Applied Mathematics, 2011, 2, 1140-1147
doi:10.4236/am.2011.29158 Published Online September 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Analytical Solutions of System of Non-Linear Differential
Equations in the Single-Enzyme, Single-Substrate
Reaction with Non-Mechanism-Based Enzyme
Inactivation
Govindhan Varadharajan, Lakshmanan Rajendran*
Department of Mathematics, The Madura College, Madurai, India
E-mail: *raj_sms@rediffmail.com
Received June 23, 2011; revised July 15, 2011; accepted July 23, 2011
Abstract
A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for
the full range of enzyme activities has been derived. The time dependent analytical solution for substrate,
enzyme-substrate complex and product concentrations are presented by solving system of non-linear differ-
ential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential
equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple ana-
lytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in
terms of dimensionless reaction diffusion parameters 12
, , and 3
 
using perturbation method. The
numerical solution of the problem is also reported using SCILAB software program. The analytical results
are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained
results are valid for the whole solution domain.
Keywords: Non-Linear Reaction Equations, Enzyme Inactivation, Homotopy Perturbation Method, Time
Dependent Analytical Solution
1. Introduction
An enzyme is a biological catalyst that regulates the rate
of chemical reaction in a living organism. Enzymes are
very essential as most chemical reactions would occur
too slowly or would lead to different product without the
activity of enzymes. Enzymes bond with a substrate to
form a transient state, an unstable intermediate complex
that requires less energy for the reaction to proceed. Like
any catalyst, the enzyme remains unaltered by the com-
pleted reaction and can therefore continue to interact
with substrates. Enzymes may speed up reactions by a
factor of many millions. Under temperature changes [1],
diluted conditions, or changes in the reaction medium
(pH or buffer) [2,3], enzyme can undergo progressive
loss of activity. Actually enzyme inactivation can result
in grievous errors in describing the behaviour of the sys-
tem, such as incorrect estimation of the kinetic parame-
ters. Therefore it is very important to be able to know
when enzyme inactivation is affecting a reaction. Under-
standing the effects of enzyme inactivation is important
for application such as predicting the behaviour of
chemical reactions in the food, chemical, and pharma-
ceutical industries [4,5]. While there are numerous mod-
els available for mechanism-based inactivation systems
or suicide substrates [6,7], there is currently no method-
ology that yields quantitative predictions for non-mecha-
nism-based enzyme inactivation. In this paper we derive
an expression for concentration of substrate, enzyme-
substrate complex and product with non-mechanism-
based enzyme inactivation, interms of dimensionless re-
action diffusion parameters 12 3
, , and
 
using
Homotopy perturbation method (HPM) and comparative
study of the same with numerical simulation.
2. Mathematical Formulation of the Problem
The model of biochemical reaction was set forth by
Michaelis and Menten in 1913 [8] and further developed
by Briggs and Haldane in 1925 [9]. This formulation
G. VARADHARAJAN ET AL.1141
considers a reaction where a substrate S binds an enzyme
E reversibly to form a complex C. The complex can then
decay irreversibly to a product P and the enzyme, which
is then free to bind another molecule of the substrate.
The enzyme is normally considered more stable while
incorporated in C than when in free E form. From this,
we can add to the Michaelis-Menten mechanism that free
E decays into its inactive form Ei. This single en-
zyme-substrate reaction system is represented as follows
[10]:
t = 0, 0
s
s
, 0c, (14)
By intrhe fowingt of d
ra
0
i
e
oducing tllo seimensionless pa-
meters
   
 
10
00
i
12
000
0
3
00
s
, , v ,
, , ,
,
SM
t ct
ket u
εse
etKK
wτ
ese
Ke
es
 


 


