Structure and physiology

Bacteria (references: Berg, 1975a, b; Adler, 1975)

The bacterial flagellum is very different from the organelle that bears that name in eukaryotic organisms (e.g., protozoans, algae, and multicellular organisms). It is a rigid or semirigid helical filament made of a single polymeric protein called fiaggelin, which has no enzymatic activity. At the base of the filament is a complicated set of components whose structure is fairly well known by now (see Figure 4.1). The basal body serves to anchor the flagellum in the cell and probably is part of the driving motor. A number of theories of motion have been suggested, and one based on experiments of Berg, Anderson, and others now seems most probable. In this model, the flagellum is a rigid or semirigid helix that is turned at the base by a rotary motor interacting with the basal body. If true, this would apparently be the only known use of the wheel and axle for locomotion in all biology! The experiments of Berg et al. involved getting the flagellum to stick to a glass surface or to tiny latex beads. In the first case, when the flagellum was immobilized, the organism was seen to rotate; in the second case, the attached spheres could be seen to rotate. From such observations, the picture of a rigidly rotating helix has emerged.

In many instances natural swimming and flying involves organisms in sufficiently close proximity to cause significant interaction via the fluid medium. In these cases, the problem emphasized in the preceding chapters – propulsion of a body through a medium at rest – is modified to the extent that a given organism actually moves through the flow field created by its neighbors. To study such interactions, some hypothesis must be made concerning the nature of the response to the modified flow field. A passive organism will not modify movements or body geometry in response to environment, even though a modified flow field is sensed. Probably the only examples of a completely passive response are to be found among the microorganisms, and we consider two possible instances below. But the most interesting aspect of this class of problems resides in the capacity of the animal to modify its swimming or flying configuration as it senses changes in the ambient flow field, presumably in order to optimize performance. As we shall see later in the chapter, it has been suggested that such considerations can account for the arrangement of fish in schools and for the formation flying of birds. The observed stability of such groups could be a result of the active orientation of individuals relative to neighbors, but it is more natural to expect that an inherent stability comes from the optimal movements sought by each individual, leading to optimal performance of the group as a whole as members fall into place in a preferred pattern.

A principal characteristic of “flight” is that a significant part of the aerodynamic force is needed to cancel the weight of the organism. Thus, certain features of flying apply to buoyant fish. In forward flight such a force can be obtained by creating horizontal bound vorticity, this being the main purpose of the lifting surfaces of the body. The soaring and gliding of birds provides a familiar example where the classical aerodynamics of fixed-wing aircraft can be applied at once, a problem we consider in Section 11.1. It is worth mentioning that there is evidence that observations of birds led Lanchester to the notions of circulation and induced drag of finite wings [see the engrossing historical summary in Durand (1963)].

However, gliding, insofar as it may be taken to be stationary in time, is a rather special instance of animal flight. The common maneuvers of natural flight – takeoff and landing, flapping flight, and hovering – are fundamentally time-dependent phenomena (see the estimate of the frequency parameter σ given in Chapter 1). In effect, natural fliers, in particular the hummingbirds and certain of the hovering insects, explore the problem of lift generation in a very different parameter range from that conventionally exploited in aeronautics. In later sections we examine some of these time-dependent problems in the context of unsteady airfoil theory.

For general references on this subject, see Gray (1968) and the review articles by Lighthill and Weis-Fogh in Wu et al. (1975, vol. 2). For well-illustrated popular accounts of insect flight, see Nachtigall (1968) and Weis-Fogh (1975b).

Introduction

This is the second basic swimming mechanism in the Stokesian realm. Although the organelles are apparently identical in ultrastructure and physiology, we use the term “flagellum” when there is only one, or a small number of these hairlike appendages on a cell, as in spermatozoans and the flagellates, and “cilia” to denote large numbers of them on the same cell, as in the ciliates. Cilia, then, are just a large number of flagella on the same cell. The ciliates tend to be larger than the cell bodies of spermatozoans by an order of magnitude. Cilia tend to be shorter than ftagella, however, and this fact puts their mechanism firmly in the Stokesian realm.

Confusion about terminology can arise from two sources: (1) in bacteria there are organelles called flagella with very different ultrastructure and physiology (Chapter 4); and (2) the term “cilia” is usually used in connection with various ciliated tissues in metazoans (manycelled animals) as in the lining of our respiratory tracts, but there is usually only one or a small number of cilia per cell in Lhese tissues.

Because of their proximity to the cell wall, we can expect to find that optimal propulsion using cilia will involve movements quite different from those of a single large flagellum. In this regard it would be valuable to understand the possible adaptive advantages that could have led to the evolution of ciliary motion. As both flagellary and ciliary modes are widespread and both are evidently successful strategies, we might suppose that there are trade-offs between them.

We now take up the mechanical principles underlying Eulerian swimming of a thin, flexible creature, with the aim of understanding morphology in terms of the mechanisms of propulsion. For summaries of the related biology see Lighthill (1975, chap. 2) and Wu (1971a).

Small-perturbation theory of slender fish

Many fish change shape rather gradually along the anterior–posterior axis. (There are, however, many exceptions: for example, angelfish.) As a first approximation, we consider the swimming of slender, neutrally bouyant fish, By slender we mean, among other things, that the cross-sectional area of the body changes slowly along its length. Notation and terminology are summarized in Figure 10.1.

