On Growth and Form

In a diagrammatic section of an Alcyonarian polype (Fig. 172), we have eight chambers set, symmetrically, about a ninth, which constitutes the “stomach.” In this arrangement there is no difficulty, _for it is obvious that, throughout the system, three boundaries meet (in plane section) in a poimt. In many corals we have as * The same is true of the insect’s wing; but in this case I do not hazard a conjectural explanation. simple, or even simpler conditions, for the radiating calcified partitions either converge upon a central chamber, or fail to meet it and end freely. But in a few cases, the partitions or “septa” converge to meet one another, there being no central chamber on which they may impinge; and here the manner in which contact is effected becomes complicated, and involves problems identical with those which we are now studying.

In the great majority of corals we have as simple or even simpler conditions than those of Aleyonium; for as a rule the calcified partitions or septa of the coral either converge upon a central chamber (or central “columella’’), or else fail to meet it and end freely. In the latter case the problem of space-partitioning does not arise; in the former, however numerous the septa be, their separate contacts with the wall of the central Fig. 173. Heterophyllia anguchamber comply with our fundamental

lata. (After Nicholson.) rule according to which three lines and no more meet in a point, and from this simple and symmetrical arrangement there is little tendency to variation. But in a few cases, the septal partitions converge to, meet one another, there being no central chamber on which they may impinge; and here the manner in which contact is effected becomes complicated, and involves problems of space-partitioning identical with those which we are now studying. In the genus Heterophyllia and in a few allied forms we have such conditions, and students of the Coelenterata have found them very puzzling. McCoy*, their first discoverer, pronounced these corals to be “totally unlike” any other group, recent or fossil; and Professor Martin Duncan, writing a memoir on Heterophyllia and its allies}, described them as “paradoxical in their anatomy.” .

The simplest or youngest Heterophylliae known have six septa (as in Fig. 174, a); im the case figured, four of these septa are conjoined two and two, thus forming the usual triple junctions together with their intermediate partition-walls: and in the case of the other two we may fairly assume that their proper and original arrangement was that of our type 6b (Fig. 158), though the central intermediate partition has been crowded out by partial coalescence. When with increasing age the septa become more numerous, their arrangement becomes exceedingly variable; for the simple reason that, from the mathematical point of view, the number of possible arrangements, of 10, 12 or more cellular partitions in triple contact, tends to increase with great rapidity, and there is little to choose between many

of them in regard to symmetry and equilibrium. But while, mathematically speaking, each particular case among the multitude of possible cases is an orderly and definite arrangement, from the purely biological point of view on the other hand no law or order is recognisable; and so McCoy described the genus as being characterised by the possession of septa “destitute of any order of arrangement, but irregularly branching and coalescing in their passage from the solid external walls towards some indefinite point near the centre where the few main lamellae irregularly anastomose.”

In the two examples figured (Fig. 174), both comparatively simple ones, it will be seen that, of the main chambers, one is in each case an unsymmetrical one; that is to say, there is one chamber which is in contact with a greater number of its neighbours than any other, and which at an earlier stage must have had contact with them all; this was the case of our type /, in the eight-celled system (Fig. 158). Such an asymmetrical chamber (which may occur in a system of any number of cells greater than six), constitutes what is known to students of the Coelenterata as a “fossula’’; and we may recognise it not only here, but also in Zaphrentis and its allies, and in a good many other corals besides. Moreover certain corals are described as having more than one fossula: this appearance being naturally produced under certain of the other asymmetrical variations of normal space-partitioning. Where a single fossula occurs, we are usually told that it is a symptom of “bilaterality”; and this is in turn interpreted as an indication of a higher grade of organisation than is implied in the purely “radial symmetry” of the commoner types of coral. The mathematical aspect of the case gives no warrant for this interpretation.

Let us carefully notice (lest we run the risk of confusing two distinct problems) that the space-partitioning of Heterophylha by no means agrees with the details of that which we have studied in (for instance) the case of the developing disc of Erythrotrichia : the difference simply being that Heterophylha illustrates the general case of cell-partitioning as Plateau and Van Rees studied it, while in Erythrotrichia, and in our other embryological and histological instances, we have found ourselves justified in making the additional assumption that each new partition divided a cell into co-equal parts. No such law holds in Heterophylla, whose case is essentially different from the others: inasmuch as the chambers whose partition we are discussing in the coral are mere empty spaces (empty save for the mere access of sea-water); while in our histological and embryological instances, we were speaking of the division of a cellular unit of living protoplasm. Accordingly, among other differences, the “transverse” or “periclnal” partitions, which were bound to appear at regular intervals and in definite positions, when co-equal bisection was a feature of the

