On Growth and Form

divide (just as the segmenting egg does), by a partition transverse to its polar axis. Such a polarity may conceivably be due to a chemical asymmetry, or anisotropy, such as we have learned of (from Professor Macallum’s experiments) in our chapter on Adsorption. Now if the chemical concentration, on which this anisotropy or polarity (by hypothesis) depends, be unsymmetrical, one of its poles being as it were deflected to one side, where a little branch or bud is being (or about to be) given off,—all in precise accordance with the adsorption phenomena described on p. 289,— then our “polar axis’’ would necessarily be a curved axis, and the partition, being constrained (again ex hypothesz) to arise transversely to the polar axis, would lie obliquely to the apparent axis of the cell (Fig. 143, B, C). And if the oblique partition be so situated that it has to meet the opposite walls (as in C), then, in order to do so symmetrically (i.e. either perpendicularly, as when the cell-wall is already solidified, or at least at equal angles on either side), it is evident that the partition, in its course from one side of the cell to the other, must necessarily assume a more or less S-shaped curvature (Fig. 143, D).

As a matter of fact, while we have abundant simple illustrations of the principles which we have now begun to study, apparent exceptions to this simplicity, due to an asymmetry of the cell itself, or of the system of which the single cell is but a part, are by no means rare. For example, we know that in cambium-cells, division frequently takes place parallel to the long axis of the cell, when a partition of much less area would suffice if it were set cross-ways: and it is only when a considerable disproportion has been set up between the length and breadth of the cell, that the balance is in part redressed by the appearance of a transverse partition. It was owing to such exceptions that Berthold was led to qualify and even to depreciate the importance of the law of minimal areas as a factor in cell-division, after he himself had done so much to demonstrate and elucidate it*. He was deeply and rightly impressed by the fact that other forces besides surface

* Cf. Protoplasmamechanik, p. 229: “Insofern liegen also die Verhaltnisse hier wesentlich anders als bei der Zertheilung hohler Kérperformen durch fliissige Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der kleinsten Flichen geforderte Lage und Kriimmung annimmt, so werden wir den Grund dafiir in andrer Weise abzuleiten haben.” tension, both external and internal to the cell, play their part in the determination of its partitions, and that the answer to our problem is not to be given in a word. How fundamentally important it is, however, in spite of all conflicting tendencies and apparent exceptions, we shall see better and better as we proceed.

But let us leave the exceptions and return to a consideration of the simpler and more general phenomena. And in so doing, let us leave the case of the cubical, quadrangular or cylindrical cell, and examine the case of a spherical cell and of its successive divisions, or the still simpler case of a circular, discoidal cell. When we attempt to investigate mathematically the position and form of a partition of minimal area, it is plain that we shall be dealing with comparatively simple cases wherever even one | dimension of the cell is much less than the other two. Where two dimensions are small compared with the third, as in a thin cylindrical filament hke that of Spirogyra, we have the problem at its simplest; for it is at once obvious, then, that the partition must lie transversely to the long axis of the thread. But even where one dimension only is relatively small, as for instance in a flattened » plate, our problem is so far simplified that we see at once that the partition cannot be parallel to the extended plane, but must cut the cell, somehow, at right angles to that plane. In short, the problem of dividing a much flattened solid becomes identical with that of dividing a simple surface of the same form.

There are a number of small Algae, growing in the form of small flattened discs, consisting (for a time at any rate) of but a single layer of cells, which, as Berthold shewed, exemplify this comparatively simple problem; and we shall find presently that it is also admirably illustrated in the cell-divisions which occur in the egg of a frog or a sea-urchin, when the egg for the sake of experiment is flattened out under artificial pressure. Fig. 144 (taken from Berthold’s Monograph of the Naples Bangiaciae) represents younger and older discs of the little alga Erythrotrichia discigera; and it will be seen that, in all stages save the first, we have an arrangement of cell-partitions which looks somewhat complex, but into which we must attempt to throw some light and order. Starting with the original single, and flattened,

