# Mathematical Modeling and Analysis of Nonlinear Enzyme Catalyzed Reaction Processes

Authors: D. Mary Celin Sharmila; T. Praveen; L. Rajendran
Journal: Journal of Theoretical Chemistry
 (2013)

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## Abstract

A mathematical model for the nonlinear enzymatic reaction process is discussed. An approximate analytical expression of concentrations of substrate, enzyme, and free enzyme-product is obtained using homotopy perturbation method (HPM). The main objective is to propose an analytical solution, which does not require small parameters and avoid linearization and physically unrealistic assumptions. Theoretical results obtained can be used to analyze the effect of different parameters. Satisfactory agreement is obtained in the comparison of approximate analytical solution and numerical simulation.


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## Body

## 1. Introduction

The importance of biocatalytic processes and reactions for organic synthesis and the pharmaceutical food and cosmetics industry has been constantly growing during the last few years [1, 2]. From a synthetic point of view, enzymes are highly efficient catalysts for an extremely broad palette of reactions [3]. Enzymes of one type, but from different origins, are specialized for substrates, positions in substrates, and products [4]. Enzyme reactions do not follow the law of mass action directly. The rate of the reaction only increases to a certain extent as the concentration of substrate increases. The maximum reaction rate is reached at high substrate concentration due to enzyme saturation. This is in contrast to the law of mass action that states that the reaction rate increases as the concentration of substrate increases [5]. Various simplified analytical models have been developed over the last 20 years. In brief, the analysis involves the construction and solution of reaction/diffusion differential equations, resulting in the development of approximate analytical expressions for [6, 7] nonlinear enzyme catalyzed reaction processes.The simplest model that explains the kinetic behaviour of enzyme reactions is the classic 1913 model of Michaelis and Menten [8] which is widely used in biochemistry for many types of enzymes. The Michaelis-Menten model is based on the assumption that the enzyme binds the substrate to form an intermediate complex which then dissociates to form the final product and release the enzyme in its original form. The schematic representation of this two-step process is given by
(1)E+S↔k-1k1C→k2E+P,
wherek1, k-1, and k2 are constant parameters associated with the rates of the reaction. Note that it is generally assumed that the second step of the reaction equation (1) is irreversible. In reality, this is not always the case. Typically, reaction rates are measured under the condition that the product is continually removed, which prevents the reverse reaction of the second step from occurring effectively.In this paper we have derived an expression for concentration of substrate, enzyme-substrate, and free enzyme-product with nonmechanism based enzyme inactivation, in terms of dimensionless reaction diffusion parametersε,λ1,λ2, andλ3using homotopy perturbation method (HPM). Comparative study of the same with numerical simulation is presented.


