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, thelubjecl of energy metabolism and body size is one ofthe best studied and understood within the extensivefield of comparative physiology. This is symbolized inFig. I, where wc can sec our old friend Gulliver, as wellas a Lilliputian who is walking down the cobblestonestreet. The immediate problem that the Lilliputianemperor had thrust upon him was how much food togive the Man Mountain. Swift (27) reported that ilwas exactly 1,7211 Lilliputian portions. Does Gulliver's Delphic expression indicate that he is looking atthe Lilliputian or at the empty space lo the left? Thesignificance of the empty space should soon becomeclear.

Comparative physiology is based on the premisethe animals arc more or less similar and thus can bccompared. This does not mean that they arc alike, andlhe deviations from the general pattern are often asmeaningful and as interesting as the similarities. Thosewho have dissected a racehorse or a greyhound mayhave noted diat these animals have larger than proportionate hearts. In proportion to their body sizemammals generally have very similar heart size, about5 or 6 g/kg body weight, and wc arc so used lo this scalethat wc immediately notice a deviation.

The fact thai most mammals arc similar in lhat theyhave just above 0.5% of their body weight as heart,may, at first glance, bc surprising, for wc know lhatthe small animal in relation to its body size has a farhigher metabolic rate than the large one. To supply lhetissues with oxygen at lhe necessary high rate obviouslycannot lie achieved by merely adjusting stroke volume,which is limited by the size of the heart; thus, theheart frequency remains lhe major variable to adjust,as evident from heart rales between 500 and 1,000/minin the smallest mammals (13).

Among mammals in general, perhaps the mostconspicuous difference is their size. A 4-ton elephant isa million times as large as a 4-g shrew, and the largest

Presented at lhe 20th Autumn Meeting ut Ihe American Physiological Society. Davit, Calif., August 25-2'J. I'J69.1 Thii work w.u supported by National bulimics o! HealthGram IIK-0222H and Research Career Award I-K6-GM-2I.522

living mammal, ftic blue whale, can bc another 25-foldlarger. Twenty-five million shrews put together arevery difficult lo imagine, and since ihcy cat abouthalf their body weight of food per day, they would bcvery hard to feed. Wc can conceive of 25 elephants puttogether, although it is a formidable mass of elephants.

If wc just consider the size of an elephant, wc caneasily make sonic serious mistakes. A few years ago, anote appeared in Science which described, very appropriately, lhe reaction of a male elephant to LSD (30).The investigators wanted to study lhe peculiar condition of the male elephant known as "muslh." A maleelephant in "musth," a word of Sanscrit origin, is violentand uncontrollable, but he is not in rut (and, althoughsome 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 calculationof lhe dose of I.SD as "an elephantine fallacy" (II).The authors had calculated the dose, based on theamount that puts a cat into a rage, and had multipliedup by weight until they arrived at 297 mg of I-SD to bcgiven to the elephant. The long description of whathappened can bc shortened by saying that after theinjection of the 297 mg (enough for 1,500 trips, for asingle human dose is about 0.2 mg), the elephant immediately Started trumpeting and running around,then he stopped and swayed, 5 min after the injectionhe collapsed, went into convulsions, defecated, anddied.

I am using this example lo illustrate the tragic eventsthat may result from a lark of appreciation of lhe problems of scaling. How should wc calculate drug dosage?Of course, if wc want to achieve equal concentrationsin the body fluids of a small and a large animal, wcshould calculate in simple proportion to their weights.If this calculation is done by extrapolation from a 2.6-kg cal and its dose of 0.1 mg LSD/kg, wc arrive al thenow obviously lethal dose of 297 mg LSD for the elephant.If instead, as the letter-writer in Science said, wc calculate on the basis of metabolic rale, wc find that a muchsmaller 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 therecould 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 complexsituation and might want to consider brain weight.

Wc could also use as a basis for the calculation ananimal which is not as notoriously tolerant lo LSD ascats; for example, wc could use man. The weight of aman is 70 kg, and a dose of only 0.2 mg LSD gives himsevere psychotic symptoms. On - a weight basis thissuggests that the elephant should receive 11 instead ofnearly 300 nig of LSD. Based on metabolic rale, whichin man is 13 liters oxygen/hoi|r and in the elephant 210,wc gel a 3-ing dose for the elephant. If wc consider thebrain size, which in man is 1,400 g and in lhe elephantabout 3,000 g, wc arrive at only 0.4 mg.

