Bioenergetics and Growth Chapter 16

Chapter 16 Time Relations of Growth of Individuals and Populations All motion of natural action is performed in time. F. Bacon Everj'thing exists not only in a frame of space but also in a pattern of time. G. E. Coghill As early as the fifth century B.C., Greek physicians developed a very clever method for the study of growth which is employed by scientists down to our day. A hen was set upon a number of eggs; each day one of these eggs was opened and the changes that took place could be observed. H. E. Sigerist

16.1: IntroductioE and definitions. The use of isotopes in the study of metaboUc processes has shown that, perhaps, all constituents of the living cell are involved in continuous chemical reactions, continuous breaking down and building up—catabolism and anabolism. It is onlj' the pattern, the life whirl pool, that endures. Biologic synthesis, that is, the interaction of exogenous material (food) in the formation of new chemical-morphological units, thus occurs not only during the period conventionally designated as growth, but throughout the entire life cycle^. The occurrence of widespread synthesis throughout life may also be ob served without refined metabolic studies. Thus blood cells and epidermis have long been known to undergo rapid destruction and renewal; there is a continuous need for growth catalysts (hormones, vitamins, etc.) and structural materials (amino acids, minerals) to compensate for the continuous losses, breaking down, or catabolism, of the body. These constructive processes are more dramatic during periods following starvation and injur}', especially regeneration of limbs in lower forms of life. The most spectacular type of directed biosynthesis is, of course, growth and development, especially during embiyonic life. Everyone has been impressed by the miraculous transformation of the sticky \\'hite and yellow mass of hen's egg into a fully dressed, befeathered, respectable chick, all in 21 days. The original egg cell must have travelled at a dizzy pace to build up so complex a mechanism—probably exceeding in complexity the astronomi cal W'Onders with their galaxies and supergalaxies. 1Schoenheimer, Rudolph, Physiol. Rev., 20, 218 (1940); Growth, Second Supplement, ). 27 (1940); "Dynamic state of body constituents," Harvard University Press, 1942 (ed. )y H. T. Clarke). 484

GROWTH RATES

485

Growth as thus defined is inseparable from metabolism, and the several chapters in

this book are merely different aspects of essentially the same problem, mef,abolismgrowth. Thus Chapter 6 is concerned with enzymes in metabolism, in biologic synthe sis; Chapter 7, with hormones; Chapters 13 to 15, with "maintenance" catabolism;

Fjg. 16.1. Photographs (X 135) of rabbit eggs during 4 days after fertilization. A, 1-cell stage with 2 polar bodies; 13, two primary blastomeres, about 25 hours after; C, 4-cell stage 29 hours after copulation; D and E, 6-ceIl and 8-cell stages 32 hours; F, 32cell morula 55 hours; G, morula 70 hours; 11, trophoblast cells, 71 hours; I, fluid collec tions in the forthcoming segmentation cavity 77hours; J, segmentation cavity 90hours; K, inner-cell mass flattening into germ-disk 92 hours. From P. W. Gregory [Plate I, Carnegie Inst. Wash., 21, 407 (1930)], arranged by G. L. Streeter [5ci. MontMy, 32, 498 (1931)].

BIOENERGF/riCS AND GROWTH

486

Chapter 20, with general nut ritional aspects, anil so on. This(Oinptei' is concerned wil.h the definitions and lime rehxtions of average development an W M g .Sauattuftpo;^ franponmattr.

k(l-f) ScalB

Fig. 16.6. Equivalent growth curves of rats, yejist populations, maize plant, oat plant, and a squash.

There is no pronounced increase in the size of the mass in the first few cleavages of the fertilized egg, but there is increasingly pronounced differentiation so that at the 16cell stage the Irophoblasts, the cells destined to furnish the fetal membrane, and the

structures of implantation of the egg on the maternal tissue, become distinctive. The trophoblast cells divide more rapidly and are, therefore, smaller than the other cells.' Next most con8|)icuous is the formation of the tluid-distended vesicle, the blastocyst (Fig, 16.1), and cupping into a hollow sphere. Fluid collects within this sphere, perhaps through the secretory activity of the trophoblast cells, and the cell mass, for the first time, enlarges by the accumulation of fluids. The outs'tanding feature of the "growth" of the blastocyst is the absorption of tre mendous quantities of water. Davenport'^ estimated that after six weeks the human " Zygote, the "yoked" first cell of the body, the union of male and female germ cells, or gametes, carriers of the genes or hereditary determiners. " Davenport, C. B., "How we came by our bodies," New York, 1936.

