On Growth and Form

Again, just as Sachs shewed that there was a limit to the amount of cytoplasm which could gather round a single nucleus, so Boveri has demonstrated that the nucleus itself has definite limitations of size, and that, in cell-division after fertilisation, each new nucleus has the same size as its parent-nucleus*. In all these cases, then, there are reasons, partly no doubt physiological, but in very large part purely physical, which set limits to the normal magnitude of the organism or of the cell. But as we have already discussed the existence of absolute and definite limitations, of a physical kind, to the possible increase in magnitude of an organism, let us now enquire whether there be not also a lower limit, below which the very existence of an organism is impossible, cr at least where, under changed conditions, its very nature must be profoundly modified.

Among the smallest of known organisms we have, for instance, Micromonas mesnili, Bonel, a flagellate infusorian, which measures about -34y, or -00034 mm., by -00025mm.; smaller even than this we have a pathogenic micrococcus of. the rabbit, M. progrediens, Schréter, the diameter of which is said to be only -00015 mm. or ‘15, or 1°5 x 10-°cem.,—about equal to the thickness of * Boveri, Zellen-studien, V. Ueber die Abhdngigkeit der Kerngrésse und Zellenzahl der Seeigellarven von der Chromosomenzahl der Ausgangszellen. Jena, 1905.

the thinnest gold-leaf; and as small if not smaller still are a few bacteria and their spores. But here we have reached, or all but reached the utmost limits of ordinary microscopic vision; and there remain still smaller organisms, the so-called “ filter-passers,”’ which the ultra-microscope reveals, but which are mainly brought within our ken only by the maladies, such as hydrophobia, foot- and-mouth disease, or the “mosaic” disease of the tobacco-plant, to which these invisible micro-organisms give rise*. Accordingly,

since it is only by the diseases which they occasion that these tiny bodies are made known to us, we might be tempted to suppose that innumerable other invisible organisms, smaller and yet smaller, exist unseen and unrecognised by man. To illustrate some of these small magnitudes I have adapted the preceding diagram from one given by Zsigmondyt. Upon the * Recent important researches suggest that such ultra-minute “ filter-passers” are the true cause of certain acute maladies commonly ascribed to the presence of much larger organisms; cf. Hort, Lakin and Benians, The true infective Agent in Cerebrospinal Fever, etc., J. Roy. Army Med. Corps, Feb. 1916.

+ Zur Erkenntniss der Kolloide, 1905, p. 122; where there will be found an interesting discussion of various molecular and other minute magnitudes, same scale the minute ultramicroscopic particles of colloid gold would be represented by the finest dots which we could make visible to the naked eye upon the paper. A bacillus of ordinary, typical size is, say, |» in length. The length (or height) of a man is about a millon and three-quarter times as great, i.e. 1-75 metres, or 1-75 x 10®w; and the mass of the man is in the neighbourhood of five million, million, million (5 x 1048) times greater than that of the bacillus. If we ask whether there may not exist organisms as much less than the bacillus as the bacillus is less than the dimensions of a man, it is very easy to see that this is quite impossible, for we are rapidly approaching a point where the question of molecular dimensions, and of the ultimate divisibility of matter, begins to call for our attention, and to obtrude itself as a crucial factor in the case.

Clerk Maxwell dealt with this matter in his articie “ Atom*,” and, in somewhat greater detail, Errera discusses the question on the following lnest. The weight of a hydrogen molecule is, according to the physical chemists, somewhere about 8-6 x 2 x 10-* milligrammes; and that of any other element, whose molecular weight is WM, is given by the equation The analysis of ordinary bacteria shews them to consist of about 85 % of water, and 15 % of solids; while the solid residue of vegetable protoplasm contains about one part in a thousand of sulphur. We may assume, therefore, that the living protoplasm contains about

woo X HS = 15 x 10-° parts of sulphur, taking the total weight as = 1. But our little micrococcus, of 0-15 uw in diameter, would, if it were spherical, have a volume of + Sur la limite de petitesse des organismes, Bull. Soc. R. des Sc méd. et nat. de Bruvelles, Jan. 1903; Rec. d awuvres (Physiol. générale), p. 325. and therefore (taking its density as equal to that of water), a weight of PepelorscdOn? = 1654 10c4 mem: But of this total weight, the sulphur represents only ees Tae 105P 2 sx NO-t aga. And if we divide this by the weight of an atom of sulphur, we have 27 x 10-17 = 275 x 10-8 = 10,000, or thereby.

