## SUMMARY

The logarithmic nature of the allometric equation suggests that metabolic
rate scaling is related to some fractal properties of the organism. Two
universal models have been proposed, based on (1) the fractal design of the
vasculature and (2) the fractal nature of the `total effective surface' of
mitochondria and capillaries. According to these models, basal and maximal
metabolic rates must scale as *M*^{3/4}. This is not what we
find. In 34 eutherian mammalian species (body mass *M*_{b}
ranging from 7 g to 500 kg) we found
*V̇*_{O2max} to scale
with the 0.872 (±0.029) power of body mass, which is significantly
different from 3/4 power scaling. Integrated structure-function studies on a
subset of eleven species (*M*_{b} 20 g to 450 kg) show that the
variation of
*V̇*_{O2max} with body
size is tightly associated with the total volume of mitochondria and of the
locomotor musculature capillaries. In athletic species the higher
*V̇*_{O2max} is linked
to proportionally larger mitochondrial and capillary volumes. As a result,
*V̇*_{O2max} is
linearly related to both total mitochondrial and capillary erythrocyte
volumes, as well as to their surface areas. Consequently, the allometric
variation of maximal metabolic rate is directly related to the scaling of the
total effective surfaces of mitochondria and capillaries, thus confirming the
basic conjecture of the second fractal models but refuting the arguments for
3/4 power scaling. We conclude that the scaling of maximal metabolic rate is
determined by the energy needs of the cells active during maximal work. The
vascular supply network is adapted to the needs of the cells at their working
limit. We conjecture that the optimization of the arterial tree by fractal
design is the result rather than the cause of the evolution of metabolic rate
scaling. The remaining question is why the energy needs of locomotion scale
with the 0.872 or 7/8 power of body mass.

- metabolic rate
- scaling
- locomotor muscle
- aerobic capacity
- mitochondria
- capillary
- fractal design
- vascular supply network
- energy demand

## Maximal metabolic rate

Maximal metabolic rate (MMR) is elicited by a well-defined dominant process: the energy needs of locomotor activity at its sustainable maximum during a run. In contrast, other forms of MR have a more complex origin. Basal metabolic rate (BMR) reflects the lowest need of energy at rest in a postprandial state under thermoneutral conditions. It is the energy used to run basic cell functions in a large variety of organs, such as maintaining cell potentials, driving the heart and circulation, synthetic and housekeeping functions of all sorts, and maintenance of body temperature. BMR and MMR thus define the span over which the metabolic rate of the organism can vary. Their definitions are unambiguous and, as such, they are reproducible. They reflect the performance of the organism under these specific test conditions, but are not conditions that occur usually in life. Natural forms of MR are intermediate. Field metabolic rate (FMR) reflects the time-averaged MR elicited by all possible organismic functions at variable levels of activity, such as foraging or hunting for food, or digestion, and is commonly found to be about 4 times BMR. What is termed sustained MR refers to the metabolic rate that allows for maintaining body mass over longer time; it is a form of FMR mostly related to nutrient supply.

Whereas BMR appears to depend essentially on body mass, MMR shows large inter-individual and inter-species variability, related to the degree of work or exercise capacity. It is typically about tenfold higher than BMR. Well-trained human athletes can achieve MMR up to 20 times higher than their BMR. Even greater variation is found in animals. Athletic species such as dogs or horses increase their MR by 30-fold from rest to maximal exercise, and in race horses this factor can rise to 50, as in the pronghorn antelope, the world's top athletic mammal.

MMR reflects the limitation of oxidative metabolism of muscle cells. It is not sustainable for more than a few minutes because, under these limiting conditions, the incipient aerobic energy deficit must be covered by anaerobic glycolysis, which leads to lactate accumulation in the blood and thus to cessation of exercise.

