A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method

doi:10.4236/ns.2011.37078

Vol.3, No.7, 556-565 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.37078

A two-parameter mathematical model for immobilized

enzymes and Homotopy analysis method

Rathinasamy Angel Joy1, Athimoolam Meena2, Shunmugham Loghambal2, Lakshmanan

Rajendran2*

1Department of Mathematics, Sri. G.V.G. Visalakshi College for women (Autonomous), Udumalpet, India;

2Department of Mathematics, The Madura College, Madurai, India; *Corresponding Author: [email protected]

Received 20 June, 2011; revised 4 July, 2011; accepted 10 July, 2011.

ABSTRACT

A two parameter mathematical model was de-

veloped to find the concentration for immobi-

lized enzyme systems in porous spherical par-

ticles. This model contains a non-linear term

related to reversible Michaelies-Menten kinetics.

Analytical expression pertaining to the sub-

strate concentration was reported for all possi-

ble values of Thiele module φ and α. In this

work, we report the theoretically evaluated

steady-state effectiveness factor for immobi-

lized enzyme systems in porous spherical par-

ticles. These analytical results were found to be

in good agreement with numerical results.

Moreover, herein we employ new “Homotopy

analysis method” (HAM) to solve non-linear re-

action/diffusion equation.

Keyw ords: Mathematical Modeling;

Michaelis-Menten Kinetics; Homotopy Analysis

Method; Reaction/Diffusion Equation; Effectiveness

factor

1. INTRODUCTION

The enzymes can be easily separated from the reaction

bulk and reused by using immobilized enzymes on a

porous support. Here the reaction occurs only inside the

particle. Hence the external diffusion processes and dif-

fusion within the particles affect the reaction rate. The

internal diffusion effects can be quantitatively expresses

by the effectiveness factor η (ratio of the average rate

inside the particle to the rate in the absence of diffu-

sional limitations). The mathematical models for esti-

mating the effectiveness factor for heterogeneous enzy-

matic systems are developed on the basis of the follow-

ing assumptions [1]. The catalytic particle is spherical

and its radius is R, the enzyme is uniformly distributed

throughout the whole catalytic particle, the enzyme reac-

tion is mono substrate, the system is at steady-state and

is isothermal, the mass transfer resistance between solu-

tion and particle external surface is negligible, the sub-

strate and product diffusion inside the catalytic particle

can be modeled by the first Fick’s law and the effective

diffusivity does not change through out the particle.

In this assumption, a two-parameter model was pro-

posed by Engasser and Horvath and this provides gener-

alized plots of the effectiveness factor as a function of

dimensionless modulus for the evaluation of simple

Michaelis-Menten and product competitive inhibition

kinetics. This model is used in the design of heteroge-

neous enzymatic reactors: fixed bed reactors [2] con-

tinuous tank reactors [3] and fluidized bed reactors [4].

The same model is used for the simulation of a packed

bed immobilized enzyme reactor performing lactose

hydrolysis.

Only numerical solutions were available for all the

above said models since substrate concentration rate is a

non-linear function of the substrate and product concen-

trations. More often finite differences [5] and orthogonal

allocation [6] methods were used to solve the boundary

value problem. Here as in enzymatic kinetics, the result

is non linear equations system whenever the mass bal-

ance equations are non-linear. The solution may not be

unique and can have convergence problem if finite dif-

ferences were used [5]. The orthogonal allocation method

is not reliable when high diffusional limitations occur

because this method uses polynomial expressions to ap-

proach the concentration profiles [6]. To solve the

boundary value problem Range Kutta method can also

be used and here the initial substrate concentration is

needed which is unknown. This can be calculated by

successive calculations (shooting method) [7].

Analytical solutions have been obtained in the limit-

ing cases of zero and first reaction order [8-10]. For the

remaining, numerical calculus has been ordinarily used,

being the different variables of the system expressed in

dimensionless form [11-17]. The calculus complexity

increases when the reaction mechanism is more complex

[18,19] (Michaelis-Menten Kinetics). When reversible

R. A. Joy et al. / Natural Science 3 (2011) 556-565

557

or product competitive inhibition mechanisms have been

considered, only external diffusional limitations [20]

have been evaluated, otherwise unsatisfactory results

were obtained [21-23].

