The Kinetics of Growth

Having constructed a curve of growth which shows as accurately as possible the weight or size of an organism at different moments of time (care being taken to maintain a constant external environment) it is possible to express the observed data in a more compact form by means of an algebraic equation, and as long as this equation is regarded simply as an empirical and convenient way of expressing known facts, the procedure is often useful if not particularly instructive. It is, however, important to go further than this and enquire what type of growth factors could conceivably result in the existence of the observed equation of growth. In other words, by differentiating the equation of a growth curve, an expression is sought for the factors controlling the rate of growth at particular instants of time. The algebraic and graphical treatments of growth data are liable to cause confusion of thought unless they are handled with considerable care, and it seems, perhaps, worth while to consider certain elementary facts associated with these methods before considering their immediate application.

To express the results of a prolonged series of observations in the form of a compact algebraic equation is always convenient and may be of importance. Thus it is of value to have an equation by which the size of a particular type of horse or cow can be calculated for any particular age. At the same time, the utility of the formula must be assessed by the degree of accuracy given by its results ; if the values given by the formula are substantially different to those of direct observation the formula is of limited value ; if the errors of the equation are of the same order of magnitude as the observational errors involved in collecting the data, the formula is of correspondingly greater value. It is clear that the value of an empirical formula depends solely on this relationship—or in other words upon its ability to reproduce observed figures.

A second desideratum of an empirical formula is that it should hold good over a maximum range of observations, or at any rate its range of utility should be accurately defined. Fig. 1 illustrates this point. Let each point on the curve indicate an experimental observation. The smooth curve drawn through all these points is a curve calculated from the formula ⁠. If for experimental reasons it had been necessary to confine observations to the period 24—36 hours, the observed points within this period could have been joined by a line which is so nearly straight that its total curvature would fall within the experimental errors of the observations. Similarly, for the opening stages of the experiment, the points can be found by an experimental curve of the type ⁠. It is clear that, of the three equations, only one applies to the whole range of observations and therefore only this particular curve provides an adequate empirical formula. As long as formulae are entirely empirical, we may simply select that particular one which is most suitable for the range in which we are immediately interested, but as a rule the wider the range of applicability the more useful is the formula.

Finally, it may be remembered that in an empirical formula of growth there are only two symbols which have any real meaning, viz. the symbol denoting time and the symbol denoting weight or size, all the others—which may be as numerous as one likes—are all meaningless and can have values assigned to them which give the best representation of the observed experimental data.

Let us now assume that, for a particular growth curve, the size of the organism can be expressed within suitable limits of error, as a function of its age by means of a particular equation. It does not follow that this equation is the only one which will fit the same facts, for as long as each observation is liable to error there must be more than one curve which will express these facts within the required limits. This is a matter of small importance as long as the desired equation is purely empirical; naturally, if one equation fits the facts better than another we select the former, but if both are equally good any selection is purely arbitrary. This is not the place to consider methods of fitting equations to particular curves, but there can be little doubt that more caution is required than is sometimes displayed in the graphical analysis of biological growth. In the first place, the method necessarily involves a known standard of experimental error since in any growth curve there is no such thing as an observed point on a true curve. All observations are liable to error and since the error in observing weight is usually greater than that when observing time, the observations of weight should strictly be recorded on a growth curve as lines and not as points. The length of each recorded line should define the limits of error of the particular observation. The shorter are the lines (i.e. the more accurate the observations) the more strictly is it possible to define the growth curve, and hence the greater is the accuracy with which it is possible to construct an empirical formula.

If, as is very occasionally the case, the size of an organism (x), when plotted against its age (t) results in the possibility of drawing a straight line through all the observational lines, then without further investigation we can construct an empirical formula of the type x + c = kt, since the straightness of a line can be tested by a suitable ruler. Most growth curves are, however, not of this nature. They are quite clearly curves and the nature of their curvature cannot be tested by such simple methods. For this reason, it is customary to re-arrange the facts before plotting them, in the hope that one particular type of re-arrangement will result in a straight line graph. If this procedure is adopted, it is essential to bear in mind that any change in the units concerned may involve an important modification of the degree of accuracy with which an equation can be tested by graphical methods. The only safe rule is to make sure that the limits of experimental error characteristic of every observation are represented in all regions of the graph by a distance which can readily be seen by the eye. This standard is by no means easy to maintain if one or more of the axes of the graph represents the logarithm of a variable.

For example, let us suppose that during the course of growth we make the four observations shown in column 2 of Table I, and that each observation is correct to the nearest whole number.

If we plot the logarithm of x against the time, we get a perfectly straight line. Let us suppose that the logarithmic scale is such that a point, 1 mm. above the point t = o, log x = 1·2041 is denoted by 1·25, then at this point x = 18. Now consider a similar point above t = 8, log x = 3·6123, then log x = 3·6523 or x = 4595. In other words, at the end of the graph 1 mm. of the logarithmic axis represents 250 times the value of the same distance at the other end of the scale. Now suppose that a series of observations from experiment give the values shown in column 3, Table I, and that once more the figures are accurate to the nearest whole number. Plotting them on the same scale as before, the last observation will be denoted by a point o-i mm. above the line joining the other three points. To judge the significance of this by eye is quite impossible even if the graph is constructed with the very greatest care. It is obvious that the errors of graphical plotting are in this case much too great to bear any real relationship to experimental errors.