I do not intend to give the answer.to bow much I.SDshould have been injected, if any; I only wish to pointout that scaling obviously is not a simple problem. Therearc, however, many situations in which deviations fromstrict similarity arc more obvious and easier to analyze.Figure 2 shows two animals that none of us has everseen. To the right is a Neohipparion, an extinct ancestorof the horse, and we can see immediately that this mustbc an animal the size of a small deer or a dog. To theleft is another extinct animal, the Mastodon, drawn loequal size, and yet, we know immediately thai theselarge and heavy bones must bc those of an animal thesize of an elephant. Our instantaneous perception ofdie true size of these animals, aldiough drawn to thesame size, expresses a well-known rule of scaling. Themass of an animal increases as the third power of thelinear dimension, and to support this mass the crosssection of the bones must bc increased beyond what isachieved by increasing their diameter in linear pro

portion. Linear scaling would just square the supporting cross section of the bone, and the bones wouldbc crushed by the weight of the cubed increase in mass.

The increase in skeletal weight with the body size ofmammals is expressed in a more general way in Fig. 3.This graph is plotted on logarithmic coordinates, andthe points fall on a straight line. If lhe increase in allbody dimensions were scaled linearly, the weight ofthe skeleton would remain as the same |>ercentagc ofthe body weight, and wc would have a line wilh a slopeof 1.0. Instead, we do find a line with a steeper slope,the large animals have relatively heavier skeletons, andthe slope of lhe line is 1.13.

A physiological variable which concerns us muchmore than the skeletal size of mammals is their metabolic rate. If metabolic rate were scaled directly relativelo the body weight wc would, of course, find a regressionline with a slope of 1.0. The fact that this is not possibleis so well known lhat I hesitate to restate it, but it willbc necessary for my further discussion. More than 100years ago French writers (23) pointed out that heatdissipation from warm-blooded animals must bc proportional to their free surface, and small animals must,

no. 3. Weight of thiproportionally lo an ii

FEDERATION PROCEEDINGS ENERGY METABOLISM AND ITS REGUI.ATION

31,20 1075021,00 8805l 'J.oO 750018,20 76620 , 6 1 6 2 8 66,50 37213 , 1 3 2 1 2 3

35,68 103610,91 111315,87 120716,20 109765,16 118366,07 116388,07 1212

bcrause of their larger relative surface, have a higherrelative rate of heat production than large animals..Similar considerations led lo lhe formulation of Bcrg-mann's rule, a rule that claims lhat animals in colderclimates have relatively small cars and oilier appendages,thus reducing lhe area of external surfaces from whichthey can suffer heat loss (4).

The first experimental examination of these problemswas made by Rubner, who in 1883 published a study ofdogs of various sizes (22). The original table from Rub-ncr's paper is reproduced in Fig. 4. The second columnshows that the dogs were from 31- lo about 3-kg size, asize range of 10:1. (Today, wc could easily find dogsof much more different sizes.) In these dogs the metabolic rate per kilogram (column 5) increased as lheliotly weight diminished. On the other hand, if themetabolic rate was calculated per body surface area(last column) there was a nearly constant relationshipbetween metabolic rale and body surface of the dogs.Rubner interpreted this as necessary for the animal lokeep warm, for heal loss lakes place from the surfaceand must bc related to the extent of this surface, lie alsovery explicitly stated that this is not due to any specificactivity of the cells, but that it is due to lhe stimulationof receptors in the skin which in turn act on the cells ofthe metabolizing tissues. Wc now consider this argumenterroneous.

These findings established the "surface law," or the"surface rule" as 1 prefer lo call il, and the extensiveuse of body surface as a basis of reference for metabolicrate. Il became evident, however, only 5 years afterRubner*! work, that the need for heat dissipation cannot bc the primary reason for the relationship. In 1888von HocSSlin (29) published a study of fish, and hefound that also in these the oxygen consumption showeda much closer relation lo surface ihan io weight. Obviously, fish have little need to keep warm in proportionlo the extent of their body surface.

