On Growth and Form

_ pressure is of the magnitude of thousands of atmospheres,—a conclusion which is supported by other physical considerations. The contraction of a liquid surface and other phenomena of surface tension involve the doing of work, and the power to do work is what we call energy. It is obvious, in such a simple case as we have just considered, that the whole energy of the system is diffused throughout its molecules; but of this whole stock of energy it is only that part which comes into play at or very near to the surface which normally manifests itself in work, and hence we may speak (though the term is open to some objections) of a specific surface energy. The consideration of surface energy, and of the manner in which its amount is increased and multiplied by the multiplication of surfaces due to the subdivision of the organism into cells, is of the highest importance to the physiologist ; and even the morphologist cannot wholly pass it by, if he desires to study the form of the cell in its relation to the phenomena of surface tension or “‘capillarity.” The case has been set forth with the utmost possible lucidity by Tait and by Clerk Maxwell, on whose teaching the following paragraphs are based: they having based their teaching upon that of Gauss,—who rested on Laplace.

Let # be the whole potential energy of a mass M of liquid; let e, be the energy per unit mass of the interior liquid (we may call it the internal energy); and let e be the energy per unit mass for a layer of the skin, of surface S, of thickness t, and density p (e being what we call the surface energy). It is obvious that the total energy consists of the internal plus the surface energy, and that the former is distributed through the whole mass, minus its surface layers. That is to say, in mathematical language,

and this is as much as to say that the total energy of the system may be taken to consist of two portions, one uniform throughout the whole mass, and another, which is proportional on the one hand to the amount of surface, and on the other hand is proportional to the difference between e and ey, that is to say to the difference between the unit values of the internal and the surface energy. It was Gauss who first shewed after this fashion how, from the mutual attractions between all the particles, we are led to an expression which is what we now call the potential energy of the system; and we know, as a fundamental theorem of dynamics, that the potential energy of the system tends to a minimum, and in that minimum finds, as a matter of course, its stable equilibrium.

We see in our last equation that the term Me, is irreducible, save by a reduction of the mass itself. But the other term may be diminished (1) by a reduction in the area of surface, S, or (2) by a tendency towards equality of e and é9, that is to say by a diminution of the specific surface energy, e. These then are the two methods by which the energy of the system will manifest itself in work. The one, which is much the more important for our purposes, leads always to a diminution of surface, to the so-called “principle of minimal areas”; the other, which leads to the lowering (under certain circumstances) of surface tension, is the basis of the theory of Adsorption, to which we shall have some occasion to refer as the modus operandi in the development of a cell-wall, and in a variety of other histological phenomena. In the technical phraseology of the day, the ‘capacity factor” is involved in the one case, and the “intensity factor”? in the other.

Inasmuch as we are concerned with the form of the cell it is the former which becomes our main postulate: telling us that the energy equations of the surface of a cell, or of the free surfaces of cells partly in contact, or of the partition-surfaces of cells in contact with one another or with an adjacent solid, all indicate a minimum of potential energy in the system, by which the system is brought, ipso facto, into equilibrium. And we shall not fail to observe, with something more than mere historical interest and curiosity, how deeply and intrinsically there enter into this whole class of problems the “principle of least action” of Maupertuis, the “lineae curvae maximi minimive proprietate gaudentes” of Euler, by which principles these old natural philosophers explained correctly a multitude of phenomena, and drew the lines whereon the foundations of great part of modern physics are well and truly laid.

In all cases where the principle of maxima and minima comes into play, as it conspicuously does in the systems of liquid films which are governed by the laws of surface-tension, the figures and conformations produced are characterised by obvious and remarkable symmetry. Such symmetry is in a high degree characteristic of organic forms, and is rarely absent in living things,—save in such cases as amoeba, where the equilibrium on which symmetry depends is likewise lacking. And if we ask what physical equilibrium has to do with formal symmetry and regularity, the reason is not far to seek; nor can it be put_better than in the following words of Mach’s*. “In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. In each deformation positive and negative work is done. One condition, therefore,

‘though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus supplied by symmetry. Regularity is successive symmetry. There is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular.” As we proceed in our enquiry, and especially when we approach the subject of t¢sswes, or agglomerations of cells, we shall have from time to time to call in the help of elementary mathematics. But already, with very little mathematical help, we find ourselves in a position to deal with some simple examples of organic forms.

