agutter_04_metabolic_793537.pdf.txt

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>1742-4682-1-13.fm</title>
<meta name="Author" content="csproduction"/>
<meta name="Creator" content="FrameMaker 7.0"/>
<meta name="Producer" content="Acrobat Distiller 5.0.5 (Windows)"/>
<meta name="CreationDate" content=""/>
</head>
<body>
<pre>
Theoretical Biology and Medical
Modelling

BioMed Central

Open Access

Review

Metabolic scaling: consensus or controversy?
Paul S Agutter*1 and Denys N Wheatley2
Address: 1Theoretical and Cell Biology Consultancy, 26 Castle Hill, Glossop, Derbyshire, SK13 7RR, UK and 2BioMedES, Hilton Campus MG7,
Aberdeen AB24 4FA, UK
Email: Paul S Agutter* - tcbc26@btopenworld.com; Denys N Wheatley - wheatley@abdn.ac.uk
* Corresponding author

Published: 16 November 2004
Theoretical Biology and Medical Modelling 2004, 1:13

doi:10.1186/1742-4682-1-13

Received: 10 July 2004
Accepted: 16 November 2004

This article is available from: http://www.tbiomed.com/content/1/1/13
© 2004 Agutter and Wheatley; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

metabolic rateallometric scalingpower lawssupply networksfluid flow

Abstract
Background: The relationship between body mass (M) and standard metabolic rate (B) among
living organisms remains controversial, though it is widely accepted that in many cases B is
approximately proportional to the three-quarters power of M.
Results: The biological significance of the straight-line plots obtained over wide ranges of species
when B is plotted against log M remains a matter of debate. In this article we review the values
ascribed to the gradients of such graphs (typically 0.75, according to the majority view), and we
assess various attempts to explain the allometric power-law phenomenon, placing emphasis on the
most recent publications.
Conclusion: Although many of the models that have been advanced have significant attractions,
none can be accepted without serious reservations, and the possibility that no one model can fit all
cases has to be more seriously entertained.

Introduction: Kleiber and metabolic scaling
In 1932, Kleiber published a paper in an obscure journal
[1] showing that standard metabolic rates among mammals varied with the three-quarters power of body mass:
the so-called "elephant to mouse curve", termed "Kleiber's
law" in this review. Since that date, this and similar allometric scaling phenomena have been widely and often
intensively investigated. These investigations have generated continuing debates. At least three broad issues
remain contentious, each compounded on the one hand
by the problem of obtaining valid data (in particular,
finding procedures by which reliable and reproducible
measures of standard metabolic rate can be obtained,
especially in poikilotherms) and on the other by statistical

considerations (in particular, the validity of fitting scattered points to a straight line on a semi-logarithmic plot).
The first issue is disagreement as to whether any consistent
relationship obtains between standard metabolic rate and
body mass. Moreover, those who acknowledge such a
relationship hold divergent opinions about its range of
application. Is it valid only for limited numbers of taxa, or
is it universal? Since the 1960s there has been a measure
of consensus: a consistent allometric scaling relationship
does exist, at least among homoiotherms. Nevertheless,
not all biologists agree, and scepticism is widespread, particularly about the alleged universality of Kleiber's law.

Page 1 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

Second, assuming that some version of Kleiber's law (a
consistent metabolic scaling relationship) applies to at
least some taxa, there are disagreements about the gradient of the semi-log plot. That is, if B = aMb, where B =
standard metabolic rate, M = body mass, and a and b are
constants, what is the value of b? Kleiber [1] and many
subsequent investigators claimed that b = 0.75, and on
this matter too a measure of consensus has obtained since
the 1960s. Once again, however, not all biologists agree.
A significant minority of investigators hold that b = 0.67;
and other values have been suggested, at least for some
organisms.
Third, assuming a consistent scaling relationship and an
agreed value of b, how is Kleiber's law to be interpreted
mechanistically? What is its physical or biological basis?
For those who claim that b = 0.67, this issue is simple:
standard metabolic rate depends on the organism's surface to volume ratio. But for proponents of the majority
view, that b = 0.75, the issue is not simple at all. Many
interpretations have been proposed, and since several of
these are of recent coinage and seem to be mutually
incompatible, a critical comparative review seems timely.
Kleiber's initial paper [1] found support within a decade.
The allometric scaling relationship B = aMb (B = standard
metabolic rate, M = body mass, a and b are constants and
b is taken to be approximately 0.75), was inferred by other
investigators during the 1930s [2,3]. Relevant data have
been reviewed periodically since then (e.g. [4-15]) and
recent developments have rekindled interest in the field.

http://www.tbiomed.com/content/1/1/13

or some other number. The question is whether these variations are deviations from a general law, or whether there
is no such law. Conflicting opinions on this fundamental
point recall the traditional philosophical difference
between physicists and biologists: the former are inclined
to see abstract mathematical generalities in any set of
numerical data, the latter to see concrete particulars. All
recent attempts to explain Kleiber's law by "universal"
models have involved physicists and mathematicians; the
sceptics are predominantly biologists.
Dodds et al. [16] re-examined published scaling data from
Kleiber's original paper onwards and concluded that the
consensus (b = 0.75) was not statistically supported. Feldman [17] found no evidence for any wide-ranging allometric power law in biology and dismissed all attempts to
explain scaling relationships by physical or mathematical
principles. Atanasov and Dimitrov [18] found evidence
that b ranges from around 0.67 to more than 0.9 over all
major animal groups, the values perhaps reflecting complexity of organisation; single values such as 0.75 emerge
only as averages over each group. Other investigators have
been less sceptical; publications by Enquist and Niklas
[19,20] give particularly impressive support to the generality of Kleiber's law because Niklas was previously
among the doubters.

Many biological variables other than standard metabolic
rate also reportedly fit quarter-power scalings (relationships of the kind V = kMb, where V is the variable in question, k is a constant and b = n/4; n = 3 for metabolic rate).
Examples include lifespans, growth rates, densities of trees
in forests, and numbers of species in ecosystems (see e.g.
[9]). Some commentators infer that Kleiber's law is, or
points to, a universal biological principle, which they
have sought to uncover. Others doubt this, not least
because it is unclear how (for example) tree densities can
be consequences of metabolic scaling or can have the
same mechanistic basis. This article focuses on the metabolic rate literature, mentioning other variables only in
passing, because most debates in the field have arisen
from metabolic rate measurements.

Whatever one's position, it is indisputable that the Kleiber
relationship has many exceptions, even among mammals.
Bartels [21] showed that some mammals, such as shrews,
have B values well above those expected from the Kleiber
curve. Andersen [22] discussed the high B values for
whales and seals and attributed them to the cold-water
habitat. Nevertheless, Kleiber's law has been extended
beyond placental mammals to birds and marsupials.
Birds have generally higher a values than placental mammals and marsupials have lower ones, but the 0.75-power
relationship is still inferred by many investigators (e.g.
[4]). McNab [13] accepted Kleiber's law as a general
approximation but emphasized species variations, which
he attributed to differences in diet, habitat and physiological adaptation. Elgar and Harvey [23] also found variability among groups of species but reasoned that standard
metabolic rates vary taxonomically rather than with temperature regulation, food intake or activity. Economos
[24] was also critical of McNab, at least in respect of mammals.