cell, we have no difficulty with the first two cell-divisions; for we know that no bisecting partitions can possibly be shorter than the two diameters, which divide the cell into halves and into quarters. We have only to remember that, for the sum total of partitions to be a minimum, three only must meet in a point; and therefore, the four quadrantal walls must shift a little, pro- - ducing the usual little median partition, or cross-furrow, instead of one common, central point of junction. This little intermediate wall, however, will be very small, and to all intents and purposes we may deal with the case as though we had now to do with four equal cells, each one of them a perfect quadrant. And so our problem is, to find the shortest line which shall divide the quadrant of a circle into two halves of equal area... A radial partition (Fig. 145, a), starting from the apex of the quadrant, is at once excluded, for a reason similar to that just referred to; our choice must lie therefore between two modes of division such - as are illustrated in Fig. 145, where the partition is either (as in B)

concentric with the outer border of the cell, or else (as in c) cuts that outer border; in other words, our partition may (B) cut both radial walls, or (c) may cut one radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) periclinal, and (c) anticlinal*. We may either treat the walls of the dividing quadrant as already solidified, or at least as having a tension compared with which that of the incipient partition film is inconsiderable. In either case the partition must meet the cell-wall, on either side, at right angles, and (its own tension and curvature being everywhere uniform) it must take the form of a circular are.

Now we find that a flattened cell which is approximately a quadrant of a circle invariably divides after the manner of Fig. 145, c, that is to say, by an approximately circular, anticlinal wall, such as we now recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let us then consider that Nature has solved our problem for us, and let us work out the actual geometric conditions. Let the quadrant OAB {in Fig. 146) be divided into two parts of equal area, by the circular are MP. It is required to determine (1) the position of P upon the are of the quadrant, that is to say the angle BOP; (2) the position of the point M on the side OA; and (3) the length of the arc MP in terms of a radius of the quadrant.

(1) Draw OP; also PC a tangent, meeting OA in C; and PN, perpendicular to OA. Let us call aa radius; and @ the angle at C, which is obviously equal to OPN, or POB. Then * There is, I think, some ambiguity or disagreement among botanists as to the use of this latter term: the sense in which I am using it, viz. for any partition which meets the outer or peripheral wall at right angles (the strictly radial partition being for the present excluded), is, however, clear.

We see accordingly that the equation is solved (as accurately as need be) when @ is an angle somewhat over 34° 38’, or say 34° 381’. That is to say, a quadrant of a circle is bisected by a circular arc cutting the side and the periphery of the quadrant at right angles, when the arc is such as to include (90° — 34° 38’), i.e. 55° 22’ of the quadrantal arc. This determination of ours is practically identical with that which Berthold arrived at by a rough and ready method, without the use of mathematics. He simply tried various ways of dividing a quadrant of paper by means of a circular arc, and went on doing so till he got the weights of his two pieces of paper approximately equal. The angle, as he thus determined it, was 34-6°, or say

(2) The position of M on the side of the quadrant OA is given by the equation OM = acosec @—acot@; the value of which expression, for the angle which we have just discovered, is -3028. That is to say, the radius (or side) of the quadrant will be divided by the new partition into two parts, in the proportions of nearly three to seven. (3) The length of the arc MP is equal to a@ cot 6; and the value of this for the given angle is -8751. This is as much as to say that the curved partition-wall which we are considering is shorter than a radial partition in the proportion of 8? to 10, or seven-eights almost exactly.

