Size and Shape in Biology
Thomas McMahon
Science, New Series, Vol. 179, No. 4079. (Mar. 23, 1973), pp. 1201-1204.
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and Society (Wiley-Interscience, New York,
(1971); Sci. Amer. 224, 224 (Sept. 1971).
7. A. Coale, Science 170, 132 (1970).
8. I. Taeuber [in Man's Place in the Island
E c o s y s t e t ? ~ ~F, . R . Fosberg, E d . (Univ. of
Hawaii Press, Honolulu, 1961). pp. 226-2621
analyzes how absorption into expanding societies affects the age and sex composition of
populations in formerly isolated social systems.
9. 0. D . Duncan, Handbook for Modern Sociolo g y , R. E . L. Faris, Ed. (Rand McNally,
Chicago, 1964), pp. 36-82.
10. K. Boulding describes the human ecosystem
as the "totality of human organizations" [The
Organizafional Re~,olution(Quadrangle, Chicago, 1952), p. xxii] and 0. D. Duncan notes
that the cycling of information is a unique
feature of the human ecosystem (9, pp. 4 M 2 ) .
1 1 . R . Freedman. Pop. Index 31, 417 (1965).
Size and Shape in Biology
Elastic criteria impose limits on biological
proportions, and consequently on metabolic rates.
Thomas McMahon
Observers o f living organisms since well as animals must be built strongly
Galileo have recognized that metabolic enough to stand under their own
activities must somehow be limited by weight. In the following, a general rule
surface areas, rather than body vol- is derived for the changing proportions
umes. Rubner ( I ) observed that heat o f idealized trees as a function o f scale,
production rate divided by total body and later the results are applied to
surface area was nearly constant in animals.
dogs o f various sizes, and proposed the
explanation that metabolically produced
heat was limited by an animal's ability Buckling
to lose heat, and thus total body surConsider a tall, slender cylindrical
face area. When more precise method?
o f measurement became available, column o f length 1 and diameter d
Kleiber ( 2 ) noticed that when rate o f loaded by the force P , representing the
heat production is plotted against body total weight o f the column, acting at
weight on logarithmic scales for ani- the center o f mass. Such a column will
mals over a size range from rats to fail in compression i f the applied stress
steers, the points fall extremely close P I A , where A = d 2 / 4 , exceeds the
to a straight line with slope 0.75 (Fig. maximum compressive stress, u,,,,,. Pro1 ) . The result has since been confirmed vided that the column is slender enough,
for animals as different in size as the it may also fail in what is known as
mouse and the elephant (3-5),and has elastic buckling, uhereby a small lateral
been verified for other metabolically displacement (caused, for example, by
related variables, s~lchas rate o f oxygen the smallest gust o f wind), allows the
consumption ( 6 ) . Excellent reviews o f weight P to apply a toppling moment
the problem are available (7-10).
which the elastic forces o f the bent
While it is often true that biological column below are not sufficient to relaws are not derivable from physical sist. In this case, "slender enough"
laws in any simple sense, Kleiber's r ~ ~ l emeans that l l d is greater than 25, a
may be one o f those fort~iitousexcep- range which includes virtually all trees
tions which D'Arcy Thompson ( 1 1 ) ( 1 2 ) . The critical length for buckling
suggests lie at the basis o f a funda- is related to the diameter b y :
mental "science o f form." Plants as
The author is assistant professor of applied
mechanics in the division of engineering and
applied physics, Harvard University, Cambridge,
Massachusetts 02138.
23 M A R C H 1973
where p is the weight per unit volume
and E is the elastic nlodulus o f the
12. S . S. Kuznets, Proc. Amer. Philos. Soc. 111,
170 (1967).
13. M. Abramowitz, A ~ n e r . Econ. Rev. 46, 8
(1965).
14. J. Krebs and J. Spengler, in Technology and
the Atnerican Econotny (Report of the National Commission on Technology, Automation,
and Economic Progress) (Government Printing
Office, Washington, D.C., 1966), vol. 2, pp.
359-360.
material. The mathematician Greenhill
( 1 3 ) showed that when the force due
to weight is distributed over the total
extent o f the column instead o f being
taken as acting at the center o f mass,
the critical height becomes:
This result is identical to Eq. 1 , with
only a change in the numerical constant. It may be demonstrated that another change in the constant occurs
when the solid cylinder is made hollow, provided that the thickness o f the
wall is proportional to the diameter.