The system of Equations (11)-(13) with initial condi-
tio
1
1
cat
k
k
k
SE C PE
 (1)
3
k
i
EE (2)
n (14) can be represented in dimensionless form as
follows:
In this mechanism, the enzyme inactivation is irre-
versible. This mechanism illustrates the binding of sub-
strate S and release of product P. E is the free enzyme
and C is the enzyme-substrate complex. The time evolu-
tion of reaction (1)-(2) are obtained by applying the law
of mass action to yield the set of system of following
non-linear differential equations.
1
d
dS
skesKc
t
(3)
1
d
dM
ekesKcKe
t
 
(4)
1
d
dM
ckesKc
t
(5)
d
dcat
pkc
t (6)
1
d
d
i
ekK e
t
(7)
Initial conditions at t = 0 is given by
0
s
s, , , (8)
0
ee0,c0p0.
i
e
In this system, S
K
is the equilibrium dissociation
constant for enzyme-substrate complex,
M
is the Mi-
chaelis-Menten constant and
K
is the constant for
enzyme inactivation. By imposing the following two
conservation of laws,
0i
eece (9)
and
0
s
sc p (10)
We describe the reaction mechanism (3)-(7) using the
following differential equations

10 1
d
di
sksecekK c
t 
1
d
d
uuεuv uwv

  (15)
2
d
d
vuuvuw v
  (16)
33 3
d
d
wvw

  (17)
The boundary conditions are
(0)1, (0)0 ,uvw(0) 0
 (18)
3. Analytical Expressions of Concentrations
Nonlinear phenomena play a crucial role in applied
to Equations (15)-(17) (Ref Appendix-A)
under Non-Steady State and Steady State
Condition
mathematics and physics. Explicit solutions to the non-
linear equations are of fundamental importance. Various
methods for obtaining explicit solution to nonlinear evo-
lution equations have been proposed. Recently, many
authors have used the HPM for solving various problems
and demonstrated the efficiency of the HPM for solving
non-linear structures and various physics and engineer-
ing problems [11-14]. This method is a combination of
topology and classic perturbation techniques. Ji Huan He
used the HPM to solve the Lighthill equation [15], the
Duffing equation [16] and the Blasius equation [17]. The
idea has been used to solve non-linear boundary value
problems, integral equations and many other problems.
In these methods [18-22], the homotopy perturbation
method is applied and the obtained results show that the
HPM is very effective and simple. The HPM is unique in
its applicability, accuracy, efficiency and uses the im-
bedding parameter p as a small parameter and only a few
iterations are needed to search for an asymptotic solution.
Using this method, we can obtain the following solution
S
(11)

10 1
d
di
cks ecekKc
t
M
(12)
10 i
d
d
i
ekKec e
t

with initial conditions at
(13)
Copyright © 2011 SciRes. AM
G. VARADHARAJAN ET AL.
Copyright © 2011 SciRes. AM
1142


3

2
1 e
e
ue
1
2
23
2
ee
ee




 




 


 
 
(19)

 



3
2
22
2
222222 23
22
ee
ee ee
v

 
  



2

 

(20)
 







323
3
333
23 23 23223
1eee
e
we
 
 


2
   
 
  
(21)
Equations (19)-(21) represent the analytical expression
f the dimensionless substrate concentration
u
o, en-
zyme-substrate concentration v
and free enzyme
concentration w
for all values of parameters
12 3
, , and
 
. .
For steady state condition, the differential Equations
(15)-(17) become as follows:
1 0 uεuv uwv

