Jttf-Alft 1970
ENERGY METABOLISM AND ITS REGULATION
Energy metabolism, body size,
and problems of scaling1
KNUT SCHMIDT-NIELSEN
Department of Zoology, Duke University, Durham, North Carolina
v_ywiNG to Max Kleibcr's lucid contributions, the
lubjecl of energy metabolism and body size is one of
the best studied and understood within the extensive
field of comparative physiology. This is symbolized in
Fig. I, where wc can sec our old friend Gulliver, as well
as a Lilliputian who is walking down the cobblestone
street. The immediate problem that the Lilliputian
emperor had thrust upon him was how much food to
give the Man Mountain. Swift (27) reported that il
was exactly 1,7211 Lilliputian portions. Does Gulli
ver's Delphic expression indicate that he is looking at
the Lilliputian or at the empty space lo the left? The
significance of the empty space should soon become
clear.
Comparative physiology is based on the premise
the animals arc more or less similar and thus can bc
compared. This does not mean that they arc alike, and
lhe deviations from the general pattern are often as
meaningful and as interesting as the similarities. Those
who have dissected a racehorse or a greyhound may
have noted diat these animals have larger than pro
portionate hearts. In proportion to their body size
mammals generally have very similar heart size, about
5 or 6 g/kg body weight, and wc arc so used lo this scale
that wc immediately notice a deviation.
The fact thai most mammals arc similar in lhat they
have just above 0.5% of their body weight as heart,
may, at first glance, bc surprising, for wc know lhat
the small animal in relation to its body size has a far
higher metabolic rate than the large one. To supply lhe
tissues with oxygen at lhe necessary high rate obviously
cannot lie achieved by merely adjusting stroke volume,
which is limited by the size of the heart; thus, the
heart frequency remains lhe major variable to adjust,
as evident from heart rales between 500 and 1,000/min
in the smallest mammals (13).
Among mammals in general, perhaps the most
conspicuous difference is their size. A 4-ton elephant is
a million times as large as a 4-g shrew, and the largest
Presented at lhe 20th Autumn Meeting ut Ihe American Physi
ological
Davit,
Calif., August
25-2'J.bulimics
I'J69. o! Health
1 ThiiSociety.
work w.u
supported
by National
Gram IIK-0222H and Research Career Award I-K6-GM-2I.522
living mammal, ftic blue whale, can bc another 25-fold
larger. Twenty-five million shrews put together are
very difficult lo imagine, and since ihcy cat about
half their body weight of food per day, they would bc
very hard to feed. Wc can conceive of 25 elephants put
together, although it is a formidable mass of elephants.
If wc just consider the size of an elephant, wc can
easily make sonic serious mistakes. A few years ago, a
note appeared in Science which described, very appro
priately, lhe reaction of a male elephant to LSD (30).
The investigators wanted to study lhe peculiar con
dition of the male elephant known as "muslh." A male
elephant in "musth," a word of Sanscrit origin, is violent
and uncontrollable, but he is not in rut (and, although
some people think so, it docs not mean lhat he "must"
have a male). Shortly after the publication of this mile,
a letter io the editor of Science described lhe calculation
of lhe dose of I.SD as "an elephantine fallacy" (II).
The authors had calculated the dose, based on the
amount that puts a cat into a rage, and had multiplied
up by weight until they arrived at 297 mg of I-SD to bc
given to the elephant. The long description of what
happened can bc shortened by saying that after the
injection of the 297 mg (enough for 1,500 trips, for a
single human dose is about 0.2 mg), the elephant im
mediately Started trumpeting and running around,
then he stopped and swayed, 5 min after the injection
he collapsed, went into convulsions, defecated, and
died.
I am using this example lo illustrate the tragic events
that may result from a lark of appreciation of lhe prob
lems of scaling. How should wc calculate drug dosage?
Of course, if wc want to achieve equal concentrations
in the body fluids of a small and a large animal, wc
should calculate in simple proportion to their weights.
If this calculation is done by extrapolation from a 2.6kg cal and its dose of 0.1 mg LSD/kg, wc arrive al the
now obviously lethal dose of 297 mg LSD for the elephant.
If instead, as the letter-writer in Science said, wc calcu
late on the basis of metabolic rale, wc find that a much
smaller dose of U0 mg is needed. This makes some sense,
for wc ran expect that detoxification of a drug or its
nn ot scaling, lhe I.illipu:ide how much rood lo give him (27).
excretion may be related to metabolic rale. Hut there
could be other considerations or special circumstances.
For example, LSD could bc concentrated in lhe brain,
and in thai event we would have a much more complex
situation and might want to consider brain weight.