Where a process of growth involves the use of a logarithmic function it is very unwise to trust to graphical methods unless it can be shown that the ratio of probable error to the mean is constant during the whole period of growth ; it is safer to make a direct comparison between observed figures and those calculated by means of a suitable formula. Graphical methods may, of course, often be used as a guide to finding a suitable formula.

If an equation derived from or expressing a growth curve is to be utilised as a guide to the fundamental processes controlling the rate of growth, a much more rigorous test must be applied than is necessary for purely empirical representation. It is now not sufficient to show that a particular equation results in a curve passing within a given distance of all the observed points, it is also necessary to show that a similar curve cannot be derived from any other type of equation. The extent to which we can satisfy this condition obviously depends on how accurately we can establish our data. If every observation were absolutely accurate there would only be one curve to fit the facts, but the greater are the observational errors the larger is the number of curves which will fit them. The question therefore arises how accurate can we make the experiments? Can they be such that the applicable curves are all of essentially the same type? This cannot be determined by eye. In Fig. 2 are drawn two sigmoid curves ; as far as one can see they both run close to all the points of one and the same series of observations; one or other curve might be adequate for empirical purposes. The two curves are, however, essentially different: one of them belongs to the type and inflects when ⁠; the other is represented by the equation and inflects when (approx.). These two curves will be considered later as they are both conceivably related to metazoan growth¹; at the moment, it should be noted that only data of very high accuracy could possibly reduce the experimental error to such small dimensions as to distinguish clearly between the two types.

As already pointed out, as long as an equation of growth is of a purely empirical nature there can be as many meaningless constants as are convenient, and they can be given any suitable values. If, however, an equation is based on concrete conceptions and thereby acquires a real meaning, there should seldom be more than one indeterminable constant, and all the others should have a real meaning and be capable of independent measurement. An example illustrating the danger of assigning arbitrary values to growth “constants” will be found in Appendix I (p. 272).

With these general principles in mind, we may now survey some of the observed data of animal growth and consider how far they can be used in framing a quantitative conception of the processes involved.

Perhaps the most fundamental type of organic growth is that found in bacteria in which each cell grows and reproduces itself at a constant speed as long as the conditions in which it lives are maintained at a satisfactory and constant level. Observations of this type of growth were made by Barber (1908), who isolated single individuals of Bacillus coli from a rapidly growing culture and observed their rate of reproduction at a constant temperature. Table II shows the time which elapsed between one generation and another at 37 ·7° C.

For a considerable period of time therefore every Bacillus reproduced itself every twenty minutes. Starting from a single bacillus it is obvious that, as long as these conditions hold, the number of bacteria present at any particular multiple of the generation time can be calculated by a process of compound interest as long as no bacterium dies, or as long as a fixed percentage of the population dies per unit of time. Since the rate of growth of each individual is the same, it follows that the rate of growth of the whole culture will be proportional to the number of bacteria (x) present at any particular moment

Starting from one bacillus, when t = o, x = 1

It is important to notice that the succesive generations of Bacillus observed by Barber were all derived from one original organism, and that as far as is known there was no material difference in the average size of each cell during the course of the experiment. It should also be noted that the constancy of the time elapsing between successive generations was established by direct observation, and is not deduced by calculations from the total number of cells produced in a given time.

Barber’s results are of peculiar importance because they are derived form a system which is truly comparable to those which exist in simple physico-chemical reactions. Many of the phenomena of chemical dynamics receive adequate analysis if the velocity of the reaction is assumed to be proportional to the active mass of the reacting substances. If a reaction is such that a molecule of a substance A is converted into other compounds none of which, interfere in any way with the activity of the original substance, then the velocity of the reaction is proportional to the number of molecules of A which are present.

This equation is so frequently applied to the kinetics of growth that it is advisable to appreciate its full significance. It implies (i) that all the molecules of A or a fixed percentage of them are taking part in the reaction during the whole period of their life, (ii) that although the kinetic energy of individual molecules may vary, yet the population as a whole does not change in this respect, but has a constant reaction velocity k. For example, at all times during the reaction the molecules of A are all disposed on the same variability curve when they are grouped according to degrees of kinetic activity (Fig. 3). In other words, although they are not all alike, yet collectively the heterogeneity of the population does not change as the reaction proceeds; the mean value for the population remains constant, and therefore the constant k in equation (3) is justified. Now exactly the same conditions hold good in Barber’s experiment. Every bacterium is growing, and although the generation time varies from 18·7−21·2 minutes, there is good reason to believe that the average of 20.0 holds good throughout the whole experiment. At the same time the change in the number of bacteria present is only comparable to the change in the number of the units in a chemical system within certain limits and under certain circumstances.

The customary unit of growth in a bacterial population is one bacterium, and if all the bacteria present were to divide synchronously, the curve relating time to number of bacteria present would consist of a series of steps, although the points denoting the number of bacteria present at successive periods of twenty minutes would all lie on a smooth logarithmic curve (Fig 4).

The apparent discontinuity of growth of such a population is due to the unit which has been chosen to denote growth—namely an individual bacterium, and such a population does not conform to the rule that at any instant all the bacteria or a fixed percentage of them are taking part in the production of a new individual. On the contrary, there will be periods when none of the bacteria are dividing, whilst every twenty minutes there would be moments at which each bacterium has just produced another separate individual. If, however, we define growth as an increase in the amount of living material, it is obvious that growth is a continuous process which must be measured not in terms of individual bacteria but in terms of smaller units within individual cells, and the velocity at which these units change in number cannot be measured in terms of individual cells except in purely arbitrary and discontinuous units—viz. an amount of new tissue equal to and present as one complete bacterium.

