On Growth and Form

In our curve of growth. (Fig. 35) we cannot safely interpolate values for the first three days, that is to say for the dates between amputation and the first actual measurement of the regenerated part. What goes on in these three days is very important; but we know nothing about it, save that our curve descended to zero somewhere or other within that period. As we have already learned, we can more or less safely interpolate between known points, or actual observations; but here we have no known starting-point. In short, for all that the observations tell us, and for all that the appearance of the curve can suggest, the curve of growth may have descended evenly to the base-line, which it would then have reached about the end of the second

day; or it may have had within the first three days a change of direction, or “point of inflection,” and may then have sprung Fig. 35. Curve of regenerative growth in tadpoles’ tails. (From M. L. Durbin’s data.) at once from the base-line at zero. That is to say, there may have been an intervening “latent period,’ during which no growth occurred, between the time of injury and the first measurement of regenerative growth; or, for all we yet know, regeneration may have begun at once, but with a velocity much less than that - which it afterwards attained. This apparently trifling difference would correspond to a very great difference in the nature of the phenomenon, and would lead to a very striking difference in the curve which we have next to draw. .

The curve already drawn (Fig. 35) illustrates, as we have seen, the relation of length to time, i.e. L/T = V. The second (Fig. 36) represents the rate of change of velocity; it sets V against T; and V/T or L/T?, represents (as we have learned) the acceleration of growth, this being simply the “differential coefficient,” the first derivative of the former curve. Now, plotting this acceleration curve from the date of the - first measurement made three days after the amputation of the tail (Fig. 36), we see that it has no point of inflection, but falls steadily, only more and more slowly, till at last it comes down nearly to the base-line. The velocities of growth are continually diminishing. As regards the missing portion at the beginning of the curve, we cannot be sure whether it bent round and came down

to zero, or whether, as in our ordinary acceleration curves of growth from birth onwards, it started from a maximum. The former is, in this case, obviously the more probable, but we cannot be sure. As regards that large portion of the curve which we are acquainted with, we see that it resembles the curve known as a rectangular hyperbola, which is the form assumed when two variables (in this case V and 7) vary inversely as one another. If we take the logarithms of the velocities (as given in the table) and plot them against time (Fig. 37), we see that they fall, approximately, into a straight line; and if this curve be plotted on the

proper scale we shall find that the angle which it makes with the base is about 25°, of which the tangent is -46, or in round numbers $. Had the angle been 45° (tan 45° = 1), the curve would have been actually a rectangular hyperbola, with V7 = constant. As it is, we may assume, provisionally, that it belongs to the same family of curves,so that V™ 7”, or V™/"T, or VT”!™, are all severally constant. In other words, the velocity varies inversely as some power of the time, or vice versa. And in this particular case, the equation V7? = constant, holds very nearly true; that is to say the velocity varies, or tends to vary, inversely as the square of

ote ae en B= 10--, 12. si aun omnes days Fig. 38. Rate of regenerative growth in larger tadpoles. the time. If some general law akin to this could be established as a general law, or even as a common rule, it would be of great importance. But though neither in this case nor in any other can the minute increments of growth during the first few hours, or the first couple of days, after injury, be directly measured, yet the most important point is quite capable of solution. What the foregoing curve leaves us in ignorance of, is simply whether growth starts at zero, with zero velocity, and works up quickly to a maximum velocity from which it afterwards gradually falls away; or whether after a latent period, it begins, so to speak, in full force. The answer

to this question-depends on whether, in the days following the first actual measurement, we can or cannot detect a daily ancrement in velocity, before that velocity begins its normal course of diminu- : tion. Now this preliminary ascent to a maximum, or point of inflection of the curve, though not shewn in the above-quoted experiment, has been often observed: as for instance, in another similar experiment by the author of the former, the tadpoles being in this case of larger size (average 49-1 mm.)*.

Here we have just what we lacked in the former case, namely a visible point of inflection in the curve about the seventh day (Figs. 38, 39), whose existence is confirmed by successive observations on the 3rd, 5th, 7th and 10th days, and which justifies to some extent our extrapolation for the otherwise unknown period up to and ending with the third day; but even here there is a short space near the very beginning during which we are not quite sure of the precise slope of the curve.