Variations in the value of b
Most debates about the value of b assume some version of
Kleiber's law: i.e. that a single allometric scaling relationship fits metabolic rates over a wide range of organisms.
However, as noted in the introduction, there are dissenters. Everyone acknowledges considerable variation both
within and among taxa, no matter whether b = 0.75, 0.67,

It is difficult to define "standard metabolic rate" in poikilotherms; ambient temperature, time since last meal and
other variables markedly affect measurements [9,13,25].
A heterogeneous array of poikilotherm data [5] revealed
an "average" b value of roughly 0.75. There were wide
divergences in some taxa; notwithstanding these, Hemmingsen [4,5] argued that over all animals, plants and

Page 2 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

protists, metabolic rate scales as the 0.75-power of body
mass. More recently published data [26,27] support this
conclusion for a wide range of organisms and body
masses. However, a careful re-evaluation of Hemmingsen's data by Prothero [28] cast further doubt on the
applicability of Kleiber's law to unicellular organisms.
Scepticism persists, mostly on the grounds of the intrinsic
variability of the data, which is too often underestimated
because it is disguised in the customary logarithmic plots
and is seldom subjected to adequate statistical analysis
[11,29]. However, this too has been debated; a suitable
choice of procedures for estimating parameters might
eliminate inconsistencies and discrepancies from the data,
giving more credence to the belief that b = 0.75 over a
wide range of taxa [30]. In the following section we shall
examine some of the more divergent data in more detail.
In short, there is a clear but by no means total consensus
that (i) Kleiber's law is widely (even universally) applicable in biology, (ii) b is approximately 0.75. Variability in
the data is generally admitted, so the consensus – and the
claim that Kleiber's law manifests a general biological
principle – can legitimately be doubted.
The mass transfer model [31]
Some of the doubts about the consensus are powerfully
supported by studies on small aquatic organisms. Reviewing a large literature on metabolic rates in aquatic invertebrates and algae, Patterson [31] deployed chemical
engineering principles to explain why the b values ranged
from about 0.3 to 1.2 in these taxa (his Table 1 provides
an excellent summary). Assuming that the delivery of
nutrients to each organism entails diffusion through a
boundary layer, Patterson showed how water movements
and organism size might affect such delivery and hence
determine metabolic rate. Using simple geometrical models of organisms (plates, cylinders and spheres), he
derived b values ranging from 0.31 to 1.25, more or less
consistent with the experimental values.

Patterson plotted two dimensionless numbers against
each other, viz. Sherwood number, Sh = hmW/D, where hm
= mass transfer coefficient, W = characteristic dimension
of organism and D = diffusivity; and Reynolds number (a
function of organism size), Re = ρUW/µ, where ρ = density, U = water flow speed and µ = coefficient of viscosity.
The graphs, which had the form Sh = c.Red, where d = 0.5
for ideal laminar flow and 0.8 for turbulent flow (c is a
constant of proportionality), revealed the relative importance of diffusion and mass transfer (convective movement) in the supply of materials. Patterson was able to
derive an expression for hm, and was thus able to relate the
supply of materials to body mass.

http://www.tbiomed.com/content/1/1/13

The two main attractions of this model are (1) good agreement with a wide range of data and (2) derivation from
basic physical principles without ad hoc biological or
other assumptions. Patterson's approach has implicit support in the literature: Coulson [32] used chemical engineering principles to argue that mammalian metabolic
rates are supply-limited, but he did not develop the argument in mathematical detail. However, Patterson's model
has drawbacks. First, it is hard to see how his reasoning
can be generalised to other taxa, notwithstanding Coulson's proposal (discussed in a later section). Second, by
focusing on diffusion and convective mass transfer, he
ignored active processes in the uptake of materials, which
are likely to dominate in many organisms. Third, he
assumed that metabolism in general is supply-limited; in
homoiotherms at least, it is more nearly demand-limited
under resting conditions, though even this is an oversimplification.
The Patterson model has not been given much attention
by other investigators in the field and perhaps it deserves
more consideration. Despite its inherent limitations (it is
exclusively concerned with small aquatic eukaryotes) it is
a potentially fruitful contribution to biophysics.
Scaling of metabolic rate with surface-to-mass ratio
Several workers accept the reality of allometric scaling but
question the value b = 0.75, which a consensus of physiologists has accepted since the 1960s. Many of these sceptics
claim that the "true" value of b is 0.66 or 0.67 because the
principal determinant of metabolic scaling is the surfaceto-volume ratio of the organism; hence, assuming constant body density, the surface-to-mass ratio. The first
study to suggest this explanation for the mass dependence
of B is attributed to Rubner [33], who studied metabolic
rates in various breeds of dog. Heusner [34] reported that
b is approximately 0.67 for any single mammalian species
and suggested that the interspecies value of 0.75 is a statistical artefact. Feldman and McMahon [35] disagreed, but
Heusner sustained his position in subsequent articles. For
instance, reviewing a substantial body of published data
[36], he argued that metabolic rate data for small and
large mammals lie on parallel regression lines, each with
a gradient of approximately 0.67 but with different intercepts (i.e. values of a, termed the "specific mass coefficients"). Hayssen and Lacy [37] found b = 0.65 for small
mammals and b = 0.86 for large ones, again suggesting
that b = 0.75 is a cross-species "average" with no biological
significance; but it is questionable whether their data were
measurements of standard metabolic rate in all cases.
McNab [13] reported lower values: 0.60 and 0.75, respectively. Heusner [36] reasoned that if a few large mammals
are added to a sample of predominantly small ones, a single regression line for all the data might have a gradient

Page 3 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

around 0.75. This, however, is misleading, as the following paragraphs will argue.
According to Heusner, the ratio B/M0.67 is a mass-independent measure of standard metabolism. Variations
indicate the effects of factors other than body mass. Other
workers broadly share Heusner's opinion (see e.g. [12] for
review and [38] for a good recent exemplar). Bartels [21]
found a value of 0.66 for mammals; Bennett and Harvey
[39] reported 0.67 for birds. Of course, if B varies as M0.67,
the interesting problem is not the index (b) in the Kleiber
equation but the allegedly constant relationship between
specific mass coefficient (a) and body size. This point was
developed by Wieser [40], who distinguished the ontogeny
of metabolism, which comprises several phases but follows the surface rule (M0.67) overall, from the phylogeny of
metabolism, which concerns the mass coefficients (a).
Following Heusner's argument, Wieser [40] wrote the
allometric power law in the form B = anM0.66 and deduced
that the specific mass coefficient an = aM0.09. Here, a is an
interspecific mass coefficient (3.34 w in mammals if M is in
kg). Another difficulty with this type of explanation lies in
the calculation of body surface area; the Meeh coefficient,
k, where surface area = kM0.67, is difficult to measure unequivocally but is generally taken as ~10 (see [3]). Yet
another possible difficulty was identified by Butler et al.
[41], who questioned Heusner's dimensional analysis
argument and concluded that no version of Kleiber's law
(i.e. no value of b that is constant over a range of species)
could be substantiated by his approach.
The claim that b = 0.67 remains a minority view. Those
who accept it are faced with the twin difficulties of (i)
establishing that their estimates of surface area are correct
and (ii) explaining why, in Wieser's notation, an = aM0.09.
Moreover, even if such arguments as Heusner's are valid
for homoiotherms, it is hard to justify their extrapolation
to poikilothermic animals, plants and unicellular organisms, all of which are held by consensus to fit Kleiber's law
(but see the two preceding sections). Why should temperature fluxes across the body surface be the main determinants of metabolic rate in poikilotherms, particularly
microorganisms? Even in mammals, maintenance of
body temperature might not be the main contributor to
energy turnover at rest (see later). Contrary to the view of
Dodds et al. [16], therefore, b = 0.67 cannot be treated as
a "null hypothesis".
Throughout the remainder of this article, the consensus
position will be assumed: Kleiber's law is valid for a wide
range of organisms, and b = 0.75. This assumption is
made tacitly and provisionally and does not imply dismissal of the foregoing sceptical arguments; but a field can
only be reviewed coherently from the consensus point of
view.

