On Growth and Form

according to what we have already learned, upon the relative magnitudes of the tensions at the surface of the cells and at the boundary between them®*. In the typical case of an equally divided cell, such as a double and co-equal soap-bubble, where the partition-wall and the outer walls are similar to one another and in contact with similar substances, we can easily determine the form of the system. For, at any point of the boundary of the partition-wall, O, the tensions being equal, the angles QOP, ROP, QOR are all equal, and each is, therefore, an angle of 120°. But OQ, OR being tangents, the centres of the two spheres (or circular arcs in the figure) lie on perpendiculars to them; therefore the radii CO, C’O meet at an

angle of 60°, and COC’ is an equilateral triangle. That is to say, the centre of each circle lies on the circumference of the other; the partition lies midway between the two centres; and the length (i.e. the diameter) of the partition-wall, PO, is times the radius, or -866 times the diameter, of each of the cells. This gives us, then, the form of an aggregate of two equal cells under uniform conditions. As soon as the tensions become unequal, whether from changes in their own substance or from differences in the substances with which they are in contact, then the form alters. If the tension

* Tn an actual calculation we must of course always take account of the tensions on both sides of each film or membrane. along the partition, P, diminishes, the partition itself enlarges, - and the angle YOR increases: until, when the tension P is very small compared to @ or R, the whole figure becomes a circle, and the partition-wall, dividing it into two hemispheres, stands at right angles to the outer wall. This is the case when the outer wall of the cell is practically solid. On the other hand, if P begins to increase relatively to Q and R, then the partition-wall contracts, and the two adjacent cells become larger and larger segments of a sphere, until at length the system becomes divided into two separate cells.

In the spores of Liverworts (such as Pellia), the first partitionwall (the equatorial partition in Fig. 109, a) divides the spore into - two equal halves, and is therefore a plane surface, normal to the surface of the cell; but the next partitions arise near to either end of the original spherical or elliptical cell. Each of these latter partitions will (like the first) tend to set itself normally to the cell-wall; at least the angles on either side of the partition will be identical, and their magnitude will depend upon the tension existing between the cell-wall and the surrounding medium. They will only be right angles if the cell-wall is already practically solid, and in all probability (rigidity of the cell-wall not being quite attained) they will be somewhat greater. In either case the partition itself will be a portion of a sphere, whose curvature will now denote a difference of pressures in the two chambers or cells, which it serves to separate. (The later stages of cell-division, represented in the figures b and c, we are not yet in a position to deal with.)

We have innumerable cases, near the tip of a growing filament, where in like manner the partition-wall which cuts off the terminal cell constitutes a spherical lens-shaped surface, set normally to the adjacent walls. At the tips of the branches of many Florideae, for instance, we find such a lenticular partition. In Dvzctyota dichotoma, as figured by Reinke, we have a succession of such ‘partitions; and, by the way, in such cases as these, where the

tissues are very transparent, we have often in optical section a puzzling confusion of lines; one being the optical section of the curved partition-wall, the other being the straight linear projection of its outer edge to which we have already referred. In the conical terminal cell of Chara, we have the same lens-shaped curve, but a little lower down, where the sides of the shoot are approximately parallel, we have flat transverse partitions, at the edges of which, however, we recognise a convexity of the outer cell-wall and a definite angle of contact, equal on the two sides

Fig. 110. Cells of Dictyota. Fig. 111. Terminal and other cells (After Reinke.) ot Chara. In the young antheridia of Chara (Fig. 112), and in the not dissimilar case of the sporangium (or conidiophore) of Mucor, we easily recognise the hemispherical form of the septum which shuts off the large spherical cell from the cylindrical filament. Here, in the first phase of development, we should have to take into consideration the different pressures exerted bys the single curvature of the cylinder and the double curvature of its spherical cap (p. 221); and we should find that the partition would have a somewhat low curvature, with a radius less than the diameter of the cylinder; which it would have exactly equalled but for the Fig. 112. Young

