The Geometry of Biological Time

Frontmatter

Introduction

Abstract

This is a story about dynamics: about change, flow, and rhythm, mostly in things that are alive. My basic outlook is drawn from physical chemistry, with its state variables and rate laws. But in living things, physical and chemical mechanisms are mostly quite complex and confusing, if known at all. So I’m not going to deal much in mechanisms, nor even in cause and effect. Instead I will adopt the attitude of a naturalist-anatomist, describing morphology. The subject matter being dynamics, we are embarked upon a study of temporal morphology, of shapes not in space so much as in time. But by introducing molecular diffusion as a principle of spatial ordering, we do come upon some consequences of temporal morphology for the more plainly visible shapes of things in space.

Arthur T. Winfree

1. Circular Logic

Abstract

My objective for this chapter is to draw your attention to a few peculiarities inherent in the logic of periodic functions. I find a visual approach the most fruitful for thinking about such matters. As the pictures involved consist mainly of mappings between circles and products of circles, I must first say a few words about the notions of topological spaces and mappings. This chapter thus has four sections:

A.

Spaces, with emphasis on rings (i.e., closed loops. To avoid the more exact connotations of the word circle I use ring, trusting the reader do not confuse my meaning with algebraic rings.)

 

B.

Mappings, with emphasis on the winding number of mappings to a ring

 

C.

Phase singularities of maps (Parts I and II), with emphasis on the consequences of a nonzero winding number

 

D.

Technical details on the application of circular logic to biological rhythms

 

Arthur T. Winfree

2. Phase Singularities (Screwy Results of Circular Logic)

Abstract

A phase singularity is a point at which phase is ambiguous and near which phase takes on all values. My purpose in this chapter is to give examples by somewhat idealized description of phase singularities observed in several experimental systems. In some cases, the phase singularity is at this writing only inferred and not yet demonstrated. Some cases of purely hypothetical and trivial nature are also thrown in to help clarify the principles that I take to be involved in the more interesting biological examples. Much is glossed over here that should disturb a thoughtful person acquainted with the physiology of any one of these systems. These details are dealt with in Chapter 10 and in the Bestiary (Chapters 11–23).

Arthur T. Winfree

3. The Rules of the Ring

Abstract

Although it may be fashionable to acknowledge that everything is connected to everything else in principle, some things are more tightly connected to each other than to all the rest. Such a little knot of causal interactions goes by the name of a system. It is also fashionable to speak of one’s playthings as systems, and I shall adhere to this convention. It will make life easier in the long run to clarify some conventional jargon at this point, as follows.

Arthur T. Winfree

4. Ring Populations

Abstract

My intent in this chapter is to direct your attention to several idealizations of rhythmic behavior in collections of many similar ring devices. It turns out that some of the peculiar limitations on the behavior of simple clocks do not apply to populations of simple clocks. Here we also encounter our first example in which a phase singularity emerges from an idealized model of the structure and mechanism of a rhythmic system. The chapter is divided into four sections:

A.

Collective rhythmicity in a population without interactions among constituent clocks. This is mainly about phase resetting by a stimulus.

 

B.

Collective rhythmicity in a population whose individuals are all influenced by the aggregate rhythmicity of the community. This is mainly about mutual synchronization and opposition to it.

 

C.

Spatially distributed simple clocks without interactions. This is mostly about patterns of phase in space.

 

D.

Ring devices interacting locally in space. This is mostly about waves.

 

Arthur T. Winfree

5. Getting off the Ring

Abstract

My purpose in this chapter is to start “putting flesh on the bones” of the simple clock metaphor. Up to this point, I’ve tried to hold your attention on “phase” and its rate of change by confining discussion to the simplest metaphor of smooth cyclic dynamics, namely, the ring device. I have studiously avoided allusion to other degrees of freedom of the “state” of any biological clock. To make the transition to a broader perspective in an orderly way, I now wish to introduce just one additional notion, i.e., that a rhythmic process might be adjustable not only in phase, our exclusive preoccupation in previous chapters, but also in some measure of its vigor, amplitude, range, or degree of variation during the cycle.

Arthur T. Winfree

6. Attracting Cycles and Isochrons

Abstract

In Section A of this chapter we associate a phase with each state of a limit cycle oscillator during dynamics in the absence of any perturbing influence. In Section B a stimulus smoothly alters the trajectories so that phase changes in peculiar ways, even discontinuously. This analysis is intended to apply to smooth dynamics. Accordingly, in Section C references are compiled to models which violate this precondition and thus do not fall within the purview of this chapter.