 (22)
uuvuw 20v
 
0
(23) 
33 3
vw

 
n obtain substrate
(24)
Solving the above equations, we c
concentration u, enzyme substrate c
fr
expression cor-
responding to the substrate concentration u, enzyme sub-
st
ussion
s (15)-(18), are also
lved using numerical methods. The functionbvp4c in
a
oncentration v and
ee enzyme concentration w, as follows:
0, 0, 1uvw 
When t tends to infinity, the analytical Figure 1. Time-dependent behaviour of the reactant con-
centration of the substrate
u
, enzyme-substrate
v
and free enzyme
w
for 6.5
,
lott 12
, 0.4

ed using Equ
4
ation
rate concentration v and free enzyme concentration w
from the Equations (19)-(21) confirms the validity of
mathematical analysis.
4. Results and Disc
and rves
ula
3
(19)-(21). The key to the graph (---) represents Equations
(19)-(21) and (….) denotes simtion.
.03
. The cu are ps
The non-linear differential Equation
so
Scilab software which is a function of solving two-point
boundary value problems (BVPs) for ordinary differen-
tial equations is used to solve this equation. Its numerical
solution is compared with our analytical result in Fig-
ures 1-5 and it gives a satisfactory agreement for all
values of time
. The Scilab program is also given in
Appendix B.
Equations (19)-(21) are the closed and simple analyti-
cal expression of concentrations. Figures 1-5 describes
the dimensionless unsteady-state concentrations versus
time
.The analytical expression of concentrations of
substrate

u
, enzyme-substrate complex
v
and
free eyme

w
nz
have been plotted for various values
of dimensss reaction parameters 12
,
ionle ,

and
3
. Recently ll and Hanson [10] obtained the fol-
lowing analytical expression of concentrayme
bstrate complex c and free enzyme ei assuming
Schne
tion of enz
Figure 2. Time-dependent behaviour of the reactant
concentration of the substrate
u
, enzyme-substrate
v
and free enzyme
w
for 6.5
, 1 . ,
002
23
. and .
001 009
.The curves are plotted using
Equations (19)-(21). The key to the grap (----) represents
Equations (19)-(21) and (….) denotes simulation.
su
G. VARADHARAJAN ET AL.1143
τ
Figure 3. Time-dependent behaviour of the reactant
concentration of the substrate
u
, enzyme-substrate
v
and free enzyme

w
for 6.5
, 1 , 0.008
23
. and .0 00908

. The curves are plotted using
Equations (19)-(21). The key he graph (---) repres
Equations (1 9) -(21 ) and (… .) d simulation.
to tents
enotes
τ
Figure 4. Time-dependent behaviour of the reactant
concentration of the substrate

u
, enzyme-substrate

v
and free enzyme

w
for 6.5
,
key
12
, 0.1 9.5

3
.and0 95
. The curves are plotted
using Equations (19)-(21). The to the graph (---)
represents Equations (19)-(21) and (….) denotes simulation.
d0
dt for t > c
t and 0
c
s
s
for c
tt
 
12
21
10
exp exp
ll

ca
blt lt
kk e



 (25)
2
(26)
where
 
01
exp exp
i
eea ltb lt 
Figure 5. Time-dependent behaviour of the reactant
concentration of the substrate

u
, enzyme-substrate
v
and free enzyme
w
for 10.5
,
12
,
3
. and
0.091.20 8

. Th
The key to
e curv
th
es are plotted
e graph (---)using Equations (19)-(21).
represents Equations (19)-(21) and (….) denotes simulation.

02 1
/
/
el lk
all


01 1
12
/
/
k k
bll k
el
12 1
k
, ,
2
1
1
0
()
M
app
M
kk k
lOt
ks

,

210
app
M
lkks
2
Ot
(27)
Figures 1-5 represent thmensionless concentration
of substrate u, enzyme-substrate complex v, and free
enzyme w for various values of parameter
e di
12
, ,

3
and
.
tration of
decay an
tia
stant. T
From these figures it is inferred that the
substrate u follows a first-orde
d it is always decreasing function
concen-
r exponential
from
f rate con-
the ini-
l value. The concentration of enzyme-substrate com-
plex v initially increases and attains its steady state
value at short intervals of time for all values o
he value of free enzyme w increases slowly
and reaches the steady state when time is very large. The
time taken the steady value depends upon the values of
parameters 12 3
, , and .
 