“4: : : : : antheridium of additional pressure inwards which it receives Chara. from the curvature of the large surrounding sphere. But as the latter continues to grow, its curvature decreases, and so likewise does the inward pressure of its surface; and accordingly the little convex partition brlges out more and more. In order to epitomise the foregoing facts let the annexed diagrams (Fig. 113) represent a system of three films, of which one is a partition-wall between the other two; and let the tensions at the three surfaces, or the tractions exercised upon a point at their meeting-place, be proportional to T, T’ and ¢t. Let a, B, y be, as in the figure, the opposite angles. Then:

(1) If T be equal to T’, and ¢ be relatively insignificant, the angles a, 8 will be of 90°. (2) If T= T’, but be a little greater than ¢, then ¢ will exert an appreciable traction, and a, 8 will be more than 90°, say, for instance, 100°. The more complicated cases, when ¢, T and T’ are all unequal, are already sufficiently explained. The biological facts which the foregoing considerations go a long way to explain and account for have been the subject of much argument and discussion, especially on the part of the botanists. Let me recapitulate, in a very few words, the history of this long discussion.

Some fifty years ago, Hofmeister laid it down as a general law that “The partition-wall stands always perpendicular to what was previously the principal direction of growth in the cell,”—or, in other words, perpendicular to the long axis of the cell*. Ten years later, Sachs formulated his rule, or principle, of “rectangular section,” declaring that in all tissues, however complex, the cell-walls cut one another (at the time of their formation) at right angles*. Years before, Schwendener had found, in the final results of cell-division, a universal system of “orthogonal trajectoriest”’; and this idea Sachs further developed, introducing complicated systems of confocal ellipses and hyperbole, and distinguishing between periclinal walls, whose curves approximate to the peripheral contours, radial partitions, which cut these at an angle of 90°, and finally anticlines, which stand at right angles to the other two.

Reinke, in 1880, was the first to throw some doubt upon this explanation. He pointed out various cases where the angle was not a right angle, but was very definitely an acute one; and he saw, apparently, in the more common rectangular symmetry merely what he calls a necessary, but secondary, result of growth. Within the next few years, a number of botanical writers were content to point out further exceptions to Sachs’s Rule§; and in some cases to show that the curvatures of the partition-walls, especially such cases of lenticular curvature as we have described, were by no means accounted for by either Hofmeister or Sachs; while within the same period, Sachs himself, and also Rauber, attempted to extend the main generalisation to animal tissues).

While these writers regarded the form and arrangement of the cell-walls as a biological phenomenon, with little if any direct relation to ordinary physical laws, or with but a vague reference to ‘“‘mechanical conditions,’ the physical side of the case was soon urged by others, with more or less force and cogency. Indeed the general resemblance between a cellular tissue and a “froth” * Sachs, Ueber die Anordnung der Zellen in jiingsten Pflanzentheilen, Verh. phys. med. Ges. Wiirzburg, x1, pp. 219-242, 1877; Ueber Zellenanordnung und Wachsthum, zbid. xu, 1878; Ueber die durch Wachsthum bedingte Verschiebung kleinster Theilchen in trajectorischen Curven, Monatsber. k. Akad. Wiss. Berlin, 1880; Physiology of Plants, chap. xxvii, pp. 431-459, Oxford, 1887.

had been pointed out long before, by Melsens, who had made an “artificial tissue” by blowing into a solution of white of egg*. In 1886, Berthold published his Protoplasmamechantk, in which he definitely adopted the principle of “minimal areas,” and, following on the lines of Plateau, compared the forms of many cell-surfaces and the arrangement of their partitions with those assumed under surface tension by a system of “ weightless films.” But, as Klebsf points out in reviewing Berthold’s book, Berthold was careful to stop short of attributing the biological phenomena to a definite mechanical cause. They remained for him, as they had done for Sachs, so many “phenomena of growth,” or “properties of protoplasm.”