Since that lime there have been innumerable studiesof tlits subject, but before I discuss these, I should likelo give sonic further attention to the body surface ofanimals. The body surface area is a surprisingly regular

function of body size. On the other hand, il is difficultlo determine wilh accuracy lhe exact size of the surfacearea, and Klcil)cr has repeatedly pointed out that anaccuracy of better than 20% cannot by any means bce x p e c t e d ( 1 6 ) . .

Surface areas of a large number of organisms, fromless than a gram lo several Ions, arc compiled in Fig. 5.In this graph the fully drawn, straight line indicates thesurface area of a sphere of a given weight and a specificgravity of 1.0. The sphere, of course, has lhe smallestpossible surface of any geometrical shape of a givenvolume, and compared with this wc sec that animalshave surface areas lhat quite regularly amount lo abouttwice the sphere of the same weight. Someone may askalxiut lhe deviating "points in the upper right hand ofthe graph—they arc not animals, they arc merely beechtrees, and the reason for including these should becomeevident later on. The trees have larger relative surfacesthan animals, but they arc on a line parallel to that ofanimals, whiclwhas a slope of 0 67. In general, then, thesurface of living organisms seems lo be an amazinglyregular function of the square of their linear dimension(or the two-thirds power of their mass or volume). It isnow important to realize lhat not only for geometricalreasons is il difficult to deviate from the regularity oflhe surface relationship lo volume bul if wc wish lodesign a workable animal, large or small, wc find ourselves inextricably enmeshed in the need for consideringsurfaces. A moment's thought gives us a long list ofsurface-related processes, heat loss, as wc discussedbefore, must bc surface related, lhe uptake of oxygenin lungs or in gills depends on the area of their surface,the diffusion of oxygen through lhe walls of capillariesmust bc related lo their surface areas, food uptake inthe intestine likewise must bc surface related, and soon. In fact, all cells have surfaces, and the surface pro-

Body surface cm'

Body weight 1kgno. 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 indisregard of surface relationships. To mention an example from problems of temperature regulation, Kleiberhas staled lhat if a steer is designed wilh the metabolicrale of a mouse, lo dissipate heat al the rate il is produced, its surface temperature would have to be wellabove the boiling point Conversely, if a mouse is designed wilh the weight-related metabolic rate of a steer,to keep warm it would need to have as surface insulationa fur at least 20 cm thick. Obviously, Surface considerations arc essential and cannot bc ignored.

After Rubncr's work several eminent physiologistsgave a great deal of attention lo the relation betweenbody surface and metabolic rate. The basic thinking inthe field, however, remained somewhat confined byundue emphasis on surface areas, until in 1932 Klcilicrpublished an article lhat has ever since influenced allour concepts of the subject (15). He published his paperin llitgaidia, a little-known journal of agriculturalscience published by the California Agricultural Experiment Station at Davis, now the University ofCalifornia at Davis, our host at this meeting. I have arare reprint of this paper, it hears a red stamp whichsays "Max Kleiber, Personal Copy, Do Not RemoveFrom Files." Kleiber obviously has realized the logicalfallacy of these directions, for if il slays in the tiles, howcan ii be used.' This must lie the reason that he per*sonally gave it to ine, and I am very grateful for it indeed.

In this paper Kleiber showed that the metabolic rateof mammals not only is an amazingly regular functionOf their body size, but also thai il is significantly differentfrom a direct surface function..He examined animalsover a size range horn rals lo steers, and published acurve lhat rould be called a "rai-lo-siccr curve."The phrase does not form easily in lhe mouth and hasnot become well known; whal is recognized, however,is Kleibcr's generalization thai on a log-log scale themetabolism of mammals as related to their body weightforms a straight line with a slope of 0.75. Two yearslater Brody et al (7) published their well-known "mouse-to-elephant curve," and 4 years later again Benedict (3)published a similar curve in his book Vital Energeticsin which he somewhat reluctantly admitted that, although animals do not know about logarithms, in alog-log plot the points fall amazingly close to a straightline.