Recently Gomez et al., [1] presented the effectiveness

factor of two-parameter model using Runge-Kutta

method. However, to the best of author’s knowledge, no

general analytical results of substrate, product concen-

trations and effectiveness factor for immobilized enzyme

on porous supports have been published. The purpose of

this article is to derive steady state analytical expression

of substrate, product concentration and the effectiveness

factor using Homotopy analysis method (HAM).

2. FORMULATION OF THE BOUNDARY

VALUE PROBLEM AND ANALYSIS

Figure 1 represents the schematic representation of

the geometry adopted by spherical catalyst particle [1].

In general, it was assumed that in steady-state system the

substrate and product diffusion inside the catalytic parti-

cle can be modeled by the first Fick’s law and the effec-

tive diffusivity does not change through out the particle.

Under the above assumptions, the coupled differential

equations for substrate and product in spherical

co-ordinates are [1]:

2

d

SS

S

DdC

rV

rdr

r









(1)

2

d

dd

PP

S

DC

rV

rr

r









(2)

Now the boundary conditions are [1]

dd

00; 0;

dd

SP

CC

rrr

 (3)

;

SSRPPR

rR CCCC  (4)

where reaction rate









mS Peq

S

mS mPP

VC CK

V

K

CKKC



 .

Here S

C and

P

C denote the dimensional substrate

and product concentration, r is the radial co-ordinate, R

denotes the radius of the particle, S

D and

P

D are the

diffusion-coefficients of the substrate and product re-

spectively, SR

C and

P

R

C denote the local substrate

and product concentration., eq

K

is the reaction equilib-

rium constant, m

K

is the Michaelies-Menten constant

and m

V defines the maximum reaction rate. The form

of S

V determines the mathematical method to solve the

above equations and its complexity. Adding Eqs.1 and 2

and using the boundary conditions given by Eqs.3 and 4

the following relationship can be established:



S

P

PRSR S

P

D

CCC C

D

  (5)

Figure 1. Schematic representation of the geometry

adopted by spherical catalyst particle.

Substituting the value of

P

C in S

V, we can obtain



1

S

mSSE

eq P

S

MM

M

PE SES SE

PPP

D

VCC

KD

VD

KK

KCC CC

KKD











 





(6)

where

P

E

eq

SE

C

KC

,







P

RSPPR

SE

eqS P

CDDC

CKDD



 and







11

P

RSPPR

PEeq SE

eqS P

CDDC

CKC

K

DD





.

We make the non-linear differential equations outlined

in Eqs.1 and 2 dimensionless by introducing the follow-

ing dimensionless parameters:

SSE

SR SE

CC

UCC



, r

R



,



2

1

S

eq P

m

SRSE SS

M

PP

D

KD

RV

CCDD

K

KD



















and



1

M

MPESE

P

S

M

SR SE

PP

K

KCC

K

D

K

CC KD













(7)

where U represents the dimensionless substrate concen-

tration,



denotes the dimensionless radius of the par-

ticle,



and



denote the dimensionless modulus.

R. A. Joy et al. / Natural Science 3 (2011) 556-565

558

Now the Eqs.1 and 2 reduces to the following dimen-

sionless form [1] :

2

1d d

dd

UU

U















 (8)

The boundary conditions are given by

0;



d0

d

U



 (9)

1;



1U (10)

The effectiveness factor is [1]



1

2

0

31 d;

U







 

 (11)

The set of expressions presented in Eqs.9, 10 and 11

define the boundary value problem.

3. HOMOTOPY ANALYSIS METHOD

Liao [24] proposed a powerful analytical method for

nonlinear problems, namely the Homotopy analysis

method (see Appendix A). Different from all reported

perturbation and non-perturbative techniques, the

Homotopy analysis method [25-30] itself provides us

with a convenient way to control and adjust the conver-

gence region and rate of approximation series, when

necessary. Briefly speaking, the Homotopy analysis

method has the following advantages: It is valid even if a

given nonlinear problem does not contain any

small/large parameters at all; It can be employed to effi-

ciently approximate a nonlinear problem by choosing

different sets of base functions. In this paper we employ

HAM to give approximate analytical solutions of cou-

pled non-linear reaction/diffusion Eq.9. Using Homo-

topy analysis method (see Appendix -A) we can obtain

the following new approximate substrate concentration

by solving the Eq.9.