Wc could also use as a basis for the calculation an
animal which is not as notoriously tolerant lo LSD as
cats; for example, wc could use man. The weight of a
man is 70 kg, and a dose of only 0.2 mg LSD gives him
severe psychotic symptoms. On - a weight basis this
suggests that the elephant should receive 11 instead of
nearly 300 nig of LSD. Based on metabolic rale, which
in man is 13 liters oxygen/hoi|r and in the elephant 210,
wc gel a 3-ing dose for the elephant. If wc consider the
brain size, which in man is 1,400 g and in lhe elephant
about 3,000 g, wc arrive at only 0.4 mg.
I do not intend to give the answer.to bow much I.SD
should have been injected, if any; I only wish to point
out that scaling obviously is not a simple problem. There
arc, however, many situations in which deviations from
strict similarity arc more obvious and easier to analyze.
Figure 2 shows two animals that none of us has ever
seen. To the right is a Neohipparion, an extinct ancestor
of the horse, and we can see immediately that this must
bc an animal the size of a small deer or a dog. To the
left is another extinct animal, the Mastodon, drawn lo
equal size, and yet, we know immediately thai these
large and heavy bones must bc those of an animal the
size of an elephant. Our instantaneous perception of
die true size of these animals, aldiough drawn to the
same size, expresses a well-known rule of scaling. The
mass of an animal increases as the third power of the
linear dimension, and to support this mass the cross
section of the bones must bc increased beyond what is
achieved by increasing their diameter in linear pro
portion. Linear scaling would just square the sup
porting cross section of the bone, and the bones would
bc crushed by the weight of the cubed increase in mass.
The increase in skeletal weight with the body size of
mammals is expressed in a more general way in Fig. 3.
This graph is plotted on logarithmic coordinates, and
the points fall on a straight line. If lhe increase in all
body dimensions were scaled linearly, the weight of
the skeleton would remain as the same |>ercentagc of
the body weight, and wc would have a line wilh a slope
of 1.0. Instead, we do find a line with a steeper slope,
the large animals have relatively heavier skeletons, and
the slope of lhe line is 1.13.
A physiological variable which concerns us much
more than the skeletal size of mammals is their meta
bolic rate. If metabolic rate were scaled directly relative
lo the body weight wc would, of course, find a regression
line with a slope of 1.0. The fact that this is not possible
is so well known lhat I hesitate to restate it, but it will
bc necessary for my further discussion. More than 100
years ago French writers (23) pointed out that heat
dissipation from warm-blooded animals must bc pro
portional to their free surface, and small animals must,
no. 3. Weight of thi
proportionally lo an ii
ENERGY METABOLISM AND ITS REGUI.ATION
FEDERATION PROCEEDINGS
31,20 10750
21,00 8805
l'J.oO 7500
18,20 7662
0,61 6286
6,50 3721
3,13 2123
35,68
10,91
15,87
16,20
65,16
66,07
88,07
1036
111 3
1207
1097
1183
1163
1212
bcrause of their larger relative surface, have a higher
relative rate of heat production than large animals.
.Similar considerations led lo lhe formulation of Bcrgmann's rule, a rule that claims lhat animals in colder
climates have relatively small cars and oilier appendages,
thus reducing lhe area of external surfaces from which
they can suffer heat loss (4).
The first experimental examination of these problems
was made by Rubner, who in 1883 published a study of
dogs of various sizes (22). The original table from Rubncr's paper is reproduced in Fig. 4. The second column
shows that the dogs were from 31- lo about 3-kg size, a
size range of 10:1. (Today, wc could easily find dogs
of much more different sizes.) In these dogs the meta
bolic rate per kilogram (column 5) increased as lhe
liotly weight diminished. On the other hand, if the
metabolic rate was calculated per body surface area
(last column) there was a nearly constant relationship
between metabolic rale and body surface of the dogs.
Rubner interpreted this as necessary for the animal lo
keep warm, for heal loss lakes place from the surface
and must bc related to the extent of this surface, lie also
very explicitly stated that this is not due to any specific
activity of the cells, but that it is due to lhe stimulation
of receptors in the skin which in turn act on the cells of
the metabolizing tissues. Wc now consider this argument
erroneous.
These findings established the "surface law," or the
"surface rule" as 1 prefer lo call il, and the extensive
use of body surface as a basis of reference for metabolic
rate. Il became evident, however, only 5 years after
Rubner*! work, that the need for heat dissipation can
not bc the primary reason for the relationship. In 1888
von HocSSlin (29) published a study of fish, and he
found that also in these the oxygen consumption showed
a much closer relation lo surface ihan io weight. Ob
viously, fish have little need to keep warm in proportion
lo the extent of their body surface.
