bioenergetics and growth chapter 16

8/13/2019 Bioenergetics and Growth Chapter 16

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Chapter 16

Time Relations of Growth of Individuals and Populations

All motion of natural action is performed i n t im e. F. BaconEverj thing exists not only in a frame of space but also in a pattern of time.

G. E. CoghillAs early as th e fifth century B .C. , G re ek physicians developed a very

clever method fo r t he s tu d y of growth which is employed by sc ien t is t s downto our day. A hen was set upon a number of eggs; each day one of these eggswas 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 ofmetaboUc processes has shown that, perhaps, all constituents of the living cellare involved in continuous chemical reactions, continuous breaking down andbuilding upcatabolism and anabolism. I t is onlj t he pat te rn , the life whirl

pool, that endures. Biologic synthesis, that is, the interaction of exogenousmaterial food) in the formation of new chemical-morphological units, thusoccurs not only during the period conventionally designated as growth, butthroughout the entire life cycle^.

The occurrence of widespread synthesis throughout life may also be observed without refined metabol ic s tudies . Thus blood cells and epidermishave long been known to undergo rapid destruction and renewal; there is acontinuous need for growth catalysts hormones, vitamins, etc.) and structuralmater ia ls amino acids, minerals) to compensate for the continuous losses,breaking down, or catabolism, of th e body. These constructive processesare more dramatic during periods following starvation and injur} , especiallyregenera tion of limbs in lower forms of life.

The most spectacular type of directed biosynthes is is, of course, growthand development, especially during embiyonic life. Everyone has beenimpressed by the miraculous transformation of the sticky \\ hite and yellowmass of hen s egg into a fully dressed, befeathered, respectable chick, all in21 days. The original egg cell must have travelled at a dizzy pace to build

up so complex a mechanismprobably exceeding in complexity the astronomical W nders 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


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G R O W T H R T S 8

Growth as thus defined is inseparable from metabolism, and the several chapters inthis book are merely different aspects of essentially the same problem, mef,abolism-growth. Thus Chapter 6 is concerned with enzymes in metabolism, in biologic synthesis; 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; 3 two primary blastomeres, about 25hours after; C,4-cell stage 29 hours after copulation; D and E, 6-ceIl and 8-cell stages 32 hours; F, 32-cell morula 55 hours; G, morula 70 hours; 11, trophoblast cells, 71hours; I, fluid collections 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)].


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48 6 B I O E N E R G F r i C S A N D GROWTH

Chapter 20, with general nut ritional aspects, anil so on. This (Oinptei is concerned wil.hthe definitions and lime rehxtions of average development an


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48 8 BIOENERGETICS AN D GROWTH

material from the environment involving neither cell multiplication nor cell enlargement. Such increase is not regarded as true growth. Ye t operationally, from th estandpoint of quantitative measurement of growth of the organism as a whole, we mustconsider these non-protoplasmic inclusionsif they are irreversibleas parts of thegrowth process.

A L L A N TO t S

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Fig. 16.3 a, b. Domestic fowl s egg and its development, from A. L. Romanoff. CornellExp. Bull. 205, 1931 and 1936. IBeginning of alimentary tract, hours; vertebral column, 20 hours; nervous system, 21 hours; head, 22 hours; blood island-vitellin circulation, 23 hours; eye, 24 hours; ear, 25 hours; heart beat, 42 hours; amnion, 50 hours; legsand wings 63 hours; allantois, 70 hours; reproductive organs, 5th day; feathers, 8th day;beak turns toward air cells, 17th day; yo k sac begins to enter body cav ity 19th day;yolk sac completely drawn into body cavity, 20th day; hatching of chick, 2l8t day.

The knowledge of the mechanism whereby the protoplasmic mass increases is in it sinitial stages of investigation is not understood.

For li terature see Gulick, A., Growth, 241 1939); Af/yances inEnzymotogy, 4 1944).Bergman, M., Chem. Rev., 22, 423 1938). Wrinch, Dorothy, Protoplasma, 25, 550 1936);Proc. Roy.Soc., 161A,505 1937). Mark, H., A aurc,140,8 1937). See also many papersby F. S. Hammett in Growth indicating the functions of various amino acids an d othersubstances in growth of the hydroid obelia; Schoenheimer, i and Supplements 2, 3, an d 4 194 M2) to Growth.


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G R O W T H R A T E S 48 9

Diffcrcnlialion (cytogcncsis, histogenesis) is transformation of mother cell, such asegg cell, into different kinds of daughter cellsbrain cells, kidney cells, liver cells, etc.This process is irreversible. Egg cells a re t rans fo rmed to liver cells, but liver cells

cannot be transformed into egg cells. There is a running down of growth potentialitieswith increasing differentiation in th e individual, analogous to running down of freeenergy in t he l arge r universe (Ch. 2).

Morphogenesis (organogenesis), another aspect of development, refers to the organization of the various cells into special organs of definite form, andthe organization of theorgan-systems into the body as a whole.

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^100X ^ risingS shapedcurvecumulativeorcourseofgrowthorth e tota l weight a t given age.

Needham^ subdivides growth into: (1) cell multiplication, (2) intussusception, orincrease in size of cells, and (3) accretion or increase in amount of non-living structuralmatter;

differentiationinto: fl

increase innumber

ofkinds

of cells,and

(2) increase inmorphological heterogeneity; meiaftoZisffi into: (1) respiration oxidation , (2) fermentat ion or glycolysis, (3) catabolism of protein, (4) civtiibqlism of fat , (5) chemical activity,as pigment-formation, glycogen synthesis, etc.

O th er a ut ho rs have olher scl iemes of classifications. We defined development toinclude growth; Hammett defined growth to include developmentgrowth is the co-

Needham, J . , Biol. Rev., 8, 180 (1933).


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4 90 B I O E N E R G E T I C S A N D G R O W T H

ordinatecl expression of incremental an d developmental factors and functions . In thischapter, concerned not w ith g ro wth mechanisms but with time r el at io ns , g ro w th isdefined operationally by increase in weight.

Tlie bird s egg is doccptively large because of its cleidoic nature, it s yolk and albumen stores and related redundant structures , such as ai r chambers for the nutrition()f t he embryo. In contrast, th e mammalian egg is a minute speck. The mammalianegg, first observed in 1827 in th e dog , appears to be nearly independent of th e size of themature animal. I t ranges* from about 70 to 8o i (/i is ro off millimeter) in rodents (mice,rats, guinea pigs) to about I40m in dogs, horses, sheep, goats, pigs, whales, and pr imates ,including man. In other words, the egg of mouse, man, or whale is of practically thesame size, about A millimeter in diameter. Yet, a given egg grows, differentiates, an ddevelops into chick, mouse, man, or whale an d goes through life according to its respect iv e inh er it ed pattern.

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G R O W T H R A T E S 491

Under g reat magnincat ion the fertilized mammal s egg Is seen to contain particles inmotion, in agitation, eventuating in cleavage of the egg cell into two daugbter cells.In this divi sion , as in perhaps th e following 3 or 4 cleavages, there is no increase in themass as a whole, but only subdivisions of th e cells.

This basic process of division of one cell into two, tlie nature of which is no t understood, is on e of th e most d ist inguish ing charac ter i st i cs of l iv ing organ isms . An evenmore remarkable example of division, inexplicable by known pliyslcal forces, is t he prod uc ti on a nd expulsion of polar bodies by th e unfertilized egg in preparation for th e reception of tlie sperm s contribution to the zijgoie^^ This anticipation of future developmental needs of both tlie individual and the race sets living processes apart fromnon-l iv ing. S imil ar ly, other s t ruc tures develop-in anticipation ofand long beforethe t ime when they will be c al le d u po n to function in the service of the organism as awhole (Ch. 10).

BBt.^-.ttasfrancateptlcn

g> W M g Sauattuftpo ^fr nponm ttr

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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 ofthe fertilized egg, but there is increasingly pronounced differentiation so that a t the 16-cell stage the Irophoblasts, the cells destined to furnish the fetal m em br an e, a nd thestructures of implantation of the egg on t he materna l tissue, become distinctive. Thetrophoblast cells divide more ra pi dl y a nd a re , therefore, smaller than the other cells.

Next most con8|)icuous is th e formation of the tluid-distended vesicle, the blastocyst(Fig, 16.1), and cupping in to a hollow sphere. Fluid collects within this sphere, perhaps

through th e secretory activity of t he t rophob la st cells, an d th e cell mass, for th e firstt ime , en la rges by th e accumulation of fluids.The outs t and ing fea ture of th e growth of the blastocyst is th e absorption of tre

mendous quantities of water. Davenport estimated tha t after six weeks th e human

Zygote, the yoked f ir st cell of the body, th e union of male an d female germ cells,or gametes, carriers of the genes or hereditary determiners.

Davenport, C. B., How we came by our bodies, New York, 1936.