## 2. Mathematical Formulation of the Problems

If a small amount of enzyme is used and all but one substrate is kept constant, then the rate of the enzymatically catalyzed reaction depends on the substrate concentration and initial rate as in the equationυ=υmax[S]/(KM+[S]), given by [9], where KM is the Michaelis constant: KM=(k-1+k2)/k1. The typical notation of the enzyme catalyzed reaction with one substrate can be given as [10]
(2)A+E↔k2k1X↔k4k3P+E,
where A is substrate, E is enzyme, X is enzyme-substrate complex, and P is free enzyme product. The kinetic equations consist of(3a)d[A]dt=k2[X]-k1[A][E],(3b)d[E]dt=(k2+k3)[X]-(k1[A]+k4[P])[E],(3c)d[P]dt=k3[X]-k4[P][E],with a conservation relation given in [6]:
(4)[E]+[X]=[E]total.
It is obvious that the derivative of a substrate with respect to time gives the rate. Thus, the rate is a function of compounds[V],[A](intracellular and extracellular), enzyme concentrations E and kinetic parameterk. However, the enzyme concentration is hidden in the kinetic constants in the parameter vectork; herewith, we can writevas a function of[V],[A], and[P]; that is,v=v([A],[V],k)[11].The more general form of (3a)–(3c) can be written in the form of(5a)d[A]dt=ν2-ν1,(5b)d[E]dt=ν2+ν3-ν1-ν4,(5c)d[X]dt=ν1+ν4-ν2-ν3,(5d)d[P]dt=ν3-ν4.The form of rate equations is as follows:
(6)ν1=k1[A][E],ν2=k2[X],ν3=k3[X],ν4=k4[P][E].
The initial condition att=0are as follows:
(7)[A]=A0,[E]=E0,[X]=0,[P]=0.
The concentration of the reactants in (5a)–(5d) is denoted by lower case letters
(8)s=[A],e=[E],c=[X],p=[P],
The law of mass action leads to the system of following non-linear kinetic equations [12]:(9a)dsdt=-k1se+k2c,(9b)dedt=-k1se+k2c+k3c-k4pe,(9c)dpdt=k3c-k4pe,(9d)dcdt=k1se-k2c-k3c+k4pe,with the boundary conditions being
(10)s(0)=s0,e(0)=e0,p(0)=0,c(0)=0.
Adding (9b) and (9c), we get
(11)dedt+dcdt=0.
Using the initial conditions (10) we obtain
(12)e(t)+c(t)=e0.
With this, the system of ordinary differential equations reduces to the following three differential equations:(13a)dsdt=-k1s(e0-c)+k2c,(13b)dcdt=k1s(e0-c)-(k2+k3)c+k4p(e0-c),(13c)dpdt=k3c-k4p(e0-c).With initial conditionss(0)=s0, c(0)=0, andp(0)=0. By introducing the following set of nondimensional variables and parameters,
(14)τ=k1e0tε,u(τ)=s(t)s0,v(τ)=c(t)s0,(15)w(τ)=p(t)s0,λ1=k2k1s0,λ2=k3k1s0,(16)λ3=k4k1,ε=e0s0,η=λ1+λ2,
the system of (13a)–(13c) and the initial conditions (10) can be represented in dimensionless form as follows:(17a)dudτ=-εu+uv+λ1v,(17b)dvdτ=εu-ηv+λ3εw-uv-λ3v,(17c)dwdτ=λ2v-λ3εw+λ3vw,with
(18)u(0)=1,v(0)=0,w(0)=0.


## 3. Implementation of the HPM

We indicate how (33)–(35) in this paper are derived. To find the solution of (14)–(16), we first construct a homotopy as follows:
(19)(1-p)[dudτ+εu]+p[dudτ+εu-uv-λ1v]=0,(20)(1-p)(dvdτ+ηv)+p(dvdτ+ηv-εu-λ3εw+uv+λ3vw)=0,(21)(1-p)(dwdτ+λ3εw)+p(dwdτ+λ3εw-λ2v-λ3vw)=0.
And the initial approximations are as follows:
(22)u(0)=1,v(0)=0,w(0)=0.
Approximate solutions of (33)–(35) are
(23)u=u0+pu1+p2u2+p3u3+⋯(24)v=v0+pv1+p2v2+p3v3+⋯(25)w=w0+pw1+p2w2+p3w3+⋯.
Substituting (23)–(25) into (19)–(21), respectively, and comparing the coefficients of like powers ofp, we can obtain the following differential equations for the concentration of substrate:
(26)p0:du0dτ+εu0=0,p1:du1dτ+εu1-u0v0-λ1v0=0,p2:du2dτ+εu2-(u0v1+u1v0)-λ1v1=0.
For enzyme substrate concentrationv,
(27)p0:dv0dτ+ηv0=0,p1:dv1dτ+ηv1-εu0-λ3εw0+u0v0+λ3v0w0=0,p2:dv2dτ+ηv2-εu1-λ3εw1+u0v1+u1v0+λ3(v0w1+v1w0)=0.
For product concentrationw(28)p0:dw0dτ+λ3εw0=0,p1:dw1dτ+λ3εw1-λ2v0-λ3v0w0=0,p2:dw2dτ+λ3εw2-λ2v1-λ3(v0w1+v1w0)=0,p3:dw2dτ+λ3εw3-λ2v2-λ3(v0w2+v1w1+v2w0)=0.
Solving (26)–(28), and using the boundary conditions (22), we can find the following results:
(29)u0(τ)=e-ετ,u1(τ)=0,u2(τ)=1(η-ε)[-e-2ετ+e-ετ]+εη(η-ε)[e-(η+ε)-e-εt]+λ1ετe-ετ(η-ε)+λ1ε(η-ε)[e-ητ-e-ετ],v0(τ)=0,v1(τ)=εη-ε[e-ετ-e-ητ],v2(τ)=ε[e-ητ-e-2ετ](η-ε)(2ε-η)+[e-(ε+η)τ-e-ητ](η-ε),w0(τ)=0,w1(τ)=0,w2(τ)=λ2ε(η-ε)[(e-ετ-e-λ3ετ)(λ3-1)ε-(e-ητ-e-λ3ετ)(λ2ε-η)],w3(τ)=λ2[e-2ετ-e-λ3ετ](ε-η)(η-2ε)(λ3-2)-λ2[e-(ε+η)τ-e-λ3ετ](ε-η)(λ3-ε-η)w3(τ)=-λ2ε[e-ητ-e-λ3ετ](ε-η)(η-2ε)(λ3ε-η)+λ2[e-ητ-e-λ3ετ](ε-η)(λ3ε-η).
According to the HPM, we can conclude that
(30)u(τ)=limp→1u(τ)=u0+u1+u2,(31)v(τ)=limp→1v(τ)=v0+v1+v2,(32)w(τ)=limp→1w(τ)=w0+w1+w2+w3.
Substitute (29) in (30)-(31) we obtain (33)–(35) in the text.