According to this estimate, then, our little Micrococcus progrediens should contain only about 10,000 atoms of sulphur, an element indispensable to its protoplasmic constitution; and it follows that an organism of one-tenth the diameter of our micrococcus would only contain 10 sulphur-atoms, and therefore only ten chemical “molecules” or units of protoplasm! It may be open to doubt whether the presence of sulphur be really essential to the constitution of the proteid or “ protoplasinic ~ molecule; but Errera gives us yet another illustration of a similar kind, which is free from this objection or dubiety. The molecule of albumin, as is generally agreed, can scarcely be less than a thousand times the size of that of such an element as sulphur: according to one particular determination*, serum albumin has a constitution corresponding to a molecular weight of 10,166, and even this may be far short of the true complexity of a typical albuminoid molecule. ‘The weight of such a molecule is

Ss0ee LONG So 10m 3-7 One mem: Now the bacteria contain about 14°, of albuminoids, these constituting by far the greater part of the dry residue; and therefore (from equation (5)), the weight of albumin in our micrococcus is about ae LS x 10-44 2-5) x 10-B mem. If we divide this weight by that which we have arrived at as the weight of an albumin molecule, we have According to the most recent estimates, the weight of the hydrogen molecule is somewhat less than that on which Errera based his calculations, namely about 16 x 10-7? mgms. and according to this value, our micrococcus would contain just about 27,000 albumin molecules. In other words, whichever determination we accept, we see that an organism one-tenth as large as our micrococcus, in linear dimensions, would only contain some thirty molecules of albumin; or, in other words, our micrococcus is only about thirty times as large, in linear dimensions, as a single albumin molecule *.

We must doubtless make large allowances for uncertainty in the assumptions and estimates upon which these calculations are based; and we must also remember that the data with which the physicist provides us in regard to molecular magnitudes are, to a very great extent, maximal values, above which the molecular magnitude (or rather the sphere of the molecule’s range of motion) is not likely to he: but below which there is a greater element of uncertainty as to its possibly greater minuteness. But nevertheless, when we shall have made all reasonable allowances for uncertainty upon the physical side, it will still be clear that the smallest known bodies which are described as organisms draw nigh towards molecular magnitudes, and we must recognise that the subdivision of the organism cannot proceed to an indefinite extent, and in all probability cannot go very much further than it appears to have done in these already discovered forms. For, even.after giving all due regard to the complexity of our unit (that is to say the albumin-molecule), with all the increased possibilities of interrelation with its neighbours which this complexity implies, we cannot but see that physiologically, and comparatively speaking, we have come down to a very simple thing. .

While such considerations as these, based on the chemical composition of the organism, teach us that there must be a definite lower limit to its magnitude, other considerations of a purely physical kind lead us to the same conclusion. For our discussion of the principle of similitude has already taught us that, long before we reach these almost infinitesimal magnitudes, the * McKendrick arrived at a still lower estimate, of about 1250 proteid molecules in the minutest organisms. Brii. Ass. Rep. 1901, p. 808.

diminishing organism will have greatly changed in all its physical relations, and must at length arrive under conditions which must surely be incompatible with anything such as we understand by life, at least in its full and ordinary development and manifestation. We are told, for instance, that the powerful force of surfacetension, or capillarity, begins to act within a range of about 1/500,000 of an inch, or say 0-05. A soap-film, or a film of oil upon water, may be attenuated to far less magnitudes than this; the black spots upon a soap-bubble are known, by various concordant methods of measurement, to be only about 6 x 10-7 cm., or about -006 » thick, and Lord Rayleigh and M. Devaux* have obtained films of oil of -002 4, or even -001 w in thickness.

But while it is possible for a fluid film to exist in these almost molecular dimensions, it is certain that, long before we reach them, there must arise new conditions of which we have little knowledge and which it is not easy even to imagine. It would seem that, in an organism of «1 p in diameter, or even rather more, there can be no essential distinction between the interior and the surface layers. No hollow vesicle, I take it, can exist of these dimensions, or at least, if it be possible for it to do so, the contained gas or fluid must be under pressures of a formidable kind}, and of which we have no knowledge or experience. Nor, I imagine, can there be any real complexity, or heterogeneity, of its fluid or semi-fluid contents; there can be no vacuoles within such a cell, nor any layers defined within its fluid substance, for something of the nature of a boundary-film is the necessary condition of the existence of such layers. Moreover, the whole organism, provided that it be fluid or semi-fluid, can only be spherical in form. What, then, can we attribute, in the way of properties, to an organism of a size as small as, or smaller than, say:05? It must, in all probability, be a homogeneous, structureless sphere, composed of a very small number of albuminoid or other molecules. Its vital properties and functions must be extraordinarily limited ; its specific outward characters, even if we could see it, must be nzl; and its specific properties must be little more than those of an ion-laden corpuscle, enabling it to perform