http://www.tbiomed.com/content/1/1/13

McMahon's model [42]
A vertical column displaced by a sufficiently large lateral
force buckles elastically. The critical length of column, lcr,
= k(E/ρ)1/3d2/3, where d = column diameter, E = Young's
modulus and ρ = density. If E and ρ are constant then lcr3
= cd2, where c is a constant of proportionality. McMahon
[42] applied this reasoning to bone dimensions for stationary quadrupeds. In a running quadruped the limbs
support bending rather than buckling loads but the vertebral column receives an end thrust that generates a buckling load. It follows that all bone proportions change in
the same way with animal size. The mass of a limb, wl, =
αld2, where α is a constant. If wl is proportional to M, as it
generally must be, then M = βld2, where β is another constant. Hence (given the above relationship between l and
d) M is proportional to l4, implying that l is proportional
to wl1/4; hence d is proportional to wl3/8, or M3/8. Empirical support for this relationship appeared in [43].

McMahon [42] also applied this argument to muscles. The
work done by a contracting muscle, W, is proportional to
σA∆l, where σ is tensile strength, A is the cross-sectional
area and ∆l is the length change during contraction. The
power developed, W/t (t = time), is therefore σA∆l/∆t.
Since σ and ∆l/∆t are roughly constant and independent
of species, W/t varies with A; and since A is proportional
to d2, W/t it is proportional to d2, and therefore to (M3/8)2
= M3/4. If this deduction applies to any skeletal muscle (as
seems plausible), then it applies to the entire set of metabolic variables supplying the muscular system with nutrients and oxygen. Hence, B varies as M3/4. A broadly
comparable but simpler argument was advanced by Nevill
[44]; large mammals have proportionately more muscle
mass than smaller ones. If the contribution of the muscle
to B (which Nevill assumes is proportional to M) is partialled out, then the residual B is proportional to M2/3.
Nevill's paper is seldom cited.
One difficulty with McMahon's model is that little of the
energy turnover under conditions of standard metabolic
rate measurement entails muscle contraction. The model
might still be valid if maximum metabolic rate followed
the same allometric scaling law as B; this has been widely
believed, and Taylor et al. [45] adduced evidence for it.
However, recent detailed studies [46-48] indicate that
maximum metabolic rate in birds and mammals scales as
M0.88, not M0.75, although there are disagreements about
whether aerobic capacity determines the allometry of
maximum metabolic rate [48,49]. Weibel [50] presented
a large set of data to this effect. (On the other hand, there
are reports that in birds the index decreases rather than
increases with increasing metabolic output, e.g. [58].)
Another drawback of the McMahon model is that it cannot apply to organisms without muscles, such as protists.
This perhaps explains why McMahon's elegant deduction

Page 4 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

has been largely ignored in recent debates about Kleiber's
law.
The Economos model [51]
An increased gravitational field increases energy metabolism in animals [52,53]. Work against gravity is proportional to M1.0. If maintenance metabolism were related to
surface area (proportional to M0.67) then a combination
of the two effects, surface-to-mass ratio and work against
gravity, might explain the observed M3/4 relationship. This
model [51] is difficult to assess: it is not clear why the two
proposed factors, surface area dependence and gravitational loading, should combine for all animals (and other
taxa) in just the right proportions to generate a 0.75power dependence on body mass. To take just one example, aquatic microbes are more affected by Brownian
motion than by gravity, so why should they show the
same balance between surface-to-mass ratio and gravitational effects as mice or elephants? Pace et al. [54] suggested that the Economos model could be critically tested
under conditions of weightlessness in space. No corroboration (or refutation) by studies on astronauts has been
reported.
Allometric scaling in cells and tissues
Before more recent models purporting to explain Kleiber's
law are discussed, some comments are needed on scaling
of metabolism at the organ, tissue and cell levels. Belief
that the Kleiber relationship can be explained in terms of
the inherent properties of the cells dates from the 1930s
[3,55] and persists (e.g. [56,57].

Standard metabolic rate (B) is usually measured as oxygen
consumption rate, which correlates with nutrient utilization [9,15] and rates of excretion of nitrogenous and other
wastes [2]; so research in the field has been dominated by
respiratory studies. Lung volume, trachaeal volume, vital
capacity and tidal volume all scale as M but respiratory frequency varies as M-0.31, ventilation rate as M0.77 and oxygen consumption rate as M0.72 [58-60]. All mammals
extract a similar percentage of oxygen (~3%) from
respired air [9]. The significance of "pulmonary diffusion
capacity" has been debated; it scales as M1.0 so it is disproportionate in bigger animals [17,61-65].
Stahl [60] described the scaling of cardiovascular and haematological data. Blood haemoglobin concentration is
the same for all mammals except those adapted to high
altitudes. Blood volume is ~6–7% of body volume for all
mammals except aquatic ones. Erythrocyte volume varies
with species but bears no obvious relationship to M. The
oxygen affinity of haemoglobin varies with body size,
being lower in smaller mammals, which unload oxygen to
their tissues more rapidly. Capillary density is more or less
constant in mammals with bodies larger than a rat's,

http://www.tbiomed.com/content/1/1/13

though it is greater in the smallest mammals [65]. The
heart accounts for ~0.6% of body mass in all mammals
[66]. Heart rate scales as M-0.25, cardiac output as M0.81
(60) and circulation time as M0.25. The energy cost of supplying the body with 1 ml of oxygen is similar for all
mammals [15].
Standard metabolic rate has two main components: service functions, e.g. the operation of heart and lungs; and
cellular maintenance functions, e.g. protein and nucleic
acid turnover (e.g. [67]). Krebs [68] elucidated this second
component by studying tissue slices; his investigation has
since been extended. Oxygen consumption per kg
decreases with increasing M in all tissues, but tissues do
not all scale identically. Horse brain and kidney have half
the oxygen consumption rates of mouse brain and kidney
but the difference between these species in respect of liver,
lung and spleen is 4-fold [68-70]. Metabolic rate in liver
scales as M0.63; for some organs the exponent is closer to
1.0; the sum of oxygen consumption rates over all tissues
gives – approximately – the expected 0.75 index [71]. The
difficulty of recalculating B from tissue-slice data is considerable, so the Martin and Fuhrman calculation [71] has
wide confidence limits. Spaargen [72] suggested that tissues that use little oxygen constitute different percentages
of body mass in large and small mammals, leading to a
distortion of the surface law (B = M2/3), which would otherwise be valid. More recently, however, Wang et al. [73]
repeated the Martin and Fuhrman calculation using
improved data, and found impressive support for the consensus B = M3/4.
Cells of any one histological type are size-invariant among
mammals but allometric scaling is reported at the cellular
level; e.g. the metabolic rate of isolated hepatocytes scales
as M-0.18 [74]. Numbers of mitochondria per gram of liver
(or per hepatocyte), however, scale as M-0.1 [75,76]. The
apparent discrepancy between these values might be illusory (cf. [77]), or it might indicate a greater proton leak in
mitochondria from livers of smaller animals [78] or
allometry in redox slip [79]. Also, larger animals have
smaller inner mitochondrial membrane surface areas (the
scaling is M-0.1) and different fatty acid compositions [71].
The discrepancy between the scalings of hepatocyte and
whole-body metabolism is probably explained by the
decrease in liver mass, which scales as M0.82 [75,80]. Combining liver mass with hepatocyte oxygen consumption,
the derived scaling for liver metabolism is M0.82.M-0.18 =
M0.64, consistent with the experimental tissue-slice data
(M0.63; see above). Combining liver mass with mitochondrial number per hepatocyte gives a similar value [77].
Cytochrome c and cytochrome oxidase contents scale
roughly as M0.75 [81-85]. The allometric scaling of mitochondrial inner membrane area, and the body-size-

Page 5 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

http://www.tbiomed.com/content/1/1/13

related differences in unsaturated fatty acid content,
remain unexplained.

unit mass of tissue vary with the quarter-power of body
size (M), implying the validity of Kleiber's law.

