Mathematical Modeling and Analysis of Nonlinear Enzyme Catalyzed Reaction Processes

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.

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 E+S↔^𝑘1_𝑘−1C−−→^𝑘2E+P, (1) 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=𝜐_max[S]/(K_𝑀+[S]), given by [9], where 𝐾_𝑀 is the Michaelis constant: 𝐾_𝑀=(𝑘₋₁+𝑘₂)/𝑘₁. The typical notation of the enzyme catalyzed reaction with one substrate can be given as [10] A+E↔^𝑘1_𝑘2X↔^𝑘3_𝑘4P+E, (2) where A is substrate, E is enzyme, X is enzyme-substrate complex, and P is free enzyme product. The kinetic equations consist of𝑑[A]𝑑𝑡=𝑘₂[X]−𝑘₁[AE],𝑑[E]][=𝑘𝑑𝑡²+𝑘₃[X]−𝑘₁[A]+𝑘₄[PE],𝑑[P]][𝑑𝑡=𝑘₃[X]−𝑘₄[P][E], (3a)(3b)(3c)with a conservation relation given in [6]: [E]+[X]=[E]_total. (4) 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 of𝑑[A]𝑑𝑡=𝜈₂−𝜈₁,𝑑[E]𝑑𝑡=𝜈₂+𝜈₃−𝜈₁−𝜈₄,𝑑[X]𝑑𝑡=𝜈₁+𝜈₄−𝜈₂−𝜈₃,𝑑[P]𝑑𝑡=𝜈₃−𝜈₄. (5a)(5b)(5c)(5d)The form of rate equations is as follows: 𝜈₁=𝑘₁[A][E],𝜈₂=𝑘₂[X],𝜈₃=𝑘₃[X],𝜈₄=𝑘₄[P][E]. (6) The initial condition at𝑡=0are as follows: [A]=A₀,[E]=E₀,[X]=0,[P]=0. (7) The concentration of the reactants in (5a)–(5d) is denoted by lower case letters 𝑠=[A],𝑒=[E],𝑐=[X],𝑝=[P], (8) The law of mass action leads to the system of following non-linear kinetic equations [12]:𝑑𝑠𝑑𝑡=−𝑘₁𝑠𝑒+𝑘₂𝑐,𝑑𝑒𝑑𝑡=−𝑘₁𝑠𝑒+𝑘₂𝑐+𝑘₃𝑐−𝑘₄𝑝𝑒,𝑑𝑝𝑑𝑡=𝑘₃𝑐−𝑘₄𝑝𝑒,𝑑𝑐𝑑𝑡=𝑘₁𝑠𝑒−𝑘₂𝑐−𝑘₃𝑐+𝑘₄𝑝𝑒, (9a)(9b)(9c)(9d)with the boundary conditions being 𝑠(0)=𝑠₀,𝑒(0)=𝑒₀,𝑝(0)=0,𝑐(0)=0. (10) Adding (9b) and (9c), we get 𝑑𝑒+𝑑𝑡𝑑𝑐𝑑𝑡=0. (11) Using the initial conditions (10) we obtain 𝑒(𝑡)+𝑐(𝑡)=𝑒₀. (12) With this, the system of ordinary differential equations reduces to the following three differential equations:𝑑𝑠𝑑𝑡=−𝑘₁𝑠𝑒₀−𝑐+𝑘₂𝑐,𝑑𝑐𝑑𝑡=𝑘₁𝑠𝑒₀−𝑐−𝑘₂+𝑘₃𝑐+𝑘₄𝑝𝑒₀−𝑐,𝑑𝑝𝑑𝑡=𝑘₃𝑐−𝑘₄𝑝𝑒₀−𝑐. (13a)(13b)(13c)With initial conditions𝑠(0)=𝑠₀, 𝑐(0)=0, and𝑝(0)=0. By introducing the following set of nondimensional variables and parameters, 𝑘𝜏=¹𝑒₀𝑡𝜀(𝜏)=𝑠(𝑡),𝑢𝑠₀(𝜏)=𝑐(𝑡),𝑣𝑠₀,𝑤(𝜏)=𝑝(𝑡)𝑠₀,𝜆₁=𝑘₂𝑘₁𝑠₀,𝜆₂=𝑘₃𝑘₁𝑠₀,𝜆₃=𝑘₄𝑘₁𝑒,𝜀=⁰𝑠₀,𝜂=𝜆₁+𝜆₂, (14)(15)(16) the system of (13a)–(13c) and the initial conditions (10) can be represented in dimensionless form as follows:𝑑𝑢𝑑𝜏=−𝜀𝑢+𝑢𝑣+𝜆₁𝑣,𝑑𝑣𝑑𝜏=𝜀𝑢−𝜂𝑣+𝜆₃𝜀𝑤−𝑢𝑣−𝜆₃𝑣,𝑑𝑤𝑑𝜏=𝜆₂𝑣−𝜆₃𝜀𝑤+𝜆₃𝑣𝑤, (17a)(17b)(17c)with 𝑢(0)=1,𝑣(0)=0,𝑤(0)=0. (18)