this or that chemical reaction, or to produce this or that pathogenic effect. Even among inorganic, non-living bodies, there must be a certain grade of minuteness at which the ordinary properties become modified. For instance, while under ordinary circumstances crystallisation starts in a solution about a minute solid fragment or crystal of the salt, Ostwald has shewn that we may have particles so minute that they fail to serve as a nucleus for crystallisation,—which is as much as to say that they are too minute to have the form and properties of a “crystal”; and again, in his thin oil-films, Lord Rayleigh has noted the striking change of physical properties which ensues when the film becomes attenuated to something less than one close-packed layer of molecules *.

Thus, as Clerk Maxwell put it, “molecular science sets us face to face with physiological theories. Jt forbids the physiologist from imagining that structural details of infinitely small dimensions ‘such as Leibniz assumed, one within another, ad infinitum] can furnish an explanation of the infinite variety which exists in the properties and functions of the most minute organisms.’ And for this reason he reprobates, with not undue severity, those advocates of pangenesis and similar theories of heredity, who would place “a whole world of wonders within a body so small and so devoid of visible structure as a germ.” But indeed it scarcely needed Maxwell’s criticism to shew forth the immense physical difficulties of Darwin’s theory of Pangenesis: which, after all, is as old as Democritus, and is no other than that Promethean particulam undique desectam of which we have read, and at which we have smiled, in our Horace.

There are many other ways in which, when we “make a long excursion into space,” we find our ordinary rules of physical behaviour entirety upset. A very familiar case, analysed by Stokes, is that the viscosity of the surrounding medium has a relatively powerful effect upon bodies below a certain size. A droplet of water, a thousandth of an inch (25 w) in diameter, cannot fall in still air quicker than about an inch and a half per second; and as its size decreases, its resistance varies as the diameter, and not (as with larger bodies) as the surface of the

drop. Thus a drop one-tenth of that size (2-5), the size, apparently, of the drops of water in a light cloud, will fall a hundred times slower, or say an inch a minute; and one again a tenth of this diameter (say -25 yu, or about twice as big, in linear dimensions, as our micrococcus), will scarcely fall an inch in two hours. By reason of this principle, not only do the smaller bacteria fall very slowly through the air, but all minute bodies meet with great proportionate resistance to their movements in a fluid. Even such comparatively large organisms as the diatoms and the foraminifera, laden though they are with a heavy shell of flint or lime, seem to be poised in the water of the ocean, and fall in it with exceeding slowness.

The Brownian movement has also to be reckoned with,—that remarkable phenomenon studied nearly a century ago (1827) by Robert Brown, facile princeps botanicorum. It is one more of those fundamental physical phenomena which the biologists have contributed, or helped to contribute, to the science of physics. The quivering motion, accompanied by rotation, and even by translation, manifested by the fine granular particles issuing from a crushed pollen-grain, and which Robert Brown proved to have no vital significance but to be manifested also by all minute particles whatsoever, organic and inorganic, was for many years unexplained. Nearly fifty years after Brown wrote, it was said to be ‘due, either directly to some calorical changes continually taking place in the fluid, or to some obscure chemical action between the solid particles and the fluid which is indirectly promoted by heat*.” Very shortly after these last words were written, it was ascribed by Wiener to molecular action, and we now know that it is indeed due to the impact or bombardment of molecules upon a body so small that these impacts do not for the moment, as it were, “average out” to approximate equality on all sides. The movement becomes manifest with particles of somewhere about 20 in diameter, it is admirably displayed by particles of about 12 in diameter, and becomes more marked the smaller the particles are. The bombardment causes our particles to behave just like molecules of uncommon size, and this

behaviour is manifested in several ways*. Firstly, we have the quivering movement of the particles; secondly, their movement backwards and forwards, in short, straight, disjointed paths; thirdly, the particles rotate, and do so the more rapidly the smaller they are, and by theory, confirmed by observation, it 1s found that particles of 1 in diameter rotate on an average through 100° per second, while particles of 13 ~ in diameter turn through only 14° per minute. Lastly, the very curious result appears, that in a layer of fluid the particles are not equally distributed, nor do they all ever fall, under the influence of gravity, to the bottom. . But just as the molecules of the atmosphere are so distributed, under the influence of gravity, that the density (and therefore the number of molecules per unit volume) falls off in geometrical progression as we ascend to higher and higher layers, so is it with our particles, even within the narrow limits of the little portion of fluid under our microscope. It is only in regard to particles of the simplest form that these phenomena have been theoretically investigated +, and we may take it as certain that more complex particles, such as the twisted body of a Spirillum, would show other and still more complicated manifestations. It is at least clear that, just as the early microscopists in the days before Robert Brown never doubted but that these phenomena were purely vital, so we also may still be apt to confuse, in certain cases, the one phenomenon with the other. We cannot, indeed, without the most careful scrutiny, decide whether the. movements of our minutest organisms are intrinsically “vital”’ (in the sense of being beyond a physical mechanism, or working model) or not.