The slope thai Brody found was 0.734, very close lo0.75 which Kleiber had suggested should bc used, infart, lhe difference between these numbers is not statistically significant. Quibbling about the validity of thesecond and third decimal of the exponent appearssomewhat meaningless when wc realize that beforeBrody calculated his rurvc he made certain "adjustments," which for the elephant consisted in "30%deducted from the original value (I0r.' for Standingand 20 ri for heat increment of feeding)" (5).

Similar studies, relating metabolic rate to body size,have been extended to numerous cold-blooded vertebrates, and, although the regression lines are lower,their slo|>es arc similar. Even the metabolic rates ofbeech trees fall on a line wilh a similar slope. This shoulddis|>cl forever the notion that temperature regulation is aprimary stimulus in the regular relationship betweenheal production and body size. Likewise, many invertebrates have metabolic rate-body size curves on similarstraight Hues, which within statistical limits can easily bcsaid io have lhe same slope. Some, however, have significantly different slopes, and a few, some snails andsome insects, have been found to have metabolic lineswilh slopes of 1.0, that is, the metabolic rate in these isdirectly proportional lo lhe weight of the animal.

The entire field has been ^reviewed by Hemmingsen(12) in a monumental summary which covers die rangefrom the smallest microorganisms lo lhe largest knownmammals (Fig. 6). The wide range covered is reflectedin 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 alower level which continues down through many invertebrates. Microorganisms again seem to bc organizedon a line with a similar slope. The slope is significantlydifferent from a surface relation (indicated by the linemarked 0.1)7) or a direct weight relationship (represented by the line marked 1.00).

The equation which describes these lines is the familiar exponential equation:

In lhe logarithmic form this equation gives a Ii

^ s j f ^ -

.- '"j Unicellular,.*]*■ organisms VOX)

no. 6. Mci.ib.4i.weight. Noic dial cfold dim-ii-ncc in 111

FEDERATION PROCEEDINGS July-Auiuit 1970 ENERGY METABOLISM AND ITS REGULATION

log 7 - a log x + log b

where a, the slope of the straight line, is lhe exponentin lhe preceding equation. Similarly, when wc refer tothe metabolic curves we have been discussing, wc havelog metabolic rale related to log body weight by aStraight line with the slope a, or:

metabolic rale - a-logbody weight + *

1 should now like to return lo the biological meaningof this exponent a, or the slopes of lhe straight regressionlines, and the statistical significance of differences between them. Is the slope 0.75 as proposed by Kleibersignificantly different from 0.67? The answer is that toestablish a significant difference between these two exponents requires animals which difTer in size by morethan 9 lo 1 (16). Wc now remember lhat this is approximately the range there was in die size of Rubncr's dogs,and therefore Rubner really did not have material toestablish with certainty a slope different from 0 67,or a simple surface relationship. In those Studies lhatinclude a large number of mammalian species of widelydifferent sizes, exponents have consistently been higherthan 0.67. For example, Brody and Procter (6) derived0.734 and Kleiber 0.756. Kleiber (16) has shown lhatthese exponents cannot bc established as significantlydifferent from 0.75 unless we examine animals thai covera range from smaller than 4 g lo larger than 000 ions,and I therefore imagine lhat wc will never have cx|>eri-mcntal observations to support a slope significantly different from the simple 0.75 lhat Kleiber has suggestedwe should use. One reason that Kleiber advocates theuse of the 0.75 power is the much greater simplicity inthe arithmetical computations that this permits.

Wc can now summarize the events that led up to thepresent-day concepts of this field. Early in the last

^ r^sammi

I. II. Observed metabolic r.body weight and plvtled

noted from Klrilyrr (17).)

century French writers realized lhat heat loss must berelated to the free surface of the animal, whatever thaiis. Rubner (22) made die firsi experimental study of theproblem, using dogs of widely different sizes. He didindeed find a close relationship to Ixiily surface, andthus established the body surface as a common referencepoint in metabolic studies. Kleiber (15), after examiningmetabolic dala from mammals over a much widersize range stated that the exponent is much closer tothe 0.75 power of the body weight.