As long as one complete new bacterium is manufactured by each member of a population in twenty minutes and all are dividing synchronously, the number present at the end of each complete period of twenty minutes can, as shown above, be expressed by a logarithmic curve comparable to that of a first order chemical reaction, but the actual velocity of growth (when this is defined as the rate of increase in both cases we must adopt a satisfactory unit for the measurement of “growing” material. So far this does not appear to have been done in the case of isolated cells growing in vitro. In other words, until the growth curve can be plotted for inter-cleavage periods we cannot determine the nature of the growth process itself¹

Although no bacterial population ever exhibits synchronous division, the above arguments hold good in other systems. In any culture of actively growing bacteria, there can be little doubt that at any instant the population could be expressed by a curve comparable to Fig. 3 if the population is divided up in terms of numbers of individuals exhibiting a given phase of division. In such cases there will always be a fixed percentage of the total cells which is about to produce new independent bacteria, and for this reason the number of bacteria present at any instant when of living matter) may follow a law of quite a different nature—bearing no obvious relationship to a logarithmic system. In a simple first order chemical reaction the units involved are of the smallest possible dimensions since only complete molecules can take part in the reaction, and until comparable units are available for the observation of growth we cannot say how far growth is controlled by a first order reaction or by some other means. The true velocity of growth can only be followed within a single cell or within a population which is dividing synchronously, and plotted as a function of time will produce a logarithmic curve quite irrespective of the way in which the actual growth process occurs within the individual cells.

The work of Lane-Clayton (1909), Slator (1913), and more recently of Richards (1928) illustrates clearly the logarithmic nature of growth within actively growing cultures of bacteria and yeast. In every case the conditions have been the same, viz. the population has been homogeneous in the sense that the average generation time of all the individuals has been constant, and has been heterogeneous in the sense that the cells at any one moment have been in different phases of the reproductive process, and this heterogeneity has been constant throughout the experiment. For these two reasons, the resultant growth curve is logarithmic although it throws no real light on the nature of the reaction which increases the amount of growing material. In other words, growth curves of this type indicate the rate of increase of a population of cells, just as a first order reaction may indicate the rate of increase or decrease in a population of chemical molecules, without in either case defining the nature of the process which goes on within the units themselves. The available data are all summed up in Barber’s observation that successive divisions do not influence the time required for the production of a new bacterium if the environmental conditions are constant.

Since these experiments illustrate to some extent the principles of graphical analysis, one of them may be considered in some detail. The results as given by Lane-Clayton are recorded in Table III.

The values so obtained are shown in Table IV, column 2. If we accept an average value of k of 0·946, we can calculate the number of bacteria present on the assumption that exactly 87 were present at the beginning. These figures are shown in column 3. It is quite clear that the validity of the compound interest law is subjected to a rigorous test if we compare the observed numbers of bacteria with the calculated figures, and are in a position to decide whether or not the discrepancies revealed do or do not fall within the limits of accuracy of the experimental counting on the one hand and the errors involved by calculating logarithmic values on the other. Presumably, in this particular case, the observed and calculated figures agree within the limits of these errors, but this could not be determined accurately simply by estimating the position of points on a graph by means of the human eye or by inspection of the calculated values of k given in Table IV, column 2. It is therefore important to avoid the presentation of facts in such a form as makes it difficult to detect divergence between observed and calculated values; they ought always to be presented in a form which clearly exhibits a real basis of comparison in terms of a measurable unit. The deceptive properties of logarithmic plotting can be realised from the fact that an error of 0·01 in the logarithm of 5·87 involves an error in the antilogarithm of approximately 17,000 bacteria.

Precisely the same analysis applies to Slator’s (1913) observations on the rate of reproduction of yeast cells. From Table V it would appear that the course of growth is very accurately represented by the formula kt = log_eN/n where N is the number of yeast cells at time t and where n is the number of cells present at the beginning of the experiment. In these cases the value of k varies between 0·105 and 0·098, and is apparently very constant about an average of 0·102. If, however, we use this average value for calculating the number of cells present at the end of each experiment, it is clear that the agreement between observed and calculated values is by no means so complete (Table VI).

The two examples given illustrate the extreme care which must be exercised in the use of formulae which involve a logarithmic term, and show that the only real test consists in a comparison of observed and calculated values. If the differences between such values are within the errors of experimental observation, we may conclude that the results of the experiments themselves are not definitely against the theoretical conceptions on which the formula is based.

It is clear that unless some information is available concerning the relationship of n_1,n₂, n₃… and k₂, k₃… it is impossible to foretell the course of growth of a heterogeneous population. From Table VII it is, however, clear that as growth goes on in a heterogeneous population, the heterogeneity becomes less because the percentage of rapidly growing cells very rapidly increases ; hence if we subculture a rapidly growing population, we very quickly obtain a homogeneous culture which will obey the compound interest law. The experiments of Lane-Clayton and of Richards are of peculiar interest as in each case the cultures were of a homogeneous type, thereby justifying the use of a growth constant k in equation (i). In almost every other biological system this condition is by no means so clearly established, for we have no evidence that we are dealing with either a population which is homogeneous or with one whose heterogeneity is always the same.