492

BIOBNERGETICS AND GROWTH

egg is nearly 500,000 times its initial weight, weighing about a gram, the increase in weight being 98 per cent water. Water is economical for growth and gives plenty of "elbow room" for the developmental processes; it is the solvent and carrier of nutrients and wastes, and there must be plenty of it prior to the development of the circulatory system. The blastocyst forms the embryonic envelope and establishes contact for the inter

change of fluids between embryo and mother. After the attachment of the egg, the inner cells begin segmentation and differentiation to form the embryo and the nmniotic and yolk-sac vesicles form. The two vesicles flatten against each other and, together with the cells between them (mesoblast cells), form a three-layered germ disk which forms the embryo. The remainder is, like the trophoblast, accessory and temporary. The germ disk is formed in man about the third week. Afc ofAni.T;ais in Months 15

Koise Eat

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Di

Days 0 2

4

6

incubation A9e

Fig. 16.23a. Prenatal weight growth of the chick embryo and of its constituents and metabolites plotted from several sources on arithlog paper with indicated slopes.

BIOENERQETICS AND GROWTH

510

It is instructive to indicate the relation betweenthe instantaneous percentage growth rates as computed by equation (8a) and the conventional percentage growth rate com

puted by equation (2), which isthe method proposed byMinot^', used almost universally

by biologists until the appearance of our 1927 paper-^. We may take for illustration the data on fetal growth of the albino rat", ages 13 to 22

days. According to Minot, the computation for evaluating relative rates of growth is carried on as follows:

Days 0

rig. 16.23b. Prenatal weight growth in man, plotted from Streeter on arithlog paper. During the period 50 to 100 days prenatal life, growth occurs at 8 per centper day (body weight is doubled once in 8.7 days); between 160 and 230 days, at 1.7 per cent per day (body weight doubled in41 days); between 240 and280 days at 1.3 percent pertiay (body weight doubled in 55 days). Eight per xjent per day is equivalent to 240 per cent per

month; 1.7 per cent per day is equivalent to 51 per cent permonth; 1.25 percent perday is equivalent to 37.5 per cent per month.

Take the weight at a given age, and the weightat the next older age for which there are observations. From these data calculate the average daily increase in weight for

the period between the two determinations of weight, thenexpress the daily increase in percentage of the weight at the beginning of the period. "Minot, C. S., "The problem of age growth and death," New York, 1908; also J. PAysioi., 12, 97 (1891).

„

.

^

.

" Univ. Mo. Agr. Exp.Sta. Res. Bull. 97,1927. See also Schmalhausen, I., Arcft.£niwicklungsmech. Organ., 109, 455 (1927); 110, 33 (1927); 124,82 (1931). 2' Stotsenburg, J. M., Anal. Rec., 9, 667 (1915).

GROWTH RATES

511

We have showa^' that our inslanlaneuus growth rate, k in equation (8), is related to Minot's/iju/c growth rate, R in equation (2), by tlic logaritlimic function k = hi(R + I)

H8 illustratedbyFig. 16.9. Thedifference is particularly strikingingrowtli rate beyond about 10 per cent, as illustrated by the following numerical oxamjjles. At age 13 days, VKi = .040 gram and at 22 days = 4.630 grams. Tlie percentage growth rate, according to the method of Minot (eq. 2), is then

X 100 = 1275 •010 X 0

per cent per day; according to our method, the true or instantaneous percentage rate, . , In 4.630 A:(eq.8a),i8only ^

In .040

X100 = 52 per cent per day; depending on the method

used, the sameset of data yields 1275 per ccnt or 52 per cent growth rate. Kos.

Lba

*70 -

60

0-

i40 130 120

50

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40 .

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20 -

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Age Fig. 16.24. Juvenile growth in man. Well-nourished children grow at an approxi mately constant percentage rate between 5 and 15years (about 10 per cent per year; the body weight is doubled once in about 7 years). The prepubertal acceleration, so con

spicuous in the literature on growth of children, is usually found only in the curves of poorly nourished children (Figs. 16.50 and 16.52).

The difference in percentage rates as determined by the two methods decrease with

decreasing time intervals between weighings. Thus, by reducing the intervals between weighings to 5 days, we have: The fetal weight at 13 days, .040 gram; at 18days, 1.000 r, 1000 - -O-IO grams; hence R = — • X 100 = 480 per cent; and k •040 X 5

In 1 00 — In 04 5

X 100 =

64.4 per cent per day.

Similarly, for a two-day interval (between 13and 15days), Minot's arithmetic method yields 90 per cent, whereas our exponential method yields 51.5per cent per day. " Univ. Mo. Agr. Exp. Sta. Res. Bull. 97, pp. 18-19, 1927.

