On Growth and Form

* On the Distribution of Potassium in animal and vegetable Cells; Journ. of Physiol. Xxxi1, p. 95, 1905. + The reader will recognise that there is a fundamental difference, and contrast, By this means Macallum demonstrated some years ago the unexpected presence of accumulations of potassium (i.e. of chloride or other salts of potassium) localised in particular parts of various cells, both solitary cells and tissue cells; and he arrived at the conclusion that the localised accumulations in question were simply evidences of concentration of the dissolved potassium salts, formed and localised in accordance with the Gibbs-Thomson law. In other words, these accumulations, occurring as they actually do in connection with various boundary surfaces, are evidence, when they appear irregularly distributed over such a surface, of inequalities in its surface tension*; and we may safely take it that our potassium salts, like inorganic substances in general, tend to raise the surface tension, and will therefore be found concentrating at a portion of the surface whose tension is weak ft.

In Professor Macallum’s figure (Fig. 98, 1) of the little green alga Pleurocarpus, we see that one side of the cell is beginning to bulge out in a wide convexity. This bulge is, in the first place, a sign of weakened surface tension on one side of the cell, which as a whole had hitherto been a symmetrical cylinder; in the second place, we see that the bulging area corresponds to the position of a great concentration of the potassic salt; while in the third place, from the physiological point of view, we call the phenomenon the first stage in the process of conjugation. In Fig. 98, 2, of Mesocarpus (a close ally of Spirogyra), we see the same phenomenon admirably exemplified in a later stage. From the adjacent cells distinct outgrowths are being emitted, where the surface tension has been weakened: just as the glass-blower warms and softens a small part of his tube to blow out the softened area into a bubble or diverticulum; and in our Mesocarpus cells (besides a certain amount of potassium. rendered visible over the boundary which between such experiments as these of Professor Macallum’s and the ordinary staining processes of the histologist. The latter are (as a general rule) purelyempirical, while the former endeavour to reveal the true microchemistry of the cell. “On peut dire que la microchimie n’est encore qu’& la période d’essai, et -que l'avenir de Vhistologie et spécialement de la cytologie est tout entier dans la microchimie”’ (Prenant, A., Méthodes et résultats de la Microchimie, Journ. de PAnat. et de la Physiol. xtvt, pp. 343-404, 1910).

t In accordance with a simple corollary to the Gibbs-Thomson law. separates the green protoplasm from the cell-sap), there is a very large accumulation precisely at the poimt where the tension of the originally cylindrical cell is weakening to produce the bulge. But in a still later stage, when the boundary between the two conjugating cells is lost and the cytoplasm of the two cells becomes fused together, then the signs of potassic concentration quickly disappear, the salt becoming generally diffused through the now symmetrical and spherical “zygospore.”

Fig. 98. Adsorptive concentration of potassium salts in (1) cell of Plewrocarpus about to conjugate; (2) conjugating cells of Mesocarpus; (3) sprouting spores of Equisetum. (After Macallum.) In aspore of Equisetum (Fig. 98, 3), while it is still a single cell, no localised concentration of potassium is to be discerned; but as ‘soon as the spore has divided, by an internal partition, into two cells, the potassium salt is found to be concentrated in the smaller one, and especially towards its outer wall, which is marked by a pronounced convexity. And as this convexity (which corresponds to one pole of the now asymmetrical, or quasi-ellipsoidal spore) grows out into the root-hair, the potassium salt accompanies its growth, and is concentrated under its wall. The concentration is,

accordingly, a concomitant of the diminished surface tension which is manifested in the altered configuration of the system. In the case of ciliate or flagellate cells, there is to be found a characteristic accumulation of potassium at and near the base o1 the cilia. The relation of ciliary movement to surface tension lies beyond our range, but the fact which we have just mentioned throws light upon the frequent or general presence of a little protuberance of the cell-surface just where a flagellum is given off (cf. p. 247), and of a little projecting ridge or fillet at the base of an isolated row of cilia, such as we find in Vorticella.

Yet another of Professor Macallum’s demonstrations, though its interest is mainly physiological, will help us somewhat further to comprehend what is implied in our phenomenon. In a normal cell of Spirogyra, a concentration of potassium is revealed along the whole surface of the spiral coil of chlorophyll-bearing, or “chromatophoral,” protoplasm, the rest of the cell being wholly destitute of the former substance: the indication being that, at this particular boundary, between chromatophore and cell-sap, the surface tension is small in comparison with any other interfacial surface within the system.