If the number of bacteria is allowed to increase in any given volume of medium, there comes a time when the compound interest law breaks down. Sooner or later the growth rate declines and eventually sinks to zero : after this, there is a rapid decline in the number of bacteria present. The factors responsible for this break-down of the logarithmic law of population growth are not completely known. Graham-Smith (1920) has shown that under certain conditions the amount of food stuffs available plays an important rôle. In a particular culture medium Graham-Smith found that Staphylococcus aureus reproduces itself until there are approximately 10 million organisms per 0·01 c.c. of medium, and this figure is independent of the number of bacteria originally inoculated into the medium.

There is therefore an upper limit to the density of bacteria which can be obtained in any given culture. This upper limit is largely dependent on the concentration of nutrient substances in the medium (Table IX)—see also Penfold (1912).

Unless the food supply is maintained at a uniform level, the maximum population for any particular medium is not stable for any significant period of time but the number of organisms rapidly declines. This is almost certainly due in part to the fact that the foodstuffs available are being steadily depleted in order to maintain the normal activities of the organism. For each concentration of food there is a characteristic maximum density of population, and unless the food is maintained at this level, the density will fall with falling concentration of food.

The far-reaching effect of food supply upon growth rate is seen when identical cultures are grown at different temperatures. The higher the temperature the more intense are the katabolic processes of the organisms, and consequently a higher concentration of food is required to maintain a maximum population. In other words, with identical cultures the food supply begins to run short sooner at a high temperature than at a low, and consequently the maximum density at a high temperature is lower. This is illustrated by Graham-Smith’s figures (Table X).

In other words, at the higher temperatures the maximum density is more rapidly reached, but the maximum itself is lower than at lower temperatures.

Even when the food supply of a bacterial culture is very rich, the growth rate of the culture will still eventually approximate to zero, and from this we may infer that other factors in addition to lack of food are also operative in reducing the growth rate. How far these factors represent toxic substances produced by the bacteria themselves has not been fully investigated but forms an important subject of research. (See Richards, 1928.)

If a + k₁x₀= A

or if x₀ is very small compared to x,

This equation gives an accurate representation of the observed course of reproduction, for there can be little doubt that it expresses the observed data within the limits of experimental error; nevertheless, it cannot possibly be accepted as an adequate representation of the known facts. Equation (8) assumes five things : (i)that at no moment will the growth be strictly logarithmic since falls steadily from the beginning of incubation, (ii) the maximum number of bacteria will not be reached until the whole of the foodstuffs have entirely disappeared, (iii) that no food material is required for the maintenance of a bacterium once it has been formed, (iv) that all the bacteria react to adverse conditions to the same extent and that the death rate when food is scarce is not higher than when food is plentiful, (v) that having approached the maximum density the population can be divided by sub-culturing into fresh medium, and each bacterium will at once begin to grow as rapidly as the original culture in its early life. Now it is extremely improbable that any of these assumptions are true; on the other hand, they are at variance with fairly well established facts. It therefore follows that equation (10) cannot be of any real value, although as an empirical equation it may have some practical use. This example illustrates the fact that although the observed growth of a population of cells can be represented by a simple equation, it by no means follows that we are justified in assuming that the factors controlling growth are equally simple. That the food supply is an important factor determining the rate of growth we already know from direct experiment, and our knowledge is not only not increased but is liable to be mis-interpreted by the use of “simple” quantitative conceptions which are not based on facts.

But the food is decreasing partly owing to the formation of new bacteria and partly owing to the nutritive needs of these bacteria, so that,

Hence

or

Where

In other words, it is possible to express the number of bacteria in terms of the quantity of food present, but it is not possible to calculate with any ease the number of bacteria present at any particular moment. For a practical application of equation (11) see Gray, 1929.

The data derived from an observation of the growth of unicellular organisms are of peculiar importance since the systems concerned are very much more simple than those which exist in the body of an organised metazoon. Not only is it possible to grow bacteria in media of known and controllable composition, but the units of growth (i.e. all the organisms in a culture) may, in special cases, be essentially similar to one another. The facts indicate that as long as the external environment is maintained at a satisfactory and constant level, each cell of a homogeneous population multiplies at a rate which is constant within certain definable limits ; there is no indication that the powers of growth are declining with the age of the culture. Old cultures exhibit a reduced growth rate because the external conditions automatically change as time proceeds. Given a constant medium, however, the rate of reproduction and the rate of “accidental” death remain constant, and it is possible to define the reproduction rate as a constant peculiar to the species.

On the other hand, if the medium varies owing to a diminution of food supply or to the accumulation of the products of growth, the effect on the growth rate is clearly of an inhibitory nature. At the same time, the available data do not as yet enable us to form a quantitative expression of this change. The relationship between reproduction and concentration of food is not linear (see Penfold, 1912), and we do not know the precise relationship between lack of food and the death rate of the population. Presumably some bacteria are more sensitive to adverse conditions than others, and the death rate will be related to adverse conditions by a variability curve comparable to that of higher organisms. The net result of our present lack of knowledge makes it impossible to construct a differential equation for reproduction which takes adequate cognisance of the known qualitative facts, and from this it follows that, although it is possible to express the final growth curve by means of a simple formula, this formula cannot throw any real light on the underlying causes of growth.

