**Meet the editor**

Dr. Muhammad Akhyar Farrukh is an associate professor and the founding director of Nano-Chemistry Lab. at GC University Lahore, Pakistan. He has been awarded three gold medals for his outstanding academic performance in physical chemistry and three gold medals for his excellent performance in research and service to society. He has been awarded many international and national

awards: selected by UNESCO as the "Representative of Pakistan" in the first Assembly of the WAYS in Morocco, awarded "Young Chemist Award" by IUPAC in Italy, selected as "Young Scientist" by TWAS in Egypt and IAP in Germany and by IAP/World Economic Forum in China, selected as "Young Researcher" at the 63rd Lindau Nobel Laureate Meeting**,** awarded the IUPAC-2015 *Award for Chemists* as an outstanding chemist from developing countries in S. Korea, received highest *Research Productivity Awards* (Category A) in 2016 and *Ranked 11th in Pakistan* in Chemistry.

Contents

**Preface VII**

**Section 1 Kinetics Modeling and Mechanism 1**

Chapter 1 **Complex Reactions and Dynamics 3**

**Enzyme Kinetics 21**

Qiuming Peng

Dongwon Jung

Martin Nielsen

Muhammad Shahzad and Faisal Sultan

Chapter 3 **Autoignition and Chemical-Kinetic Mechanisms of**

Wemerson Daniel C. dos Santos

Rozina Khattak and Hasan M. Khan

**Acceptorless Dehydrogenation 91**

Chapter 5 **Competition Kinetics: An Experimental Approach 79**

Chapter 6 **Catalyst Kinetics and Stability in Homogeneous Alcohol**

Chapter 2 **Mathematical Modeling and Simulation of Nonlinear Process in**

**the Fuels with Various Autoignition Reactivity 37**

Chapter 4 **New Materials to Solve Energy Issues through Photochemical and Photophysical Processes: The Kinetics Involved 57** Tatiana Duque Martins, Antonio Carlos Chaves Ribeiro, Geovany Albino de Souza, Diericon de Sousa Cordeiro, Ramon Miranda Silva, Flavio Colmati, Roberto Batista de Lima, Lucas Fernandes Aguiar, Leandro Lima Carvalho, Renan Gustavo Coelho S. dos Reis and

Lakshmanan Rajendran, Mohan Chitra Devi, Carlos Fernandez and

**Homogeneous Charge Compression Ignition Combustion for**

Murtaza Sayed, Luqman Ali Shah, Javed Ali Khan, Noor S. Shah,

## Contents

### **Preface XI**

Chapter 6 **Catalyst Kinetics and Stability in Homogeneous Alcohol Acceptorless Dehydrogenation 91** Martin Nielsen

#### **X** Contents

### **Section 2 Kinetics of Nanomaterials 111**

Preface

convenient for readers.

"Kinetics of Nanomaterials."

throughout the process of the publication of this book.

nism of reactions.

The purpose of the book on *Advanced Chemical Kinetics* is to provide insight into different aspects of chemical reactions both at the bulk and nanoscale level. The book covers topics from the basic to advanced level so that readers can get maximum benefits for teaching,

This book has been divided into three sections, respectively (i) "Kinetics Modeling and Mechanism," (ii) "Kinetics of Nanomaterials," (iii) "Kinetics Techniques," to make it more

The first section "Kinetics Modeling and Mechanism" covers six chapters with a wide range of topics. It focuses on activation energy and complexity arising during the chemical reac‐ tion. The chapters in this section will provide a measurement of reaction routes with de‐ tailed mechanism, presented with the help of modern techniques. Mathematical modeling analysis and simulation of enzyme kinetics are discussed in the form of differential equa‐ tions. These models provide analytical understanding due to their potential in predicting kinetic processes. One of the chapters demonstrates the autoignition and chemical kinetic mechanisms of homogeneous charge compression ignition combustion for the fuels like methane, dimethyl ether, isooctane, and n-heptane as the single-stage and two-stage ignition fuels. A zero-dimensional single-zone engine model of "CHEMKIN" in Chemkin-Pro is used to study mechanisms for high- and low-temperature reactions. Photophysical process‐ es and photochemical changes, presented by new materials and devices to provide a control of energy transfer processes, are discussed in detail. The mechanism of hydroxyl radical, hydrate electron, and hydrogen atom is given with the examples of ciprofloxacin, norfloxa‐ cin, and bezafibrate. The last chapter of this section covers the acceptorless alcohol dehydro‐ genation as an effective tool for organic syntheses for economic production of a wide range of organic compounds, such as aldehydes, ketones, ester, amides, carboxylic acids, etc.

Nanomaterials nowadays are playing an important role in catalysis, degradation of organic pollutants, wastewater treatment, etc. The understanding of the kinetics of nanomaterials is important. To cover this gap in knowledge, the second section of this book is dedicated to

The third section highlights an overview of experimental techniques like electrothermal ex‐ plosion, differential thermal analysis, electro-thermography, X-ray diffraction, ultrasound, gas chromatography, and hydrogen nuclear magnetic resonance used to study the mecha‐

I would like to thank Publishing Process Manager Ms. Vais Ana for her cooperation

**Muhammad Akhyar Farrukh** Founding Director/In-Charge Nano-Chemistry Laboratory GC University Lahore, Pakistan

learning, and doing research in the area of chemical kinetics.

## Preface

**Section 2 Kinetics of Nanomaterials 111**

**VI** Contents

Ayşe Neren Ökte

**Section 3 Kinetics Techniques 165**

**Reactions 167**

**Water/Alcohol Solvent 133** Mahabubur Chowdhury

**Ash-Sepiolite Ternary Catalyst 143**

Chapter 11 **Ultrasound as a Metrological Tool for Monitoring Transesterification Kinetics 197**

V. Alvarenga and Rodrigo P.B. Costa-Félix

Chapter 7 **Oxidation of Glycerol to Lactic Acid by Gold on Acidified Alumina: A Kinetic and DFT Case Study 113** Thabang A. Ntho, Pumeza Gqogqa and James L. Aluha

Chapter 8 **Hydrothermal Precipitation of β-FeOOH Nanoparticles in Mixed**

Chapter 9 **Adsorption, Kinetics and Photoactivity of ZnO-Supported Fly**

Chapter 10 **Kinetics of Heterogeneous Self-Propagating High-Temperature**

Raphaela M. Baêsso, Pâmella A. Oliveira, Gabriel C. Moraes, André

Christopher E. Shuck and Alexander S. Mukasyan

The purpose of the book on *Advanced Chemical Kinetics* is to provide insight into different aspects of chemical reactions both at the bulk and nanoscale level. The book covers topics from the basic to advanced level so that readers can get maximum benefits for teaching, learning, and doing research in the area of chemical kinetics.

This book has been divided into three sections, respectively (i) "Kinetics Modeling and Mechanism," (ii) "Kinetics of Nanomaterials," (iii) "Kinetics Techniques," to make it more convenient for readers.

The first section "Kinetics Modeling and Mechanism" covers six chapters with a wide range of topics. It focuses on activation energy and complexity arising during the chemical reac‐ tion. The chapters in this section will provide a measurement of reaction routes with de‐ tailed mechanism, presented with the help of modern techniques. Mathematical modeling analysis and simulation of enzyme kinetics are discussed in the form of differential equa‐ tions. These models provide analytical understanding due to their potential in predicting kinetic processes. One of the chapters demonstrates the autoignition and chemical kinetic mechanisms of homogeneous charge compression ignition combustion for the fuels like methane, dimethyl ether, isooctane, and n-heptane as the single-stage and two-stage ignition fuels. A zero-dimensional single-zone engine model of "CHEMKIN" in Chemkin-Pro is used to study mechanisms for high- and low-temperature reactions. Photophysical process‐ es and photochemical changes, presented by new materials and devices to provide a control of energy transfer processes, are discussed in detail. The mechanism of hydroxyl radical, hydrate electron, and hydrogen atom is given with the examples of ciprofloxacin, norfloxa‐ cin, and bezafibrate. The last chapter of this section covers the acceptorless alcohol dehydro‐ genation as an effective tool for organic syntheses for economic production of a wide range of organic compounds, such as aldehydes, ketones, ester, amides, carboxylic acids, etc.

Nanomaterials nowadays are playing an important role in catalysis, degradation of organic pollutants, wastewater treatment, etc. The understanding of the kinetics of nanomaterials is important. To cover this gap in knowledge, the second section of this book is dedicated to "Kinetics of Nanomaterials."

The third section highlights an overview of experimental techniques like electrothermal ex‐ plosion, differential thermal analysis, electro-thermography, X-ray diffraction, ultrasound, gas chromatography, and hydrogen nuclear magnetic resonance used to study the mecha‐ nism of reactions.

I would like to thank Publishing Process Manager Ms. Vais Ana for her cooperation throughout the process of the publication of this book.

> **Muhammad Akhyar Farrukh** Founding Director/In-Charge Nano-Chemistry Laboratory GC University Lahore, Pakistan

**Section 1**

**Kinetics Modeling and Mechanism**

**Kinetics Modeling and Mechanism**

**Chapter 1**

Provisional chapter

**Complex Reactions and Dynamics**

Complex Reactions and Dynamics

Muhammad Shahzad and Faisal Sultan

Muhammad Shahzad and Faisal Sultan

http://dx.doi.org/10.5772/intechopen.70502

Abstract

invariant manifold

1. Introduction

reactions.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Starting from the general idea of reaction kinetics, their classification, concentrations, and chemical equilibrium, we will focus on their activation energy and complexity arising during the chemical reaction. As in complex and higher-dimensional chemical problems, we need special arrangements, specifically, in the case when a system attains different completion paths or several routes. The stiffness of the system can be removed if we distinctly measure their available reaction routes and get a comparison between them and overall reactions. Secondly, the construction and comparison of the invariant region of the manifold based on the modern decomposition techniques in different available reaction routes allow us to discuss the dynamical properties of the system.

DOI: 10.5772/intechopen.70502

Keywords: chemical equilibrium, detailed mechanism, model reduction, reaction routes,

The chemical kinetics or reaction kinetics is the branch of physical chemistry that deals with the study of chemical processes, their rates, rearrangement of atoms, the effect of various variables, the formation of intermediates, etc. In fact, the chemical kinetics is the study of different factors affecting the speed of a chemical process and gives information about the mechanism of reaction and transition states. At the macroscopic level, the chemical kinetics deals with the study of amount reacted, formed, and the rates of their formation. While at the microscopic or molecular level, we study the mechanism of a chemical reaction, i.e., atomic

The chemical kinetics is classified into three types, mathematical, detailed, and applied kinetics, while their elementary reactions are described as unimolecular, bimolecular, and termolecular

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

collision, activation energy at different stages during the reaction.

**Chapter 1**

Provisional chapter

### **Complex Reactions and Dynamics**

Complex Reactions and Dynamics

Muhammad Shahzad and Faisal Sultan

Additional information is available at the end of the chapter Muhammad Shahzad and Faisal Sultan Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70502

#### Abstract

Starting from the general idea of reaction kinetics, their classification, concentrations, and chemical equilibrium, we will focus on their activation energy and complexity arising during the chemical reaction. As in complex and higher-dimensional chemical problems, we need special arrangements, specifically, in the case when a system attains different completion paths or several routes. The stiffness of the system can be removed if we distinctly measure their available reaction routes and get a comparison between them and overall reactions. Secondly, the construction and comparison of the invariant region of the manifold based on the modern decomposition techniques in different available reaction routes allow us to discuss the dynamical properties of the system.

DOI: 10.5772/intechopen.70502

Keywords: chemical equilibrium, detailed mechanism, model reduction, reaction routes, invariant manifold

### 1. Introduction

The chemical kinetics or reaction kinetics is the branch of physical chemistry that deals with the study of chemical processes, their rates, rearrangement of atoms, the effect of various variables, the formation of intermediates, etc. In fact, the chemical kinetics is the study of different factors affecting the speed of a chemical process and gives information about the mechanism of reaction and transition states. At the macroscopic level, the chemical kinetics deals with the study of amount reacted, formed, and the rates of their formation. While at the microscopic or molecular level, we study the mechanism of a chemical reaction, i.e., atomic collision, activation energy at different stages during the reaction.

The chemical kinetics is classified into three types, mathematical, detailed, and applied kinetics, while their elementary reactions are described as unimolecular, bimolecular, and termolecular reactions.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The mathematical kinetics deals with the analysis of different mathematical models used in direct and inverse chemical kinetics. These models represent a set of ordinary/partial differential equations and a set of algebraic equations. Further, direct kinetic problems deal with the analysis of steady state or nonsteady state kinetic models consisting of known kinetic parameters. On the other hand, an inverse kinetic problem reconstructs kinetic dependencies and estimates their parameters based on experimental kinetic data, either steady or nonsteady state.

products but it does depend on the temperature), and ρ = 1,…,m are the reaction numbers,

The reactions in which the reactants are in different phases and their rates are affected by surface areas are called heterogeneous reactions, i.e., the reaction between gases and liquids, solids and liquids, etc. As in the case of gas solid catalytic reactions, reactants at elementary steps will be gas phase component or surface intermediate. Thus, Eq. (1) can now be written as

> X i βρi

Bi þX j βρj

Yj, (2)

Complex Reactions and Dynamics

http://dx.doi.org/10.5772/intechopen.70502

i ð Þ a � x αρi (forward rate of reaction) and

i ð Þx βρi ¼ 0, (4)

: (3)

j αρjXj ������ �!� kþ i k� i

βB

5

αρjXj ������ �!� k þ i k� i

here again Ai and Bi are the reactant and products in the gas phase and Xj and Yj are the

With an assumption that α and β are either zero or one, it implies that only one molecule in an

At initial space: a 0 time t > 0: (a�x) x

i ð Þx βρi (backward rate of reaction) and the product formation is the difference between the

i ð Þ a � x αρi � k

The equilibrium is a dynamical process, and when a system goes to an equilibrium, the left-hand

while the conversions of products to reactants and reactants to products are still going on,

According to Le-Chatelier's principle, if a system at equilibrium state is disturbed by an

The system becomes complex when a reaction undergoes more than one pathways when more than one products are formed from the same reactants or different reactants produce the same

� i ð Þx βρi

dt ¼ 0

�

while the stoichiometric coefficients αρi,βρi are the nonnegative integers.

X i

þP j βρj Yj.

k � αρiAi þX

elementary reaction from the gas phase reacts or zero at all.

