VOL. 32., NO.
3
September,
I957
THE QUARTERLY REVIEW
of BIOLOGY
QUANTITATIVE LAWS IN METABOLISM AND GROWTH
T
BYLUDWIGVON BERTALANFFY
Los Angeles
Mt. Sinai Hospital,and University
BiologicalResearch,
ofSouthern
California,
INTRODUCTION
_ HE work reviewed in this paper is
aimed at establishingconnectionsbeaspectsofliving
tweentwofundamental
and growth.
their
metabolism
organisms,
What we call growthof even a simple
complexphenomenon
organismis a tremendously
fromthe biochemical,physiological,cytological,
viewpoints.Thereare,however,
and morphological
certainaspects that are amenableto quantitative
analysis,and such an approach appears to lead
to some insight into the connectionsbetween
metabolismand growth,and to some answer to
the seeminglytrivial,but in fact hardlyexplored
question, "Why does an organismgrow at all,
and why, after a certain time, does its growth
come to a stop?"
QUANTITATIVE RELATIONS BETWEEN BODY
SIZE AND METABOLIC RATE
it may be
In orderto begin this investigation,
emphasizedthat,in manyphysiologicalactivities,
the absolutesize of the body is a mostimportant
the rate of processes.Whether
factordetermining
we take total metabolism,heart or respiratory
rate, the chemicalcompositionof the organism,
excretion,or the enzymecontentof the cells-we
vary
always will findthat theycharacteristically
with body size, this being true even thoughthe
the organismscomparedin such respectsshow a
tremendous
diversityin theiranatomy,physiological mechanisms,adaptations to certainenvironments,and so forth(cf. Adolph, 1949). To give
just one example:pulse rate in mammalsclosely
correspondsto the Y powerof body weightover
seven orders of magnitude,from a dwarf bat
weighingsome four grams, to an elephant of
2000 kilograms,in spite of the fact that the
animals under comparison belong to different
orders,and are adapted to all sorts of climate
and ways ofliving(Fig. 1).
The relationbetweenmetabolicrate and body
size belongsto the classical topicsof physiology.
It goes back over more than a hundredyears to
the time when Sarrus and Rameaux, Bergmann
and Leuckart, and Richet noticed that weightspecificmetabolicrate, that is, the intensityof
metabolismas measuredby oxygenconsumption
or calorieproductionper kilogramofbodyweight,
decreases with increasingbody size. A classical
example is provided by Rubner's experiments
size (Table 1). It appears
with dogs of different
that metabolic rate per kilogramdecreases. If,
however,metabolismis calculated per unit of
body surface, approximatelyconstant values
appear. The comparisonof metabolic rates in
mammals led Rubner to the contentionthat
warm-blooded animals produce daily about
1000 Cal. per square meterof body surface.This
is the originof the famous surfacerule, which
was explainedbyRubnerin termsofhomeothermy.
All warm-bloodedanimals heat theirbodies to a
of ca. 37?C. Heat outputtakesplace
temperature
throughthebodysurface.Hence, thesamenumber
of caloriesmust be producedper unit surfacein
orderto maintainthe body temperature
constant.
There are, however, considerable difficulties
217
218
THE QUARTERLY REVIEW OF BIOLOGY
1000
~~~~80O
L
IL
3
=
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OVS~~~~~~~MQMTO-'
P
co
DD
o
-
X
C
ED6Eh'0G
4W"
.39 5Sg log
!50Q ioOs
RABBIT
-
-A
SS
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EUOSELjEP
509
gI
kg SW
I
tk!g
X
,
50k(s l0kg
200O kg
500kq
IOQOV9
BODY WEIGHT
FIG. 1. ALLomETRic DZEPENDENCE OF PULS5E FEREQuENcY ONBODY WEIGHTIN MAMMLS
It
rate,
pulse
basal
may be assumed that the volume of blood transported per minute is proportional to the basal metabolic
as the oxygen consumed must be transported by the blood. This volume is equal to stroke volume (S) X
frequency (F). In a rough first approximation, S may be taken as proportional to body weight (W). The
metabolic rate follows interspecifically, in the series of mammals, the W! rule. Hence:
S.F. = CW.', and
= C'W
=
Thefigmreshows that the allometryconstantof pulse frequency,a =-.28.
Notwithstandingthe gross over
whichneglectsanatomical,physiological,ecologicaloand otherdifferencesabsolute body size is the
simplirication
dominatingfactorin the controlof pulse frequency,in a rangefromthe dwarfbat (4 g. body weight)to the elephant (2000 kg.). ModifiedafterBertalanffy(1951a).
TABLE 1
Metabolismin dogs
AfterRubner (1902).
in kg.
Weight
3.1
6.5
11.0
17.7
19.2
23.7
30.4
Cal.per
production
Cal.m.
production
per
kg.
sq.
bodysurface
85.8
61.2
57.3
45.3
44.6
40.2
34.8
1909
1073
1191
1047
1141
1082
984
in measuring the outer surfaces of animals
exactly,but a simplemathematicaldevice can be
applied. If two bodies are reasonablysimilarin
shape, their surfacescan be expressedas a 23
powerofweight,sincethecubicrootofthevolume
or weightis a lineardimension,and therefore
its
square has the dimensionof a surface.Hence, the
surfaceareas of geometrically
similarbodies can
be obtained by multiplyingthe 23 power of the
weightby a suitableconstant.This is seen in the
well-known
formulaofMeeh:
S =bW
(1)
The surface rule of metabolism accordingly
states that the basal metabolic rate is proportional to the /% powerof the weight.In the case
of man, the determinationof the basal metabolism is a clinicalroutine,in orderto diagnose
thyroiddisordersand the like.Here thesomewhat
more complicated Dubois formula is applied.