If we define growth as an increase in the amount of living material going on within individual cells, we cannot determine its nature or velocity from a study of the number of cells present in an ordinary culture, or from a study of the total activity of the whole culture. The process of growth in a suitable medium may or may not follow the course of a first order reaction, but until it has been observed within a single cell its nature will remain unknown.

The essential features exhibited by a growing culture of bacteria, are also seen in metazoon cells grown in vitro, although the requisite conditions for growth are more specific, and the known facts are even less complete than in the case of bacteria (see Carrel (1923), and Fischer (1925) for numerous references). The rate of growth of tissue cells in vitro depends in part upon the concentration of embryonic extract present in the medium; as long as this is constant (in addition to other environmental factors) the cells will grow and multiply at a uniform rate. Until we know, however, the precise manner in which the embryonic extract is destroyed or absorbed by growing cells it is premature to express the facts as a differential equation, or to apply them to growth in situ.

We may conclude, therefore, that as long as a population of cells is of constant homogeneity, and as long as the external conditions remain at a constant and favourable level, the rate of reproduction, like other biological characteristics, also remains constant. When, however, the population is not homogeneous, but changes its heterogeneity as time proceeds, or whenever the conditions of the environment are changing automatically owing to the process of growth itself, then our data are as yet an inadequate basis for formulating a quantitative conception of the changes in reproduction rate which ensue. Differential equations which are derived from an observed reproduction curve can only be regarded as empirical expressions and throw no real light on the underlying factors controlling the growth rate.

The growth of a cell inside the body of a living organism is very different to that of a similar cell isolated in a nutrient medium containing embryonic extract. As long as a cell is in its natural position in the body, its growth is determined both qualitatively and quantitatively not only by the size or age of the organism but also by the particular position occupied by the cell and the particular relationships of its neighbours. The normal environment of a cell inside a rapidly growing embryo is therefore extremely complex and quite unlike that of a cell grown in vitro. With one or two exceptions differentiated cells cannot be cultivated as such in vitro and since the bulk of an animal’s body is composed of differentiated cells, it follows that evidence derived from in vitro experiments does not as yet throw much light on the growth which normally occurs in the body as a whole. When we are able to cultivate differentiated cells (e.g. muscle and glandular epithelia) as such in vitro, then it will be possible to see how far the presence of one type of cell is able to modify the rate of growth of others, and thereby form some real conception of the processes of growth in vivo. At present we have to restrict our analysis to points of comparative detail. A priori, it is extremely improbable that a system so complex should conform in any fundamental way to the behaviour of a simple system of strictly definable heterogeneity.

As shown above, the so-called logarithmic growth curve of unicellular populations is observed so long as there is a fixed percentage of the total number of units undergoing change at any instant; this fixed percentage can be expressed as a fraction of the total number of cells present. In the case of a metazoon the total number of cells present (except in certain specified tissues) is no indication of the amount of material present; for this reason it is customary to dispense with the cell as a unit of metazoon growth.

The simplest possible conception of a growing metazoon suggests a number of different units (e.g. tissues) each growing at its own characteristic rate. The total growth of such a system would represent the summation of the growths of the various parts. Some of these parts may grow at the expense of others and some of the growing units may give rise to material which is not capable of growth. Unless therefore there is some factor which controls the total amount of growth made by all the units at each particular moment, it is impossible to apply to metazoon growth the simple algebraic treatment applicable to a homogeneous population of bacteria growing in a constant environment. It is very doubtful whether evidence in favour of such a controlling factor or “master reaction” can be derived legitimately from the apparent ease with which an equation with only two variables can be made to express the size of a growing organism. The conception of a master reaction as one possessing a simple chemical nature is of vital importance in any quantitative study of growth as it enables us to ignore to some extent the heterogeneous nature of its ultimate products. If such a reaction exists, it will have a definable velocity constant, since the units involved will comprise a molecular population to which chemical principles can be applied legitimately. From a purely biological point of view, it seems reasonable to regard the whole process of organised growth as the result of a coordinating mechanism whereby the velocity of individual processes are mutually dependent on each other to a marked degree (see Gray, 1929). At the same time it is by no means easy to express the biological facts in terms of chemical units. Even in their simplest form these facts suggest a degree of complexity far beyond anything yet investigated in inanimate systems. We might conceivably depict the course of growth by the system illustrated in Fig. 6; the raw materials entering the growing body are converted first into a substance X, and from this there are derived a series of compounds a, b, c, d, each of which can be converted into the other or into specific and permanently differentiated tissues. In such a system the total production of specific tissue might be proportional to X, whereas the amount of any one tissue would also depend on the amount of other compounds of the series a − d. The conception of a master reaction of this type restricts our conception of growth to quantitative principles, and would usually imply that if one organ were excised (and were not regenerated) the rest would grow with increased speed. In actual practice the effect of removing parts of the body in some cases has no effect on the growth rate of the remainder although, in others, compensating hypertrophy or regeneration may occur. To some extent the heterogeneity of the whole body can be eliminated by investigating the growth of specific organs (Scammon, 1925) ; in such cases, however, we are liable to ignore the fact that the rate of growth of one organ depends on the behaviour of another, so that we are still dealing with an extremely complicated system. If the existence of a master reaction is regarded as intrinsically probable, we are still faced with the difficulty of determining the amount of the active principle present at a given time. Unless the active mass of X bears some simple relationship to a measurable unit, we are no nearer to a quantitative estimate of growth.