When we melt a stick of sealing-wax in the flame, surface tension (which was ineffectively present in the solid but finds play in the now fluid mass), rounds off its sharp edges into curves, so striving towards a surface of minimal area; and in like manner, by melting the tip of a thin rod of glass, Leeuwenhoek made the little spherical beads which served him for a microscopet. When any drop of protoplasm, either over all its surface or at some free end, as at the extremity of the pseudopodium of an amoeba, is

+ Similarly, Sir David Brewster and others made powerful lenses by simply dropping small drops of Canada balsam, castor oil, or other strongly refractive liquids, on to a glass plate: On New Philosophical Instruments (Description of a new Fluid Microscope), Edinburgh, 1813, p. 413. seen likewise to “round itself off,” that is not an effect of “vital contractility,’ but (as Hofmeister shewed so long ago as 1867) ~ a simple consequence of surface tension; and almost immediately afterwards Engelmann* argued on the same lines, that the forces which cause the contraction of protoplasm in general may “be just the same as those which tend to make every non-spherical drop of fluid become spherical!”” We are not concerned here with the many theories and speculations which would connect the phenomena of surface tension with contractility, muscular movement or other special physiological functions, but we find ample room to trace the operation of the same cause in producing, under conditions of rest and equilibrium, certain definite and inevitable forms of surface.

It is however of great importance to observe that the living cell is one of those cases where the phenomena of surface tension are by no means limited to the owter surface; for within the heterogeneous substance of the cell, between the protoplasm and its nuclear and other contents, and in the alveolar network of the cytoplasm itself (so far as that “alveolar structure” is actually present in life), we have a multitude of interior surfaces; and, especially among plants, we may have a large, inner surface of “interfacial”? contact, where the protoplasm contains cavities or “vacuoles” filled with a different and more fluid material, the “cell-sap.” Here we have a great field for the development of surface tension phenomena: and so long ago as 1865, Nageli and Schwendener shewed that the streaming currents of plant cells might be very plausibly explained by this phenomenon. Even ten years earlier, Weber had remarked upon the resemblance between these protoplasmic streamings and the streamings to be observed in certain inanimate drops, for which no cause but surface tension could be assigned f.

The case of amoeba, though it is an elementary case, is at the same time a complicated one. While it remains “amoeboid,” it is never at rest or in equilibrium; it is always moving, from one to another of its protean changes of configuration; its surface tension is constantly varying from point to point. Where the surface tension is greater, that portion of the surface will contract into spherical or spheroidal forms; where it is less the surface will correspondingly extend. While generally speaking the surface energy has a minimal value, it is not necessarily constant. It may be diminished by a rise of temperature; it may be altered by contact with adjacent substances*, by the transport of constituent materials from the interior to the surface, or again by actual chemical and fermentative change. Within the cell, the surface energies developed about its heterogeneous contents will constantly vary as these contents are affected by chemical metabolism. As the colloid materials are broken down and as the particles in suspension are diminished in size the “free surface energy” will be increased, but the osmotic energy will be diminished f. Thus arise the various fluctuations of surface tension and the various phenomena of amoeboid form and motion, which Biitschli and others have reproduced or imitated by means of the fine emulsions which constitute their “artificial amoebae.” A multitude of experiments shew how extraordinarily delicate is the adjustment of the surface tension forces, and how sensitive they are to the least change of temperature or chemical state. Thus, on a plate which we have warmed at one side, a drop of alcohol runs towards the warm area, a drop of oil away from it; and a drop of water on the glass plate exhibits lively movements when

* Hayceraft and Carlier pointed out (Proc. R.S.H. xv, pp. 220-224, 1888) that the amoeboid movements of a white blood-corpuscle are only manifested when the corpuscle is in contact with some solid substance: while floating freely in the plasma or serum of the blood, these corpuscles are spherical, that is to say they are at rest and in equilibrium. The same fact has recently been recorded anew by Ledingham (On Phagocytosis from an adsorptive point of view, Journ. of Hygiene, xu, p. 324, 1912). On the emission of pseudopodia as brought about by changes in surface tension, see also (int. al.) Jensen, Ueber den Geotropismus niederer Organismen, Pfliiger’s Archiv, Lim, 1893. Jensen remarks that in Orbitolites, the pseudopodia issuing through the pores of the shell first float freely, then as they grow longer bend over till they touch the ground, whereupon they begin to display amoeboid and streaming motions. Verworn indicates (Allg. Physiol. 1895, p. 429), and Davenport says (Hxperim. Morphology, 11, p. 376) that “this persistent clinging to the substratum is a ‘thigmotropic’ reaction, and one which belongs clearly to the category of ‘response.’”’ (Cf. Piitter, Thigmotaxis bei Protisten, A. f. Physiol. 1900, Suppl. p. 247.) But it is not clear to my mind that to account for this simple phenomenon we need invoke other factors than gravity and surface-action.