BIOENERGETICS AND GROWTH

512

If a set of data followsan exponential course (eqs. 7, 8, 9), then, knowing the numeri cal value of k, it is possible to compute the time required for doubling body weight or population size. The time required for a growing body or population to double itself in size is the ratio of the natural logarithm of 2, that is, 0.693 ... to the value of k. The reason for this is given in the following derivation.

B=.650

,

Mm.

0

Fig. 16.25. Graphic method for evaluating the constants of equations (12) and (14) for a rat. The correct weight value of the mature weight. A, is 350 gms.; if (350— IK) is plotted against age a straight line results. If a larger or smaller value of A is assumed, the curve deviates from a straight line as indicated. The value of B is read from the curve at the point when f = 0; i* = 1.76, the age when {A —W) —A = 350 gm. Solve equation (7) InW^ = InA + kt for I, \uW t =

k

InA

(7)

GROWTH RATES

513

When the original weight, A, is doubled, W becomes 2A and

_ In2A —InA _

k

2A

k^ A

In2

k •KgS 600 500 -

400

'•k

1;

1'

t

1

)

; )

:

7

?

4

7^

Fig. 16.30. Relative speeds of approach to the mature weight of albino and Norway Both are plotted in terms of the percentage of mature weiglit. One month in tlie albino rat Is equivalent in speed of approach to mature weight to 3.3 months in tlie Norway rat. The upper curvcs represent albino rats respectively of Donaldson and of Greenman; the lower curve represents Norway rats of King, rats.

The fact that the paper is divided on a decimal scale gives the slope, k, of equation (7), a value in terms of common logarithms. This value is converted into natural logarithms by dividing the observed slope by 2.3 (In 10 = 2.302. . .).

The equation may be fitted by the method of least squares (appendix to Ch. 13), but only after the data have been plotted on arithlog paper to as certain that there are no breaks in the curve and that tlie distribution of the

data on the arithlog grid is linear. Equation (7) may be fitted only to data distributed linearly on the arithlog paper. Fig. IG.IO shows the same (Stotsenburg) rat-growth data on arithlog paper (upper left) and also on arithmetic paper (lower right). The arithmetic curve

BIOENERGETICS AND GROWTH

518

has an increasing slope. The curve drawn through the arithlog cui-ve has a constant slope, k. Equations (7) or (8) thus represent satisfactorily the course of groA\-th of the rat fetus from 14 days after conception to birth, and the in stantaneous growth rate is constant—53 per cent per day as given by the equation fitted to the data ir =

.000065eO-"'

If this reasoning is sound, we have reached the important conclusion that the instantaneous percentage growth rate remains constant during the physio logically enormous period from 14 days after conception to birth. /

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Fig. 16.31. Relative approach to the mature weight of rats on normal diets (numbers 3, 4, 5, and 6) and "maximum-growth" diets (numbers 1 and 2). Curve 1 represents rat 3414 of Osborne and Mendel, curve 2, the average of rats B2135, B2132, B2164, B2161, B3380, B3432, B3414, B3441, B581, BG93, B1978, BI974, B2264, B226, B3218 raised on improved diets described in J. Biol. Chem., 69, 668, (1926). Curve 3 represents Osborne and Mendel's 1925 averages. The points connected by broken curves in Fig. 16.31 represent 98 per cent of mature weight.

The instantaneous percentage growth rate duiing the 10 days following birth is only about 12 per cent; but it is constant, which is the essential new fact. The constancy of percentage growth rate during the first 10 postnatal days is also indicated in Fig. IG.l 1, in which the values of lOO^*, for 3 sets of rats, are plotted against age; th{> resulting curves are horizontal, that is, the

percentage rate of growth is practically constant. The break in the curve at birth is, among other factoi-s, associated with a radical change in the mode of life.

519

GROWTH RATES

111 Fig. 16.12 the data during the first 10 days following birth are plotted again, together with the remaining data for the phase of growth preceding the inflection. From conccption age 32 days (10 daj's after birth) to 52 days, the instantaneou.s growth rate appears to be 4 per cent per day; from 52 days up to the inflection (about 85 days after conception, 65 days after birth) 3 per cent per (lay. (Wc arc not certain of the presence of later breaks.) The breaks arc also illustrated in the increment curve'^® of Fig. 16.13. The conclusion is, then, that while the percentage growth rate decHnes with age, the decline is much slower than has ever been thought before, and the decline does not appear to be continuous. The percentage growth rate remains relatively constant between rather wide limits, and then declines relatively abruptly to a new low level.

an in

6) ai Id

biioCa

Fig. 16.32. See legend for Fig. 16.31.