Isolated mammalian cells reportedly attain the same
mitochondrial numbers and activities after several generations in culture, irrespective of the tissue of origin or the
organism's body mass [86-88]. If allometric scaling is lost
at the cellular level after several generations in vitro, then
presumably mitochondrial densities, inner membrane
areas and cytochrome levels somehow become "normalized". This is a readily testable prediction [see [89]], but it
does not appear to have been subjected to critical experiments. If it is corroborated there will be interesting mechanisms to investigate.

The most detailed account of this argument [95] begins
with the reasonable assumption that M scales with LD,
where L is the physical length of the organism and D is its
dimensionality. It proceeds with a theorem: the sum of
flows through all parts of the network, F, is proportional
to the (dimensionless) length multiplied by the metabolic
rate. A quantity measuring the total flow of metabolites
per unit mass of organism is then defined: r1 = F/M. r1
(which has units of inverse time) measures the dependence of the network's geometry on body mass, so it indicates the energy cost of metabolite delivery. Another
parameter, r2, measures the metabolite demand by the tissues: r2 = the dimensionless length of the "service volume"
(the amount of tissue that consumes one unit of metabolite per unit time). It is then deduced that B is proportional to (Mr1/r2)D/(D+1). Provided that r1 and r2 change
proportionately – i.e. supply always matches demand –
then for a three-dimensional organism, Kleiber's law follows. According to Banavar et al. [94], deviations from
Kleiber's law indicate inefficiency or some physiological
compensation process.

The main conclusions from this section are: (a) different
organs make different contributions to the scaling of
whole-organism metabolic rates; (b) differences at the cellular level make relatively small contributions to scaling at
the organ level; (c) these differences at cellular level might
disappear altogether after several generations in culture.
The most striking conclusion is (b). It implies that allometric scaling of metabolic rate does not after all, for the
most part, reside in cellular function but at higher levels
of physiological organisation. If this is the case, then the
alleged applicability of Kleiber's law to unicellular organisms is called into question.
Resource-flow models
Coulson's flow model [42] was mentioned earlier. It
relates tissue or organ oxygen consumption rates to circulation times, i.e. to the rate of supply of oxygen and nutrients, and these scale as M0.25 (see previous section).
Coulson's approach contrasts with traditional biochemical measurements: the principal variable is not the concentration of a resource but the supply rate; metabolic
activity depends on encounter frequency not concentration.
This perspective merits further development, particularly
by extension to the cell internum [89-93]. Obviously, it is
within the cell that the reactant molecules are passed over
the catalysts; and the flow rate increases with the cell's
metabolic activity, as Hochachka [93] cogently described.

However, flow theories advanced to explain Kleiber's law
have not followed this line of argument. Banavar et al.
[94,95] and Dreyer and co-workers [27,96] have shown
that the Kleiber relationship can be deduced from the
geometries of transport networks, without reference to
fluid dynamics. Broadly, these authors argue that as a supply network with local connectivity branches from a single
source (in a mammalian circulatory system, the heart is
the source), the number of sites supplied by the network
increases. Natural selection has optimized the efficiency
of supply. A general relationship can be derived between
body size and flow rate in the network: delivery rates per

This model has been criticized [97] because the assumed
network does not resemble (e.g.) the mammalian circulatory system, where only terminal nodes (capillaries), not
all nodes (as the model implies), are metabolite exchange
sites. Also, the model seems to predict that r1/r2 will
decrease as B rises from standard to maximal; but the best
data suggest the opposite trend (see earlier discussion:
[46-48]). Banavar et al. do not explicitly allow for differences among tissue types, which are considerable (see
above), except perhaps in terms of rather implausible variations among r1/r2 ratios. On the other hand, the model
is simple and flexible and it reflects recent developments
in the physics of networks. If it could be applied to flow at
the cellular level, it might accord with the requirements
discussed at the beginning of this section; though it is difficult to see how this can be achieved.
Rau [98] also advanced a fluid-flow model, but his conception is physical not geometrical. Assuming Pouseille
flow through an array of similar tubes, such as capillaries,
and a roughly constant flow speed, Rau used scaling arguments to derive the relationship t = kM1/4, where t is the
transport time and k is a constant. If the fluid transport
rate (essentially the reciprocal of t) is proportional to B/M,
Kleiber's law follows. However, Rau's model appears to
assume that because metabolic rate is energy per unit
time, it can be equated with the product of fluid volume
flow rate and pressure (since energy is equal to pressure
times volume). This assumption, which appears to be
based exclusively on dimensional analysis, is fallacious.

Page 6 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

Four-dimensional models
Blum [99] observed that the "volume" of an n-dimensional sphere of radius r is V = πn/2rn/Γ(n/2 + 1), and that
A = dV/dr = nπn/2rn-1/Γ(n/2 + 1). Here, Γ(n) is the gammafunction such that Γ(n + 1) = nn, Γ(2) = 1 and Γ(3/2) = π1/
2/2. Suppose two objects have "volumes" V and V and
1
2
"areas" A1 and A2. From the foregoing, A1/A2 = (V1/V2)(n1)/n; so if n = 4, a 3/4-power relationship between "volumes" (hence, masses?) emerges from a familiar mathematical principle. Might Kleiber's law therefore follow
from a four-dimensional description of organisms?
Speakman [100] pointed out that if n = 4, then A is volume (it has three dimensions) and V is hypervolume, the
biological significance of which is obscure. However,
West et al. [88,101,102] have indeed proposed a fourdimensional model to explain the Kleiber relationship,
and considerable claims have been made for their
account.