But we must also compare the length of this curved “ antielinalt partition-wall (MP) with that of the concentric, or periclinal, one (RS, Fig. 147) by which the quadrant might also be bisected. The length of this partition is obviously equal to the arc of the quadrant (i.e. the peripheral wall of the cell) divided by 1/2;- (such as we actually find in nature) notably the best, but the periclinal one, when it comes to dividing an entire quadrant, is

The two cells into which our original quadrant is now divided, while they are equal in volume, are of very different shapes; the one is a triangle (MAP) with two sides formed of circular arcs, and the other is a four-sided figure (WOBP), which we may call approximately oblong. We cannot say as yet how the triangular portion ought to divide; but it is obvious that the least possible partition-wall which shall bisect the other must run across the long axis of the oblong, that is to say periclinally. This, also, 1s precisely what tends actually to take place. In the following diagrams (Fig. 148) of a frog’s egg dividing under pressure, that is to say when reduced to the form of a flattened plate, we see, firstly, the division into four quadrants (by the partitions 1, 2); secondly, the division of each quadrant by means of an anticlinal circular arc (3, 3), cutting the peripheral wall of the quadrant approximately in the proportions of three to seven; and thirdly,

Fig. 148. Segmentation of frog’s egg, under artificial compression. (After Roux.) we see that of the eight cells (four triangular and four oblong) into which the whole egg is now divided, the four which we have called oblong now proceed to divide by partitions transverse to their long axes, or roughly parallel to the periphery of the egg. The question how the other, or triangular, portion of the divided qudarant will next divide leads us to another well-defined problem, ; which is only a slight extension, making allowance for the circular ares, of that elementary problem of the triangle we have already considered. We know now that an entire quadrant must divide (so that its bisecting wall shall have the least possible area) by means of an anticlinal partition, but how about any smaller sectors of circles? It is obvious in the case of a small prismatic

sector, such as that shewn in Fig. 149, that a periclinal partition. is the smallest by which we can possibly bisect the cell; we want, accordingly, to know the limits below which the periclinal partition — is always the best, and above which the anticlinal arc, as in the case of the whole quadrant, has the advantage in regard to small- This may be easily determined; for the preceding investigation is a perfectly general one, and the results hold good for sectors of any other arc, as well as for the quadrant, or are of 90°. That is to say, the length of the partition-wall MP is always determined by the angle 0, according to our equation MP = a6 cot @; and the angle @ has a definite relation to a, the angle of arc.

Moreover, in the case of the periclinal boundary, RS (Fig. 147) (or ab, Fig. 149), we know that, if it bisect the cell, In the accompanying diagram (Fig. 150), I have plotted the various magnitudes with which we are concerned, in order to exhibit the several limiting values. Here we see, in the first place, the curve marked a, which shews on the (left-hand) vertical scale the various possible magnitudes of that angle (viz. the angle of arc of the whole sector which we wish to divide), and on the horizontal scale the corresponding values of 0, or the angle which

Angle (@) determining the intersection of the partition-wall with the outer border of the cell. Angle of are (a) of the prismatic sector, or cell, undergoing division. nN Do determines the point on the periphery where it is cut by the partition-wall, MP. Two limiting cases are to be noticed here: (1) at 90° (point A in diagram), because we are at present only Lengths of the several partitions, in terms of a radius (r=1). dealing with arcs no greater than a quadrant; and (2), the point (B) where the angle 6 comes to equal the angle a, for after that point the construction becomes impossible, since an anticlinal bisecting partition-wall would be partly outside the cell. The only partition which, after the poimt, can possibly exist, is a periclinal one. This point, as our diagram shews us, occurs when the angles (a and 6) are each rather under 52°.

Next I have plotted, on the same diagram, and in relation to the same scales of angles, the corresponding lengths of the two partitions, viz. RS and MP, their lengths being expressed (on the right-hand side of the diagram) in relation to the radius of the circle (a), that is to say the side wall, OA, of our cell. The limiting values here are (1), C, C’, where the angle of are is 90°, and where, as we have already seen, the two partition-walls have the relative magnitudes of MP: RS = 0-875: 1-111; (2) the point D, where RS equals unity, that is to say where the periclinal partition has the same length as a radial one; this occurs when a is rather under 82° (cf. the pomts D, D’); (3) the point #, where RS and MP intersect; that is to say the point at which the two partitions, periclinal and anticlinal, are of the same magnitude; this is the case, according to our diagram, when the angle of arc is just over 623°. We see from this, then, that what we have called an anticlinal partition, as MP, is only likely to occur in a triangular or prismatic cell whose angle of arc les between 90° and 623°. In all narrower or more tapering cells, the periclinal partition will be of less area, and will therefore be more and more likely to occur. :