Dimensionally,however,the Dubois formulais
identical with the surface rule. The Dubois
formulais: S = kW0425X L0 725.As, presupposing
geometricalsimilarity,
lengthL = cW', this can
be written:S = kWO-420.
cW0 725(033) = bW
The relationbetweenmetabolicrate and body
size can be studiedeitherintraspecifically,
i.e., by
comparinganimalsofthesamespeciesand different
body size, or interspecifically,
i.e., by comparing
adult animals of differentspecies. We are at
present mainly concerned with intraspecific
comparison.
A grave objection can be raised against the
surfacerule as foundin textbooksof physiology.
In consideringthe quantitativerelationbetween
metabolic rate and body size, homeothermic
vertebrates and, in particular, mammals are
almost solely taken into account (e.g., Brody,
1945; Kleiber, 1947; Krebs, 1950). However,the
case ofmammalsis by no meanssimplebut rather
QUANTITATIVE LAWS IN METABOLISM AND GROWTH
is intricate.Moreover,as we shall see presently,
many familiar conclusions and explanatory
hypothesesfall flatif not onlymammalsbut also
vertebratesand invertebratesare
poikilothermic
necessary
It is therefore
takeninto consideration.
to considerthe problemon the broaderbasis of
comparativephysiology.A considerablepart of
thisworkhas been carriedthroughin the author's
laboratories.
In orderto understandtheseresults,one more
mathematicalformulais necessary.The dependence of the metabolicrate of an animal on body
size can be expressedin the equation:
219
TABLE 2
CO2production
pallasii
ofArmadillidium
(Temperature
21?C.)
AfterMUller (1943b).
15 33 50 100 160
Weightin mg.
3.0 5.2 7.2 11.2 15.2
Cmm.C02/hr.
200 174 144 112 94
Perg./hr.
Perunitsurface
(WI)/ 48.5 54.2 53.0 49.8 51.6
hr.
of animals in which the surfacerule does not
hold.
3. Thus we come to the statementthat several
M = bWG,
(2) metabolictypesexistwithrespectto the relation
whereM is the metabolic rate per unit time, betweenmetabolicrate and body size.
In view of what was said previously,three
W the body weight,and a and b are constants.
metabolic
types,that is, three different
ways of
This is a special case of the so-called allometric
fornula (Huxley, 1932) which expresses the dependenceof the metabolicrate on body size
this classificationapplying,
dependenceon body size foran enormousamount can be distinguished,
of morphological,
biochemical,physiological,and as was emphasized, to intraspecificallometry,
evolutionarydata. This formulacan furtherbe that is, to individuals of differentsizes or to
growinganimalswithinone species.
way:
writtenin thefollowing
In thefirsttype,metabolicrate is proportional
(3)
log M = log b +o? log W
to a surfaceor the 23 powerof theweight.RepreThat is to say, if metabolic rate is plotted sentatives of this type include fishesbut also
such as crustaceans,clams,
we certaininvertebrates,
against body weight double-logarithmically,
obtaina straightline the slope of whichindicates and ascaris. Table 2 presentsone example, the
theconstanta. Ifa = 3, themetabolicratefollows metabolic rate in the sowbug, Armadillidium.
the surfacerule. If a = 1 or the slope is 450, the As can be seen, its oxygenconsumptionper unit
metabolic rate is proportionalto weight. With weightdecreaseswith increasingbody size, but
remainsconstantper unit surface.Subsequently
1 > a > 23, an intermediary
case obtains.
values are taken, that is, if it will be seen that sowbugand companyreveal
If weight-specific
metabolicrate per unit weightis plottedinstead quite a bit about human growthas a central
of thatof the total animal,the equationbecomes: problemofphysiology.
Here the
The second type is quite different.
M =bW
(4) metabolic rate is proportionalnot to surface
area, but to weightitself,so oxygenconsumption
metabolicrates, in an animal of double size is simplydoubled,in
weight-specific
Correspondingly,
as a generalrule,decreasewithincreasingweight, an animal fourtimesas large is quadrupled,etc.
and the slope of thelogarithmic
plot is negative.
Direct proportionality
of metabolicrate to weight
After these preliminaries,we can summarize is found in growinginsect larvae and hemimethe experimentalresults in the followingway tabolous insects, as well as interspecifically,
in
1941b,et seq.):
(Bertalanffy,
comparingimagos of differentrelated species.
1. The surfacerulealso holdsforpoikilothermic Table 3 shows metabolic rates in the walking
The rule is, stick, Dixippus morosus.Oxygen consumption
vertebratesand certaininvertebrates.
of a wide application;but the explana- per gramand hour appears to be constantover a
therefore,
tion given by Rubner is too restricted,for in wide range,coveringall body sizes and the entire
animals thereis no thermoregula- development.Other groups belonging to this
poikilothermic
tion, and thus the latter cannot be the basic type are land snails of the familyHelicidae,
factor in the relation between body size and intraspecifically
as well as interspecifically,
and
metabolicrate.
annelidssuch as the earthworm.
2. On the otherhand, there are many classes
Finally, in the thirdtypemetabolicrates are
220
THIE QUARTERLY REVIEW OF BIOLOGY
TABLE 3
Oxygen
consumption
ofDixippusmorosus
(Temperature
20?C.)
AfterMuller(1943a).
8 130 250 450 630 850
Weightin mg.
2.0 30.6 60.7 113.2 154.8206.6
Cmm.02/hr.
Perg./hr.
250 236 243 252 245 242
TABLE 4
Oxygen
consumption
ofPlanorbis
sp.
(Temperature
23?C.)
AfterBertalanffy
and Muller(1943).
Weightin mg.
Cmm. 02/hr.
30-3558-6290-100140 190-200
2.3
3.9
5.4
69 65 56
Perg./hr.
Per unit surface 22.9 25.1 26.1
(WI)/hr.