It now becomes evident what Gulliver was looso intently. In Fig. 7 he is contemplating the 1967volume of Annual Review of Physiology, the same volumethai Dr. Black mentioned in his introduction. Kleibcr'spreface IO this volume (17) was entitled "An Old Professor of Animal Husbandry Ruminates." Among otherinteresting 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 Nil-gnrilia paper, and thai the Lilliputians thus bad anticipated Kleiber by 233 years. Kleiber .also pointed milthat if they had used Rubncr's surface rule of 18113,Gulliver would have received only 675 portions andwould have starved miserably.

Can wc analyze what the very common slope of 0.75really means? I should like lo look at some of the dilli-cullics encountered in the scaling of an organism beforeI try lo answer this question. The way metabolic rate isplotted in Fig. 8 gives us a better visual impression ofhow rapidly metabolic rate increases wilh decreasingbody size. The metabolic rale, when calculated pergram body weight and plotted on a linear scale, points

l careful reading of Gulliver's Travcli will reveal someuiscrepancy between Kleiber'i auumptiom and Swift's statementsregarding Lilliputian dimensions.

out the tremendous increase in the small animal. Thisagain means that wc must supply to the cells of the smallest mammal, oxygen and nutrients at rates that aresome 100-fold as greaf as in the largest mammal. Ishall therefore discuss for a moment the parameters thatgovern the oxygen supply lo the tissues, and afterwardsconsider the gas exchange in the lungs, both beingessential components in lhe oxygen supply.

The rate of diffusion of oxygen from the capillaryto die metabolizing cell is determined by a) the diffusiondistance, and 4) the diffusion head (or difference inPo, from capillary to cell). The former is simply afunction of capillary density, while lhe latter is influencedby lhe oxygen dissociation curve for blood and Implemented by the Bohr effect, die Bohr effect in turn needing the presence of carbonic anhydrasc to bc effectivewithin the short time the blood remains in the capillary.

Krogh (18) was aware of lhe need for a higher capillary density in small animals, and he confirmed this byquantitative determinations of capillary numbers inmuscle from horse, dog, and guinea pig. This generalircnd has been reported by several later investigators,but differences in technique make comparisons betweenspecies uncertain. We, therefore, decided to use uniform techniques and examine several different musclesfrom a large number of mammals (25). Wc found widevariations from muscle to muscle within one animal,which only in pan were related to the proportion ofred and white muscle fibers, and a much less consistentbody size relationship than had been assumed to exist(Fig. 9). True enough, the very smallest animals hadvery high capillary densities, but the larger ones, fromthe rabbit up, displayed no certain trend in their capillar., rb-nsilifa*. As far as I can sec, this should mean lhatoth^. scaling considerations arc more important thancapillary distance, probably for entirely different reasons.The muscle capillaries arc, of course, interspersed bymuscle fibers, and in any event the muscle filler is theprimary functional element of the muscle. Perhaps il is

mttmaEsmSBtm

, w, j. body :,d In relatlor

largethe contractile elements or lo the conduction system lorthe action potential. If this is so, there will be constraints on how far capillary density can bc decreasedin die large animal's muscles.

Let us next look at die unloading tension for oxygenin the capillary blood, as expressed by the dissociationcurve (Fig. 10). It is necessary that for these purposeswc compare whole, unaltered blood at the normal pHof the organism, for il is whole blood and not dilutehemoglobin solutions in phosphate buffer that :uns inour blood vessels. The trend is unmistakable, the largeanimals have dissociation curves to the left and smallanimals to the right. Wc often refer to the half-saturation pressure (PM) as the "unloading tension." but wedo not have to establish the exact unloading tension tosec that in the smaller animal the hemoglobin givesup its oxygen at a higher oxygen pressure than in thelarger one. Figure 10 also shows the effect of acid on thedissociation curve for mouse, indicated by the dottedcurve to the right of lhe normal curve for the sameanimal (8). This shift to the right when acid is added(Bohr cUcrt) increases the unloading tension for oxygen.