+ Cf. Pauli, Allgemeine physikalische Chemie d. Zellen u. Gewebe, in Asher-Spiro’s Ergebnisse der Physiologie, 1912; Przibram, Vitalitdt, 1913, p. 6. we bring into its neighbourhood a heated wire, or a glass rod dipped in ether. When we find that a plasmodium of Aethalium, for instance, creeps towards a damp spot, or towards a warm spot, or towards substances that happen to be nutritious, and again creeps away from solutions of sugar or of salt, we seem to be dealing with phenomena every one of which can be paralleled by ordinary phenomena of surface tension*. Even the soap-bubble itself is imperfectly in equilibrium, for the reason that its film, like the protoplasm of amoeba or Aethalium, is an excessively heterogeneous substance. Its surface tensions vary from point to poimt, and chemical changes and changes of temperature increase and magnify the variation. The whole surface of the bubble is in constant movement as the concentrated portions of the soapy fluid make their way outwards from the deeper layers; it thins and it thickens, its colours change, currents are set up in it, and little bubbles glide over it; it continues in this state of constant movement, as its parts strive one with another in all their interactions towards equilibrium fF.

In the case of the naked protoplasmic cell, as the amoeboid phase is emphatically a phase of freedom and activity, of chemical and physiological change, so, on the other hand, is the spherical form indicative of a phase of rest or comparative inactivity. In the one phase we see unequal surface tensions manifested in the creeping movements of the amoeboid body, in the rounding off of the ends of the pseudopodia, in the flowing out of its substance over a particle of “food,” and in the current-motions in the interior of its mass; till finally, in the other phase, when internal homogeneity and equilibrium have been attained and the potential

* The surface-tension theory of protoplasmic movement has been denied by many. Cf. (e.g.), Jennings, H. §., Contributions to the Study of the Behaviour of the Lower Organisms, Carnegie Inst. 1904, pp. 130-230; Dellinger, O. P., Locomotion of Amoebae, ete. Journ. Exp. Zool. m1, pp. 337-357, 1906; also various papers by Max Heidenhain, in Anatom. Hefte (Merkel und Bonnet), ete. + These various movements of a liquid surface, and other still more striking movements such as those of a piece of camphor floating on water, were at one time ascribed by certain physicists to a peculiar force, swi generis, the force épipolique of Dutrochet: until van der Mensbrugghe shewed that differences of surface tension were enough to account for this whole series of phenomena (Sur la tension superficielle des liquides considérée au. point de vue de certains mouvements observés a leur surface, Mém. Cour. Acad. de Belgique, xxxtv, 1869; cf. Plateau, p. 283).

energy of the system is for the time being at a minimum, the cell assumes a rounded or spherical form, passing into a state of “rest,” and (for a reason which we shall presently see) becoming at the same time “encysted.”’ In a budding yeast-cell (Fig. 60), we see a more definite and restricted change of surface tension. When a “bud” appears, whether with or without actual growth by osmosis or otherwise of the mass, it does so because at a certain part of the cell-surface the surface tension has more or less suddenly diminished, and the area of that portion expands accordingly; but in turn the surface tension of the expanded area will \:° 3) make itself felt, and the bud will be rounded off Fig. 60. into a more or less spherical form.

The yeast-cell with its bud is a simple example of a principle which we shall find to be very important. Our whole treatment of cell-form in relation to surface-tension depends on the fact (which Errera was the first to point out, or to give clear expression to) that the incipient cell-wall retains with but little impairment the properties of a liquid film*, and that the growing cell, in spite of the membrane by which it has already begun to be surrounded, behaves very much like a fluid drop. But even the ordinary yeast-cell shows, by its ovoid and non-spherical form, that it has acquired its shape under the influence of some force other than that uniform and symmetrical surface-tension which would be productive of a sphere; and this or any other asymmetrical form, once acquired, may be retained by virtue of the solidification and consequent rigidity of the membranous wall of the cell. Unless such rigidity ensue, it is plain that such a conformation as that of the cell with its attached bud could not be long retained, amidst the constantly varying conditions, as a figure of even partial equilibrium. But as a matter of fact, the cell in this case is not in equilibrium at all; it is in process of budding, and is slowly altering its shape by rounding off the bud. It is plain that over its surface the surface-energies are unequally distributed, owing to some heterogeneity of the substance; and to this matter we shall afterwards return. In like manner the developing egg