In the same year, but while still apparently unacquainted with Berthold’s work, Errerat published a short but very lucid article, in which he definitely ascribed to the cell-wall (as Hofmeister had already done) the properties of a semi-liquid film and drew from this as a logical consequence the deduction that it must assume the various configurations which the law of minimal areas imposes on the soap-bubble. So what we may call Errera’s Law is formulated as follows: A cellular membrane, at the moment of its formation, tends to assume the form which would be assumed, under the same conditions, by a liquid film destitute of weight.

Soon afterwards Chabry, in discussing the embryology of the Ascidians, indicated many of the points in which the contacts between cells repeat the surface-tension phenomena of the soapbubble, and came to the conclusion that part, at least, of the embryological phenomena were purely physical§; and the same line of investigation and thought were pursued and developed by Robert, in connection with the embryology of the Mollusca|]. Driesch again, in a series of papers, continued to draw attention to the presence of capillary phenomena in the segmenting cells

of various embryos, and came to the conclusion that the mode of segmentation was of little importance as regards the final result*. Lastly de Wildemant, in a somewhat wider, but also vaguer generalisation than Errera’s, declared that “The form of the cellular framework of vegetables, and also of animals, in its essential features, depends upon the forces of molecular physics.” Let us return to our problem of the arrangement of partition films. When we have three bubbles in contact, instead of two as in the case already considered, the phenomenon 1s strictly analogous to our former case. The three bubbles will be separated by three partition surfaces, whose curvature will depend upon the relative

size of the spheres, and which will be plane if the latter are all of the same dimensions; but whether plane or curved, the three partitions will meet one another at an angle of 120°, in an axial line. Various pretty geometrical corollaries accompany this arrangement. For instance, if Fig. 114 represent the three associated bubbles in a plane drawn through their centres, ¢, c’, c’’ (or what is the same thing, if it represent the base of three bubbles resting on a plane), then the lines wc, ue”, or sc, sc’, etc., drawn to the

* “Dass der Furchungsmodus etwas fiir das Zukiinftige unwesentliches ist,” Z. f. w. Z. wv, 1893, p. 37. With this statement compare, or contrast, that of Conklin, quoted on p. 4; cf. also pp. 157, 348 (footnotes). centres from the points of intersection of the circular arcs, will _always enclose an angle of 60°. Again (Fig. 115), if we make the angle c”uf equal to 60°, and produce uf to meet cc” in f, f will be the centre of the circular arc. which constitutes the partition Ow; and further, the three points f,g, h, successively determined in this

manner, will lie on one and the same straight line. In the case of coequal bubbles or cells (as in Fig. 114, B), it is obvious that the lines joining their centres form an equilateral triangle; and consequently, that the centre of each circle (or sphere) lies on the circumference of the other two; it is also obvious that uf is now parallel to cc”, and accordingly that the centre of curvature of the partition is now infinitely distant, or (as we have already said), that the partition itself is plane.

When we have four bubbles in conjunction, they would seem to be capable of arrangement in two symmetrical ways: either, as in Fig. 116 (A), with the four partition-walls meeting at right angles, or, as in (B), with five partitions meeting, three and three, at angles of 120°. This latter arrangement is strictly analogous to the arrangement of three bubbles in Fig. 114. Now, though both of these figures, from their symmetry, are apparently figures of equilibrium, yet, physically, the former turns out to be of unstable

and the latter of stable equilibrium. If we try to bring our four bubbles into the form of Fig. 116, A, such an arrangement endures only for an instant; the partitions glide upon each other, a median wall springs into existence, and the system at once assumes the form of our second figure (B). This is a direct consequence of the law of minimal areas: for it can be shewn, by somewhat difficult mathematics (as was first done by Lamarle), that, in dividing a closed space into a given number of chambers by means of partitionwalls, the least possible area of these partition-walls, taken together, can only be attained when they meet together in groups of three, at equal angles, that is to say at angles of 120°.