After the relationship between dissociation curve andbody size of mammals had been pointed out and Interpreted as being related to the greater need for oxygenin the small mammal (24), there has been an increasedinterest in studies of whole blood, and additional mammals have been found to adhere to lhe same generalpatlcrn. Of particular interest is, of course, the elephant(2), which has a curve to the left of all the other mammals.

As already mentioned, the Bohr elfect is of great

ENERGY METABOLISM AND ITS REGULATION

help because it increases the unloading pressure foroxygen. Figure 11 shows a compilation made by Riggsof the Bohr effects of various mammals (21). He foundthai from the elephant to the mouse there is a greatincrease in the magnitude of the Bohr effect; die effectof acidification of the blood has a greater effect on theunloading tension for oxygen in the smaller animal.The Chihuahua is interesting, il deviates considerablyfrom the regression line and thus seems anomalous.Perhaps the fact that its Bohr effect is lhe same as thatof a normal dog indicates that in spite of its half-kilogram size and peculiar appearance, the Chihuahua stillis a dog.

To uulizc die Bohr effect for an increase in oxygendelivery it is quite necessary that the blood is acidifiedbefore it leaves the capillary. The lime a red cell remains in the capillary is only a fraction of a second, andwithout carbonic anhydrase the CO, from the tissueswould not Ik hydratcd and form carbonic acid in thisshort lime With Larimer I have studied lhe concentration of carbonic anhydrase in lhe red cells of variousmammals, and indeed, the small animals have a significantly higher concentration of this enzyme in theirred cells than the large animals (19). Carbonic anhydrase is not considered lo bc essential in respiration,and inhibition of its funclion with Diamox has little effect. It has been described by Davenport as an enzymein search of a function. We suggested an alternateviewpoint, that carbonic anhydrase is not necessaryin CO- transport, and that its primary role is lo aid inthe adequate delivery of oxygen lo die tissues.

I shall now briefly deal with the opposite clementin the oxygen transport system, the uptake of oxygen in

£ltfif,ar,t Blo.t H.m.tl.bini -- J It) Animal IV,.■;.', t

B./lr Cl/tct. C o t t l . « . „ .

^ - J

» \. CmfUi* S.tt.ri., N.x N.1

E 0C A M m W X * o . s . r

S . - I

\ifi*r It

1

— i — i — i — i - 1 1 1 1 1 1 1 1 1 J _

*M, 12. Lung volume of ma■ to body size, as indicated L,ly 10 (From Tcnncy and Re

the lungs. Here the scaling problems are a great dealeasier io deal with, and wc can therefore carry ouranalysis further.

First of all, wc find lhat the lung volume in mammalsis directly related 10 the body weight (Fig. 12). By plotting lung volume against body weight wc obtain astraight line with a slope that is very close to 1.0, in otherwords, as a general pattern all mammals arc scaled similarly and have a lung volume of 6.3 % of the body weight.As I said before, there arc always deviations from suchregression lines. These may bc due lo errors or to varia t i o n s i n m e t h o r * " ' 'particularly huetion than the general patucome clearly known.

If next wc look at the diffusion area of the lung, wefind thai ibis variable is directly related lo oxygenconsumption of animals, rather than to their bodysize (Fig. 13). This does indeed make physiological sense.Oxygen consumption, of course, is related to bodyweight wilh a power of 0.75, and therefore lhe diffusion

size of die mammalian lung and its diffusion area arcindeed adjusted lo the metabolic needs for oxygen.

Another example will show that wc can further extend ihe use of allomclric equations and their predictivevalue. The vital capacity of mammals is very nearly thesame funclion of body size as the lung volume in Fig. 12,and can be described by the equation

Viial capacity = 0.063 It' •

KIO. 13. Diffusion area of llnple proportion lo lhe rale ofy and Rcmme.. (211).)

long is scaledunion. (From Te

The tidal volume is similarly related lo body size,

Tidal volume - 0.0063 II' "

If we now divide the tidal volume by the vital capacity,we obtain lhe diinciisinnlcss number 0.1. This predictslhat in mammals in general, die lidal volume is onc-lenih of die vital capacity, irrespective of their bodysize. Although there are deviations from ibis generalpattern, the statement has a predictive va.ue which isuseful. Il has practical use in lhat wc can take a rator a horse and predict iis expected physiological parameters, or we can look at the results of our studiesand sec how ihcy fit the general pattern, or deviatefrom it. In this case we ended up with a nondimensionalnumb T which describes the similarity of respiratoryventilation in all mammals. In this regard, therefore,mammals have a simply scaled similarity without anyresidual body size-dependent exponent.