Since that lime there have been innumerable studies
of tlits subject, but before I discuss these, I should like
lo give sonic further attention to the body surface of
animals. The body surface area is a surprisingly regular
function of body size. On the other hand, il is difficult
lo determine wilh accuracy lhe exact size of the surface
area, and Klcil)cr has repeatedly pointed out that an
accuracy of better than 20% cannot by any means bc
expected (16). .
Surface areas of a large number of organisms, from
less than a gram lo several Ions, arc compiled in Fig. 5.
In this graph the fully drawn, straight line indicates the
surface area of a sphere of a given weight and a specific
gravity of 1.0. The sphere, of course, has lhe smallest
possible surface of any geometrical shape of a given
volume, and compared with this wc sec that animals
have surface areas lhat quite regularly amount lo about
twice the sphere of the same weight. Someone may ask
alxiut lhe deviating "points in the upper right hand of
the graph—they arc not animals, they arc merely beech
trees, and the reason for including these should become
evident later on. The trees have larger relative surfaces
than animals, but they arc on a line parallel to that of
animals, whiclwhas a slope of 0 67. In general, then, the
surface of living organisms seems lo be an amazingly
regular function of the square of their linear dimension
(or the two-thirds power of their mass or volume). It is
now important to realize lhat not only for geometrical
reasons is il difficult to deviate from the regularity of
lhe surface relationship lo volume bul if wc wish lo
design a workable animal, large or small, wc find our
selves inextricably enmeshed in the need for considering
surfaces. A moment's thought gives us a long list of
surface-related processes, heat loss, as wc discussed
before, must bc surface related, lhe uptake of oxygen
in lungs or in gills depends on the area of their surface,
the diffusion of oxygen through lhe walls of capillaries
must bc related lo their surface areas, food uptake in
the intestine likewise must bc surface related, and so
on. In fact, all cells have surfaces, and the surface pro-
Body surface cm'
Body weight 1kg
no. 5. Body surface of vcnebraics in relation to body weigf
The fully drawn line repre-cnu the surface of a iphcre of a dc
«ity of 1.0. The larger |K>inu in the upper right-hand corner rcpr
■cut beech l-rcs (K.om Hemmiug-cn (12).)
cesses, or membrane processes as wc call them now,
nuisl be related to the areas of these surfaces.
Obviously, wc could hardly design an animal in
disregard of surface relationships. To mention an ex
ample from problems of temperature regulation, Kleiber
has staled lhat if a steer is designed wilh the metabolic
rale of a mouse, lo dissipate heat al the rate il is pro
duced, its surface temperature would have to be well
above the boiling point Conversely, if a mouse is de
signed wilh the weight-related metabolic rate of a steer,
to keep warm it would need to have as surface insulation
a fur at least 20 cm thick. Obviously, Surface considera
tions arc essential and cannot bc ignored.
After Rubncr's work several eminent physiologists
gave a great deal of attention lo the relation between
body surface and metabolic rate. The basic thinking in
the field, however, remained somewhat confined by
undue emphasis on surface areas, until in 1932 Klcilicr
published an article lhat has ever since influenced all
our concepts of the subject (15). He published his paper
in llitgaidia, a little-known journal of agricultural
science published by the California Agricultural Ex
periment Station at Davis, now the University of
California at Davis, our host at this meeting. I have a
rare reprint of this paper, it hears a red stamp which
says "Max Kleiber, Personal Copy, Do Not Remove
From Files." Kleiber obviously has realized the logical
fallacy of these directions, for if il slays in the tiles, how
can ii be used.' This must lie the reason that he per*
sonally gave it to ine, and I am very grateful for it in
deed.
In this paper Kleiber showed that the metabolic rate
of mammals not only is an amazingly regular function
Of their body size, but also thai il is significantly different
from a direct surface function..He examined animals
over a size range horn rals lo steers, and published a
curve lhat rould be called a "rai-lo-siccr curve."
The phrase does not form easily in lhe mouth and has
not become well known; whal is recognized, however,
is Kleibcr's generalization thai on a log-log scale the
metabolism of mammals as related to their body weight
forms a straight line with a slope of 0.75. Two years
later Brody et al (7) published their well-known "mouseto-elephant curve," and 4 years later again Benedict (3)
published a similar curve in his book Vital Energetics
in which he somewhat reluctantly admitted that, al
though animals do not know about logarithms, in a
log-log plot the points fall amazingly close to a straight
line.