2

42

2

() 11

6

1

20 6

6

7

160

h

U

hh h

hh

h

hh







 











 















 









(12)

Using Eq.12, we can obtain the effectiveness factors





 

23

1

(,)

18108 1

hA

hAA hBB



 



 







 



(13)

where



22

61

and arctanh

Ahh

h

BA





 







(14)

Eqs.12 and 13 represent the new approximate ana-

lytical expression of substrate and effectiveness factor

for all values of parameters

4. LIMITING CASES

4.1. Case I: First Order Catalytic Kinetics

In this case U



. Now the above Eq.9 reduces to

2

1d d

dd

UU















 (15)

Using reduction of order method we can obtain the

substrate concentration as



sinh

U



























(16)

The effectiveness factor is



31 coth 1



 













(17)

4.2. Case II: Zero Order Catalytic Kinetics

In this case U



. Now the Eq.9 reduces to

2

1d d

dd

U

















(18)

Solving Eq.18, we can obtain the concentration of the

substrate as follows:



2

1

11

6

U





 (19)

The effectiveness factor is







1181 arctanh

108 arctanh1

lll l

l



 



 















(20)

where





261l





.

5. RESULTS AND DISCUSSION

Eqs.12 and 13 represent the analytical solution of the

concentration of substrate and effectiveness factor re-

spectively. The Thiele modulus



can be varied by

R. A. Joy et al. / Natural Science 3 (2011) 556-565

559

changing either the particle radius or the amount of con-

centration of substrate. This parameter describes the

relative importance of diffusion and reaction in the par-

ticle radius. When



is small, the kinetics are the

dominant resistance; The overall uptake of substrate in

the enzyme matrix is kinetically controlled. Under these

conditions, the substrate concentration profile across the

membrane is essentially uniform. In contrast, when the

Thiele modulus



is large, diffusion limitations are the

principal determining factor.

5.1. Numerical Simulation

The HAM provides an analytical solution in terms of

an infinite power series. However, there is a practical

need to evaluate this solution and to obtain numerical

values from the infinite power series. The consequent

series truncation and the practical procedure conducted

to accomplish this task, together transforms the other-

wise analytical results into an exact solution, which is

evaluated to a finite degree of accuracy. In order to in-

vestigate the accuracy of the HAM solution with a finite

number of terms, the system of differential equation

were solved. To show the efficiency of the present

method for our problem in comparison with the numeri-

cal solution (SCILAB program) we report our results

graphically. The SCILAB program is also given in Ap-

pendix (C).

5.2. Comparison of Analytical and

Numerical Results

Figures 2(a)-(c) show the dimensionless steady-state

substrate concentration for the different values of



calculated using Eq.13. From these figures, we can see

that the value of the concentration increases when Thiele

modulus



decreases. The concentration of substrate



U



increases slowly and rises abruptly when

0.4



 and all values of



. When 1



 and 5





the concentration of substrate



1U



 (steady- state

value). When



is small, the overall uptake of sub-

strate in the enzyme matrix is kinetically controlled and

the substrate concentration profile across the membrane

is identical.

Figure 3 represents the effectiveness factor



versus

dimensionless Thiele modulus



for different values of

dimensionless module



. From this figure, it is in-

ferred that, a constant value of dimensionless module



, the effectiveness factor decreases quite rapidly as

dimensionless module



increases, approaching zero

at high values, which corresponds to internal diffusion

controlled processes. Moreover, it is also well known

that, a constant value of dimensionless module



, the

(a)

(b)

(c)

Figure 2. Comparison of normalized substrate concentration U

versus normalized distance ρ for various values of Thiele

moduilli φ. (a) α = 2 (b) α = 5 (c) α = 10. The curves are plot-

ted using Eq. (13). (–) denotes the analytical results (+) de-

notes the numerical results. Here h = –0.87.

R. A. Joy et al. / Natural Science 3 (2011) 556-565

560

Figure 3. The normalized effectiveness factor η versus Thiele

moduilli φ for various values of parameter α. The curves are

plotted using Eq.13. Here h = –0.135.

effectiveness factor increases with increasing values of



.

6. CONCLUSIONS

A non-linear time independent equation has been

formulated and solved analytically using Homotopy

analysis method. The primary result of this work is the

first approximate calculations of substrate concentrations

and effectiveness factor for non-linear Michaelis-Menten

kinetic scheme. A simple closed form of analytical ex-

pressions of steady-state substrate and effectiveness fac-

tor are given. The analytical expressions for the substrate

concentration profiles for all values of parameters



and



are derived using Homotopy analysis method.

This method is an extremely simple method and it is also

a promising method to solve other non-linear equations.

The extension of this procedure to other direct reaction

of substrate at underlying microdisc electrode surface

seems possible.

