On Growth and Form

transverse direction will be very difficult, while flexure in a longitudinal direction will be comparatively easy; just as, in the case of a long narrow strip of iron, we may easily bend it into folds running transversely to its long axis, but not the other way. The manner in which our little Lagena-shell tends to fold or wrinkle, longitudinally in its wider part, and transversely or annularly in its narrow neck, is thus completely and easily explained.

An identical phenomenon is apt to occur in the little flaskshaped gonangia, or reproductive capsules, of some of the hydroid zoophytes. In the annexed drawings of these gonangia in two species of Campanularia, we see that in one case the little vesicle has the flask-shaped or unduloid configuration of a Lagena; and here the walls of the flask are longitudinally fluted, just after the manner we have witnessed in the latter genus. But in the other Campanularian the vesicles are long, narrow and tubular, and here a transverse folding or pleating takes the place of the longitudinally fluted pattern. And the very form of the folds or pleats is enough to suggest that we are not dealing here with a simple phenomenon of surface-tension, but with a condition in which surface-tension and stiffness are both present, and play their parts in the resultant form. . Passing from the solitary flask-shaped cell of Lagena, we have, in another series of forms, a constricted cylinder, or succession

Fig. 89. Various Foraminifera (after Brady). a, Nodosaria simplex; 6, N. pygmaea; c, N. costulata; e, N. hispida; f, N. elata; d, Rheophax (Lituola) distans; g, Sagrina virgata. of unduloids; such as are represented in Fig. 89, illustrating certain species of Nodosaria, Rheophax and Sagrina. In some of these cases, and certainly in that of the arenaceous genus Rheophax, we have to do with the ordinary phenomenon of a segmenting or partially segmenting cylinder. But in others, the structure is not developed out of a continuous protoplasmic cylinder, but as we can see by examining the interior of the shell, it has been formed in successive stages, beginning with a simple unduloid “Lagena,” about whose neck, after its solidification, another drop of protoplasm accumulated, and in turn assumed the unduloid, or lagenoid, form. The chains of interconnected bubbles which

Morey and Draper made many years ago of melted resin are a very similar if not identical phenomenon*. There now remain for our consideration, among the Protozoa, the great oceanic group of the Radiolaria, and the little group of their freshwater allies, the Heliozoa. In nearly all these forms we have this specific chemical difference from the Foraminifera, that when they secrete, as they generally do secrete, a hard skeleton, it is composed of silica instead of lime. These organisms and the various beautiful and highly complicated skeletal fabrics which they develop give us many interesting illustrations of physical phenomena, among which the manifestations of surface-tension are very prominent. But the chief phenomena connected with their skeletons we shall deal with in another place, under the head of spicular concretions.

In a simple and typical Heliozoan, such as the Sun-animaleule, Actinophrys sol, we have a “drop” of protoplasm, contracted by its surface tension into a spherical form. Within the heterogeneous protoplasmic mass are more fluid portions, and at the surface which separates these from the surrounding protoplasm a similar surface tension causes them also to assume the form of spherical “vacuoles,” which in reality are little clear drops within the big one; unless indeed they become numerous and closely packed, in which case, instead of isolated spheres or droplets they will constitute a “froth,” their mutual pressures and tensions giving rise to regular configurations such as we shall study in the next chapter. One or more of such clear spaces may be what is called a “contractile vacuole”: that is to say, a droplet whose surface tension is in unstable equilibrium and is apt to vanish altogether, so that the definite outline of the vacuole suddenly disappears f. Again, within the protoplasm are one or more nuclei, whose own surface tension (at the surface between the nucleus and the surrounding protoplasm), has drawn them in turn into the shape

+ The presence or absence of the contractile vacuole or vacuoles is one of the chief distinctions, in systematic zoology, between the Heliozoa and the Radiolaria. As we have seen on p. 165 (footnote), it is probably no more than a physical consequence of the different conditions of existence in fresh water and in salt. of spheres. Outwards through the protoplasm, and stretching far beyond the spherical surface of the cell, there run stiff linear threads of modified or differentiated protoplasm, replaced or reinforced in some cases by delicate siliceous needles. In either case we know little or nothing about the forces which lead to their production, and we do not hide our ignorance when we ascribe their development to a “radial polarisation” of the cell. In the case of the protoplasmic filament, we may (if we seek for a hypothesis), suppose that it is somehow comparable to a viscid stream, or “liquid vein,” thrust or squirted out from the body of the cell. But when it is once formed, this long and comparatively rigid filament is separated by a distinct surface from the neighbouring protoplasm, that is to say from the more fluid surfaceprotoplasm of the cell; and the latter begins to creep up the filament, just as water would creep up the interior of a glass tube, or the sides of a glass rod immersed in the liquid. It is the simple case of a balance between three separate tensions: (1) that between the filament and the adjacent protoplasm, (2) that between the filament and the adjacent water, and (3) that between the water and the protoplasm. Calling these tensions respectively T;,, Ty», and T',,,, equilibrium will be attained when the angle of contact between the fluid protoplasm and the filament is such that