Figuratively .speaking, the medium in which the body ceils grow has buffer properties analogous to those of body fluids against acid or alkali. When, for example, acid is added to blood, the blood pH remains constant because of its buffer properties (Ch. 10). It is only after a certain fraction of the buffer is "spent" that the acidity exceeds a certain threshold, or critical value, and affects the welfare of the oiganism. May not an analogous situation exist with respect to the growth-retarding substances in the body? " Weight increments increase at the same percentage rate, lOOfc, as the body weight itself.

520

BIOENKRGETICS AND GROWTH

Indeed, this a])pears to be the situation for groAVth of a population of lactioacid producing oi-ganisms in milk, measured by the rate of accumulation of lactic acid in milk, as shown in Fig. 16.14. Neutralizing the lactic acid in the milk is followed by a new cycle of exponential growth (Fig. 18.1). Fig. 16.15 shows the constancy in the percentage growth rate of B. coli

(also an acid-pioducing organism) in broth. This constancyin the percentage growth rate of acid-producing bacteria Ls due to the high buffer value of the

medium, which neutralizes the growth-retarding lactic acid as it is produced, thus keeping the culture in the same state for a relatively long period. When the threshold pH is exceeded, the percentage growth rate declines.

Asimilar situation prevails in humanpopulation growth. When the density of the population is very low during its early histoiy and much more fertile land is available than the population can utilize, the natural increase of the human population occurs at a constant percentage rate. This is illustrated by Figs. 16.16 and 16.56 for the growth of the American Colonies and United States population. This population greAv at the instantaneous rate of 2.9 per cent per year from 1660 to 1890.

The critical or ."threshold value" was

reachedin 1880, when the growth of the population began to decline (Ch. 25). Kg

1

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50 -

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40

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Fig. 16.33. Comparison of growth of English children, "laboring" and "nonlaboring." Data by Roberts, compiled by

B. T. Baldwin, Univ. Iowa Studies in Child Wcljare, 1, 1 (1921).

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W

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Fig. 16.17 represents the percentage growth rate (100^•) of the rat as a func tion of age; it indicates the manner of decline in the "growth potential" with in creasing age. The graph reminds one of a series of water pipes, each of which is i-elatively horizontal, has a relatively constant head piessure, and is below

its predece.ssor, finally fading to zero.

This is the e.ssential history of a running

stream to its ultimate end, and of growth to its ultimate end.

The most striking age curve obtained in this analysis of early growth is shown in Fig. 16.18a, which relates CO2 production with age in tlie chick embryo. The age increase in CO2 production is perhaps a better index of growth than tho age increase of weight, since Aveight increase may be due to

increase in relatively inert, or even non-living, matter whereas CO2 production represents definitely living, metabolizing tissue.

521

GROWTH RATES

We prepared Fig. 16.18a from data by Atwood and Weakley.^® Each egg was incubated individually in a glass tube. The data points represent the average daily CO2 output of the 63 eggs which hatched normal chicks. The circles represent observed values. The relatively high CO2 output during the first day in comparison to the second is apparent only because the eggs were kept, as is customary, in a cool cellar before incubation. When, there fore, the eggs were subjected to the relatively high incubator temperature, there was in addition to the metabolic COj output an expulsion of the excess

f

He ifih

Fig. 16.34. The data in Fig. 16.33 in terms of per centage of mature weight, including data on growth in height.

'

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M01 rL bcs 'in( or

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ijh

CO2 because of its lower solubility at the incubator temperature. This ex planation is substantiated by the data for the CO2 excretion of a control, infertile, egg represented in Fig. 10.18aby crosses. The CO2 production during the first day is virtually the same foi' the control and incubating egg. The CO2 production associated with incubation is properly represented by the difference between the fertile and infertile control. The distribution of this difference between the CO2 production of fertile and infertile eggs is fairly linear on the arithlog grid: the instantaneous in crease in CO2 production during the first Jour days of inciihation is seen to be of the order of 100 per cent per day. It i.s interesting to note that the cleavage " Atwood, H., and Weakley, C. E., W. Virginia Agr. Exp. Sta. Bull. 185, 1924.

522

lilOENEHGETICS AND GROWTH

rate of the rabbit and rat egg« during the first three days of inciibutiun (Fig. 1G.2) is of the order of 120 (rat) to 140 per ('ent (rabbit) per day. Fig. 16.19 illustrates in a more striking manner the drop in CO2 production during the second day. Ilere the percentage increases, 100 (lnir2 — lnT7i), were plotted against age. The greater drop in ]3ohr and Hasselbulch'.s curve is probably due to a lower pi-eincul)ation tempei-ature than in Atwood and Weakley's.

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Age Fig. 16.35. Relative approach to maximum milk yield under Advanced-Registry test and ordinary conditions of management. The better-fed test animals approach the mature level at a more rapid rate.