7.3
9.5
52
48
27.0 28.2
intermediatebetween proportionalityto weight
to surfacearea. To this type
and proportionality
belongsuchpond snailsas Planorbisand Lymnaea
like Planaria. Table 4 gives data
and filatworms
fortheramshorn
snail,and showsthatitsmetabolic
rate decreases with respect to its weight,but
increaseswithrespectto its surfacearea.
The relationsmentionedare typicaland characteristicof the species concerned.Table 5 gives a
survey of available observations.A few minor
discrepanciesneed elucidation,but in generalit
can be said that the "metabolictype," i.e., the
relationofmetabolicrate to bodysize,is a physiological characteristicof the species or group of
speciesconcerned.
INTERPRETATIONS OF THE SIZE
DEPENDENCE
OF METABOLISM
We now come to the question,what is at the
basis of the relationbetweenmetabolicrate and
body size and, in particular,of its most familiar
form,the surfacerule? (Cf. Kleiber, 1947; Bertalanffy,1951a; Bertalanffyand Pirozynski,1953;
We must admit that we
withfurtherreferences).
do notknow.What can be shown,however,is that
the explanationsusuallygivenare insufficient.
There seemsto be, first,thealternativewhether
the dependenceof metabolicrate on body size is
based upon cellular or upon organismic factors.
That is to say, thedecreaseof metabolicrate with
increasingbody size may be due to intrinsic
in themetabolismofthecellsofsmaller
differences
and larger individuals,in which case it should
also be foundin the respirationof tissuestaken
out of the organism;or else, it may be regulated
by factorspresentand active onlyin theorganism
as a whole. Let us start with the organismic
hypotheses.
The most familiarone has already been mentioned, namely, thermoregulation.
There is no
doubt that energyexpense for thermoregulation
formsa considerablepart of the total metabolism
in homeothermic
animals.However,this explanation cannot be generalsince the surfacerule also
applies,and in factis moreaccuratelyestablished,
in cold-bloodedvertebratesand even in certain
invertebrates
wherethereis no thermoregulation.
Anotherinterpretation
assumes that the surfaceruleis based upon theanatomyand physiology
of the circulatory
system.The supply of oxygen
and nutritivematerialsto the tissuesis naturally
a functionof the intensityof the blood current.
The latterdependson factorssuch as the size and
stroke volume of the heart, the frequencyof
heart beat, the diameterof the blood vessels,the
degree of capillarization,and the like. As has
already been indicated, there are rather strict
quantitativerelationsbetween body size, metabolic rate, and pulse rate. Thus, in interspecific
comparison"fromthe mouse to the elephant",
pulse rate decreases approximatelyproportional
to the % power of the weight(Fig. 1). So does
basal metabolicratein theinterspecific
comparison
of mammals,if adult specimensof corresponding
species are plotted (Brody, 1945; Kleiber, 1947).
However, hemodynamicscannot offera general
explanation.Remember,for example,the clams,
where the circulatory system is completely
fromthat foundin vertebrates,or recall
different
ascaris, which has no blood circulationat allanimalswhosemetabolicrate nevertheless
follows
the surfacerule.
Recent investigations
of the Ludwiglaboratory
(Ludwig, 1956; Kienle and Ludwig, 1956; Sattel,
1956) give some supportto the hypothesisproposed by Ludwig and by Bertalanffy(1951a, p.
252f.) that the "metabolictypes" are connected
withtypesof respiratory
apparatus.Gill-breathing
animalsappearto followthe surfacerule;henceits
validityin fishand certaininanimateclasses. On
the otherhand, the surfaceof tracheasin insect
larvae developsproportionalto body volume,as
was shownby Sattel (1956) in Bombyxmori;hence
the proportionality
of metabolicrate to weight.
TABLE 5
Relationbetween
metabolicrateand bodysize
Species
PLATYHELMINTHES
Dugesia gonocephala
NEMATHELMINTHES
Ascaris lumbricoides
ANNELIDA
Lumbricussp.
Eisenia foetida
MOLLUSCA
Lamellibranchiata
Anodontacygnaea
Dreissensiapolymorpha
Prosobranchia
Lithoglyphus,Paludina fasciata and
P. vivipara
Pulmonata
Lymnaeastagnalis
Lymnaeastagnalis
Lymnaeaacuricularia
Planorbissp.
Planorbiscorneus
Planorbiscorneus
Isidora proteus
Pulmonata and Operculata,15 species
intra-andinterspecific
Helicidae
Helix, Chilotrema,and Cepaea (interspecific)
Cepaea vindobonensis
CRUSTACEA
Branchiopoda
Daphnia pulex
Artemiasalina
Isopoda
Asellus aquaticus
Asellus aquaticus
Armadillidiumpallasi
Porcellioscaber
Oniscusasellus
Ligia oceanica
Decapoda
Astacus astacus
Potamobiustorrentiuin
Pugettiaproducta
Homarusvulgaris
INSECTA
Hemimetabola
Dixippus morosus
Holometabola
Various species,intra-and interspecific
Tenebriomolitor
PISCES
Lebistesreticulatus
Various species (Scorpaena,Abramis,
Cyprinus,etc.)
REPTILIA
Lacerta
Reference
to
Respirationproportional
W2'8(surface),W (weight)
or intermediate
Bertalanffyand Muller, 1943
Intermediate
Kruger,1940
Surface
Muller, 1943b
Kruger,1952
Weight
Surface?
Weinland,1919
Ludwig and Krywienczyk,1950
Surface
Surface
Krywienczyk,1952a
Surface
and Muller, 1943
Bertalanffy
Fusser and Kruger,1951
Krywienczyk,1952b
and Maller, 1943
Bertalanffy
Fusser and Kruger,1951
Krywienczyk,1952b
Krywienczyk,1952b
v. Brand, Nolan and Mann, 1948
Intermediate
Intermediate
Weight?
Intermediate
Intermediate
Intermediate
Intermediate
Surface (high tenperature[30?C.]?)