This model addresses the supply of materials (particularly
oxygen) through space-filling fractal networks of branching tubes. It assumes that as a result of natural selection,
organisms maximize their use of resources. The initial
account [101] assumed that energy dissipation is minimised at all branch-points in the network and that the terminal branches are size-invariant (for instance, blood
capillaries are the same lengths and diameters in mice and
elephants). Kleiber's law and analogous scalings were
deduced from these assumptions. In particular, the threequarters-power exponent was shown to be inherent in the
geometry of a branching network that preserves total
cross-sectional area at each branch point. The circulatory
systems of large animals such as mammals are not exactly
area-preserving, but West et al. [101] reasoned that this
objection could be circumvented by considering the pulsatile flow generated in the larger arteries by the action of
the heart.
A second, simpler account [102] developed the model
from a geometrical basis. The crucial feature of the
branching network is the size-invariance of the terminal
units. The effective exchange area, a, is a function of the
element lengths at each level of the hierarchy, but one of
these, the terminal one (l0), is invariant. Writing Φ as a
dimensionless function of the (dimensionless) ratio l1/l2
leads to
a (l0, l1, l2,...) = l12Φ(l0/l1, l2/l1...)
Introducing a scaling factor, λ, leads to
a (l0, l1, l2,...) = λ2l12 Φ(l0/λ l1, l2/l1...)
which is not proportional to λ2 because l0 is fixed. The
dependence of Φ on λ is not known a priori, but it can be

http://www.tbiomed.com/content/1/1/13

parameterized as Φ(l0/λ l1, l2/l1...) = λεΦ(l0/l1, l2/l1...),
where ε is between 0 and 1. This power law reflects the
fractal character of the network's hierarchical organization. Similar reasoning is applied to body volume, hence
body mass, and the following expression for the exchange
surface area is derived:a = kMr, r = (2 + ε)/(3 + ε + ζ),
where k is a constant and ζ (0 < ζ < 1) is an arbitrary exponent of length, just as ε is an arbitrary exponent of area. If
natural selection has acted to maximize the scaling of a,
then ε must tend to 1 and ζ to 0. This gives r = 0.75. If a
limits the supply of oxygen and nutrients, and hence
determines standard metabolic rate, then B is proportional to a and Kleiber's law follows.
The model has several attractions: it derives from wellestablished physical principles, invokes natural selection
and is mathematically impeccable. It implies that cells
and organelles transport materials internally along spacefilling fractal networks rather than by "diffusion", which
seems correct [83,85,86,103]. The self-similarity of these
transport networks is emphasized particularly in [88]. The
dimensionalities of effective exchange surfaces, a, are predicted to be closer to 3 than 2; empirically, the microscopic convolutions of surfaces such as the mammalian
intestinal mucosa are well known. The mass of the smallest possible mammal is deduced and shown to be close to
the mass of the shrew. Other approaches to exchange networks, assuming minimum energy expenditure and scaleinvariance, have led to similar models [104]. The model
can be adapted, with no loss of rigour, to new data: Gillooly et al. [105] showed that the fractal supply network
principle can be combined with simple Boltzmann kinetics to explain the effects of both body mass and temperature on metabolic rates. Since mass and temperature are
the primary determinants of many physiological and ecological parameters, this work suggests that the model [88]
could revolutionize biology.
This is an impressive range of successes. However, West
and his co-workers make claims that are less compelling.
The observation that cytochrome oxidase catalytic rates fit
the same allometric curve as whole-organism metabolic
rates is claimed as corroboration. However, cytochrome
oxidase is not an organism, or a cell: it does not have a
metabolic rate. It is also debatable whether mitochondria
can be said to have "metabolic rates". (In contrast, Hochachka and Somero [106] noted that oxygen turnover in
the whole biosphere can be fitted to the same curve; but
they recognized this as "a contingent fact with no biological significance".) Also, the explanation derived by West
and his colleagues for the alleged body-mass-invariance of
the metabolic rates of cultured cells (see earlier) is mathe-

Page 7 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

matically neat, but it leads to no experimentally testable
predictions, and the heterogeneous data sources cited in
this context make the explicandum itself unconvincing.
Finally, the model is said to explain the quarter-power
scalings of a wide range of biological variables other than
metabolic rate, including population densities of trees
[19] and carnivorous animals [107], plant growth rates,
vascular network structure and maturation times
[18,108], and life-spans [88]. It is not clear why any of
these variables should depend on the fractal geometries of
space-filling supply networks, still less on metabolic rates;
though there is widespread interest in the application of
scaling laws in ecology, for instance in modelling biodiversity [109] and food webs [110].
Moreover, there are definite flaws in the model:(1) If West et al. were correct, maximal and standard metabolic rates should both scale as M0.75. The weight of evidence suggests that maximum rate in homoiotherms
scales as M0.88 (see earlier discussion [46-49] and following section).
(2) During maximal energy output by an organism, the
supply of material is likely to be limiting. For example, in
mammals, muscle contraction is responsible for most of
the energy turnover at maximum output and it is generally
believed that the rate is limited by oxygen supply (if
anaerobic capacity is ignored). However, under standard
metabolic rate conditions, energy demand is generally
more significant, i.e. for the service and cellular maintenance functions mentioned previously. Therefore, it is not
clear why the geometry and physics of the supply system
should predict the allometric scaling of standard rather
than maximal metabolic rate. ("Supply" and "demand"
under conditions of maximal aerobic metabolism are
complex terms because many physiological steps are
involved. The extent to which each step limits the maximum metabolic rate might be quantifiable by a suitable
extension of metabolic control analysis [111]; this
remains an active research area to which West et al.
scarcely refer.)
(3) The mathematical derivations given in [101] are idealisations, but they do not seem to allow for large deviations
from b = 0.75. However, there are often wide differences
among empirical b values, as discussed earlier; these were
addressed in, for example, [18] and [31]. Also, the model
does not account, or allow, for the differences in allometric scaling among mammalian tissues and organs
[66,73,80].
(4) West et al. accept that some of their proposed hierarchical supply networks might be "virtual" (as in mitochondria) rather than explicit (as in mammalian blood

http://www.tbiomed.com/content/1/1/13

circulation), but it is not clear why such networks must
always have the same geometry. For instance, why should
the intracellular network discussed by Hochachka [93]
show area-preserving branching? There is no evidence that
it does. Moreover, the "flow" of reductants through mitochondria presumably takes place in the plane of the inner
membrane, which has one dimension fewer than (say)
the mammalian circulatory system, so even if mitochondria can be said to have "metabolic rates", the 0.75-power
law cannot apply here; yet, allegedly, it does apply.
These difficulties show that the West et al. model, despite
its impressive economy, elegance, consistency and range,
cannot be accepted unreservedly in its present form. The
very generality, or "universality", of this model has made
it suspect for some biologists [25]. The implication that it
reveals a long-suspected universal biological principle
implicit in Kleiber's law has ensured its attraction for others [14].
The model of Darveau and co-workers [112]
This group elaborated a multi-cause rather than a singlecause account of allometric scaling. Their "allometric cascade" model holds that each step in the physiological and
biochemical pathways involved in ATP biosynthesis and
utilization has its own scaling behaviour and makes its
own contribution (defined by a control coefficient
between 0 and 1) to the whole-organism metabolic rate.
Thus, many linked steps rather than a single overarching
principle account for Kleiber's law.

This idea is inherently plausible, and the model is attractive because it draws upon recent advances in metabolic
control analysis in biochemistry [111] and physiology
[113]. It emphasises that standard metabolic rate is determined by energy demand, not supply; and it predicts an
exponent for maximal metabolic rate in mammals
between 0.8 and 0.9, rather than 0.75, which agrees with
experimental findings [46-49] and the data cited by
Weibel [50]. Implicitly – though the authors do not
emphasize this – it seems capable of explaining b values
that are far from 0.75 (cf [31]). It is hardly surprising,
therefore, that many responses to the Darveau et al. model
have been positive [e.g. [114]].
However, Darveau et al. made no attempt to explain why
the values of b are typically around 0.75, as West et al. and
others have done. The model is phenomenological, not
physical and mathematical; their equations are not
derived from any fundamental principle(s). Moreover,
their data cover only some three orders of magnitude of
body mass, whereas many studies have involved much
wider ranges. This might make their overall b values misleading [103] or, alternatively, more credible [18]. When
their equations are applied to a mass range of eight orders