The slope thai Brody found was 0.734, very close lo
0.75 which Kleiber had suggested should bc used, in
fart, lhe difference between these numbers is not sta
tistically significant. Quibbling about the validity of the
second and third decimal of the exponent appears
somewhat meaningless when wc realize that before
Brody calculated his rurvc he made certain "adjust
ments," which for the elephant consisted in "30%
deducted from the original value (I0r.' for Standing
and 20 ri for heat increment of feeding)" (5).
Similar studies, relating metabolic rate to body size,
have been extended to numerous cold-blooded verte
brates, and, although the regression lines are lower,
their slo|>es arc similar. Even the metabolic rates of
beech trees fall on a line wilh a similar slope. This should
dis|>cl forever the notion that temperature regulation is a
primary stimulus in the regular relationship between
heal production and body size. Likewise, many inverte
brates have metabolic rate-body size curves on similar
straight Hues, which within statistical limits can easily bc
said io have lhe same slope. Some, however, have signifi
cantly different slopes, and a few, some snails and
some insects, have been found to have metabolic lines
wilh slopes of 1.0, that is, the metabolic rate in these is
directly proportional lo lhe weight of the animal.
The entire field has been ^reviewed by Hemmingsen
(12) in a monumental summary which covers die range
from the smallest microorganisms lo lhe largest known
mammals (Fig. 6). The wide range covered is reflected
in the fact dial each division on the coordinates of Fig.
6 stands, not for a 10-fold, but a 1,000-fold difference.
Wann-bloodcd animals fall on a very nice straight line,
and cold-blooded vertebrates ou a similar line but at a
lower level which continues down through many in
vertebrates. Microorganisms again seem to bc organized
on a line with a similar slope. The slope is significantly
different from a surface relation (indicated by the line
marked 0.1)7) or a direct weight relationship (repre
sented by the line marked 1.00).
The equation which describes these lines is the fa
miliar exponential equation:
In lhe logarithmic form this equation gives a Ii
^sjf^-
.- '"j Unicellular,
.*]*■ organisms VOX)
no. 6. Mci.ib.4i.
weight. Noic dial c
fold dim-ii-ncc in 111
FEDERATION PROCEEDINGS
log 7 - a log x + log b
where a, the slope of the straight line, is lhe exponent
in lhe preceding equation. Similarly, when wc refer to
the metabolic curves we have been discussing, wc have
log metabolic rale related to log body weight by a
Straight line with the slope a, or:
metabolic rale - a-logbody weight + *
1 should now like to return lo the biological meaning
of this exponent a, or the slopes of lhe straight regression
lines, and the statistical significance of differences be
tween them. Is the slope 0.75 as proposed by Kleiber
significantly different from 0.67? The answer is that to
establish a significant difference between these two ex
ponents requires animals which difTer in size by more
than 9 lo 1 (16). Wc now remember lhat this is approxi
mately the range there was in die size of Rubncr's dogs,
and therefore Rubner really did not have material to
establish with certainty a slope different from 0 67,
or a simple surface relationship. In those Studies lhat
include a large number of mammalian species of widely
different sizes, exponents have consistently been higher
than 0.67. For example, Brody and Procter (6) derived
0.734 and Kleiber 0.756. Kleiber (16) has shown lhat
these exponents cannot bc established as significantly
different from 0.75 unless we examine animals thai cover
a range from smaller than 4 g lo larger than 000 ions,
and I therefore imagine lhat wc will never have cx|>erimcntal observations to support a slope significantly dif
ferent from the simple 0.75 lhat Kleiber has suggested
we should use. One reason that Kleiber advocates the
use of the 0.75 power is the much greater simplicity in
the arithmetical computations that this permits.
Wc can now summarize the events that led up to the
present-day concepts of this field. Early in the last
^r^sammi
noted from Klrilyrr (17).)
I. II. Observed metabolic r.
body weight and plvtled
century French writers realized lhat heat loss must be
related to the free surface of the animal, whatever thai
is. Rubner (22) made die firsi experimental study of the
problem, using dogs of widely different sizes. He did
indeed find a close relationship to Ixiily surface, and
thus established the body surface as a common reference
point in metabolic studies. Kleiber (15), after examining
metabolic dala from mammals over a much wider
size range stated that the exponent is much closer to
the 0.75 power of the body weight.