Returning to Fig. 16.18a, from 4 to 15 days the data points are distributed in a remarkably imiform manner around a straight line, indicating an in stantaneous increase of 31 per cent per day.

The second remarkable feature

of this graph is the pause between 16 and 19 days. The chick no doubt passes a critical period, a "metamorphosis", at this sta^. This statement relating to a critical period is substantiated by the mortality curve, Fig. 16.20, which passes a peak at this time. Tlie trigger mechanism in the break may be a change in the mode of respiration: the respiratory function is transferred from the chorioallantoic membrane to the lungs; the chick "metamorphoses" from an aquatic to a terrestrial mode of respiration.

GROWTH RATES

523

The smaller peak in the mortality curve at about five days may perhaps be correlated with the peak in the lactic-acid curve sho\Mi in Fig. 16.20. The mechanism of lactic-acid oxidation apparently does not begin to function efficiently until this time. Fig. 10.21 represents data of Hasselbalch and of Murray. The distx'ibution of the data points Is less regular becau.se of the smaller number of embryos. D5 -

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Age Fig. 16.36. Relative lactation slopes for Jersey cattle.

These charts do not show the last stages of growth; otherwi.se, the general features of the charts are the same. The values of k for the data of Hassel balch are the same as those for the data of Atwood and Weakley. The value

of k for Murray's data is higher, but this is probably due to a higher incubation temperature. Fig. 10.18b, which represents the age curve of nitrogen storage in the silkworm embryo, is remarkably similar in respect to the growth pause to the age curve of CO2 excretion in Figs. 16.18a and 10.21. Fig. 1().22 represents the growth of the fowl during 12 weeks of postnatal life. There appears to be a break in the curve at 3 weeks. The major in flection occurs at the age of about 12 weeks. The values of h during this period are of the same order as those found for the rat.

524

BIOENERGETICS AND GROWTH

Humans have a very slow prenatal growth rate in contrast to other species. While the instantaneous prenatal growth rate in the rat is about 53 per cent per day (Fig. 10.10), that of man^^ ranges from a maximum of 8 per cent to a minimum of 1.3 per cent per day (Fig. 10.23b). Therefore, given percentage rates of growth do not indicate equivalent developmental stages. It was already noted (Fig. 16.7) that the age curve of man is distinguished from those of other species by a very long juvenile period. Fig. 16.24 shows that growth during the juvenile period is about 10 per cent per year, that is, only 0.83 per cent (10/12) per month or 0.03 per cent (0.83/30) per day. 05

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Age .•I = 55Ui h = liolstcin A = 515; k Guernsey, KM A = 460; k = Ayrshire A =-- 436; k = Jersey, RM A = 420; k = Jersey Fig. 16.39. Growt.li of

-.04U; I* = 8.3 .044; t* = 0.2 -.050; t' = 9.1 -.050; t* = 0.2 -.054; I* = 8.9 dairy cuttle.

available food supply, available freedom from tiie deleterious products of growth, and so on. These ideas may be foi-mulated in the conventional terms of the physical chemist, with the symbols used for the self-accelerating phase of growth. " Cohn, A. E., and Murray, H. A., Jr., Qttarl. Hcv. Biol., 2, 469 (1927).

GROWTH RATES

527

Let A represent the limiting food i-etjiiired to attain maximum individual or population size, and IF the food supply at the given time; (/I — W) then represents the concentI'ation of tho limiting food supply at the moment just

sufficient to permit attainment of maximum indivichial or population size. It is rea.'ionable to assume that tlie instantaneous growth velocity, dW/d(, at the given time will ho proportional to tho concentration of the limiting food supply, that is, to tho value (/I — 11') = -kiA -

W)

(9)

dt

K S! DIP:

I'ig. 1(5.40, (ii'owlii of Horsfis.

Instead of food, tho gi-owth-Hmiting fact or in tho environment may be some gi'owth product, as lactic acid to growing hurtic-acid bacteria in milk, or alcohol to alcohol-producing yeast in fruit juice. Let A be the concentration of lactic acid or alcohol suppi-essing completely the growth force residing in the cells, and W its concentration at the present time. Then, as before (A — W) i-epresents the amount of growth which the environment will permit in order dW

to bring the population to the maximum size, A, and — = at

(A — W),

the instantaneous velocity of growth at the given time. Analogous reasoning applies to the growth of multicollular animals, which are cell populations, after all.

528

BIOENERGETICS AND GROWTH

The numerical values of the constants are estimated as follows: A, which represents the concentration of -the growth-limiting factor when growth is completely inhibited, may be used to repicsent the mature weight of the animal (or the maximum size of the population) under a given set of con ditions; W may be used to represent the weight of the animal (or size of the population) at the given time; (A — IF) then represents the amount of growth yet to be made to reach the mature weight.