Liebsch, 1929
Weight
and Muller, 1943
Bertalanffy
Weight
1948
Jan6aroik,
and Krywienczyk,1953
Bertalanffy
Surface
Surface
Muller, 1943b
Will, 1952
Muller, 1943b
Will, 1952
Will, 1952
Ellenby, 1951
Surface
Surface
Surface
Intermediate
Surface
Probably surface
Kalmus, 1930
Wolsky,1934
Weymouthet al., 1944
Thomas, 1954
Surface?
Weight?
Intermediate
Surface
Muller, 1943a
Weight
Kittel, 1941
Weight
and Muller, 1943
Bertalanffy
Weight
and Muller, 1943
Bertalanffy
Jost,1928
Surface
Surface
Kramer,1934
Surface
221
222
THE QUARTERLY REVIEW OF BIOLOGY
Intermediatecases wouldresultfromthe presence and Campbell, 1952), of succinodehydraseand
malicodehydrase
(Fried and Tipton,1953).
apparatus.
oftwo typesof respiratory
The picture, however, is differentin intraStill anotherexplanationof the surfacerule is
based upon anatomical or chemicalchanges in specificcomparisons,as betweenrat tissuesfrom
body size and age. Seven
compositionwithincreasingbody size. "Metaboli- animals of different
callyactive" organssuchas theviscera,thebrain, main organsof the rat have been investigatedby
etc., are relativelylargerin small as comparedto Bertalanffyand Pirozynski (1951, 1953), and
large animals. So it can be assumed that they skeletalmusculature,whichis particularlyimporconsumerelativelymore oxygenand are respon- tant because it formsa highpercentageof body
metabolic mass, was studied by Bertalanffyand Estwick
sible for the higher weight-specific
decline
rate in smallerorganisms.However,the relative (1953). As Fig. 2 illustrates,no significant
fromone of average Qo2withincreasingbody size is found
growthof innerorgansis verydifferent
organto the other,and so it is improbablethat it in brain, lung, and kidney; a slight decline in
can yield the simplerelationof the surfacerule skeletal muscle,liver, and heart; and a marked
1951a). A quantita- declinein thediaphragm.So thereis no systematic
ofmetabolism(cf.Bertalanffy,
and Pirozynski,1953) decrease of Qo, in the various organs consistent
tive estimate(Bertalanffy
to account with, and responsiblefor the decrease of the
shows that this factoris not sufficient
weight-specific
metabolic rate with increasing
fortheactual variationsofbasal metabolicrate.
in termsof body size. These results have been essentially
Now we come to the interpretations
intracellularfactors.This amountsto sayingthat confirmedby other workers and with other
metabolic rate materials: in growingchicken by Crandall and
the decrease of weight-specific
with increasingsize, as expressedin the surface Smith (1952), in the heart muscle of the guinea
decrease in the pig by Wollenberger
and Jehl(1952), in theteleost
rule, is due to a corresponding
is measured brain by Vernbergand Gray (1953), and in rat
oftissues.Tissuerespiration
respiration
as Qo,, that is, ,l 02/mg. dry weight/hr.,as testes by Homma (1953). Similarly,Fried and
determinedwith the Warburg apparatus. A Tipton (1953) did not find a decrease in the
enzymes.
considerableamount of work has recentlybeen contentof respiratory
From this it would appear that genetic,and
done along these lines,partlystimulatedby our
in the enzymatic
differences
own work now to be presented,just as we may hencespecies-specific,
also say that the interestin comparativemetab- activityand Qo, oftissuesare foundin interspecific
however,in animals of
olism as classifiedin the metabolictypes men- comparisons.Differences,
weightare irregular
tionedhas been stimulatedby the investigations the same speciesand different
The ques- withrespectto thevarioustissuesor are absent.
on growthlawsto be explainedhereafter.
is a
So we have to assume factorswhich,within
tionofthe size dependenceoftissuerespiration
controversialone, but the statementsto follow the intact organism,regulate the respirationof
the tissues,the sum total of whichis the metabappear to be a fairpresentationof the case.
In interspecificcomparison of mammalian olism of the entire animal, but which do not
species of different
sizes, rangingfromthe mouse show up in the isolated tissueused forWarburg
and Pirozynski,1951,
(Bertalanffy
to the horse, a decrease of Qo, with increasing determination
body size is generallyfound, as a number of Schmidt-Nielsen,Bertalanffy,and Pirozynski,
observershave established(Kleiber, 1941; Wey- 1951). We have alreadysaid that the organismic
mouth, Field, and Kleiber 1942; Krebs, 1950; factors usually contemplateddo not offer a
Martin and Fuhrmann, 1955). This decrease, satisfactoryexplanation.What one may expect
however,is not parallelin the variousorgansand, can be illustratedby the action of thyroxin.It is
as a generalrule,is less thanwouldcorrespondto easy to induce an increase of metabolismby
into the animalin vivo; but
thesurfaceruleor the 3 powerruleofmetabolism. injectionof thyroxin
way,a decreasewithincreasing in spite of many efforts
In a corresponding
made, nobodyhas been
this effectby an
body size was found in enzymaticsystemscon- able to reproducesatisfactorily
such as in the concentra- administrationof thyroxinto a tissue in vitro.
nectedwithrespiration,
tion of glutathione (Gregoryand Goss, 1933; On the otherhand, chronichormonalconditions
Patru'sev,1937), of cytochromec (Rosenthaland are manifestedby significantchanges of the
Drabkin, 1943), of cytochromeoxidase (Kunkel tissue Qo, as has been shownwith tissuesfrom
QUANTITATIVE
LAWS IN METABOLISM
AND GROWTH
223
15
7.~~~~~~~~~~~~~~z
6 t--Y?US
4
31
a
1o
20
30
40
50
60
eo
MO
l_
no
300
J
400
BODY WEi6rT IN G.
