylabel('u1(x,2)')
%---------------------------------------------------------------
figure
plot(x,u2(end,:))
title('u2(x,t)')
xlabel('Distance x')
ylabel('u2(x,2)')
% --------------------------------------------------------------
function [c,f,s] = pdex4pde(x,t,u,DuDx)
c = [1; 1];
f = [1; 1] .* DuDx;
a=0.1;
r=1;
F =(r*u(1))/(1+a*u(1));
s=[-F; F];
% --------------------------------------------------------------
function u0 = pdex4ic(x); %create a initial conditions
u0 = [0; 1];
% --------------------------------------------------------------
function [pl,ql,pr,qr]=pdex4bc(xl,ul,xr,ur,t) %create a boundary conditions
pl = [ul(1)-1; ul(2)];
ql = [0; 0];
pr = [ur(1)-1; ur(2)];
qr = [0; 0]