It now becomes evident what Gulliver was loo
so intently. In Fig. 7 he is contemplating the 1967
volume of Annual Review of Physiology, the same volume
thai Dr. Black mentioned in his introduction. Kleibcr's
preface IO this volume (17) was entitled "An Old Pro
fessor of Animal Husbandry Ruminates." Among other
interesting subjects he discussed Lilliputian physiology,
and after making some assumptions about their size,'
'lc found that the Lilliputians had used the slope 0.76
■rlicn they calculated Gulliver's need for food, that is,
a slope not significantly different from that in the Nilgnrilia paper, and thai the Lilliputians thus bad antici
pated Kleiber by 233 years. Kleiber .also pointed mil
that if they had used Rubncr's surface rule of 18113,
Gulliver would have received only 675 portions and
would have starved miserably.
Can wc analyze what the very common slope of 0.75
really means? I should like lo look at some of the dillicullics encountered in the scaling of an organism before
I try lo answer this question. The way metabolic rate is
plotted in Fig. 8 gives us a better visual impression of
how rapidly metabolic rate increases wilh decreasing
body size. The metabolic rale, when calculated per
gram body weight and plotted on a linear scale, points
l careful reading of Gulliver's Travcli will reveal some
uiscrepancy between Kleiber'i auumptiom and Swift's statements
regarding Lilliputian dimensions.
July-Auiuit 1970
ENERGY METABOLISM AND ITS REGULATION
out the tremendous increase in the small animal. This
again means that wc must supply to the cells of the small
est mammal, oxygen and nutrients at rates that are
some 100-fold as greaf as in the largest mammal. I
shall therefore discuss for a moment the parameters that
govern the oxygen supply lo the tissues, and afterwards
consider the gas exchange in the lungs, both being
essential components in lhe oxygen supply.
The rate of diffusion of oxygen from the capillary
to die metabolizing cell is determined by a) the diffusion
distance, and 4) the diffusion head (or difference in
Po, from capillary to cell). The former is simply a
function of capillary density, while lhe latter is influenced
by lhe oxygen dissociation curve for blood and Imple
mented by the Bohr effect, die Bohr effect in turn need
ing the presence of carbonic anhydrasc to bc effective
within the short time the blood remains in the capillary.
Krogh (18) was aware of lhe need for a higher capil
lary density in small animals, and he confirmed this by
quantitative determinations of capillary numbers in
muscle from horse, dog, and guinea pig. This general
ircnd has been reported by several later investigators,
but differences in technique make comparisons between
species uncertain. We, therefore, decided to use uni
form techniques and examine several different muscles
from a large number of mammals (25). Wc found wide
variations from muscle to muscle within one animal,
which only in pan were related to the proportion of
red and white muscle fibers, and a much less consistent
body size relationship than had been assumed to exist
(Fig. 9). True enough, the very smallest animals had
very high capillary densities, but the larger ones, from
the rabbit up, displayed no certain trend in their capil
lar., rb-nsilifa*. As far as I can sec, this should mean lhat
oth^. scaling considerations arc more important than
capillary distance, probably for entirely different reasons.
The muscle capillaries arc, of course, interspersed by
muscle fibers, and in any event the muscle filler is the
primary functional element of the muscle. Perhaps il is
, w, j. body :
,d In relatlor
mttmaEsmSBtm
large
the contractile elements or lo the conduction system lor
the action potential. If this is so, there will be con
straints on how far capillary density can bc decreased
in die large animal's muscles.
Let us next look at die unloading tension for oxygen
in the capillary blood, as expressed by the dissociation
curve (Fig. 10). It is necessary that for these purposes
wc compare whole, unaltered blood at the normal pH
of the organism, for il is whole blood and not dilute
hemoglobin solutions in phosphate buffer that :uns in
our blood vessels. The trend is unmistakable, the large
animals have dissociation curves to the left and small
animals to the right. Wc often refer to the half-satura
tion pressure (PM) as the "unloading tension." but we
do not have to establish the exact unloading tension to
sec that in the smaller animal the hemoglobin gives
up its oxygen at a higher oxygen pressure than in the
larger one. Figure 10 also shows the effect of acid on the
dissociation curve for mouse, indicated by the dotted
curve to the right of lhe normal curve for the same
animal (8). This shift to the right when acid is added
(Bohr cUcrt) increases the unloading tension for oxygen.
After the relationship between dissociation curve and
body size of mammals had been pointed out and In
terpreted as being related to the greater need for oxygen
in the small mammal (24), there has been an increased
interest in studies of whole blood, and additional mam
mals have been found to adhere to lhe same general
patlcrn. Of particular interest is, of course, the elephant
(2), which has a curve to the left of all the other mam
mals.
As already mentioned, the Bohr elfect is of great
ENERGY METABOLISM AND ITS REGULATION
help because it increases the unloading pressure for
oxygen. Figure 11 shows a compilation made by Riggs
of the Bohr effects of various mammals (21). He found
thai from the elephant to the mouse there is a great
increase in the magnitude of the Bohr effect; die effect
of acidification of the blood has a greater effect on the
unloading tension for oxygen in the smaller animal.