The reaction rates measured on either side are kþ

side will become equal to the right-hand side, i.e., dx

dx dt ¼ k þ

kþ

i ð Þ a � x αρi � k

although there is no net change in the number of reactant and product molecules.

external force, then the system tries to offset the force and attains a new position.

products. Such types of reactions are called parallel reactions or side reactions, i.e.,

rate of forward and backward reactions, i.e.,

j

surface intermediate. In a more typical form, it can be written as αA þ P

The concentration of the involved species can be measured as (single step reaction);

The construction of the mathematical model is the key part of chemical kinetics, which gives a complete description of reaction mechanism and its rates. It provides a working tool to better understand and design chemical processes, i.e., food decomposition and the complex chemistry of biological systems, etc. These models are also used in designing the fast and slow trajectories of complex chemical reactions and modification of chemical reactors to optimize product yield, more efficiently separate products, and eliminate environmentally harmful byproducts.

In detailed kinetics, we study the reconstruction of detailed mechanism of reaction either based on kinetic or nonkinetic data. These mechanisms consist of a set of elementary steps having forward and reverse reactions along with the governing equation of mass-action law for the kinetic dependencies. In catalyst reactions, it covers the reactant, products, intermediate, surface properties, reaction steps, reaction routes, adsorption properties, etc.

The goal of applied kinetics is to study the kinetic dependence of the rate of chemical reactions on their involved or related conditions, i.e., temperature, pressure, concentration, and so on. This dependence can be related to a single or a series of mathematical models usually called kinetic models. These kinetic models are necessary to represent the hierarchy of models at each stage, i.e., initial, intermediate, and final levels as well as to develop an easy way for mathematical simulations of a chemical process.

The reactions in which a single molecule rearranges itself to make one or more products are called unimolecular reactions or a first-order reaction (A!B), like radioactive decay in which particles are emitted from a single atom. The reactions in which two molecules take part to form a product are called bimolecular reactions or second order (2A!B or A + B!C), like cycloaddition reaction. The reactions in which three particles collide at the same place and time to form a product are called termolecular reactions or third order (3A!B or A + 2B!C). The third-order reactions are not very common as all the three reactants must have to collide simultaneously to form a product.

The chemical reactions in which the reactants are in the same phases are called homogeneous reactions, i.e., the reaction between two gases, two solids, or two liquids. Let us consider a reversible chemical reaction represented as

$$\sum\_{i} \alpha\_{\rho i} A\_i \xleftarrow[k\_i^+] \xrightarrow[k\_i^-]{k\_i^+} \sum\_{i} \beta\_{\rho i} B\_{i\nu} \tag{1}$$

here Ai and Bi are the reactants and products, ki is the rate constants for forward k þ i and backward directions k� i (that does not depend on the initial concentration of the reactants and products but it does depend on the temperature), and ρ = 1,…,m are the reaction numbers, while the stoichiometric coefficients αρi,βρi are the nonnegative integers.

The reactions in which the reactants are in different phases and their rates are affected by surface areas are called heterogeneous reactions, i.e., the reaction between gases and liquids, solids and liquids, etc. As in the case of gas solid catalytic reactions, reactants at elementary steps will be gas phase component or surface intermediate. Thus, Eq. (1) can now be written as

$$\sum\_{i} \alpha\_{\rho i} A\_i + \sum\_{j} \alpha\_{\rho j} X\_j \xleftarrow[k\_i^-] \xrightarrow[i]{k\_i^+} \sum\_{i} \beta\_{\rho i} B\_i + \sum\_{j} \beta\_{\rho j} Y\_{j\succ} \tag{2}$$

here again Ai and Bi are the reactant and products in the gas phase and Xj and Yj are the surface intermediate. In a more typical form, it can be written as αA þ P j αρjXj ������ �!� kþ i k� i βB

$$+\sum\_{j} \beta\_{\rho j} Y\_{j}.$$

The mathematical kinetics deals with the analysis of different mathematical models used in direct and inverse chemical kinetics. These models represent a set of ordinary/partial differential equations and a set of algebraic equations. Further, direct kinetic problems deal with the analysis of steady state or nonsteady state kinetic models consisting of known kinetic parameters. On the other hand, an inverse kinetic problem reconstructs kinetic dependencies and estimates their

The construction of the mathematical model is the key part of chemical kinetics, which gives a complete description of reaction mechanism and its rates. It provides a working tool to better understand and design chemical processes, i.e., food decomposition and the complex chemistry of biological systems, etc. These models are also used in designing the fast and slow trajectories of complex chemical reactions and modification of chemical reactors to optimize product yield, more efficiently separate products, and eliminate environmentally harmful by-

In detailed kinetics, we study the reconstruction of detailed mechanism of reaction either based on kinetic or nonkinetic data. These mechanisms consist of a set of elementary steps having forward and reverse reactions along with the governing equation of mass-action law for the kinetic dependencies. In catalyst reactions, it covers the reactant, products, intermedi-

The goal of applied kinetics is to study the kinetic dependence of the rate of chemical reactions on their involved or related conditions, i.e., temperature, pressure, concentration, and so on. This dependence can be related to a single or a series of mathematical models usually called kinetic models. These kinetic models are necessary to represent the hierarchy of models at each stage, i.e., initial, intermediate, and final levels as well as to develop an easy way for mathe-

The reactions in which a single molecule rearranges itself to make one or more products are called unimolecular reactions or a first-order reaction (A!B), like radioactive decay in which particles are emitted from a single atom. The reactions in which two molecules take part to form a product are called bimolecular reactions or second order (2A!B or A + B!C), like cycloaddition reaction. The reactions in which three particles collide at the same place and time to form a product are called termolecular reactions or third order (3A!B or A + 2B!C). The third-order reactions are not very common as all the three reactants must have to collide

The chemical reactions in which the reactants are in the same phases are called homogeneous reactions, i.e., the reaction between two gases, two solids, or two liquids. Let us consider a

> X i βρi

i (that does not depend on the initial concentration of the reactants and

Bi, (1)

þ i and

αρiAi ������! k þ i k � i

here Ai and Bi are the reactants and products, ki is the rate constants for forward k

X i

parameters based on experimental kinetic data, either steady or nonsteady state.

ate, surface properties, reaction steps, reaction routes, adsorption properties, etc.

matical simulations of a chemical process.

simultaneously to form a product.

backward directions k�

reversible chemical reaction represented as

products.

4 Advanced Chemical Kinetics

With an assumption that α and β are either zero or one, it implies that only one molecule in an elementary reaction from the gas phase reacts or zero at all.

The concentration of the involved species can be measured as (single step reaction);

The reaction rates measured on either side are kþ i ð Þ a � x αρi (forward rate of reaction) and k � i ð Þx βρi (backward rate of reaction) and the product formation is the difference between the rate of forward and backward reactions, i.e.,

$$\frac{d\mathbf{x}}{dt} = k\_i^+ (a - \mathbf{x})^{a\_{\mu i}} - k\_i^- (\mathbf{x})^{\mathcal{S}\_{\mu i}}.\tag{3}$$

The equilibrium is a dynamical process, and when a system goes to an equilibrium, the left-hand side will become equal to the right-hand side, i.e., dx dt ¼ 0

$$k\_i^+ (a - \mathbf{x})^{a\_{\mid i}} - k\_i^- (\mathbf{x})^{\beta\_{\mid i}} = \mathbf{0},\tag{4}$$

while the conversions of products to reactants and reactants to products are still going on, although there is no net change in the number of reactant and product molecules.

According to Le-Chatelier's principle, if a system at equilibrium state is disturbed by an external force, then the system tries to offset the force and attains a new position.

The system becomes complex when a reaction undergoes more than one pathways when more than one products are formed from the same reactants or different reactants produce the same products. Such types of reactions are called parallel reactions or side reactions, i.e.,

The energy required to pass the reactant ER to activated complex EΘ is called activation energy or energy of activation EAct:EΘ–ER. It may be supplied in any form, mechanical, chemical, or

or different depending on the type of reactions. In thermos, the neutral reaction and ΔH=0, the energy of activation in both the directions are same. While in endothermic reactions,

The activated complex is a separate entity and there exists an equilibrium between reactants (products, under reversible reactions) and activated complex (Figure 1).  It is also understood that the higher

Ai (7)

i , i.e.,

N i¼1 αSi Ai ⇔

Act reactions must be the same

Complex Reactions and Dynamics

7

Ai ↦ EF Act \_

thermal, to enable the reactant to convert into the product, i.e.,

Act, holds and in exothermic reactions, EF

The activation energy during the forward EF

the activation energy, the slower the reaction.

anism can be defined as

EF Act > EB

Θ∓ i ) P N i¼1 βSi Ai. X N

i¼1 αSi

X N

i¼1 αSi

Figure 1. A complex reaction mechanism involving energy barriers and transition states.

But still, there is some activation going on between the activated complexes Θ∓

Θ∓ i : ΘR

Therefore, a complete complex chemical reaction mechanism can be defined as P

Here a is the initial concentration of the species A and after some time t > 0, it dispersed into C and D. Similarly, sometimes it happens that different chemical species give the same products

$$\begin{array}{ccccc} \xrightarrow{n\_1} & \mathbb{C} \\ A \overset{\mathcal{N}^t}{\mathop{\scalebox{-0.5pt}{ $\mathbf{a}\succ\mathbf{b}$ }}} & \xrightarrow{\mathbf{c}\_1} & B \overset{\mathcal{N}^t}{\mathop{\scalebox{-0.5pt}{ $\mathbf{a}\succ\mathbf{b}$ }}} \\ A \overset{\mathcal{N}^t}{\mathop{\scalebox{-0.5pt}{ $\mathbf{a}\succ\mathbf{b}$ }}} & B \\ \xrightarrow{n\_2} & D \underset{(\mathbf{a}\succ\mathbf{x})^{n\_2}}{\mathop{\longrightarrow}} & \xrightarrow{n\_2} & D \\ \end{array}$$

or a system is reversible at different stages. In all these cases, we need to follow all their paths to get the detailed mechanisms,

The rate of reaction (Wρ(c)) is proportional to the number of collisions per unit time between the reactants but only a small fraction of the total is effective, i.e., not every collision between the reactants gives the result.

There may be a few reasons behind its ineffectiveness [1], i.e.:

The reactant molecules may attain insufficient energy (<EAct, i.e., activation energy, J/ mol) at different stages during the reaction.

The molecules may not get aligned properly or orientate during the collision (depending upon the geometry of the particles and kind of reaction that is taking place), etc.

If Z is the effective collision in which molecules have energy ≥EAct, then

$$Z\_E = \frac{Z\_0 e^{E\_{\text{Act}}}}{RT},\tag{5}$$

and e EAct/RT gives the fraction of collisions with energy ≥EAct.

In a complex chemical reaction, the reactant molecules (intermediates, complex Θi) pass through different transition states due to their bond breaking and energy redistribution factors. Here the species stays for a very short period, usually called transition period of activated complex (where the hidden reactions between the chemical species are still going on very fast).

The energy required to pass the reactant ER to activated complex EΘ is called activation energy or energy of activation EAct:EΘ–ER. It may be supplied in any form, mechanical, chemical, or thermal, to enable the reactant to convert into the product, i.e.,

! n1 k þ 1

! n2 kþ 2

Here a is the initial concentration of the species A and after some time t > 0, it dispersed into C and D. Similarly, sometimes it happens that different chemical species give the same products

> B b ↗t ↘t

or a system is reversible at different stages. In all these cases, we need to follow all their paths

 n1 kþ 1

The rate of reaction (Wρ(c)) is proportional to the number of collisions per unit time between the reactants but only a small fraction of the total is effective, i.e., not every collision between

The reactant molecules may attain insufficient energy (<EAct, i.e., activation energy, J/ mol) at

The molecules may not get aligned properly or orientate during the collision (depending upon

ZE ¼ Z0eEAct

In a complex chemical reaction, the reactant molecules (intermediates, complex Θi) pass through different transition states due to their bond breaking and energy redistribution factors. Here the species stays for a very short period, usually called transition period of activated complex (where the hidden reactions between the chemical species are still going on very fast).

RT , (5)

A a ↙t ↖t n2 kþ 2

C ð Þ a�x n1

D ð Þ a�x n2

A a ↗t ↘t

! n1 kþ 1

! n2 kþ 2

A a ↗t ↘t

There may be a few reasons behind its ineffectiveness [1], i.e.:

the geometry of the particles and kind of reaction that is taking place), etc.

If Z is the effective collision in which molecules have energy ≥EAct, then

EAct/RT gives the fraction of collisions with energy ≥EAct.

to get the detailed mechanisms,

6 Advanced Chemical Kinetics

the reactants gives the result.

and e

different stages during the reaction.

C ð Þ a�x n1

D ð Þ a�x n2

C ð Þ a�x n1

D ð Þ a�x n2

> ! n1 kþ 1

! n2 kþ 2

C ð Þ b�x n1

D ð Þ b�x n2

$$\sum\_{i=1}^{N} \alpha\_{S\_i} A\_i \mapsto \underline{\boldsymbol{E}}\_{\text{Act}}^{\widehat{\boldsymbol{F}}} \mapsto \sum\_{i=1}^{N} \beta\_{S\_i} A\_i \tag{6}$$

The activation energy during the forward EF Act and backward EB Act reactions must be the same or different depending on the type of reactions. In thermos, the neutral reaction and ΔH=0, the energy of activation in both the directions are same. While in endothermic reactions, EF Act > EB Act, holds and in exothermic reactions, EF Act < EB Act. It is also understood that the higher the activation energy, the slower the reaction.

The activated complex is a separate entity and there exists an equilibrium between reactants (products, under reversible reactions) and activated complex (Figure 1). Thus, a reaction mechanism can be defined as

$$\sum\_{i=1}^{N} \alpha\_{\mathbb{S}\_i} A\_i < = > \Theta\_i = > \sum\_{i=1}^{N} \beta\_{\mathbb{S}\_i} A\_i \tag{7}$$

But still, there is some activation going on between the activated complexes Θ∓ i , i.e.,

$$
\Theta\_i^{\mp} : \quad \Theta\_i^{\mathbb{R}} = > \Theta\_i^{\mathbb{P}} \, .
$$

Therefore, a complete complex chemical reaction mechanism can be defined as P N i¼1 αSi Ai ⇔

Figure 1. A complex reaction mechanism involving energy barriers and transition states.

Θ∓

In case of reversible complex chemical reactions,

$$\sum\_{i=1}^{N} \alpha\_{\mathbb{S}i} A\_i \Leftrightarrow \Theta\_i^{\mathbb{F}} \Leftrightarrow \sum\_{i=1}^{N} \beta\_{\mathbb{S}i} A\_i \tag{8}$$

3. Linear algebra and graph theory

and +2, or +1 and �3),

relation between σ and Nint is

provide a basis for the overall reactions [2].

they are important to be measured?