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4 9 B I O B N E R G E T I C S A N D G R O W T H

egg is nearly 500,000 times its ini tial weight, weighing about a gra m, the increase inweigh t being 98 pe r cent water. Water is economica l for growth and gives plenty of elbow room for th e developmental processes; i t is the solvent and carrier of nutrientsand wastes, an d there must be plenty of i t prior to th e development of the circulatorysystem.

The blastocyst forms the embryonic enve lope and establishes contact for th e in te rchange of fluids between embryo and mother. After t he a tt ac hm e nt of the egg, theinner cells begin segmentat ion and differentiation to form the embryo and the nmniotican d yolk-sac vesicles form. The two vesicles flatten against each other and, togetherwith th e cells between them (mesoblast cells), form a three-layered germ disk whichforms th e embryo. The remainder is, like the trophoblast, accessory and temporary.The germ disk is formed in m an about the third week.

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G R O W T H R T S 4 9 3

week in man, and th e organs are formed by the t hi rd m on th . In other words, the ground plan is l aid very early in life an d rounded out l at er through enlargement an dremodeling of the parts. Age changes in shape are due to differences in growth rates ofthe constituent parts Cli. 17).

There is an orderly sequence, or gradient^*, in organ formation. The head has precedence in development over the tail end, and so on in cephalochordal sequence for theother organs. The sequences may be associa ted with organizer and hormone action C h . 7 ) .

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AQ6Fig. 16.8. Comparison of weight ga in an d specific mortality in children.

With t he se b ri ef definitions and introduction we proceed with our m ajorproblem, an analy.sis of the t ime relations of growth in weight.

Many of our gigantic industrial organizations are extremely complex; noone can know all their details. Yet accountan ts , qui te ignorant of thesedetails, render itifolligible and useful corporate statements. ikewise the

Cf., Chi ld , C. M., Individuality in organisms, Univ. Chicago Press, 1915. Cf. Bertalanffy, L. V., Theoretische Biologie, Berlin, 1932; also Roux Arch., 181,

613 1934), and Human Biology, 10, 181 1938).


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9 l i l O E N E R G E T I C S A N D G R O W T H

animal bodj is extremely complex; no one can know all its details. Yetas we shall show in this chapter the time relations of growth can be repre

sented by intelligible useful and rational statements or laws of growth.

ao 120 160 200 24 0

Tpue Pepcentape Fate[ioo^Y

Fig 16.9. The relation Ijctwccii the percentage growth rate computed by the instsintaneouslogarithmic and b y t he finite arithmetic method


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G R O W T H R AT E S 9

I t ma y be recal led in this connection, that some of the great Jaws of thephysical scionccs, such as Newton s la w of gravitation, say nothing about

detailed mechanisms involved; they are only intelligible, useful, and more orless rational descriptive statements of the phenomenon. There is, of course,a wide range in i-ationality in many so-called laws of nature. We hopethat the following growth equations partake more of laws of nature and lessof the accountant s purely empirical rendering of a financial statement.

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X tfi O 23 25 7? 29 31

Age

Fig. 16.10. Growth of th e white rat between 13 days after conception an d 10 days afterbirth, plotted on arithlog and on arithmetic coordinate paper.

16.2: The shape of the age curve of growth of individuals and populations

There is no bound to prolific nature in plants and animals but what ismade by their crowding an d interfering with each other s means of sub-

sistence. T . R. Mallhus 1798


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496 I O E N E R G E T I S A N D G R O W T H

Fig. 16.5 represents age curves of growth of populatioiis of yeast and flies,and of individual pumpkins and rats. They all have the same s-like shape.Indeed, all these curves may be made to coincide, as shown in Fig. 16.C.

Fig. 16.7 demonstrates the close similarity between the age curves of diffe re nt a nim al species. The human ag e curve, however, differe from theotheis in having a very long juveni le per iod, a long interval between weaningand puberty approximately 3 to 13 yeai-s); this period is almost absent inlaboratory and farm animals. In these animals, weaning merges into adolescence w it ho ut th e intervention of the juvenile phase found in man. Theuniquely long juvenile period in man should be of particular intei est to stud e n t s of educat ion.^

9 Donaldson)

2 Gpeenman)

cj Gpeenman)

Bipth Age

Fig. 16.11. T he t ru e percentage growth rate of three sets of rats during th e first te ndays of postnatal life.

The general similarity between the curves of growth of individuals and ofpopulations is not surprising, since ultimately both are collections of in

dividuals. Our bodies are made of cells, and our bodies, in turn, are cells inthe social body. Individuals are organisms and also units of a larger organism, an epiorganism Sect . 10.9).

Cf. Fiske , John, in The Meaning of Infancy ; Boston, 1883: I f there is an y onething in which the human race is signally dis tinguished from other mammals, i t is theenormous duration of their infancy . . . this period of helplessness . . . is a period of plasticity a door through which the capacity for progress can enter . . . power to modify. . . i n h er i te d t e n d en c i es .


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G R O W T H R T S 497

It may be seenfrom Fig. 16.4 that the age curve of growth may be dividedinto two principal segments, the first of increasing slope, which may be designated as the self-acccleraling phase of growth and the second of decreasingslope, which may be designated as the self-inhibiting phase of growth.

The general shape of the age curve may thus be said to be determined byt\N o opposing forces: a growth-accelerating force and a growth-retarding force.

G w e n m a i

G n c n m a i

Pig 16.12. Postnatal growth of ra t to t he puber ta l inflection

The formermanifests

itself in the tendency of the reproducing units to reproduce at a constant percentage rate indefinitely^ when permitted to do so. Inthe absence of inhibiting forces, the number of new individuals produced perunit t ime is always proportional to the number of reproducing units. Thatis, the percentage growth rate tends to remain constant. The potentiallyinfinite growth abili ty and consequent immortali ty of somatic cells wasestablished by Leo Loeb by in vivo experiments with cancer tissue trans-

Loeb, L., J Med. Res., 6, 28 1901), an d J Gen. Physiol 8, 417 1926).


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4 9 8 H I O E S E H a E T l C S A N D G R O W T H

plantation oi the cancer tissue to successive generations of animals , and verified by Cai rel^ by in vitro experiments wi th normal connective and other

tissues, by cultivating t issue cells in vitro under condi tions of continuousirrigation, tiuis preventing change in environment which would result in thedevelopment of growth-inhibiting forces.

But there comes a time, marked especially by the inflection in the growthcurve, when the increase in the population tends to be proportional no t to thenumber of reproducing individuals in the iôpulation but to the availableresources necessary for growth; the i-esources may be in the form of space,

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16/91

GHOWTII R AT E S 499

Th e inflection in the age curve Figs. 10.4 to 10.8 an d 16.13 represents thepo.sition at which the increase in growth velocity ceases, and the decrease in

velocity has not yet begun; therefore, the inflection represents the positionat which gains arc most rapid, and perhaps most economical Figs. 16.4and 16.8).

At the point of inflection tlie change in the lime rate of growth {i.e., inacceleration is the same in all animals or populations the numerical valueof the acceleration at this time is zero . This i > therefore, a point of geometric and physiologic age equivalence. There is at least one physiological

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pH 4.73 is reached. The acid increiises at 47 per cent per hour, or is doubled in 0.471 5 houi - s .

.stage thi ough which all animals seem to pass at this time, i.e., pubert3^ Thusin Fig. 16.4 the inflection in the curve of the female rat occurs at about 65days 80 days after conception , and this is the usual age at which the vaginaopens. In childien the inflection t ccurs between 12 and 15 year s the age ofpubei ty.


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5 0 0 B I O E M C R G E T I C S A N D G R O W T H

A third fact relating to children, and possiblj to animals also, is that thecurve of specific mortality {i.e., the ratio of the number dying to the number

living of the same age) passes through a minimum at approximately the sameage as the growth curve passes through its inflection. This is shown in Fig.16.8. The specific mortality decreases to this age, an d increases thereafter.

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The point of inflection, then indicates: (1) the time of maximum velocityof growth (transition from increasing to decreasing growth velocity); (2)the age of puberty; (3) the lowest specific mortality, the beginning of theperiod of increasing specific mortality; and 4 a geometric referent for thedetermination of equivalenceof age in different animals (and also equivalenceof age in the growth of populations . The point of inflection is thus an important growth constant Figs. 10.7 and 16.8 show that the inflection inman occurs at the age of about 14, which corresponds on tlie axis of ordinatesto somewhat over 60 per cent of the mature weiglit. In chimpanzees theinflection at age seven yeai-s) oocui-s at slightly below per cent ofthe mature


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GROWTH H AT E S 501

body weight In laboratory and farm animals (Fig. 16.7),on the other hand,the inflection takes place when about 30 per cent of th e m atu re weight is

reached, corresponding to about six months in cattle, or two months in sheep.Summarizing, the shape of all age curves of growth, whether of individuals

or populations, is sigmoid; the early phase is of rising slope and the later

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Fig. 16 16 Growth ofthe human population in the North American Colonies and in theUnited Sta tes k = .029; the population increased at 2.9 per cent per year, doubled

0 09 3itself in 24 years. (From data in A century of population growth in the United

States by W. S. Rossiter, Bureau of the Census, United States Department of Commerce and Labor Washington 1909).

phase of declining slope. The junction between the two phases occurs duringpuberty in animals, flowering in plants, and coming of age in populations,when, because of the back pressure of the environment, the reproductive ratedeclines or the excess population sets out on migrations, such as the dramaticmigration of the lemming to the sea.