case, are comparatively few and irregular in the earlier stages of Heterophyllia, though they begin to appear in numbers after the main, more or less radial, partitions have become numerous, and when accordingly these radiating partitions come to bound narrow and almost parallel-sided interspaces; then it is that the transverse or periclinal partitions begin to come in, and form what the student of the Coelenterata calls the “dissepiments” of the coral. We need go no further into the configuration and anatomy of the corals; but it seems to me beyond a doubt that the whole question of the complicated arrangement of septa and dissepiments throughout the group (including the curious vesicular or bubble-like tissue of the Cyathophyllidae and the general structural plan of

the Tetracoralla, such as Streptoplasma and its allies) is well worth investigation from the physical and mathematical point of view, after the fashion which is here slightly adumbrated. The method of dividing a circular, or spherical, system into eight parts, equal as to their areas but unequal in their peripheral boundaries, is probably of wide biological application; that is to say, without necessarily supposing it to be rigorously followed, the typical configuration which it yields seems to recur again and again, with more or less approximation to precision, and under widely different circumstances. I am inclined to think, for instance, that the unequal division of the surface of a Ctenophore by its

meridian-like ciliated bands is a case in point (Fig. 175). Here, if we imagine each quadrant to be twice bisected by a curved anticline, we shall get what is apparently a close approximation to the actual position of the ciliated bands. The case however is complicated by the fact that the sectional plan of the organism is never quite circular, but always more or less elliptical. One point, at least, is clearly seen in the symmetry of the Ctenophores; and that is that the radiating canals which pass outwards to correspond in position with the ciliated bands, have no common centre, but diverge from one another by repeated bifurcations, in a manner comparable to the conjunctions of our cell-walls.

In like manner I am inclined to suggest that the same principle may help us to understand the apparently complex arrangement Fig. 176. Diagrammatic arrangement of partitions, represented by skeletal rods, in larval Echinoderm (Ophiura). of the skeletal rods of a larval Echinoderm, and the very complex conformation of the larva which is brought about by the presence of these long, slender skeletal radii. In Fig. 176 I have divided a circle into its four quadrants, and have bisected each quadrant by a circular arc (BC), passing from radius to periphery, as in the foregoing cases of cell-division; and I have again bisected, in a similar way, the triangular halves of each quadrant (DD). I have also inserted a small circle in the middle of the figure, concentric with the large one. If now we imagine those lines in the figure which I have drawn black to be replaced by solid rods we shall have at once the frame-work of an Ophiurid (Pluteus) larva. Let us imagine all these arms to be

bent symmetrically downwards, so that the plane of the paper is transformed into a spheroidal surface, such as that of a hemisphere, or that of a tall conical figure with curved sides; let a membrane be spread, umbrella-like, between the outstretched skeletal rods, and let its margin loop from rod to rod in curves which are possibly catenaries, but are more probably portions of an “elastic curve,’ and the outward resemblance to a Pluteus larva is now complete. By various slight modifications, by altering the relative lengths of the rods, by modifying their curvature or by replacing the curved rod by a tangent to itself, we can ring the changes which lead us from one known type of Pluteus to another. The case of the Bipinnaria larvae of Echinids is certainly analogous, but it becomes very much more complicated; we have to do with a more

complex partitioning of space, and I confess that | am not yet able to represent the more complicated forms in so simple a way. There are a few notable exceptions (besides the various unequally segmenting eggs) to the general rule that in cell-division the mother-cell tends to divide into equal halves; and one of these exceptional cases is to be found in connection with the development of “stomata” in the leaves of plants. The epidermal cells by which the leaf is covered may be of various shapes; sometimes, as in a hyacinth, they are oblong, but more often they have an irregular shape in which we can recognise, more or less clearly, a distorted or imperfect hexagon. In the case of the oblong cells, a transverse partition will be the least possible, whether the cell be equally or unequally divided, unless (as we have already seen

the space to be cut off be a very small one, not more than about three-tenths the area of a square based on the short side of the original rectangular cell. As the portion usually cut off is not nearly so small as this, we get the form of partition shewn in Fig. 179, and the cell so cut off is next bisected by a partition at right angles to the first; this latter partition splits, and the two last-formed cells constitute the so-called “guard-cells” of the stoma. In other cases, as in Fig. 178, there will come a point where the minimal partition necessary to cut off the required fraction of the cell-content is no longer a transverse one, but is a portion of a cylindrical wall (2) cutting off one corner of the

is now a certain segment of a 2b circle, with an arc of approximately 120°; and its next division will be by means of a curved wall Fig. 179. Diagrammatic development and at length (for a reason which of stomata in Hyacinth. is not yet clear) the partition wall between the new-formed cells splits, and again we have the phenomenon of a “stoma” with its attendant guard-cells. In Fig. 179 are shewn the successive stages of division, and the changing curvatures of the various walls which ensue as each subsequent partition appears, introducing a new tension into the system.