7. ACKNOWLEDGEMENTS

This work was supported by the Department of Science and Tech-

nology (DST) Government of India. The authors also thank Mr.M.S.

Meenakshisundaram, Secretary, The Madura College Board, Principal

and S.Thiagarajan Head of the Department of Mathematics, The

Madura College, Madurai, India for their constant encouragement. It is

our pleasure to thank the referees for their valuable comments.

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APPENDIX A

Basic Idea of Liao’s [31] Homotopy Analysis

method

Consider the following differential Eq.31:



0Nut 

 (A1)

where, Ν is a nonlinear operator, t denotes an independ-

ent variable, u(t) is an unknown function. For simplicity,

we ignore all boundary or initial conditions, which can

be treated in the similar way. By means of generalizing

the conventional homotopy method, Liao constructed the

so-called zero-order deformation equation as:



0

1; ;pLtpu tphHtNtp









(A2)

where





0,1p is the embedding parameter, h ≠ 0 is a

nonzero auxiliary parameter, H(t) ≠ 0 is an auxiliary

function, L is an auxiliary linear operator,





0

ut

is an

initial guess of u(t) and



:tp



is an unknown func-

tion. It is important, that one has great freedom to choose

auxiliary unknowns in HAM. Obviously, when 0p



and 1p, it holds:

 

0

;0tut



and

 

;1tut



 (A3)

respectively. Thus, as p increases from 0 to 1, the solu-

tion



;tp



varies from the initial guess



0

ut to the

solution



ut . Expanding



;tp



in Taylor series

with respect to p, we have:

 

0

1

;m

m

tputut p









 (A4)

where

 

0

;

1

!

m

mp

m

tp

ut mp







 (A5)

If the auxiliary linear operator, the initial guess, the

auxiliary parameter h, and the auxiliary function are so

properly chosen, the series Eq.A4 converges at p =1

then we have:

 

0

1

m

utu tut







. (A6)

Define the vector



01

,, ,

nn

uu uu (A7)

Differentiating Eq.A2 for m times with respect to the

embedding parameter p, and then setting p = 0 and fi-

nally dividing them by m!, we will have the so-called

mth-order deformation equation as:







11mmm mm

LuuhH t





u (A8)

where



1

10

1

;

1

1!

m

mm p

m

Ntp

mp

















u (A9)

and

0, 1,

1, 1.

m









 (A10)

Applying 1

L



on both side of Eq.A8 , we get











1

11

mmm mm

ututhL Ht









 



u (A11)

In this way, it is easy to obtain m

u for 1,m at

th

M

order, we have

 

0

M

m

utu t



 (A12)

when M, we get an accurate approximation of

the original Eq.A1 . For the convergence of the above

method we refer the reader to Liao [31]. If Eq. A1 admits

unique solution, then this method will produce the

unique solution. If Eq.A1 does not possess unique solu-

tion, the HAM will give a solution among many other

(possible) solutions.

APPENDIX B

2

1d d

dd

UU

U















 (B1)

Now the boundary conditions become

d

0,0

d

U





 (B2)

1, 1U





 (B3)

In order to solve Eq.B1 by means of the HAM, we

first construct the Zeroth-order deformation equation by

taking





1Ht



,



2

d2d

1d

d

d2d

d

p

ph









 















 





(B4)

subject to the boundary conditions





0; 0p



,





1; 1p



 (B5)

where





0, 1p is an embedding parameter and h ≠ 0

is the so-called convergence control parameter. When

0p







2

d;0 d;0

20

d

 







(B6)

R. A. Joy et al. / Natural Science 3 (2011) 556-565

563

From Eq.B6 we get

01





(B7)

When 1p the Eq. B4 is equivalent to Eq.B1, thus

it holds







;1 U





 (B8)

Expanding



;p



in Taylor series with respect to

the embedding parameter p, we have,

 

0

1

;m

m

pUU p

 







 (B9)

where







0;0U



 (B10)

and





m

U





1,2, m will be determined later.

Note that the above series contains the convergence con-

trol parameter h. Assuming that h is chosen so properly

that the above series is convergent at 1p. We have

the solution series as

 

0

1

m

UU U











 (B11)

where







0

;

1

!

m

mp

m

Up

Ut mp







 (B12)

substituting Eq .B 9 into the zeroth-order deformation

Eqs.B4 and B5 and equating the co-efficient of the like

powers of p we have,



2

111

0

2

00

0

2

dd

2

:1

d

dd

20

d

phh

h









































(B13)



2

222

0

2

00

11

22

dd

2

:1

d

dd

22

0

dd

phh

hh













 

















 





(B14)

and so on. From Eqs.B13 and B14 we get



2

11

6

h







 (B15)



42

221

20 6

6

7

160

hh h

hh

h

hh







































(B16)

Adding Eqs.B7 , B15 and B1 6 we get Eq.13 in the

text.