Gretber, W. F. and Yerkes, R. M., Am. J . Phys. Anthrop., 27, 181 (1940).


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502 B I O E N H R G E T I C S A N D GROWTH

16.3: Definitions an d quantitative representations of growth rates

If nature were our banker, she would not add the interest to the principalevery year; rather would the interest be added to the capital continuouslyf rom m om en t to m o m e n t . J W . Mellor

For purposes of quantitative analysis, growth may be defined as relativelyirreversible time change in magnitude of tlic measured dimension or function.

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Fig. 16.17. The percentage rales lOOi) of grouih of th e albino ra t plotted againstage. These arc , of course, computed values, computed by the grapliic method explainedin the text. The curve illustrates in a s tr ik ing manne r the discontinuous nature of thegrowth process.

The concept of irreversibility i s emphasized to excludc iiuctuating time changesof a fortuitous nature, such as tho.se occasioned by fluctuating food supplyAvith consequent fat tening and leaning, with gestation, lactation, and so on.

Growth in weight is usually represented in one or all of the three waj s shownin Fig. 16.4: 1) absolute gain in the given magnitude per unit time; 2)relative rate (or -percentage when mul tipl ied by 100) gain per unit time; 3)cumulative, or course-of-growth weight up to, or the weight at , a given time.All these forms of representation may be made in conventional mathematicalterminology.


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G R O W T H R A T E S 5 03

Thus the absolute gain in the observed-Aveight difference, Wi, for thecorresponding time difference, k may be represented by the equation

Average nhnolule growth rate =

C m s

Days

W2 Wi larger w ei gh t l es s smaller weight*

2 U larger time less smaller time

apt l e

f iptj It k K i

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I n c u b a t i o n

1)

Fig. IG.lSa. Carbon dioxide excroUoii in th e c hi ck embryo. From 0 to 4 days therate of increase is 08 per ccnt per day doubled once in 0.7 day or in 17 hours); between4 and 14 days, th e rate is 31 pe r cent per day doubled in 2.2 days). The pause in thecurve coincides with th e maximum in the mortalitj curve Fig. 16.20), and with th echange in the mode of respiration see text).

Equation 1) represents average absohite growth rate, in contrast to Iniegrowth rate applicable to extremely short intervals of time onlj . If a Hol-stein cow weighs 1000 pounds at age 1000 d ay s from conception, she gainedon he average one pound a d ay ; b ut there was no day w^hen she actua lly gained

Cf. Hayes, F. R., and Armstrong, F. II., Can. J . Res ., 21, 23 1043).


21/91


exactly one pound To cite extreme illustrations at the en d of the first weekafter conception, she gained onlj^ about 0.0001 pound a day; a t 5 months afterbirth 2 pounds a day; at 1000 days, only one-fourth a day T hu s t he conceptof average rate, when applied to growth is an abstraction and when theaverage extends over a considerable per iod of time as in the example cited,it gives no idea of the actual rate at any given age. The shorter the intervalof time for which the average is computed, the more nearly does it approach

G m s

Aige

Fig 16.18b. Nitrogen storage in tiie silk worm, from data by Luciani and Lo MonacoThe nitrogen curve of t he s il k worm embryo is similar to the CO2 e xcretion curve of thechick embryo

the true value; and when reduced to an interval dl, so short that there is notime for the velocity of growth to change, the true growth rate dW/dt isobtained True growth rate is then ijistanlaneous growth rate dW/dt

Sirm\aY\y, relative or when multiplied by 100, percentage growth rate is conventionally repi-esented by the weight gain during a given t ime interva ldivided by the weight of the organism, T i a t the beginning of the time int e r v a l

Wi - WiAverage relative growth rate R

Wi 2


22/91


Here again we have not true or instantaneous growth rate, but the conventional Minot s) growth rate. The conventional and true percentage growth rates

a re n ea rly identical when the weight gain, W2 Wi, is very small in comparison to the weight of the organism. But when the weight gain is relatively

B o h P a n d H a s s e l b a l c h

Mxippay

Atwood and V/eakley

Dayso

IncuDationAge

Fig. 16 19 Carbon dioxide excretion in the chick embryo, from three sources. Thepercentage rates were obtained by subtracting the natural logarithms of the successivevalues, and multiplying the result by 1 The fluctuations between 4 and 14 days areslight and there is no systematic change in t he rate.

large comparedwith the body weight, the conventional growth rate computedfrom equation 2 may be misleadingly exaggerated, because the weightat the


23/91

6 0 6 B I O E N E R G E T I C S A N D G R O W T H

beginning of the time interval, existed a relatively long time ago, notat the t ime of observation. Thus the computed percentage grow th rate of the

pojDulation in this coimtrj would be enormousl.y exaggerated if the popiilationgain from 1 6G to 1940 were rela ted 1 o the population size in 16G0. Thepopulation gain diu ing 1940 must be rola led to the population size during1940, not during J66G. Likewise, the relative growth rate of an animal ala given age must properly be related to the bod} size at the given age, no t thebody size, Wi, of some ear lier age , t\.

Another objection to Minot s equation (2) fo r computing percentage rateis its failure to recognize that the physiologic significance of a physical unitof time, such as a day, changes rapidly with age. The growth rates computed by equation (2) are for continuously decreasing physiological t imeintervals, even though they are for constant astronomic or physical time intervals. The use of instantaneous rates eliminates the discrepancy betweenphysiologic and physical time (Ch. 19).

ID V/ \/

8 1 \V , / \

y i \0 n \ V

j / / f \

\ \E ysO 2 4 6 iO 12 14 i6 V i

Incu t j a t ion AigeFig. 16.20. Percentage morli ili ty in the chick embryo. Circlos represent embryos in

an incubator; crosses, incubation under hens, Hie first peak in the mortality curve corresponds to the peak in concentration of lactic acid as reported by Tomita. The secondpeak in the mortality curve coincides approximately with the pause in the growth ctirves.

A more appropriate dent)minator would be the a\>era


24/91


on the assumption that the growth rate occui-s in a Hnear manner, which inpractice is true only for short intervals.

W2 -In place of finite weight gain,(2 i l

dW/dt, and divide the instantaneous gain by the weight a t the time of thegain, thus:

, we use instantaneous weight gain,

Instantaneous ( true ) rel at ive g rowth ratedW/dt

W(4)

In this case, the denominator, W, represents the weight of the animal at theinstant the rate dW/dt is measured, not the weight, W\, a t some earl ier age.

CcpepsogP9p24npa

200

n m

d t

Cnmsptpepo pwd

.5

pci>3}hFS4 0 0

28 -egj^ t O]200 6

160 o o a

teysO 2 4 6 6 12 W 16 Ifi 20 D ^O 2 4 6 3 10 J2 14 16 16 20Incubation Ape Incubation Age

Fig. 16.21. The coursc of COj excretion in the chick embryo plot ted from data byHasselbalch and by Murray.

I t is, of course, imiJossible to measure the inslantaneous I ate of growth inthe laboratoiy, because of the finite time interval required for making a measurement; but oven if it were possible, tlic experimental erroi^s of measurementwould be greater than the instantaneous gains. Here abst rac t mathematicsis used to solve a pract ical problem. The infinite number of infinitesimalinstantaneous rates are added up, oi integrated. (The method of integratingis explained in elementary calculus text-books.) The method is outlinedsymbolically in the following equations.

k W

dw , r

L iInJF = \nA + k t

W =

d t

K u

GiTisppew^day.

(6)

(6)

(7)

(8)


25/91

5 08 I O E N E R G E T I S A N D G R O W T H

In the above equations dW/di represents instantaneous absolute growthdW/di

ra te ;W

and k represent instantaneous relative, or, when multiplied by 100,

percentage growth rate; e is the base of natural logarithms, \nW is the naturallogarithm of weight W at time t, and A is the natural logarithm of W when t =0 .

. For purposes of computing the numerical value of the instantaneous relativegrowth rate, k in equation 8) is written in the forni

k =InlFj InWi

ti ti 8a)

That is, the instantaneous relat ive growth rate, k, is the difference betweenthe natural logarithms of weights and W\, divided by th e t ime interva l

i

V.^ks 0K) 12

Fig. 16.22. Pos tna ta l growth of the domesti c fowl du ring the self-accelerating phase ofgrowth. During the first month , t he growthrate is 5 p er c en t per day body doubl ed in13.8 days . D uring th e following 7 weeks,t he g ro wt h rate is 3 per cent per day bodyis doubled in 23 days .

h. Thus the practically impossible task of measuring the instantaneousgrowth rate is made possible by a mathematical device.