Page 8 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

of magnitude, different b values are obtained, not necessarily consistent with published data; but on the other
hand, the published data might not be correct.
In the first published account of this model [112] the
mathematical argument was flawed. The basic equation
was given in the form B = aΣciMb(i), where a is a constant
coefficient, cI is the control coefficient of the ith step in the
cascade and b(i) is the exponent of the ith step. By definition, the sum of all the cI values is unity. Darveau et al. did
not derive this equation; they stated it. They also stated
that the overall exponent, the b term in the Kleiber equation, is a weighted average of all the individual b(i) values,
the weighting being determined by the relevant control
coefficients. It has been suggested that this leads to untenable inferences. For example, since the units of B and a are
fixed, the units of cI must depend on those of b(i); but by
definition, both b(i) and cI must be dimensionless. Also,
according to the basic equation, the contribution made by
each step to the overall metabolic rate depends on the
units in which body mass is measured. If this criticism is
valid then it is impossible to evaluate the model as it
stands, because any attempt to align its predictions with
experimental data would be meaningless. Another reservation about this model is that it does not purport to
apply to all taxa, as the West et al. model does; it relates
only to metazoa, and in particular to homoiotherms.
However, most of the relevant data in the literature concern homoiotherms.
A subsequent publication from this group [115] re-stated
the basic equation in the form B = aΣcI(M/m)b(i), where
the constant a is described as the "characteristic metabolic
rate" of an animal with characteristic body mass m. This
eliminates the problem of mass units, because the mass
term has been rendered dimensionless; and it is mathematically simple to express control coefficients in dimensionless form. The revised equation might therefore be
immune to some of the criticisms levelled at its predecessor. However, some of the earlier reservations remain: the
equation remains phenomenological, not physical or geometrical; and the restriction in its range of application is
explicit. Nevertheless, these considerations by no means
invalidate the model. Indeed, it is supported by data from
experiments in exercise physiology [116].
The models of Darveau et al. [112,115], Banavar et al.
[94,95] and West et al. [88,102] all have attractive features; but they all have flaws, and they cannot be reconciled with one another. If the positive contributions to
biology that these models represent could be further
developed, and their defects eliminated, could they be
harmonized? If so, the advancement of our understanding
would be considerable.

http://www.tbiomed.com/content/1/1/13

Conclusions
Several explanatory or quasi-explanatory models have
been proposed for the allometric scaling of metabolic rate
with body mass. Most of them have significant attractions,
particularly the most recent ones, but none of them can be
unreservedly accepted. The variability of experimental
data leaves room for doubt that Kleiber's law is universally
or even widely applicable in biology [17,117], yet most
workers in the field presume that it is. Even if such doubts
are set aside, no model has yet addressed every relevant
issue. For example, the biochemical reasons for the allometric scalings of mitochondrial inner membrane areas and
unsaturated fatty acid contents, and the direct proportionality of "pulmonary diffusion capacity" to body mass,
remain unexplained. Despite the continuing controversy
in the field, the consensus remains, and practical use has
been made of Kleiber's law, for example in making
numerical predictions of anatomical and physiological
parameters for veterinary applications [118]. Perhaps the
last word should be given to Bokma [119], whose most
recent paper explores the power-scaling of metabolic rate
to body mass (b) on an intra-specific basis from a total of
113 species. He came to the conclusion that there was no
single universal value of b. This evidence alone must make
us more sceptical of there being some unifying law
involved that demands that b holds close to 0.75. There is
clearly no consensus otherwise Nature, Science and the
Proceedings of the National Academy of Sciences USA would
cease to publish so regularly many of the articles to which
we have referred. The subject is not only unresolved, but
remains very much within the general interest of biologists.
Kleiber's law remains a fascinating mystery; possibly a
delusion, possibly a widespread or even ubiquitous biological phenomenon for which no entirely satisfactory
account has yet been offered. Recent developments,
though mutually conflicting as they stand, have the
potential to lead to new insights and to uncover one or
more general biological principles that will have a profound impact on our understanding of the living world.

Acknowledgements
We are indebted to Raul Suarez, Jim Clegg, John Porteous and George
Somero for their critical comments, helpful discussions and encouragement.

References
1.
2.

3.
4.

Kleiber M: Body size and metabolism. Hilgardia 1932, 6:315-353.
Brody S, Procter RC, Ashworth US: Basal metabolism, endogenous nitrogen, creatinine and neutral sulfur excretion as
functions of body weight. Univ Missouri Agric Exp Sta Res Bull 1934,
220:1-40.
Benedict FG: Vital Energetics: a Study in Comparative Basal
Metabolism. Washington DC: Carnegie Institute of Washington;
1938.
Hemmingsen AM: The relationship of standard (basal) energy
metabolism to total fresh weight of living organisms. Rep
Steno Mom Hosp Copenhagen 1950, 4:1-58.

Page 9 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.

Hemmingsen AM: Energy metabolism as related to body size
and respiratory surfaces, and its evolution. Rep Steno Mom Hosp
Copenhagen 1960, 9:1-110.
Zeuthen E: Oxygen uptake as related to body size in organisms. Quart Rev Biol 1953, 28:1-12.
Kleiber M: The Fire of Life. An Introduction to Animal Energetics. New York: Wiley; 1961.
Schmidt-Nielsen K: Scaling in biology: the consequences of size.
J Exp Zool 1975, 194:287-308.
Schmidt-Nielsen K: Scaling: Why is Animal Size So Important?
Cambridge: Cambridge University Press; 1984.
Peters RH: The Ecological Implications of Body Size. Cambridge: Cambridge University Press; 1983.
McMahon TA, Bonner JT: On Size and Life. New York: Scientific
American Books; 1983.
Calder WA: Size, Function, and Life History. Cambridge, Ma:
Harvard University Press; 1984.
McNab BK: Complications inherent in scaling the basal rate of
metabolism in mammals. Quart Rev Biol 1988, 63:25-54.
Niklas KJ: Plant Allometry: the Scaling of Form and Process.
Chicago: University of Chicago Press; 1994.
Brown JH, West GB, eds: Scaling in Biology. New York: Oxford
University Press; 2000.
Dodds PS, Rothman DH, Weitz JS: Re-examination of the '3/4law' of metabolism. J Theoret Biol 2001, 209:9-27.
Feldman HA: On the allometric mass exponent, when it exists.
J Theoret Biol 1995, 172:187-197.
Atanasov AT, Dimitrov BD: Changes of the power coefficient in
the 'metabolism-mass' relationship in the evolutionary process of animals. Biosystems 2002, 66:65-71.
Niklas KJ, Enquist BJ: Invariant scaling relationships for interspecific plant biomass production rates and body size. Proc
Natl Acad Sci USA 2001, 98:2922-2927.
Enquist BJ, Niklas KJ: Invariant scaling relationships across treedominated communities. Nature 2001, 410:655-660.
Bartels H: Metabolic rate in mammals equals the 0.75 power
of their body weight. Exp Biol Med 1982, 7:1-11.
Andersen HT, ed: The Biology of Marine Mammals. New York: Academic Press; 1969.
Elgar MA, Harvey PH: Basal metabolic rates in mammals:
allometry, phyogeny and ecology. Funct Ecol 1987, 1:25-36.
Economos AC: Elastic and/or geometric similarity in mammalian design? J Theoret Biol 1983, 103:167-172.
Secor SM, Diamond J: Determinants of the postfeeding metabolic response of Burmese pythons, Python molurus. Physiol
Zool 1997, 70:202-212.
McKenzie D: New clues as to why size equals destiny. Science
1999, 284:1607-1609.
Dreyer O: Allometric scaling and central source systems. Phys
Rev Lett 2001, 87(03810):1-3.
Prothero J: Scaling of energy metabolism in unicellular organisms: a re-analysis. Comp Biochem Physiol A 1986, 83:243-248.
Smith RJ: Allometric scaling in comparative biology: problems
of concept and method. Am J Physiol 1984, 246:R152-R160.
Kaitaniemi P: Testing the allometric scaling laws. J Theoret Biol
2004, 228:149-153.
Patterson MR: A mass transfer explanation of metabolic scaling relationships in some aquatic invertebrates and algae. Science 1992, 255:1421-1423.
Coulson RA: Metabolic rate and the flow theory: a study in
chemical engineering. Comp Biochem Physiol 1986, 84:217-229.
Rubner M: Über die Einflus der Körpergrösse auf Stroff und
Kraftwechsel. Z Biol 1883, 19:535-562.
Heusner AA: Energy metabolism and body size. I. Is the 0.75
mass exponent of Kleiber's equation a statistical artifact?
Respir Physiol 1982, 48:1-12.
Feldman HA, McMahon TA: The 3/4 mass exponent for energy
metabolism is not a statistical artifact:. Respir Physiol 1983,
52:149-163.
Heusner AA: Size and power in mammals. J Exp Biol 1991,
160:25-54.
Hayssen V, Lacy RC: Basal metabolic rates in mammals: taxonomic differences in the allometry of BMR and body mass.
Comp Biochem Physiol 1985, 81A:741-754.