We have now learned that, according to these experiments, with which many others are in substantial agreement, the rate of growth in the regenerative process is as follows. After a very short latent period, not yet actually proved but whose existence is highly probable, growth commences with a velocity which very rapidly increases to a maximum. The curve quickly,—almost suddenly,—changes its direction, as the velocity begins to fall; and the rate of fall, that is, the negative acceleration, proceeds at a slower and slower rate, which rate varies inversely as some power of the time, and is found in both of the above-quoted experiments to be very approximately as 1/T?. But it is obvious that the value which we have found for the latter portion of the curve (however closely it be conformed to) is only an empirical value; it has only a temporary usefulness, and must in time give

Fig. 39. Daily increment, or amount regenerated, corresponding to Fig, 38. place to a formula which shall represent the entire phenomenon, from start to finish. While the curve of regenerative growth is apparently different from the curve of ordinary growth as usually drawn (and while this apparent difference has been commented on and treated as valid by certain writers) we are now in a position to see that it only looks different because we are able to study it, if not from the beginning, at least very nearly so: while an ordinary curve of growth, as it is usually presented to us, is one which dates, not

from the beginning of growth, but from the comparatively late, and unimportant, and even fallacious epoch of birth. A complete curve of growth, starting from zero, has the same essential characteristics as the regeneration curve. Indeed the more we consider the phenomenon of regeneration, the more plainly does it shew itself to us as but a particular case of the general phenomenon of growth*, following the same lines, obeying the same laws, and merely started into activity by the special stimulus, direct or indirect, caused by the infliction of a wound. Neither more nor less than in other problems of physiology are we called upon, in the case of regeneration, to indulge in metaphysical speculation, or to dwell upon the beneficent purpose which seemingly underlies this process of healing and restoration.

It is a very general rule, though apparently not a universal one, that regeneration tends to fall somewhat short of a complete restoration of the lost part; a certain percentage only of the lost tissues 1s restored. This fact was well known to some of those old investigators, who, like the Abbé Trembley and hke Voltaire, found a fascination in the study of artificial injury and the regeneration which followed it. Sir John Graham Dalyell, for instance, says, in the course of an admirable paragraph on regeneration T : “The reproductive faculty...is not confined to one portion, but may extend over many; and it may ensue even in relation to the regenerated portion more than once. Nevertheless, the faculty gradually weakens, so that in general every successive regeneration is smaller and more imperfect than the organisation preceding it; and at length it is exhausted.”

In certain minute animals, such as the Infusoria, in which the capacity for “regeneration” is so great that the entire animal may be restored from the merest fragment, it becomes of great interest to discover whether there be some definite size at which the fragment ceases to display this power. This question has * The experiments of Loeb on the growth of Tubulazia in various saline solutions, referred to on p. 125, might as well or better have been referred to under the heading of regeneration, as they were performed on cut pieces of the zoophyte. (Cf. Morgan, op. cil. p. 35.)

been studied by Lillie*, who found that in Stentor, while still smaller fragments were capable of surviving for days, the smallest portions capable of regeneration were of a size equal to a sphere of about 80 u in diameter, that is-to say of a volume equal to about one twenty-seventh of the average entire animal. He arrives at the remarkable conclusion that for this, and for all other species of animals, there is a “minimal organisation mass,” that is to say a “minimal mass of definite size consisting of nucleus and cytoplasm within which the organisation of the species can just find its latent expression.”’ And in like manner, Boverit has shewn that the fragment of a sea-urchin’s egg capable of growing up into a new embryo, and so discharging the complete functions of an entire and uninjured ovum, reaches its limit at about one-twentieth of the original egg,—other writers having found a limit at about one-fourth. These magnitudes, small as they are, represent objects easily visible under a low power of the microscope, and so stand in a very different category to the minimal magnitudes in which life itself can be manifested, and which we have discussed in chapter IT.

A number of phenomena connected with the linear rate of regeneration are illustrated and epitomised in the accompanying diagram (Fig. 40), which I have constructed from certain data given by Ellis in a paper on the relation of the amount of tail regenerated to the amount removed, in Tadpoles. These data are summarised in the next Table. The tadpoles were all very much of a size, about 40 ynm.; the average length of tail was very near to 26 mm., or 65 per cent. of the whole body-length; and in four series of experiments about 10, 20, 40 and 60 per cent. of the tail were severally removed. The amount regenerated in successive intervals of three days is shewn in our table. By plotting the actual amounts regenerated against these three-day intervals of time, we may interpolate values for the time taken to regenerate definite percentage amounts, 5 per cent., 10 per cent., etc. of the

* Lillie, F. R., The smallest Parts of Stentor capable of Regeneration, Journ. of Morphology, xu, p. 239. 1897. + Boveri, Entwicklungsfahigkeit kernloser Seeigeleier, etce., Arch. f. Entw. Mech. tm, 1895. See also Morgan, Studies of the partial larvae of Sphaerechinus, ibid. 1895; J. Loeb, On the Limits of Divisibility of Living Matter, Biol. Lectures, 1894, etc. Fig. 40. Relation between the percentage amount of tail removed, the percentage restored, and the time required for its restoration. (From M. M. Ellis’s data.)

amount removed; and my diagram is constructed from the four sets of values thus obtained, that is to say from the four sets of experiments which differed from one another in the amount of tail amputated. To these we have to add the general result of a fifth series of experiments, which shewed that when as much as 75 per cent. of the tail was cut off, no regeneration took place at all, but the animal presently died. In our diagram, then, each curve indicates the time taken to regenerate n per cent. of the amount removed. All the curves converge towards infinity, when the amount removed (as shewn by the ordinate) approaches 75 per cent.; and all of the curves start from zero, for nothing is regenerated where nothing had been removed. Hach curve approximates in form to a cubic parabola.