fp a very small one. The precise form of the curve is somewhat different from that which, under ordinary circumstances, is assumed by a liquid which creeps up a solid surface, as water in contact with air creeps up a surface of glass; the difference being due to the fact that here, owing to the density of the protoplasm being practically identical with that of the surrounding medium, the whole system is practically immune from gravity. Under normal circumstances the curve is part of the “elastic curve”’ by which that surface of revolution is generated which we have called, after Plateau, the nodoid; but in the present case it is apparently a catenary. Whatever curve it be, it obviously forms a surface of revolution around the filament.

Since the attraction exercised by this surface tension is symmetrical around the filament, the latter will be pulled equally in all directions; in other words it will tend to be set normally to the surface of the sphere, that is to say radiating directly outwards from the centre. If the distance between two adjacent filaments be considerable, the curve will simply meet the filament at the angle a already referred to; but if they be sufficiently near together, we shall have a continuous catenary curve forming a hanging loop between one filament and the other. And when this is so, and the radial filaments are more or less symmetrically interspaced, we may have a beautiful system of honeycomb-like depressions over the surface of the organism, each cell of the honeycomb having a strictly defined geometric configuration. In the simpler Radiolaria, the spherical form of the entire organism is equally well-marked; and here, as also in the more complicated Heliozoa (such as Actinosphaerium), the organism is

Fig. 90. A, Trypanosoma tineae (after Minchin); B, Spirochaeta anodontae (after Fantham). differentiated into several distinct layers, each boundary surface tending to be spherical, and so constituting sphere within sphere. One of these layers at least is close packed with vacuoles, forming an “alveolar meshwork,” with the configurations of which we shall attempt in another chapter to correlate the characteristic structure of certain complex types of skeleton.

An exceptional form of cell, but a beautiful manifestation of surface-tension (or so I take it to be), occurs in Trypanosomes, those tiny parasites of the ‘blood that are associated with sleepingsickness and many other grave or dire maladies. These tiny organisms consist of elongated solitary cells down one side of which runs a very delicate frill, or “undulating membrane,” the free edge of which is seen to be slightly thickened, and the whole of

which undergoes rhythmical and beautiful wavy movements. When certain Trypanosomes are artificially cultivated (for instance ° T. rotatorvum, from the blood of the frog), phases of growth are witnessed in which the organism has no undulating membrane, but possesses a long cilium or “flagellum,” springing from near the front end, and exceeding the whole body in length*. Again, in T. lewisw, when it reproduces by “multiple fission,” the products of this division are likewise devoid of an undulating membrane, but are provided with a long free flagellum}. It is

Fig. 91. A, Trichomonas muris, Hartmann; B, Trichomastix serpentis, Dobell; C, Trichomonas. angusta, Alexeieff. (After Kofoid.) a plausible assumption to suppose that, as the flagellum waves about, it comes to lie near and parallel to the body of the cell, and that the frill or undulating membrane is formed by the clear, fluid protoplasm of the surface layer springing up in a film to run up and along the flagellum, just as a soap-film would be formed in similar circumstances.

This mode of formation of the undulating membrane or frill appears to be confirmed by the appearances shewn in Fig. 91. Here we have three little organisms closely allied to the ordinary Trypanosomes, of which one, Trichomastix (B), possesses four flagella, and the other two, Trichomonas, apparently three only: the two latter possess the frill, which is lacking in the first*. But it is impossible to doubt that when the frill is present (as in A and C), its outer edge is constituted by the apparently missing flagellum (a), which has become attached to the body of the creature at the point c, near its posterior end; and all along its course, the superficial protoplasm has been drawn out into a film, between the flagellum (a) and the adjacent surface or edge of the body (0). Moreover, this mode of formation has been actually witnessed and described, though in a somewhat exceptional case. The little flagellate monad Herpetomonas is normally destitute of an undulating membrane, but possesses a single long terminal flagellum. According to Dr D. L. Mackinnon, the cytoplasm in a certain stage of growth becomes somewhat “‘sticky,’ a phrase which we may in all probability interpret to mean that its surface tension is being reduced. For this stickiness is shewn in two ways. In the first place, the long body, in the course of its various bending movements, is apt to adhere head to tail (so to speak), giving a rounded or sometimes annular form to the organism, such as has also been described in certain species or stages of Trypanosomes. But again, the long flagellum, if it get bent backwards upon the body, tends to adhere to its surface. ‘‘ Where the flagellum was pretty long and active, its efforts to continue movement under these Pale dndanioe tou aneeed abnormal conditions resulted in the Trypanosome. (After D. L. gradual lifting up from the cytoplasm Mackinnon.) of the body of a sort of pseudoundulating membrane (Fig. 92).