For example, Schaudinn has suggested that the undulating movements of Spirochaete pallida must be due to the presence of a minute, unseen, “undulating membrane’; and Doflein says of the same species that “sie verharrt oft mit eigenthiimlich zitternden Bewegungen zu einem Orte.” Both movements, the trembling or quivering * The modern literature on the Brownian Movement is very large, owing to the value which the phenomenon is shewn to have in determining the size of the atom. For a fuller, but still elementary account, see J. Cox, Beyond the Atom, 1913, pp. 118-128; and see, further, Perrin, Les Atomes, pp. 119-189.

+ Cf. R. Gans, Wie fallen Stabe und Scheiben in einer reibenden Flissigkeit ? Miinchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown’sche Bewegung nicht kugelférmiger Teilchen, Wiener Ber. 1912, p. 2339. movement described by Doflein, and the undulating or rotating movement described by Schaudinn, are just such as may be easily and naturally interpreted as part and parcel of the Brownian phenomenon. While the Brownian movement may thus simulate in a deceptive way the active movements of an organism, the reverse statement also to a certain extent holds good. One sometimes lies awake of a summer's morning watching the flies as they dance under the ceiling. It is a very remarkable dance. The dancers do not whirl or gyrate, either in company or alone; but they advance and retire; they seem to jostle and rebound; between the rebounds they dart hither or thither in short straight snatches of hurried flight; and turn again sharply in a new rebound at the end of each little rush. Their motions are wholly “erratic,” independent of one another, and devoid of common purpose. This is nothing else than a vastly magnified picture, or simulacrum, of the Brownian movement; the parallel between the two cases lies in their complete irregularity, but this in itself implies a close resemblance. One might see the same thing in a crowded market-place, always provided that the bustling crowd had no business whatsoever. In like manner Lucretius, and Epicurus before him, watched the dust-motes quivering in the beam, and saw in them a mimic representation, rez simulacrum et imago, of the eternal motions of the atoms. Again the same phenomenon may be witnessed under the microscope, in a drop of water swarming with Paramoecia or suchlike Infusoria ; and here the analogy has been put to a numerical test. Following with a pencil the track of each little swimmer, and dotting its place every few seconds (to the beat of a metronome), Karl Przibram found that the mean successive distances from a common base-line obeyed with great .exactitude the “Einstein formula,” that is to say the particular form of the “law of chance” which is applicable to the case of the Brownian movement*.

The phenomenon is (of course) merely analogous, and by no means identical with the Brownian movement; for the range of motion of the little active organisms, whether they be gnats or infusoria, is vastly greater than that of the minute particles which are * Ueber die ungeordnete Bewegung niederer Thiere, Pfliiger’s Archiv, cit, p. 401, 1913. passive under bombardment; but nevertheless Przibram is inclined to think that even his comparatively large infusoria are small enough for the molecular bombardment to be a stimulus, though not the actual cause, of their irregular and interrupted movements.

There is yet another very remarkable phenomenon which may come into play in the case of the minutest of organisms; and this is their relation to the rays of light, as Arrhenius has told us. On the waves of a beam of light, a very minute particle (on vacuo) should be actually caught up, and carried along with an immense velocity; and this “radiant pressure” exercises its most powerful influence on bodies which (if they be of spherical form) are just about -00016 mm., or -16 ~ in diameter. This is just about the size, as we have seen, of some of our smallest known protozoa and bacteria, while we have some reason to believe that others yet unseen, and perhaps the spores of many, are smaller still. Now we have seen that such minute particles fall with extreme slowness in air, even at ordinary atmospheric pressures: our organism measuring -16 py would fall but 83 metres in a year, which is as much as to say that its weight offers practically no impediment to its transference, by the slightest current, to the very highest regions of the atmosphere. Beyond the atmosphere, however, it cannot go, until some new force enable it to resist the attraction of terrestrial gravity, which the viscosity of an atmosphere is no longer at hand to oppose. But it is conceivable that our particle may go yet farther, and actually break loose from the bonds of earth. For in the upper regions of the atmosphere, say fifty miles high, it will come in contact with the rays and flashes of the Northern Lights, which consist (as Arrhenius maintains) of a fine dust, or cloud of vapour-drops, laden with a charge of negative electricity, and projected outwards from the sun.