Our approximate ana-
lytical expressions of substrate, enzyme substrate and
free enzyme concentration are compared with simulation
results in Figures 1-5. A satisfactorygreement is noted.
5. Conclusions
The time dependent non-linear reaction-diffusion equa-
tion has been formulated and solved analytically and
numerically. Analytical expression of substrate, enzyme-
a
substrate and free enzyme concentrations interms of di-
n diffusion parameters mensionless reactio12
, ,

Copyright © 2011 SciRes. AM
G. VARADHARAJAN ET AL.
1144
3
an d
are derived using the HPM. The primary result
of this work is simple approximate calculations of con-
centrations for all values of dimensionless parameters
12 3
, , and
 
. 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
nds of system of coupled non-linear equations in
multi-substrate systems and networks of coupled enzyme
reactions.
6. Acknowledgements
This work was supported by the Council of Scientific
and Industrial Research (CSIR No.01 (2442)/10/EMR-II),
Governmen
to all ki
urai, In
t of Indi
dia for their
ui
ens, J. C. Ma
a. The authors also thank Secretary,
and Principal, Head of the
The Madura College, Ma-
constant encouragement.
, pp. 1123-
z, M. B. Amao, A. N. P Hiner, F.
rtins, E. Brosens, K. Van Belle, D. M.
The Madura College Board
Department of Mathematics,
d
7. References
[1] M. P. Deutscher, “Rat Liver Glutamyl Ribonucleic Acid
Synthetase. I. Purification and Evidence for Separate En-
zymes for Glutamic Acid and Glutamine,” Journal of
Biological Chemistry, Vol. 242, No. 6, 1967
1131.
2] J. Hernadez-R[ Gar-
cia-Canovas and M. Acosta, “Catalase-Like Activity of
Horseradish Peroxidase: Relationship to Enzyme Inacti-
vation by H2O2,” Biochemical Journal, Vol. 354, No. 2,
2001, pp. 107-114.
[3] J. Mess
Jacobs, R. Willem and L. Wyns, “Kinetics and Active
Site Dynamics of Staphylococcus Aureus Reductase,”
Journal of Biological Inorganic Chemistry, Vol. 7, 2002,
pp. 146-156. doi:10.1007/s007750100282
[4] S. Ho and G. S. Mittal, “High Voltage Pulsed Electrical
Field for Liquid Food Pasteurization,” Food Reviews In-
ternational, Vol. 16, No. 4, 2000, pp. 395-434.
doi:10.1081/FRI-100102317
[5] C. P. Yao and R. H Levy, “Inhibition-Based Metabolic
Drug-Drug Interactions: Predictions from in Vitro Data,”
Journal of Pharmaceutical Sciences, Vol. 91, No. 9, 2002,
pp. 1923-1935. doi:10.1002/jps.10179
[6] S. G. Waley, “Kinetics of Suicide Substrates,”
mical Journal, Vol. 185, 1980
Bioche-
, pp. 771-773.
e Kinetik der In-
2-3, 2007, pp. 269-274.
[7] S. G. Waley, “Kinetics of Suicide Substrates. Practical
Procedures for Determining Parameters,” Biochemical
Journal, Vol. 227, 1985, pp. 843-849.
[8] L. Michaelis and M. L. Menten, “Di
vertinwirking,” Biochemische Zeitschrift, Vol. 49, 1913,
pp. 333-369.
[9] G. E. Briggs and J. B. S. Haldane, “A Note on the Kinet-
ics of Enzyme Action,” Biochemical Journal, Vol. 19, No.
2, 1925, pp. 338-339.
[10] S. Schnell and S. M. Hanson, “A Test for Measuring the
Effects of Enzyme Inactivation,” Biophysical Chemistry,