It is obvious that in the case of the oblong cells of the epidermis in the hyacinth the stomata will be found arranged in regular rows, while they will be irregularly distributed over the surface of the leaf in such a case as we have depicted in Sedum. While, as I have said, the mechanical cause of the split which constitutes the orifice of the stoma is not quite clear, yet there can be little or no doubt that it, like the rest of the phenomenon, is related to surface tension. It might well be that it is directly due to the presence underneath this portion of epidermis of the hollow air-space which the stoma is apparently developed “for the purpose” of communicating with; this air-surface on both. sides of the delicate epidermis might well cause such an alteration of tensions that the two halves of the dividing cell would tend to part company. In short, if the surface-energy in a cell-air contact were half or less than half that in a contact between cell and cell, then it is obvious that our partition would tend to split, and give us a two-fold surface in contact with air, instead of the original boundary or interface between one cell and the other. In Professor ‘Macallum’s experiments, which we have briefly discussed in our short chapter on Adsorption, it was found that large quantities of potassium gathered together along the outer walls of the guardcells of the stoma, thereby indicating a low surface-tension along these outer walls. The tendency of the guard-cells to bulge outwards is so far explained, and it is possible that, under the existing conditions of restraint, we may have here a force tending, or helping, to split the two cells asunder. It is clear enough, however, that the last stage in the development of a stoma, is, from the physical point of view, not yet properly understood.

In all our foregoing examples of the development of a “tissue” we have seen that the process consists in the successive division of cells, each act of division being accompanied by the formation of a boundary-surface, which, whether it become at once a solid or semi-solid partition or whether it remain semi-fluid, exercises in all cases an effect on the position and the form of the boundary which comes into being with the next act of division. In contrast to this general process stands the phenomenon known as “free cell-formation,”’ in which, out of a common mass of protoplasm, a number of separate cells are simultaneously, or all but simu!- taneously, differentiated. In a number of cases it happens that, to begin with, a number of “mother-cells” are formed simultaneously, and each of these divides, by two successive divisions,

into four “daughter-cells.”” These daughter-cells will tend to group themselves, just as would four soap-bubbles, into a “tetrad,” the four cells corresponding to the angles of a regular tetrahedron. For the system of four bodies is evidently here in perfect symmetry ; the partition-walls and their respective edges meet at equal angles: three walls everywhere meeting in an edge, and the four edges converging to a point in the geometrical centre of the system. This is the typical mode of development of pollengrains, common among Monocotyledons and all but universal . among Dicotyledonous plants. By a loosening of the surrounding tissue and an expansion of the cavity, or anther-cell, in which

they lie, the pollen-grains afterwards fall apart, and their individual form will depend upon whether or no their walls have solidified before this hberation takes place. For if not, then the separate grains will be free to assume a spherical form as a consequence of their own individual and un- © restricted growth; but if they become solid or rigid prior to the separation of the Go tetrad, then they will conserve more or less completely the plane interfaces and sharp Fig.181. Dividingspore angles of the elements of the tetrahedron. Cee pany CENCE: The latter is the case, for instance, in the pollen-grains of Epilobium (Fig. 180, 1) and in many others. In the Passion-flower (2) we have an intermediate condition: where we can still see an indication of the facets where the grains abutted on one another in the tetrad, but the plane faces have been swollen by growth into spheroidal or spherical surfaces. It is obvious that there may easily be cases where the tetrads of daughter-cells are prevented from assuming the tetrahedral form: cases, that is to say, where the four cells are forced and crushed into one plane. The figures given by Goebel of the development of the pollen of Neottia (3, a-e: all the figures referring to grains taken from a single anther), illustrate this to perfection; and it will be seen that, when the four cells he in a plane, they conform exactly to our typical diagram of the first four cells in a segmenting ovum. Occasionally, though the four cells lie in a plane, the diagram seems to fail us, for the cells appear to meet in a simple cross (as in 5); but here we soon perceive that the cells are not in complete interfacial contact, but are kept apart by a little intervening drop of fluid or bubble of air.