The amount regenerated varies also with the age of the tadpole and with other factors, such as temperature; in other words, for any given age, or size, of tadpole and also for various, specific temperatures, a similar diagram might be constructed. The power of reproducing, or regenerating, a lost limb is particularly well developed in arthropod animals, and is sometimes accompanied by remarkable modification of the form of the regenerated limb. A case in point, which has attracted much attention, occurs in connection with the claws of certain Crustacea *.

In many Crustacea we have an asymmetry of the great claws, one being larger than the other and also more or less different in form. For instance, in the common lobster, one claw, the larger of the two, is provided with a few great “crushing” teeth, while the smaller claw has more numerous teeth, small and serrated. Though Aristotle thought otherwise, it appears that the crushingclaw may be on the right or left side, indifferently; whether it be on one or the other is a problem of “‘chance.” It is otherwise in many other Crustacea, where the larger and more powerful claw is always left or right, as the case may be, according to the species: where, in other words, the “probability” of the large or the small claw being left or being right is tantamount to certaintyT.

The one claw is the larger because it has grown the faster ; + Lobsters are occasionally found with two symmetrical claws: which are then ‘ usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. Calman, P.Z.S. 1906, pp. 633, 634, and reff. it has a higher “coefficient of growth,” and accordingly, as age advances, the disproportion between the two claws becomes more and more evident. Moreover, we must assume that the characteristic form of the claw is a “function” of its magnitude; the knobbiness is a phenomenon coincident with growth, and we never, under any circumstances, find the smaller claw with big crushing teeth and the big claw with little serrated ones. There are many other somewhat similar cases where size and form are manifestly correlated, and we have already seen, to some extent, that thé phenomenon of growth is accompanied by certain ratios of velocity that lead inevitably to changes of form. Meanwhile, then, we must simply assume that the essential difference between the two claws is one of magnitude, with which a certain differentiation of form is inseparably associated.

If we amputate a claw, or if, as often happens, the crab “casts it off,” it undergoes a process of regeneration,—it grows anew, and evidently does so with an accelerated velocity, which acceleration will cease when equilibrium of the parts is once more attained : the accelerated velocity being a case in point to illustrate that vis revulsionis of Haller, to which we have already referred. With the help of this principle, Przibram accounts for certain curious phenomena which accompany the process of regeneration. As his experiments and those of Morgan shew, uf the large or knobby claw (A) be removed, there are certain cases, e.g. the common lobster, where it is directly regenerated. In other cases, e.g. Alpheus*, the other claw (B) assumes the size and form of that which was amputated, while the latter regenerates itself in the form of the other and weaker one; A and B have apparently changed places. In a third case, as in the-crabs, the A-claw regenerates itself as a small or B-claw, but the B-claw remains for a time unaltered, though slowly and in the course of repeated moults it later on assumes the large and heavily toothed A-form.

Much has been written on this phenomenon, but in essence it is very simple. It depends upon the respective rates of growth, upon a ratio between the rate of regeneration and the rate of growth of the uninjured limb: complicated a little, however, by the possibility of the uninjured limb growing all the faster for a time after the animal has been relieved of the other. From the time of amputation, say of A, A begins to grow from zero, with a high “regenerative” velocity; while B, starting from a definite magnitude, continues to increase, with its normal or perhaps somewhat accelerated velocity. The ratio between the two velocities of growth will determine whether, by a given time, A has equalled, outstripped, or still fallen short of the magnitude of B.

That this is the gist of the whole problem is confirmed (if confirmation be necessary) by certain experiments of Wilson’s. It is known that by section of the nerve to a crab’s claw, its growth is retarded, and as the general growth of the animal proceeds the claw comes to appear stunted or dwarfed. Now in such a case as that of Alpheus, we have seen that the rate of regenerative growth in an amputated large claw fails to let it reach or overtake the magnitude of the growing little claw: which latter, in short, now appears as the big one. But if at the same time as we amputate the big claw we also sever the nerve to the lesser one, we so far slow down the latter’s growth that the other is able to make up to it, and in this case the two claws continue to grow at approximately equal rates, or in other words continue of coequal size.