Arthur T. Winfree

7. Measuring the Trajectories of a Circadian Clock

Abstract

If, in context of real laboratory experiments, we wish to seriously contemplate models with more than one degree of freedom, then we must find two or more independent empirical measures corresponding to the movements of the system in its state space. We must seek to plot a trajectory in a space of two or more measureable quantities. If we can find a way to do this, then we can distinguish the quickly attracting cycle of Chapter 6 from the orbitally stable kinetic schemes of Chapters 4 and 5.

Arthur T. Winfree

8. Populations of Attractor Cycle Oscillators

Abstract

Chapter 4 provides a preliminary look at the phenomena to which we now turn : the phenomena typical of aggregates of oscillators. Just as the oscillator populations of physics comprise a very special case with very special properties (associated with linearity, energy conservation, etc.), so did the simple clocks of Chapter 4 comprise another very special case with very special properties (associated with the one-dimensionality of their state space). My objective in this chapter is to organize under the same four headings as in Chapter 4 some discussions and examples of what I take to be the characteristic behavior of attractor cycle oscillators in populations and communities. Such oscillators can have any number (> 2) of variables mutually determining their rates of change in nonlinear ways. Linear oscillators, conservative oscillators, and simple clocks are special limiting cases of the attractor cycle oscillators considered in this chapter.

Arthur T. Winfree

9. Excitable Kinetics and Excitable Media

Abstract

Chapter 3 on ring devices is followed by Chapter 4 on populations and communities of such single-variable units: both simple clocks and the nonoscillating hourglasses. Chapter 6 on oscillators with more than one variable of state is followed by populations and communities in Chapter 8. What about nonoscillatory kinetics with more than one variable? That case is taken up briefly here, together with consequences of interaction in spatially distributed communities. The upshot is a new kind of oscillator and a new kind of phase singularity, both of which are apparently exhibited in diverse chemical and physiological systems. Even though no isolated piece of it may oscillate, an excitable medium can organize itself spatially in a way that stabilizes oscillation at a characteristic period. Architecturally, this configuration more resembles a clock than anything encountered in previous chapters: It consists of crossed concentration gradients, any one of which might be taken as the clock’s “hand”, a pointer that physically rotates about a fixed pivot once in each cycle of oscillation. At the pivot, nothing changes; the pivot is a phase singularity and all the rest is built around it.

Arthur T. Winfree

10. The Varieties of Phaseless Experience: In which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways

Abstract

In Chapter 1 we dwelt on the notion of smooth maps from one space to another. In Chapter 2 it emerged that certain kinds of mapping involving circles cannot be contrived smoothly. As an application we saw that certain kinds of experimentally observed continuity and smoothness involving measures that are periodic in space or time inescapably imply an unobserved (but observable) discontinuity. A phase singularity is one way to resolve this crisis of continuity implicit in the observation of nonzero winding number.

Arthur T. Winfree

11. The Firefly Machine

Abstract

A machine once existed (Figure 1) in which 71 flickering neon lamps were each coupled electrically to all the others. (There were 71 because out of 100 constructed, 29 drifted outside the intended range of autonomous period during initial “wearing in”.) The purpose of building this machine was just to “look and see what would happen”, on a hunch that groups of oscillators might synchronize together in fleeting alliances. One hope was that by plotting the output of this population of interacting oscillators in the same format as biologists use to plot activity rhythms of multicellular animals, enough resemblances might be noticed to suggest some interpretation of the tantalizingly complex biological records.

Arthur T. Winfree

12. Energy Metabolism in Cells

Abstract

Cells have three alternative means of procuring energy for digestion and biochemical synthesis, for maintaining concentration gradients, for muscular contractions and cell division, and for maintaining body heat:

1.

Photosynthesis : The chloroplasts of green plants capture photons to convert ADP to ATP. Water is split to reduce NADP to NADPH, releasing oxygen.

 

2.

Respiration: The mitochondria use that oxygen and convert ADP to ATP. In the process, NADH is oxidized to NAD and water.

 

3.

Glycolysis: Lacking illuminated chloroplasts or lacking oxygen, cells metabolize sugars by fermentation to make a little ATP from ADP. Historically, this was probably the first way to make the high-energy pyrophosphate bond of ATP. All cells maintain this pathway. Most cells fall back on glycolysis only when they have no better alternative, but it is common to have no better alternative. The microorganisms of yogurt, sauerkraut, gangrene, and food poisoning, for example, subsist wholly on glycolysis, as do faculative anaerobes such as intertidal bivalves (e.g., oysters) and parasitic helminths (e.g., schistosomes) and diving vertebrates (e.g., green sea turtles) during their prolonged periods of contented abstinence from respiration. Red blood cells have no other energy supply. Poorly vascularized tissue such as the cornea of the eye, compact tumors, and embryos rely heavily on glycolysis for their energy needs.