The numbers of key components Nkc are given by the equation

key components + number of nonkey reactions = number of reactions

For this, the reaction route Nrr of the system can be measured as

forward N f

trees Nc are

In chemical engineering, the mathematical methods of graph theory have found wide applications in complex chemical reactions and in a sequence of uni (or multi) or parallel reacting events. A graph is a combination of nodes (points) and edges (lines) [2], while a cyclic graph involves finite sequences of edges with the single node (from where it begins and ends).

Similarly, related to any combination of reaction, a tree can be defined as a sequence of noncyclic graph edges. In a spanning tree, certain intermediate may form from other intermediates after a sequence of transformations but does not agree to counter any two reactions with the same step (e.g., +1 and �1) nor two reactions started with the same intermediates (e.g., �1

Spanning trees can be described in terms of "forward" (generated by a sequence of forwarding reactions), "backward" (generated by a sequence of reverse reactions), and "combined" spanning trees (generated by a sequence of both forward and backward reactions). A single-route, n-steps Ns (edges) reaction mechanism has Nint (intermediates) nodes, such as Nint = Ns = N. The total numbers of spanning trees are N 2 in any reaction, while the

In a chemical reaction, the overall reaction can be found by multiplying the reactions with certain coefficients, the so-called Horiuti numbers σ, and then adding the results. While the

Horiuti number allows us to distinguish the short-lived intermediate and long-lived components, i.e., to eliminate the intermediates using an RREF of the stoichiometric matrix S, the intermediates must be listed first, not last. Then the rows in which all intermediates vanish

and the number of key components equals the number of key reactions. Also, the number of

In Figure 2, their curves represent two different solution curves of their respective reaction routes lying at different phase space, i.e., one lies in 2D while the second lies within 3D.

Now the question arises, if a complex reaction adopts different completion routes before giving the product, then how can one relate (or distinguish) such available routes and why

and backward N b spanning trees are N and the numbers of combined spanning

Nc ¼ N Nð Þ¼ � 2 N2 � 2N (14)

σ:Nint ¼ 0 (15)

Complex Reactions and Dynamics

9

http://dx.doi.org/10.5772/intechopen.70502

Nkc ¼ Nc � rank Mð Þ (16)

The concentration of activated complex can thus be obtained by applying the equilibrium conditions, i.e.,

$$\left[\Theta\_i^{\mp}\right] = k\_i^{\mp} \sum\_{i=1}^{N} \alpha\_{\mathbb{S}\_i} A\_i \tag{9}$$

where ∓ refers to the activated complex.

#### 2. Reaction rate

A stoichiometric vector γρ of the reaction mechanism (1) is an n� dimensional vector with coordinates γρi = βρi � αρi, that is, "gain minus loss" in the ρth elementary reaction. In matrix form, it takes a form

$$\mathcal{S} = \begin{bmatrix} \mathcal{V}\_{\rho 1}, \ \mathcal{V}\_{\rho 2}, \dots, \mathcal{V}\_{\rho i} \end{bmatrix} \tag{10}$$

The chemical composition of the substances is given by the molecular matrix M, with the element mij as a number of atoms of the jth element in the ith component. M is a (Nc � Ne) matrix, while Nc is the number of reacting components lying in the mixture consisting of Ne; the chemical elements and the law of conservation of atoms say

$$M n\_{\varepsilon} = V\_{\varepsilon} \text{ (constant vector)}.\tag{11}$$

The total number of any moles of ci atoms can be measured by using the relation

$$M^T n\_c = n\_e \tag{12}$$

Here, MT is the transposed molecular matrix and nc and ne are the component amount (mol) and the amount of the chemical elements (mol) in the column vector form.

The dynamics of the involved concentration species can be measured when we measure the rate of formation of the products or deformation and disappearance of the reactants.

Finally, the rate of reaction will take a form

$$X(\mathcal{C}) = \dot{\mathcal{C}} = \frac{d\mathcal{C}}{dt} = \mathcal{S}\_{\rho} \, W\_{\rho}(\mathcal{C}) \tag{13}$$

Here Wρ(c): Wρð Þ¼ c kρð Þ T Q i c αρi i is the reaction rate function of the ρth step (i.e., the difference between the rate of forward Wþ ρ ð Þc and backward W� ρ ð Þc reactions).

### 3. Linear algebra and graph theory

In case of reversible complex chemical reactions,

where ∓ refers to the activated complex.

conditions, i.e.,

8 Advanced Chemical Kinetics

2. Reaction rate

form, it takes a form

X N

Ai ⇔ Θ∓

The concentration of activated complex can thus be obtained by applying the equilibrium

A stoichiometric vector γρ of the reaction mechanism (1) is an n� dimensional vector with coordinates γρi = βρi � αρi, that is, "gain minus loss" in the ρth elementary reaction. In matrix

> S ¼ γρ1; γρ2;…; γρi h i

The chemical composition of the substances is given by the molecular matrix M, with the element mij as a number of atoms of the jth element in the ith component. M is a (Nc � Ne) matrix, while Nc is the number of reacting components lying in the mixture consisting of Ne;

Here, MT is the transposed molecular matrix and nc and ne are the component amount (mol)

The dynamics of the involved concentration species can be measured when we measure the

rate of formation of the products or deformation and disappearance of the reactants.

X cð Þ¼ c\_ ¼ dc

ρ ð Þc and backward W�

Mnc ¼ Vc ð Þ constant vector : (11)

MTnc ¼ ne (12)

dt ¼ Sρ Wρð Þc (13)

i is the reaction rate function of the ρth step (i.e., the difference

ρ ð Þc reactions).

the chemical elements and the law of conservation of atoms say

Finally, the rate of reaction will take a form

i c αρi

Here Wρ(c): Wρð Þ¼ c kρð Þ T Q

between the rate of forward Wþ

The total number of any moles of ci atoms can be measured by using the relation

and the amount of the chemical elements (mol) in the column vector form.

i ⇔ X N

> i¼1 αSi

i¼1 βSi Ai (8)

Ai (9)

(10)

i¼1 αSi

> Θ∓ i � � ¼ k ∓ i X N

In chemical engineering, the mathematical methods of graph theory have found wide applications in complex chemical reactions and in a sequence of uni (or multi) or parallel reacting events. A graph is a combination of nodes (points) and edges (lines) [2], while a cyclic graph involves finite sequences of edges with the single node (from where it begins and ends).

Similarly, related to any combination of reaction, a tree can be defined as a sequence of noncyclic graph edges. In a spanning tree, certain intermediate may form from other intermediates after a sequence of transformations but does not agree to counter any two reactions with the same step (e.g., +1 and �1) nor two reactions started with the same intermediates (e.g., �1 and +2, or +1 and �3),

Spanning trees can be described in terms of "forward" (generated by a sequence of forwarding reactions), "backward" (generated by a sequence of reverse reactions), and "combined" spanning trees (generated by a sequence of both forward and backward reactions). A single-route, n-steps Ns (edges) reaction mechanism has Nint (intermediates) nodes, such as Nint = Ns = N. The total numbers of spanning trees are N 2 in any reaction, while the forward N f and backward N b spanning trees are N and the numbers of combined spanning trees Nc are

$$N^{\varepsilon} = N(N-2) = N^2 - 2N \tag{14}$$

In a chemical reaction, the overall reaction can be found by multiplying the reactions with certain coefficients, the so-called Horiuti numbers σ, and then adding the results. While the relation between σ and Nint is

$$
\sigma.N\_{\text{int}} = 0 \tag{15}
$$

Horiuti number allows us to distinguish the short-lived intermediate and long-lived components, i.e., to eliminate the intermediates using an RREF of the stoichiometric matrix S, the intermediates must be listed first, not last. Then the rows in which all intermediates vanish provide a basis for the overall reactions [2].

The numbers of key components Nkc are given by the equation

$$N\_{kx} = N\_c - rank(M) \tag{16}$$

and the number of key components equals the number of key reactions. Also, the number of key components + number of nonkey reactions = number of reactions

In Figure 2, their curves represent two different solution curves of their respective reaction routes lying at different phase space, i.e., one lies in 2D while the second lies within 3D.

Now the question arises, if a complex reaction adopts different completion routes before giving the product, then how can one relate (or distinguish) such available routes and why they are important to be measured?

For this, the reaction route Nrr of the system can be measured as

Figure 2. A complex chemical reaction passes through different transition states and adopts different completion routes before giving the products.

$$N\_{rr} = N\_s - N\_{\text{int}} + N\_{\text{as}} \tag{17}$$

While the overall reaction evolves no intermediates, i.e., A2 þ 2B Ð

�1 �2 2 0 00 0 �1 0 �1 10 0 2 �1 0 �1 1 0 1 �1 �1 01

Nint ¼

Atomic balance constraints are given by Eq. (11)

Nkc=Nc�rank(M)=6�3=3,

Note that the relation between them is orthogonal, i.e., Eqs. (15) and (18) hold. While the stoichiometric matrix of intermediates and Horiuti matrix are Ns = 4, Nint = 3(Z,AZ,BZ), Nas = 1(Z),

ηt,A ¼ 2ηc1 þ ηc3 þ ηc6

ηt,Z ¼ 2ηc2 þ ηc3 þ ηc5

ηt,B ¼ 2ηc4 þ ηc5 þ ηc6

whereas, nt,A, nt,Z, and nt,B are the total number of moles of A, Z, and B atoms, respectively. The key components Nkc and reaction route Nrr of the system are given by Eqs. (16) and (17)

�22 0 �10 1 2 � 1 � 1 1 � 1 0 , M ¼

matrix (10) and molecular matrix (11) infer

Figure 3. Four-step reversible reaction having two routes.

S¼

kþ

1

8 >>>>>>>><

>>>>>>>>:

A2 Z AZ B BZ AB

CCCCCCCCA

0

BBBBBBBB@

,

,

:

k� 2AB, the stoichiometric

Complex Reactions and Dynamics

11

http://dx.doi.org/10.5772/intechopen.70502

(19)

(20)

whereas, Ns is the number of steps in the detailed mechanism and Nas is the number of active sites in the mechanism. Based on the molecular matrix, the molar masses of the components can be determined from the atomic masses of the elements. The product of the stoichiometric matrix S and molecular matrix M gives

$$SM = 0\tag{18}$$

To answer the second part of the above question, we need to consider all its available routes to get the detailed reaction mechanism. Then a comparison of these route solutions with the whole reaction mechanism allows us to give any concluding remarks, but we believe that the result obtained through different routes may be similar or vary depending on the type of reactions.

#### 4. Multiroute reactions mechanism

To understand this idea, let us discern the four-step reversible complex chemical reaction [3] defined over a closed system having two available routes. The mechanism involves six chemical substances (species Ci) represented as (Figure 3),

Figure 3. Four-step reversible reaction having two routes.

Nrr ¼ Ns � Nint þ Nas (17)

SM ¼ 0 (18)

whereas, Ns is the number of steps in the detailed mechanism and Nas is the number of active sites in the mechanism. Based on the molecular matrix, the molar masses of the components can be determined from the atomic masses of the elements. The product of the stoichiometric

Figure 2. A complex chemical reaction passes through different transition states and adopts different completion routes

To answer the second part of the above question, we need to consider all its available routes to get the detailed reaction mechanism. Then a comparison of these route solutions with the whole reaction mechanism allows us to give any concluding remarks, but we believe that the result obtained through different routes may be similar or vary depending on the type of reactions.

To understand this idea, let us discern the four-step reversible complex chemical reaction [3] defined over a closed system having two available routes. The mechanism involves six chem-

matrix S and molecular matrix M gives

before giving the products.

10 Advanced Chemical Kinetics

4. Multiroute reactions mechanism

ical substances (species Ci) represented as (Figure 3),

While the overall reaction evolves no intermediates, i.e., A2 þ 2B Ð kþ k� 2AB, the stoichiometric matrix (10) and molecular matrix (11) infer

$$S = \begin{bmatrix} -1 & -2 & 2 & 0 & 0 & 0\\ 0 & -1 & 0 & -1 & 1 & 0\\ 0 & 2 & -1 & 0 & -1 & 1\\ 0 & 1 & -1 & -1 & 0 & 1 \end{bmatrix}, \quad M = \begin{pmatrix} A & Z & B\\ 2 & 0 & 0\\ 0 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1\\ 0 & 1 & 1\\ 1 & 0 & 1 \end{pmatrix} \begin{cases} A\_2\\ Z\\ AZ\\ AB\\ AB\\ AB \end{cases} \tag{19}$$

Note that the relation between them is orthogonal, i.e., Eqs. (15) and (18) hold. While the stoichiometric matrix of intermediates and Horiuti matrix are Ns = 4, Nint = 3(Z,AZ,BZ), Nas = 1(Z),

$$N\_{\rm int} = \begin{bmatrix} -2 & 2 & 0 \\ -1 & 0 & 1 \\ 2 & -1 & -1 \\ 1 & -1 & 0 \end{bmatrix} \\ \\ \sigma = \begin{bmatrix} -1 & 0 \\ 0 & 1 \\ -2 & -1 \\ 0 & 1 \end{bmatrix}$$

Atomic balance constraints are given by Eq. (11)

$$\begin{aligned} \eta\_{t,A} &= 2\eta\_{c\_1} + \eta\_{c\_3} + \eta\_{c\_6}, \\ \eta\_{t,Z} &= 2\eta\_{c\_2} + \eta\_{c\_3} + \eta\_{c\_5}, \\ \eta\_{t,B} &= 2\eta\_{c\_4} + \eta\_{c\_5} + \eta\_{c\_6}. \end{aligned} \tag{20}$$

whereas, nt,A, nt,Z, and nt,B are the total number of moles of A, Z, and B atoms, respectively. The key components Nkc and reaction route Nrr of the system are given by Eqs. (16) and (17) Nkc=Nc�rank(M)=6�3=3,

This means we can reduce this system into three components

Nrr=NsNint+Nas=43+1=2.

Hence, this reaction mechanism has two independent routes Nrr. Also, when we multiply step-1 and step-3 by its Horiuti numbers, all the intermediates vanished and we get an overall reaction. The same is the case with steps 1, 2, and 4. The dimension of these two routes can be determined by their respective Horiuti numbers. Sets of Horiuti numbers for the first route and second route are (1,0,0,2) and (1,2,2,0), respectively. This implies both the routes are nonlinear.

