Mathematical Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that Follow the Michaelis – Menten Mechanism

In this paper, mathematical models of immobilized enzyme system that follow the Michaelis-Menten mechanism for both reversible and irreversible reactions are discussed. This model is based on the diffusion equations containing the nonlinear term related to Michaelis-Menten kinetics. An approximate analytical technique employing the modified Adomian decomposition method is used to solve the non-linear reaction diffusion equation in immobilized enzyme system. The concentration profile of the substrate is derived in terms of all parameters. A simple expression of the substrate concentration is obtained as a function of the Thiele modulus and the Michaelis constant. The numerical solutions are compared with our analytical solutions for slab, cylinder and spherical pellet shapes. Satisfactory agreement for all values of the Thiele modulus and the Michaelis constant is noted. Graphical results and tabulated data are presented and discussed quantitatively to illustrate the solution.


Introduction
Many problems in theoretical and experimental biology involve reaction diffusion equations with nonlinear chemical kinetics. Such problems arise in the formulation of substrate and product material balances for enzymes immobilized within particles [1] in the description of substrate transport into microbial cells [2], in membrane transport, in the transfer of oxygen to respiring tissue and in the analysis of some artificial kidney systems [3]. For such cases, the problem is often well poised as a two-point nonlinear boundary-value problem because of the saturation, Michaelis-Menten, or Monod expressions which are used to describe the consumption of the substrate.
Mireshghi et al [4] provide a new approach for estimation of mass transfer parameters in immobilized enzymes systems. Benaiges et. al [5] studied the isomerzation of glucose into fructose using a commercial immobilized glucose-isomerase. The Michaelis-Menten equation is the most common rate expression used for enzyme reactions. This equation can also be used for immobilized enzymes [6,7]. Many authors discussed the application of immobilized enzyme reactors extensively, but immobilized enzyme engineering is still in its infancy. Several general categories It is further assumed that: (1) The kinetics of the free enzyme are described by the Michaelis-Menten equation for irreversible reactions and by the modified Michaelis-Menten equation for reversible reactions; (2) No partition effect exists between the particle surface and the interior; (3) The temperature, density, and effective diffusivity of reactants inside the particle are constant; (4) A quasi-steady-state condition is attained; (5). The partition effect between the support and bulk fluid phase is neglected; (6) Enzyme deactivation is neglected. Based on these above assumptions, the governing differential equations with boundary conditions for irreversible and reversible reactions are

Irreversible Reactions
A differential mass balance equation for the substrate for irreversible reactions in dimensionless form can be represented as follows [15]: The boundary conditions are given by (with external mass transfer resistance) (4) where C represents the dimensionless substrate concentration, X represents the dimensionless distance to the center or the surface of symmetry of the pellet, φ , b β and Bi represents the Thiele module, dimensionless parameter for bulk fluid phase and Biot number respectively. The g characterizes the shape of the immobilized catalyst with g =1, 2, 3 for a slab, cylindrical, and spherical pellets respectively. So it can be regarded as a 'shape factor' for the particle. The dimensionless variables are defined as follows: In the above expressions, the parameters 1, , , , and e b x R K D S S represent the distance to the center, the half-thickness of the pellet, the external mass transfer coefficient, the effective diffusivity of the substrate in the pellet, the substrate concentration inside the pellet and substrate concentration in the bulk fluid phase respectively. and m m K V are the kinetic parameters. The equation (1) also describes the temperature or concentration variation in many fields of physics, chemistry, biology, biochemistry, and many others [7][8][9][10][11][12][13][14]. The effectiveness factor, Ef is given by The initial substrate reaction rate 0 v is given by where 0 b S denotes the initial substrate concentration.

Reversible Reactions
For reversible reactions, the governing differential equation for the dimensionless substrate concentration in the pellets is the same as Eq. 1 except the following dimensionless parameters.
where , , and where α represent the ratio of the catalyst volume to the volume of the fluid phase reactor, ( 0.0252) a = is the ratio of the catalyst volume to the volume of fluid phase in the reactor. The Eqn. (9) is solved with the initial condition ( 0) 1 Y t = = .

