Short description:

From cell division to heartbeat, clocklike rhythms pervade the activities of living organisms. The cycles of life are ultimately biochemical in mechanism but many of the principles that dominate their orchestration are common to diverse mechanisms. They are in essence mathematical principles. The first half of this book explains periodic processes in living systems and in their non-living analogues in the abstract terms of systems theory. Emphasis is given to phase singularities, waves, and mutual synchronization in tissues composed of many clock-like units. No theoretical background is assumed: the required notions are introduced with copious pictures and examples.
The second half of the book is devoted to descriptions of the most commonly used experimental systems such as an electrical device built from many coupled oscillators; a chemical reagent which gives rise to colorful self-organizing periodic wave patterns; apparent singularities in the periodic molecular architecture of arthropod cuticle; the short-period biochemical rhythms that organize metabolism and development in lower organisms; electrical oscillations and waves underlying nervous coordination, e.g. the heartbeat and its pathologies; circadian (24-hour) clocks in fungi, fruitflies, and higher plants; medically-motivated attempts to control the cell division cycle and the human female cycle; and the "clockface" interpretation of crystal-like regularities observed in regeneration of severed limbs.

Long description:

Geometry of Biological Time deals with dynamics of processes that repeat themselves regularly. Such rhythmic return through a cycle of change is an ubiquitous principle of organization in living systems. In this revised and updated edition the author plans to extend the thread from 1980 to the present concentrating on areas which he personally feels have been interesting and where he feels there will be much activity in the future. This involves going through spatial biochemical, electrophysiological, and organismic dynamical systems and patterns that were discovered by pursuing the theme of phase singularities that the original book introduced. In particular the work on excitability in cell membranes has been thoroughly updated as have the references throughout the book.

From the reviews:

"This book is a wonderful exposition about his life?s work and is presented with such clarity that the reader gains an insight into details of his manner of thinking?the reader is taken on a quest to solve biological mysteries and travels from the intricate beginnings of formulating the questions, through the apices and troughs of analyzing, and on to the discovery of the holy grail, that which yields biological predictions and answers. One leaves with a sense of understanding that hopefully can be applied to one?s own work?this book could easily serve a dual purpose. First, through the introduction of basic mathematical concepts such as topology combined with modeling, the book serves as an introduction helping the non-applied reader to see the beauty of mathematical modeling. Second, because Winfree explains the modeling process very thoroughly, instead of just presenting equations, the book can serve as a medium for the medical professional to acquire an understanding of the modeling process. A wonderful accomplishment...I am proud to have this book on my shelf and consider it?a seminal text in mathematical modeling. This is a text that every first year graduate student should look at in some detail." (Mathematical Reviews)

"The new edition of The Geometry of Biological Time is a fascinating update of the delightful original. This new edition contains thoughtful commentary on new developments in the field, adding a historical and sociological dimension to the original book's elegant and unifying treatment of biological problems involving processes that repeat themselves regularly, i.e. involve 'rhythmic return through a cycle of change.' The book is an enjoyable page-turner, even for those readers with only a passing interest in biology, and demonstrates well the synergistic effect between biology and mathematics. ...
The term 'page-turner' may seem unusual in mathematics; however, it is appropriate here.Not only is the reader continually tantalized by the figures appearing on the ensuing pages, but the new commentary lends a mystery-novel feeling to the book. This second edition was created by inserting new text boxes into the original, mostly intact, edition. This style leads to a fascinating historical picture. For example, the text, '(...In the latter cases the periodicity approximation gets worse closer to the pivot. I wish here to sweep such matters under the rug (in 1978))' is followed by a new text box that begins, 'The bulge under the rug grew and grew...', continuing with a description of developments over the last two decades. The plot continues even now, with descriptions of the last twenty years often followed in the book by descriptions of current puzzles. In summary, the original book is good and the second edition is even better; the historical commentary is fascinating, and there are also a few reorganized and new chapters presenting additonal biological examples." (MAA Online)

Table of Contents:

1. Circular Logic.- 2. Phase Singularities (Screwy Results of Circular Logic).- 3. The Rules of the Ring.- 4. Ring Populations.- 5. Getting off the Ring.- 6. Attracting Cycles and Isochrons.- 7. Measuring the Trajectories of a Circadian Clock.- 8. Populations of Attracting-Cycle Oscillators.- 9. Excitable Kinetics and Excitable Media.- 10. The Varieties of Phaseless Experience: In which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways.- 11. The Firefly Machine.- 12. Energy Metabolism in Cells.- 13. The Malonic Acid Reagent (?Sodium Geometrate?).- 14. Electrical Rhythmicity and Excitability in Cell Membranes.- 15. The Aggregation of Slime Mold Amoebae.- 16. Numerical Organizing Centers.- 17. Electrical Singular Filaments in the Heart Wall.- 18. Pattern Formation in the Fungi.- 19. Circadian Rhythms in General.- 20. The Circadian Clocks of Insect Eclosion.- 21. The Flower of Kalanchoë.- 22. The Cell Mitotic Cycle.- 23. The Female Cycle 650 A: Women, Hormones, and Eggs.- Index of Author Citations and Publications.- Index of Subjects.