As soon as our particle acquires a charge of negative electricity it will begin to be repelled by the similarly laden auroral particles, and the amount of charge necessary to enable a particle of given size (such as our little monad of -16 2) to resist the attraction of gravity may be calculated, and is found to be such as the actual conditions can easily supply. Finally, when once set free from the entanglement of the earth’s atmosphere, the particle may be propelled by the “radiant pressure” of hight, with a velocity which will carry it.—like Uriel gliding on a sunbeam,—as far as the orbit of Mars in twenty days, of Jupiter in eighty days, and as far as the nearest fixed star in three thousand years! This, and much more, is Arrhenius’s contribution towards the acceptance of Lord Kelvin’s hypothesis that life may be, and may have been, disseminated across the bounds of space, throughout the solar system and the whole universe !

It may well be that we need attach no great practical importance to this bold conception; for even though stellar space be shewn to be mare liberum to minute material travellers, we may be sure that those which reach a stellar or even a planetary bourne are infinitely, or all but infinitely, few. But whether or no, the remote possibilities of the case serve to illustrate in a very vivid way the profound differences of physical property and potentiality which are associated in the scale of magnitude with simple differences of degree.

When we study magnitude by itself, apart, that is to say, from the gradual changes to which it may be subject, we are dealing with a something which may be adequately represented by a number, or by means of a line of definite length; it 1s what mathematicians call a scalar phenomenon. When we introduce the conception of change of magnitude, of magnitude which varies as we pass from one direction to another in space, or from one instant to another in time, our phenomenon becomes capable of representation by means of a line of which we define both the length and the direction; it is (in this particular aspect) what is called a vector phenomenon.

When we deal with magnitude in oreliniag to the dimensions of space, the vector diagram which we draw plots magnitude in one direction against magnitude in another,—length against height, for instance, or against breadth; and the result is simply what we call a picture or drawing of an object, or (more correctly) a “plane projection” of the object. In other words, what we call Form is a ratio of magnitudes, referred to direction in space. When in dealing with magnitude we refer its variations to successive intervals of time (or when, as it is said, we equate it with time), we are then dealing with the phenomenon of growth; and it is evident, therefore, that this term growth has wide meanings. For growth may obviously be positive or negative; that is to say, a thing may grow larger or smaller, greater or less; and by extension of the primitive concrete signification of the word, we easily and legitimately apply it to non-material things, such as temperature, and say, for instance, that a body “grows” hot or cold. When in a two-dimensional diagram, we represent a magnitude (for instance length) in relation to time (or “plot”

length against time, as the phrase is), we get that kind of vector diagram which is commonly known as a “curve of growth.” We perceive, accordingly, that the phenomenon which we are now studying is a velocity (whose ‘‘ dimensions”’ are a or 7) ; and _this phenomenon we shall speak of, simply, as a rate of growth. In various conventional ways we can convert a two-dimensional into a three-dimensional diagram. We do so, for example, by means of the geometrical method of “perspective” when we represent upon a sheet of paper the length, breadth and depth of an object in three-dimensional space; but we do it more simply, as a rule, by means of “‘contour-lines,” and always when time is one of the dimensions to be represented. If we superimpose upon one another (or even set side by side) pictures, or plane projections, of an organism, drawn at successive intervals of time, we have such a three-dimensional diagram, which is a partial] representation (limited to two dimensions of space) of the organism’s gradual change of form, or course of development; and in such a case our contour-lines may, for the purposes of the embryologist, be separated by intervals representing a few hours or days, or, for the purposes of the palaeontologist, by interspaces of unnumbered and innumerable years*.

Such a diagram represents in two of its three dimensions form, and in two, or three, of its dimensions growth; and so we see how intimately the two conceptions are correlated or inter-related to one another. In short, it is obvious that the form of an animal is determined by its specific rate of growth in various directions ; accordingly, the phenomenon of rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form, and, mathematically speaking, organic form itself appears to us as a function of time.

* Sometimes we find one and the same diagram suffice, whether the intervals of time be great or small; and we then invoke “Wolff's Law,” and assert that the life-history of the individual repeats, or recapitulates, the history of the race. + Our subject is one of Bacon’s “Instances of the Course,” or studies wherein we “measure Nature by periods of Time.” In Bacon’s Catalogue of Particular Histories, one of the odd hundred histories or investigations which he foreshadowed is precisely that which we are engaged on, viz. a ‘‘ History of the Growth and_ Increase