Analysis and Control of Polynomial Dynamic Models with Biological Applications

Gabor Szederkenyi, Attila Magyar, Katalin M. Hangos

• Publisher: Academic Press
• Year: 2018
• Print ISBN: 9780128154953
• eBook ISBN: 9780128154960
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Analysis and Control of Polynomial Dynamic Models with Biological Applications synthesizes three mathematical background areas (graphs, matrices and optimization) to solve problems in the biological sciences (in particular, dynamic analysis and controller design of QP and polynomial systems arising from predator-prey and biochemical models). The book puts a significant emphasis on applications, focusing on quasi-polynomial (QP, or generalized Lotka-Volterra) and kinetic systems (also called biochemical reaction networks or simply CRNs) since they are universal descriptors for smooth nonlinear systems and can represent all important dynamical phenomena that are present in biological (and also in general) dynamical systems.- Describes and illustrates the relationship between the dynamical, algebraic and structural features of the quasi-polynomial (QP) and kinetic models- Shows the applicability of kinetic and QP representation in biological modeling and control through examples and case studies- Emphasizes the importance and applicability of quantitative models in understanding and influencing natural phenomena

Chapter 1

Introduction

Abstract

The background and motivation of the book is given in this chapter. The importance of developing and using dynamic models for dynamic analysis and control is emphasized, and the appropriate model forms, namely the kinetic and quasipolynomial model classes, are briefly and informally introduced.

Keywords

Kinetic systems; Chemical reaction networks; Biochemical reaction networks; Quasipolynomial systems; Lotka-Volterra models

Chapter Outline

• 1.1 DYNAMIC MODELS FOR DESCRIBING BIOLOGICAL PHENOMENA
• 1.2 KINETIC SYSTEMS
  • 1.2.1 Chemical Reaction Networks With Mass Action Law
  • 1.2.2 Chemical Reaction Networks With Rational Functions as Reaction Rates
• 1.3 QP MODELS
  • 1.3.1 Original Lotka-Volterra Equations
  • 1.3.2 Generalized Lotka-Volterra Equations

The application of dynamical models describing the change of measured or computed quantities in time and/or in space has been indispensable not only in science but also in everyday life. It is also commonly accepted that dynamics plays a key role in understanding and influencing numerous complex processes taking place in living systems [1].

The deep understanding and the targeted manipulation of dynamical models’ behavior are in the focus of systems and control theory that now provides us with really powerful methods for model analysis and controller synthesis in numerous engineering application fields. The efficient treatment of nonlinear and uncertain models is a well-developed field of control theory that has recently been a promising foundation for biological applications. Both system classes studied in the book, namely quasipolynomial (QP, or generalized Lotka-Volterra) systems and kinetic systems (also called [bio]chemical reaction networks, or simply CRNs) are so-called universal descriptors for smooth nonlinear systems. This means that they can represent all important dynamical phenomena that are present in biological (and also in general) dynamical systems. Moreover, both system classes are really natural descriptors of biological and biochemical processes: QP systems are the generalizations of Lotka-Volterra models originally used for modeling general population dynamics and related phenomena, while kinetic systems composed of elementary reaction steps come from the description of (bio)chemical processes. The main practical advantage of QP and kinetic systems is their relatively simple matrix-based algebraic structure that allows the development of efficient computational (e.g., optimization-based) methods for their dynamical analysis and control. Moreover, the direct physical interpretation of many important system properties is often possible for these models.

Therefore, our aims with writing the book are the following: (1) To describe and illustrate the relation between dynamical, algebraic, and structural features of the studied model classes in a unique way not known by the authors in the literature. (2) To show the applicability of kinetic and QP representation in biological modeling and control through examples and case studies. (3) To show and emphasize the importance of quantitative models in understanding and influencing natural phenomena.

The target audience include graduate students in computer science, electrical, chemical, or bioengineering, applied mathematicians, and engineers, as well as researchers who are interested in a brief summary of the analysis and control of QP and kinetic systems.

1.1 The Notion and Significance of Dynamical Models in the Description of Natural and Biological Phenomena

Although the descriptive power of mathematical models is necessarily limited, their widespread utilization not only in research and development but also in the everyday life of today’s technical civilization is clearly indispensable. When we are interested in the evolution of certain quantities, usually in time and/or space, we use dynamical models. The intensive use of dynamical models originates from classical physics for the description of the motion of objects. During the last century, by generalizing the concept of motion, the use of dynamical models became essential in other applied fields like electrical, chemical, and process engineering as well.

The key role of dynamics in the explanation of important phenomen...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• About the Authors
• Preface
• Acknowledgments
• Chapter 1: Introduction
• Chapter 2: Basic Notions
• Chapter 3: Model Transformations and Equivalence Classes
• Chapter 4: Model Analysis
• Chapter 5: Stabilizing Feedback Control Design
• Chapter 6: Case Studies
• Appendix A: Notations and Abbreviations
• Appendix B: Mathematical Tools
• Bibliography
• Index