The case (Ff) where the angle a is just 60° is of some interest. Here, owing to the curvature of the peripheral border, and the consequent fact that the peripheral angles are somewhat greater than the apical angle a, the perichnal partition has a very slight and almost imperceptible advantage over the anticlinal, the relative proportions being about as MP: RS = 0-73: 0-72° Butit the equilateral triangle be a plane spherical triangle, i.e. a plane triangle bounded by circular arcs, then we see that there is no longer any distinction at all between our two partitions; MP and RS are now identical.

On the same diagram, I have inserted the curve for values of cosec 6 — cot d = OM, that is to say the distances from the centre, along the side of the cell, of the starting-point (1) of the anticlinal partition. The point C” represents its position in the case of a quadrant, and shews it to be (as we have already said) about 3/10 of the length of the radius from the centre. If, on the other hand, our cell be an equilateral triangle, then we have to read off the point on this curve corresponding to a = 60°, and we find it at the point F’” (vertically under F), which tells us that the partition now starts 4-5/10, or nearly halfway, along the radial wall.

The foregoing considerations carry us a long way in our investigations of many of the simpler forms of cell-division. Strictly speaking they are limited to the case of flattened cells, in which we can treat the problem as though we were simply partitioning a plane surface. But it is obvious that, though they do not teach us the whole conformation of the partition which divides a more complicated solid into two halves, yet they do, even in such a case, enlighten us so far, that they tell us the appearance presented in one plane of the actual solid. And as this is all that we see in a microscopic section, it follows that the results we have arrived at will greatly help us in the interpretation of microscopic appearances, even in comparatively complex cases of cell-division.

Let us now return to our quadrant cell (OAPB), which we B' have found to be divided into a triangular and a quadrilateral portion, as in Fig. 147 or Fig. 151; and let us now suppose the whole _ system to grow, in a uniform fashion, as a prelude to further subdivision. The whole quadrant, growing uniformly (or with equal radial increments), will still remain a quadrant, and it is a cual obvious, therefore, that for every new increment of size, more will be added to the margin of its triangular portion than to the

narrower margin of its quadrilateral portion; and these increments will be in proportion to the angles of arc, viz. 55° 22’: 34° 38’, or as -96:-60, 1.e. as 8:5. And accordingly, if we may assume (and the assumption is a very plausible one), that, just as the quadrant itself divided into two halves after it got to a certain size, so each of its two halves will reach the same size before again dividing, it is obvious that the triangular portion will be doubled in size, and therefore ready to divide, a considerable time before the quadrilateral part. To work out the problem in detail would lead us into troublesome mathematics; but if we simply assume that the increments are proportional to the increasing radii of the circle, we have the following equations :—

Let us call the triangular cell 7, and the quadrilateral, Q@ (Fig. 151); let the radius, OA, of the original quadrantal cell =a=1; and let the increment which is required to add on a portion equal to 7 (such as PP’A’A) be called xz, and let that required, similarly, for the doubling of Q be called 2’. This is as much as to say that, supposing each cell tends to divide into two halves when (and not before) its original size is doubled, then, in our flattened disc, the triangular cell 7 will tend to divide when the radius of the disc has increased by about a third (from 1 to 1-345), but the quadrilateral cell, Y, will not tend to divide until the linear dimensions of the disc have increased by about a half (from | to 1-517).