through all its successive phases of form is never in complete equilibrium; but is merely responding to constantly changing conditions, by phases of partial, transitory, unstable and conditional equilibrium. It is obvious that there are innumerable solitary plant-cells, and unicellular organisms in general, which, like the yeast-cell, do not correspond to any of the simple forms that may be generated under the influence of simple and homogeneous surface-tension ; and in many cases these forms, which we should expect to be unstable and transitory, have become fixed and stable by reason of the comparatively sudden or rapid solidification of the envelope. This is the case, for instance, in many of the more complicated forms of diatoms or of desmids, where we are dealing, in a less striking but even more curious way than in the budding yeast-cell, not with one simple act of formation, but with a complicated result uf successive stages of localised growth, interrupted by phases of partial consolidation. The original cell has acquired or assumed a certain form, and then, under altering conditions and new distributions of energy, has thickened here or weakened there, and has grown out or tended (as it were) to branch, at particular points. We can often, or indeed generally, trace in each particular stage of growth or at each particular temporary growing point, the laws of surface tension manifesting themselves in what is for the time being a fluid surface; nay more, even in the adult and completed structure; we have little difficulty in tracing and recognising (for instance in the outline of such a desmid as Euastrum) the rounded lobes that have successively grown or flowed out from the original rounded and flattened cell. What we see in a many chambered foraminifer, such as Globigerina or Rotalia, is just the same thing, save that it is carried out in greater completeness and perfection.

The little organism as a whole is not a figure of equilibrium or of minimal area; but each new bud or separate chamber is such a figure, conditioned by the forces of surface tension, and superposed upon the complex aggregate of similar bubbles after these latter have become consolidated one by one into a rigid system. which, under the influence of surface tension, a surface can possibly assume. In doing so, we are obviously limited to conditions under which other forces are relatively unimportant, that is to say where the “surface energy” is a considerable fraction of the whole energy of the system; and this in general will be the case when we are dealing with portions of liquid so small that their dimensions come within what we have called the molecular range, or, more generally, in which the “specific surface” is large*: in other words it will be small or minute organisms, or the small cellular elements of larger organisms, whose forms will be governed by surface-tension; while the general forms of the larger organisms will be due to other and non-molecular forces. For instance, a large surface of water sets itself level because here gravity is predominant; but the surface of water in a narrow tube is manifestly curved, for the reason that we are here dealing with particles which are mutually within the range of each other’s molecular forces. The same is the case with the cell-surfaces and cell-partitions which we are presently to study, and the effect of gravity will be especially counteracted and concealed when, as in the case of protoplasm in a watery fluid, the object is immersed in a liquid of nearly its own specific gravity.

We have already learned, as a fundamental law of surfacetension phenomena, that a liquid film in equilibrium assumes a form which gives it a minimal area under the conditions to which it is subject. And these conditions include (1) the form of the boundary, if such exist, and (2) the pressure, if any, to which the film is subject; which pressure is closely related to the volume, of air or of liquid, which the film (if it be a closed one) may have to contain. In the simplest of cases, when we take up a soapfilm on a plane wire ring, the film is exposed to equal atmospheric pressure on both sides, and it obviously has its minimal area in the form of a plane. So long as our wire ring lies in one plane (however irregular in outline), the film stretched across it will still be in a plane; but if we bend the ring so that it lies no longer in a plane, then our film will become curved into a surface which may be extremely complicated, but is still the smallest possible

surface which can be drawn continuously across the uneven boundary. The question of pressure involves not only external pressures acting on the film, but also that. which the film itself is capable of exerting. For we have seen that the film is always contracting to its smallest limits; and when the film is curved, this obviously leads to a pressure directed inwards,—perpendicular, that is to say, to the surface of the film. In the case of the soap-bubble, the uniform contraction of whose surface has led to its spherical form, this pressure is balanced by the pressure of the air within; and if an outlet be given for this air, then the bubble contracts with perceptible force until it stretches across the mouth of the tube, for instance the mouth of the pipe through which we have blown the bubble. A precisely similar pressure, directed inwards, is exercised by the surface layer of a drop of water or a globule of mercury, or by the surface pellicle on a portion or “drop” of protoplasm. Only we must always remember that in the soapbubble, or the bubble which a glass-blower blows, there is a twofold pressure as compared with that which the surface-film exercises on the drop of liquid of which it is a part; for the bubble consists (unless it be so thin as to consist of a mere layer of molecules*) of a liquid layer, with a free surface within and another without, and each of these two surfaces exercises its own independent and coequal tension, and corresponding pressure f.