Putting these difficulties on one side for the moment, we may examine the possibility of expressing the growth rate of the whole metazoon body in terms of simple measurable units. It is universally admitted that the factors controlling the growth rate are not as yet capable of direct measurement. In a chemical system, on the other hand, the factors controlling the rate of reaction are, under standard conditions of environment, the active masses of the reacting substances, and these are at times proportional to the weight of the substances in unit volume of medium : these can be determined with accuracy. When, therefore, we express the number of molecules undergoing change in terms of the total number of molecules present, we are dealing with measurable quantities. In the same way we can express the number of growing bacteria in terms of the total number of organisms present. When, however, we consider metazoon growth we cannot do this, for it involves the power of expressing the amount of new tissue formed in terms of units which cannot directly be observed. We are, in fact, trying to express the amount of new tissue formed per unit of time per unit of growing tissue. It is impossible to do this in most cases of metazoon growth because there is no obvious way of knowing how far all the organism is growing. In practice we assume that all the organism or a fixed percentage of it is growing, and therefore express the growth rate in terms of new tissue formed per unit weight of the whole organism. Minot (1908), was perhaps the first to use this method, and his formula for the growth rate was

Where W₁ is the weight of the organism at the beginning of the observational unit of time, and W₂ is the weight of the end of this period. As pointed out by Brody (1927), this method gives erroneous values unless the period of time is extremely short, and unless the growth rate is changing very slowly. In practice it is very difficult to fulfil the conditions necessary for accurate results. Minot’s formula is however an attempt to express the growth rate in terms of measurable quantities ; it leads to erroneous results because growth is a continuous and cumulative process. All attempts to overcome this latter difficulty are based on the assumption that the amount of growing tissue in an organism is directly proportional to the whole weight of the organism. In some cases (embryos, larvae, etc.) this assumption does not appear to be altogether unreasonable, but we usually encounter difficulties.

If the amount of growing tissue is to be assumed to be proportional to the weight of the whole animal, should this weight include the water content of the body or not? In the case of young fish both these standards lead to the same result because the percentage of water in the organism does not vary materially during a prolonged period of growth (Gray, 1926). In the chick, on the other hand, an estimate of the growth rate based on wet weight figures may differ materially from that based on dry weight figures (Murray, 1925) ; in this case we do not know whether the percentage of water in the “growing” tissues changes, or whether the percentage of water in the whole animal changes with the intensity of fat storage, bone formation and similar processes.

In a few cases, it is tempting to adopt a purely physiological standard for the estimation of growth. For example, we find that for a significant period of time the amount of oxygen consumed per gram of growing fish (Salmo fario) remains constant (Gray, 1927), and from this we might conclude that the amount of living respiring material is actually directly proportional to the weight of the animal. Sooner or later, however, this criterion breaks down and the intensity of respiration falls. In the chick, Murray found still more irregular phenomena.

If we are prepared to accept the rate of respiration as a measure of the amount of living tissues in very young embryos, we reach an interesting conclusion by studying the rate of growth during the segmentation stages of Echinus. It will be seen from Fig. 7 that the specific growth rate as measured by (δR/δt)/R (where R is the rate of respiration of the whole organism) rises during the early stages of development; in other words there is a “lag” phase during which the specific growth rate is increasing to a maximum. It will be remembered that where a bacterium is removed from a medium in which growth is very slow, and is placed in a favourable medium, there is also a lag phase. It looks as though the same principle applies to a metazoon egg, since after the conditions for growth again become favourable (by fertilisation), there elapses a period of time before the specific growth rate reaches its maximum. Later on, in both cases, the rate falls off unless the medium is artificially maintained at a satisfactory level. These data from Echinus eggs obviously provides another example of the way in which more than one differential equation of growth can be derived from one and the same set of data (see Gray, 1927) and how each differential equation gives a different conception of the nature of the growing system.

We now have two main difficulties in obtaining a satisfactory expression for the total growth rate of a metazoon, (i) the heterogeneity of the system undergoing change, (ii) the difficulty of expressing the growth rate in terms of rational and measurable units. Nevertheless, it is claimed (Robertson, Brody, and others) that the form of the observed growth curve is sufficiently simple to justify the belief that the whole process of growth is controlled by a master reaction involving two variables, each of which is a linear function of the size of the whole organism. “Incidentally, but not primarily, the applicability of the equation of the chemist to the time relations of growth may be taken as further substantiating evidence that growth is limited by a chemical reaction” (Brody, 1926, p. 234). It is at this point that the principles of graphical analysis, discussed in the first section of this paper, come into prominence. Both Brody and Robertson (1923) claim that a typical growth curve can be expressed by particular equations, but the equations they use differ from each other. Robertson claims that the curves are of a symmetrical sigmoid type, although more than one such curve may be included during the whole life cycle. Brody (1925℃7), on the other hand, divides each sigmoid curve into a number of distinct exponential phases, each of which is presumably more or less independent of the other.

As already pointed out, it is impossible to say how far a particular sigmoid curve is the only one capable of expressing the observed facts unless the experimental data are known with extreme accuracy, and it is almost certain that this degree of accuracy is not forthcoming for many of the cases quoted by Robertson in support of his equation. Robertson’s equation in its differential form is of the type

and, if this be true, the integrated curve is symmetrical and must inflect when the animal’s growth is half completed. By selecting suitable values of k₁ and k and by using an appropriate number of superimposed curves there can be no doubt that an equation of this type can be shown to express the facts. Unless, however, there are definite experimental reasons for adopting this procedure, the equation has no real meaning unless its advocates can prove that no other equation will fit the facts.