Necessary conditions for the validity of slender-body theory are such as to ensure that velocity perturbations caused by the fish are a small fraction of its swimming speed. It is certainly sufficient that the body surface S be smooth and that tangent planes always make a small angle with the x axis. However, it turns out that if these conditions are exactly met and the cross-sectional area is zero at the extremities, the theory predicts zero mean thrust. Fortunately, the model also allows us to treat “slender” fish with sharp downstream edges, at the caudal fin, for example, even though s(x) may be discontinuous there. Also, it is not necessary that surface slope always be small, at the nose of the fish, for example, provided that this occurs over at most a fraction of its length.

This monograph developed from a course given at the Courant Institute in the spring of 1976. At that time a preliminary set of notes was prepared for the class by M. Lewandowsky, who also contributed Chapter 4 and made numerous helpful suggestions. My purpose at the time was to provide a brief complement to the relevant portions of Lighthill's book, Mathematical Biofluiddynamics, as well as a guide to the papers contained in the symposium proceedings Swimming and Flying in Nature, edited by Wu, et al, and to incorporate background and supplementary material as needed. The present book has been changed only by the addition of Chapter 12 on interactions and a number of revisions and corrections.

Although, as a result of these origins, this monograph falls far short of a definitive treatise, I hope that this glimpse of a fascinating and rapidly evolving body of research will be useful to students of fluid mechanics seeking a compact introduction to biological modeling. Those familiar with the aerodynamics of fixed-wing aircraft or the hydrodynamics of ships, to take two examples from many, are well aware that in these applications the theory is concerned with design as well as analysis. The products of our technology are both the subjects and the results of our mathematical modeling. In “natural” swimming and flying the situation is no longer the same. Nature has already provided the answers.

The applications in this chapter represent the opposite end of the spectrum to the physical systems of Chapter 5. When we are trying to analyse the behaviour of an individual, or of a group, we cannot write down a set of equations of motion for the system based on known quantitative laws, and then look to see what catastrophe theory has to say about the solutions of these equations. What we must do is quite different. If we observe in a system some or all of the features which we recognize as characteristic of catastrophes – sudden jumps, hysteresis, bimodality, inaccessibility and divergence – we may suppose, at least as a working hypothesis, that the underlying dynamic is such that catastrophe theory applies. We then choose what appear to be appropriate state and control variables and attempt to fit a catastrophe model to the observations.

Right from the start, we see one of the advantages of catastrophe theory in this sort of problem. The data which are available are often not quantifiable. We can generally order our observations; for example, we can tell whether a person has become more angry or less angry. And we can usually say whether or not a variable is continuous and whether it changes smoothly. On the other hand, algebraic concepts such as addition and multiplication generally have no meaning: it makes little real sense to say that someone has become twice as angry.

In this chapter we derive Thorn's famous list of seven elementary catastrophes. We have already seen that we need only consider Taylor series in one or two variables; we now have to find all the different cases which can arise with codimension no greater than 4.

We begin with some mathematical preliminaries. First, we have to be clear about what we mean by ‘different’ cases. The usual statement is that there are seven qualitatively different catastrophes, and when we come to discuss the applications of the theory it will be clear enough what this means. But we need a more precise definition if we are to know what sorts of mathematical operations we may use in the derivations, so we say that two catastrophes are equivalent if one can be transformed to the other by (i) a diffeomorphism of the control variables, and (ii) at each point in the control space a diffeomorphism of the state variables. The resulting family of state-variable diffeomorphisms must be smooth when considered as a function of the control variables.

A diffeomorphism is a one-to-one continuous differentiable transformation. It is sometimes useful (though not strictly accurate) to think of two geometric objects as being topologically equivalent, or homeomorphic, if one can be continuously deformed into the other without any tearing or pasting together. To the same degree of accuracy we may think of two geometric objects as being diffeomorphic if they are homeomorphic and if, in addition, the deformation involved no creasing or flattening of creases.

In this chapter we bring together the different ways in which we have seen that catastrophe theory can be applied, and we discuss how catastrophe theory compares with other methods as a means of explaining nature.

Applications of catastrophe theory

We began our study of the applications of catastrophe theory by quoting Thorn's observation that they form a spectrum of different types. Now that we have seen a number of applications we can describe the spectrum in more detail.

At the extreme ‘physical’ end, catastrophe theory is used much like any other mathematical method, to help us discover the properties of a known, or at any rate postulated, dynamic. For example, when we are studying the buckling of elastic structures or the stability of ships (Zeeman, 1977a), the concept of a universal unfolding and the techniques for finding one can warn us when our analysis is structurally unstable and can be used to suggest what further effects are to be expected. Catastrophe theory can also be used to establish results which are true for a large class of systems, each with a known dynamic; Berry's (1976) work on caustics, which we discussed in Chapter 5, is a good example.

In the middle of the spectrum are the applications to the study of systems whose mechanisms are not known in sufficient detail for us to write down and solve the equations that describe them, but for which we are reasonably confident that we know the sorts of equations that are involved.

One of the most interesting – and difficult – problems in biology is that of understanding development, the process by which a fertilized egg becomes first an embryo and then a fully formed organism. And an important aspect of development is morphogenesis, the creation of the various forms characteristic of the organism and its constituent parts. Of course the problem of form and the succession of forms is encountered in other branches of science as well, but it is in developmental biology that it is especially important. How is it that out of a single cell there can develop an organism which is to such a large extent the same as all others of the same species? How is it that this can happen even though different individuals within a species may be quite different in size and in certain details of their shape? And how is it that the process is so stable, allowing considerable variation in the environment and resisting many (though by no means all) perturbations?

In seeking an answer to these questions, the first step is to try to establish exactly what it is that the developmental process accomplishes. In the language of the earlier chapters of this book, what do we mean when we say that two individuals are ‘of the same form’? It is clearly impossible to give a completely satisfactory definition, but the mathematical relation that most nearly captures the essential idea is that of topological equivalence.