FIG. 2. TIssuE RESPIRATIONor VARious ORGANSOF TIE WHITE RAT IN RELATIONTO BODY WEIGHT
Qo2 =
.dl02/mg.drywt./hr.
Only regressionlines are shown; for individual data and statisticalevaluation cf. the originalpaper. After
and Pirozynski(1953).
Bertalanffy
hypophysectomizedanimals and in pituitary thegrowthcurvesoftheseveralspecies.It appears
dwarfmice,whichlack somatotrophin
(Bertalanffy that we have been successfulin establishinga
and Estwick,1954).
definiteand strictconnectionbetweenmetabolic
The writerdoes not feelhappy about thisstate types and growth types, in consequence of a
of affairs,and the situationwould be much more generaltheoryofgrowthwhichestablishesrational
satisfactoryif a straightforward
relationbetween quantitative laws of growthand indicates the
the decrease of the weight-specific
metabolic physiologicalmechanismupon which growthis
rate and the tissue respirationcould be found. based.
Indeed, the Ottawa study was startedwith this
Let us startwitha ratherobviousdeliberation,
was not borne firstindicatedby Putter (1920). Animal growth
expectation,which,unfortunately,
can be considereda resultof a counteractionof
out by thefacts.
The explanationof the surfacerule and of the synthesisand destruction,of the anabolismand
size-dependenceof metabolismin general thus catabolismof the buildingmaterialsof the body.
remainsratherunsatisfactory.
We have,at present, There will be growthso long as buildingup preto take the metabolictype,in the sense defined, vails over breakingdown; the organismreachesa
as an empiricaldatum of the species concerned. steady state if and when both processes are
However, even this cautious attitude leads to equal. We may expressthisin a generalformula:
certainremarkableinferences
with respectto the
-KWn.
dWIdt = nWWm
(5)
problemof growth.
METABOLIC TYPES AND GROWTH TYPES
It has already been stated that, among the
various animal classes, so-calledmetabolictypes
can be distinguishedby virtue of the relation
between the metabolicrate and the body size.
Now as thereare different
metabolictypes,there
are also different
growthtypeswhich are distinguishedby the course of growthas expressedin
In words: The change of body weight W is
given by the difference
betweenthe processesof
building up and breakingdown; v and K are
constants of anabolism and catabolism respectively,and the exponentsm and n indicatethat
the latterare proportionalto some power of the
bodyweightW.
Obviously the growthof any organismis of
an enormouscomplexity,whetherwe considerit
224
THE QUARTERLY REVIEW OF BIOLOGY
froma biochemical,physiological,morphological, physiologicalfacts (cf. Bertalanify,1951a), that
of it is directlyproportionalto weight.On the other
or any otheraspect. However,the justification
an overallformulaand thesimplemodelit implies hand, mathematicalconsiderations(Bertalanffy,
Our equationstates that the 1941b) show that our basic equation is rather
lies in the following.
grossresultofsyntheticand degradativeprocesses insensitiveto smallerdeviationsof the exponent
withinthe organismfollowsthe law of allometry, n fromunity.So we may put, withoutany conthat is, that the rate of these processescan be siderableloss of generality,the exponentn equal
expressedas a power functionof body mass. to 1. This makesthesolutionofourbasic equation
But this assumptionis justified,because at least mucheasier.
therateofall physiological The solutionofequation5 (n = 1) is (Bertalanffy,
in a firstapproximation
processeshithertoinvestigatedcan be expressedin 1941b):
allometric or power formulas (Adolph, 1949).
W = {87/K - [vq/K- Wo (1-m)]e-(1_-tn)Kt}1-m (6)
The intrinsiccomplexityof the phenomenon
concerneddoes not preclude it from following withWo = weightat timet = 0.
forexample,
sucha simple,generallaw. Remember,
The case is somewhatdifferent
withrespectto
what has been found in the dependenceof the anabolism. The synthesisof high-molecular
cell
basal or restingmetabolismof the intact animal. componentsneeds, on the one hand, building
Of course,what is called the basal metabolicrate blocks such as amino acids, sugars,phosphates,
is, in fact,the outcomeof innumerableand to a and so forth,and on theotherhand energywhich,
large extentunknownprocessesof intermediary in aerobic animals, is provided by oxidative
metabolism. Not only this, but the growing processes. Both can be taken into account as
organismundergoeschanges at the biochemical, limitingfactors.The experimental
resultsindicate
physiological,cellular,and morphologicallevels. that,so faras higheranimalsare concerned,there
that a is a lawfulconnectionbetweenrespiration,anabNevertheless,we can state quite definitely
certain organismobeys, let us say, the surface olism,and growthwhich works out in the folrule; that is, that the rate of metabolismof the lowingway.
entireanimal,whateverits size or developmental The exponentn in our basic equation denotes
age, can be expressedas a functionof the -'
the dependenceof anabolismon body weight.If
powerofits respectivebodyweight.
we insertform thatvalue whichis experimentally
We have nowmorecloselyto definetheprocesses foundforthe size dependenceof restingmetaboappearing in our basic equation. The catabolic lism,thegrowthlaws forthe organismin question
processesmean, of course,the continuousloss of follow automatically.Thus we can predict the
buildingmaterialas it takes place in any living growth type of an animal from its metabolic
organism.Biochemically,thismeans the turnover type,and thispredictionhas provedto be correct
of buildingmaterialsand particularlyof proteins, in a largenumberof cases, oftenin a quite unexas demonstratedby the isotope techniques. pectedway.
Cytologically,it means the renewal of cells, as
In a firsttype,respirationis proportionalto the
found in many tissues and organs,oftenat an 23 power of weight,according to the surface
the law of growthassumes the
unexpectedlyhighrate (cf. Leblond and Stevens, rule. Accordingly,
1948; Storeyand Leblond,1951; F. D. Bertalanffy following
form:
and Leblond, 1953; Leblond and Walker,1956; a
dW/dt - W2 -KW.