The Chihuahua is interesting, il deviates considerably
from the regression line and thus seems anomalous.
Perhaps the fact that its Bohr effect is lhe same as that
of a normal dog indicates that in spite of its half-kilo
gram size and peculiar appearance, the Chihuahua still
is a dog.
To uulizc die Bohr effect for an increase in oxygen
delivery it is quite necessary that the blood is acidified
before it leaves the capillary. The lime a red cell re
mains in the capillary is only a fraction of a second, and
without carbonic anhydrase the CO, from the tissues
would not Ik hydratcd and form carbonic acid in this
short lime With Larimer I have studied lhe concentra
tion of carbonic anhydrase in lhe red cells of various
mammals, and indeed, the small animals have a sig
nificantly higher concentration of this enzyme in their
red cells than the large animals (19). Carbonic anhy
drase is not considered lo bc essential in respiration,
and inhibition of its funclion with Diamox has little ef
fect. It has been described by Davenport as an enzyme
in search of a function. We suggested an alternate
viewpoint, that carbonic anhydrase is not necessary
in CO- transport, and that its primary role is lo aid in
the adequate delivery of oxygen lo die tissues.
I shall now briefly deal with the opposite clement
in the oxygen transport system, the uptake of oxygen in
£ltfif,ar,t
Blo.t H.m.tl.bini -
It)Animal IV,.■;.,' t
-J
.Cottl. «.„.
B./lr Cl/tct
E
*M,
Lung
of ma L,
■
to 12.
body
size,volume
as indicated
ly 10 (From Tcnncy and Re
N.
N.
CAMmW X*o.s.r
0
S.-I
\ifi*r It
1
— i — i — i — i -
1
1
1
1
1
1
1
1
1
J_
KIO. 13. Diffusion area of ll
nple proportion lo lhe rale of
y and Rcmme.. (211).)
long is scaled
union. (From Te
The tidal volume is similarly related lo body size,
the lungs. Here the scaling problems are a great deal
easier io deal with, and wc can therefore carry our
analysis further.
First of all, wc find lhat the lung volume in mammals
is directly related 10 the body weight (Fig. 12). By plot
ting lung volume against body weight wc obtain a
straight line with a slope that is very close to 1.0, in other
words, as a general pattern all mammals arc scaled simi
larly and have a lung volume of 6.3 % of the body weight.
As I said before, there arc always deviations from such
regression lines. These may bc due lo errors or to vari
ations
in
methor*"
'
'
tion than the general patu
come clearly known.
If next wc look at the diffusion area of the lung, we
find thai ibis variable is directly related lo oxygen
consumption of animals, rather than to their body
size (Fig. 13). This does indeed make physiological sense.
Oxygen consumption, of course, is related to body
weight wilh a power of 0.75, and therefore lhe diffusion
\. CmfUi* S.tt.r
1
Specific compliance - 0.019 (cm H.O)-'
particularly hue
^-J
»
i.,
x
obtain the specific compliance
size of die mammalian lung and its diffusion area arc
indeed adjusted lo the metabolic needs for oxygen.
Another example will show that wc can further ex
tend ihe use of allomclric equations and their predictive
value. The vital capacity of mammals is very nearly the
same funclion of body size as the lung volume in Fig. 12,
and can be described by the equation
Viial capacity = 0.063 It' •
Tidal volume - 0.0063 II' "
If we now divide the tidal volume by the vital capacity,
we obtain lhe diinciisinnlcss number 0.1. This predicts
lhat in mammals in general, die lidal volume is onclenih of die vital capacity, irrespective of their body
size. Although there are deviations from ibis general
pattern, the statement has a predictive va.ue which is
useful. Il has practical use in lhat wc can take a rat
or a horse and predict iis expected physiological pa
rameters, or we can look at the results of our studies
and sec how ihcy fit the general pattern, or deviate
from it. In this case we ended up with a nondimensional
numb T which describes the similarity of respiratory
ventilation in all mammals. In this regard, therefore,
mammals have a simply scaled similarity without any
residual body size-dependent exponent.