Some years ago, Drorbaugh (9) studied mice, rats,rabbits, and dogs in some further detail, and amongoilier parameters he discussed lung compliance. Hefound compliance to bc directly related 10 the bodysize, expressed as die equation

Compliance - 0.00121 B'.-inl/em II.O

This equation says dial die number of milliliters the lungwill bc expanded for a change of 1 cm water pressure isdirectly related 10 die body weight of the animal, andlhe lung compliance of a large animal is thereforescaled simply in proportion 10 its size. The vital capacity, as you remember, is related in the same way 10body size. By dividing one equation with the other, we

obtain the specific compliance

Specific compliance - 0.019 (cm H.O)-'

To express this in words, for a change in pressure of 1cm II.O, all mammals have the same change in lungvolume relative 10 their vital capacity; that is, a changeof 1 cm water pressure causes a change of 1.9% in lungvolume. The prediction which Drorbaugh correctlymade from these considerations is that the pressure perintake of one tidal volume should bc the same in allmammals.

The scaling of a large number of physiological variables and the derivation of nondimensional numberswhich describe their interrelations have become increasingly important. Several years ago Adolph published an important j>.i|,f-r in this field (I), and WalterStahl, who unfortunately died recently, had before hisdeath completed a major monograph on FlrysiuhgicalSimilarity und Modeling which is now about to be published (26).

Before I conclude I find it appropriate to restate thatKleiber initialed this way of analyzing physiological

of the metabolic rate problem in 1932. Wc were thenrelieved of die constraining demand 10 fit metabolicrate 10 body surface, and the healed discussions of howto determine "free" or "true" surface have thereforesubsided. The "surface law" as such docs not even survive as a "surface rule," but the analysis of function inrelation lo body size has in itself become an Immenselyinteresting and productive field.

To illustrate this last point I should like to refer to adiagram from Kleibcr's book, The Fire oj Life, whichshows animal productivity in relation 10 body size(Fig. 14). In this diagram Kleiber showed lhat on IIon of hay, wc can maintain one steer for 120 days, or300 rabbits for only 30 days. However, in meat production, the efficiency is the same for the two animals, i|ispite 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"i t i n t h e s e w o r d s : " T h e r a t i o o f f o o d c o n s u m p t i o n t o ( 2 0 ) . M a y e r m o s t a p p r o p r i a t e l y h o n o r e d K l e i b e r * !b a s a l m e t a b o l i s m a n d m a i n t e n a n c e i s i n d e p e n d e n t o f p r o d u c t i v i t y a n d l u c i d a n a l y t i c a l c o n t r i b u t i o n s b ybody 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 Waslon, 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. MissouriUniv. 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. f-18. Krooii. A. Thi An.

New Haven: Yale Univ. Pre19. 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. ."

IU38-1839. (Quoted from Kayser anil Hci24. Schmidt-Nielsen, K., and J. I.. Larimi

195:424. 1958.25. Schmidt-Nielsen. K . and P. Pennycuik. An

746, 1961.2 6 . S t - — " ' D • » — • - ' — • -■ e : _ . ; . . : , . .

Paris 3: 1091.(14).)

Am. J. Physiol.

1970.27. Swirr. J. T,

ture 197: 54. 1903.29. Von Hoessun. H. Du Hois-Riymond Auh. Anat. Physiol. 323.

I Will30. Wur, l_ J., C. M. Pierce and VV. I). Thomas. Seitml 138:

1100, 1962.

Relation of structure to energy couplingin rat liver mitochondria

L E S T E R P A C K E R

Department oj Physiology, University oj California, Berkeley, andThe Physiology Research Uboralory, Veterans Administration Hospital,Martinet:, California

n n i t c o n t e x t i n w h i c h P r o f e s s o r K l c i l > c r w r o t e

about, The Fire oj UJe, or the energy system of theorganism, I would like to discuss the role of the intactfunctioning membranes of die mitochondria, the site ofprimary energy transduction in lhe eukaryolic cells o!animals and plants.