(7)
table of the rates of cell renewalas foundby the
Leblond school is given in Bertalanffy,1957).
We shall not bother with the mathematical
The isotopeand othertechniqueshave shownthat elaboration,but show immediatelythe results.
theanimalorganismmaintainsitselfin a so-called Fig. 3 gives metabolismand growthin the small
dynamic or steady state (Schoenheimer,1947), aquariumfish,Lebistesreticulatus.
Metabolicrates,
chemicalcomponentsas well as cells being con- measuredas oxygenconsumption,are presented
tinuallywornout or degraded,and on the other in the log-logor allometricplot. As will be rehand being replaced by way of resynthesisand membered,in the case of the surfacerule the
the formationof new cells. So far as the rate of allometricregressionline has a slope of 23. So
catabolism is concerned,we may assume, as a far as the growthcurvesare concerned,the solufirst approximationand based upon various tion of the growth equation gives theoretical
QUANTITATIVE
LAWS IN METABOLISM
AND GROWTH
225
200
__
_
curves with the followingmain characteristics.
First, growthrates are decreasingand growth
eventuallyattains a steady state. Secondly,the
a)
-0
curves for weightgrowthand linear growthare
4o 0~
The curve of weight
characteristically
different.
growthis sigmoid,with a point of inflexionat
- ---40 ---about one-thirdof the finalweight.The curveof
-e00
30
lineargrowthis a decayingexponentialwithouta
- 20
60 80 100
200 300 500 800
30 40
turningpoint. This is what we actually find
Weight In mg.
experimentally.
This is themostcommonformofgrowthcurves,
foundin fish,in a numberof invertebrateclasses
and also, with certainrestrictions,
in mammals.
160
40
-The validityof these growthequations has been
shown in many examples (Putter, 1920; Berta_
140
5
3
lanffy,1934, 1951a), and theyhave been adopted
120
6-30
in applied biology.
It appears that the "Bertalanffy
growthequaE
E
00
b) 125----_It
tion" is widelyappliedin international
fisheries.
has been foundto fitthe commercially
exploited
fishspeciesstudiedby the FisheriesLaboratoryof
80
4~~~~the Ministryof Agriculture,
Fisheriesand Food at
-0
?' 2 I 0
Lowestoft(England), withthe possible exception
ofthehake (Wimpenny,
pers.commun.).A comprehensivetheoreticalmodel of the dynamicsof exploitedfishpopulationshas beendeveloped,where-0
0
in growthof the speciesconcernedis represented
12
10
6
0
4
2
8
by equation 7 (Beverton, 1954; Beverton and
Time in weeks
Holt, 1957). Discussionof this populationmodel FIG. 3. IHE FIRST METABOLIC AND GROWTH TYPE
(which,apartfromfisheries,
maywellbe adaptable
curves(b) in the
Metabolicrate (a) and growtlh
to otherpopulations)is beyondthe scope of the
Growthcurvesfor,d:
Guppy(Lebistesreticutatus).
presentreview.It shouldbe mentioned,however,
lengthgrowth;- - - - weightgrowth;calculated
to equation(7). AfterBertalanffy
and Millthat examinationof the variousgrowthfunctions according
proposed led to the conclusionthat "von Ber- ler (1943).
talanify'sgrowthequationis the mostsatisfactory
ofanythathave hithertobeen developed"(Bever- weightis continuallyshiftedin disfavorof the
ton and Holt, l.c.). Ampledata as well as descrip- surface.Consequently,so long as the animal is
tionof mathematicalanalysiscan be foundin this small, surface-proportional
anabolism prevails
work.
over weight-proportional
catabolism, and the
A relationsimilarto that stated by Bertalanffy animal grows.The largerit grows,the more the
et al. for the surfacedependenceof respiration surplus remaining for growth decreases, and
was found by Yoshida (1956) in food intake. eventuallya steady state will be reached where
The quantityofplanktonconsumedby thesardine anabolism and catabolism balance each other,
is proportionalto the square of body length,and and growthcomesto an end.
the same appears to be true for assimilating Now we come to the second type. We have
and the gut.
organs,such as the gill-rakers
said that in certain animals, for example, in
This characteristiccourse of growthis easily insects, respirationdoes not follow the surface
understood.If a body,withnot too muchchange rule but ratheris proportionalto weightitself.
of shape, increasesin size, its surfacesincrease Let us see whathappensin thiscase (Fig. 4). The
approximatelywith the second power of the log-logplot of metabolicrate againstbody weight
length,but its volume and mass with the third will give a line witha slope of 45?. On the other
power. Hence, the ratio between surface and hand, we have to insert1 for the exponentm in
C',j
--
226
THE QUARTERLY REVIEW OF BIOLOGY
80
60
a)~~
0-0
-j
Of course,an insect larva does not grow to any
indefinitesize. However,growthis stopped here
by an altogether differentmechanism. It is
metamorphosiswhich abruptly intercepts the
exponentialincrease,even causing a decrease in
body weightas largeamountsof tissueare broken
down in order to develop the imago. The same
metabolicand growthtype also applies to hemimetabolousinsectslike the walking-stick,
where
there is no apparent metamorphosis,but the
hormonal mechanismsresponsiblefor development appear to be similar.Again,in land snails,
whichalso belongto thistype,exponentialgrowth
is interceptedby seasonalcycles.
Finally, we have described a thirdmetabolic
type,one where metabolic rate is intermediate
betweensurfaceand weightproportionality,
and
whichis exemplified
by pond snails.Again we can
calculate what growthcurvesare theoretically
to
be expected.If we inserta value 23 < m < 1 into
the basic equation, it appears that the growth
should followa thirdtype. The curve of weight
growthdoes not differvery much fromthat in
-
-
-
0
-A
10
5--20
60 80 100
30 40
WeightIn mg.