5. The measuring methods

initial parameters are defined as

¼ 0:1, ceq

d dt

initial parameters are defined as

c eq ¼ 0:1, ceq

c eq

kþ ¼ 1, kþ

given by Eq.  For comparison, we refer the readers to [28].

cA2 cB cAB cZ cAZ cBZ

The kinetic equations of the above reaction mechanism (R-1) can be measured by using Eq. (13)

� c2c5 � 2k

� c2c5

A reduced form of the system (R-1) can be achieved by using the Eqs. (12) and (13). While

Similarly, (R-2) implies that we can reduce this system into three components, while Ns = 3,

Thus, a single reaction route is available. The kinetic equations for the involved species are

By using Eqs. (12) and (13), a system can be reduced into three numbers of species and their

Figures 6 and 7 clear the idea of the slow invariant manifold (SIM), i.e., decomposing the system into their fast and slow motion. Their solution trajectories (during their relaxation time) quickly move toward the low-dimensional manifold and after that start moving along it [4–13]. That is the easy way of getting an idea of the SIM. Otherwise, by using the different available methods of SIM, i.e., [14–27] we will get their initial approximations lying on it or

¼ :4, ceq

eq c eq , ceq

¼ 0:5:

¼ 1, kþ

¼ 0:1, kþ

� c2 c6 � k þ c2c5 þ k

� c2 c6 þ 2k þ c1c2 � 2k � c2

� c2 c6 þ k þ c2c5 � k

� c4

� c2 c6

¼ 0:1, ceq

¼ 0:5, kþ

� c2c5 þ 2k

� c2c5 þ c1c2 þ 2k � c2

> þ c1c2 � 2k � c2

¼ 0:5.

� c4 � 2k þ c1c2 þ 2k � c2

� c4

¼ 0:500000000c

¼ 0:2500000000c

eq c eq =c eq c eq ,

eq c eq =c eq :

Complex Reactions and Dynamics

http://dx.doi.org/10.5772/intechopen.70502

First-route: two-step mechanisms (Figure 4).

Second-route: three-step mechanisms (Figure 5).

Figure 4. R-1: The first route of the reaction mechanism is a two-step reversible reaction involving five chemical species, while Nkc = 2.

Figure 5. R-2: The second route of the reaction mechanism is a three-step reversible reaction involving six chemical species, while Nkc = 3.

### 5. The measuring methods

This means we can reduce this system into three components

Hence, this reaction mechanism has two independent routes Nrr. Also, when we multiply step-1 and step-3 by its Horiuti numbers, all the intermediates vanished and we get an overall reaction. The same is the case with steps 1, 2, and 4. The dimension of these two routes can be determined by their respective Horiuti numbers. Sets of Horiuti numbers for the first route and second route are (1,0,0,2) and (1,2,2,0), respectively. This implies both the routes are nonlinear.

Figure 4. R-1: The first route of the reaction mechanism is a two-step reversible reaction involving five chemical species,

Figure 5. R-2: The second route of the reaction mechanism is a three-step reversible reaction involving six chemical

Nrr=NsNint+Nas=43+1=2.

12 Advanced Chemical Kinetics

while Nkc = 2.

species, while Nkc = 3.

First-route: two-step mechanisms (Figure 4).

Second-route: three-step mechanisms (Figure 5).

The kinetic equations of the above reaction mechanism (R-1) can be measured by using Eq. (13)

$$\frac{d}{dt}\begin{bmatrix}c\_{A\_2} \\ c\_B \\ c\_{AB} \\ c\_Z \\ c\_{AZ}\end{bmatrix} = \begin{bmatrix}-k\_1^+c\_1c\_2^2 - k\_1^-c\_3^2 \\ k\_2^+c\_3c\_4 - k\_2^-c\_2c\_5 - 2k\_1^+c\_1c\_2^2 + 2k\_1^-c\_3^2 \\ -k\_2^+c\_3c\_4 + k\_2^-c\_2c\_5 + 2k\_1^+c\_1c\_2^2 - 2k\_1^-c\_3^2 \\ -k\_2^+c\_3c\_4 + k\_2^-c\_2c\_5 \\ k\_2^+c\_3c\_4 - k\_2^-c\_2c\_5 \end{bmatrix}$$

while Ns=2, Nint=2(Z,AZ),Nas=1(Z), and a reaction route Nrr is 1.

A reduced form of the system (R-1) can be achieved by using the Eqs. (12) and (13). While initial parameters are defined as

$$c\_1^{eq} = 0.5, c\_2^{eq} = 0.1, c\_3^{eq} = 0.1, c\_4^{eq} = 0.4, c\_5^{eq} = 0.1, k\_1^+ = 1, k\_2^+ = 0.5.$$

Similarly, (R-2) implies that we can reduce this system into three components, while Ns = 3, Nint = 3(Z,ZO,ZCO),Nas = 1(Z), and Nrr = 1.

Thus, a single reaction route is available. The kinetic equations for the involved species are given by Eq. (13)

$$\frac{d}{dt}\begin{bmatrix}c\_{A\_{2}}\\c\_{B}\\c\_{AB}\\c\_{Z}\\c\_{Z}\\c\_{AZ}\\c\_{AZ}\\c\_{BZ}\end{bmatrix} = \begin{bmatrix}-k\_{1}^{+}c\_{1}c\_{2}^{2}-k\_{1}^{-}c\_{3}^{2}\\2k\_{3}^{+}c\_{3}c\_{4}-2k\_{3}^{-}c\_{2}^{2}c\_{6}-k\_{2}^{+}c\_{2}c\_{5}+k\_{2}^{-}c\_{4}-2k\_{1}^{+}c\_{1}c\_{2}^{2}+2k\_{1}^{-}c\_{3}\\-k\_{3}^{+}c\_{3}c\_{4}+k\_{3}^{-}c\_{2}^{2}c\_{6}+2k\_{1}^{+}c\_{1}c\_{2}^{2}-2k\_{1}^{-}c\_{3}\\-k\_{3}^{+}c\_{3}c\_{4}+k\_{3}^{-}c\_{2}^{2}c\_{6}+k\_{2}^{+}c\_{2}c\_{5}-k\_{2}^{-}c\_{4}\\-k\_{2}^{+}c\_{2}c\_{5}+k\_{2}^{-}c\_{4}\\k\_{3}^{+}c\_{3}c\_{4}-k\_{3}^{-}c\_{2}^{2}c\_{6}\end{bmatrix}.$$

By using Eqs. (12) and (13), a system can be reduced into three numbers of species and their initial parameters are defined as

$$\begin{aligned} c\_2^{\epsilon \eta} &= 0.1, c\_3^{\epsilon \eta} = 0.1, c\_5^{\epsilon \eta} = .4, & c\_1^{\epsilon \eta} &= 0.500000000 c\_3^{\epsilon \eta} c\_3^{\epsilon \eta} / c\_2^{\epsilon \eta} c\_2^{\epsilon \eta}, \\ c\_4^{\epsilon \eta} &= 0.25000000000 c\_2^{\epsilon \eta} c\_5^{\epsilon \eta}, & c\_6^{\epsilon \eta} &= 0.2500000000 c\_3^{\alpha \eta} c\_5^{\alpha \eta} / c\_2^{\alpha}. \\ k\_1^+ &= 1, k\_2^+ = 0.5, k\_3^+ = 0.5. \end{aligned}$$

Figures 6 and 7 clear the idea of the slow invariant manifold (SIM), i.e., decomposing the system into their fast and slow motion. Their solution trajectories (during their relaxation time) quickly move toward the low-dimensional manifold and after that start moving along it [4–13]. That is the easy way of getting an idea of the SIM. Otherwise, by using the different available methods of SIM, i.e., [14–27] we will get their initial approximations lying on it or near to it. For comparison, we refer the readers to [28].

6. The routes comparison

represent their equilibrium point.

In Figure 9, the curves lie in the plane c1 and c4 are not the projected image of the above curves. Instead, it is the behavior of the species measured near the equilibrium point in the first reaction route mechanism, whose Nkc = 2. Similarly, the above lines describe the behavior of the

Complex Reactions and Dynamics

15

http://dx.doi.org/10.5772/intechopen.70502

Figure 8. Two-dimensional view of both the reaction routes solutions and both squares represent their equilibrium point.

Figure 9. First-route vs. second-route. The three-dimensional view of both the reaction routes solutions and both squares

Figure 6. The behavior of the reduced species c1 and c4 near the equilibrium point (square). While their solution trajectories approaching toward the equilibrium (during their path) give the region where slow invariant manifold (SIM) lies.

Figure 7. The equilibrium point (square) and behavior of the reduced species near to it.

### 6. The routes comparison

Figure 6. The behavior of the reduced species c1 and c4 near the equilibrium point (square). While their solution trajectories approaching toward the equilibrium (during their path) give the region where slow invariant manifold (SIM)

Figure 7. The equilibrium point (square) and behavior of the reduced species near to it.

lies.

14 Advanced Chemical Kinetics

In Figure 9, the curves lie in the plane c1 and c4 are not the projected image of the above curves. Instead, it is the behavior of the species measured near the equilibrium point in the first reaction route mechanism, whose Nkc = 2. Similarly, the above lines describe the behavior of the

Figure 8. Two-dimensional view of both the reaction routes solutions and both squares represent their equilibrium point.

Figure 9. First-route vs. second-route. The three-dimensional view of both the reaction routes solutions and both squares represent their equilibrium point.

Author details

References

Muhammad Shahzad\* and Faisal Sultan

McGraw-Hill; 2012. p. 352

s11144-017-1163-5

2011. p. 428

1.3171613

\*Address all correspondence to<EMAIL_ADDRESS>
York: Springer; 2005. p. 469-489. DOI: 10.1007/b98103

putational Ecology and Software. 2015;5(3):254-270

Chemical Society of Pakistan. 2015;37(2):207-216

space Exposition. January 08; 2009.

Pakistan. 2016;38(5):828-835

j.jcp.2008.02.006

Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

[1] Houston PL. Chemical Kinetics and Reaction Dynamics. 3rd ed. Courier Corporation/

Complex Reactions and Dynamics

17

http://dx.doi.org/10.5772/intechopen.70502

[2] Constales D, Yablonsky GS, D'hooge DR, Thybaut JW, Marin GB. Advanced Data Analysis and Modelling in Chemical Engineering. 2nd ed. Elsevier; 2016. p. 414. DOI: 10.1007/

[3] Marin GB, Yablonsky GS. Kinetics of Chemical Reactions. 1st ed. John Wiley & Sons;

[4] Al-Khateeb AN, Powers JM, Paolucci S, Sommese AJ, Diller JA, Hauenstein JD, Mengers JD. One-dimensional slow invariant manifolds for spatially homogenous reactive systems. The Journal of Chemical Physics. 2009;131(2):024118. DOI: http://dx.doi.org/10.1063/

[5] Chiavazzo E, Gorban AN, Karlin IV. Comparison of invariant manifolds for model reduction in chemical kinetics. Communications in Computational Physics. 2007;2(5):964-992 [6] Gorban AN, Karlin IV. Invariant Manifolds for Physical and Chemical Kinetics. New

[7] Shahzad M et al. Measuring the complex behavior of the SO2 oxidation reaction. Com-

[8] Shahzad M et al. Initially approximated quasi equilibrium manifold. Journal of the

[9] Al-Khateeb, Ashraf, et al. Calculation of slow invariant manifolds for reactive systems. In: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aero-

[10] Chiavazzo E, Karlin IV. Quasi-equilibrium grid algorithm: Geometric construction for model reduction. Journal of Computational Physics. 2008;227(11):5535-5560. DOI: 10.1016/

[11] Shahzad M et al. Slow manifolds in chemical kinetics. Journal of the Chemical Society of

Figure 10. Variation of chemical species concentration of overall reaction mechanism with respect to time, while Nkc=1.

species near the equilibrium point measured in second reaction route mechanism, whose Nkc = 3. Note that it's invariant region and equilibrium point exactly lie over the invariant region of the first route, i.e., Figures 8, 9.

Now, the overall reaction mechanism involves no intermediate, and the variations of the concentration of involved chemical species are given in Figure 10.

### 7. Summary

In this chapter, both the physicochemical conceptual assumptions (used for species behavior and activated complex) and a set of mathematical tools (for their dynamical behavior and simplification) are presented. Mathematically, simplification can be done by "model reduction," that is, the rigorous way of approximating and representing a complex model in simplified form.

Here, we have considered a complex problem having a common step: conferred their available routes then allied graphically. Although we have not applied any numerical or analytical technique to measure the SIM but one can easily examine (by applying such techniques) that their solution trajectories will also lie in the same invariant regions that can also be correlated with each other and even with the whole reaction mechanism.

Thus, the idea initiated here can easily be correlated with the method used for the construction of slow manifold in a complex chemical reaction based on the decomposition techniques of entropy maximum along with certain constraints (lies on the manifold or given by slowest eigenvectors) at the equilibrium point. This will allow us to bring together different available mathematical ideas and methods, commonly used to transform the complex chemical problems from one way to the other, to enhance progress in understanding.

### Author details

Muhammad Shahzad\* and Faisal Sultan

\*Address all correspondence to<EMAIL_ADDRESS>
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

### References

species near the equilibrium point measured in second reaction route mechanism, whose Nkc = 3. Note that it's invariant region and equilibrium point exactly lie over the invariant region of

Figure 10. Variation of chemical species concentration of overall reaction mechanism with respect to time, while Nkc=1.

Now, the overall reaction mechanism involves no intermediate, and the variations of the

In this chapter, both the physicochemical conceptual assumptions (used for species behavior and activated complex) and a set of mathematical tools (for their dynamical behavior and simplification) are presented. Mathematically, simplification can be done by "model reduction," that is, the rigorous way of approximating and representing a complex model in simpli-

Here, we have considered a complex problem having a common step: conferred their available routes then allied graphically. Although we have not applied any numerical or analytical technique to measure the SIM but one can easily examine (by applying such techniques) that their solution trajectories will also lie in the same invariant regions that can also be correlated

Thus, the idea initiated here can easily be correlated with the method used for the construction of slow manifold in a complex chemical reaction based on the decomposition techniques of entropy maximum along with certain constraints (lies on the manifold or given by slowest eigenvectors) at the equilibrium point. This will allow us to bring together different available mathematical ideas and methods, commonly used to transform the complex chemical prob-

concentration of involved chemical species are given in Figure 10.

with each other and even with the whole reaction mechanism.

lems from one way to the other, to enhance progress in understanding.

the first route, i.e., Figures 8, 9.

7. Summary

16 Advanced Chemical Kinetics

fied form.

[12] Kooshkbaghi, M, et al. The global relaxation redistribution method for reduction of combustion kinetics. The Journal of Chemical Physics. 2014;141(4):044102. DOI: dx.doi. org/10.1063/1.4890368.