Analytical Expression of the Concentration for Irreversible and Reversible Reactions Using MADM
In the recent years, much attention is devoted to the application of the modified Adomian decomposition method (MADM) to the solution of various non-linear problems in physical and chemical sciences. This method is used to find the approximate analytical solution in terms of a rapidly Follow the Michaelis -Menten Mechanism convergent infinite power series with easily computable terms [17][18]. In other words, the zeroth component used in the standard ADM can be divided into the two functions [19][20][21]. The ADM is unique in its applicability, accuracy and efficiency and only a few iterations are needed to find the asymptotic solution. The basic concept of the method is given in Appendix A.

Irreversible Reaction Without External Mass Transfer Resistance
Solving equation (1) using this method (see Appendix B), we obtain the concentration of the immobilized catalyst with g =1, 2, 3 for a slab, cylindrical, and spherical pellets as follows: Effective factor Ef for slab, cylindrical and spherical is ( ) Using Eqns. (7) and (12) the initial substrate reaction rate 0 v can be obtained as follows:

Irreversible Reaction with External Mass Transfer Resistance
The substrate concentration with external mass transfer resistance for the initial and boundary conditions (Eqns. (2) and (4)) is obtained (see Appendix C) from Eqn. (1) as follows: The change in reaction rate can be expressed quantitatively by introducing the effectiveness factor, Ef . Using Eqn. (6), effective factor for slab, cylindrical and spherical is ( ) Using Eqns. (7) and (15), the initial substrate reaction rate 0 v can be obtained as follows: Summary of all the expression of substrate concentration and effectiveness factor for with and without external mass transfer resistance are also given in Table 1.

Reversible Reaction
By replacing the variables , and by , , the Eqns. (11 -14), we can obtain the concentration of substrate and effective factor of reversible reaction.

Analytical Expression of the Concentration of Substrate Using the New Approach of HPM
The advantage of the new homotopy perturbation method (HPM) is that it does not need a small parameter in the system [32]. Recently, many authors have used HPM for various problems and reported the efficiency of the HPM to handling nonlinear engineering problems [33][34][35]. Recently, a new approach to HPM is introduced to solve the nonlinear problem, in which one will get better simple approximate solution in the zeroth iteration [19]. In this paper, a new approach to the Homotopy perturbation method is applied (Appendix D) to solve the nonlinear differential equation (9). Using this method, the analytical expressions of the substrate concentrations can be obtained as follows:

Numerical Simulation
The non-linear differential equations (1) and (7) for the given initial boundary conditions are solved numerically using the Matlab program [36]. The numerical values of parameters used in this work are given in Table 2 and Table  3. Its numerical solution is compared with our analytical results in Tables (3) -(5) and it gives satisfactory agreement. In all the case, the average relative error is less than 1.3%. The Matlab program is also given in Appendix-E and F.
Dimensionless parameter in reaction for bulk fluid phase
Plot of effectiveness factor Ef against Thiele modulus φ and Michaelis -Menten constant b β is shown in Fig. 9 (a -b).
Effectiveness factor is a dimensionless pellet production rate that measures how effectively the catalyst is being used. For η near unity, the entire volume of the pellet is reacting at the same high rate because the reactant is able to diffuse quickly through the pellet. For η near zero, the pellet reacts at low rate. The reactant is unable to penetrate significantly into the interior of the pellet and the reaction rate is small in a large portion of the pellet volume. The effectiveness factor decreases from its initial value, when the diffusional restriction or b β increases. The effectiveness factor is maximum (   13)) initial substrate reaction rate with the numerical result [24] and experimental data [24] (Refer Table 3) for case 1.
Our analytical expression (Eqn. (17)) for the concentration of substrate is compared with the experimental results in Fig. (11). Good agreement with the experimental data is noted. From this figure, it is inferred that reversible substrate   (17)) for substrate concentration with the numerical result [24] and experimental data [24] for the time courses of substrate consumption in a batch reactor model (Refer Table 3).