TheVonstant k has a perfectly definite meaning. It is the instantaneousrelative rate of growth for a given unit of time. Thus, for the growth of thefetus of the albino rat, from 14 days to birth, the value of k is 0.53; this meansthat the instantaneous percentage rate of growth is about 53 per cent per day

or 53 X 30 = 159 per cent per month; or 53 X 7 = 371 per cent per week,etc. As regards the constant A in equation 8), theoretically it has the valueof W when t = 0. For the rat, A was found to be G5 gram. This doesnot imply that at conception the fertilized egg weighs G5 gram, for theconstant was obtained on the bas is of data from 14 to 22 days only, and oneis not justified in extrapolating this value to conception. While, therefore,A has a definite theoretical meaning, practicall} no significance should be


26/91

GROWTH R AT E S 5 9

attached to it. Until data become available which indicate the justificationof extrapolating the curve to conception it need be considered merely as aparameter of the equation

a

C C

I I ID o fS

Mpms Gma^ .2 2.0

.1 1oa .S

5 5

^ 03 .3 2 .2

;f 1 ^.008 06

D 5 5

MomaGms.

2 2

1 15 a

DaysO 2 4 6 6 10 12 14 16 iO 20 22 24Incubation AgeNitpopcn Solids

G m s

.1 1.0 6 .6

5 5

8 02 .2

.01 .1DOS 8

5 9

3 3 2 2

1 1Days 0 2 4 6 6 10 12 14 16 16 20

incubation A9e

Fig. 16 23a Prenatal weight growth of the chick embryo and of its constituents andmetabolites plotted fromseveral sources on arithlog paper with indicated slopes.

p >a^

ue A shICalaGms

i 2 2

8 6

5 J05

3 32 2

i D i


27/91

5 1 0 BIOENERQETICS AND GROWTH

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

puted by equation 2), whichis the method proposedby Minot^ ,used almost universallyby biologists until the appearance of our 1927 paper-^.We may take for illustration the data on fetal growth of the albinorat , ages 13 to 22

days. According to Minot, the computation for evaluating relative rates of growth isc a r r i e d on a s fo l lows:

Days 0

rig. 16.23b. Prenatal weight growth in man,plotted from Streeter on arithlog paper.During the period 5 to 100 days prenatal life, growth occurs at 8 per centperday bodyweight is doubled once in 8.7 days); between 160 and 230 days, at 1.7 per cent per day body weight doubled in 4 days); between 240 and 280 days at 1.3 per cent pertiay bodyweight doubled in 55 days). Eight per xjent per day is equivalent to 240 per cent permonth; 1.7 percentperdayis equivalent to 5 percentper month; 1.25 percentperdayis equivalent to 37.5 per cent per month.

Take the weight at a givenage,and the weightat the next older age for which thereare observations. From these data calculate the average daily increase in weight forthe period between the two determinations of weight, then express the daily increasein percentage of the weight at the beginning ofthe 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.ni-wicklungsmech. Organ., 109, 455 1927); 110, 33 1927); 124,82 1931).

2 Stotsenburg, J. M., Anal. Rec., 9, 667 (1915).


28/91


We have showa^ that our inslanlaneuus growth rate, k in equation (8), is related toMinot s/iju/c growth rate, R in equation (2), by tlic logaritlimic function

k = hi R + I)H8 illustratedbyFig. 16.9. The difference isparticularly strikingin growtli rate beyondabout 10 per cent, as illustrated by the following numerical oxamjjles.

At age 13 days, i = .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 = 1275010 X 0

per cent per day; according to our method, the true or instantaneous percentage rate,

. , In 4.630 - In .040A: eq.8a),i8only ^ X 1 = 52 per cent perday; dependingon the method

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

Kos .*70

6

5

4

L b a

i 4

1 30 2

li O

10 0

9

eo^

-

.

,

-

-

B aN e

s i n l

3P>

D P ic e M a

)dD

n n t

a

>ch

in )

-

jcO S

|30

20

VSYps .

A g e

7

so^

5

O e

Fig. 16.24. Juvenile growth in man. Well-nourished children grow at an approximately constant percentage rate between 5 and 15years (about 10per cent per year; thebody weight is doubled once in about 7 years). The prepubertal acceleration, so conspicuous in the literature on growth of children, is usually found only in the curves ofpoorly nourished children (Figs. 16.50 and 16.52).

Th e difference in percentage rates as determined by th e two methods decrease withdecreasing time intervals between weighings. Thus, by reducing the intervals between

weighings to 5 days, we have: The fetal weight at 13days, .040 gram; at 18days, 1.000r, 1000 - O IO In 1 00 In 04grams; hence R = X 100 = 480 per cent; an d k X 100 =

040 X 5 564.4 per cent per day.

Similarly, for a two-day interval (between13and 15days), Minot s arithmetic methodyields 90per cent, whereas our exponential method yields 51.5per cent per day.

Univ. Mo. Agr. Exp. Sta . Res. Bull. 97, pp. 18-19, 1927.


29/91

5 2 B I O E N E R G E T I C S AN D GROWTH

If a set of data followsan exponential course eqs. 7, 8, 9 , then, knowing the numerical value of k, it is possible to compute the time required for doubling body weight orpopulation size. The time required for a growing body or population to doubleitself insize is the ratio of the natural logarithm of 2, that is, 0.693 to the value of k. Thereason for this is given in the following derivation.

B=.650

,

M m 0

Fig. 6 25 Graphic method for evaluating the constants of equations 12 and 14for a rat The correct weight value of the mature weight. A, is 350gms.; if 350IK isplotted 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 thecurve at the point when f = 0; i* = 1.76, the age when {A W A = 350 gm.

Solve equation 7

fo r I,

InW^ = In A + kt

\ u - In At =

k

7


30/91

G R O W T H R A T E S

When th e original weight A is doubled W becomes 2A and

_ In2A InA _ 2A In2

k k ^ A k

513

Fig 16.26 Graphic method f or e va luating the constants for a Jersey cow.A s t ra igh t line results if A is assumedto be 420 kg but no t 450 or 300 kg; fis the age when A W = A whichis 420 kg.

KgS6

5

4

3

2

k i

.

Vi r I

\ V \\ N

1

y

\ \? \a

A

K.

s e Co

B j

P1

Mbso e o 4 6

iOO

^70 I60 / 4 1 r Ofc 3 ^ 9 1 ) 67 3B5 4 I

^2 1 rA 1 1 i:

i J-

f hAge

Fig. 16.31. Relative approach to th e mature weight of rats on normal d ie ts (numbers3, 4, 5, and 6) and maximum-growth diets (numbers 1 and 2). Curve 1 represents rat3414 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 onimproved diets described in J . Biol. Chem., 69, 668, (1926). Curve 3 represents Osborneand Mendel s 1925 averages. The poin ts connected by broken curves in Fig. 16.31represent 98 pe r cent of mature weight.

Th e instantaneous percentage growth rate duiing the 10 days followingbirth is only about 12 pe r cent; but it is constant, which is the essential newfact. The constancy of percentage growth rate during th e first 10 postnataldays is also indicated in Fig. IG.l 1, in which the values of lOO^ for 3 sets ofrats, are pl ot ted against age; th{> resulting curves are horizontal, that is, thepercentage rate of growth is practically constant. The break in the curveat birth is, among other factoi-s, associated with a radical change in the modeof l i fe .


36/91

G R O W T H R T E S 519

Fig. 16.12 the data during the first 10 days following birth are plottedagain, together with the remaining data for the phase of growth preceding

the inflection. From conccption age 32 days 10 daj s af ter b i rth) to 52 days,the instantaneou.s growth rate appears to be 4 per cen t per day; from 52 daysup 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. Theconclusion is, then, that while the percentage growth rate decHnes with age,the decline is much slower than has ever been thought before, and the declinedoes not appear to be continuous. The percentage growth rate remainsrelatively constant between rather wide limits, and then declines relativelyabruptly to a new low level.

a n

6) a i Id

b i ioCa

Fig. 16.32. See legend fo r Fig. 16.31.

Figuratively .speaking, the medium in which the body ceils grow has bufferproperties 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). I t is only after a certain fraction of the bufferis spent that the acidity exceeds a certain threshold, or critical value, andaffects the welfare of the oiganism. M ay not an analogous situation existwith respect to the growth-retarding substances in the body?

Weight increments increase a t the same percentage rate, lOOfc as the body weighti t s e l f


37/91

5 2 0 B I O E N K R G E T I C S A N D G R O W T H

Indeed, this a] pears to be the situation for groAVth of a population of lactio-acid 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 th emilk 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 percentagegrowth rate of acid-producing bacteria due to the high buffer value of th emedium which neutralizes the growth-retarding lactic acid as it is produced,thus keeping the culture in the same state for a relatively long period. Whenthe threshold pH is exceeded, the percentage growth rate declines.