http://www.tbiomed.com/content/1/1/13

38.
39.
40.

41.
42.
43.
44.
45.

46.
47.
48.

49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.

62.
63.

64.

White CR, Seymour RS: Mammalian basal metabolic rate is
proportional to body mass2/3. Proc Natl Acad Sci USA 2003,
100:4046-4049.
Harvey PH, Bennett PM: Evolutionary biology. Brain size, energetics, ecology and life history patterns. Nature 1983,
306:314-315.
Wieser W: A distinction must be made between the ontogeny
and the phylogeny of metabolism in order to understand the
mass exponent of energy metabolism. Respir Physiol 1984,
55:1-9.
Butler JP, Feldman HA, Fredberg JJ: Dimensional analysis does not
determine a mass exponent for metabolic scaling. Am J Physiol
1987, 253:R195-R199.
McMahon TA: Using body size to understand the structural
design of animals: quadrupedal locomotion. J Appl Physiol 1975,
39:619-627.
Brody S: Bioenergetics and Growth, with Special Reference to the Efficiency
Complex in Domestic Animals New York: Reinholt; 1945.
Nevill AM: The need to scale for differences in body size and
mass: an explanation of Kleiber's 0.75 mass exponent. J Appl
Physiol 1994, 77:2870-2873.
Taylor CR, Maloiy GMO, Weibel ER, Longman VA, Kamau JMZ, Seeherman HJ, Heglund NC: Design of the mammalian respiratory
system. III Scaling maximum aerobic capacity to body mass:
wild and domestic mammals. Respir Physiol 1981, 44:25-37.
Hinds DS, Baudinette RV, MacMillen R, Halpern EA: Maximum
metabolism and the aerobic factorial scope of endotherms. J
Exp Biol 1993, 182:41-56.
Bishop CM: The maximum oxygen consumption and aerobic
scope of birds and mammals: getting to the heart of the matter. Proc Roy Soc London B 1999, 266:2275-2281.
Weibel ER, Bacigalupe LD, Schmitt B, Hoppeler H: Allometric scaling of maximal metabolic rate in mammals: muscle aerobic
capacity as determinant factor. Respir Physiol Neurobiol 2004,
140:115-132.
Koteja P: On the relation between basal and maximum metabolic rate in mammals. Comp Biochem Physiol 1987, 87:205-208.
Weibel ER: Symmorphosis: on Form and Function in Shaping Life Cambridge, Ma: Harvard University Press; 2000.
Economos AC: Gravity, metabolic rate and body size of mammals. Physiologist 1979:S71.
Smith AH: Physiological changes associated with long-term
increases in acceleration. COSPAR: Life Sci Space Res 1976,
14:91-100.
Smith AH: The roles of body mass and gravity in determining
the energy requirements of homoiotherms. COSPAR: Life Sci
Space Res 1978, 16:83-88.
Pace N, Rahlmann DF, Smith AH: Scaling of metabolic rate on
body mass in small laboratory animals. Physiologist
1981:S115-S116.
Holmes E: The Metabolism of Living Tissues. Cambridge: Cambridge University Press; 1938.
Altman PL, Dittmer DS: Metabolism Bethesda, Maryland: Fed Am Soc
Exp Biol Press; 1968.
Kozlowski J, Konarzewski M, Gawelczyk AT: Cell size as a link
between noncoding DNA and metabolic rate scaling. Proc Natl
Acad Sci USA 2003, 100:14080-14085.
Lasiewski RC, Calder WA: A preliminary allometric analysis of
respiratory variables in resting birds. Respir Physiol 1971,
11:152-166.
Tenney SM, Bartlett D: Comparative quantitative morphology
of the mammalian lung: trachea. Respir Physiol 1967, 3:130-135.
Stahl WR: Scaling of respiratory variables in mammals. J Appl
Physiol 1967, 22:453-460.
Gehr P, Mwangi DK, Ammann A, Maloiy GMO, Taylor CR, Weibel
ER: Design of the mammalian respiratory system. V. Scaling
morphometric pulmonary diffusing capacity to body mass:
wild and domestic mammals. Respir Physiol 1981, 44:61-86.
O'Neil JJ, Leith DE: Lung diffusing capacity scaled in mammals
from 25 g to 500 kg. Fed Proc 1980, 39:972.
Weibel ER, Taylor CR, Gehr P, Hoppeler H, Mathieu O, Maloiy GMO:
Design of the mammalian respiratory system. IX. Functional
and structural limits for oxygen flow. Respir Physiol 1981,
44:151-164.
Taylor CR, Weibel ER: Design of the mammalian respiratory
system. I. Problem and strategy. Respir Physiol 1981, 44:1-10.

Page 10 of 11
(page number not for citation purposes)

Theoretical Biology and Medical Modelling 2004, 1:13

65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.

77.
78.
79.
80.
81.

82.
83.
84.

85.

86.
87.

88.
89.
90.