The phenomenon of regeneration goes some way towards helping us to comprehend the phenomenon of “multiplication by fission,’ as it is exemplified at least in its simpler cases in many worms and worm-like animals. For physical reasons which we shall have to study in another chapter, there is a natural tendency for any tube, if it have the properties of a fluid or semi-fluid substance, to break up into segments after it comes to a certain length; and nothing can prevent its doing so, except the presence of some controlling force, such for instance as may be due to the pressure of some external support, or some superficial thickening or other intrinsic rigidity of its own substance. If we add to this natural tendency towards fission of a cylindrical or tubular worm, the ordinary phenomenon of regeneration, we have all that is essentially implied in “reproduction by fission.” And in so far

as the process rests upon a physical principle, or natural tendency, we may account for its occurrence in a great variety of animals, zoologically dissimilar; and also for its presence here and absence there, in forms which, though materially different in a physical sense, are zoologically speaking very closely allied. But the phenomena of regeneration, like all the other phenomena of growth, soon carry us far afield, and we must draw this brief discussion to a close.

For the main features which appear to be common to all curves of growth we may hope to have, some day, a physical explanation. In particular we should like to know the meaning of that point of inflection, or abrupt change from an increasing to a decreasing velocity of growth which all our curves, and especially our acceleration curves, demonstrate the existence of, provided only that they include the initial stages of the whole phenomenon: just as we should also like to have a full physical or physiological explanation of the gradually diminishing velocity of growth which follows, and which (though subject to temporary interruption or abeyance) is on the whole characteristic of growth in all cases whatsoever. In short, the characteristic form of the curve of growth in length (or any other linear dimension) is a phenomenon which we are at present unable to explain, but which presents us with a definite and attractive problem for future solution. It would seem evident that the abrupt change in velocity must be due, either to a change in that pressure outwards from within, by which the “forces of growth” make themselves manifest, or to a change in the resistances against which they act, that is to say the tension of the surface; and this latter force we do not by any means limit to “‘surface-tension” proper, but may extend to the development of a more or less resistant membrane or “skin,” or even to the resistance of fibres or other histological elements, binding the boundary layers to the parts within. I take it that the sudden arrest of velocity is much more likely to be due to a sudden increase of resistance than to a sudden diminution of internal energies: in other words, I suspect that it is coincident with some notable event of histological differentiation, such as

the rapid formation of a comparatively firm skin; and that the dwindling of velocities, or the negative acceleration, which follows, is the resultant or composite effect of waning forces of growth on the one hand, and increasing superficial resistance on the other. This is as much as to say that growth, while its own energy tends to increase, leads also, after a while, to the establishment of resistances which check its own further increase, Our knowledge of the whole complex phenomenon of growth is so scanty that it may seem rash to advance even this tentative suggestion. But yet there are one or two known facts which seem to bear upon the question, and to indicate at least the manner in which a varying resistance to expansion may aftect the velocity of growth. For instance, it has been shewn by Frazee* that electrical stimulation of tadpoles, with small current density and low voltage, increases the rate of regenerative growth. As just such an electrification would tend to lower the surface-tension, and accordingly decrease the external resistance, the experiment would seem to support, in some slight degree, the suggestion which I have made.

Delaget has lately made use of the principle of specific rate of growth, in considering the question of heredity itself. We know that the chromatin of the fertilised egg comes from the male and female parent alike, in equal or nearly equal shares; we know that the initial chromatin, so contributed, multiplies many thousand-fold, to supply the chromatin for every cell of the offspring’s body; and it has, therefore, a high “coefficient of growth.” If we admit, with Van Beneden and others, that the initial contributions of male and female chromatin continue to be transmitted to the succeeding generations of cells, we may then conceive these chromatins to retain each its own coefficient of growth; and if these differed ever so little, a gradyal preponderance of one or other would make itself felt in time, and might conceivably explain the preponderating influence of one parent or the other upon the characters of the offspring. Indeed O. Hertwig is said (according to Delage’s interpretation) to have actually shewn that we can artificially modify the rate of growth of one or other chromatin, and so increase or diminish the influence of the maternal or paternal heredity. This theory of Delage’s has its fascination, but it calls for somewhat large assumptions; and in particular, it seems (like so many other theories relating to the chromosomes) to rest far too much upon material elements, rather than on the imponderable dynamic factors of the cell.

We may summarise, as follows, the main results of the foregoing discussion : | (1) Except in certain minute organisms and minute parts of organisms, whose form is due to the direct action of molecular forces, we may look upon the form of the organism as a “function of growth,” or a direct expression of a rate of growth which varies according to its different directions. (2) Rate of growth is subject to definite laws, and the velocities in different directions tend to maintain a ratio which is more or less constant for each specific organism; and to this regularity is due the fact that the form of the organism is in general regular and constant.

(3) Nevertheless, the ratio of velocities in different directions is not absolutely constant, but tends to alter or fluctuate in a regular way; and to these progressive changes are due the changes of form which accompany “development,” and the slower changes of form which continue perceptibly in after life. (4) The rate of growth is a function of the age of the organism , it has a maximum somewhat early in life, after which epoch of maximum it slowly declines.