A similar situation prevails inhumanpopulation growth. When the density

of the population is very low during its early histoiy and much more fertileland is available than the population can utilize, the natural increase of thehuman population occurs at a constant percentage rate. This is illustratedby Figs. 16.16 and 16.56 for the growth of the American Colonies and UnitedStates population. This population gr v at the instantaneous rate of 2.9 percent per year from 1660 to 1890. The critical or . threshold value wa sreachedin 1880 when the growth of the population began to decline Ch. 25 .

Kg

6

50

40

20Yra B

1 -. -

-

-

-

w-

/-

0 No i l TIK \ : La O P -

d

W 12 U iO 18 20

Fig. 16.33. Comparison of growth ofE ng lis h ch ild re n laboring and non-laboring. Data by Roberts compiled byB. T. Baldwin, Univ. Iowa Studies in ChildWcljare 1 1 1921 .

Fig. 16.17 represents the percentage growth rate 100^ of the rat as a function of age; it indicates t he manner of decline in t he g rowth poten ti al with increasing age. The graph reminds one of a series of water pipes, each of whichis i-elatively horizontal, has a relat ively constant head piessure, and is belowits predece.ssor,finally fading to zero. This is the e.ssentialhistory ofa runningstream to its ultimate end and of growth to its ultimate end.

The most striking age curve obtained in this analysis of ear ly g rowth isshown in Fig. 16.18a, which relates CO2 production w ith ag e in t li e chi ckembryo. The age increase in CO2 production is perhaps a better index ofgrowth than tho age increase of weight, since Aveight increase may be due toincrease in relatively inert, or even non-living, matter whereas CO2 productionrepresents definitely living, metabolizing tissue.


38/91

GROWTH R AT E S 5 2 1

We prepared Fig. 16.18a from data by Atwood and Weakley ^ Each eggwas incubated individually in a glass tube. The data points represent the

average daily CO output of the 63 eggs which hatched normal chicks. Thecircles represent observed values. The relatively high CO output duringthe first day in comparison to the second is apparent only because the eggswere kept, as is customary, in a cool cellar before incubation. When, therefore, the eggs were subjected to the relatively high incubator temperature,there was in addition to the metabolic COj output an expulsion of the excess

Fig 16.34. The data inFig 16.33 in terms of percentage of mature weightincluding data o n g ro wt h inheight

f

He ifih

/,/ //

/ 0 M 1 bcs in(/ -5

w ijh

CO2 because of its lower solubility at the incubator temperature. This explanation is substantiated by the data for the CO excretion of a control,infertile, egg represented in Fig. 10.18aby crosses. The CO production duringthe first day is virtually the same foi the control and incubating egg. TheCO production associated with incubation is properly represented by thedifference between the fertile and infertile control.

The distribution of this difference between the CO2 production of fertileand infertile eggs is fairly linear on the arithlog grid: the instantaneous increase in CO production during thefirst Jour days of inciihation is seen to beof 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.


39/91

5 22 l i l O E N E H G E T I C S A N D G R O W T H

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 productionduring the second day. Ilere the percentage increases, 100 (lnir2 lnT7i),were plotted against age. The greater drop in ]3ohr and Hasselbulch'.s curve isprobably due to a lower pi-eincul)ation tempei-ature than in Atwood and Weak-ley s.

P

0

0

y0

^

-

//

/

0

0

- / //

-

//

-

0

/ //

; 0 j

jj?e,

ii?e

Biy

n e t

sL

o n t ^

-

///

Age

Fig. 16.35. Relative approach to maximum milk yield under Advanced-Registry testand ordinary conditions of management. The bet ter- fed test animals approach th emature level at a more rapid rate .

Returning to Fig. 16.18a, from 4 to 15 day s th e data points are distributedin a remarkably imiform manner around a straight line, indicating an instantaneous increase of 31 per cent p er d ay. The second remarkable featureof this graph is the pause between 16 and 19 days. The chick no doubt passesa critical period, a metamorphosis , at this s ta^ . This statement relatingto a critical period is substantiated by the mortality curve, Fig. 16.20, whichpasses a peak at this time. Tlie trigger mechanism in the break may be achange in the mode of respiration: the respiratory function is transferred fromthe chorioallantoic membrane to the lungs; the chick metamorphoses froman aquatic to a terrestrial mode of respiration.


40/91

GROWTH R AT E S 5 23

The smaller peak in th e mortality curve at about five days may perhaps becorrelated with the peak in the lactic acid curve sho\Mi in Fig. 16.20. The

mechanism of lact ic-acid oxidation apparently does not begin to functionefficiently un til th is time.Fig. 10.21 represents data of Hasselbalch and of Murray. The distx ibution

of the data points Is less regular becau.se of th e smaller number of embryos.

D 5

fn n

-

ce

-

-

//

//

x

0

s

dO

7 S

TO

6 3

-

J 70 0

0

- hi 1 o o>14-1 1913? 2 t

Vf

AgeFig. 16.36. Relative lactat ion slopes for Jersey cattle .

These charts do not show the l as t s tages of growth; otherwi.se the generalfeatures of th e charts are the same. Th e values of k for the data of Hasselbalch are the same as those fo r the data of Atwood and Weakley. The valueof k for Murray s data is higher but this is probably due to a higher incubationtemperature. Fig. 10.18b which represents the age curve of nitrogen storage

in the silkworm embryo is remarkably s imilar in respect to th e growth pauseto 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 inflection occurs at the age of about 12 weeks. The values of h during this periodare of th e sa me o rde r as those found for the ra t .


41/91

5 2 4 B I O E N E R G E T I C S A N D G R O W T H

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 centper day Fig. 10.10 , that of man^^ ranges from a maximum of 8 per cent to aminimum of 1.3 per cent per day Fig. 10.23b . Therefore, given percentagerates of growth do no t indicate equivalent developmental stages. I t wasalready noted Fig. 16.7 that th e age curve of man is distinguished from thoseof other species by a very long juvenile period. Fig. 16.24 shows that growthduring the juvenile period is about 10 per cent per year, that is, only 0.83 percent 10/12 per month or 0.03 per cent 0.83/30 per day.

iDO

0 5

9 0 //:

3 5

SO

7 5

7

6 51

C l u e ; i n s e j

. e i n s e y5uU r o n ld i e t 2Sti tpfl11

Fig. 16.38. Uolative lactation curves of Hols tcin ca tt l e.

indefinitely because of th e restricting limitation of the universe. Each livesunder a condition not onlj - of pressure f rom within tending toward expansionof self or kind to multiply and fill every occupiable niche, but also* under apressure from without which keeps exj)ansive pressure under control.

Mention was made of the restrictive effect of the population increase ofDrosophila yeast bacteria on further increase of the population. Analogousrestrictive effects m a y c on tr ib ut e to decline in growth rate of multicellularanimals in later growth stages. For instance, the decrease of the ratio of

Coker R. E. Scientific MonUi ly, 48, 61, 121 1939).


43/91

5 2 6 B I O E N E R G E T I C S A N D GHOWTH

surfacc to body weight during growth^ results in progressive congestion in thetransportation of nutrients and wa.stcs. One may also mention progressive

interference with th e transportation of nutrients and wastes through progressive accumulation of inert materials in the body and progressive dehydrationand decline in permeability.

The mass law equation for tlie self-inhibiting phase of growth may be fornm-lated just as for th e self-accelerating phase, with this difference: during theself-accelerating phase, th e growth rate, ilW/dt, is proportional to th e size ofthe population or to that of the multicelhilar individual; during the self-in-hibiting phase it is proportional to available living space , available land,

w

.

* 4 J _ I ^

a

> 70

V 7^

~ - 1 1 ~- ~

j

t t -j K

7*-

J

u f/ t

-

Zi y

T

2fir -

N f1

J-

f *

fxl / _D 1^ 5 S 5?a

U i

uoo

m

900

00 0

700

600

500

40 0

300

200

100

D

l i o l s t c i n

Guernsey K MAyrshireJersey RMJersey

Age

. I = 55Ui h = . 04U; I* = 8.3A = 515; k .044; t* = 0. 2A = 460; k = .050; t = 9.1A 436; k = . 0 5 0 ; t* = 0.2A = 420; k = . 054 ; I* = 8. 9

Fig. 16.39. Growt .l i of dairy cuttle.

available food supply available freedom from tiie deleterious products ofgrowth and so on. These ideas may be foi-mulated in the conventional termsof 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).


44/91

G R O W T H R T E S 527

Let A represent the limiting food i-etjiiired to attain maximum individualor 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 justsufficient 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 limitingfood supply, that is, to tho value (/I 11 )

d t= k i - W) (9)

K S DIP:

I ig. 1 5.40, ii owlii of Horsfis .

Instead of food, th o gi-owth-Hmiting fact or in tho environment may be somegi owth product, as lactic acid to growing hurtic-acid bacteria in milk, or alcoholto alcohol-producing yeast in fruit juice. Let A be the concentration of lacticacid 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 t he amount of growth which the environment will permit in order

d Wto bring the population to the maximum size, A, and = (A W),

a t

the instantaneous velocity of growth at the given t ime. Analogous reasoningapplies to the growth of multicollular animals, which are cell popula tions,a f t e r a l l .