## 4. Analytical Solution of Substrate, Enzyme, Enzyme-Substrate Complex, and Free Enzyme Product Using HPM

Non-linear phenomena play a crucial role in applied mathematics and chemistry. Construction of particular exact solutions for these equations remains an important problem. Finding exact solutions that have a physical, chemical, or biological interpretation is of fundamental importance. The investigation of exact solution of non-linear equation is interesting and important. In the past, many authors mainly had paid attention to study solution of non-linear equations by using various methods, variational iteration method [13], and homotopy perturbation method [14–17].The homotopy perturbation method has been extensively worked out over a number of years by numerous authors. The idea has been used to solve nonlinear boundary value problems [15], integral equations [18–20], Klein-Gordon and Sine-Gordon equations [21], Emden-Flower type equations [22], and many other problems. This wide variety of applications shows the power of the HPM to solve functional equations. The HPM is unique in its applicability, accuracy and efficiency. The HPM [23] 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, we can obtain the following solution to (14)–(16) (see the appendix):
(33)u(τ)=e-ετ+[-e-2ετ+e-ετ](η-ε)+ε[e-(η+ε)-e-ετ]η(η-ε)u(τ)=+λ1ετe-ετ(η-ε)+λ1ε[e-ητ-e-ετ](η-ε),(34)v(τ)=εη-ε[e-ετ-e-ητ]+ε[e-ητ-e-2ετ](η-ε)(2ε-η)v(τ)=+[e-(ε+η)τ-e-ητ](η-ε),(35)w(τ)=λ2ε(η-ε)[(e-ετ-e-λ3ετ)(λ3-1)ε-(e-ητ-e-λ3ετ)(λ2ε-η)]w(τ)=+λ2[e-2ετ-e-λ3ετ](ε-η)(η-2ε)(λ3-2)-λ2[e-(ε+η)τ-e-λ3ετ](ε-η)(λ3-ε-η)w(τ)=-λ2ε[e-ητ-e-λ3ετ](ε-η)(η-2ε)(λ3ε-η)+λ2[e-ητ-e-λ3ετ](ε-η)(λ3ε-η).Equations (33)–(35) represent the analytical expression of the dimensionless substrate concentrationu(τ), dimensionless enzyme-substrate concentrationv(τ), and dimensionless free enzyme product concentrationw(τ)for all values of parametersε,λ1,λ2 and λ3. For steady condition, the differential equations (17a)–(17c) become as follows:
(36)-εu+uv+λ1v=0,εu-ηv+λ3εw-uv-λ3v=0,λ2v-λ3εw+λ3vw=0.
Solving the above equations, we can obtain the concentrations of substrate u, enzyme substrate complexv, and product free enzymewas follows:u=0,v=0,w=0. Whenτtends to infinity, the analytical expression corresponding to the substrate concentrationu, enzyme substrate concentrationvand free enzyme product concentrationwfrom (33)–(35) confirm the validity of mathematical analysis.