(Tenebrio)
200 300
5
E
1:
.43
>
-
-?-
20
b)
2--f
a)
0 8jII7
6
=
30 40
o
20
40
60
80
60 80 100
200 300
Weight in mg.
500 700
Timqe in hours
(Drosophila)
METABOLICAND GROWnTYasPE
FIG. 4. THE SECOND
(b) culrve(exponentMetabolicrate(a) and growth
and Muller
ial) in insectlarvae. AfterBertalanify
(1943).
E8
=
'-
--V4
{>
the growthequation, and thereuponget a comcm e
_
_
growthcurveand a secondgrowth
pletelydifferent
40
type.In contrastto the firsttype,anabolismand
0
run
catabolism,both being weight-proportional,
at the same pace. The morecatabolismincreases,
the more does anabolism. Therefore,growth
4
12
24
8
0
16
20
rates will not decrease but always increase,and
Time in weeks
the largertheanimalbecomes,the fasterit grows. FIG. 5. THE THIRD METABOLIC AND GROWTH TYPE
Growthis not limitedbut exponential,and no
rate (a) and lineargrowth
(b) (diameter
steady state is reached.This seems to be a para- ofMetabolic
snail (Planorbissp.). After
shell)in the ramshorn
butis exactlywhathappenls. Bertalanffy
stateofaffairs,
doxrical
and Muller(1943).
QUANTITATIVE
LAWS IN METABOLISM
AND GROWTH
227
TABLE 6
Metabolictypesand growth
types
Metabolictype
Growthtype
Examples
I. Respirationsurface-proportional (a) Linear growthcurve: attainingwithout Lamellibranchs,fish,
mammals
inflexiona steady state.
(b) Weight growthcurve: sigmoid,attaining,with inflexionat ca. 13 of final
weight,a steady state.
II. Respirationweight-proportional Linear and weightgrowthcurves exponen- Insect larvae, Orthoptera,
Helicidae
tial, no steadystate attained,but growth
or seasonal
interceptedby metamorphosis
cycles.
III. Respirationintermediatebetween (a) Linear growthcurve: attainingwithin- Planorbidae
flexiona steady state.
surface- and weight-propor(b) Weight growthcurve: sigmoid,similar
tionality
to I(b).
thefirsttype.Lineargrowth,however,is different, developmentrevisions and the introductionof
as its curveis S-shapedwith an inflexion.Again, complicatingfactorswill be necessary(forrecent
discussions of the theory,cf. Duspiva, 1955;
ourpredictionis verified(Fig. 5).
animalclasses thereare different Harms, 1955; Linzbach,1955; Zeuthen,1955; BerSo in different
1957).
metabolic types and differenttypes of growth, talanffy,
A case in point is growthin mammals.From
agreeing with theoreticalexpectation.Table 6
givesa surveyof examplesinvestigated,and may the viewpointof metabolism,it may be said that,
be consideredas a firstdraftfora new chapterin in a firstand crude approximation,mammals
physiologyof appear to belong to our firsttype, where the
physiology,namely, a comparative
surfacerule of metabolismapplies. Indeed, the
growth.
From the theoryof growthjust outlined,many surfacerule was firststated by Rubnerformamconsequences can be derived which have been mals, and it was already mentionedthat the
verifiedempirically(cf. Bertalanffy1957). Only clinicallyimportantcase, the determinationof
one furtherexamplewill be given. We can com- basal metabolismin man, applies the surface
pare the values of the catabolic constantK which rulein the somewhatmodifiedformof the Dubois
mammalian
werecalculatedfromthe growthcurves,withthe standard formula.Correspondingly,
pattern
values of proteinturnoveras directlydetermined growthroughlyfollowsthe characteristic
In a numberofcases, theconstants of the firstgrowthtypediscussed.
by experiment.
witha degreeofcorrespondence In detail, however, there are complications.
have been verified
and somewhatparadoxically,there
sincethe theoreticalmodel Unfortunately
whichis quite striking,
and, on the other are relativelyfew good data suitable for this
is admittedlyoversimplified
hand,theerrorin thephysiologicaldeterminations typeof analysis.
If the basal or restingmetabolismin the rat
of proteinturnoveris considerable.For example,
in 1938 the authorcalculatedthe turnoverrate of (Fig. 6) is measuredover the entirelife span, it
the
protein from the growth curve of man. The appears that as a crudeoverallapproximation
value found was 1.165 g./kg. body weight/day. surfaceruleobtains,thatis, the overallallometric
Eleven years later, Sprinson and Rittenberg regressionline has a slope of about 23. In more
(1949) calculated protein turnoverfrom their detail,however,thereis a breakin thisline,such
withN15,and founda value of 1.3g./ that the firstpart of the allometricline is steeper,
experiments
kg. body weight/day.Only a sound theorycould and the second part much flatter,than would
correspondto a slope of 23. The breaktakesplace
have permittedsuchprediction.
at a body weightof about 100 g., that is, preGROWTH IN MAMMALS AND MAN
cedingsexual maturation.As we have found in
It is, however,obvious that the theoryrepre- otherinvestigations(Bertalanifyand Pirozynski,
and that withfurther 1952, 1953), similar breaks appear in quite a
sentsa firstapproximation,
THE QUARTERLY REVIEW OF BIOLOGY
228
curveis different
fromall others.Since our growth
formulas
to a large number of species,
apply
1000 _
the shapes of theirgrowthcurves are the same,
800 _
and the same curve can be used to representthe
600
growthof various species, simplyby takingdifE 400
Wt1nYmuS
*th
r g.
ferentscales fortimeand body size. Fig. 8 shows
the growthcurves of a fish and a mouse. The
latter, similar to that of the rat, also shows a
300
growthcycle in detailedanalysis,whichdoes not
CP 100
much alter the picture.If, however,the growth
curve of man is entered,it appears to be unique.