Wherever we have a true cellular complex, an arrangement of cells in actual physical contact by means of a boundary film, we find this general principle in force; we must only bear in mind that, for its perfect recognition, we must be able to view the object in a plane at right angles to the boundary walls. For instance, in any ordinary section of a vegetable parenchyma, we recognise the appearance of a “froth,” ‘precisely resembling that which we can construct by imprisoning a mass of soap-bubbles in a narrow vessel with flat sides of glass; in both cases we see the cell-walls everywhere meeting, by threes, at angles of 120°, irrespective of the size of the individual cells: whose relative size, on the other hand, determines the curvature of the partition-walls. On the surface of a honey-comb we have precisely the same conjunction, between cell and cell, of three boundary walls, meeting at 120°. In embryology, when we examine a segmenting egg, of four (or more) segments, we find in like manner, in the great majority of cases, if not in all, that the same principle is still exemplified; the four segments do not meet in a common centre, but each cell is in contact with two others, and the three, and only three, common boundary walls meet at the normal angle of 120°. A so-called polar furrow*, the visible edge of a vertical partitionwall, joins (or separates) the two triple contacts, precisely as in Fig. 116, B.

In the four-celled stage of the frog’s egg, Rauber (an exceptionally careful observer) shews us three alternative modes in which the four cells may be found to be conjoined (Fig. 117). In (A) we have the commonest arrangement, which is that which we have just studied and found to be the simplest theoretical one; that namely where a straight “polar furrow” intervenes, and where, at its extremities, the partition-walls are conjoined three by three. In (B), we have again a polar furrow, which is now seen to be a portion of the first “segmentation-furrow” (cf. Fig. 155 ete.) by which the egg was originally divided into two; the four-celled stage being reached by the appearance of the transverse furrows

* It was so termed by Conklin in 1897, in his paper on Crepidula (J. of Morph. xi, 1897). It is the Querfurche of Rabl (Morph. Jahrb. v, 1879); the Polarfurche of O. Hertwig (Jen. Zeitschr. xiv, 1880); the Brechungslinie of Rauber (Neue and their corresponding partitions. In this case, the polar furrow is seen to be sinuously curved, and Rauber tells us that its curvature gradually alters: as a matter of fact, it (or rather the partition-wall corresponding to it) is gradually setting itself into a position of equilibrium, that is to say of equiangular contact with its neighbours, which position of equilibrium is already attained or nearly so in Fig. 117, A. In Fig. 117, C, we have a very different condition, with which we shall deal in a moment.

According to the relative magnitude of the bodies in contact, this “polar furrow” may be longer or shorter, and it may be so minute as to be not easily discernible; but it is quite certain that no simple and homogeneous system of fluid films such as we are dealing with is in equilibrium without its presence. In the accounts given, however, by embryologists of the segmentation of the egg, while the polar furrow is depicted in the great majority Fig. 117. Various ways in which the four cells are co-arranged in the four-celled stage of the frog’s egg. (After Rauber.)

of cases, there are others in which it has not been seen and some in which its absence is definitely asserted*. The cases where four cells, lying in one plane, meet a a point, such as were frequently figured by the older embryologists, are very difficult to verify, and I have not come across a single clear case in recent literature. Considering the physical stability of the other arrangement, the great preponderance of cases in which it is known to occur, the difficulty of recognising the polar furrow in cases where it is very small and unless it be specially looked for, and the natural tendency of the draughtsman to make an all but symmetrical structure appear wholly so, I am much inclined to attribute to