Conclusions
In this paper an approximate analytical solutions of the nonlinear initial boundary value problem in Michaelis-Menten kinetics have been derived. The modified Adomian decomposition method (MADM) is used to obtain the solutions for the non-linear model of an immobilized biocatalyst enzyme. Approximate analytical expressions for the concentration of substrate and the effective factor in immobilized biocatalyst enzymes are derived. The analytical solutions agree with the experimental results and numerical solutions (Matlab program) with and without external mass resistance for a slab, cylindrical and spherical pellets. These analytical results are more descriptive and easy to visualize and optimize the kinetic parameters of immobilized enzymes.

Appendix A: Basic Concept of Modified Adomian Decomposition Method
Consider the singular boundary value problem of 1 n + order nonlinear differential equation in the form Where , 1 m n n ≤ ≥ , so, the problem can be written as The inverse operator 1 L − is therefore considered a 1 n + fold integral operator, as below where the components ( ) n y x of the solution ( ) y x will be determined recurrently. Specific algorithms were seen in [8,12] to formulate Adomian polynomials. The following algorithm: can be used constant Adomian polynomials, when ( ) can be used to approximate, the exact solution. The approach presented above can be validated by testing it on a variety of several linear and nonlinear initial value problems.

Appendix B: Analytical Solution of Substrate Concentration Without External Mass Transfer Resistance
The solutions of Eq. (1) for 3 g = allow us to predict the concentration profiles of dimensionless substrate concentration in immobilized enzymes. In order to solve Eq.
(1) can be written with the operator form Where A and B are the constants of integration. We let, We identify the zeroth component as And the remaining components as the recurrence relation where n A are the Adomian polynomials of 0, 1, ... , n C C C .
We can find the first few n A as follows: Apply the boundary conditions in (B. 1) we get, Again to find 1 C Using (B. 9) in (B. 10), Again using this formula to find 1 C , Integrating Eqn. (B. 12), Where A and B are integrating constants. Again using boundary conditions Eqn. (B. 13) becomes, ( ) ( ) Now, consider ( ) 120 36 Where C and D are integrating constants. Apply boundary conditions we get the value for C and D, Therefore Eqn. (B. 17) in the form, Adding the Eqns. (B. 9), (B. 14) and (B. 19) we get the solution Eqn. (7). Similarly, to apply the above method for 1, 2 g g = = to find the solution.

Appendix C: Analytical Solution of Substrate Concentration with External Mass Transfer Resistance
The solutions of Eq. (1) for 3 g = allow us to predict the concentration profiles of dimensionless substrate concentration in immobilized enzymes. In order to solve Eq.
(1) can be written with the operator form , Applying the inverse operator 1 L − on both sides of Eq. (C.1) yields ( ) Where A and B are the constants of integration. We let, We identify the zeroth component as And the remaining components as the recurrence relation where n A are the Adomian polynomials of 0, 1, ... , n C C C .
We can find the first few n A as follows: Apply the boundary conditions in (C. 9) we get, 0 1 C = (C.10) Again to find 1 C Using (C. 10) in (C. 11), Again using this formula to find 1 C , Integrating Eqn. (C. 13), Where A and B are integrating constants. Again, using boundary conditions Eqn. (C. 14) becomes, Where C and D are integrating constants. Apply boundary conditions we get the value for C and D, Therefore Eqn. (C. 11) in the form, ( ) Adding the Eqns. (C. 10), (C. 15) and (C. 20) we get the solution Eqn. (13). Similarly, to apply the above method for 1, 2 g g = = to find the solutions. β , (ii). Thiele modulus φ Figure 13. Plot of the three-dimensional dimensionless concentration C against the dimensionless distance X for the three pellets calculated using Eqn. (14) (with mass-transfer resistance).