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An introduction to the mathematical, computational, and analytical techniques used for modeling biological rhythms, presenting tools from many disciplines and example applications. All areas of biology and medicine contain rhythms, and these behaviors are best understood through mathematical tools and techniques. This book offers a survey of mathematical, computational, and analytical techniques used for modeling biological rhythms, gathering these methods for the first time in one volume. Drawing on material from such disciplines as mathematical biology, nonlinear dynamics, physics, statistics, and engineering, it presents practical advice and techniques for studying biological rhythms, with a common language. The chapters proceed with increasing mathematical abstraction. Part I, on models, highlights the implicit assumptions and common pitfalls of modeling, and is accessible to readers with basic knowledge of differential equations and linear algebra. Part II, on behaviors, focuses on simpler models, describing common properties of biological rhythms that range from the firing properties of squid giant axon to human circadian rhythms. Part III, on mathematical techniques, guides readers who have specific models or goals in mind. Sections on “frontiers” present the latest research; “theory” sections present interesting mathematical results using more accessible approaches than can be found elsewhere. Each chapter offers exercises. Commented MATLAB code is provided to help readers get practical experience. The book, by an expert in the field, can be used as a textbook for undergraduate courses in mathematical biology or graduate courses in modeling biological rhythms and as a reference for researchers.
This book, based on a selection of invited presentations from a topical workshop, focusses on time-variable oscillations and their interactions. The problem is challenging, because the origin of the time variability is usually unknown. In mathematical terms, the oscillations are non-autonomous, reflecting the physics of open systems where the function of each oscillator is affected by its environment. Time-frequency analysis being essential, recent advances in this area, including wavelet phase coherence analysis and nonlinear mode decomposition, are discussed. Some applications to biology and physiology are described. Although the most important manifestation of time-variable oscillations is arguably in biology, they also crop up in, e.g. astrophysics, or for electrons on superfluid helium. The book brings together the research of the best international experts in seemingly very different disciplinary areas.
Biological Oscillators: Their Mathematical Analysis introduces the main features of the dynamic properties of biological oscillators and the mathematical techniques necessary for their investigation. It is not a comprehensive description of all known biological oscillators, since this would require a much bigger volume as well as a different type of expertise. Instead certain classes of biological oscillators are described, and then only in as much detail as required for the study of their dynamics. The opening chapter reviews fundamental mathematical concepts and techniques which will be used in the remainder of the book. These include phase plane techniques; asymptotic techniques of Krylov, Bogoliubov, and Mitopolski; and the describing function. Subsequent chapters discuss examples of biological oscillators; phase shifts and phase response curves; the entrainment of oscillators by external inputs; the dynamics of circadian oscillators; effects of changing environment on the dynamics of biological oscillators; the features peculiar to populations of interacting oscillators; and biological phenomena attributable to populations of oscillators.
Fundamentals of the mathematical theory of oscillators. Examples of biological rhythms. Phase shifts and phase response curves. Entrainment of oscillators by external inputs. The dynamics of circadian oscillators. Effects of changing environment on the dynamics of biological oscillators. Populations of interacting oscillators. Biological phenomena attributable to populations of oscillators.
As 1 review these pages, the last of them written in Summer 1978, some retrospec tive thoughts come to mind which put the whole business into better perspective for me and might aid the prospective reader in choosing how to approach this volume. The most conspicuous thought in my mind at present is the diversity of wholly independent explorations that came upon phase singularities, in one guise or another, during the past decade. My efforts to gather the published literature during the last phases of actually writing a whole book about them were almost equally divided between libraries of Biology, Chemistry, Engineering, Mathematics, Medicine, and Physics. A lot of what 1 call "gathering " was done somewhat in anticipation in the form of cönjecture, query, and prediction based on analogy between developments in different fields. The consequence throughout 1979 was that our long-suffering publisher re peatedly had to replace such material by citation of unexpected flurries of papers giving substantive demonstration. 1 trust that the authors of these many excellent reports, and especially of those I only found too late, will forgive the brevity of allusion I feIt compelled to observe in these substitutions. A residue of loose ends is largely collected in the index under "QUERIES. " It is c1ear to me already that the materials I began to gather several years ago represented only the first flickering of what turns out to be a substantial conflagration.