Greenhill further showed that i f the
shape o f the column is taken as a
cone, or a paraboloid o f revolution, the
result is again only to change the numerical constant. Recently, Keller and
Niordson ( 1 4 ) have derived that the
tallest self-supporting homogeneous tapering column is 2.034 times as tall as
a cylindrical column made o f the same
volume o f the same material, and that
the distance to the top o f such a tapering column above any cross section is
proportional to the diameter o f that
cross section raised to the 2/3 power.
The rule requiring height to go as
diameter to the % power is thus independent o f many details o f the model
proposed for the elastic stability o f
tree trunks.
The limbs o f trees must also be proportioned to endure the bending forces
produced by their own weight. I f a
branch i s considered to be a cantilever
beam built into the trunk, there exists
a particular beam length l,, for which
the tip o f the branch extends the greatest horizontal distance away from the
trunk ( 1 5 ) . Branches longer than I,,
droop so much that their tips actually
come closer to the trunk. Suppose that
the purpose o f branches is to carry
their leaves out o f the shadow o f higher
branches, and therefore to achieve a
maximum lateral displacen~ent from
the trunk. Then the limb should grow
n o longer than I,,., where
and C dcpcnds only o n the droop angle
O,,, which in turn dcpcnds only o n the
angle at which the limb leaves the
trunk ( 1 5 ) . The result may be made
general for a tapered o r hollow limb
exactly as was done for the buckling
problem. Comparing Eqs. 1, 2, and 3,
it is apparent that elastic criteria set
lcngth proportional to the 2/3 power
of diameter i n both the trunk and the
branches.
I t should be possible to check the
validity of thcse results by measuring
the proportions of trees of different
scale. Such a check would be arduoils
if it wcre necessary to know E and p
for each species; fortunately, the ratio
E l p is quite accurately constant in
green woods (16, 17). I n Fig. 2, the
trunk diameter 1.525 meters from the
ground is plotted against the total
hcight for 576 individual trees, representing nearly every species found in
the United States. T h e data, taken primarily from the American Forestry
Association's 'Social register of big
trees" ( I s ) , include specimens both
very slender and very stout, since trees
arc eligible for lhis list according to
their bigness, a n index depending o n
the sum of thcir circumference and
height (19). A solid line representing
Eq. 2 is also shown in Fig. 2; it was
calci~lated for E = 1.05 x 105 kilograms per square meter and p = 6.18 x
1 0 ~ i l o g r a m sper cubic meter ( 1 6 ) .
Body weight (kg)
T h e broken line, which fits near the
center of the data points, has the same
slope as the solid line bist rcprcscnts a
sequence of trces \\hose hcight in cach
case is only one-fourth of the critical
buckling hcight. T h e conclusion seems
to be that the proportions of trees are
limited by elastic criteria. since there
arc no data points to the lclt s f the
solid line.
Animal Proportions
Just as trees must assunle thicker
proportions with increasing size, so
must animals adjust thcir shape with
scale. T h e argument has long been
offered that animals could not remain
geometrically similar from the snlall to
the large because thcir limbs, whose
cross-sectional area increases as the
square of characteristic body dimension
L, must then support a weight which
increases as L:' ( 7 ) . Thc dillisuity with
these arguments bascd on strength criteria is the inevitable concl~tsion that
animals may grow no larger than s size
which makes the applied stress squal
to the yield stress of their materials.
Animals larger than this size woilld
have to increase supporting areas directly with weight, so that no increases
in height could be tolerated, only increases in width. I f yield stress wcre
the only criterion, an animal with
slender proportions like the bobcat
shorlld be capable of attaining the same
absolute height as ihe lion. In fact,
it is widely found that some animals
grow larger than others, and animals of
,.oL-----J-LL
.o1
small scale are relatively more slender
than thosc of large scale (scc cover).
Perhaps this transformation occurs, as
in differently sized trees, for rcasor~s
bascd o n elastic rathcr than strength
criteria.
In the following, M C consider comparisons between animals of the same
family, so that thcir shape is grossly
similar. 'l'hc only change in shape permitted is I'or lengths to bear a specified
relationship to diameters: all lcngths
will be proportional to one another, as
will be all diameters. Each limb, bone,
o r muscle will thus have a length 1
and diameter tl, where length will be
taken as it measurement parallel to the
direction of tension o r con~prcssionand
diameter will be measured perpendicular to this direction. Thus, the lcngth
of the trunk is the distance between
shoillder and hip whether the animal
is bipedal o r yuadrapedal (Fig. 3a,
bottom).
When a q~ladruped is standing a t
rest, the four limbs will be exposed primarily to buckling loads, but the vcrtebra1 colum~nand its nlusculature must
withstand bcnding loads. \Vhen the
same animal runs, the situation is substantially rcversed in those phases of
the motion where the limbs are providing their maximum propulsive cfrort.