 

Arthur T. Winfree

13. The Malonic Acid Reagent (“Sodium Geometrate”)

Abstract

A new chemical reaction with either excitable or periodic dynamics appears every month in the theoretical journals. But only one has been widely studied experimentally in ways that reveal wave-like organization in space. There are already two entire books about it: Zhabotinsky (1974, in Russian) and Tyson (1976b).

Arthur T. Winfree

14. Electrical Rhythmicity and Excitability in Cell Membranes

Abstract

Every cell has a plasma membrane. The plasma membrane is a thin film, less than a hundred angstroms thick, which maintains a difference between inside and outside by gatekeeping the passage of molecules and ions. Every cellular membrane is freely permeable to some substances (e.g., water) and essentially impermeable to others (e.g., proteins and certain ions). Nerve cells and some secretory cells are distinguished from most other kinds of cell chiefly in that the selective permeability of their plasma membranes depends sharply on an electric field. All cells experience an electric potential difference between inside and outside, ultimately because amino acids bear an ionic charge and, once polymerized inside the cell, they can’t get out.¹ This potential difference is typically about one-tenth of a volt, so the thin plasma membrane is stressed by an electric field in the order of 10 million volts/m. In nerve cells, molecular anatomy within the plasma membrane is believed to readjust when this field is reduced to less than a certain threshold. With its selective permeability altered, the membrane passes certain ions that it had formerly restrained, resulting in a further decrease in the field maintained, and a self-catalyzing breakdown of membrane potential quickly ensues. But things are so arranged that a recovery promptly follows in which electrical imbalance is restored.

Arthur T. Winfree

15. The Aggregation of Slime Mold Amoebae

Abstract

Two kinds of slime mold play central roles in this book. Later on we will meet the “true” slime mold (Myxomycetes), an acellular jelly remarkable for the regularity and synchrony of mitosis in its many nuclei. Topologically, the true slime mold is one single monstrous cell. But in the present chapter, our concern is with the cellular slime molds (Acrasiales), the best studied of which is Dictyostelium discoideum (Bonner, 1967; Gerisch, 1968). This creature is more conventional in its cellular structure but is equally astonishing topologically in that its cells wander independently, like the individual workers of an ant colony. Like the ant hive, Dictyostelium is a “superorganism”, a genetically homogeneous being composed of autonomous individuals, nevertheless organized altruistically for the collective good. The life cycle runs as follows.

Arthur T. Winfree

16. Growth and Regeneration

Abstract

Many kinds of living organisms regrow appendages that are crushed or torn off in the mishaps of an active life. People have scarcely any abilities of this sort, a fact which contributes to their jealous curiosity about the mechanisms of regeneration in more resilient organisms. This curiosity runs deeper than mere jealousy would motivate because regeneration in many ways resembles the initial normal development of an animal’s structures. Normal development plus regeneration, collectively called morphogenesis, presumably operates by some general rules that we might at least elucidate empirically as a prelude to ferreting out deeper mechanisms. Yet for all the imaginative and meticulous efforts of at least four generations of developmental biologists, few general rules have stood the test of time. If principles of widespread applicability exist, they remain tantalizing obscure.

Arthur T. Winfree

17. Arthropod Cuticle

Abstract

Experiments with the limbs of amphibians, roach legs, and fly wing disks suggested to French et al. (1976) some simple rules governing growth and pattern formation in the insects and in higher animals. To apply these rules (Chapter 16), we must first find a point on a ring associated with each point on the animal’s two-dimensional surface. French et al. argue that morphogenesis is conducted primarily within two-dimensional sheets of cells and that within these sheets cells know their identity in part as a point on a ring, which we might think of as an angle or a phase. The pattern of phase (together with a second, independent quantity) across the two-dimensional sheet determines the qualitative pattern of growth and differentiation. In particular, phase singularities play a crucial role, for example determining the number and handedness of limbs (Glass, 1977).