The movements of this structure were so exactly those of a true undulating membrane that it was difficult to believe one was not dealing with a small, blunt trypanosome*.” This in short is a precise description of the mode of development which, from theoretical considerations alone, we should conceive to be the natural if not the only possible way in which the undulating membrane could come into existence. There is a genus closely allied to Trypanosoma, viz. Trypanoplasma, which possesses one free flagellum, together with an undulating membrane; and it resembles the neighbouring genus Bodo, save that the latter has two flagella and no undulating membrane. In like manner, Trypanosoma so closely resembles Herpetomonas that, when individuals ascribed to the former genus exhibit a free flagellum only, they are said to be in the “Herpetomonas stage.” In short all through the order, we have pairs of genera, which are presumed to be separate and distinct, viz. Trypanosoma-Herpetomonas, Trypanoplasma-Bodo, Trichomastix- Trichomonas, in which one differs from the other mainly if not

solely in the fact that a free flagellum in the one is replaced by an undulating membrane in the other. We can scarcely doubt that the two structures are essentially one and the same. ‘ The undulating membrane of a Trypanosome, then, according to our interpretation of it, is a liquid film and must obey the law of constant mean curvature. It is under curious limitations of freedom: for by one border it is attached to the comparatively motionless body, while its free border is constituted by a flagellum which retains its activity and is bemg constantly thrown, like the lash of a whip, nto wavy curves. It follows that the membrane, for every alteration of its longitudinal curvature, must at the same instant become curved in a direction perpendicular thereto; it bends, not as a tape bends, but with the accompaniment of beautiful ‘but tiny waves of double curvature, all tending towards the establishment of an “equipotential surface” ; and its characteristic undulations are not originated by an active mobility of the membrane but are due to the molecular tensions which produce the very same result in a soap-film under similar circumstances.

In certain Spirochaetes, S. anodontae (Fig. 90) and S. balbiani (which we find in oysters), a very similar undulating membrane exists, but it is coiled in a regular spiral round the body of the cell. It forms a “screw-surface,” or helicoid, and, though we might think that nothing could well be more curved, yet its mathematical properties are such that it constitutes a ‘‘ruled surface” whose “mean curvature” is everywhere nil; and this property (as we have seen) it shares with the plane, and with the plane alone. Precisely such a surface, and of exquisite beauty, may be produced by bending a wire upon itself so that part forms an axial rod and part a spiral wrapping round the axis, and then dipping the whole into a soapy solution.

These undulating and helicoid surfaces are exactly reproduced among certain forms of spermatozoa. The tail of a spermatozoon consists normally of an axis surrounded by clearer and more fluid protoplasm, and the axis sometimes splits up into two or more slender filaments. To surface tension operating between these and the surface of the fluid protoplasm (just as in the case of the flagellum of the Trypanosome), I ascribe the formation of the undulating membrane which we find, for instance, in the spermatozoa of the newt or salamander; and of the helicoid membrane, wrapped in a far closer and more beautiful spiral than that which we saw in Spirochaeta, which is characteristic of the spermatozoa of many birds.

Before we pass from the subject of the conformation of the solitary cell we must take some account of certain other exceptional forms, less easy of explanation, and still less perfectly understood. Such is the case, for instance, with the red blood-corpuscles of man and other vertebrates; and among the sperm-cells of the decapod crustacea we find forms still more aberrant and not less perplexing. These are among the comparatively few cells or cell-like structures. whose form seems to be incapable of explanation by theories of surface-tension.

In all the mammalia (save a very few) the red blood-corpuscles are flattened circular discs, dimpled in upon their two opposite sides. This configuration closely resembles that of an indiarubber ball when we pinch it tightly between finger and thumb; and we may also compare it with that experiment of Plateau’s (described on p. 223), where a flat cylindrical oil-drop, of certain relative dimensions, can, by sucking away a little of the contained oil, be made to assume the form of a biconcave disc, whose periphery is part of a nodoidal surface. From the relation of the nodoid to the “elastic curve,’ we perceive that these two examples are closely akin one to the other.