A t thcse moments, the limbs are supporting bcnding loads, while the vertebral colunln is receiving an end
thrust and thus a buckling load. The
fact that the loads are dynanlic rather
than static is not a consideration: the
nlaxinlunl deflection of a structure sliddenly loaded ~ ~ n d eitsr own weight is
I
0.1
1.0
I
I0
Diameter (m)
Fig. 1 (left). bfetabolic heat production plotted against body weight on logarithl-r~icscales. The solid line has slope %. The broken
line, which does not fit the data, has slope % and represents the sray surf;~cenrca increases with weight for geonlelrically similar
Fig. 2 (right). Tree height plotted against trunk base diameter on logarithmic scales for record trees
shapes [adapted from ( 2 ) )
representing nearly every American species. The trunk proportions are limited by elastic buckling criteria, since no points lie to the
left of the solid line. Data from (18, 19).
1202
SCIENCE, VQL. 179
sectional area is A , shortens a length
A1 against force oA in time At. T h e
power this muscle expends is oAAllAt,
where o is the tensile stress developed,
and is in general a function of the
shortening velocity AllAt. Hill ( 2 3 )
reported that "the inherent strength of
a contracting voluntary muscle fiber is
roughly constant, being of the order of
a few kilograms per square centimeter
of cross-section." Me also presented
arguments and experimental data to
prove that the speed of shortening,
AllAt, is a constant in any particular
muscle from species to species. If we
understand from the work of Hill and
others that both u and Allat may be
taken as constant, then the power output of a particular muscle and hence
all the metabolic variables involved in
maintaining the flow of energy to that
muscle depend only o n its crosssectional area. But this area is proportional to d', and hence
maximal power output a ( W3&1' = W 0i6
(7)
This is precisely the statement of
Kleiber's law we were looking for, provided we have some confidence that
maximal energy metabolism exceeds
basal metabolic rate by a factor, the
metabolic "scope," which is invariant
with respect to scale. Hemmingsen (8)
has presented evidence to this effect.
According to the model proposed
here, if lung volume goes as W (4, 2 1 )
but alveolar ventilation goes as W 0 , i 5 ,
then respiratory frequency must scale
as W-"."? The identical argument may
be made for ventricular stroke volume,
cardiac output, and heart rate. In fact,
Adolph ( 2 4 ) reported that b for respiratory frequency in mammals is
- 0.28 [Tenney ( 1 0 ) independently
gave the same number]. For heart
rate, b has been reported as - 0.27
( 2 5 ) and - 0.25 ( 2 2 ) . Stahl (9) observed that the ratio of many physiological periods to one another is found
to be nearly constant, independent of
scale. Thus, the ratio of gut pulsation
time to pulse time is nearly the same
in all mammals, and each animal lives
for approximately the same number of
heartbeats o r breath cycles. Other authors have discussed the importance of
this conclusion in arriving at the
"physiological age" of living organisms.
Summary and Corrclusions
Arguments based on elastic stability
and flexure, as opposed to the more
conventional ones based on yield
strength, require that living organisms
adopt forms whereby lengths increase
as the % power of diameter. The
somatic dimensions of several species
of animals and of a wide variety of
trees fit this rule well.
It is a simple matter to show that
energy metabolism during maximal
sustained work depends on body crosssectional area, not total body surface
area as proposed by Rubner (1) and
many after him. This result and the
result requiring animal proportions to
change with size amount to a derivation of Kleiber's law, a statement
only empirical until now, correlating
the metabolically related variables with
body weight raised to the % power.
I n the pr~ssent model, biological frequencies are predicted to go inversely
as body weight to the % power, and
total body surface areas should correlate with body weight to the 96 power.
All predictions of the proposed model
are tested by comparison with existing
data, and the fit is considered satisfactory.
In T h e Fire o f Life, Kleiber ( 5 )
wrote "When the concepts concerned
with the relation of body size and
metabolic rate are clarified, . . . then
compartive physiology of metabolism
will be of great help in solving one of
the most intricate and interesting problems in biology, namely the regulation
of the rate of cell metabolism." Although Will ( 2 3 ) realized that "the essential point about a large animal is
that its structure should be capable of
bearing its own weight and this lcaves
less play for other factors," he Mas
forced to use an oversimplified "geo-
metric similarity" hypothesis in his important work on animal locomotion
and muscular dynamics. It is my hope
that the model proposed here promises
useful answers in comparisons of living
things on both the microscopic and the
gross scale, as part of the growing science of form, which asks precisely how
organisms are diverse and yet again
how they are alike.