Now as Macallum points out, the presence of potassium is known to be a factor, in connection with the chlorophyll-bearing protoplasm, in the synthetic production of starch from CO, under the influence of sunlight. But we are left in some doubt as to the consecutive order of the phenomena. For the lowered surface tension, indicated by the presence of the potassium, may be itself a cause of the carbohydrate synthesis; while on the other hand, this synthesis may be attended by the production of substances (e.g. formaldehyde) which lower the surface tension, and so conduce to the concentration of potassium. All we know for certain is that the several phenomena are associated with one another, as apparently inseparable parts or inevitable concomitants of a certain complex action.

And now to return, for a moment, to the question of cell-form. When we assert that the form of a cell (in the absence of mechanical pressure) is essentially dependent on surface tension, and even when we make the preliminary assumption that protoplasm is essentially a fluid, we are resting our belief on a general consensus of evidence, rather than on compliance with any one crucial definition. The simple fact is that the agreement of cell-forms with the forms which physical experiment and mathematical theory assign to liquids under the influence of surface tension, is so frequently and often so typically manifested, that we are led, or driven, to accept the surface tension hypothesis as generally applicable and as equivalent to a universal law. The occasional difficulties or apparent exceptions are such as call for further enquiry, but fall short of throwimg doubt upon our hypothesis. Macallum’s researches introduce a new element of certainty, a “nail in a sure place,” when they demonstrate that, in certain movements or changes of form which we should naturally attribute to weakened surface tension, a chemical concentration which would naturally accompany such weakening actually takes place. They further teach us that in the cell a chemical heterogeneity may exist of a very marked kind, certain substances being accumulated here and absent there, within the narrow bounds of the system.

Such localised accumulations can as yet only be demonstrated in the case of a very few substances, and of a single one in particular; and these are substances whose presence does not produce, but whose concentration tends to follow, a weakening of surface tension. The physical cause of the localised inequalities of surface tension remains unknown. We may assume, if we please, that it is due to the prior accumulation, or local production, of chemical bodies which would have this direct effect; though we are by no means limited to this hypothesis.

But in spite of some remaining difficulties and uncertainties, we have arrived at the conclusion, as regards unicellular organisms, that not only their general configuration but also their departures from symmetry may be correlated with the molecular forces manifested in their fluid or semi-fluid surfaces. We now pass from the consideration of the solitary cell to that of cells in contact with one another,—to what we may call in the first instance “cell-aggregates,’—through which we shall be led ultimately to the study of complex tissues. In this part of our subject, as in the preceding chapters, we shall have to give some consideration to the effects of various forces; but, as in the case - of the conformation of the solitary cell, we shall probably find, and we may at least begin by assuming, that the agency of surface tension is especially manifest and important. The effect of this surface tension will chiefly manifest itself in the production of surfaces minimae areae: where, as Plateau was always careful to point out, we must understand by this expression not an absolute, but a relative minimum, an area, that is to say, which approximates to an absolute minimum as nearly as circumstances and the conditions of the case permit.

There are certain fundamental principles, or fundamental equations, besides those which we have already considered, which we shall need in our enquiry. For instance the case which we briefly touched upon (on p. 265) of the angle of contact between the protoplasm and the axial filament in a Heliozoan we shall now find to be but a particular case of a general and elementary theorem. Let us re-state as follows, in terms of Energy, the general principle which underlies the theory of surface tension or capillarity.

When a fluid is m contact with another fluid, or with a solid or a gas, a portion of the energy of the system (that, namely, which we call surface energy), is proportional to the area of the surface of contact: it is also proportional to a coefficient which is specific for each particular pair of substances, and which is constant for these, save only in so far as it may be modified by changes of temperature or of electric charge. The condition of munimum potential energy in the system, which is the condition of equilibrium, will accordingly be obtained by the utmost possible diminution in the area of the surfaces in contact. When we have three bodies in contact, the case becomes a little more complex. Suppose for instance we have a drop of some fluid, A, floating on another fluid, B, and exposed to air, C. The whole surface energy of the system may now be considered as divided into two parts, one at the surface of the drop, and the other outside of the same; the latter portion is inherent in the surface BC, between the mass of fluid B and the superincumbent air, C; but the former portion consists of two parts, for it is divided between the two surfaces AB and AC, that namely which separates the drop from the surrounding fluid and that which separates it from the atmosphere. So far as

the drop is concerned, then, equilibrium depends on a proper balance between the energy, per unit area, which is resident in its own two surfaces, and that which is externa! thereto: that is to say, if we call #,, the energy at the surface between the two fluids, and so on with the other two pairs of surface energies, the condition of equilibrium, or of maintenance of the drop, is that - If, on the other hand, the fluid A happens to be oil and the fluid B, water, then the energy per unit area of the water-air surface is greater than that of the oil-air surface and that of the oil-water surface together ; i.e.