Vol. 125, No.
doi:10.1016/j.bpc.2006.08.010
[11] S. J. Li and Y. X. Liu, “An Improved Approach to
Nonlinear Dynamical System Identification Using PID
Z. Nturforsch, “Applica-
pplied Mechanics and Engineering, Vol.
Neural Networks,” International Journal of Nonlinear
Sciences and Numerical Simulation, Vol. 7, No. 2, 2006,
pp. 177-182.
[12] M. M. Mousa, S. F. Ragab and
tion of the Homotopy Perturbation Method to Linear and
Nonlinear Schrödinger Equations,” Zeitschrift für Natur-
forschung, Vol. 63, 2008, pp. 140-144.
[13] J. H. He, “Homotopy Perturbation Technique,” Computer
Methods in A
178, 1999, pp. 257-262.
doi:10.1016/S0045-7825(99)00018-3
[14] J. H. He, “Homotopy Perturbation Method: A New
Nonlinear Analytical Technique,” Applied Mathematics
and Computation, Vol. 135, No. 1, 2003, pp. 73-79.
doi:10.1016/S0096-3003(01)00312-5
[15] J. H. He, “A Simple Per
Equation,” Applied Mathematics and
turbation Approach to Blasius
Computation, Vol.
140, No. 2-3, 2003, pp. 217-222.
doi:10.1016/S0096-3003(02)00189-3
[16] J. H. He, “Some Asymptotic Methods for Stro
Nonlinear Equations,” International J
ngly
ournal of Modern
Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199.
doi:10.1142/S0217979206033796
[17] J. H. He, G. C. Wu and F. Austin
tion Method Which Should Be Fol
, “The Variational Itera-
lowed,” Nonlinear
lems,”
r Mechanics, Vol. 35,
Science Letters A, Vol. 1, No. 1, 2010, pp. 1-30.
[18] J. H. He, “A Coupling Method of a Homotopy Technique
and a Perturbation Technique for Non-Linear Prob
International Journal of Non-Linea
No. 1, 2000, pp. 37-43.
doi:10.1016/S0020-7462(98)00085-7
[19] D. D. Ganji, M. Amini and A. Kolahdooz, “Analytical
jendran, “Mathematical Model-
erometric Immobi-
Investigation of Hyperbolic Equations via He’s Methods,”
American Journal of Engineering and Applied Sciences,
Vol. 1, No. 4, 2008, pp. 399-407.
[20] S. Loghambal and L. Ra
ing of Diffusion and Kinetics of Amp
lized Enzyme Electrodes,” Electrochimica Acta, Vol. 55,
No. 18, 2010, pp. 5230-5238.
doi:10.1016/j.electacta.2010.04.050
[21] A. Meena and L. Rajendran, “Mathematical Modeling of
Amperometric and Potentiometric Biosensors and System
of Non-Linear Equations—Homotopy Perturbation Ap-
proach,” Journal of Electroanalytical Chemistry, Vol.
644, No. 1, 2010, pp. 50-59.
doi:10.1016/j.jelechem.2010.03.027
[22] A. Eswari and L. Rajendran, “Analytical Solution of
Steady State Current an Enzyme Modified Microcylinder
Electrodes,” Journal of Electroanalytical Chemistry, Vol.
648, No. 1, 2010, pp. 36-46.
doi:10.1016/j.jelechem.2010.07.002
Copyright © 2011 SciRes. AM
G. VARADHARAJAN ET AL.
Copyright © 2011 SciRes. AM
1145
tion method. In this Appendix, we indicate how Equations
topy was constructed to determine the solution of Equations
APPENDIX A
Solution of the Equations (15)-(18) using Homotopy perturba
(19)-(21) in this paper are derived. Furthermore, a Homo
(15)-(17).
1
(1 ) 0
dd
pu puvuvuw






(A1)
dduu

22
ddvv

(1) 0
dd
pvpvuuvuw






(A2)
3333
dd
(1 ) 0
dd
ww
pwpwv


 

 
 