Although equation (9) appears to differ from equation (5), both represent the mass law (for a first-oi'der process); both represent a direct proportionality between gi-owth velocity, dW/dl, and some growth-hmiting factor. In equa-

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Fig. 16.41. Growth of Sheep.

tion (5) the growth-limiting factor is W, the growth already made; in equation (9) it is {A — W). In equation (5), k is the relative growth rate with respect to the growth already made, k =

dW/dt W

(5)

while in equation (9) k is the relative growth rate with respect to the growth yet to be made dW/dt -k « A-W

(9)

GROWTH RATES

529

Before applj'ing equation (9) to data, it is integrated, for reasons explained in the preceding section: dW/dl « -feU - If)

(9)

dW

In (A —

= —kt + InS (integration constant)

(10)

A -

ir = Be-^'

(11)

W

A -

(12)

The significance of the constanc.v of the exponent k is that the growth veloc ity declines at a constant percentage rate 100/c, illustrated by the following numerical example.

K0 <

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17

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Fig. 16.55a. Growth of tlic luiiniin brain in weight, IK, atid in frequency, /, of brain potentials (or alpha frequency or alplui waves, or liergor rliythms). The value of —fc (decline in the relative growth rate) is —0.4G8 for the alpha waves and —0.485 for brain weight (i.e., 48.5 per cent per year); l.lie mature values, Wa and f,i are 1333 gm. for the . W f Wa brain and 10 for the alpha waves. Since rfj- ~ -i- = W = f-p—; tiiat is, the brain weight, A

J.A

W, is 133.3times the alpha wave frequency, / (Weinbach).

Ja

GROWTH RATES

543

Summarizing this discussion, prior to puberty in animals, flowering in plants, and "coming ofage" in populations, the growth rate tends to be pro portional to the growth already made, indicated by the equation dW kW

dt

(5) (8)

Following this age growth rate tends to be proportional to the growth yet to be made dW

-HA - W)

dt

W

A-

(9) (12)

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Fig. 16.55b. The equivalence chart between brain and wave frequency shows their excellentcorrespondence. Courtesy of A. P. Weinbach, Growth, 2, 247 (1938).

[The numerical value ofk in equation (9) or (12) is,ofcourse, different and has an opposite sign from the k in equation (5) or (8)].

Animals rapidly attaining the maximal body weight. A, have a high slope, k, as in equations (9) or (12); those attaining it slowly, have a low slope, k. The value ofk forgiven guinea pigs is 0.22; for cattle, which approach mature weight much more slowly, the value of A; is much lower, 0.04; for mice, ap proaching the mature weight much more rapidly, k is much higher, 0.71; and so on.

BIOENERGETICS AND GROWTH

544

16.6: Genetic growth constants. The value of A (mature weight) and of the slope, k, on the growthcurve arc intrinsic or genetic characteristics of the animal under given environmental conditions, in the same sense that the equilibrium and velocity constants of a chemi(^al reaction in vitro are intrinsic characteristics of the chemical system under given conditions. Growth, like a chemical reaction in vitro, is by definition increase in the mass of one com

ponent at the expense of another, and the rates of approach to the equilib rium level are analogous. Table 16.1 (appendix) presents numerical values of the intrinsic or genetic growth constants, A, k, and related derived values.

Figs. 16.30 to 16.54 present the constants graphically, together -with age curves of growth in w'eight of several animal species. The charts are for the most part self-explanatol•}^ The values of A (mature

weight), k (speed of approach to mature weight during the self-inhibiting 2cx> tco

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Fig 16 56a The growth of the liunian popuhitioii in the U.S.A., plotted ou semilog

paper. The line represents the equation P = .0842e'' «»', meaning that the population, growsat 2.9% per year (instantaneous basis) or is doubled in 23.4 years.

phase of growth), and i* (age at which the extrapolated curve meets the age axis) are given for each species. The differences within the species are due mostly to difTerences in environmental conditions, especially food supply. Before proceeding with the discussion of the influence of environment on the numeri cal values of the constant, it is interesting to note that the mature size. A, of different

species tends to be related to the growth constant, k, or to the time requiredto reach a

GROWTH RATES

545

given percentage of the mature weight (Fig. 16.29 plotted from Table 16.1). As might be expected from the long juvenile period, the data points for man deviate from the general curve. Also, as might be cxpected, there is considerable scatter of the data because A and k are not influenced by environmental conditions in the same direction and the environmental conditions for the different animals were not the same. More

over, males, females, and castrates were lumped together in Fig. 16.29. Nevertheless, the index of correlation is relatively satisfactory.