APPENDIX C

Determining the Region of h for Validity

The analytical solution should converge. It should be

noted that the auxiliary parameter h controls the conver-

gence and accuracy of the solution series. The analytical

solution represented by Eq.13 contains the auxiliary

parameter h, which gives the convergence region and

rate of approximation for the homotopy analysis method.

In order to define region such that the solution series is

independent of h, a multiple of h-curves are plotted. The

region where the distribution of U and U versus h is a

horizontal line is known as the convergence region for

the corresponding function. The common region among

the )(



U and its derivatives are known as the overall

convergence region. To study the influence of h on the

convergence of solution, the h-curves of U(0.5) and





0.5U are plotted in Figures 4(a) and (b) respectively,

for



=5 and 10





. These figures clearly indicate that

the valid region of h is about –1 < h < –0.8. Similarly we

can find the value of the convergence control parameter h

for different values of constant parameters.

APPENDIX D

function pdex1

m = 2;

x = linspace(0,1);

t = linspace(0,100);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

% Extract the first solution component as u.

u = sol(:,:,1);

% A surface plot is often a good way to study a solution.

surf(x,t,u)

title(Numerical solution computed with 20 mesh points.)

x label(Distance x’)

y label(Time t’)

% A solution profile can also be illuminating.

figure

plot(x,u(end,:))

title(‘Solution at t = 2’)

x label(Distance x')

y label(u(x,2)')

% --------------------------------------------------------------

function [c,f,s] = pdex1pde(x,t,u,DuDx)

c = 1;

f = DuDx;

k = 0.01;

alpha = 0.5;

s = –k*u/(alpha + u);

% --------------------------------------------------------------

function u0 = pdex1ic(x)

u0 = 1;

% --------------------------------------------------------------

R. A. Joy et al. / Natural Science 3 (2011) 556-565

564

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)

pl = 0;

ql = 1;

pr = ur–1;

qr = 0;

APPENDIX E

Consider

U

d

dU

d



















2

2

1 (E1)

(a)

(b)

Figure 4. The h curves to indicate the convergence region, for

φ = 5 and α = 10.

Boundary conditions are:

.0 ;0

,1 U;1







d

dU (E2)

Using reduction of order, we have

2; ; 0.PQ R









(E3)

Let

Uuv (E4)

Substitute Eq.E4 in Eq.E1 and choose u such that

d

20

d

uPu







Substituting the value of P, we obtain

1

u





(E5)

Now the Eq.E1 reduces to

11

vQvR





 (E6)

2

11

1d

Q, 0

2d 4

PR

QR

u







  (E7)

Substituting Eq.E7 in Eq.E6 we obtain

0.vv









 (E8)

Solving we obtain

AeBe- .v







 (E9)

Substituting Eq.E5 and Eq.E9 in Eq.E4 we have

1AeBe- U



















(E10)

Using the boundary conditions we obtain the value of

the constants as

11

; B

2sinh 2sinh

A







 

 

 

(E11)

Substituting in Eq.E10 we obtain the solution as

sinh

U

























(E12)

R. A. Joy et al. / Natural Science 3 (2011) 556-565

565

Appendix F

Nomenclature and Units

Symbols

S

C Concentration of the substrate (mol·cm–3)

P

C Concentration of the product (mol·cm–3)

r Radial co-ordinate (cm)

SR

C Local substrate concentration (mol·cm–3)

P

R

C Local product concentration (mol·cm–3)

eq

K

Reaction equilibrium constant (none)

m

K

Michaelis-Menten constant (mol·cm–3)

S

D

Diffusion coefficient of the substrate (cm2·sec–1)

P

D

Diffusion coefficient of the product (cm2·sec–1)

R Radius of the particle (cm)

S

V Local reaction rate per unit of catalytic particle volume

(mol·cm–3·s–1)

m

V Maximum reaction rate per unit of catalytic particle

volume (mol·cm–3·s–1)

U Normalized substrate concentration (none)

,





Normalized Thiele modulus (none)



Effectiveness factor (none)

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