The spores of ferns (7) develop in very much the same way as pollen-grains; and they also very often retain traces of the shape which they assumed as members of a tetrahedral figure. Among the “tetraspores” (8) of the Florideae, or Red Seaweeds, we have a phenomenon which is in every respect analogous. Here again it is obvious that, apart from differences in actual magnitude, and apart from superficial or “accidental” differences (referable to other physical phenomena) in the way of colour,

texture and minute sculpture or pattern, it comes to pass, through the laws of surface-tension and the principles of the geometry of position, that a very small number of diagrammatic figures will sufficiently represent the outward forms of all the tetraspores, four-celled pollen-grains, and other four-celled aggregates which are known or are even capable of existence. We have been dealing hitherto (save for some shght exceptions) with the partitioning of cells on the assumption that the system either remains unaltered in size or else that growth has proceeded uniformly in all directions. But we extend the scope of our enquiry very greatly when we begin to deal with unequal growth, with growth, that is to say, which produces a greater extension along some one axis than another. And here we come close in touch with that great and still (as I think) insufficiently appreciated generalisation of Sachs, that the manner in which the cells divide is the result, and not the cause, of the form of the dividing structure: that the form of the mass is caused by its growth as a whole, and is not a resultant of the growth of the cells individually considered*. Such asymmetry of growth may be easily imagined, and may conceivably arise from a variety of causes. In any individual cell, for instance, it may arise from molecular asymmetry of the structure of the cell-wall, giving it greater rigidity in one direction than another, while all the while the hydrostatic pressure within the cell remains constant and uniform. In an aggregate of cells, it may very well arise from a greater chemical, or osmotic, activity in one than another, leading to a localised increase in the fluid pressure, and to a corresponding bulge over a certain area of the external surface. It might conceivably occur as a direct result of the preceding cell-divisions, when these are such as to produce many peripheral or concentric walls in one part and few or none in another, with the obvious result of strengthening the common boundary wall and resisting the outward pressure of growth in parts where the former is the case; that is to say, in our dividing quadrant, if

its quadrangular portion subdivide by periclines, and the triangular portion by oblique anticlines (as we have seen to be the natural tendency), then we might expect that external growth would be more manifest over the latter than over the former areas. As a direct and immediate consequence of this we might expect a tendency for special outgrowths, or “buds,” to arise from the triangular rather than from the quadrangular cells; and this turns out to be not merely a tendency towards which theoretical considerations point, but a widespread and important factor in the morphology of the cryptogams. But meanwhile, without enquiring further into this complicated question, let us simply take it that, if we start from such a simple case as a round cell which has divided into two halves, or four quarters (as the case may be), we shall at once get bilateral symmetry about a main axis, and other secondary results arising therefrom, as soon as one of the halves, or one of the quarters, begins to shew a rate of growth in advance of the others; for the more rapidly growing cell, or the peripheral wall common to two or more such rapidly growing cells, will bulge out into an ellipsoid form, and may finally extend into a cylinder with rounded or ellipsoid end.

This latter very simple case is illustrated in the development of a pollen-tube, where the rapidly growing cell develops into the elongated cylindrical tube, and the slow-growing or quiescent part remains behind as the so-called “vegetative” cell or cells. Just as we have found it easier to study the segmentation of a circular disc than that of a spherical cell, so let us begin in the same way, by enquiring into the divisions which will ensue if the dise tend to grow, or elongate, in some one particular direction, instead of in radial symmetry. The figures which we shall then obtain will not only apply to the disc, but will also represent, in all essential features, a projection or longitudinal section of a solid body, spherical to begin with, preserving its symmetry as a solid of revolution, and subject to the same general laws as we have studied in the disc*.

* In the following account I follow closely on the lines laid down by Berthold; Protoplasmamechanik, cap. vii. Many botanical phenomena identical and similar to those here dealt with, are elaborately discussed by Sachs in his Physiology of Plants (chap. xxvii, pp. 431-459, Oxford, 1887); and in his earlier papers, Ueber die Anordnung der Zellen in jiingsten Pflanzentheilen, and Ueber Zellenanordnung (1) Suppose, in the first place, that the axis of growth lies symmetrically in one of the original quadrantal cells of a segmenting disc; and let this growing cell elongate with comparative rapidity before it subdivides. When it does divide, it will necessarily do so by a transverse partition, concave towards the apex of the cell: and, as further elongation takes place, the cylindrical structure which will be developed thereby will tend to be again and again subdivided by similar concave transverse partitions. If at any time, through this process of concurrent elongation and subdivision, the apical cell become equivalent to, or less than, a hemisphere, it will next divide by means of a longitudinal, or