Here there is no equilibrium, and in order to obtain it the water-air surface must always tend to decrease and the other two interfacial surfaces to increase; which is as much as to say that the water tends to become covered by a spreading film of oil, and the waterair surface to be abolished. The surface energy of which we have here spoken is manifested in that contractile force, or “tension,” of which we have already had so much to say*. In any part of the free water surface, for instance, one surface particle attracts another surface particle, and the resultant of these multitudinous attractions is an equilibrium of tension throughout this particular surface. In the case of our three bodies in contact with one another, and within a small area

very near to the point of contact, a water particle (for instance) will be pulled outwards by another water particle; but on the opposite side, so to speak, there will be no water surface, and no water particle, to furnish the counterbalancing pull; this counterpull, which is necessary for equilibrium, must therefore be provided by the tensions existing in the other two surfaces of contact. In short, if we could imagine a single particle placed at the very point of contact, it would be drawn upon by three different forces, whose directions would he in the three surface planes, and whose magnitude would be proportional to the specific tensions characteristic of the two bodies which in each case combine to form the “interfacial” surface. Now for three forces acting at a point to be in equilibrium, they must be capable of representation, in magnitude and direction, by the three sides of a triangle, taken in order, in accordance with the elementary theorem of the Triangle of Forces. So, if we know the form of our floating drop (Fig. 100), then by drawing tangents from O (the point of mutual contact),

* It can easily be proved (by equating the increase of energy stored in an increased surface to the work done ia increasing that surface), that the tension measured per unit breadth, 7',,, is equal to the energy per unit area, L,,,. we determine the three angles of our triangle (Fig. 101), and we therefore know the relative magnitudes of the three surface tensions, which magnitudes are proportional to its sides; and conversely, if we know the magnitudes, or relative magnitudes. of the three sides of the triangle, we also know its angles, and these determine the form of the section of the drop. It is scarcely necessary to mention that, since all points on the edge of the drop are under similar conditions, one with another, the form of the drop, as we look down upon it from above, must be circular, . and the whole drop must be a solid of revolution.

The principle of the Triangle of Forces is expanded, as follows, by an old seventeenth-century theorem, called Lami’s Theorem : “Tf three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle contained between the directions of the other two.” That is to say And from this, in turn, we derive the equivalent formulae, by which each force is expressed in terms of the other two, and of the From this and the foregoing, we learn the following important and useful deductions:

(1) The three forces can only be in equilibrium when any one of them is less than the sum of the other two: for otherwise, the triangle is impossible. Now in the case of a drop of olive-oil upon a clean water surface, the relative magnitudes of the three tensions (at 15° C.) have been determined as follows: Water-air surface Am my. 15 Oil-air surface ... aks yet 32 Oil-water surface oie Be: 21 No triangle having sides of these relative magnitudes is possible ; and no such drop therefore can remain in equilibrium.

(2) The three surfaces may be all alike: as when a soapbubble floats upon soapy water, or when two soap-bubbles are joined together, on either side of a partition-film. In this case, the three tensions are all equal, and therefore the three angles are all equal; that is to say, when three similar liquid surfaces meet together, they always do so at an angle of 120°. Whether our two conjoined soap-bubbles be equal or unequal, this is still the invariable rule; because the specific tension of a particular surface is unaffected by any changes of magnitude or form.

(3) If two only of the surfaces be alike, then two of the angles will be alike, and the other will be unlike; and this last will be the difference between 360° and the sum of the other two. A particular case is when a film is stretched between solid and ~ parallel walls, like a soap-film within a cylindrical tube. Here, so long as there is no external pressure applied to either side, so long as both ends of the tube are open or closed, the angles on either side of the film will be equal, that is to say the film will set itself at right angles to the sides.

Many years ago Sachs laid it down as a principle, which has become celebrated in botany under the name of Sachs’s Rule, that one cell-wall always tends to set itself at right angles to another ° cell-wall. This rule applies to the case which we have just illustrated; and such validity as the rule possesses is due to the fact that among plant-tissues it very frequently happens that one cell-wall has become solid and rigid before another and later partition-wall is developed in connection with it.