We may begin by pointing to the difference in approach to mature weight in Wistar Norway and albino rats^" (Fig. 16.30). The mature weight is virtually the same" in both, but the albinos attained mature weight much earlier. Dr. King informed us that the Norway curves approach the albinos 200

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Fig. 16.56b. Growth of the human population in the U.S.A. showing the increment

curve (2) and cumulative curve (1), extrapolated to year 2100.

with successive generations of cage life. Cage life apparently does not agree ^ith fii-st-generation Nor^\ay rats.

Figs. 16.31 and 16.32 represent growth curves ofalbino rats fed on ordinary and "improved" diets. The animals on the "maximum-growth" ration reach the mature size much earlier than the controls.

The data for Osborne

and Mendel "maximum groAvtli" rats were given the writer by Dr. Osborne during a personal visit. Many published paperefrom Osborne and Mendel's " Data in Donaldson, H. H., "The rat," 1915, 1924 (Wistar Institute, Philadelphia).

BIOENERGETICS AND GROWTH

546

laboratory on exceptional!}' rapid growth of rats®^ substantiate the curves in Fig. 16.31 and 16.32.

Similar results were observed on other species. Thus, Figs. 16.33 and 16.34 indicate that the average English "laboring" individual appears to be smaller than the aveiage "non-laboring" individual, and that cliildren of the laboring class take longer to reach o• a given growth stage than those of the non-laboring. This difference may be due in part to differences in environmental conditions.®^ This difference in weight-growth rate (see also Figs. 16.50 to 16.54) may partially account for diffei'ences in intelligence quotients. It is not unreason able to assume that mental and physical development are associated.®® 6

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Since the maintenance cost of animals is the largest item in the growth cost, rapid growth and early maturity imply some saving, and therefore increased efficiency not onlj- of growth but also of productive processes (such as milk and egg production, muscular work), since the growth expense is charged to the

productive jjroeess during maturity.

(For exceptions, see Chs. 3 and 18.)

There are similar environmental influences on the maturation speed of

pliysiologi(^al functions, such as milk production, illu.strated in Figs. 16.35 to Osbonic, T. B., and Mendel, L. B., el al., J. Biol. Clicin., 69, 611 (11)2G), and 75, 776 (1927). Smith, A. H., and Biug, b\ C., J. Nutrition, 1, 179 (11)28-29). Anderson, W. E., and Smith, A. H., Avi. J. Physiol., 100, 511 (1932). Bryan, A. H., and (ijviscr, D. W.,

"Diet and pituitary hormone on growth," Am. J. Physiol., 99, 379 (1931-32). Cf. Pal,on, D. N., and Findlay, L., "Poverty, nutrition and growth," Medical Re

search Council (English) Special Report 101, 1926.

Cf. Bichl, W. C., "Early inanition and tlie developmental schedule in the rat," J. Comp. Psychol., 28, 1 (1939).

mond, Proc. Roy. Soc. London, 125B, 311 (1938).

Fig. 16.57. Parents and offspring of reciprocal Shetland-Shire crosses. Courtesy of ArthurW. Walton and John Ham

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548

BIOENERGETICS AND GROWTH

16.38. Advanced Registry dairy cattle receive better care and feed, and they approach their maximum milk yield and weight at an earlier age.

Employing chemical terminology, increasing the effective concentration of

the gro\\'th-limiting constituents (food) in the system increases the velocity,

k, of the process (growth). The growth velocity, k, of the poorly fed rat of

Donaldson (Figs. 16.31 and 16.32) is 0.0135, but that of the well-fed rat of Osborne and Mendel is0.0266; thus the speed ofapproach to the mature value is twice as great in the rats on the better diet.

Nutritional aspects of growth arediscussed in Chapter 20; endocrine aspects, in Chapter 7;seasonal aspccts, in Chapter 8; temperature aspects, in Chapter 11.

Fig. 16.57a. Robertson's theo retical autocatalytic monomo-

lecular cycle representing the time rate of gain, dW/dl, as function of time. Note its sym

metry about the center. Actual "growth cycles" are not so symmetrical.

"Growth Cycle

Fig. 16.57, from Walton and Hammond," is a dramatic illustration of

spatial effect on growth—how the size of the mother influences the prenatal size of the offspring. The parents were the giant (1800 - 2400-lb or 800 1000-kg) Shire and the dwarf (540-lb or 200-kg) Shetland. The cross foals from the giant mother and dwarf father were three times the size of the cross foals from the dwarf mother and the giant father. The maternal-environ mental limitationmasked the genetic differences duringthe prenatalperiod of

growth. After weaning, however, the genetic influence came into its own. The cross foals of the dwarf mothers grew much more rapidly than Shetlands

and the growth curves of the two crosses tended to converge to a common mature weight. Bj'^erly^'' reported similar results by hybridizing bantam silkies and Rhode Island Red chickens.