Some years ago, Drorbaugh (9) studied mice, rats,
rabbits, and dogs in some further detail, and among
oilier parameters he discussed lung compliance. He
found compliance to bc directly related 10 the body
size, expressed as die equation
Compliance - 0.00121 B'.-inl/em II.O
This equation says dial die number of milliliters the lung
will bc expanded for a change of 1 cm water pressure is
directly related 10 die body weight of the animal, and
lhe lung compliance of a large animal is therefore
scaled simply in proportion 10 its size. The vital ca
pacity, as you remember, is related in the same way 10
body size. By dividing one equation with the other, we
To express this in words, for a change in pressure of 1
cm II.O, all mammals have the same change in lung
volume relative 10 their vital capacity; that is, a change
of 1 cm water pressure causes a change of 1.9% in lung
volume. The prediction which Drorbaugh correctly
made from these considerations is that the pressure per
intake of one tidal volume should bc the same in all
mammals.
The scaling of a large number of physiological vari
ables and the derivation of nondimensional numbers
which describe their interrelations have become in
creasingly important. Several years ago Adolph pub
lished an important j>.i|,f-r in this field (I), and Walter
Stahl, who unfortunately died recently, had before his
death completed a major monograph on Flrysiuhgical
Similarity und Modeling which is now about to be pub
lished (26).
Before I conclude I find it appropriate to restate that
Kleiber initialed this way of analyzing physiological
of the metabolic rate problem in 1932. Wc were then
relieved of die constraining demand 10 fit metabolic
rate 10 body surface, and the healed discussions of how
to determine "free" or "true" surface have therefore
subsided. The "surface law" as such docs not even sur
vive as a "surface rule," but the analysis of function in
relation lo body size has in itself become an Immensely
interesting and productive field.
To illustrate this last point I should like to refer to a
diagram from Kleibcr's book, The Fire oj Life, which
shows animal productivity in relation 10 body size
(Fig. 14). In this diagram Kleiber showed lhat on I
Ion of hay, wc can maintain one steer for 120 days, or
300 rabbits for only 30 days. However, in meat produc
tion, the efficiency is the same for the two animals, i|i
spite of die difference in their metabolic rates. This
FEDERATION PROCEEDINGS
conclusion was confirmed by Jean Mayer, who staled for productive processes is independent of body weight"
it in these words: "The ratio of food consumption to (20). Mayer most appropriately honored Kleiber*!
basal metabolism and maintenance is independent of productivity and lucid analytical contributions by
body weight, and therefore, the excess feed lhat may go referring to this Statement as "Kleibcr's law."
-"-■—4 100: 579, 19"9.
pert, K. Barhey, K. I
E.M. Lano and J. Metcalf. Am. J. Physiol. 205: 331.
F. G. Vital Eiuigitics. A Study in Compaiativi
. . Washington, D. C: Carnegie IiuL of Was
lon, 1938.
**-*&■ 1: 595-708. 1847.
»'/-"«"
i in Oomistu Animals. New York: Rein
6. Hhodv, S., and R. C. Procter. Mutual Unit. Agr. Expt. Sia.
His. Hull. 166: 89, 1932.
7. Broov. S., R. C Procter and U. S. Ajhwortu. Missouri
Univ. Agr. ExfL Sia. His. Hull. 220: I, 1034.
8. DoUOLAS. C C, J. S. HaLDANE AND J. II. S. HaLDANE. J.
Physiol.. London 44: 275, 1912.
9. DaONMUOH, J. E. J. Appl. Physiol. 15: 1069, 1960.
10. Creoory. W. K. Ann. A'. Y. Acad. StL 22: 267, 1912.
11. Harwood. P. D. Seumi 139: 684. I9G3.
12. Himmincsen. A. M. Kept. St
Lot. 9: I. 1960.
13. Holt. J. P., E. A. Rhode and H. Kini
704, "**
14. Kayi_...
15. Ki.eiuer. M. llileardia ti: SI5, 1932.
16. Kleiber. M. Thi Fur oj l.iji. New York: WUey. 1961.
17. Kleiber. M. Ann. f18. Krooii. A. Thi An.
New Haven: Yale Univ. Pre
19. Larimer. J. I... and K. Scum
Physiol. I: 19. I960.
20. Mayer. J. Yali J. Hiol. Mid. 21:415. 1949.
21. Rioos, A. J. Ctn. Physiol. 43: 737. 1'JoO.
22. Ru«ner. M. Z. Biol. 19: 535. 1883.
23. Sarrus and Rameaux. Bull. Acad. Hoy. ."
Paris 3: 1091.
IU38-1839. (Quoted from Kayser anil Hci
(14).)
Am. J. Physiol.
24. Schmidt-Nielsen, K., and J. I.. Larimi
195:424. 1958.
25. Schmidt-Nielsen. K . and P. Pennycuik. An
746, 1961.
26.
St-
—
"'
D
• » — • - ' — • -■
e:_.;..:,.