That the same group of observed facts can be subjected to entirely different types of graphical analysis is illustrated by the growth curve of the rat. It can be resolved into a series of curves of the type selected by Robertson (1923) or it can be regarded as a single curve inflecting when the animal is not one-half grown but one-third its full size (Daven-port, 1927).

The whole curve can, within the limits of reasonable error, either be expressed by an equation of the type advocated by Crozier (1926) or by that illustrated in equation (15)

or if

From equation (16) it might be inferred that the rate of growth is determined by three factors.

1. It is directly proportional to the weight of the whole organism (x).

2. It is directly proportional to the amount of essential foodstuffs conveyed to each unit of tissue, if the total of such foodstuffs available is determined by the surface of a digestive or absorptive organ, this surface varying with the two-thirds power of the weight of the animal.

3. A continuous process of wear and tear (whose intensity has an average constant value (k₂) per unit weight of tissue), going on during the whole period of life.

A priori, none of these assumptions is clearly improbable, on the other hand they are not established facts. The only means of assessing the value of the hypothesis lies in further experimental analysis and not by an elaborate selection of arbitrary values of constants which will reproduce the data already known. For this particular example some experimental analysis is available; if equation (16) be based on reality then the amputation of a limb or tail should lead to an increase in the weight of the remaining organism by further growth, since the limit of growth is partly determined by the factor k₂x. Such compensating growth does not occur, and consequently the equation cannot be valid, however nicely it represents the course of normal growth when expressed in an integrated form.

Returning to Brody’s exponential equations (1925), it is sufficient to point out that any curve can be expressed as a series of straight lines, or exponential curves if suitable limits are selected. Unless, therefore, there is good independent evidence that the whole growth cycle is divisible into a finite number of successive and different processes, the process of subdivision of the growth curve is purely arbitrary.

We therefore reach the conclusion that it is by no means easy to deduce from known growth curves a proper understanding of the factors controlling the rate of metazoon growth. We are compelled to examine these factors by direct methods.

Direct investigation of internal growth factors is very far from complete, but from a study of the growth of tissues in vitro it seems fairly clear that young embryos contain a supply of a growth promoting substance which is greater than that possessed by older embryos. In the adult animal this substance appears to be absent except in certain cells. Now if the whole of an embryo possesses the same potential powers of growth in all its parts, it is not unreasonable to suppose that the growth rate is proportional to the weight of the embryo (x) and to the concentration or amount of the growth promoting substance (y)

If y decreases from a finite value to zero, it follows that the integrated growth curve will be sigmoidal in form quite independently of the manner in which the decrease in y occurs. The rate of disappearance of the growth promoting substance will only determine the point of inflection and symmetry of the curve ; the essential point is that it will always be sigmoidal. The simplest possible conception of the way in which y disappears is embodied in the hypothesis that each gram of newly formed tissue uses up in its formation the same amount of growth promoting substance. In other words, if there are a units of this substance present at any time, then after x grams of new tissue have been formed there will be left a − kx units of growth promoting substance. If we also assume that every unit of new tissue formed has the same potential powers of growth as any of the pre-existing tissue, then if x is the total number of grams of tissue at any moment, the potential powers of growth are k₁x. If the rate of growth is proportional to the amount of growing tissue and to the amount of growth providing substance, then

In this way it is possible to reach Robertson’s fundamental equation, basing it to some extent on established facts. There is however no real evidence to show the precise manner in which the growth promoting substance disappears during development, and there is consequently no real justification for writing equation (17) in the form of equation (18).

It is perhaps worth noting that a sigmoidal growth curve would result if the growth rate were reduced owing to the formation of a growth inhibiting substance which combines reversibly with the tissues and slows their growth. Such a hypothesis, if based on experimental facts, would harmonise with the observed powers possessed by tissues to recover from wounds. (See Appendix II.)

From the point of view elaborated in this paper, the comparison of metazoon growth with the behaviour of comparatively simple chemical reactions meets with three main difficulties. Firstly, a series of observations which approximate to a sigmoid curve can only be expressed in the form of a specific differential equation when the accuracy of the observations reaches a very high level. Until such data are available it is impossible to determine how far they can only be expressed by the highly specific curves applicable to chemical systems. Secondly, there is no direct method of determining the active mass of the growing substance or of the other factors involved in the reaction : these may be proportional to the weight of the organism although no definite proof exists. Thirdly, the growing system is known to be statistically heterogeneous, and in the absence of reliable evidence to the contrary, it is intrinsically improbable that the system will behave like a system whose heterogeneity is constant.

