Analysis and Control of Polynomial Dynamic Models with Biological Applications

By Gabor Szederkenyi, Attila Magyar and Katalin M. Hangos

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

• Language: English
• Publisher: Academic Press
• Release date: Mar 30, 2018
• ISBN: 9780128154960

Gabor Szederkenyi

Gábor Szederkényi received the M.Eng degree in computer engineering (University of Veszprém, 1998), his PhD in information sciences (University of Veszprém, 2002), and the DSc title in engineering sciences (Hungarian Academy of Sciences, 2013). Currently, he is a full professor at PPKE and the head of the Analysis and Control of Dynamical Systems research group. His main research interest is the computational analysis and control of nonlinear systems with special emphasis on reaction networks and kinetic models. He is the co-author of one book, several book chapters, more than 40 journal papers, and more than 60 conference papers on the theory and applicaton of the analysis and control of nonlinear systems. His education record includes BSc and MSc level courses on linear systems theory, nonlinear control and its application in robotics and in biological systems. The history of the scientific cooperation of the proposed three authors dates back to 2003. Since then, they have published more than 25 joint scientific papers in international journals and conference proceedings mostly related to the topic of the proposed book. The scientific background of the three authors are really complementary: Prof. Katalin Hangos is an internationally known expert in the modelling and control of thermodynamical and (bio)chemical systems, Dr. Attila Magyar has significant experience in the analysis, application and control of quasi-polynomial systems, while Prof. Gábor Szederkényi has results on the optimization-based structural analysis and synthesis of kinetic systems. Moreover, all three authors have had continuous education and supervising experience both on the MSc and PhD levels at different universities.

Book preview

2017

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

1.1DYNAMIC MODELS FOR DESCRIBING BIOLOGICAL PHENOMENA

1.2KINETIC SYSTEMS

1.2.1Chemical Reaction Networks With Mass Action Law

1.2.2Chemical Reaction Networks With Rational Functions as Reaction Rates

1.3QP MODELS

1.3.1Original Lotka-Volterra Equations

1.3.2Generalized 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 phenomena occurring in living systems is now also a commonly accepted view. However, the possible complexity of processes and their interactions in living systems and the difficulties in measuring important quantities in vivo may still be an obstacle for the application of quantitative models of moderate size in life sciences. This situation has been continuously improving, since the accumulation of well-structured biological knowledge about molecular and higher level processes mainly in the form of reliable models, and the recent fast development in computer and computing sciences have converged, resulting in the birth of a new discipline called systems biology, that is now able to address important challenges in the field of life sciences. Dynamical models, for example, in the form of differential equations, are also routinely needed in synthetic biology, which is focused on the engineering design and construction of biological devices.

As indicated earlier, the theory of dynamical systems is an important link connecting different scientific fields with common tools, techniques, and a unified point of view in the convergence paradigm between life sciences, physics, and engineering [2]. This fact is clearly supported by numerous recent groundbreaking results heavily involving advanced computational models in biology, such as efficient defibrillation with a low shock intensity [3], the artificial pancreas operating under a wide range of conditions [4], or drug dosage control in cancer therapies [5], just to mention a