All forms of MR in mammals and other taxa show some scaling with body mass
*M*_{b} according to the allometric equation
MR∝*aM*_{b}^{b}. One of the tasks of experimental
biology is to sort out the effects of body size and of other determinants,
such as athletic prowess, on this relationship, separating variations in the
exponent *b* and in the coefficient *a*. This will be considered
in setting up models for this analysis.

## Models and hypotheses of MR scaling

When Max Kleiber stated that the standard metabolic rate of mammals scaled
`*close to the three-fourth power of body mass*'
(Kleiber, 1947), he argued on
the basis of empirical studies in which he had first found a scaling exponent
of 0.739±0.03 (Kleiber,
1932), and later one of 0.756±0.004. He suggested that 3/4
would be easier to remember. This was not meant to be `Kleiber's law', as it
is now often called, but rather an empirical observation that was confirmed
repeatedly and eventually broadly accepted as a fact
(Schmidt-Nielsen, 1984).
Kleiber, however, attempted no mechanistic explanation of the 3/4 power
exponent.

The logarithmic nature of the allometric equation
log*B*=(3/4)log*M*_{b}*+*log*a*, where
*B=*metabolic rate, suggests that it may be based on some
scale-invariant or self-similar features of the organism, and therefore it
appeared attractive to search for the possibility that some fractal properties
of the organism were related to metabolic rate scaling. This possibility was
first suggested and explored by West et al.
(1997). In an attempt to
explain the `1/4 power law' from first principles they proposed two different
models.

In their first model (West et al.,
1997) they considered the fractal design of the network for
nutrient and O_{2} supply, the vascular system, and introduced two
premises: (1) that blood vessels form a self-similar fractal network with the
volume of blood *V*_{bl} proportional to body mass:
*V*_{bl}∝*M*_{b}; and (2) the capillaries
as the terminal units of this network are invariant of fixed size whereas
their number (*N*_{(cap)}) is proportional to the metabolic
rate *B* of the cells with which they interact:
*N*_{(cap)}∝*B*∝*M*_{b}^{b}.
Applying the principles that (a) the network is volume-filling and (b) the
energy needed to transport nutrients and O_{2} is minimized, they
derived that the conditions were fulfilled by *b*=3/4. This model has
recently been criticized by Kozlowski and Konarzewski
(2004), primarily on the basis
of the internal inconsistency of the premises.

The second model of West et al.
(1999) considers the fractal
nature of the structural surfaces across which metabolic activity takes place:
the capillaries and the mitochondria. The authors conjecture that metabolic
rate *B* is limited by the geometry and scaling behaviour of what they
call the total effective surface of the organism, *a*, which they take
to be either the total capillary surface or the total mitochondrial surface
(probably the area of their inner membranes where oxidative phosphorylation
takes place). Their prediction that *B*∝*a* thus assumes
that metabolic rate scales like the capillary and the inner mitochondrial
surface areas. This tacitly implies that the rate of `specific activity' of
these membranes is size-independent.

The scaling argument for *B* against body size runs as follows. If
the effective surfaces were built on Euclidean geometry their surface area
would scale with body volume *V* or body mass *M*_{b}
as:
(1)

Assuming that these surfaces show fractal design, however, the scaling exponents must be modified to include a factor related to the fractal dimension of the structure so that: (2)

where 2*+*ϵ_{a} is the fractal dimension of the surface
whereby 0<ϵ_{a}<1; ϵ_{a}=0 means Euclidean
surface, whereas ϵ_{a}=1 is a space-filling fractal surface. West
et al. (1999) also considered
the fractal nature of the containing volume and of body mass with the result
that the exponent of the volume takes the form
3*+*ϵ_{a}+ϵ_{l}, where ϵ_{l} is
a fractal dimension factor for the linear vascular tree. This new model is
linked to the first one by two assumptions: that the terminal capillary units
are invariant of fixed size across the size range of species, and that the
vascular tree is fractal in nature. The authors then conjecture that organisms
have evolved to maximize the scaling of the total effective surface
*a*, which is achieved when ϵ_{a}=1 andϵ
_{l}=0. On these premises one finds:
(3)

`*regardless of the details of branching architecture and dynamics
governing the metabolic process and distribution of resources*'
(West et al., 1999). This then
is the basis of the `universal scaling law' that holds for mammalian
organisms, cells, mitochondria, and molecules
(West et al., 2002). This is
because `*although living things occupy three-dimensional space, their
internal physiology and anatomy operate as if they were four-dimensional*'
(West et al., 1999).