Schmidt-Nielsen K, Pennycuik P: Capillary density in mammals in
relation to body size and oxygen consumption. Am J Physiol
1961, 200:746-750.
Prothero J: Heart weight as a function of body weight in mammals. Growth 1980, 43:139-50.
Buttgereit F, Brand MD: A hierarchy of ATP-consuming processes in mammalian cells. Biochem J 1995, 312:163-167.
Krebs HA: (1950) Body size and tissue respiration. Biochim Biophys Acta 1950, 4:249-269.
Kleiber M: Body size and metabolism of liver slices in vitro.
Proc Soc Exp Biol Med 1961, 48:419-423.
Couture P, Hulbert JA: On the relationship between body mass,
tissue metabolic rate and sodium pump activity in mammalian liver and kidney cortex. Am J Physiol 1995, 268:R641-R650.
Martin AW, Fuhrman FA: The relationship between summated
tissue respiration and metabolic rate in the mouse and dog.
Physiol Zool 1955, 28:18-34.
Spaargen DH: Metabolic rate and body size: a new view on the
'surface law' for basic metabolic rate. Acta Biotheor 1994,
42:263-269.
Wang Z, O'Connor T, Heshk S, Heymsfield SB: The reconstruction of Kleiber's law at the organ-tissue level. J Nutr 2001,
131:2967-2970.
Porter RK, Brand MD: Cellular oxygen consumption depends
on body mass. Am J Physiol 1995, 269:R226-R228.
Smith RE: Quantitative relations between liver mitochondria
metabolism and total weight in mammals. Ann New York Acad
Sci 1956, 62:403-422.
Schwertzmann K, Hoppeler H, Kayar SP, Weibel EP: Oxidative
capacity of muscle and mitochondria: correlation of physiological, biochemical and morphological characteristics. Proc
Natl Acad Sci USA 1989, 86:1583-1587.
Else PL, Hulbert AJ: Mammals: an allometric study of metabolism at tissue and mitochondrial level. Am J Physiol 1985,
248:R415-R421.
Porter RK, Hulbert AJ, Brand MD: Allometry of a mitochondrial
proton leak: influence of surface area and fatty acid composition. Am J Physiol 1996, 271:R1550-R1560.
Brand MD, Chien L-F, Diolez P: Experimental discrimination
between proton leak and redox slip during mitochondrial
electron transport. Biochem J 1994, 310:477-481.
Stahl WR: Organ weights in primates and other mammals. Science 1965, 150:1039-1042.
Drabkin DL: The distribution of the chromoproteins, hemoglobin, myoglobin, and cytochrome c, in the tissues of different species, and the relationship of the content of each
chromoprotein to body mass. J Biol Chem 1950, 182:317-333.
Kunkel HO, Spalding JF, Franciscis G, Futrell MF: Cytochrome oxidase activity and body weight in rats and in species of large
animals. Am J Physiol 1956, 186:203-206.
Jansky L: Total cytochrome oxidase activity and its relation to
basal and maximal metabolism. Nature 1961, 189:921-922.
Guan M-X, Fischel-Ghodsian N, Attardi G: Biochemical evidence
for nuclear gene involvement in phenotype of non-syndromic deafness associated with mitochondrial 12S rRNA mutation. Hum Molec Genet 1996, 5:963-971.
Villani G, Attardi G: In vivo control of respiration by cytochrome c oxidase in wild-type and mitochondrial DNA
mutation-carrying human cells. Proc Natl Acad Sci USA 1997,
94:1166-1171.
Terroine EF, Roche J: La respiration des tissus. I. Production
calorique des homeotherms et intensité de la respiration in
vitro des tissu homologues. Arch Intern Physiol 1925, 24:356-399.
Rumsey WL, Schloss C, Nuutinen EM, Robiolo M, Wilson DF: Cellular energetics and the oxygen dependence of respiration in
cardiac myocytes isolated from adult rat. J Biol Chem 1990,
265:15392-15399.
West GB, Woodruff WH, Brown JH: Allometric scaling of metabolic rate from molecules and mitochondria to cells and
mammals. Proc Natl Acad Sci USA 2002, 99:2473-2478.
Wheatley DN, Clegg JS: What determines the basal metabolic
rate of vertebrate cells in vivo? BioSystems 1994, 32:83-92.
Wheatley DN: On the possible importance of an intracellular
circulation. Life Sci 1985, 36:299-307.

http://www.tbiomed.com/content/1/1/13

91.
92.

93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.

109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.

Clegg JS, Wheatley DN: Intracellular organization: evolutionary
origins and possible consequences to metabolic rate control
in vertebrates. Am Zool 1991, 31:504-513.
Wheatley DN: On the vital role of fluid movement in organisms and cells: a brief historical account from Harvey to
Coulson, extending the hypothesis of circulation. Medical
Hypotheses 1999, 52:275-284.
Hochachka PW: The metabolic implications of intracellular
circulation. Proc Natl Acad Sci USA 1999, 96:12233-12239.
Banavar JR, Naritan A, Rinaldo A: Size and form in efficient transportation networks. Nature 1999, 399:130-132.
Banavar JR, Dalmuth J, Maritan A, Rinaldo A: Supply-demand balance and metabolic scaling. Proc Natl Acad Sci USA 2002,
99:10506-10509.
Dreyer O, Puzio R: Allometric scaling in animals and plants. J
Math Biol 2001, 43:144-156.
Haff PK: Rivers, blood and transportation networks. Nature
2000, 408:159-160.
Rau AR: Biological scaling and physics. J Biosci 2002, 27:475-478.
Blum JJ: On the geometry of four-dimensions and the relationship between metabolism and body mass. J Theoret Biol 1977,
64:599-602.
Speakman JR: On Blum's four-dimensional geometric explanation for the 0.75 exponent in metabolic allometry. J Theoret
Biol 1990, 144:139-141.
West GB, Brown JH, Enquist BJ: A general model for the origin
of allometric scaling laws in biology. Science 1997, 276:122-126.
West GB, Brown JH, Enquist BJ: The fourth dimension of life:
fractal geometry and allometric scaling of organisms. Science
1999, 284:1677-1679.
Agutter PS, Wheatley DN: Random walks and cell size. BioEssays
2001, 22:1018-1023.
Santillan M: Allometric scaling law in a simple oxygen
exchanging network: possible implications on the biological
allometric scaling laws. J Theor Biol 2003, 223:249-257.
Gillooly JF, Brown JH, West GB, Savage VM, Charnov EL: Effects of
size and temperature on metabolic rate. Science 2001,
293:2248-2251.
Hochachka PW, Somero GN: Biochemical Adaptation: Mechanism and Process in Physiological Evolution. Oxford: Oxford
University Press; 2001.
Carbone C, Gittleman JL: A common rule for the scaling of carnivore density. Science 2002, 295:2273-2276.
Enquist BJ: Universal scaling in tree and vascular plant allometry: towards a general quantitative theory linking plant
form and function from cells to ecosystems. Tree Physiol 2002,
22:1045-1064.
Richie ME, Olff H: Spatial scaling laws yield a synthetic theory
of biodiversity. Nature 1999, 400:557-560.
Garlaschelli D, Caldarelli G, Pietronero L: Universal scaling relations in food webs. Nature 2003, 423:165-168.
Fell D: Understanding the Control of Metabolism London: Portland Press;
1999.
Darveau C-A, Suarez RK, Andrews RD, Hochachka PW: Allometric
cascade as a unifying principle of body mass effects on
metabolism. Nature 2002, 417:166-170.
Jones JH: Optimization of the mammalian respiratory system: symmorphosis versus single-species adaptation. Comp
Biochem Physiol 1998, 120B:125-138.
Burness GP: Ecology. Elephants, mice, and red herrings. Science
2002, 296:1245-1247.
Hochachka PW, Darveau CA, Andrews RD, Suarez RK: Allometric
cascade: a model for resolving body mass effects on metabolism. Comp Biochem Physiol 2003, 134:675-691.
Batterham AM, Jackson AS: Validity of the allometric cascade
model at submaximal and maximal metabolic rates in exercising men. Respir Physiol Neurobiol 2003, 135:103-106.
Heusner AA: Body size, energy metabolism, and the lungs. J
Appl Physiol 1983, 5:867-873.
Lindstedt L, Schaeffer PJ: Use of allometry in predicting anatomical and physiological parameters of mammals. Lab Anim 2002,
36:1-19.
Bokma F: Evidence against universal metabolic allometry. Func
Ecol 2004, 18:184-187.

Page 11 of 11
(page number not for citation purposes)

</pre>
</body>
</html>