## 5. Results and Discussion

The substrate concentration versus time is plotted in Figures1 and 2 using (33). From Figures 1(a) and 1(b), it is observed that the dimensionless substrate concentrationudecreases. When parametersλ1=λ2=λ3=0.001, the dimensionless substrate concentrationugradually decreases as the value of parameterεincreases and reaches the steady state whenτ≥0.5. Whenλ1=λ2=λ3=1, the concentration decreases rapidly asεincreases.u≈1whenε≤0.1. In Figure 2, it is inferred that the concentration decreases as dimensionless timeτincreases and reaches its minimum whenτ=1. The graph is shown for various values ofλ1whenλ2=λ3=0.001, ε=1.Profile of the normalized concentrations of substrateuis calculated using (33) for various values of dimensionless parametersε,λ1,λ2, and λ3. Solid line: (33); dotted line: numerical simulation.
(a)
(b)Figure 2
Profile of the normalized concentrations of substrateu is calculated using (33) for various values of dimensionless parameterλ1. Solid line: (33); dotted line: numerical simulation.Figures3 and 4 are plotted using (34), dimensionless timeτas abscissa, and enzyme concentrationvas ordinate. From Figures 3(a) and 3(b), it is observed that the enzyme concentration increases whenτ≤0.2and reaches the steady state value whenτ>0.2. The graph is shown for various values of the parameterε, whenλ1=λ2=λ3=0.001 and 0.01. From Figure 4, it is inferred that enzyme concentrationvreaches its maximum between the time 0.03–0.09, and reaches the steady state whenτ≥0.5.Profile of dimensionless enzyme concentrationv is calculated using (34) for various values of dimensionless parametersε,λ1,λ2andλ3. Solid line: (34); dotted line: numerical simulation.
(a)
(b)Figure 4
Profile of dimensionless enzyme concentrationv is calculated using (34) for various values of dimensionless parameterλ2. Solid line: (34); dotted line: numerical simulation.Figures5 and 6 shows the free enzyme product concentration versus time for various values of parameter λusing (35). From Figures 5(a)-5(b), it is inferred that product concentrationwincreases very slowly as the time increases. The graphs are shown for various values of the parameterε, whenλ1=λ2=λ3=0.001 and 0.01. In Figure 6, it is noted that the product concentration increases slowly and reaches the steady state atτ≥0.3. Profile of dimensionless concentrations u, v, and w versus the dimensionless timeτusing (33), (34), and (35) for the fixed values of the parameters is plotted in Figure 7. From this figure, it is inferred that the concentration of substrate decreases, whereas the concentration of enzyme increases. But for all time the concentration of free enzyme-product have at most constant value.Profile of dimensionless free enzyme product concentrationwis calculated using (35) for various values of dimensionless parametersε,λ1,λ2andλ3. Solid line: (35); dotted line: numerical simulation.
(a)
(b)Figure 6
Profile of dimensionless free enzyme product concentrationv is calculated using (35) for various values of dimensionless parameterλ3. Solid line: (35); dotted line: numerical simulation.Figure 7
Profile of dimensionless concentrationsu, v, and w versus the dimensionless timeτusing (33), (34), and (35) for the fixed values of the parametersλ1,λ2,λ3, and ε. Solid line: (33), (34), and (35); dotted line: numerical simulation.


## 6. Conclusion

Approximate analytical solutions to the system of non-linear reaction equations in enzyme reaction mechanism are presented using homotopy perturbation method. A simple, straight forward, and a new method of estimating the concentrations of substrate, enzyme-substrate, and product are derived. This solution procedure can be easily extended to all kinds of system of coupled non-linear equations with various complex boundary conditions in enzyme-substrate nonlinear reaction diffusion processes.



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