* Thus Wilson (J. of Morph. vit, 1895) declared that in Amphioxus the polar furrow was occasionally absent, and Driesch took occasion to criticise and to throw doubt upon the statement (Arch. f. Hntw. Mech. 1, 1895, p. 418). error or imperfect observation all those cases where the junctionlines of four cells are represented (after the manner of Fig. 116, A) as a simple cross*. But while a true four-rayed intersection, or simple cross, is theoretically impossible (save as a transitory and highly unstable condition), there is another condition which may closely simulate it, and which is common enough. There are plenty of representations of segmenting eggs, in which, instead of the triple junction and polar furrow, the four cells (and in like manner their more numerous successors) are represented as rounded off, and separated from one another by an empty space, or by a little drop of an extraneous fluid, evidently not directly miscible with the fluid surfaces of the cells. Such is the case in the obviously accurate figure which Rauber gives (Fig. 117, C) of the third mode of conjunction in the four-celled stage of the frog’s egg. Here Rauber is most careful to point out that the furrows do not simply “cross,” or meet in a point, but are separated by a little space, which he calls the Polgriibchen, and asserts to be constantly present whensoever the polar furrow, or Brechungslinie, is not to be discerned. This little interposed space, with its contained drop of fluid, materially alters the case, and implies a new condition of theoretical and actual equilibrium. For, on the one hand, we see that now the four intercellular partitions do not meet one another at all; but really impinge upon four new and separate partitions, which constitute interfacial contacts, not between cell and cell, but between the respective cells and the intercalated drop. And secondly, the angles at which these four little surfaces will meet the four cell-partitions, will be determined, in the usual way, by the balance between the respective tensions of these several surfaces.

In an extreme case (as in some pollen-grains) it may be found that the cells under the observed circumstances are not truly in surface contact: that they are so many drops which touch but do not “wet” one another, and which are merely held together by the pressure of the surrounding envelope. But even supposing, * Precisely the same remark was made long ago by Driesch: “Das so oft schematisch gezeichnete Vierzellenstadium mit zwei sich in zwei Punkten scheidende Medianen kann man wohl getrost aus der Reihe des Existierenden streichen,”

as 1s in all probability the actual case, that they are in actual fluid contact, the case from the point of view of surface tension presents no difficulty. In the case of the conjoined soap-bubbles, we were dealing with semilar contacts and with equal surface tensions throughout the system; but in the system of protoplasmic cells which constitute the segmenting egg we must make allowance for an inequality of tensions, between the surfaces where cell meets cell, and where on the other hand cell-surface is in contact with the surrounding medium,—ain this case generally water or one of the fluids of the body. Remember that our general condition is that, in our entire system, the sum of the surface energies is a minimum; and, while this is attained by the sum of the surfaces being a minimum in the case where the energy is uniformly distributed, it is not necessarily so under non-uniform conditions. In the diagram (Fig. 118) if the energy per unit area be greater along the contact surface cc’, where cell meets cell, than along ca or cb, where cellsurface is in contact with the surrounding medium, these latter surfaces will tend to increase and the surface of cell-contact to diminish. In short there will be the usual balance of forces between the tension along the surface cc’, and the two opposing tensions along ca and cb. If the former be greater than either of the other two, the outside angle will be less than 120°; and if the tension along the surface cc’ be as much or more than the sum of the other two, then the drops will stand in contact only, save for the possible effect of external pressure, at a point. This is the explanation, in general terms, of the peculiar conditions obtaining in Nostoc and its allies (p. 300), and it also leads us to a consideration of the general properties and characters of an “epidermal” layer.

While the inner cells of the honey-comb are symmetrically situated, sharing with their neighbours in equally distributed pressures or tensions, and therefore all tending with great accuracy to identity of form, the case is obviously different with the cells at the borders of the system. So it is, in hke manner, with our froth of soap-bubbles. The bubbles, or cells, in the interior of the mass are all alike in general character, and if they be equal in size are alike in every respect: their sides are uniformly flattened*, and tend to meet at equal angles of 120°. But the bubbles which constitute the outer layer retain their spherical surfaces, which however still tend to meet the partition-walls connected with them at constant angles of 120°. This outer layer of bubbles, which forms the surface of our froth, constitutes after a fashion what we should call in botany an “epidermal” layer. But in our froth of soap-bubbles we have, as a rule, the same kind of contact (that is to say, contact with air) both within and without the bubbles; while in our living cell, the outer wall of the epidermal cell is exposed to air on the one side, but is in contact with the