The form of the corpuscle is symmetrical, and its surface is a surface of revolution; but it is obviously not a surface of Ek 1G constant mean curvature, nor of constant pressure. For we see at once that, in the sectional diagram (Fig. 93), the pressure inwards due to surface tension is positive at A, and negative at C; at B there is no curvature in the plane of the paper, while perpendicular to it the curvature is negative, and the pressure therefore is also negative. Accordingly, from the point of view of surface tension alone, the blood-corpuscle is not a surface of equilibrium; or in other words, it is not a fluid drop suspended in another liquid. It is obvious therefore that some other force or forces must be at work, and the simple effect of mechanical pressure is here excluded, because the blood-corpuscle exhibits its characteristic shape while floating freely in the blood. In the lower vertebrates the blood-corpuscles have the form of a flattened oval disc, with rather sharp edges and ellipsoidal surfaces, and this .again is manifestly not a surface of equilibrium.

Two facts are especially noteworthy in connection with the form of the blood-corpuscle. In the first place, its form is only maintained, that is to say it is only in equilibrium, in relation to certain properties of the medium in which it floats. If we add a little water to the blood, the corpuscle quickly loses its characteristic shape and becomes a spherical drop, that is to gay a true surface of minimal area and of stable equilibrium. If on the other hand we add a strong solution of salt, or a little glycerine, the corpuscle contracts, and its surface becomes puckered and uneven. In these phenomena it is so far obeying the laws of diffusion and of surface tension.

In the second place, it can be exactly imitated artificially by means of other colloid substances. Many years ago Norris made the very interesting observation that in an emulsion of glue the drops assumed a biconcave form resembling that of the mammalian corpuscles*. The glue was impure, and doubtless contained lecithin ; and it is possible (as Professor Waymouth Reid tells me) to make a similar emulsion with cerebrosides and cholesterin oleate, in which the same conformation of the drops or particles is beautifully shewn. Now such cholesterin bodies have an important place among those in which Lehmann and others have shewn and studied the formation of fluid crystals, that is to say of bodies in which the forces of crystallisation and the forces of surface tension are battling with one another}; and, for want of a better explanation, we may in the meanwhile suggest that some such cause is at the bottom of the conformation the explanation of which presents so many difficulties. But we must not, perhaps, pass from this subject without adding that the case is a difficult and complex one from the physiological point of view. For the surface of a blood-corpuscle consists of a “semi-permeable membrane,” through which certain substances pass freely and not others (for the most part anions and not cations), and it may be, accordingly, that we have in life a continual state of osmotic inequilibrium, of negative osmotic tension within, to which comparatively simple cause the imperfect distension of the corpuscle may be also duet. The whole phenomenon would be comparatively easy to understand if we might postulate a stiffer peripheral region to the corpuscle, in the form for instance of a peripheral elastic ring.

Such an annular thickening or stiffening, like the “ collapse-rings”” which an engineer inserts in a boiler, has been actually asserted to exist, but its presence is not authenticated. . But it is not at all improbable that we have still much to learn about the phenomena of osmosis itself, as manifested in the case of minute bodies such as a blood-corpuscle; and (as Professor Peddie suggests to me) it is by no means impossible that curvature t Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the Red Blood Corpuscle, Biochem. Journal, 1, p. 55, 1908.

of the surface may itself modify the osmotic or perhaps the adsorptive action. If it should be found that osmotic action tended to stop, or to reverse, on change of curvature, it would follow that this phenomenon would give rise to internal currents; and the change of pressure consequent on these would tend to intensify the change of curvature when once started*. — The sperm-cells of the Decapod crustacea exhibit various singular shapes. In the Crayfish they are flattened cells with ‘stiff curved processes radiating outwards like a St Catherine’s wheel; in Inachus there are two such circles of stiff processes ; in Galathea we have a still more complex form, with long and

Fig. 94. Sperm-cells of Decapod Crustacea (after Koltzoff). a, Inachus scorpio; b, Galathea squamifera; c, do. after maceration, to shew spiral fibrillae. slightly twisted processes. In all these cases, just as in the case of the blood-corpuscle, the structure alters, and finally loses, its characteristic form when the nature or constitution (or as we may assume in particular—the density) of the surrounding medium is changed. Here again, as in the blood-corpuscle, we have to do with a very important force, which we had not hitherto considered in this connection,—the force of osmosis, manifested under conditions similar to those of Pfeffer’s classical experiments on the plant-cell. The surface of the cell acts as a “semi-permeable membrane,”