The physico-chemical conception of growth enables us to depict certain facts against a simple and attractive background, but the practice of expressing observations in their simplest apparent form is undoubtedly liable to cause confusion, particularly when we know that the systems providing these observations are themselves extremely complex. There seems little doubt that our knowledge of the kinetics of growth can be advanced only by a direct study of factors which directly control growth rate and by expressing these factors in a quantitative form. In this way, a real differential equation will be made available, and this, in its integrated form, will harmonise with the results obtained by observing the size of the organism at selected moments of time. This position has not yet been reached. For the present, the whole of the facts indicate that the size and form of a growing organism are the resultants of a large number of obscure factors all dependent on each other and not adequately represented in any system of unorganised growth, and still less so in a simple physico-chemical system. A premature attempt to express the growth rate by a differential equation involves the grave danger of confusing rather than of clarifying our knowledge of the facts. At the same time, a real advantage is gained by attempting to put together the known facts concerning growth factors in such a way as will indicate whether or not they are capable of giving an adequate picture of observed facts, or whether a search must be made for further factors. In other words, it is legitimate to work from a differential equation in which the terms represent real entities ; it is not very profitable to base our ideas on a differential equation which is solely derived from an observed curve of growth.

The known facts of growth in vivo and in vitro seem to indicate quite clearly that as an organism increases in size or age the environment for growth becomes less favourable for those tissues still capable of growth. Until the cause of this phenomenon has been subjected to direct quantitative study, it is unlikely that we shall find an equation for any particular growth curve which is more than an empirical representation of observed data.

A quantitative expression for the growth rate of living systems can only be obtained when the systems are homogeneous (or of known heterogeneity), and when the external conditions are constant. When more than one type of cell is present, or when the external conditions are varying, the known data (concerning even unorganised growth), are insufficient for the construction of a real equation defining the rate of growth in a given set of conditions existing at a particular instant of time. An equation representing the size of a population of cells or of an organism in terms of age, yields, on differentiation, a quantitative but empirical representation of the factors controlling the rate of growth, but since more than one equation can always represent a typical growth curve within the limits of probable error, a selection of one particular equation rests solely on the intrinsic probability of its differentiated form. The degree of probability can only be established by direct experiment.

In the growing body of a metazoon the conditions of growth are extremely complex, and it is difficult to express the growth rate of the whole organism in terms of rational units. Graphical treatment of the data underlying a typical growth curve is liable to produce errors of considerable magnitude, and often tends to confuse the facts. The units which compose a metazoon’s body form a very heterogeneous system, in which the rate of growth of one organ is dependent on that of others. It is, therefore, intrinsically improbable that the behaviour of such a system should conform to that of a simple chemical system in which the variables are few in number and capable of accurate analysis. The conception of growth as a simple physico-chemical process should not be accepted in the absence of very rigid and direct proof ; at present, it rests on the results of a process of graphical analysis which is often, if not always, of a relatively inaccurate nature.

The danger involved by the assignment of arbitrary values to the constants of an equation of growth may be illustrated by the following example: Ledingham and Penfold (1914) obtained the following figures of the growth rate of bacteria during such conditions as exhibited a marked lay-phase (Table XI).

To account for these figures we might make the following hypothesis. At the beginning of the experiment let us imagine that all the 217 bacteria are unable to reproduce, but as time goes on, they gradually recover from the effects of their past history. All the bacteria will not recover at the same rate, some will recover after a short time and some after a longer time; at the end of three hours, however, the whole culture is known to be growing at a steady rate. Now, it is reasonable to suppose that the course of recovery of the whole population will follow a sigmoid probability curve such as is shown in Fig. 8. As a first approximation we can replace this curve by the dotted line AC. Let the number of inactive or non-growing bacteria at any time be y.

Then

Let the number of growing bacteria at time t be x,

Then

The total number of bacteria at time t is x + y

Giving k the arbitrary value of 0 ·0285, the calculated values of x + y are those shown in Table XII, column 3.

It might well be that the calculated figures are sufficiently near those actually observed to fall within the limits of experimental error. If however h = 0·0285, then, when the whole population is growing at its full maximal rate, the number of bacteria should be doubled in a period of time equal to By actual experiment, however, the generation time at the end of three hours was found to be 18 mins, and if this is true k must equal 0-0385, and if this be so the calculated figures are shown in Table XII, column 4, and it is quite obvious that the original hypothesis breaks down.

The sigmoid nature of a typical metazoon growth curve expresses the fact that up to a given point the increase in weight of the organism per unit time at first increases, after which it declines. This can be expressed algebraically by the equation

where represents the rate of increase in of the organism; x represents a factor weight which increases with the weight (w) of the organism, and y is a factor which decreases as x increases. If the weight of an organism reaches a maximum value, then y must approach zero. The integrated growth curve will be sigmoidal whatever be the rate or manner in which x and y change in value ; the precise form of the curve will however depend on these processes.

The principles outlined in this paper suggest that x may increase and y may decrease in a variety of ways and yet in every case the integrated growth curve will express the observed size of an animal within the limits of experimental error. The advocacy of one specific differential equation of growth over others must therefore depend on independent external evidence. For example, Richards (1928) has shown that under certain conditions the sigmoidal form of the growth curve for yeast is directly associated with the generation of alcohol. Since the effect of the alcohol is reversible, we might imagine that it unites with actively growing cells in a reversible manner, rendering them incapable of growth. Under such circumstances it might be possible to advocate a differential equation of the type given below.

where c is a constant involving the threshold value of alcohol inhibition. The integrated growth curve would then be

The sigmoid nature of this curve can be appreciated by re-differentiating (24) and by putting a − bc = a

The first term will decrease and when the daily increments in weight will cease to increase, and at this point the growth curve will show an inflection. The limit of growth will be reached when ⁠.

It would be interesting but laborious to apply an equation of this type to Richards’ data.