[26] Constales D et al. Thermodynamic time-invariances: Theory of TAP pulse-response experiments. Chemical Engineering Science. 2011;66(20):4683-4689. DOI: doi.org/10.1016/

Complex Reactions and Dynamics

19

http://dx.doi.org/10.5772/intechopen.70502

[27] Yablonsky GS, Constales D, Marin GB. New types of complexity in chemical kinetics: intersections, coincidences, and special symmetrical relationships. In: Proceedings of the 240 Conference: Science's Great Challenges, Vol. 157; 12 December 2014; John Wiley &

[28] Muhammad Shahzad, 'Different Available Completion Routes in Complex Chemical Reactions'. International Conference on Mathematics in (bio) Chemical Kinetics and Engineering – MaCKiE2017 held on 25-27 May 2017 in Budapest, Hungary. http://static.

akcongress.com/downloads/mackie/mackie2017-book-of-abstracts.pdf

Sons, Inc.; 2014. DOI: 10.1002/9781118959602.ch6

j.ces.2011.06.033

[26] Constales D et al. Thermodynamic time-invariances: Theory of TAP pulse-response experiments. Chemical Engineering Science. 2011;66(20):4683-4689. DOI: doi.org/10.1016/ j.ces.2011.06.033

[12] Kooshkbaghi, M, et al. The global relaxation redistribution method for reduction of combustion kinetics. The Journal of Chemical Physics. 2014;141(4):044102. DOI: dx.doi.

[13] Constales D, Yablonsky GS, Marin GB. Thermodynamic time invariances for dual kinetic experiments: Nonlinear single reactions and more. Chemical Engineering Science. 2012;

[14] Bongers H, Van Oijen JA, De Goey LPH. Intrinsic low-dimensional manifold method extended with diffusion. Proceedings of the Combustion Institute. 2002;29(1):1371-1378.

[15] Bykov V et al. On a modified version of ILDM approach: Asymptotic analysis based on integral manifolds. IMA Journal of Applied Mathematics. 2006;71(3):359-382. DOI: doi.

[16] Gorban AN, Karlin IV. Thermodynamic parameterization. Physica A: Statistical Mechanics and its Applications. 1992;190(3):393-404. DOI: doi.org/10.1016/0378-4371(92)90044-Q

[17] Gorban AN, Shahzad M. The Michaelis-Menten-Stueckelberg theorem. Entropy. 2011;

[18] Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combustion and Flame. 1992;88(3):239-264. DOI: doi.org/10.1016/

[19] Gorban AN, Karlin IV, Zinovyev AY. Invariant grids for reaction kinetics. Physica A: Statistical Mechanics and its Applications. 2004;333:106-154. DOI: doi.org/10.1016/j.physa.

[20] Gorban AN, Karlin IV. Method of invariant manifold for chemical kinetics. Chemical Engineering Science. 2003;58(21):4751-4768. DOI: doi.org/10.1016/j.ces.2002.12.001 [21] Gorban AN, Karlin IV, Zinovyev AY. Constructive methods of invariant manifolds for kinetic problems. Physics Reports. 2004;396(4):197-403. DOI: doi.org/10.1016/j.physrep.

[22] Shahzad M et al. Computing the low dimension manifold in dissipative dynamical

[23] Yablonsky GS, Constales D, Marin GB. Coincidences in chemical kinetics: Surprising news about simple reactions. Chemical Engineering Science. 2010;65(23):6065-6076. DOI:

[24] Yablonsky GS, Constales D, Marin GB. Equilibrium relationships for non-equilibrium chemical dependencies. Chemical Engineering Science. 2011;66(1):111-114. DOI: doi.org/

[25] Yablonsky GS et al. Reciprocal relations between kinetic curves. EPL (Europhysics Let-

org/10.1063/1.4890368.

org/10.1093/imamat/hxh100

0010-2180(92)90034-M

2003.10.043

2004.03.006

DOI: doi.org/10.1016/S1540-7489(02)80168-7

13(5):966-1019. DOI: 10.3390/e13050966

systems. The Nucleus. 2016;53(3):107-113

doi.org/10.1016/j.ces.2010.04.007

10.1016/j.ces.2010.10.014

ters). 2011;93(2):20004

73:20-29

18 Advanced Chemical Kinetics

**Chapter 2**

Provisional chapter

**Mathematical Modeling and Simulation of Nonlinear**

DOI: 10.5772/intechopen.70914

Mathematical Modeling and Simulation of Nonlinear

A deep and analytical understanding of the enzyme kinetics has attracted a great attention of scientists from biology, medicine, chemistry, and pharmacy. Mathematical models of enzyme kinetics offer several advances for this deep and analytical understanding due to their in compensable potential in predicting kinetic processes and anticipating appropriate interventions when required. This chapter concerns mathematical modeling analysis and simulation of enzyme kinetics. Experimental data and available knowledge on enzyme mechanics are used in constituting a mathematical model. The models are either in the form of linear or nonlinear ordinary differential equations or partial differential equations. These equations are composed of kinetic parameters such as kinetic rate constants, initial rates, and concentrations of enzymes. The nonlinear nature of enzymatic reactions and a large number of parameters have caused major issues with regard to efficient simulation of those reactions. In this work, an enzymatic system that includes Michaelis-Menten and Ping Pong kinetics is modeled in the form of differential equations. These equations are solved numerically in which the system parameters are estimated. The numerical results

are compared with the results from an existing work in literature.

Keywords: mathematical modeling, enzyme kinetics, chemical kinetics, nonlinear reaction-diffusion equation, amperometric, cyclic voltammetry, chronoamperometric

Enzyme kinetics is a challenging research field nowadays incorporating modern applied mathematics into biotechnology, engineering science, and pharmacy. Moreover, in medical studies, scientists work on human metabolism to improve the capabilities of some metabolites or enzymes in metabolic pathways. In industrial applications, kinetics methods are also widely used to

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Process in Enzyme Kinetics**

Process in Enzyme Kinetics

Carlos Fernandez and Qiuming Peng

Carlos Fernandez and Qiuming Peng

http://dx.doi.org/10.5772/intechopen.70914

Abstract

1. Introduction

Lakshmanan Rajendran, Mohan Chitra Devi,

Lakshmanan Rajendran, Mohan Chitra Devi,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Provisional chapter

### **Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics** Mathematical Modeling and Simulation of Nonlinear

DOI: 10.5772/intechopen.70914

Lakshmanan Rajendran, Mohan Chitra Devi, Carlos Fernandez and Qiuming Peng Lakshmanan Rajendran, Mohan Chitra Devi,

Additional information is available at the end of the chapter Carlos Fernandez and Qiuming Peng Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70914

Process in Enzyme Kinetics

#### Abstract

A deep and analytical understanding of the enzyme kinetics has attracted a great attention of scientists from biology, medicine, chemistry, and pharmacy. Mathematical models of enzyme kinetics offer several advances for this deep and analytical understanding due to their in compensable potential in predicting kinetic processes and anticipating appropriate interventions when required. This chapter concerns mathematical modeling analysis and simulation of enzyme kinetics. Experimental data and available knowledge on enzyme mechanics are used in constituting a mathematical model. The models are either in the form of linear or nonlinear ordinary differential equations or partial differential equations. These equations are composed of kinetic parameters such as kinetic rate constants, initial rates, and concentrations of enzymes. The nonlinear nature of enzymatic reactions and a large number of parameters have caused major issues with regard to efficient simulation of those reactions. In this work, an enzymatic system that includes Michaelis-Menten and Ping Pong kinetics is modeled in the form of differential equations. These equations are solved numerically in which the system parameters are estimated. The numerical results are compared with the results from an existing work in literature.

Keywords: mathematical modeling, enzyme kinetics, chemical kinetics, nonlinear reaction-diffusion equation, amperometric, cyclic voltammetry, chronoamperometric

### 1. Introduction

Enzyme kinetics is a challenging research field nowadays incorporating modern applied mathematics into biotechnology, engineering science, and pharmacy. Moreover, in medical studies, scientists work on human metabolism to improve the capabilities of some metabolites or enzymes in metabolic pathways. In industrial applications, kinetics methods are also widely used to

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

develop certain methods for improving functionality of some molecules in a cell. Many problems in theoretical and experimental biology/chemistry involve the solution of the steady-state reaction diffusion equation with nonlinear chemical kinetics. Such problems also arise in the formulation of substrate and product material balances for enzymes immobilized within particles [1, 2], in the description of substrate transport into microbial cells [3–5], in membrane transport, in the transfer of oxygen to respiring tissue [6, 7], and in the analysis of any artificial kidney system [8].

chemical reaction. The substances are transformed into each other in local chemical reaction, whereas the substances are spread out over a surface in space in diffusion. Reaction-diffusion (RD) systems arise in many branches of physics, chemistry, biology, ecology, etc. Reviews of the theory and applications of reaction-diffusion systems can be found in books and numerous articles (see, for example [20–23]). These arise in a large variety of application areas, such as flow in porous media [24], heat conduction in plasma [25], combustion problems [26], liquid evaporation [27], and of more recent interest, image processing [28]. A great effort is being made in the development of the mathematical theory of nonlinear diffusion equations and to obtain exact solutions for special cases. Their significance not only relies on the huge number of their applications but also on the fact that they provide with a rather general class of linear and nonlinear differential operators. In mathematical analysis, it has shown to be a milestone for the development of applied, abstract, and numerical analysis as well as for algebra, geometry, and topology.

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

http://dx.doi.org/10.5772/intechopen.70914

23

The modern theory of the nonlinear reaction diffusion process is an important field in today's science. The nonlinear system and coherent structures represent an interdisciplinary area with many nonlinear applications in various fields. Those applications can be divided into six disciplines: chemistry (autocatalytic chemical and enzyme reactions), physics (nonlinear optics and electric circuits, plasmas and states of solid, condensed atomic gases, hydrodynamics, galaxy dynamics and cosmology, fluid dynamics, and celestial mechanics), general relativity, biology (biofuel cell, bioreactor and biosensor, atmosphere and oceans, and animal dispersal), random media, and modern telecommunications. A great variety of phenomena in physics, chemistry, or biology can be described by nonlinear ODE/PDEs and particularly by reaction-diffusion equations. For these reasons, the theory of the analytical solutions of the reaction-diffusion equations

In reaction diffusion systems, nonlinear phenomena play a crucial role in applied mathematics and chemistry. Exact (closed-form) solution of nonlinear reaction diffusion equations plays an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. The main result obtained from reaction and diffusion systems is that nonlinear phenomena include diversity of stationary and spatiotemporary dissipative patterns, oscillations, different types of waves, excitability, biostability, etc. But it is difficult for us to obtain the exact solution for these problems. The investigation of exact solution of nonlinear equation is interesting and important. In general, this results in the need to solve linear and nonlinear reaction diffusion equations with complex boundary conditions. The enzyme kinetics in biochemical systems have usually been modeled by differential equations, which are based only on reaction without spatial dependence of the various concentrations.  Nonlinear phenomena

is considered.

To impose the functionality of some molecules in a cell, a mathematical model of such metabolic systems must be constructed and simulated. Most of the dynamical systems can be approximated by various types of differential and integral equations involving finite number of variables and parameters. Thus, the future behavior of the system can be predicted if model kinetics parameters and initial states of the variables are available. In particular, ordinary and partial differential equations (ODEs and PDEs) are popular in modeling of the metabolic pathways or enzyme kinetics.

Releasing enzyme-substrate reactions under single-molecule kinetics was reported by Shlomi et al. [9]. An integral equation method with Michaelis-Menten kinetics to solve nonlinear diffusion problems in spherical coordinates was stated by Tosaka and Miyale [10]. Maalmi et al. [11] reported numerical and semianalytical solutions of nonlinear equations, which covered diffusivity, size, bulk concentration of reactant, binding constant of Michaelis-Menten kinetics, and site reactivity values. Merchant [12] stated the M-M decay reaction terms and the Gray-Scott scheme along with the semianalytical method to nonlinear reaction-diffusion systems. Indira and Rajendran [13] described a homotopy perturbation method to obtain substrate and product concentrations within the enzymatic layers. Removal of substrate from Michaelis-Menten kinetics governed the extravascular partition in which the analytical solution for the steady-state condition was investigated by Bucolo and Tripathi [14]. Dang Do and Greenfield [15] utilized the finite integral transform method to elucidate the problem based on the nonlinear reaction diffusion coupled with the chemical kinetics of a general shape solid. Chapwanya et al. [16] conveyed an epidemiological model with the Michaelis-Menten contact rate formulation to investigate variations in the enzyme kinetics with a simple susceptible infected recovered (SIR) model. Napper [17] proposed the Michaelis-Menten kinetics model to investigate the oxygen transport to heart tissue. Regalbuto et al. [18] presented an analytical methodology for obtaining solutions based on the maximum principle to nonlinear reaction-diffusion boundary value problems.

Rajendran and Saravanakumar [19] discussed mediated bioelectrocatalysis in order to build bioreactors, bio fuel cells, and biosensors.

Due to the difficulties in solving nonlinear differential equations in enzyme kinetics, some recent advanced analytical and numerical simulation techniques are used to solve the problems in chemical kinetics. Thus, in this review, all analytical and numerical works in enzyme kinetics are summarized.

### 2. Reaction diffusion systems

Reaction diffusion system is a mathematical model based on how the concentration of substances/products is disseminated over space changes under the influence of diffusion and a local chemical reaction. The substances are transformed into each other in local chemical reaction, whereas the substances are spread out over a surface in space in diffusion. Reaction-diffusion (RD) systems arise in many branches of physics, chemistry, biology, ecology, etc. Reviews of the theory and applications of reaction-diffusion systems can be found in books and numerous articles (see, for example [20–23]). These arise in a large variety of application areas, such as flow in porous media [24], heat conduction in plasma [25], combustion problems [26], liquid evaporation [27], and of more recent interest, image processing [28]. A great effort is being made in the development of the mathematical theory of nonlinear diffusion equations and to obtain exact solutions for special cases. Their significance not only relies on the huge number of their applications but also on the fact that they provide with a rather general class of linear and nonlinear differential operators. In mathematical analysis, it has shown to be a milestone for the development of applied, abstract, and numerical analysis as well as for algebra, geometry, and topology.