45/91


The numerical values of the constants ar e estimated as follows: A, whichrepresents the concentration of -the growth-limiting factor when growth is

completely inhibited, may be used to repicsent the mature weight of theanimal (or the maximum size of the population under a given set of co nditions; W may be used to represent the weight of the animal (or size of thepopulation) at the given time; (A IF) then represents the amount of growthyet to be made to reach th e m atu re weight.

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

7

64

5 6

4 9

3224

16

nf 2 - 175

15 0

125

. 1

5

100

c r

/ 7 ^ E 2 4]J

5 .

7 5

5

2 5

0

a Zt z -

Z. ~ i n

le n a v

_a lS C .1

-

i -

M o s

c

i Q Age

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 respectto the growth a lr eady made

k =dW/dt

W (5)

while in equation (9) k is the relative growth rate with respect to the growthyet to be made

dW/dt k

A W(9)


46/91

G R O W T H R AT E S 5 29

Before applj ing equation (9) to data, it is integrated, for reasons explainedin the preceding section:

dW/dl - f e U - If)

d W

In (A = kt + InS integrat ion constant)

A - i r = Be-^

W A -

(9)

(10)

(11)

(12)

Thesignificance of

theconstanc.v of

th e exponentk is

that the growthvelocity declines at a constant percentage rate 100 c illustrated by the following

numerical example.

K0

200

17 5

150

Lb.

4 50

40 0

35 0

30 0

250

20 0

BO

100

50

0

fos.

\V ^ A - Ac-^^ = A 1 -

r 1

1

i

Si.

~z -y

z .j

t

t r 3h

r

= :

7

f 2

E

r~ A7

A2 ' 1 P e o i 3 4 r i _5

T r E 5 fi 2T 2> W m X B dsnf~ ~ / 1 r I]

m71

z tn 1 1 1 _

Age

12a)

U S.

,8

.7

.6

,5

4

.3

2

.1

0

Fig. 16.48. Growth of pigeons, doves Riddle) an d rats.

Equation 12a) offers a simpler method of determining the weight at a given age. From 12a

A - W

In

^ . - J t t

A

A - IF)= fcl

1 , A - W)

- H - ' ) 17a)

Equation 17a) shows that, when the value of k is known, one can easily determinethe age a t which a given fract ion of the mature weight is reached. Thus if i t is desiredto determine th e age at which half of th e mature weight is reached, W is replaced by 0.5A,a n d

1 , / 0.5A\ 1 , ^ , 69315 i i I T j = = T -


53/91


In a similar manner, the age when any other fractionof the mature weight is reachedmay be found by substituting the desired value in equation 17a). The following table,

inwhichaseriesofnumerical valuesofnaturallogarithms ^~ j given facilitates numerical computations.

Equation 14 may similarly be used for evaluating the age when a desired fractionof the mature weight is reached.

Fig. 16.49. Growth of domestic fowl.

= 1 _ g Ht fA ^

1 - - c Ht fA

14)

18)

t iseasilydetermined when kand I are known. Thus if it isdesired to find the age, t,when half of the mature weight is reached, VKis replaced by 0.5^4:

k k


54/91

\ K 4 a 12 B

a i i O W T H R T S

MMmtMwmm^>?9 4uuum u m m u m m m m m d m m m m

5 37

laiittdiiiniif f rvvipPFin ms mmt i 2

6 12 IS

Fig. 16.50. Growth of children preceding puberty 16.50 and following puberty 16.51 . Note th e considerable differences in mature weights and in th e shape of thea ge c ur ve s in these figures as also in Figs. 16.52 to 16.54, probably due to differences inenv i ronmenta l cond i t ions .


55/91

5 3 8

tsza i

B I O E N E R G E T I C S N D G R O W T H

j55;v;se?5^

^ t

- J-

r pt >-

J -Clas 7 2

RlHi

^ -

7 In

= gf

-

-J l

cAtoJ

iU14 }

m

100

20 24 12

AgeFig. 16.51. See legend for Fig. 10.50.

20 21 12 20 24


56/91


T a b l e t o F a c i l i t a t e E s t i m a t i n g A g e s W h i c h D i f f e r e n t F r a c t i o n s o f t h e

Fraction of M ature Weig h t

. f)0 2 5

3 0

.3 5 4 0

4 5

5 0

5 5

6 0 6 5

M a t u r e W e i g h t A r e R e a c h e dFraction of Mature Weight

10 2 8 7 6 8

0 3 5 6 6 7 0 4 3 0 7 8 0 5 1 0 8 3 0 5 9 7 8 4 0 6 9 3 1 5 0 7 9 8 5 1 0 9 1 6 2 9 1 0 4 9 8

7 0

7 5 8 0

8 5 9 0

9 5

.9 8

.9 9

.999

Chart for b oy s

^ o 0 0 6

52

1 2 0 4 0 1 3 8 6 3 1 6 0 9 4 1 8 9 7 1 2 3 0 2 6 2 9 9 5 7 3 9 1 2 0 4 6 0 5 2 6 9 0 7 8

Char t f or g ir lsFig. 16.52. The iuven ile course of growth of children same data as in Fig. 16.50

plotted on an arithlog grid. I t appears t ha t the pubertal acceleration representscompensating growth in ch ild ren who were relatively under nourished d u ri ng t he earlieryears, because Fig. 16.52 indicates tha t the lower the value of k percentage growth rate)the greater the pubertal acceleration.


57/91


The followiiig transformation of equation (14) gives a straight-line function for th egrowth curve . Transforming

W

I r

O

i o

lo

In

^ A

( - z) =Age


58/91

G R O W T H R AT E S

W = A - Be-^

d W

d t= = Ce-**

5 4 1

(12)

, dWIn r- = In A:S kt.

d t

d WHence, plotting the velocities or in practice the successive increments per unit

time against the corresponding ages on arithlog paper will result in a straight linehaving the slope k. Ihis method is inferior to plotting A W agiiinst age on accountof fluctuations due to experimental errors.

The fact that the numerical value of the velocity constant, k, is the same in thedifferential and integral equations makes it possible to derive equation 12 from aknowledge of growth velocities {i.e., gains in weight per unit time at successive ages .

W = A -

d t

. - k l= kBe- ^ Ce

(12)

0 g i s g o 9 31 I I I I I r I I I 1 1 1 1 *f~

A g e ln Vfear>s

U


59/91

54 2 BIOENERGETICS AND GROWTH

Integrating t he l as t equation,

W =

Ce-

k + A integration constant .

T h e r e f o r e

When t 16 17 16 10

12

Fig. 16.55a. Growth of tlic luiiniin brain in weight, IK atid in frequency, /, of brainpotentials 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 brainweight i.e., 48.5 per cent per year ; l.lie mature values, Wa and f,i are 1333gm. for the

. W f Wabrain and 10for the alpha waves. Since rfj- ~ -i- = W = f-p; tiiat is, the brain weight,

A J.A J aW, is 133.3times the alpha wave frequency, / Weinbach .


60/91


Summarizing this discussion, prior to puberty in animals, flowering inplants,and coming ofage in populations, the growth rate tends to be pro

portional to the growth already made, indicated by the equation

flOQjSJ| 9

4

d Wk W

d t

d W

d t H A - W)

W

Alpha Frequencij AqeBir th

O 1 2 3

O

K . A - A )

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

[The numericalvalue ofkin equation 9) or 12) is,of course, different andhasan 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 ofkfor given guinea pigs is 0.22; forcattle, which approach matureweight much more slowly, the value of is much lower, 0.04; for mice, approaching the mature weight much more rapidly, k is much higher, 0.71;a n d so o n .

OBirh

Brain Welqhh Aqe

5)

8)

Following this age growth rate tends to be proportional to the growth yet tob e m a d e

9)

12)


61/91

5 4 4 B IO E N ER G ET IC S A N D G R O W T H

16.6: Genetic growth constants. The value of A mature weight) and ofthe slope, k,onthegrowth curve arc intrinsic or genetic characteristics of the

animal under given environmental conditions, in the same sense that theequilibrium and velocity constants ofa chemi(^al reaction in vitro are intrinsiccharacteristics of the chemical system under given conditions. Growth, likea chemical reaction in vitro, is by definition increase in the mass of one component at the expense of another, and the rates of approach to the equilibrium level are analogous. Table 16.1 appendix presents numerical valuesof 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 curvesof growth in w eight of several animal species.

The charts are for the most part self-explanatol}^ The values of A matureweight), k (speed of approach to mature weight during the self-inhibiting

2cx>

tc o

1

o i

o s e i i v c o

X PCARL RIIO S CaUATIOH

/ o U . S . A . P O P U I A T I O Ml6CO

Y E A R

1 9 C 0 2 C 0 0t A o o 17CO

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

(ageat

whichthe extrapolated

curve meetsthe

ageaxis) are given for each species. The differences within the species are duemostly to difTerences in environmental conditions, especially food supply.