vertical partition; and similar longitudinal partitions will arise in the other segments of the cylinder, as soon as it comes about that their length (in the direction of the axis) is less than their breadth. But when we think of this structure in the solid, we at once perceive that each of these flattened segments of the cylinder, into which our cylinder has divided, is equivalent to a flattened circular disc; and its further division will accordingly tend to proceed like any other flattened disc, namely into four quadrants, and afterwards by anticlines and periclines in the usual way. und Wachsthum (Arb. d. botan. Inst. Wurzburg, 1878, 1879). But Sachs’s treatment differs entirely from that which I adopt and advocate here: his explanations being based on his “law” of rectangular succession, and involving complicated

systems of confocal conics, with their orthogonally intersecting ellipses and hyperbolas. A section across the cylinder, then, will tend to shew us precisely the same arrangements as we have already so fully studied in connection with the typical division of a circular cell into quadrants, and of these quadrants into triangular and quadrangular portions, and so on. But there are other possibilities to be considered, in regard to the mode of division of the elongating quasi-cylindrical portion, as it gradually develops out of the growing and bulging quadrantal cell; for the manner in which this latter cell divides will simply depend upon the form it has assumed before each successive act of division takes place, that is to say upon the ratio between its rate of growth and the frequency of its successive divisions. For, as we have already seen, if the growing cell attain a markedly oblong or cylindrical form before division ensues, then the partition will arise transversely to the long axis; if it be but a little more than a hemisphere, it will divide by an oblique partition; and if it be less than a hemisphere (as it may come to be after successive _ transverse divisions) it will divide by a vertical partition, that is to say by one coinciding with its axis of growth. An immense number of permutations and combinations may arise in this way, and we must confine our illustrations to a small number of cases. The important thing is not so much to trace out the various conformations which may arise, but to grasp the fundamental principle: which is, that the forces which dominate the form of each cell regulate the manner of its subdivision, that is to say the form of the new cells into which it subdivides; or in other words, the form of the growing organism regulates the form and number of the cells which eventually constitute it. The complex cell-network is not the cause but the result of the general configuration, which latter has its essential cause in whatsoever physical and chemical processes have led to a varying velocity of growth in one direction as compared with another.

In the annexed figure of an embryo of Sphagnum we see a mode of development almost precisely corresponding to the hypothetical case which we have just described,—the case, that is to say, where one of the four original quadrants of the mothercell is the chief agent in future growth and development. We see at the base of our first figure (a), the three stationary, or undivided quadrants, one of which has further slowly divided in the stage b. The active quadrant

; has grown quickly into a cylindrical structure, which inevitably divides, in the next place, into a series of trans- a 5 verse partitions; and accordingly, this mode of development carries with it the presence of a single “apical cell,” whose lower wall is a spherical surface with its convexity downwards. Hach cell of the subdivided cylinder now appears as a more or less flattened disc, whose mode of further sub-division we may prognosticate according to our former investigation, to which subject we shall presently return. (2) In the next place, still keeping to the case where only one of the original quadrant-cells continues to grow and develop, let us suppose that this growing cell falls to be divided when by growth it has become just a little greater than a hemisphere; it

will then divide, as in Fig. 184, 2, by an oblique partition, in the usual way, whose precise position and inclination to the base will depend entirely on the configuration of the cell itself, save only, of course, that we may have also to take into account the possibility of the division being into two unequal halves. By our hypothesis, the growth of the whole system is mainly in a vertical direction, which is as much as to say that the more actively growing protoplasm, or at least the strongest osmotic force, will be found near the apex; where indeed there is obviously more external surface for osmotic action. It will therefore be that one of the two cells which contains, or constitutes, the apex which will grow more rapidly than the other, and which therefore will be the first to divide, and indeed in any case, it will usually be this one of the two which will tend to divide first, inasmuch as the triangular and not the quadrangular half is bound to constitute the apex*. It is obvious that (unless the act of division be so long postponed that the cell has become quasi-cylindrical) it will divide by another oblique partition, starting from, and running at right angles to, the first. And so division will proceed, by oblique alternate partitions, each one tending to

triangular form. The annexed figure of the developing antheridium of a Liverwort (Riccia) is a typical example of such a case. In Fig. 185 which represents a a as “gemma” of a Moss, we see just the same thing; Aiki with this addition, that here the lower of the two sisal original cells has grown even more quickly than the other, constituting a long cylindrical stalk, and dividing in accordance with its shape, by means of transverse septa. In all such cases as these, the cells whose development we have studied will in turn tend to subdivide, and the manner in which they will do so must depend upon their own proportions; and in all cases, as we have already seen, there will sooner or later be a tendency to the formation of periclinal walls, cutting off an “epidermal layer of cells,” as Fig. 186 illustrates very well.