If we stretch a tape upon a flat table, whatever be the tension of the tape it obviously exercises no pressure upon the table below. But if we stretch it over a curved surface, a cylinder for instance, it does exercise a downward pressure; and the more curved the surface the greater is this pressure, that is to say the greater is this share of the entire force of tension which is resolved in the downward direction. In mathematical language, the pressure (p) varies directly as the tension (7), and inversely as the radius of curvature (R): that is to say, p = T/R, per unit of surface.

* Or, more strictly speaking, unless its thickness be less than twice the range of the molecular forces. + It follows that the tension, depending only on the surface-conditions, is independent of the thickness of the film. If instead of a cylinder, which is curved only in one direction, we take a case where there are curvatures in two dimensions (as for instance a sphere), then the effects of these must be simply added to one another, and the resulting pressure p is equal to T/R+T/R' or p=T (1/R + 1/R’)*.

And if in addition to the pressure p, which is due to surface tension, we have to take into account other pressures, p’, p’’, etc., which are due to gravity or other forces, then we may say that the total pressure, P= p' + p’ + T (1/R+1/R’). While in some cases, for instance in speaking of the shape of a bird’s egg, we shall have to take account of these extraneous pressures, in the present part of our subject we shall for the most part be able to neglect them.

Our equation is an equation of equilibrium. The resistance to compression,—the pressure outwards,—of our fluid mass, is a constant quantity (P); the pressure inwards, T (1/R + 1/R’), is also constant; and if (unlike the case of the mobile amoeba) the surface be homogeneous, so that 7 is everywhere equal, it follows that throughout the whole surface 1/R + 1/R’ = C (a constant). Now equilibrium is attained after the surface contraction has done its utmost, that is to say when it has reduced the surface to the smallest possible area; and so we arrive, from the physical side, at the conclusion that a surface such that 1/R + 1/R’ = C, in other words a surface which has the same mean curvature at all points, is equivalent to a surface of minimal area: and to the same conclusion we may also arrive through purely analytical mathematics. It is obvious that the plane and the sphere are two examples of such surfaces, for in both cases the radius of curvature is everywhere constant, being equal to infinity in the case of the plane, and to some definite magnitude in the case of the sphere.

From the fact that we may extend a soap-film across a ring of wire however fantastically the latter may be bent, we realise that there is no limit to the number of surfaces of minimal area which may be constructed or may be imagined; and while some of these are very complicated indeed, some, for instance a spiral helicoid screw, are relatively very simple. But if we limit ourselves to * This simple but immensely important formula is due to Laplace (Mécanique Céleste, Bk x. suppl. Théorie de Vaction capillaire, 1806).

surfaces of revolution (that is to say, to surfaces symmetrical about an axis), we find, as Plateau was the first to shew, that those which meet the case are very few in number. They are six in all, namely the plane, the sphere, the cylinder, the catenoid, the unduloid, and a curious surface which Plateau called the nodoid. These several surfaces are all closely related, and the passage from one to another is generally easy. Their mathematical interrelation is expressed by the fact (first shewn by Delaunay*, in 1841) that the plane curves by whose rotation they are generated are themselves generated as “roulettes” of the conic sections.

Let us imagine a straight line upon which a circle, an ellipse or other conic section rolls; the focus of the conic section will describe a line in some relation to the fixed axis, and this line (or roulette), rotating around the axis, will describe in space one or other of the six surfaces of revolution with which we are dealing. If we imagine an ellipse so to roll over a line, either of its foci’ will describe a sinuous or wavy line (Fig. 618) at a distance

alternately maximal and minimal from the axis; and this wavy line, by rotation about the axis, becomes the meridional line of the surface which we call the wnduloid. The more unequal the two axes are of our ellipse, the more pronounced will be the sinuosity of the described roulette. If the two axes be equal, then our ellipse becomes a circle, and the path described by its rolling centre is a straight line parallel to the axis (A); and obviously the solid of revolution generated therefrom will be a cylinder. Tf one axis of our ellipse vanish, while the other remain of finite length, then the ellipse is reduced to a straight line, and its roulette will appear as a succession of semicircles touching one another upon the axis (C); the solid of revolution will be a series of equal spheres. If as before one axis of the ellipse vanish, but the other be infinitely long, then the curve described by the rotation

* Sur la surface de révolution dont la courbure moyenne est constante, Journ. de M. Liouville, v1, p. 309, 1841. of this latter will be a circle of infinite radius, i.e. a straight line infinitely distant from the axis; and the surface of rotation is now a plane. If we imagine one focus of our ellipse to remain at a given distance from the axis, but the other to become infinitely remote, that is tantamount to saying that the ellipse becomes transformed into a parabola; and by the rolling of this curve along the axis there is described a catenary (D), whose solid of revolution is the catenoid.