(A3)
The initial approximations are as follows:
and (A4)
Approximate solutions of (A1), (A2) and (A3) are
(A5)
Substituting Equations (A5)-(A7) in Equations (A1)-(A3) and comparing t
can obtain the following differential equations.
(0)1, (0)0uv (0 )=0w
23
12 3
...................pupu pu 
0
uu
23
01 2 3
.........vv pvpv.........pv (A6)
and
23
01 2 3
..................ww pwpwpw...... (A7)
he coefficients of like powers of p, we
00
0
d
: 0
d
u
pu

(A8)
11
1
:
du
pu

100000
0vuvuw
d
 
  (A9)
22
21110110
d
: 0
d
u
puuvuwuw
 
  (A10)
and
00
20
d
: 0
d
v
pv
(A11)
11
2100000
d
: 0
d
v
pvuuvuw
  (A12)
22
22101100110
d
: 0
d
v
pvuuvuvuwuw

(A13)
and
00
30
d
: 0
d
w
pw
 (A14)
11
31 330
d
: 0
d
w
pwv


(A15)
22
32 31
d
:0
d
w
pwv

 (A16)
Solving the Equations (A8)-(A16), and using the boundary conditions (A4) , we can find the following results.
G. VARADHARAJAN ET AL.
1146
0 () ue
(A17)
1() 0u
(A18)

3
ee


 
2
1
1
ee
e
ue
 
 



22
23
2


 


  (A19)
and
0 ()0v
(A20)
2
1
2
()ee
v

(A21)


3
1
22 22
2
2
222232
22
 
1
()
eee eeeee
v



 
 


 




 





(A22)
and

0=0w
(A23)

3
11w e
 (A24)
 







323
333
2
23 23 23223
eee
e
w
 
 

2
   

  
(A25)
According to the HPM, we can conclude that
01
1
()lim ().........
p
uuuuu
2

(A26)
012
1
()lim ()........
p
vvvvv

(A27)
012
1
()lim ().........
p
wwwww

(A28)
26), Equations (A20)-
Equation (A28), we obtain the final results as described in Equations (1
APPENDIX B
Scilab program to find the solutions of the Equations (15)-(18)
fu
;
;
s);
')
'x')
)
TestFunction(t,x)
Using Equations (A17)-(A19) in Equation (A(A22) in Equations (A27) and (A23)-(A25) in
9)-(21).
nction main123456
options= odeset('RelTol',1e-6,'Stats','on');
initial conditions %
x0 = [1; 0;0]
span = [0 1]t
tic
[t,x] = ode45(@TestFunction,tspan,x0,option
toc
figure
hold on
plot(t, x(:,1))
plot(t, x(:,2),'.
plot(t, x(:,3))
legend('x1','x2')
ylabel(
xlabel('t'
return
function [dx_dt]=
Copyright © 2011 SciRes. AM
G. VARADHARAJAN ET AL.
Copyright © 2011 SciRes. AM
1147
;
a*x(1)*x(2)+a*x(1)*x(3)+a*b*x(2);
*x(2)-x(1)*x(3)-c*x(2);
*x(2)-d*x(3);
';
Dimensionless Substrate concentation (None)
Dimensionless Enzyme-substrate concentration (None)
Dimensionless free enzyme concentration (None)
a=6.5;b=0.4;c=4;d=0.3
dx_dt(1)=-a*x(1)+
dx_dt(2) =x(1)-x(1)
dx_dt(3) =d-d
dx_dt = dx_dt
return
APPENDI X C
re Nomenclatu
u
v
w
M
K
Michaelis-Menten constant (
M )
S
K
Equlibrium dissociation constant (
M )
K
Enzyme inactivation constant (
M )
cat Unimolecular rate constant (
k
M )
1
Dimensionless Equlibrium dissociation constant (None)
2
Dimensionless Michaelis-Menten constant (None)
3
Dimensionless enzyme inactivation nstant co(None)
Dimensionless reaction diffusion c nstanto (None)
Dimensionless time (None)