.
1970.
27. Swirr. J. T,
ture 197: 54. 1903.
29. Von Hoessun. H. Du Hois-Riymond Auh. Anat. Physiol. 323.
I Will
30. Wur, l_ J., C. M. Pierce and VV. I). Thomas. Seitml 138:
1100, 1962.
Relation of structure to energy coupling
in rat liver mitochondria
L E S T E R PA C K E R
Department oj Physiology, University oj California, Berkeley, and
The Physiology Research Uboralory, Veterans Administration Hospital,
Martinet:, California
n nit context in which Professor Klcil>cr wrote
about, The Fire oj UJe, or the energy system of the
organism, I would like to discuss the role of the intact
functioning membranes of die mitochondria, the site of
primary energy transduction in lhe eukaryolic cells o!
animals and plants.
We have recently been asking a scries of questions
concerning the relation of structure to energy coupling in
rat liver mitochondria. I would like to bring nine of these
questions forward for discussion together with the
answers, such as they arc. At present, our strategy in
approaching this problem has been to altcinpt lo develop,
on die one hand, a scries of probes of macromolecular
and molecular structure in the membranes of mito
chondria and suliinitochondrial vesicles and, on die
other hand, to relate these observations on atructurc to
activities which reflect energy couplings.
The system wc have chosen for special study is the
oscillatory stale, because die oscillatory state accentuates
relationships between structure and function.
7) UO CONFKIURATIONAl. CHANCES
OCCUR IN HIIOCIIONOKIA?
To answer this question, wc have investigated lightscattering changes and electron microscopy of mito
chondria under oscillatory state conditions for ion
transport (12, 19). Figure I shows lhe light-scattering
changes lhat occur in a suspension of ral liver mito
chondria during energized ion accumulation. As ions
accumulate, light scattering decreases and when ions
are lost, the light scattering increases. 1 he respiratory
changes also suggest uncoupling on swelling. Figure I
s h o w s t h e g l u l a r a l d e h y d c - fi x a l i o n t e c h n i q u e u s e d t o
trap the structure at lhe desired phases of the oscillation
The dramatic changes in ultraslruclurc which occur
during die oscillatory stale arc illustrated in Fig. 3. I his
shows the extremes of morphology which occur in the
entire population of mitochondria within 20 sec when the
Presented at the 20th Autumn Meeting of lhe American Physiological Society, Davis, Calif. August 25-29. 1969.
i n i ti a l a c a o b i c e n e r g y - s ta r v e d c o n d i ti o n , c h a r a c te r i z e d
by mitochondria with contracted inner membranes, u
compared with expanded mitochondria examined at the
peak of the first oscillation following ion accumulation.
Figure 4 shows that the main changes are: u) an expan
sion of the inner membrane compartment; 4) a change in
the appearance of the matrix material; and c) an altera
tion in the folding of die membranes.
A l th o u g h c o n v e n ti o n a l c h e m i c a l fi x a ti o n g i v e s s o m e
information on lhe changes in configuration, wc were
anxious lo examine the structure in unfixed material.
Freczc-etcliing electron microscopy affords this opportunity
(Wrigglcsworlh. Packer, and Branton. personal communi
cation).
.
Figure 5 shows that resuspending contracted and
expanded mitochondria in 20% glycerol gives good
contrast between background ice and mitochondria and
gives good detail. Cross sections verify thai contraction
and expansion of the inner membrane compartment and
lhe matrix arc not artifacts of chemical fixation.
Therefore, the answer lo die first question is lhat gross
c o n fi g u r a l i o n a l c h a n g e s o b v i o u s l y d o o c c u r i n m i t o
chondria. In a general sense, these results arc in accord
with similar studies by Hackcnbrock (7) and Green
et al. (5).
2) DO MOLECULAR CONFORMATIONAL CHANGES OCCUR?
To answer this question, optical rotatory dispersion
(ORD) and circular dichroism (CD) studies have been
u n d e r t a k e n ( 2 2 ) . To p e r f o r m t h e s e s t u d i e s , i l w a s
necessary to use a) mitochondria which were uniform
wilh respect to population such as is observed in die
contracted and expanded phases of the oscillatory stale
as shown before, b) gluiaraldchydc-fixed mitochondria
to stabilize the configuralional state, and c) as shown in
Fig. 6 low levels of light scattering obtained by rcsuspension of the fixed mitochondria in 90% glycerol to
eliminate optical artifacts. The different ORD patterns
lhat have been correlated with the contracted and
expanded states of mitochondrial ultraslruclurc are
shown in Fig. 7. It can bc seen thai expansion of the