References and Notes
1. M . Rubner, Z. Biol. Munich 19. 535 (1883).
2. M. Kleiber, Hilgardia 6, 315 (1932).
3. F. G. Benedict, Vital Energetics: A Studj, in
Conzparative Basal Melabolisnl (Carnegie Institution of Washington, Washington, D.C.,
1938).
4. S. Brody, Bioenergetics and Growth (Reinhold, New York, 1945).
5. M. Kleiber. The Fire o f Life (Wiley, New
York, 1961).
6. S. M. Tenney and J. E. Remners, N a t ~ l r e
197, 54 (1963); A. C. Guyton, Anzer. I .
Phvsiol. 150. 70 (1947): E. Zeuthen, Quart.
~ e ; .BioL 2'8 (NO: I), 'I (1953).
7. K. Schmidt-Nielsen, Proc. Amer. Physiol.
Soc. 29, 1524 (1970); A. M. Hemmingsen,
R e p . Steno M e m . Hosp. N o r d . Insulin Lab.
4, 1 (1950); S. J. Gould, Anler. Natur. 105,
113 (1971): W. R. Stahl, Science 137, 205
(1962).
'.
8 . A. M. Hemmingsen, Rep. Steno Menl. Hosp.
Nord. Insulin Lab. 9, 1 (1960).
9. W. R. Stahl. Advan. Biol. Med. Phys. 9, 355
(1963).
10. S. M. Tenney, Circ. Res. 20-21 (Suppl. I), 1-7
(1967).
11. D . W. Thompson, On Growth and Form
(Cambridge Univ. Press, London, 1917).
12. S. Timoshenko, Elements o f Strengtil o f
Materials (Van Nostrand, Princeton, N.J.,
1962).
13. G. Greenhill, Proc. Canzbrirlge Phil. Soc. 4.
65 (1881).
14. J. B. Keller and F. I. Niordson, I. Matlz.
Mech. 16 (No. 5), 433 (1966).
15. T . A. McMahon, in preparation.
16. T. A. McElhanney and R. S. Perry, Forest
Service Bulletin N o . 78 (Forest Products
Laboratories of Canada, Ottawa, 1927).
17. G. A. Garratt, The Mechanical Properties o f
W o o d (Wiley, New York, 1931).
18. "Social register of big trees," Anler. Forests
72, 16 (May 1966); ibid. 77, 25 (January 1971).
19. Other data used in Fig. 2 are from the
following books: Royal Horticultural Society,
Conifers in Cultivation (Royal Horticultural
Society, London, 1932), p. 316; W. Fry and
J . R. White, Big Trees (,Stanford Univ. Press,
Stanford, Calif., 1930).
Mathenlatical
Bioph~jsicr
20. N. Rashevsky,
(Dover, New York, 1960), vol. 2.
21. W. R. Stahl and J. Y. Gummerson, Growth
37. 21 (1967).
22. W: R. ~ t a h l ;I . Appl. Physiol. 22, 453 (1967).
23. A. V. Hill, Sci. Progr. London 38 (No. 150),
209 (1950).
24. E. F. Adolph, Science 109, 579 (1949).
25. A. J. Clark, Comparative Physiology o f the
Heart (Cambridge Univ. Press, Cambridge,
1927).
26. The author is grateful to B. Budiansky, K. E.
Kronauer, S. J. Gould, C. R. Taylor, and
G. P. DeWolf for assistance through helpful
discussions.
SCIENCE, VOL. I f 9
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Size and Shape in Biology
Thomas McMahon
Science, New Series, Vol. 179, No. 4079. (Mar. 23, 1973), pp. 1201-1204.
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References and Notes
6
Oxygen Uptake as Related to Body Size in Organisms
Erik Zeuthen
The Quarterly Review of Biology, Vol. 28, No. 1. (Mar., 1953), pp. 1-12.
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http://links.jstor.org/sici?sici=0033-5770%28195303%2928%3A1%3C1%3AOUARTB%3E2.0.CO%3B2-A
7
Geometric Similarity in Allometric Growth: A Contribution to the Problem of Scaling in the
Evolution of Size
Stephen Jay Gould
The American Naturalist, Vol. 105, No. 942. (Mar. - Apr., 1971), pp. 113-136.
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7
Similarity and Dimensional Methods in Biology
Walter R. Stahl
Science, New Series, Vol. 137, No. 3525. (Jul. 20, 1962), pp. 205-212.
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24
Quantitative Relations in the Physiological Constitutions of Mammals
E. F. Adolph
Science, New Series, Vol. 109, No. 2841. (Jun. 10, 1949), pp. 579-585.
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