(4) There is another important principle which arises not out of our equations but out of the general considerations by which we were led to them. We have seen that, at and near the point of contact between our several surfaces, there is a continued balance of forces, carried, so to speak, across the interval; in other words, there is physical continuity between one surface and another. It follows necessarily from this that the surfaces merge one ihto another by a continuous curve. Whatever be the form of our surfaces and whatever the angle between them, this small intervening surface, approximately spherical, is always there to bridge over the line of contact* ; and this little fillet, or “ bourrelet,”’

* The presence of this little liquid “ bourrelet,”’ drawn from the material of which as Plateau called it, is large enough to be a common and conspicuous feature in the microscopy of tissues (Fig. 102). For instance, the so-called “splitting” of the cell-wall, which is conspicuous at the angles of the large “ parenchymatous”’ cells in the succulent tissues of all higher plants (Fig. 103), is nothing more than a manifestation of Plateau’s “bourrelet,” or surface of continuity *.

We may now illustrate some of the foregoing principles, before we proceed to the more complex cases in which more bodies than three are in mutual contact. But in doing so, we must constantly bear in mind the principles set forth in our chapter on the forms of cells, and especially those relating to the pressure exercised by a curved film. Let us look for a moment at the case presented by the partitionwall in a double soap-bubble. As we have just seen, the three films in contact (viz. the outer walls of the two bubbles and the partition-wall between) being all composed of the same substance

the partition-walls themselves are composed, is obviously tending to a reduction of the internal surface-area. And it may be that it is as well, or better, accounted for on this ground than on Plateau’s assumption that it represents a “surface of continuity.” : * A similar “bourrelet” is admirably seen at the line of junction between a floating bubble and the liquid on which it floats; in which case it constitutes a “masse annulaire,” whose mathematical properties and relation to the form of the nearly hemispherical bubble, have been investigated by van der Mensbrugghe (cf. Plateau, op. cit., p. 386). The form of the superficial vacuoles in Actinophrys or Actinosphaerium involves an identical problem.

and all alike in contact with air, the three surface tensions must be equal; and the three films must therefore, in all cases, meet at an angle of 120°. But, unless the two bubbles be of precisely equal size (and therefore of equal curvature) it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles, and therefore that the partition-wall must be a curved surface: it is only plane when it divides two equal and symmetrical cells. It is also obvious, from the symmetry of the figure, that the centres of the spheres, the centre of the partition, and the centres of the two spherical surfaces are all on one and the same straight line.

Now the surfaces of the two bubbles exert a pressure inwards which is inversely proportional to their radii: that is to say p:p::ifr:1/r; and the partition wall must, for equilibrium, exert a pressure (P) which is equal to the difference between these two pressures, that is to say, P=1/R=1/r’ —1fr=(r—r')/rr’. It follows that the curvature of the partition wall must be just such a curvature as is capable of exerting this pressure, that is to say, R=rr'/(ry —7’). The partition wall, then, is always a portion of a spherical surface, whose radius is equal to the product, divided by the difference, of the radii of the two vesicles. It follows at once from this that if the two bubbles be equal, the radius of curvature of the partition is infinitely great, that.1s to say the partition is (as we have already seen) a plane surface.

The geometrical construction by which we obtain the position of the centres of the two spheres and also of the partition surface is very simple, always provided that the surface tensions are uniform throughout the system. If p be a point of contact between the two spheres, and cp be a radius of one of them, then make the angle cpm = 60°, and mark off on pm, pe’ equal to the radius of the other sphere; in like manner, make the angle c'pn = 60°, cutting the line cc’ in c’’; then c’ will be the centre of the second sphere, and c’’ that of the spherical partition. Whether the partition be or be not a plane surface, it is obvious that its line of yunction with the rest of the system lies in a plane,

and is at right angles to the axis of symmetry. The actual curvature of the partition-wall is easily seen in optical section; but in surface view, the line of junction is projected as a plane (Fig. 106), perpendicular to the axis, and this appearance has also helped to lend support and authority to “Sachs’s Rule.” Protococcus, divide into two equal halves, which are therefore separ- COOGOGE5 ated by a plane partition. Among the other lower Algae, akin to

and Oscillatoriae, in which the ; cells are imbedded in a gelatinous matrix, we find a series of forms such as are represented in Fig. 107. D PLL ores Sometimes the cells are solitary ; or disunited; sometimes they run Fig. 107. Filaments, or chains of . ; a : cells, in various lower Algae. 10 pal’s OF 1n TOWS, separated one (A) Nostoc; (B) Anabaena; (C) 3 pie é Rivularia; (D) Oscillatoria. from another by flat partinien= and sometimes the conjoined cells are approximately hemispherical, but at other times each half is more than a hemisphere. These various conditions depend,