16.7: Note on the relation between average and individual growth curves. A fundamental characteristic of living organisms is that they are alike in

general plan and different in detail. There .are two corresponding schools of biologists, one of which emphasizes the average and general and the other the individual and particular. The rapid development and diffusion of statistical techniques in the biologic aswell as in the physical sciences tends to

emphasize average and the general; on the other hand, great discoveries and Walton, A., and Hammond, J., Proc. Roy. Soc. London, 125B, 311 (1938), and in several other publications.

^

" Byerly, T. C., el al., J. Exp. Zool., 78, 210 (1938).

549

GROWTH RATES

concepts of biology have resulted from scrutiny of the individual, of the detail. Thus the concept of natural selection emphasized differences between in dividuals; the notion ofgene is an individual one; so is the concept of mutation. Both approaches are useful, depending on the opportunities and aims. Davenport®® investigated the relative significance of individual and average growth curves of children with special reference to the adolescent gro\\i.h sptirt.

He found that adolescent growth acceleration is often explosive, occurring at any time between 11 and 17 years. It may be sharp or fluctuating, high or low.

Environment could not be the chief factor because great variations

were observed between brothers reared in the same home.

Fig. 16.57b. The time course of oxi dation of linseed oil under various

conditions, a familiar autocatalytic

process resembling growth curves.

Timp Davenport believes that the average increment curve not only resembles the Gaussian (probability) distribution curve, but is a Gaussian curve. The peak of the average increment curve is at 14.5 years; individual growth spurts are distributed around this average. The frequency and amplitude of the spurts diminish on each side of the peak. Similar results were reported by Boas,®^ Gray,'® and others (Fig. 16.60a). MerrelF® reached a similar conclusion for rabbits. The Pearl-Reed logistic was fitted to individual and average growth curves of rabbits. The average differed in fundamental characteristics from the separate curves. Significant undulations observed in the average were absent in the individual growth curve. Undulations and changes in skewness are often the result of the averaging process, unrelated to the biology of growth. In brief, the growth process in an individual is not the smooth sigmoid curve represented by average curves. The average curve represents properties of the mathematical averaging process often absent in individuals. A consider able literature is being developed on individual growth by the "longitudi nal" method consisting of seriatim observations^® of the same individuals. Davenport, C. B., Proc. Am. Philosop. Soc., 70, 381 (1931).

Boas, F., Science, 72, 44 (1930). Gray, A. H., Roy. Phil. Soc. Glasgow, 40, 139 (1909). Merrell, M., Human Biology, 3, 37 (1931).

Scammon, R. E., "Seriatim study of human growth," Avi. J. Physical Anthrop., 10, 329 (1927).

550-

BIOENERGETICS AND GROWTH

16.8: Growth of the human population in the United States. In the next 25years the puhition of tlic Island will be 28 million, and the means of subsisence only equal to the support of 21 million.

In the next

period, the population would be 56 million, and the means of subsistence just sufficient for half that number. T. R. MallhMs, 1798.

It ma}' be .surprising to leaiii that the growth of the human population has the same shape and may he represented ])y the same equations as the growth curves of individual animals and of populations of bacteria, yeast, and flies confined to a limited space. Yet this appears to be true, and attention has already been called to this similarity (Figs. 1G.4 to 16.5, 16.15 to 16.16, 16.56a to c, and 19.32). aoo

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As we have seen, the time curve of growth—of individuals or populations— is composed of a segment of increasing slope and a segment of decreasing slope, with the inflection occurring at a critical period, flo.wering in plants, pubertyin animals, and the end of a "free frontier" or some other critical era in population growth. Now Malthus' statement of the tendency of human population— in common with other populations—to increase exponentially, that is in a geometric progression, in this case doubling itself every 25 yeai-s, is correct for the period of unrestricted growth, for the segment of increasing slope. This, indeed, is the history of the population growth in the United States

551

GROWTH RATES

between 1660 and 1880, as shown in Figs. 16.16 and 16.56. During these 220-odd years the population doubled itself every 23^ years, quite close to Malthus' estimate of 25 years. The rate of this doubling is also shown by Curve 3, Fig. 56c; the doubling curve is quite horizontal, near 25 years be tween 1660 and 1880; the population increased at the rate of 2.96 per cent— about 3 per cent—^per year.

Malthus, however, committed a common extrapolation error in saying that "in the next period, the population would be 56 million". The realization of the inherent tendency to grow, to increase in a geometric progression, at a constant percentage rate, occurs only when the environment does not restrict growth, a condition offeredby the United States up until about 1880; There after the expansive urge for geometric-progression increase is held in check i5oq *N •-0—