Before proceeding with the discussion of the influence of environment on the numerical values of the constant , it is interes ting to note that the mature size. A, of differentspecies tends to be related to the growth constant, k, or to the time requiredto reach a


62/91

G R O W T H R AT E S 5 4 5

given percentage ofthe mature weight (Fig. 16.29 plotted from Table 16.1). As mightbe expected from the long juvenile period, the data points for man deviate from thegeneral curve. Also, as might be cxpected, there is considerable scatter of the data

becauseA and k are not influenced by environmental conditions in the same directionand the environmental conditions for the different animals were not the same. Moreover, 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 weightin Wistar Norway and albino rats^ (Fig. 16.30). The mature weight isvirtually the same in both, but the albinos attained mature weight muchearlier. Dr. King informed us that the Norway curves approach the albinos

2 0 0

= 1 5 0

5 0

(1R

X /

i

1

11

VA

/ \y t11

1x /

/

1 I f /' /

/ \9 *

\ 2\\\\\\

/r J

/ ?

* --O

\

U S P O P U L

\\

\\

\

A T I O H

Y E A R

2 0 r ;

lO

5 2

Per-c e n t a g e

o f

monthlydecline

t*

5urw i ht8

ge* f ro

ception

a

s-g,s-ss

98urw i ht

kg. i lb . kg. mos . mos. mos. mos.

1100 2425 1 6 0 0 3 4 7 1 7 31 51 1 2 4

55 0 1215 8 0 5 4 6 8 3 23 39 9 346 0 1014 725 5 9 1 23 3 7 8742 0 92 6 680 5 4 8 9 2 2 3 5 81

43 6 961 4 41 5 2 1 4 2 8 78

6 7 8 1493 1677 8 2 1 1 1 9 5 2 8 5 8 7636 1400 1501 7 8 11 .0 19 .9 2 8 8 61 .2

20 0 441 260 6 2 4 4 1 5 2 6 6 7 . 0

9 0 198 170 1 3 6 4 7 9 8 1 5 3 3 2

8 0 176 200 1 8 5 5 8 7 1 2 5 2 6

5 0 110 120 1 8 8 4 7 8 3 1 2 2 2 5 3

3 9 6 5 4 6 1 5 3 1 1 5 6 1 1 2 6 6

3 6 5 3 9 2 5 3 1 3 9 6 6 1 6 6

3 2 7 1 8 4 3 4 2 7 3 7 5 3 1 1 1

2 8 6 2 9 2 4 4 7 2 2 4 2 5 8 1 1 4

2 5 5 5 1 1 4 9 2 3 4 4 5 8 1 1

2 4 4 7 4 7 4 2 6 4 1 5 6 1 9

1 4 3 1 3 3 4 3 4 2 3 6 5 2 1 1

3 5 6 7 8 5 5 2 1 6 6 2 2 8 6 5 1 6 2 5 9

3 4 7 5 5 4 3 1 9 5 2 4 6 9 5 2 2 5

3 6 . 61 4 6 5 2 6 2 1 5 5 8 9 2 1 12 2 4 8 5 3 2 5 1 8 2 1 7 6 9 9 2 3 9

1 6 5 3 6 4 4 3 4 4 2 5 7 4 6 6 6 1 5


85/91

5 68 B IO E N ER G E TI CS A N D G R O W T H

Animal

Domestic fowlCont. J ull, U . S. D . A.)Rhode Island Reds,

m a l e s

Rhode Is land Reds,m a l e s

Rhode Island R ed s, c ast r a t e d m a l e s

Rhode Island Reds,f ema les

M ay R . I . StationCornish, males. . .Cornish, femalesHamburg, malesHamburg, females M ay Waters, R. I.

Stat ionBrahma, males

B r a h m a , femalesL e gh o rn , m a le sLeghorn, femalesL X B m a l e sB X L m a l e sL X B f e m a l e sB X L females

Guinea -pig CastleFi, Arequipa x Race

B, mF2, Arequipa x Race

B, mFi, Cav ia cu tl e ri x Race

B m a l e sRace B, malesRace B, femalesFi, C. cutleri XRace B,

f e m a l e sFa, C. cutleri x Race B,

m a l e s

F2, cutleri x Race B,f ema les

C., cutleri, malesC ., cu tl er i, females

Norway rat Kingf e m a l e sm a l e s

A l b i n o r a tSpecially well fe d

cared fo r males Greenman Duhring

Inbred, 7-15 generations er ie s K in g , males . .f ema les

Ta b l e 16.1 {Continued

A Mature Wt. B

lOOiii P er

c e n t a g eo f

monthlydecline

f

50turw i htg7tur w i ys0 98turw i h

i. m kg.) mos. mos.) mos. mos.)

4 . 6 1 0 . 1 6 . 7 21 1 . 8 5 . 1 8 . 4 2 0 . 4

4 . 2 9 . 3 5 . 6 2 4 1 . 2 4 . 1 7 . 0 1 7 . 5

4 . 2 9 . 3 5 . 6 2 4 1 . 2 4 . 1 7 . 0 1 7 . 5

3 . 6 7 . 9 5 . 3 17 2 . 3 6 . 4 1 0 . 5 2 3 . 2

2 . 8 6 . 2 4 . 6 2 4 2 . 1 5 . 0 7 . 9 1 8 . 42 . 1 4 . 6 3 . 2 22 1 . 9 5 . 1 8 . 2 1 9 . 71 . 7 3 . 7 2 . 3 21 1 . 5 4 . 8 8 . 1 2 0 . 11 . 5 3 . 3 1 . 9 17 1 . 4 5 . 5 9 . 6 2 4 . 4

4 . 1 9 . 0 8 . 1 2 5 . 1 2 . 7 5 . 5 8 . 2 1 8 . 3

3 . 4 7 . 5 7 . 0 2 4 . 8 2 . 9 5 . 7 8 . 5 1 8 . 72 . 2 4 . 9 4 . 9 3 3 . 4 2 . 4 4 . 5 6 . 6 1 4 . 11 . 8 3 . 9 4 . 1 3 5 . 0 2 . 3 4 . 3 6 . 3 1 3 . 53 . 3 7 . 2 1 3 . 7 4 5 . 0 3 . 2 4 . 7 6 . 3 1 1 . 93 . 1 6 . 9 8 . 3 3 5 . 6 2 . 8 4 . 7 6 . 7 1 3 . 82 . 3 5 . 0 1 3 . 6 5 7 . 5 3 . 1 4 . 3 5 . 5 9 . 92 . 5 5 . 4 6 . 4 3 5 . 5 2 . 6 4 . 6 6 . 5 13 .6

1 . 3 0 2 . 8 7 1 . 9 5 1 9 . 8 2 . 0 5 5 . 6 9 .1 2 1 . 8

1 . 0 4 2 . 2 9 1 . 4 6 20.9 1 . 6 2 4 . 9 8 . 3 2 0 . 5

. 9 3 0 2 . 0 5 1 . 6 0 27 .4 . 2 . 0 0 4 . 5 7 . 0 1 6 . 3

. 8 7 0 1 . 9 2 1 . 4 5 2 1 . 7 2 .35 5 . 5 5 8 . 7 6 2 0 . 4

. 8 0 0 1 . 7 6 1 . 5 9 2 4 . 7 2 . 6 5 . 7 8 . 4 1 8 . 5

. 7 8 5 1 . 7 3 1 . 2 6 2 6 . 1 1 . 8 4 . 5 7 . 1 1 7 . 0

. 7 2 5 1 . 6 0 1 1 0 2 5 . 1 1 . 6 6 4 . 4 7 . 2 1 7 . 5

. 5 9 0 1 . 3 0 1 . 4 5 4 0 . 5 2 . 2 2 3 . 9 5 . 6 1 1 . 9

. 4 0 0 . 882 1 . 7 5 2 . 9 2 . 7 4 4 . 0 5 . 4 1 0 . 1

. 3 3 3 . 734 . 7 8 0 4 3 . 4 1 . 9 6 5 . 9 5 6 . 9 5 1 1 . 0

. 2 9 0 . 638 . 415 1 2 . 9 . 8 9 6 . 3 1 1 . 7 3 1 . 2

. 3 8 5 . 849 . 4 3 1 2 . 2 . 9 0 6 6 . 6 1 2 . 2 3 2 . 4

. 3 5 0 .772 .6 5 3 5 . 0 1 . 7 7 3 . 7 5 5 . 7 3 1 2 . 9

. 3 3 5 . 7 3 8 .5 7 3 8 . 3 1 . 4 3 . 2 1 5 . 0 0 1 1 . 5

. 2 1 5 . 4 7 4 . 4 2 4 9 . 0 1 . 4 2 . 6 4 . 2 9 . 3


86/91

A n i m a l

O n whole m i l k a n d wholew heat diet Sherman MacLeod).

m a l e sf ema les

inbred, 16-26 generationser ie s K ing) .M a l e sF e m a l e sStock rats Donaldson

et al.)M a l e s S t o c k r a t s NFemales \ Ferry) / C o n t r o l r a t s fo r i n b r e e d

ing experiments King)M a l e s .F e m a l e sStock rats King)

F e m a l e sStock rats Hoskins)M a l e s

F e m a l e s Run t (Series 1, No. 1)

King)F e m a l eN o r m a l l i t t e r m a t e t o

runt (Series 1, No. 3)F e m a l e

W h i t e m o u s e Robertson)MalesF e m a l e s

A l b i n o M o u s e Robertson Ray, 1925)M a l e sF e m a l e s

Pigeon Riddle Frey)Common pigeon, male

f e m a l eRing dove , male female

G R O W T H R A T E S

Tab le 16.1 (Continued)

A Mature Wt.)