### 3. Nonlinear phenomena

develop certain methods for improving functionality of some molecules in a cell. Many problems in theoretical and experimental biology/chemistry involve the solution of the steady-state reaction diffusion equation with nonlinear chemical kinetics. Such problems also arise in the formulation of substrate and product material balances for enzymes immobilized within particles [1, 2], in the description of substrate transport into microbial cells [3–5], in membrane transport, in the transfer

To impose the functionality of some molecules in a cell, a mathematical model of such metabolic systems must be constructed and simulated. Most of the dynamical systems can be approximated by various types of differential and integral equations involving finite number of variables and parameters. Thus, the future behavior of the system can be predicted if model kinetics parameters and initial states of the variables are available. In particular, ordinary and partial differential equations (ODEs and PDEs) are popular in modeling of the metabolic

Releasing enzyme-substrate reactions under single-molecule kinetics was reported by Shlomi et al. [9]. An integral equation method with Michaelis-Menten kinetics to solve nonlinear diffusion problems in spherical coordinates was stated by Tosaka and Miyale [10]. Maalmi et al. [11] reported numerical and semianalytical solutions of nonlinear equations, which covered diffusivity, size, bulk concentration of reactant, binding constant of Michaelis-Menten kinetics, and site reactivity values. Merchant [12] stated the M-M decay reaction terms and the Gray-Scott scheme along with the semianalytical method to nonlinear reaction-diffusion systems. Indira and Rajendran [13] described a homotopy perturbation method to obtain substrate and product concentrations within the enzymatic layers. Removal of substrate from Michaelis-Menten kinetics governed the extravascular partition in which the analytical solution for the steady-state condition was investigated by Bucolo and Tripathi [14]. Dang Do and Greenfield [15] utilized the finite integral transform method to elucidate the problem based on the nonlinear reaction diffusion coupled with the chemical kinetics of a general shape solid. Chapwanya et al. [16] conveyed an epidemiological model with the Michaelis-Menten contact rate formulation to investigate variations in the enzyme kinetics with a simple susceptible infected recovered (SIR) model. Napper [17] proposed the Michaelis-Menten kinetics model to investigate the oxygen transport to heart tissue. Regalbuto et al. [18] presented an analytical methodology for obtaining solutions based on

the maximum principle to nonlinear reaction-diffusion boundary value problems.

Rajendran and Saravanakumar [19] discussed mediated bioelectrocatalysis in order to build

Due to the difficulties in solving nonlinear differential equations in enzyme kinetics, some recent advanced analytical and numerical simulation techniques are used to solve the problems in chemical kinetics. Thus, in this review, all analytical and numerical works in enzyme

Reaction diffusion system is a mathematical model based on how the concentration of substances/products is disseminated over space changes under the influence of diffusion and a local

of oxygen to respiring tissue [6, 7], and in the analysis of any artificial kidney system [8].

pathways or enzyme kinetics.

22 Advanced Chemical Kinetics

bioreactors, bio fuel cells, and biosensors.

2. Reaction diffusion systems

kinetics are summarized.

The modern theory of the nonlinear reaction diffusion process is an important field in today's science. The nonlinear system and coherent structures represent an interdisciplinary area with many nonlinear applications in various fields. Those applications can be divided into six disciplines: chemistry (autocatalytic chemical and enzyme reactions), physics (nonlinear optics and electric circuits, plasmas and states of solid, condensed atomic gases, hydrodynamics, galaxy dynamics and cosmology, fluid dynamics, and celestial mechanics), general relativity, biology (biofuel cell, bioreactor and biosensor, atmosphere and oceans, and animal dispersal), random media, and modern telecommunications. A great variety of phenomena in physics, chemistry, or biology can be described by nonlinear ODE/PDEs and particularly by reaction-diffusion equations. For these reasons, the theory of the analytical solutions of the reaction-diffusion equations is considered.

In reaction diffusion systems, nonlinear phenomena play a crucial role in applied mathematics and chemistry. Exact (closed-form) solution of nonlinear reaction diffusion equations plays an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. The main result obtained from reaction and diffusion systems is that nonlinear phenomena include diversity of stationary and spatiotemporary dissipative patterns, oscillations, different types of waves, excitability, biostability, etc. But it is difficult for us to obtain the exact solution for these problems. The investigation of exact solution of nonlinear equation is interesting and important. In general, this results in the need to solve linear and nonlinear reaction diffusion equations with complex boundary conditions. The enzyme kinetics in biochemical systems have usually been modeled by differential equations, which are based only on reaction without spatial dependence of the various concentrations. The dimensionless nonlinear reaction diffusion equations are described below:

$$\frac{\partial S}{\partial \tau} = \nabla^2 S - f(R, \tau, S, P) \tag{1}$$

$$\frac{\partial P}{\partial \tau} = \nabla^2 P + g(\mathbf{R}, \tau, S, P) \tag{2}$$

where S and P represent the dimensionless concentrations of substrate and product, τ represents the dimensionless time, and R is the dimensionless radial co-ordinate of the particle. The first term on the right-hand side of the above equation accounts for active species (substrate or product) diffusion, whereas the second term f(R, τ, S, P) and g(R, τ, S, P) represents the homogeneous reaction term (nonlinear term), generally polynomial in the concentrations and time.

### 4. Common geometries and nonlinear reaction

Most commonly used electrodes/microelectrodes consist of a conducting metal/glassy carbon or semiconducting surface embedded in an insulating wall. When the conducting surface is a rectangle or disc of a few millimeters, this is known as a "planar" electrode. Diffusion to this surface is effectively planar (the effects of the edges are negligible), hence the nonlinear onedimensional reaction diffusion equation is given by:

$$\frac{\partial[\mathbb{C}]}{\partial t} = D \frac{\partial^2[\mathbb{C}]}{\partial x^2} + f([\mathbb{C}]) \tag{3}$$

Two other electrode geometries where diffusion occurs in only one spatial dimension are the hemispherical and hemicylindrical electrodes. The nonlinear two-dimensional (hemispherical or spherical) reaction diffusion equation is:

$$\frac{\partial[\mathbb{C}]}{\partial t} = D \left( \frac{\partial^2[\mathbb{C}]}{\partial x^2} + \frac{2}{r} \frac{\partial[\mathbb{C}]}{\partial r} \right) + f([\mathbb{C}]) \tag{4}$$

and for the latter is:

$$\frac{\partial[\mathbb{C}]}{\partial t} = D \left( \frac{\partial^2[\mathbb{C}]}{\partial x^2} + \frac{1}{r} \frac{\partial[\mathbb{C}]}{\partial r} \right) + f([\mathbb{C}]) \tag{5}$$

Author

G.

et al.

R. G.

L. Rajendran A. Meena et al.

A. Eswari, L. Rajendran

P. Manimozhi

S. Logambal,

 L.

Electrochemical

 Acta

55(2010)5230–5238

Amperometric

A

E1 þ S

!

E2 þ P

K

E

þ E !2

B þ E1

KA

Rajendran

A. Meena, L. Rajendran S. Anitha, L. Rajendran

P. Manimozhi,

Rajendran

A. Eswari, L. Rajendran A. Eswari, L. Rajendran A. Eswari, L. Rajendran

 Russian Journal of

47(2011)205–212

Electroanalytical

 Chemistry

Cyclic

voltammetry

EA

C2 B þ B k1

!Products

þ e \$ B

 Russian Journal of

47(2011)195–204

Electroanalytical

 Chemistry

Cyclic

voltammetry

EA

C B k

!

1 Products

þ e \$ B

 Journal of

36–46

Electroanalytical

 Chemistry 648(2010)

Amperometric

S þ E1

K

\$1

E1S ½ �! P þ E2

Homotopy

method (HPM)

Laplace Homotopy

method (HPM)

perturbation

25

Transformation

perturbation

http://dx.doi.org/10.5772/intechopen.70914

K�1

 L.

Journal of

87–92

Electroanalytical

 Chemistry 647(2010)

Amperometric

S þ E K

\$a

ES

Kα

ES Kcat

!S

þ E

0

 Journal of Physical Chemistry

7037

 Journal of

(2010)50–59

Electroanalytical

 Chemistry,

 6411 114(2010)7030–

Amperometric

B

!E

B þ S K

!

A

þ

Z

!

A

! B

D

Amperometric

Potentiometric

 and

E +

S\$[ES]!E + P

 et al.

 Sensors and Actuators B

147(2010)290–297

Amperometric

E þ S k

!

ES k

!

E þ P

c

k�1

ES + S\$ES3

1

 Journal of

35–44

Electroanalytical

 Chemistry 641(2010)

Amperometric

S þ E \$1

K1

E1S ½ �!Kcat

P þ E2

K2

 Journal

186

Mathematical

 Chemistry,

48(2010)179–

Amperometric

E þ S K

!1

ES

! E þ P

K�1

Rahamathunissa

Journal

801

Mathematical

 Chemistry

44(2008)849–

Amperometric

E þ S

KM

\$ES K

!2

E þ P

Variation iteration method

(VIM)

He's variation iteration

method

Homotopy

method (HPM)

Variational

homotopy method (VIM & HPM)

Homotopy

method (HPM)

Homotopy

method (HPM) Reduction of order

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

method

Analytical

perturbation

perturbation

perturbation

 iteration and

perturbation

Senthamarai

 et al.

Rahamathunissa

Journal of theoretical and

Chemistry,

Electrochemical

 Acta

53(2008)3566–3578

Chronoamperometric

A + e!B

B þ

Z k

!

A

þ product

7(1)(2008)113–138

Computational

Amperometric

S þ

CKM

! SC½ �! PC0 h i k

!

P þ CC

c

0

Danckwort's

Analytical

 expression

> k 0 E

!

C

Reference

Experimental

Enzymatic

 scheme

Modeling method

technique

Analytical solutions

The hemisphere can be achieved experimentally via a small drop of mercury positioned over a smaller conducting disc. A soft polymer, rubber, or other similar materials are usually employed to fabricate a hemicylinder. The electrodes are usually employed in theoretical studies due to the low dimensionality of the mass-transport equation. Additional terms such as diffusion and nonlinear reaction allow the equation to be solved analytically. Furthermore, the electrodes are not accurately or easily fabricated for practical geometries.

The corresponding nonlinear reaction-diffusion issues in enzyme kinetics are focused on the mathematical resolution. Table 1 shows the response of particular electrodes with special emphasis on earlier theoretical works in the field.

#### Example 1: Michaelis-Menten kinetics and microcylinder electrodes

The model is written for an enzyme reaction to generate an electro-active product (e.g., hydrogen peroxide from an oxidase enzyme) that reacts at an immobilization matrix, which

where S and P represent the dimensionless concentrations of substrate and product, τ represents the dimensionless time, and R is the dimensionless radial co-ordinate of the particle. The first term on the right-hand side of the above equation accounts for active species (substrate or product) diffusion, whereas the second term f(R, τ, S, P) and g(R, τ, S, P) represents the homogeneous reaction term (nonlinear term), generally polynomial in the concentrations and time.

Most commonly used electrodes/microelectrodes consist of a conducting metal/glassy carbon or semiconducting surface embedded in an insulating wall. When the conducting surface is a rectangle or disc of a few millimeters, this is known as a "planar" electrode. Diffusion to this surface is effectively planar (the effects of the edges are negligible), hence the nonlinear one-

∂½ � C

∂t ¼ D ∂2½ � C

∂t ¼ D ∂2½ � C

the electrodes are not accurately or easily fabricated for practical geometries.

Example 1: Michaelis-Menten kinetics and microcylinder electrodes

∂½ � C

∂½ � C

∂t ¼ D ∂2½ � C

Two other electrode geometries where diffusion occurs in only one spatial dimension are the hemispherical and hemicylindrical electrodes. The nonlinear two-dimensional (hemispherical

> 2 r ∂½ � C ∂r

> 1 r ∂½ � C ∂r

The hemisphere can be achieved experimentally via a small drop of mercury positioned over a smaller conducting disc. A soft polymer, rubber, or other similar materials are usually employed to fabricate a hemicylinder. The electrodes are usually employed in theoretical studies due to the low dimensionality of the mass-transport equation. Additional terms such as diffusion and nonlinear reaction allow the equation to be solved analytically. Furthermore,

The corresponding nonlinear reaction-diffusion issues in enzyme kinetics are focused on the mathematical resolution. Table 1 shows the response of particular electrodes with special

The model is written for an enzyme reaction to generate an electro-active product (e.g., hydrogen peroxide from an oxidase enzyme) that reacts at an immobilization matrix, which

∂x2 þ

∂x2 þ

∂x2 þ f Cð Þ ½ � (3)

þ f Cð Þ ½ � (4)

þ f Cð Þ ½ � (5)

4. Common geometries and nonlinear reaction

dimensional reaction diffusion equation is given by:

or spherical) reaction diffusion equation is:

emphasis on earlier theoretical works in the field.

and for the latter is:

24 Advanced Chemical Kinetics

Author

S. R. Baronas et al.

R. Baronas V. Ašerisa et al.

V. Flexer et al. R. Baronas et al.

R. Baronas R. Baronas et al. R. Baronas et al.

R. Baronas et al.

L. Rajendran

 J.

Series on chemical sensors and biosensors (2009)

Biosensor: Modeling and Simulation

Diffusion-Limited

Simulation

 and Modeling,

GhenadiiKorotcenkov

Sensors, Vol. 5, Momentum

York (2013)

> Table 1.

Contributions

 to the theoretical

 modeling of enzymatic electrodes.

 (Ed.),

 Press, LLC, New

Electrochemical

 Process, Chemical Sensors:

 of

Amperometric

 All enzyme reactions

Mathematical

 Modeling of Biosensors,

 Springer

Amperometric

 All enzyme reactions

Mathematical

 Chemistry 32

(2)(2002)225–237

Amperometric

S E!P

 Sensors

12(2012)9146–9160

Amperometric

EOX þ S k!1 Ered þ P

Ered k2 !EOX þ ne e�

Systems Nonlinear Analysis: Modeling and Control 9(3)

(2004)203–218

Chemometrics

 and Intelligent Laboratory

126(2013)108–116

Bioelectrochemistry

74(2008)201–209

Cyclic Amperometric

Amperometric

S E!P

E þ Sik1i \$ ES !i k2i E þ Pi, i ¼ 1,…, k

voltammetry

S þ EOX \$k1k�1 ESkcat !P þ Ered

 Journal of

63–71

Electroanalytical

 Chemistry 685(2012)

Amperometricparallel

S !1 E1 12 P1

S1 þ S !2 E2 P2

substrates conversion

 Biosensors

 and

Electrochimica

 Acta

240(2017)399–407

Amperometric

S þ

S E!P

E\$k1k�1ES ! P þ E

biosensor

Bioelectronics

19(2004)915–922

Amperometric

S ! P E!S

Finite-difference

Numerical

analytical solution

Digital difference technique

Numerical

Numerical

Digital difference technique.

Finite-difference

Numerical

Analytical and numerical

methods

Analytical,

VIM,ADM,

 etc.

