N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The spatiotemporal expression of genes in an organism is determined by regulatory systems that involve a large number of genes connected through a complex network of interactions. As an intuitive understanding of the behavior of these systems is hard to obtain, computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This report reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, ordinary and partial differential equations, stochastic equations, Boolean networks and their generalizations, qualitative differential equations, and rule-based formalisms. In addition, the report discusses how these formalisms have been used in the modeling and simulation of regulatory systems.

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Modeling and simulation of genetic regulatory systems: a literature review.

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

/pdf/stability-criteria-for-switched-and-hybrid-systems-cmu7vlfa6a.pdf

Stability Criteria for Switched and Hybrid Systems

''Model-free control'' and the corresponding ''intelligent'' PID controllers (iPIDs), which already had many successful concrete applications, are presented here for the first time in an unified manner, where the new advances are taken into account. The basics of model-free control is now employing some old functional analysis and some elementary differential algebra. The estimation techniques become quite straightforward via a recent online parameter identification approach. The importance of iPIs and especially of iPs is deduced from the presence of friction. The strange industrial ubiquity of classic PID's and the great difficulty for tuning them in complex situations is deduced, via an elementary sampling, from their connections with iPIDs. Several numerical simulations are presented which include some infinite-dimensional systems. They demonstrate not only the power of our intelligent controllers but also the great simplicity for tuning them.

https://hal-polytechnique.archives-ouvertes.fr/hal-00828135/document

Model-free control

http://www.math.uha.fr/sari/papers/BMB2004.pdf

Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models

We investigate the properties of the solutions of a class of piecewise-linear differential equations. The equations are appropriate to model biological systems (e.g. genetic networks) in which there are switch-like interactions between the elements. The analysis uses the concept of Filippov solutions of differential equations with a discontinuous right-hand side. It gives an insight into the so-called singular solutions which lie on the surfaces of discontinuity. We show that this notion clarifies the study of several examples studied in the literature.

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A class of piecewise linear differential equations arising in biological models

Invented by J. Monod, and independently by A. Novick and L. Szilard, in 1950, the chemostat is both a micro-organism culturing device and an abstracted ecosystem managed by a controlled nutrient flow. This book studies mathematical models of single species growth as well as competition models of multiple species by integrating recent work in theoretical ecology and population dynamics. Through a modeling approach, the hypotheses and conclusions drawn from the main mathematical results are analyzed and interpreted from a critical perspective. A large emphasis is placed on numerical simulations of which prudent use is advocated. The Chemostat is aimed at readers possessing degree-level mathematical knowledge and includes a detailed appendix of differential equations relating to specific notions and results used throughout this book.

The Chemostat: Mathematical Theory of Microorganism Cultures

The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for the development of appropriate computer tools for the modeling and simulation of gene regulation processes. We present a method for the hybrid modeling and simulation of genetic regulatory networks, based on a class of piecewiselinear (PL) differential equations that has been well-studied in mathematical biology. Distinguishing characteristics of the method are that it makes qualitative predictions of the behavior of regulatory systems and that it deals with discontinuities in the right-hand side of the differential equations. The simulation method has been implemented in Java in the computer tool Genetic Network Analyzer (GNA). The method and the tool have been used to analyze several networks of biological interest, including the network underlying the initiation of sporulation in Bacillus subtilis.

Hybrid Modeling and Simulation of Genetic Regulatory Networks: A Qualitative Approach

The aim of this paper is to show that the representation with the help of Nonstandard Analysis of a real phenomenon, presenting different observation scales, allows an important simplification of language. Indeed, it is convenient to have available the concept of infinitely small and infinitely large quantities in dealing with the macroscopic effects of microscopic phenomena. This is illustrated on two examples : the representation of two time scales systems and the representation of noise.

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Tewfik Sari

Papers

Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models

A class of piecewise linear differential equations arising in biological models

The Chemostat: Mathematical Theory of Microorganism Cultures

Hybrid Modeling and Simulation of Genetic Regulatory Networks: A Qualitative Approach

Non-standard analysis and representation of reality