The case here illustrated is of no small general importance. For it shews us that a uniform and symmetrical growth of the organism (symmetrical, that is to say, under the limitations of a plane surface, or plane section) by no means involves a uniform or symmetrical growth of the individual cells, but may, under certain conditions, actually lead to inequality among these; and this inequality may be further emphasised by differences which arise out of it, in regard to the order of frequency of further ‘subdivision. This phenomenon (or to be quite candid, this hypothesis, which is due to Berthold) is entirely independent of any change or variation in individual surface tensions; and accordingly it is essentially different from the phenomenon of unequal segmentation (as studied by Balfour), to which we have referred on p. 348.

In this fashion, we might go on to consider the manner, and the order of succession, in which the subsequent cell-divisions would tend to take place, as governed by the principle of minimal areas. But the calculations would grow more difficult, or the results got by simple methods would grow less and less exact. At the same time, some of these results would be of great interest, _and well worth the trouble of obtaining. For instance, the precise manner in which our triangular cell, 7, would next divide would be interesting to know, and a general solution of this problem is certainly troublesome to calculate. But in this particular case we can see that the width of the triangular cell near P is so obviously less than that near either of the other two angles, that a circular are cutting off that angle is bound to be the shortest possible bisecting line; and that, in short, our triangular cell will tend to subdivide, just lke the original quadrant, into a triangular and a quadrilateral portion.

But the case will be different next time, because in this new triangle, PRQ, the least width is near the innermost angle, that at Q; and the bisecting circular are will therefore be opposite to Q, or (approximately) parallel to PR. The importance of this fact is at once evident; for it means to say that there soon comes a time when, whether by the division of triangles or of quadrilaterals, we find only quadrilateral cells adjoiming the periphery of our circular disc. In the subsequent division of these quadrilaterals, the partitions will arise transversely to their long axes, that is to say, radially (as U, V); and we shall consequently have a superficial or peripheral layer of quadrilateral cells, with sides approximately parallel, that is to say what we are accustomed to call an_ epidermis. And this epidermis or superficial layer will be’in clear contrast with the more irregularly shaped cells, the products of triangles and quadrilaterals, which make up the deeper, underlying layers of tissue.

In following out these theoretic principles and others like to them, in the actual division of living cells, we must always bear in mind certain conditions and qualifications. In the first place, the law of minimal area and the other rules which we have arrived at are not absolute but relative: they are links, and very important links, in a chain of physical causation; they are always at work, but their effects may be overridden and concealed by the operation of other forces. Secondly, we must remember that, in the great majority of cases, the cell-system which we have in view is constantly increasing in magnitude by active growth; and by this means the form and also the proportions of the cells are continually liable to alteration, of which phenomenon we have already had an example. Thirdly, we must carefully remember that, until our cell-walls become absolutely solid and rigid, they are always apt to be modified in form owing to the tension of the adjacent

walls; and again, that so long as our partition films are fluid or semifluid, their points and lines of contact with one another may shift, like the shifting outlines of a system of soap-bubbles. This is the physical cause of the movements frequently seen among segmenting cells, like those to which Rauber called attention in ‘ the segmenting ovum of the frog, and like those more striking movements or accommodations which give rise to a so-called “spiral” type of segmentation.

Bearing in mind, then, these considerations, let us see what our flattened disc is likely to look like, after a few successive Fig. 153. Diagram of flattened or discoid cell dividing into octants: to shew gradual tendency towards a position of equilibrium. divisions into component cells. In Fig. 153, a, we have a diagrammatic representation of our disc, after it has divided into four quadrants, and each of these in turn into a triangular and a quadrilateral portion; but as yet, this figure scarcely suggests to us anything like the normal look of an aggregate of living cells. But let us go a little further, still limiting ourselves, however, to the consideration of the eight-celled stage. Wherever one of our radiating partitions meets the peripheral wall, there will (as we know) be a mutual tension between the three convergent films, which will tend to set their edges at equal angles to one another, angles that is to say of 120°. In consequence of this, the outer wall of each individual cell will (in this surface view of our disc)