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: 1−𝑝𝑑𝑢𝑑𝜏+𝜀𝑢+𝑝𝑑𝑢𝑑𝜏+𝜀𝑢−𝑢𝑣−𝜆₁𝑣=0,1−𝑝𝑑𝑣𝑑𝜏+𝜂𝑣+𝑝𝑑𝑣𝑑𝜏+𝜂𝑣−𝜀𝑢−𝜆₃𝜀𝑤+𝑢𝑣+𝜆₃𝑣𝑤=0,1−𝑝𝑑𝑤𝑑𝜏+𝜆₃𝜀𝑤+𝑝𝑑𝑤𝑑𝜏+𝜆₃𝜀𝑤−𝜆₂𝑣−𝜆₃𝑣𝑤=0. (19)(20)(21) And the initial approximations are as follows: 𝑢(0)=1,𝑣(0)=0,𝑤(0)=0. (22) Approximate solutions of (33)–(35) are 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+𝑝³𝑣₃+⋯𝑤=𝑤₀+𝑝𝑤₁+𝑝²𝑤₂+𝑝³𝑤₃+⋯. (23)(24)(25) 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: 𝑝⁰∶𝑑𝑢₀𝑑𝜏+𝜀𝑢₀𝑝=0,₁∶𝑑𝑢₁𝑑𝜏+𝜀𝑢₁−𝑢₀𝑣₀−𝜆₁𝑣₀𝑝=0,₂∶𝑑𝑢₂𝑑𝜏+𝜀𝑢₂−𝑢₀𝑣₁+𝑢₁𝑣₀−𝜆₁𝑣₁=0. (26) For enzyme substrate concentration𝑣, 𝑝⁰∶𝑑𝑣₀𝑑𝜏+𝜂𝑣₀𝑝=0,₁∶𝑑𝑣₁𝑑𝜏+𝜂𝑣₁−𝜀𝑢₀−𝜆₃𝜀𝑤₀+𝑢₀𝑣₀+𝜆₃𝑣₀𝑤₀𝑝=0,₂∶𝑑𝑣₂𝑑𝜏+𝜂𝑣₂−𝜀𝑢₁−𝜆₃𝜀𝑤₁+𝑢₀𝑣₁+𝑢₁𝑣₀+𝜆₃𝑣₀𝑤₁+𝑣₁𝑤₀=0. (27) For product concentration𝑤𝑝⁰∶𝑑𝑤₀𝑑𝜏+𝜆₃𝜀𝑤₀𝑝=0,₁∶𝑑𝑤₁𝑑𝜏+𝜆₃𝜀𝑤₁−𝜆₂𝑣₀−𝜆₃𝑣₀𝑤₀𝑝=0,₂∶𝑑𝑤₂𝑑𝜏+𝜆₃𝜀𝑤₂−𝜆₂𝑣₁−𝜆₃𝑣₀𝑤₁+𝑣₁𝑤₀𝑝=0,₃∶𝑑𝑤₂𝑑𝜏+𝜆₃𝜀𝑤₃−𝜆₂𝑣₂−𝜆₃𝑣₀𝑤₂+𝑣₁𝑤₁+𝑣₂𝑤₀=0. (28) Solving (26)–(28), and using the boundary conditions (22), we can find the following results: 𝑢₀(𝜏)=𝑒^−𝜀𝜏,𝑢₁(𝜏)𝑢=0,₂(𝜏)=1𝜂−𝜀−𝑒^−2𝜀𝜏+𝑒^−𝜀𝜏+𝜀𝜂𝑒𝜂−𝜀^{−(𝜂+𝜀)}−𝑒^−𝜀𝑡+𝜆₁𝜀𝜏𝑒^−𝜀𝜏+𝜆𝜂−𝜀¹𝜀𝑒𝜂−𝜀^−𝜂𝜏−𝑒^−𝜀𝜏,𝑣₀(𝜏)𝑣=0,₁(𝜏)=𝜀𝑒𝜂−𝜀^−𝜀𝜏−𝑒^−𝜂𝜏,𝑣₂(𝜏)=𝜀𝑒^−𝜂𝜏−𝑒^−2𝜀𝜏+𝑒𝜂−𝜀2𝜀−𝜂^{−(𝜀+𝜂)𝜏}−𝑒^−𝜂𝜏,𝑤𝜂−𝜀₀(𝜏)𝑤=0,