​ This volume brings together a unique collection of protocols that cover standard, novel, and specialized techniques. Cell Cycle Oscillators: Methods and Protocols guides readers through recent progress in the field from both holistic and reductionist perspectives, providing the latest developments in molecular biology techniques, biochemistry, and computational analysis used for studying oscillatory networks. Written in the highly successful Methods in Molecular Biology series format, chapters include introductions to their respective topics, lists of the necessary materials and reagents, step-by-step, readily reproducible laboratory protocols, and tips on troubleshooting and avoiding known pitfalls. Authoritative and cutting-edge, Cell Cycle Oscillators: Methods and Protocols will serve as an invaluable reference to gain further insight into the complex and incompletely understood processes that are involved in the cell cycle and its regulation by oscillatory networks.
How can we understand the complexity of genes, RNAs, and proteins and the associated regulatory networks? One approach is to look for recurring types of dynamical behavior. Mathematical models prove to be useful, especially models coming from theories of biochemical reactions such as ordinary differential equation models. Clever, careful experiments test these models and their basis in specific theories. This textbook aims to provide advanced students with the tools and insights needed to carry out studies of signal transduction drawing on modeling, theory, and experimentation. Early chapters summarize the basic building blocks of signaling systems: binding/dissociation, synthesis/destruction, and activation/inactivation. Subsequent chapters introduce various basic circuit devices: amplifiers, stabilizers, pulse generators, switches, stochastic spike generators, and oscillators. All chapters consistently use approaches and concepts from chemical kinetics and nonlinear dynamics, including rate-balance analysis, phase plane analysis, nullclines, linear stability analysis, stable nodes, saddles, unstable nodes, stable and unstable spirals, and bifurcations. This textbook seeks to provide quantitatively inclined biologists and biologically inclined physicists with the tools and insights needed to apply modeling and theory to interesting biological processes. Key Features: Full-color illustration program with diagrams to help illuminate the concepts Enables the reader to apply modeling and theory to the biological processes Further Reading for each chapter High-quality figures available for instructors to download
Recent years have seen spectacular advances in the field of circadian biology. These have attracted the interest of researchers in many fields, including endocrinology, neurosciences, cancer, and behavior. By integrating a circadian view within the fields of endocrinology and metabolism, researchers will be able to reveal many, yet-unsuspected aspects of how organisms cope with changes in the environment and subsequent control of homeostasis. This field is opening new avenues in our understanding of metabolism and endocrinology. A panel of the most distinguished investigators in the field gathered together to discuss the present state and the future of the field. The editors trust that this volume will be of use to those colleagues who will be picking up the challenge to unravel how the circadian clock can be targeted for the future development of specific pharmacological strategies toward a number of pathologies.
This textbook provides an introduction to dynamic modeling in molecular cell biology, taking a computational and intuitive approach. Detailed illustrations, examples, and exercises are included throughout the text. Appendices containing mathematical and computational techniques are provided as a reference tool.
Interest in biological rhythms has been traced back more than 2,500]ears to Archilochus, the Greek poet, who in one of his fragments suggests ",,(i,,(VWO'KE o'olos pv{}J.tos txv{}pW7rOVS ~XH" (recognize what rhythm governs man) (Aschoff, 1974). Reference can also be made to the French student of medicine J. J. Virey who, in his thesis of 1814, used for the first time the expression "horloge vivante" (living clock) to describe daily rhythms and to D. C. W. Hufeland (1779) who called the 24-hour period the unit of our natural chronology. However, it was not until the 1930s that real progress was made in the analysis of biological rhythms; and Erwin Bunning was encouraged to publish the first, and still not outdated, monograph in the field in 1958. Two years later, in the middle of exciting discoveries, we took a breather at the Cold Spring Harbor Symposium on Biological Clocks. Its survey on rules considered valid at that time, and Pittendrigh's anticipating view on the temporal organization of living systems, made it a milestone on our way from a more formalistic description of biological rhythms to the understanding of their structural and physiological basis.