We have recently been asking a scries of questionsconcerning the relation of structure to energy coupling inrat liver mitochondria. I would like to bring nine of these

q u e s t i o n s f o r w a r d f o r d i s c u s s i o n t o g e t h e r w i t h t h eanswers, such as they arc. At present, our strategy inapproaching this problem has been to altcinpt lo develop,on die one hand, a scries of probes of macromolecularand mo lecu la r s t ruc tu re i n the membranes o f m i tochondr ia and su l i in i tochondr ia l ves ic les and, on d ieother hand, to relate these observations on atructurc toactivities which reflect energy couplings.

The system wc have chosen for special study is theoscillatory stale, because die oscillatory state accentuatesrelationships between structure and function.

7) UO CONFKIURATIONAl. CHANCESOCCUR IN HIIOCIIONOKIA?

To answer this question, wc have investigated light-scat ter ing changes and e lect ron microscopy of mi toc h o n d r i a u n d e r o s c i l l a t o r y s t a t e c o n d i t i o n s f o r i o ntransport (12, 19). Figure I shows lhe l ight-scatteringchanges lhat occur in a suspension of ral l iver mitochondr ia dur ing energ ized ion accumulat ion. As ionsaccumulate, l ight scatter ing decreases and when ionsare lost, the light scattering increases. 1 he respiratorychanges also suggest uncoupling on swelling. Figure Ishows the g lu la ra ldehydc-fixa l ion techn ique used to

trap the structure at lhe desired phases of the oscillation

The dramatic changes in ul t raslruclurc which occurduring die oscillatory stale arc illustrated in Fig. 3. I hisshows the extremes of morphology which occur in theentire population of mitochondria within 20 sec when the

Presented at the 20th Autumn Meeting of lhe American Physio-logical Society, Davis, Calif. August 25-29. 1969.

in i t ia l acaobic energy-starved condi t ion, character ized

by mi tochondr ia wi th contracted inner membranes, ucompared with expanded mitochondria examined at thepeak of the first oscillation following ion accumulation.Figure 4 shows that the main changes are: u) an expansion of the inner membrane compartment; 4) a change inthe appearance of the matrix material; and c) an alteration in the folding of die membranes.

Al though convent ional chemical fixat ion gives someinformation on lhe changes in configurat ion, wc wereanxious lo examine the structure in unfixed mater ia l .Freczc-etcliing electron microscopy affords this opportunity(Wrigglcsworlh. Packer, and Branton. personal communic a t i o n ) . .

F igure 5 shows tha t resuspend ing con t rac ted andexpanded m i tochondr ia i n 20% g l yce ro l g i ves goodcontrast between background ice and mitochondria andgives good detail. Cross sections verify thai contractionand expansion of the inner membrane compartment andlhe matrix arc not artifacts of chemical fixation.

Therefore, the answer lo die first question is lhat grossconfigura l iona l changes obv ious ly do occur in m i tochondria. In a general sense, these results arc in accordw i th s im i la r s tud ies by Hackcnbrock (7 ) and Greenet a l . (5 ) .

2) DO MOLECULAR CONFORMATIONAL CHANGES OCCUR?

To answer this quest ion, opt ical rotatory dispersion(ORD) and circular dichroism (CD) studies have beenu 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 snecessary to use a) mitochondria which were uniformwilh respect to population such as is observed in diecontracted and expanded phases of the oscillatory staleas shown before, b) gluiaraldchydc-fixed mitochondriato stabilize the configuralional state, and c) as shown inFig. 6 low levels of light scattering obtained by rcsus-pension of the fixed mitochondria in 90% glycerol toel iminate optical art i facts. The different ORD patternsl ha t have been co r re l a t ed w i t h t he con t r ac ted and

expanded s ta tes o f m i t ochond r i a l u l t r as l r uc l u r c a reshown in Fig. 7. It can bc seen thai expansion of the