₁(𝜏)𝑤=0,₂(𝜏)=𝜆₂𝜀𝑒𝜂−𝜀^−𝜀𝜏−𝑒^{−𝜆3𝜀𝜏}𝜆₃𝜀−𝑒−1^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜆₂,𝑤𝜀−𝜂₃(𝜏)=𝜆₂𝑒^−2𝜀𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜂−2𝜀𝜆₃−𝜆−2²𝑒^{−(𝜀+𝜂)𝜏}−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜆₃−𝜆−𝜀−𝜂₂𝜀𝑒^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜂−2𝜀𝜆₃+𝜆𝜀−𝜂²𝑒^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜆₃.𝜀−𝜂 (29) According to the HPM, we can conclude that 𝑢(𝜏)=lim_𝑝→1𝑢(𝜏)=𝑢₀+𝑢₁+𝑢₂,𝑣(𝜏)=lim_𝑝→1𝑣(𝜏)=𝑣₀+𝑣₁+𝑣₂,𝑤(𝜏)=lim_𝑝→1𝑤(𝜏)=𝑤₀+𝑤₁+𝑤₂+𝑤₃. (30)(31)(32) 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): 𝑢(𝜏)=𝑒^−𝜀𝜏+−𝑒^−2𝜀𝜏+𝑒^−𝜀𝜏+𝜀𝑒𝜂−𝜀^{−(𝜂+𝜀)}−𝑒^−𝜀𝜏𝜂+𝜆𝜂−𝜀₁𝜀𝜏𝑒^−𝜀𝜏+𝜆𝜂−𝜀¹𝜀𝑒^−𝜂𝜏−𝑒^−𝜀𝜏,𝑣(𝜏)=𝜀𝜂−𝜀𝑒𝜂−𝜀^−𝜀𝜏−𝑒^−𝜂𝜏+𝜀𝑒^−𝜂𝜏−𝑒^−2𝜀𝜏+𝑒𝜂−𝜀2𝜀−𝜂_{−(𝜀+𝜂)𝜏}−𝑒^−𝜂𝜏,𝑤(𝜏)=𝜆𝜂−𝜀₂𝜀𝑒𝜂−𝜀^−𝜀𝜏−𝑒^{−𝜆3𝜀𝜏}𝜆₃𝜀−𝑒−1^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜆₂+𝜆𝜀−𝜂₂𝑒^−2𝜀𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜂−2𝜀𝜆₃−𝜆−2²𝑒^{−(𝜀+𝜂)𝜏}−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜆₃−𝜆−𝜀−𝜂₂𝜀𝑒^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜂−2𝜀𝜆₃+𝜆𝜀−𝜂²𝑒^−𝜂𝜏−𝑒^{−𝜆3𝜀𝜏}𝜀−𝜂𝜆₃.𝜀−𝜂 (33)(34)(35)