* For an attempt to explain the form of a blood-corpuscle by surface-tension alone, see Rice, Phil. Mag. Nov. 1914; but cf. Shorter, ibid. Jan. 1915. permitting the passage of certain dissolved substances (or their “ions”’) and including or excluding others; and thus rendering manifest and measurable the existence of a definite “osmotic pressure.” In the case of the sperm-cells of Inachus, certain quantitative experiments have been performed*. ‘The sperm-cell exhibits its characteristic conformation while lying in the serous fluid of the animal’s body, in ordinary sea-water, or in a 5 per cent. solution of potassium nitrate; these three fluids being all “isotonic” with one another. As we alter the concentration of potassium nitrate, the cell assumes certain definite forms corresponding to definite concentrations of the salt; and, as a further and final proof that the phenomenon is entirely physical, it is found that other salts produce an identical effect when their concentration is proportionate to their molecular weight, and

Fig. 95. Sperm-cells of Inachus, as they appear in saline solutions of varying density. (After Koltzoff.) whatever identical effect is produced by various salts in their respective concentrations, a similarly identical effect is produced when these concentrations are doubled or otherwise proportionately changed f. Thus the following table shews the percentage concentrations of certain salts necessary to bring the cell into the forms a and ¢ of Fig. 95; in each case the quantities are proportional to the molecular weights, and in each case twice the quantity is necessary to produce the effect of Fig. 95¢ compared with that which gives rise to the all but spherical form of Fig. 95a.

% concentration of salts in which the sperm-cell of Inachus assumes the form of If we look then, upon the spherical form of the cell as its true condition of symmetry and of equilibrium, we see that what we call its normal appearance is just one of many intermediate phases of shrinkage, brought about by the abstraction of fluid from its interior as the result of an osmotic pressure greater outside than inside the cell, and where the shrinkage of volume is not kept pace with by a contraction of the swrface-area. In the case of the blood-corpuscle, the shrinkage is of no great amount, and the resulting deformation is symmetrical; such structural inequality as may be necessary to account for it need be but ‘small. But in the case of the sperm-cells, we must have, and we actually do find, a somewhat complicated arrangement of more or less rigid or elastic structures in the wall of the cell, which like the wire framework in Plateau’s experiments, restrain and modify the forces acting on the drop. In one form of Plateau’s experiments, instead of supporting his drop on rings or frames of wire, he laid upon its surface one or more elastic coils; and then, on withdrawing oil from the centre of his globule, he saw its uniform shrinkage counteracted by the spiral springs, with the result that the centre of each elastic coil seemed to shoot out into a prominence. Just such spiral coils are figured (after Koltzoff) in Fig. 96; and they may be regarded as precisely akin to those local thickenings, spiral and other, to which we have already ascribed the cylindrical form of the Spirogyra cell. In all probability we must in like manner attribute the peculiar spiral and other forms, for instance of many Infusoria, to the

presence, among the multitudinous other differentiations of their protoplasmic substance, of such more or less elastic fibrillae, which play as it were the part of a microscopic skeleton*. But these cases which we have just dealt with, lead us to another consideration. In a semi-permeable membrane, through which water passes freely in and out, the conditions of a liquid surface are greatly modified; and, in the ideal or ultimate case, there is neither surface nor surface tension at all. And this would lead us somewhat to reconsider our position, and to enquire whether the true surface tension of a liquid film is actually responsible for all that we have ascribed to it, or whether certain of the phenomena which we have assigned to that cause may not in part be due to the contractility of definite and elastic membranes. But to investigate this question, in particular cases, is rather for the physiologist: and the morphologist may go on his way, paying little heed to what is no doubt a difficulty. In surface tension we have the production of a film with the properties of an elastic membrane, and with the special peculiarity that contraction continues with the same energy however far the process may have already gone; while the ordinary elastic membrane contracts to a certain extent, and contracts no more. But within wide limits the essential phenomena are the same in both cases. Our fundamental equations apply to both cases alike. And accordingly, so long as our purpose is morphological, so long as what we seek to explain is regularity and definiteness of form, it matters little if we should happen, here or there, to confuse surface tension with elasticity, the contractile forces manifested at a liquid surface with those which come into play at the complex internal surfaces of an elastic solid.

* As Bethe points out (Zellgestalt, Plateausche Fliissigkeitsfigur und Neurofibrille, Anat. Anz. xu. p. 209, 1911), the spiral fibres of which Koltzoff speaks must lie in the surface, and not within the substance, of the cell whose conformation is affected by them, A very important corollary to, or amplification of the theory of surface tension is to be found in the modern chemico-physical doctrine of Adsorption*. In its full statement this subject soon becomes complicated, and involves physical conceptions and mathematical treatment which go beyond our range. But it is | necessary for us to take account of the phenomenon, though it be in the most elementary way.