(^g-)

. 3 3 0

. 2 2 3

.320

.217

. 280

. 2 0 3

.270

. 1 72

.255

.189

2 1 0

. 2 3 0

. 1 6 6

.147

. 1 7 0

.0276

.0236

, 0 2 6. 0228

. 3 4 0

. 1 6 0

(lb.)

.728

.492

.706

.478

.617

. 4 4 7

. 6 9 6

.379

. 5 62

. 4 1 7

.463

.607

.366

. 3 2 4

. 3 7 6

.061

. 0 6 2

. 0 5 7

. 0 6 0

.760

.363

ig-)

.8 0

.8 8

lOOAi Per

c e n t a g eo f

monthlydecline)

6 2 0

7 0 . 7

. 4 7 0. 350

29 .13 8 3

5 9

.7 5

. 4 5 0

. 2 8 0

4 0 0

6 4 4

3 4 1

4 1 8

4 0 0 3 3 0

.280 3 5 7

.3 9

8 6

6 0

. 3 3

4 8 6

8 1 0

8 8 6

6 2 1

4 6 0 7 0 7

044 6 2 0

. 060 8 2 0

053 6 7 2

. 0 4 1

1 4 0

. 0 0 6

6 3 . 9

8 0102^

mos.)

1 7

1 9 4

1 3 3

1 .25

1 8 6

2 0 3

1 .601 1 7

1 3 6

1 1 0

1 2 8

1 6 3

1 6

1 3

1 4

.76 9 2

1 0 3

.8 8

. 139

. 0 6

5 6 9

Age (fromconception) a t

3 ^

C. M

mos.)

3 0 3

2 9 2

3 7 0

3 0 5

3 6 9

3 1 1

3 5 5

2 8

3 4 6

3 0

2 7

2 5

2 2

2 4

2 4

1 8 8

1 7 6

2 1

2 0

2 6

7 4

2

(mos.)

4 3 7

3 9 0

6 .084 .87

6 3 3

4 1 8

6 5 6

4 5

6 .676 0

4 1

3 3

3 0

3 6

3 4

3 . 0 12 6 0

3 1

3 1

3 6

1 4

65^

mo).

9 3

7 6

1 4 7

1 1 3

1 1 6

8 1

1 3 0

1 0 5

1 3 1

1 0 9

9 3

6 5

5 8

7 5

6 9

7 1 0

5 7

6 9

7 0

6 7

3 9


87/91

6 70 BIOENERGETICS A N D GROWTH

M a n

M a n

M a l e sEnglish non-laboring

classes (StephensonEnglish all classes

(Roberts)S we di sh ( Ke y)U . S . n a v a l c a d e t s

(Beyer)U. S. Amherst College

N e b s t u d e n t s(Hastings)

English laboringclasses (Stephenson)

Polish R adom Gymnasium (Suligowski)

English ar t i san classes(Roberts)

R u s s i a n S t P e t e r sburg School (Wia-zemsky)

J e w s i n S o u t h R u s s i a .(Weissenberg).

C h i n e s e s t u d e n t s inWuchang School(Mer r ins )

Phil ippine (Bobbit t) . .Japanese (Miwa)

F e m a l e s

English (all classes)(Roberts)

American well to do,New York (Baldwin)

R u s s i a n I n s t i t u t i o n s

(Diek)German (Camere r ) . . , .Japanese (Miwa)

Ta b l e 16.1 {Concluded)

lOOi^i{Per-

ceutageof

monthlydecline)

Age (fromconception) a t

A(Mature W t.) B L

a3 ^

e u

K

u

9

5sS

{kt.) m (*.) {mos.) (y . ) (.yrs.) (yrs.)

6 9 2 1 5 2 6 183808 4 8 1 3 . 6 8 1 2 3 1 6 1 2 5

6 86 8

1 4 9 91 4 9 9

504322851

2 9 8

3 6 2

1 2 3

1 3 4

1 2 4

1 2 4

1 5 9

1 6 62 32 2 4

6 5 . 8 1 4 5 1 2156 3 6 8 13 . 1 1 6 2 2 2

6 5 4 1 4 4 2 7 11 3 3 1 4 1 2 4 4 1 2 7 1 6 1 2 2 . 8

6 5 3 1 4 4 2 0 3 7 2 3 8 1 2 1 1 2 3 1 7 2 5 8

6 2 . 0 1 3 6 . 7 15782 3 7 6 1 2 2 8 1 1 4 1 5 4 2 1

6 1 6 1 3 5 8 13309 3 4 6 1 2 . 9 1 1 3 1 6 . 3 2 2 4

6 1 5 1 3 5 . 6 79447 4 4 5 1 3 4 1 2 3 1 6 2 0 . 7

5 8 1 2 7 . 9 171302 4 7 6 1 4 1 2 . 3 1 6 5 2 9

55 .05 2 .35 1 4

1 2 1 3

1 1 5 3

1 1 3 3

6301153263 9 6 9 4

3 3 1

3 8 3

4 4 6

1 1 9

1 2 4

1 2 4

1 2 6

1 2

11 . 7

1 5 4

1 5 4

1 5

2 1 8

2 9

1 9 , 7

57 .4 1 2 6 6 1331 2 6 1 1 1 1 1 1 4 5 2 2 . 6

5 6 4 1 2 4 4 7944 3 .97 1 4 1 1 1 3 3 1 8 6

5 4

5 1 6

4 6

1 1 9 1

1 1 3 8

1 1 4

13981 2 4 3 0

2751

2 6 6

4 8

3 3 6

1 2

1 1 2

1 2

i6 41 4 5

1 4 1

1 3 6

2 2 5

1 9 2

1 9 9

The percentage of monthly persistency of growth, l P may be obtained by subtracting the percentage decline, 100k, from 100; i.e., 100 P = 100 100 k.

' By age is meant, in all cases, are as counted from conception. The followingages of the animals a t birth are given should the reader desire to convert the concep-tional ages to birth ages: Cattle 9.4 mos. , horse 11 mos., swine 4.0 mos., sheep 5,0 mos.;rabbit 1.0 mos.; fowl 0.7 mos. (21 days); guinea pig 2.2 mos. (67 days); mouse 0.66 raos.(20 days); pigeon 0.6 mos. (18 days); man 0.79 yrs. (9.5 mos.).


88/91

Tbe

2Growtho

DaryC

e

HL

mF

e

Ag

Mo

Weg

H

gw

ah

C

cm

ee

oc

Widh

hp

A Mo

Noo

a

ma

lb

Noo

a

mas

in

Noo

a

ms

in

Noo

am

s

in

B

h

2

9

1

2

1

8

3

8

8

67

B

h

2

1

2

3

6

8

3

9

1

73

2

2

1

2

3

3

8

3

1

84

2

3

2

1

2

3

3

8

3

9

1

92

3

4

2

2

2

3

2

8

4

9

1

1

1

4

5

2

2

2

3

7

8

4

1

1

1

5

6

2

3

2

3

7

8

4

7

1

1

9

6

7

2

4

2

4

1

8

5

1

1

1

7

7

8

2

4

2

4

3

7

5

2

1

1

4

8

9

2

5

1

4

5

7

5

6

1

1

9

1

2

5

1

4

4

7

5

3

1

1

5

1

1

2

5

1

4

3

6

5

2

1

1

1

1

1

2

6

1

4

5

5

9

1

1

5

1

1

1

6

1

4

7

5

5

8

1

1

8

1

1

1

7

1

4

3

5

6

4

1

1

2

1

1

1

7

1

4

9

5

6

6

9

1

5

1

1

1

7

1

4

5

5

6

6

9

1

9

1

1

1

8

1

4

9

4

6

2

9

1

2

1

1

1

8

1

4

3

4

6

9

8

1

6

1

1

1

8

1

4

8

4

6

5

8

1

8

1

2

1

9

1

5

2

4

6

7

8

1

1

2

2

1

9

1

5

6

3

6

9

8

1

3

2

2

1

9

1

5

4

6

8

8

1

7

2

2

1

1

1

5

3

3

7

1

7

1

2

2

2

1

1

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i Oi


89/91

4 5 5 5 6 6 6 6 7 7 7 8 8 8 9 9 9

T

e1

2{C

n

4

1

3

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3

1

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2

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4

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3

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1

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7

3

4

1

3

5

5

1

7

6

3

2

bioenergetics and growth chapter 16

Documents