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: −𝜀𝑢+𝑢𝑣+𝜆₁𝑣=0,𝜀𝑢−𝜂𝑣+𝜆₃𝜀𝑤−𝑢𝑣−𝜆₃𝜆𝑣=0,₂𝑣−𝜆₃𝜀𝑤+𝜆₃𝑣𝑤=0. (36) 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 Figures 1 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.

(a)

(b)

Figure 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.

Figure 2 

Profile of the normalized concentrations of substrate 𝑢 is calculated using (33) for various values of dimensionless parameter𝜆1. Solid line: (33); dotted line: numerical simulation.

Figures 3 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.

(a)

(b)

Figure 3 

Profile of dimensionless enzyme concentration 𝑣 is calculated using (34) for various values of dimensionless parameters𝜀,𝜆1,𝜆2and𝜆3. Solid line: (34); dotted line: numerical simulation.

Figure 4 

Profile of dimensionless enzyme concentration 𝑣 is calculated using (34) for various values of dimensionless parameter𝜆2. Solid line: (34); dotted line: numerical simulation.

Figures 5 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.

(a)

(b)

Figure 5 

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.

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 concentrations 𝑢, 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.

Appendix

Numerical Simulation Program for (14)–(16)

 function graphmain3 options = odeset (“RelTol”, 1𝑒−6, “Stats”, “on”); % initial conditions 𝑋0=[1;0;0]; 𝑡span=[0,5]; tic [𝑡,𝑋]=ode45(@TestFunction,𝑡span,𝑋0,options); toc figure hold on plot(𝑡,𝑋(∶,1), “−” ) plot(𝑡,𝑋(∶,2),“−”) plot(𝑡,𝑋(∶,3),“∗”) legend (“𝑥1”,“𝑥2”,“𝑥3”) 𝑦 label (“𝑥”) 𝑥 label (“𝑡”) return function[𝑑𝑥_𝑑𝑡]=TestFunction(𝑡,𝑥) 𝑒=1;𝑐1=0.1;𝑐2=0.001;𝑐3=0.1; 𝑑𝑥_𝑑𝑡(1)=−𝑒∗𝑥(1)+𝑥(1)∗𝑥(2)+𝑐1∗𝑥(2); 𝑑𝑥_𝑑𝑡(2)=𝑒∗𝑥(1)−𝑐1∗𝑥(2)−𝑐2∗𝑥(2) + 𝑐3∗𝑒∗𝑥(3)−𝑥(1)∗𝑥(2)−𝑐3∗𝑥(2)∗𝑥(3); 𝑑𝑥_𝑑𝑡(3)=𝑐2∗𝑥(2)−𝑐3∗𝑒∗𝑥(3)+𝑐3∗𝑥(2)∗𝑥(3); 𝑑𝑥_𝑑𝑡=𝑑𝑥_𝑑𝑡, return

Nomenclature and Units

[E]:	Enzyme concentration (μM)
[C]:	Enzyme-substrate complex (μM)
[S]:	Substrate concentration (μM)
[E0]:	Initial enzyme concentration (μM)
𝐾𝑀:	Michaelis-Menten constant
[S0]:	Initial substrate concentration (μM)
𝑘1,𝑘2,𝑘3:	Positive rate constants (none)
𝜆1,𝜆2,𝜆3,𝜀:	Reaction diffusion parameter (none)
𝑢:	Dimensionless substrate concentration (none)
𝑣:	Dimensionless enzyme substrate concentration (none)
𝑤:	Dimensionless product concentration (none)
𝑡:	Time (sec)
𝜏:	Dimensionless time (none).

Acknowledgment

This work was supported by CSIR Project (no. 01 (2442)/10/EMR-II). The authors are thankful to the Principal and the Secretary, the Madura College, Madurai, for their encouragement.