HPM&HAM,

http://dx.doi.org/10.5772/intechopen.70914

27

 simulation

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

simulation-finite-

 simulation

simulation-finite-

 simulation and

 technique

Sevukaperumal

 et al. Applied

Mathematics

3(2012)373–381

Chronoamperometric

Glucose þ O2

H2O2

Numerical

 solution

!CatalaseH2O2 þ 12 O2

!

Glucoseoxidasegluconicacid

 þ H2O2

Homotopy

method (HPM)

 analysis

Reference

Experimental

Enzymatic

 scheme

Modeling method

technique

Author A. Eswari, L. Rajendran

G.

et al. S. Logambal, L.

Rajendran

S. Anitha et al. K. Indra, L. Rajendran

S. Thiagarajan M. Uma Maheswari,

Rajendran

P. Rijiravanich

A. Eswari, L. Rajendran

G.

Varatharajan,

Rajendran

K. Venugopal et al.

K. Indra, L. Rajendran V. Margret Ponrani, L.

Journal of

Mathematical

10.1007/s10910-011-9973-6

 Chemistry DOI:

Amperometric

G

þ E

k�

\$1

Xk�

!2

F þ E

K1

k2

Rajendran

 Journal of

Mathematical

10.1007/s10910-011-9968-3

 Chemistry DOI:

 Journal of Biomedical

(2011)631–641

 Science and Engineering

 4

Chronoamperometric

O2 þ

o � quinone

Chronoamperometric

A

B � e� !

products

\$ B þ C

þ 2Hþ þ 2e� !

2catechol

! 2o � quinone

þ 2H2O

Homotopy method (HPM)

Homotopy method (HPM)

Homotopy method (HPM)

perturbation

perturbation

perturbation

catechol

 L.

Applied

Mathematics

2(2011)1140–1147

Amperometric

S þ E K

\$1

Ckcat

Homotopy method (HPM)

perturbation

!P þ E

k�1

E k3

!Ei

 Journal of

200–208

Electroanalytical

 Chemistry 660(2011)

Amperometric

O2 þ

o � quinone

þ 2Hþ þ 2e� !

2catechol

! 2o � quinone

þ 2H2O

VIM

catechol

 et al.

Electroanalytical

 Chemistry

589(2006)249

Amperometric

O2 þ

o � quinone

þ 2Hþ þ 2e� !

2catechol

! 2o � quinone

þ 2H2O

Theory and experiment

catechol

 L.

Journal of

Mathematical

10.1007/s10919-011-9853-0

 Chemistry DOI:

 et al.

 Journal of

Mathematical

10.1007/s10919-011-9854-z

 Chemistry DOI:

Chronoamperometric

Chronoamperometric

 E þ S K

\$1

k�1ES k

!

 P2 þ

H2O V2

2

\$kr

S2 E0

O2PPO

! P2 þ

H2O V1

Electrochimica

 Acta

56(2011)3345–3352

Amperometric

S þ E \$1

KM E1S ½ �!kcat

P þ E2A

! B

Homotopy method (HPM)

Homotopy method (HPM)

perturbation

perturbation

Rahamathunissa

Journal of

474

Journal of Membrane Sciences

373(2011)20–28

Amperometric

EOX

þ S

k�

\$1

E S k

!

E þ P

2

þ Z k

!

O

þ

Products

þ ne� \$

R

Homotopy method (HPM)

VIM

perturbation

26 Advanced Chemical Kinetics

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics http://dx.doi.org/10.5772/intechopen.70914 27

Table 1. Contributions to the theoretical modeling of enzymatic electrodes.

is metallically conducting sites/particles. The reaction within the film under the Michaelis-Menten kinetics may be written as follows:

$$S + E\_1 \overset{k\_1}{\underset{k\_{-1}}{\Leftrightarrow}} [E\_1 S] \xrightarrow{k\_{\text{cut}}} P + E\_2 \tag{6}$$

produced an enzyme-reactant complex. Eq. (11) illustrates the Michaelis-Menten kinetics, in which the enzyme-substrate complex is formed after the enzyme is combined with the sub-

As can be seen from Eq. (11), the product P is released by the binding of substrate S with enzyme E. The product released is not reversible; however, the substrate binding is reversible. The

The law of mass action leads to the system of following nonlinear reaction equations [31],

dp

where k1 is the forward rate of ES complex formation and k�1 is the backward rate constant.

Figure 1 illustrates Michaelis-Menten reaction kinetics scheme for co-substrate and substrate. Limoges et al. [33] reported for a redox enzymatic homogenous system along with one-

When the enzyme is being solubilized, the electrochemical signal that is produced during the

∂x2 � C0

1 k1½ � S þ 1

∂x2 � C0

1 k1½ � S þ 1

where DP , DS are the diffusion coefficients of co-substrate and substrate, respectively; Q , S are the concentrations of co-substrate and substrate, respectively; x is the distance from the

E

k2, 2 þ 1 k2½ � Q

k2,2 þ 1 k2½ � Q (14)

(15)

k1, 2 þ 1

E

k1, 2 þ 1

reaction is governed by the following set of nonlinear partial differential equations.

∂2½ � Q

∂2½ � S

E þ P (11)

http://dx.doi.org/10.5772/intechopen.70914

29

s ¼ ½ � S , e ¼ ½ � E , c ¼ ½ � SE , p ¼ ½ � P (12)

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

dt ¼ �k1es þ k�1c (13a)

dt ¼ �k1es þ ð Þ k�1 þ k2 c (13b)

dt ¼ k1es � ð Þ k�1 þ k2 c (13c)

dt ¼ k2c (13d)

E þ S\$ k1 k�1 ES! k2

reactants' concentrations in Eq. (11) are represented by the following letters:

ds

de

dc

Example 3: Michaelis-Menten mechanism for co-substrate and substrate

dimensional mass transport equation a concise discussion and derivation.

∂½ � Q ∂t ¼ DP

∂½ � S ∂t ¼ DS

The above problem is discussed theoretically by Meena et al. [32].

strate.

The consumption rate of S is given by k1cScE � k�1cES, where ci denotes the concentration of species i. The rate is equivalent to (kcat/KM) cScE, where KM is the Michaelis constant, defined as KM = (k�1 + kcat)/k1. The consumption rate of S in the film is compensated by diffusion. If the solution is stirred uniformly, so that S is constantly supplied to the film, the mass balance for S can be written in cylindrical coordinates:

$$\frac{D\_S}{r}\frac{d}{dr}\left(r\frac{dc\_S}{dr}\right) - \frac{k\_{\rm cut}c\_EC\_S}{c\_S + K\_M} = 0\tag{7}$$

where cS is the concentration profile of substrate, cE is the concentration profile of enzyme, DS is its diffusion coefficient, and KM is the Michaelis constant. The rate of consumption will be v(r) = k cH, where k is the rate constant for the hydrogen peroxide reaction and cH is the peroxide concentration. Then, the equation of continuum for hydrogen peroxide is generally expressed in the steady-state by

$$\frac{D\_H}{r}\frac{d}{dr}\left(r\frac{dc\_H}{dr}\right) + \frac{k\_{\rm cat}c\_EC\_S}{c\_S + K\_M} - \upsilon(r) = 0\tag{8}$$

At the electrode surface (r0) and at the film surface (r1), the boundary conditions are [29]:

$$\begin{aligned} r &= r\_0: & \quad \frac{dc\_S}{dr} = 0, \quad c\_H = 0\\ r &= r\_1: & \quad c\_S = c\_{S'}^\* & \quad c\_H = 0 \end{aligned} \tag{9}$$

where c∗ S is the bulk concentration of S scaled by the partition coefficient of the film. The current is provided by the consumption rate at each site. Thus, the total current at an electrode of length L is expressed by [29]

$$I/nF = 2\pi L \int\_{r\_0}^{r\_1} rv \, dr \tag{10}$$

The analytical results of the problem are discussed by Eswari and Rajendran [30].

#### Example 2: enzyme catalysis reaction

The reactions without spatial dependence on various concentrations have modeled the enzyme kinetics in biochemical systems. Nonlinear systems of ordinary differential equations are solely based on that. Michaelis and Menten were pioneers in explaining the enzyme reaction model. In addition, they also reported the free enzyme binding to the reactant, which produced an enzyme-reactant complex. Eq. (11) illustrates the Michaelis-Menten kinetics, in which the enzyme-substrate complex is formed after the enzyme is combined with the substrate.

is metallically conducting sites/particles. The reaction within the film under the Michaelis-

½ � E1S �����!kcat

The consumption rate of S is given by k1cScE � k�1cES, where ci denotes the concentration of species i. The rate is equivalent to (kcat/KM) cScE, where KM is the Michaelis constant, defined as KM = (k�1 + kcat)/k1. The consumption rate of S in the film is compensated by diffusion. If the solution is stirred uniformly, so that S is constantly supplied to the film, the mass balance for S

where cS is the concentration profile of substrate, cE is the concentration profile of enzyme, DS is its diffusion coefficient, and KM is the Michaelis constant. The rate of consumption will be v(r) = k cH, where k is the rate constant for the hydrogen peroxide reaction and cH is the peroxide concentration. Then, the equation of continuum for hydrogen peroxide is generally

� kcatcEcS cS þ KM

dcS dr � �

dcH dr � �

r ¼ r0 : dcS

r ¼ r1 : cS ¼ c∗

þ

At the electrode surface (r0) and at the film surface (r1), the boundary conditions are [29]:

kcatcEcS cS þ KM

dr ¼ 0, cH ¼ 0

S is the bulk concentration of S scaled by the partition coefficient of the film. The

ðr1

r0

current is provided by the consumption rate at each site. Thus, the total current at an electrode

The reactions without spatial dependence on various concentrations have modeled the enzyme kinetics in biochemical systems. Nonlinear systems of ordinary differential equations are solely based on that. Michaelis and Menten were pioneers in explaining the enzyme reaction model. In addition, they also reported the free enzyme binding to the reactant, which

I=nF ¼ 2πL

The analytical results of the problem are discussed by Eswari and Rajendran [30].

S, cH ¼ 0

P þ E2 (6)

¼ 0 (7)

� v rð Þ¼ 0 (8)

rv dr (10)

(9)

S þ E1 ⇔ k1 k�1

DS r d dr r

DH r d dr r

Menten kinetics may be written as follows:

28 Advanced Chemical Kinetics

can be written in cylindrical coordinates:

expressed in the steady-state by

of length L is expressed by [29]

Example 2: enzyme catalysis reaction

where c∗

$$E + S \underset{k\_{-1}}{\overset{k\_1}{\underset{k\_{-1}}{\rightleftharpoons}}} ES \overset{k2}{\rightarrow} E + P \tag{11}$$

As can be seen from Eq. (11), the product P is released by the binding of substrate S with enzyme E. The product released is not reversible; however, the substrate binding is reversible. The reactants' concentrations in Eq. (11) are represented by the following letters:

$$\mathbf{s} = [\mathbf{S}] \quad \mathbf{e} = [\mathbf{E}] \quad \mathbf{c} = [\mathbf{S} \mathbf{E}] \quad \mathbf{p} = [\mathbf{P}] \tag{12}$$

The law of mass action leads to the system of following nonlinear reaction equations [31],

$$k\frac{d\mathbf{s}}{dt} = -k\_1\mathbf{e}\mathbf{s} + k\_{-1}\mathbf{c}\tag{13a}$$

$$\frac{de}{dt} = -k\_1 \text{es} + (k\_{-1} + k\_2)c \tag{13b}$$

$$\frac{dc}{dt} = k\_1 \text{es} - (k\_{-1} + k\_2)c \tag{13c}$$

$$\frac{dp}{dt} = k\_2 c \tag{13d}$$

where k1 is the forward rate of ES complex formation and k�1 is the backward rate constant. The above problem is discussed theoretically by Meena et al. [32].

#### Example 3: Michaelis-Menten mechanism for co-substrate and substrate

Figure 1 illustrates Michaelis-Menten reaction kinetics scheme for co-substrate and substrate. Limoges et al. [33] reported for a redox enzymatic homogenous system along with onedimensional mass transport equation a concise discussion and derivation.

When the enzyme is being solubilized, the electrochemical signal that is produced during the reaction is governed by the following set of nonlinear partial differential equations.

$$\frac{\partial[Q]}{\partial t} = D\_P \frac{\partial^2[Q]}{\partial x^2} - \frac{C\_E^0}{\frac{1}{k\_1[S]} + \frac{1}{k\_{1,2}} + \frac{1}{k\_{2,2}} + \frac{1}{k\_2[Q]}}\tag{14}$$

$$\frac{\partial[\mathcal{S}]}{\partial t} = D\_S \frac{\partial^2[\mathcal{S}]}{\partial \mathbf{x}^2} - \frac{C\_E^0}{\frac{1}{k\_1[\mathcal{S}]} + \frac{1}{k\_{1,2}} + \frac{1}{k\_{2,2}} + \frac{1}{k\_2[\mathcal{Q}]}} \tag{15}$$

where DP , DS are the diffusion coefficients of co-substrate and substrate, respectively; Q , S are the concentrations of co-substrate and substrate, respectively; x is the distance from the

Figure 1. Reaction scheme for substrate and co-substrate.

electrode surface; C0 S is the bulk concentration of substrate; C0 E is the total concentration of enzyme; k1, k2,2, and k2 are the reaction rate constants; and t is the time. The initial and boundary conditions for Eqs. (14) and (15) are given by:

$$\text{Let } t = 0, \mathbf{x} \ge 0 \text{, and}\\
\mathbf{x} = \* , \mathbf{x} \ge \mathbf{0}, \left[\mathbf{Q}\right] = \mathbf{0}, \left[\mathbf{S}\right] = \mathbf{C}\_{\mathcal{S}}^{0} \tag{16}$$

$$\forall \mathbf{x} = \mathbf{0}, t \ge \mathbf{0}: [\mathbf{Q}] = \frac{\mathbf{C}\_p^0}{1 + \exp\left[\frac{F}{RT}\left(E - E\_{PQ}^0\right)\right]}, \frac{\partial [\mathbf{S}]}{\partial \mathbf{x}} = \mathbf{0} \tag{17}$$

$$\mathbf{x} = \lnot \partial[\mathbf{Q}]/\partial \mathbf{x} = \mathbf{0} \tag{18}$$

addition, nonlinear differential equations can also assist to investigate the stability of these solutions as well as checking the simulation analysis. Nonlinear partial differential equations govern a significant variety of phenomena including physical, chemical, and biological. The development of techniques aimed at exact solutions of nonlinear differential equations with nonsteady and steady state [35] has been one of the most exciting advances of nonlinear science and theoretical physics/chemistry. An important role in nonlinear science is played by exact solutions of differential equations. Furthermore, this can be especially observed in nonlinear physical chemistry science. This can be attributed to the provision of physical information as well as more insight into the physical aspects of the problem, which could lead to further applications. Over the past few decades, different methods have been reported to solve analytical solutions such as Tanh-sech [36], extended tanh [37], Jacobi elliptic function expansion [39], hyperbolic function [38], F-expansion [40], and the First integral [41]. To solve different types of nonlinear systems of PDEs, the sine-cosine method [42] has been employed. A variety of powerful analytical methods such as homotopy perturbation method [43–45], homotopy analysis method [46, 47], Adomian decomposition method [48, 49], wavelet transform method [50], etc. are applied to solve the nonlinear problems (e.g., Eqs. (8) and (13)–(15))

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

http://dx.doi.org/10.5772/intechopen.70914

31

Many differential equations cannot be solved analytically. For practical purpose, however, such as in physical engineering sciences, a numerical approximation to the solution is often sufficient. The numerical method is mainly to solve complex problem physically or geometrically. It is also used to validate the experimental results. Some of the nonlinear equations in

Most mathematical models of enzyme kinetics are based on reaction diffusion equations or rate equations containing nonlinear terms related to the kinetics of the enzyme reaction. Powerful and accurate analytical (HPM, HAM, ADM, etc.) and numerical mathematical methods have been employed for their resolution under steady and nonsteady state conditions. The theoretical results provide very useful insight into the effects on the performance of the thickness and structure of the enzymatic film, the loading of the different species, the diffusivity of the mediator, etc. Also, the theoretical modeling and simulation of these systems enable us to characterize the enzymatic reactions (i.e., rate constant, turnover rate, and

In spite of the above-mentioned benefits, there are only limited theoretical studies addressing kinetics of enzyme reaction and most of them include a number of simplifying assumptions mainly related to the mass and charge transport inside and outside the biocatalyst film, the enzymatic kinetic scheme, and the electrode morphology. Experimental validation of proposed

chemical kinetics were solved using numerical methods [52–56].

in chemical kinetics [51].