8e0
ib.
verQor2 n0
The second part of the curve, beginningwith
is
t
fn
5
50
puberty,
follows the general pattern. The first
4
40
In infancyand
part, however,is very different.
I
I I
11II
1II1II1
30
400~
10 15 20 30 40 60
100
200
childhood,the curve is enormouslyprotracted.
Body weighti'n g.
A new growthcycle is added, as it were,to the
or RELATivE GRowTHI
typicalpatternof growth.Althoughthis change
FIG. 6. DISCONTINUITIES
IN THlE ALBiNo RAT
is heraldedin thegrowthcyclesoflowermammals,
!FAIIdiscontinuities
appearat a bodyweightof ap- onlyin man does it lead to a singularshape of the
100g.,i.e.,beforepuberty.
A correspond-growthcurve.This growthcurveofman,abnormal
proximately
of theentire
is foundin thegrowth
ingdiscontinuity
animal(see Fig. 7). Onlyregression
linesareshownin as it were,is of courseconnectedwithchangesin
data and statisticalanalysisare the hormonalbalance. This is demonstratedby
individual
thefigure;
and pathological cases, such as pubertaspraecox in
givenin the originalpapers.AfterBertalanffy
Piozynski(1952,1953)and Racine(1953).
pituitarydysfunction,
when pubertytakes place
an
at
in othermammalsand
early
age,
as
it
does
number of physiologicalcharacteristics,at the
in
still
apes.
The
singular
growth
curveofman is a
same age and body size: in the allometricgrowth
of the liver, the involutionof the thymus,the quantitative expression of the retardation of
ofliverand thymus,
and certainly human developmentwhich, according to Bolk
tissuerespiration
(1926), is one of the basic factorsin the evolution
thereare others(Fig. 6). There is small wonder
of
man. At the same time, it is an important
that these breaks and shiftsare found,as the
factor
forhis uniquenessin nature.Animalsrun
cominginto play of the sex hormonesentails a
theirdevelopmentalperiodspeedily,and
through
deep-reachingchange in the entire metabolic
pattern.
_
_
_
__ ___300
300
On the other hand, the growthcurve of the
rat was analyzed long before the physiological E
__E
studiesjust mentioned(Bertalanify,1938, 1951a; c200
-200
_
_
_ _- _
a discussionof recentliteratureon rat growthis
given in Bertalanffy,1957). The result was
foundthat the growthof the rat followsthe first
too
R100
_______
growthtype, with a characteristicchange,however,in the values of the constants,again at the
criticalpoint of around 100 g. body weight.If
these growthcycles are taken into account, an
200
300
400
0
100
Time in days
excellentfit of the empiricalgrowthcurves by
FIG. 7. GROWTH OF THE ALBINO RAT
meansof our formulascan be obtained(Fig. 7).
So it wouldappear thatmammalsbelongto the
Donaldson'sdata (d1). Length growth
calculated according to equafirsttype, with the qualification,however,that weight growth----,
(7). The calculation
showstwogrowth
cycles,with
two growth cycles must be distinguished,the ation
breakat approximately
100g. bodyweight.
Donaldtransitionfromthefirstto thesecondcycletaking son'sdataaretodaynotconsidered
tobe optimal,
since
dietshave increasedthegrowth
in
place at the timeof sexual maturation.This is to modernlaboratory
the rat.The figure
shows,however,
the excellentnube consideredno morethan a firstapproximation. mericalfitthat
can be obtainedwiththe theoretical
However,thereis one organismwhose growth formulas.
AfterBertalanffy
(1941b).
1500
_
.
\P
QUANTITATIVE
LAWS IN METABOLISM
80~~~~~~~~~~
-
-______
-
3: (;0
~
0
_
_
40
_
_
20
20
5
o0
5
Mouse x
lo
years
lO weeks
Abramis bramao
5
~~0
0
5
~~
0-
7V
o0
37
SUMMARY
Man*
<
c
40
z-
________x
_
229
and the
searchCouncil,theNationalCancerInstitute,
ResearchCouncilof Canada.A surveyof
Humanities
the problemof animalgrowthin generalis givenin
1951a,and 1957.
Bertalanffy,
100
_80
AND GROWTH
_____
years______
Time in days
FIG.8. CO1PARISON
O0FTHE GROWTH CURVES
OF A FISH,MOUSE,AND MAN
AfterBertalanffy
(1951a).
soon they reach puberty and the adult stage.
Man, on the otherhand,is givena longperiodof
youth,and is thusenabled to learnand to collect
experience.Thus the characteristic
humangrowth
curve is a prerequisitefor that mental development and civilizationwhich so sharply distinguishesman fromall otherbeings.
ACKNOWLEDGMENT
Thisessayis baseduponworkcarriedthrough
in the
author'slaboratories
at theUniversity
ofViennaand
theUniversity
ofOttawa(Canada).Partofthiswork
was aided by researchgrantsfromthe NationalRe-
between
connections
Workaimedat establishing
metabolismand growthis reviewed.In thevarious
animal classes, three "metabolic types," i.e.,
formsof dependenceof metabolicrate on body
size can be distinguished:proportionalityof
metabolicrate to surfacearea, or to weight,or
one intermediatebetween surface and weight
proportionality.The various theoriesregarding
the size dependenceof metabolismare discussed,
with particularconsiderationof the relation of
to
tissue respirationto body size. Corresponding
the "metabolic types" mentioned, there are
growth
by different
"growthtypes" distinguished
curvesof the speciesconcerned.A generaltheory
ofanimalgrowth,advancedby theauthor,permits
explanationof the connectionbetweenmetabolic
typesand growthtypes.The theoryis illustrated
by examplestaken frominvertebrateand vertebrate classes. Mammalian and human growth,
whilegenerallyfollowingthe "firstgrowthtype,"
show breaksin the growthcurve connectedwith
puberty.The data and conceptspresentedherald
a new chapter of physiology,the comparative
physiologyofgrowth.
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