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Über dieses Buch

This edited volume provides insights into and tools for the modeling, analysis, optimization, and control of large-scale networks in the life sciences and in engineering. Large-scale systems are often the result of networked interactions between a large number of subsystems, and their analysis and control are becoming increasingly important. The chapters of this book present the basic concepts and theoretical foundations of network theory and discuss its applications in different scientific areas such as biochemical reactions, chemical production processes, systems biology, electrical circuits, and mobile agents. The aim is to identify common concepts, to understand the underlying mathematical ideas, and to inspire discussions across the borders of the various disciplines.

The book originates from the interdisciplinary summer school “Large Scale Networks in Engineering and Life Sciences” hosted by the International Max Planck Research School Magdeburg, September 26-30, 2011, and will therefore be of interest to mathematicians, engineers, physicists, biologists, chemists, and anyone involved in the network sciences. In particular, due to their introductory nature the chapters can serve individually or as a whole as the basis of graduate courses and seminars, future summer schools, or as reference material for practitioners in the network sciences.



Chapter 1. Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes

We give an introduction to the geometric theory of ordinary differential equations (ODEs) tailored to applications to biochemical reaction networks and chemical separation processes. Quite often, the ordinary differential equations under investigation are “reduced” partial differential equations (PDEs) as in the search of traveling wave solutions. So, we also address ODE topics that have their origin in the PDE context.
We present the mathematical theory of invariant and integral manifolds, in particular, of center and slow manifolds, which reflect the splitting of variables and/or processes into slow and fast ones. The invariance of a smooth manifold is characterized by a quasilinear partial differential equation, and the widely used approximations of invariant manifolds are derived from such PDEs. So we also offer, to some extent, an introduction to quasilinear PDEs. The basic ideas and crucial tools are illustrated with numerous examples and exercises. Concerning the proofs, we confine ourselves to outline the crucial steps and refer, especially in the first three sections, to the literature.
The final Sects. 1.4 and 1.5 on reaction–separation processes and on chromatographic separation present new results, including their proofs. They are the outcome of many fruitful discussions with my colleagues Malte Kaspereit and Achim Kienle.
Dietrich Flockerzi

Chapter 2. Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits

We give a basic and self-contained introduction to the mathematical description of electrical circuits that contain resistances, capacitances, inductances, voltage, and current sources. Methods for the modeling of circuits by differential–algebraic equations are presented. The second part of this paper is devoted to an analysis of these equations.
Timo Reis

Chapter 3. Interacting with Networks of Mobile Agents

How should human operators interact with teams of mobile agents, whose movements are dictated by decentralized and localized interaction laws? This chapter connects the structure of the underlying information exchange network to how easy or hard it is for human operators to influence the behavior of the team. “Influence” is understood both in terms of controllability, which is a point-to-point property, and manipulability, which is an instantaneous influence notion. These two notions both rely on the assumption that the user can exert control over select leader agents, and we contrast this with another approach whereby the agents are modeled as particles suspended in a fluid, which can be “stirred” by the operator. The theoretical developments are coupled with multirobot experiments and human user-studies to support the practical viability and feasibility of the proposed methods.
Magnus Egerstedt, Jean-Pierre de la Croix, Hiroaki Kawashima, Peter Kingston

Chapter 4. Combinatorial Optimization: The Interplay of Graph Theory, Linear and Integer Programming Illustrated on Network Flow

Combinatorial optimization is one of the fields in mathematics with an impressive development in recent years, driven by demands from applications where discrete models play a role. Here, we intend to give a comprehensive overview of basic methods and paradigms, in particular the beautiful interplay of methods from graph theory, geometry, and linear and integer programming related to combinatorial optimization problems. To understand the underlying framework and the interrelationships more clearly, we illustrate the theoretical results and methods with the help of flows in networks as running example. This includes, on the one hand, a combinatorial algorithm for finding a maximum flow in a network, combinatorial duality and the max-flow min-cut theorem as one of the fundamental combinatorial min–max relations. On the other hand, we discuss solving the network flow problem as a linear program with the help of the simplex method, linear programming duality and the dual program for network flow. Finally, we address the problem of integer network flows, ideal formulations for integer linear programs and consequences for the network flow problem.
Annegret K. Wagler

Chapter 5. Stoichiometric and Constraint-Based Analysis of Biochemical Reaction Networks

Metabolic network analysis based on stoichiometric and constraint-based methods has become one of the most popular and successful modeling approaches in network and systems biology. Although these methods rely solely on the structure (stoichiometry) of metabolic networks and do not require extensive knowledge on mechanistic details of the involved reactions, they enable the extraction of important functional properties of biochemical reaction networks and deliver various testable predictions. This chapter gives an introduction on basic concepts and methods of stoichiometric and constraint-based modeling techniques. The mathematical foundations of the most important approaches—including graph-theoretical analysis, conservation relations, metabolic flux analysis, flux balance analysis, elementary modes, and minimal cut sets—will be presented, and applications in biology and biotechnology will be discussed. It will be shown that network problems arising in the context of metabolic network modeling are related to different fields of applied mathematics such as graph and hypergraph theory, linear algebra, linear programming, and combinatorial optimization. The methods presented herein are discussed in light of biological applications; however, most of them are generally applicable and useful to analyze any chemical or stoichiometric reaction network.
Steffen Klamt, Oliver Hädicke, Axel von Kamp

Chapter 6. A Petri-Net-Based Framework for Biomodel Engineering

Petri nets provide a unifying and versatile framework for the synthesis and engineering of computational models of biochemical reaction networks and of gene regulatory networks. Starting with the basic definitions, we provide an introduction into the different classes of Petri nets that reinterpret a Petri net graph as a qualitative, stochastic, continuous, or hybrid model. Static and dynamic analysis in addition to simulative model checking provide a rich choice of methods for the analysis of the structure and dynamic behavior of Petri net models. Coloring of Petri nets of all classes is powerful for multiscale modeling and for the representation of location and space in reaction networks since it combines the concept of Petri nets with the computational mightiness of a programming language. In the context of the Petri net framework, we provide two most recently developed approaches to biomodel engineering, the database-assisted automatic composition and modification of Petri nets with the help of reusable, metadata-containing modules, and the automatic reconstruction of networks based on time series data sets. With all these features the framework provides multiple options for biomodel engineering in the context of systems and synthetic biology.
Mary Ann Blätke, Christian Rohr, Monika Heiner, Wolfgang Marwan

Chapter 7. Hybrid Modeling for Systems Biology: Theory and Practice

Whereas bottom-up systems biology relies primarily on parametric mathematical models, which try to infer the system behavior from a priori specified mechanisms, top-down systems biology typically applies nonparametric techniques for system identification based on extensive “omics” data sets. Merging bottom-up and top-down into middle-out strategies is confronted with the challenge of handling and integrating the two types of models efficiently. Hybrid semiparametric models are natural candidates since they combine parametric and nonparametric structures in the same model structure. They enable to blend mechanistic knowledge and data-based identification methods into models with improved performance and broader scope. This chapter aims at giving an overview on theoretical fundaments of hybrid modeling for middle-out systems biology and to provide practical examples of applications, which include hybrid metabolic flux analysis on ill-defined metabolic networks, hybrid dynamic models with unknown reaction kinetics, and hybrid dynamic models of biochemical systems with intrinsic time delays.
Moritz von Stosch, Nuno Carinhas, Rui Oliveira
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