7. Summary

6. Numerical solutions

Michaelis-Menten constants).

The analytical expressions corresponding to the concentration of co-substrate for steady and nonsteady state conditions have been obtained by solving the above nonlinear equation using a new approach to homotopy perturbation method (HPM). Analytical expressions of the plateau current are also presented for steady and nonsteady state conditions:

$$\text{si } i = FSD\_P \left( \frac{\partial [Q]}{\partial \mathbf{x}} \right)\_{\mathbf{x} = \mathbf{0}} \tag{19}$$

where E is the electrode potential, E0 PQ is the standard potential of the P/Q couple, F is the Faraday constant, and S is the surface area of the electrode. The above problem is discussed theoretically by Rasi et al. [34].

#### 5. Analytical solutions

To study many of the physical phenomena, the exact solutions of nonlinear partial or ordinary differential equations play an important role. In order to understand the mechanism of complicated dynamical processes and physical phenomena modeled by nonlinear differential equations, the existence of approximate analytical and exact solutions is very important. In addition, nonlinear differential equations can also assist to investigate the stability of these solutions as well as checking the simulation analysis. Nonlinear partial differential equations govern a significant variety of phenomena including physical, chemical, and biological. The development of techniques aimed at exact solutions of nonlinear differential equations with nonsteady and steady state [35] has been one of the most exciting advances of nonlinear science and theoretical physics/chemistry. An important role in nonlinear science is played by exact solutions of differential equations. Furthermore, this can be especially observed in nonlinear physical chemistry science. This can be attributed to the provision of physical information as well as more insight into the physical aspects of the problem, which could lead to further applications. Over the past few decades, different methods have been reported to solve analytical solutions such as Tanh-sech [36], extended tanh [37], Jacobi elliptic function expansion [39], hyperbolic function [38], F-expansion [40], and the First integral [41]. To solve different types of nonlinear systems of PDEs, the sine-cosine method [42] has been employed. A variety of powerful analytical methods such as homotopy perturbation method [43–45], homotopy analysis method [46, 47], Adomian decomposition method [48, 49], wavelet transform method [50], etc. are applied to solve the nonlinear problems (e.g., Eqs. (8) and (13)–(15)) in chemical kinetics [51].

### 6. Numerical solutions

Many differential equations cannot be solved analytically. For practical purpose, however, such as in physical engineering sciences, a numerical approximation to the solution is often sufficient. The numerical method is mainly to solve complex problem physically or geometrically. It is also used to validate the experimental results. Some of the nonlinear equations in chemical kinetics were solved using numerical methods [52–56].

### 7. Summary

electrode surface; C0

30 Advanced Chemical Kinetics

S is the bulk concentration of substrate; C0

x ¼ 0, t ≥ 0 : ½ �¼ Q C0

plateau current are also presented for steady and nonsteady state conditions:

i ¼ FSDP

boundary conditions for Eqs. (14) and (15) are given by:

Figure 1. Reaction scheme for substrate and co-substrate.

where E is the electrode potential, E0

theoretically by Rasi et al. [34].

5. Analytical solutions

enzyme; k1, k2,2, and k2 are the reaction rate constants; and t is the time. The initial and

t ¼ 0, x ≥ 0, andx ¼ ∞, x ≥ 0, Q½ �¼ 0, S½ �¼ C0

The analytical expressions corresponding to the concentration of co-substrate for steady and nonsteady state conditions have been obtained by solving the above nonlinear equation using a new approach to homotopy perturbation method (HPM). Analytical expressions of the

Faraday constant, and S is the surface area of the electrode. The above problem is discussed

To study many of the physical phenomena, the exact solutions of nonlinear partial or ordinary differential equations play an important role. In order to understand the mechanism of complicated dynamical processes and physical phenomena modeled by nonlinear differential equations, the existence of approximate analytical and exact solutions is very important. In

∂½ � Q ∂x � �

x¼0

1 þ exp F

P

RT E � E0

PQ h i � � , ∂½ � S

x ¼ ∞, ∂½ � Q =∂x ¼ 0 (18)

PQ is the standard potential of the P/Q couple, F is the

E is the total concentration of

S (16)

∂x ¼ 0 (17)

(19)

Most mathematical models of enzyme kinetics are based on reaction diffusion equations or rate equations containing nonlinear terms related to the kinetics of the enzyme reaction. Powerful and accurate analytical (HPM, HAM, ADM, etc.) and numerical mathematical methods have been employed for their resolution under steady and nonsteady state conditions. The theoretical results provide very useful insight into the effects on the performance of the thickness and structure of the enzymatic film, the loading of the different species, the diffusivity of the mediator, etc. Also, the theoretical modeling and simulation of these systems enable us to characterize the enzymatic reactions (i.e., rate constant, turnover rate, and Michaelis-Menten constants).

In spite of the above-mentioned benefits, there are only limited theoretical studies addressing kinetics of enzyme reaction and most of them include a number of simplifying assumptions mainly related to the mass and charge transport inside and outside the biocatalyst film, the enzymatic kinetic scheme, and the electrode morphology. Experimental validation of proposed models is even more seldom. Therefore, more effort in the future research is needed in this direction in order to develop more detailed models and accurate simulations that can assist the rational development and optimization of enzyme electrodes.

[10] Tosaka N, Miyale S. Analysis of a nonlinear diffusion problem with Michaelis-Menten kinetics by an integral equation method. Bulletin of Mathematical Biology. 1982;44(6):841-849

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

http://dx.doi.org/10.5772/intechopen.70914

33

[11] Maalmi M, Strieder W, Varma A. Ligand diffusion and receptor mediated internalization: Michaelis-Menten kinetics. Journal of Chemical Engineering Science. 2001;56(19):5606-5616

[12] Merchant TR. Cubic autocatalysis with Michaelis-Menten kinetics: Semi-analytical solutions for the reaction-diffusion cell. Journal of Chemical Engineering Science. 2004;59(16):

[13] Indira K, Rajendran L. Analytical expression of the concentration of substrates and product in phenol-polyphenol oxidase system immobilized in laponite hydrogels; Michaelis-Menten formalism in homogeneous medium. Electrochimica Acta. 2011;56(18):6411-6419

[14] Bucolo J, Tripathi K. Steady-state analysis of a two-compartment barrier-limited capillary-tissue model with Michaelis-Menten saturation kinetics. Bulletin of Mathemat-

[15] Dang Do D, Greenfield F. A finite integral transform technique for solving the diffusionreaction equation with Michaelis-Menten kinetics. Journal of Mathematical Biosciences.

[16] Chapwanya M, Lubuma S, Mickens E. From enzyme kinetics to epidemiological models with Michaelis-Menten contact rate design of nonstandard finite difference schemes.

[17] Napper A, Schubert RW. Michaelis–Menten kinetics as a modelling assumption in a model of oxygen transport in heart Proceedings of the First Southern Biomedical Engi-

[18] Regalbuto C, Strieder W, Varma A. Approximate solutions for nonlinear diffusionreaction equations from the maximum principle. Journal of Chemical Engineering Sci-

[19] Rajendran L, Saravanakumar K. Analytical expression of transient and steady-state catalytic current of mediated bioelectrocatalysis. Journal of Electrochimica Acta. 2014;147:678-687 [20] Nicolis G, Prigogine I. Self-Organization in Non-equilibrium Systems. New York: Wiley; 1977

[22] Kerner BS, Osipov VV. A New Approach to Problems of Self-Organization and Turbu-

[23] Lubashevskii A, Gafiychuk VV. The projection dynamics of highly dissipative system.

[24] Vázquez JL. The Porous Medium Equation. Mathematical Theory. Oxford Mathematical

[25] Bertsch M. Asymptotic behavior of solutions of a nonlinear diffusion equation. SIAM

[21] Mikhailov AS. Foundations of Synergetics. Berlin: Springer-Verlag; 1990

lence. Autosolitons. Dordrecht: Kluwer; 1994

Monographs. Oxford: Oxford University Press; 2006

Journal on Applied Mathematics. 1982;42(1):66

Physical Review E. 1994;50(1):171

Journal of Computers and Mathematics with Applications. 2012;64(3):201-213

3433-3440

ical Biology. 1980;42(5):691-700

neering Conference. 1981;201-204

ence. 1988;43(3):513-518

1981;54(1–2):31-47

### Author details

Lakshmanan Rajendran1 \*, Mohan Chitra Devi1 , Carlos Fernandez2 and Qiuming Peng3

\*Address all correspondence to<EMAIL_ADDRESS>

3 State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao, China

### References

[10] Tosaka N, Miyale S. Analysis of a nonlinear diffusion problem with Michaelis-Menten kinetics by an integral equation method. Bulletin of Mathematical Biology. 1982;44(6):841-849

models is even more seldom. Therefore, more effort in the future research is needed in this direction in order to develop more detailed models and accurate simulations that can assist the

, Carlos Fernandez2 and Qiuming Peng3

rational development and optimization of enzyme electrodes.

\*Address all correspondence to<EMAIL_ADDRESS>
\*, Mohan Chitra Devi1

2 School of Pharmacy and Life Sciences, Robert Gordon University, UK

1 Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti, India

3 State Key Laboratory of Metastable Materials Science and Technology, Yanshan University,

[1] Aris R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vol.

[2] Engasser JM, Horvath C. Effect of internal diffusion in heterogeneous enzyme systems: Evolution of true kinetic parameters and substrate diffusity. Journal of Theoretical Biol-

[3] Cheviollotte P. Relation between the reaction cytochrom oxidase-oxygen and oxygen uptake in cells in vivo: The role of diffusion. Journal of Theoretical Biology. 1973;39:277-295

[4] McElwain DLS. A re-examination of oxygen diffusion in a spherical cell with Michaelis-

[5] Lin SH. Oxygen diffusion in a spherical shell with nonlinear oxygen uptake kinetics.

[6] Ho SP, Kostin MD. Diffusion with irreversible chemical reaction in heterogeneous media: Application to oxygen transport in respiring tissue. Journal of Theoretical Biology.

[7] Pope AS. Diffusion in tissue slices with metabolism obeying Michaelis-Menten kinetics.

[8] Lim SH. A modified model for predicting the performance of a compact artificial kidney.

[9] Shlomi R, Michael U, Joseph K. Role of substrate unbinding in Michaelis-Menten enzymatic reactions. Proceedings of the National Academy Sciences. 2012;111:4391-4396

Menten oxygen uptake kinetics. Theoretical Biology. 1978;71:205-263

Journal of Theoretical Biology. 1976;60:449-457

Journal of Theoretical Biology. 1979;80:325-332

Journal of Theoretical Biology. 1972;77:441-451

Author details

32 Advanced Chemical Kinetics

Lakshmanan Rajendran1

Qinhuangdao, China

1. Clarendon: Oxford; 1975

ogy. 1973;42:137-155

1997;64:237-251

References

[26] Aronson DG, Weinberger HF. Nonlinear diffusion in population genetics, combustion and nerve pulse propagation. In: Goldstain JA, editor. Partial Differential Equations and Related Topics, Lecture Notes in Mathematics. Berlin/New York: Springer-Verlag; 1975; pp. 446

[42] Mitchell AR, Griffiths DF. The Finite Difference Method in Partial Differential Equations.

Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics

http://dx.doi.org/10.5772/intechopen.70914

35

[43] Shanthi D, Ananthaswamy V, Rajendran L. Analysis of nonlinear reaction-diffusion processes with Michaelis-Menten kinetics by a new homotopy perturbation method. Natural

[44] Saranya J, Rajendran L, Wang L, Fernandez C. A new mathematical modelling using homotopy perturbation method to solve nonlinear equations in enzymatic glucose fuel

[45] Meena A, Rajendran L. Mathematical modeling of amperometric and potentiometric biosensors and system of nonlinear equation homotopy perturbation approach. Journal

[46] Angel Joy R, Meena A, Loghambal S, Rajendran L. A two-parameter mathematical model for immobilized enzymes and homotopy analysis method. Natural Science. 2011;37:

[47] Kirthiga M, Chitra Devi M, Meena A, Rajendran L. Mathematical modelling and kinetics of micro channel reactor. Applied and Computational Mathematics. 2016;56:234-246

[48] Sivasankari MK, Rajendran L. Analytical expression of the concentration of species and effectiveness factors in porous catalysts using the Adomian decomposition method.

[49] Renuga Devi M, Sevukaperumal S, Rajendran L. Nonlinear reaction diffusion equation with Michaelis-Menten Kinetics and adomian decomposition method. Applied Mathe-

[50] Mahalakshmi M, Hariharan G, Kannan K. The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Journal of Mathematical

[51] Rajendran L. Biosensor: Modeling and simulation of diffusion-limited process, chemical sensors: simulation and modeling. Ghenadiikorotcenkov ed. Electrochemical sensors.