Arthur T. Winfree

18. Pattern Formation in the Fungi

Abstract

In Chapter 12 we dwelt on a biochemical clock in an ascomycete, the yeast cell. The familiar bread molds and their relatives are also ascomycetes. They are called colonial ascomycetes because of their habits of growth. Like plants, they grow where the seed falls and feed through roots. An ascomycete colony starts when a spore falls on a food surface. It germinates and extends a fine web of hair-like filaments, called hyphae, across the food as an expandinh disk. This two-dimensional disk of hyphae is called a mycelium. It is not really a cellular organism since the septa dividing hyphae onto tiny compartments typically have holes in them, so that cytoplasm flows freely between the compartments.

Arthur T. Winfree

19. Circadian Rhythms in General

Abstract

One doesn’t have to look at many living organisms before noticing that a lot of behavioral physiology is temporally organized in periodic patterns. In fact, if I had to decide what impresses me as the single most conspicuous feature of natural ecosystems, I would say that it is the daily and seasonal periodism and the consequent temporal organization of niche structure, food webs, and behavior. Of course, it would seem natural to assume that any given daily rhythm or seasonal rhythm is a response to the environmental cycle of days and nights, summer and winter. This is often the case. It is, for example, in the case of “deep scattering layers”. These are layers of diverse fauna in all the world’s oceans that show up very clearly on sonar. They go deeper in the daytime and rise back toward the surface by night. My job on an Indian Ocean Expedition cruise from Woods Hole in summer of 1964 was to study these deep scattering layers using sonar and tow nets during their diurnal up-and-down migrations. Investigators in Cousteau’s diving saucer found that the deep scattering layer consists largely of small fish. They seem to simply follow light intensity, e.g., they come up during eclipses. Enright and Hamner (1967) later found a substantial role of endogenous rhythmicity in the vertical diurnal migrations of invertebrates in the scattering layers.

Arthur T. Winfree

20. The Circadian Clocks of Insect Eclosion

Abstract

Understanding the circadian timing of eclosion in insects is a pretty big undertaking. A lot of technical detail is essential and a lot of close reasoning from meticulous experiments stands in the place of direct observations on the “clock’s” unknown physiological mechanism. In that respect clockology has some of the intellectual delight of the earlier years of genetics. However, the whole argument has never been spelled out for publication in one place. This chapter once again attempts only an outline of the essentials. The story presented here seems to be generally valid for butterflies and moths, flies and mosquitos, wasps and bees (i.e., Lepidoptera, Diptera, and Hymenoptera), but I emphasize my own experimental beast, the fruitfly. For a review of insect eclosion systems from the viewpoint of physiology and ecology, see Remmert (1962).

Arthur T. Winfree

21. The Flower of Kalanchoe

Abstract

The tiny red flowers of Kalanchoe blossfeldiana open and close at 23-hour intervals. They do this for a week even when plucked from the plant and placed in a vial of sugar water under constant green light, at a constant temperature. Though blind to green, the flower’s clock is sensitive to red light. By exposing the flower to red light of intensity several watts per square meter for minutes to hours, one disrupts the normal rhythmicity. In most cases, it recovers sufficiently within four days so that a phase shift can be measured.

Arthur T. Winfree

22. The Cell Mitotic Cycle

Abstract

In no case is the process well understood whereby the growing cell “decides” to replicate its genome, segregate its chromosomes into two nuclei, and wall them off from each other by cell fission. Such a fundamental biological process presumably has some universal aspects. Its appeal as an object of investigation is further enchanced by the seductive belief that the mechanism of replication is constrained to some kind of simplicity these facts:

1.

The end state is close to the initial state (namely, a freshly divided cell); and

 

2.

The whole process repeats at fixed intervals of time, at least in certain kinds of cells in optimal growth conditions.

 

Arthur T. Winfree

23. The Female Cycle

Abstract

Monthly bleeding may have been commonplace among nuns, spinsters, and the infertile centuries ago, but it could hardly have been common among the women to whose uteri we all owe our existence. When they were not pregnant, their breast feeding encouraged lactation, which suppressed ovarian cycling. Short (1976) estimates that it may have been uncommon to experience three consecutive menstrual cycles in a lifetime under these conditions. Accordingly, the female endocrine system’s menstrual cycle has not been subjected to selection pressure for its clock-like attributes. In fact, there are diverse clues that some fraction of women are reflex ovulators, not spontaneous cyclers at all (Clark and Zarrow, 1971).¹ In a reflex ovulator, mature follicles await rupture by a surge of hormone which is elicited only by sexual stimulation. The ovum then starts its journey down the fallopian tube, and pregnancy (or, less likely, recycling) ensues.

Arthur T. Winfree

Backmatter