It appears to us that this model development is mistaken on two counts. (1)
The volume containing the effective surface *a* is a real world volume
of dimension 3, and therefore the denominator in Equation 2 should be 3
without fractal complement; and (2) in the real world, `fractal surfaces' are
geometric properties of membranes, physical structures of finite thickness
that function in conjunction with other structures (in this case the
mitochondrial matrix), and it is therefore questionable whether they can be
totally space-filling, i.e. of dimension 3. If it is justified to consider
inner mitochondrial membranes as fractals then their fractal dimension should
be 2<*D*_{f}<3. Considering this, the area-to-volume
relationship of mitochondrial membrane surfaces becomes
*a*∝*V*^{Df/3}. If the exponent 3/4 should be
explained by fractal properties of the effective surface its fractal dimension
could be *D*_{f}=2+ϵ_{a}=2.25. This discussion
will be resumed below.

These attractive models have been criticized on several grounds, but largely with the argument that fractal geometry is not necessary to explain the 3/4 power scaling of metabolic rate. Alternative explanations are found in network structures (Banavar et al., 1999), quantum mechanics (Demetrius, 2000) and topology (Bejan, 2000). It has also been pointed out that the metabolic rate of an organism is not a simple coherent process, but rather appears as the sum of many partial contributions of various compartments and steps that may show very different scaling behaviour under different functional conditions (Darveau et al., 2002; Hochachka et al., 2003).

One important critique is that 3/4 power scaling is not a general or
universal law (Darveau et al.,
2002; Weibel,
2002). It is rather a frequently occurring empirical result
pertaining to measurements of `standard' or perhaps `basal' metabolic rates in
mammals of varying body size. But different scaling may be obtained if the
conditions of basal metabolic rate are tightly applied
(White and Seymour, 2003), or
when the effects of intra-specific size variations are considered
(Heusner, 1984). Even though
approximate 3/4 power scaling may be `preponderant' when considering the
metabolic scaling under resting or field conditions
(Nagy et al., 1999;
Savage et al., 2004) there are
consistent deviations from 3/4 power scaling. One case of special interest is
the maximal metabolic rate, MMR, or
*V̇*_{O2max}, which was
repeatedly found to scale with a power larger than 3/4
(Koteja, 1987;
Taylor et al., 1988;
Bishop, 1999).

There has been less effort towards deriving a theory for the scaling of
MMR. Some early studies had suggested that MMR achieved in exercise was about
10 times BMR and would therefore scale also with
*M*_{b}^{0.75}
(Hemmingsen, 1960;
Pasquis et al., 1970;
Lechner, 1978), so that no
special consideration of this condition seemed warranted. There were, however,
some observations that suggested the possibility of a different scaling of MMR
induced by locomotion. Taylor et al.
(1980) estimated the energetic
cost of generating muscular force by running animals on a treadmill with or
without a load of up to 27% of body mass and found that O_{2}
consumption increased in direct proportion to the mass supported by the
muscles (Fig. 1). This
observation suggested that
*V̇*_{O2max} should
scale with *M*_{b}^{1}. One of the main differences
between BMR and MMR is that the latter is a limiting condition and may depend
on O_{2} supply from the lung and circulation of blood
(Fig. 2). Studies on the design
of the pathway for oxygen have shown that the pulmonary diffusing capacity,
the potential limit for O_{2} uptake, scales in fact with
*M*_{b}^{1}, thus differently from BMR (Weibel,
1972,
1973). With these two
observations in mind Taylor and Weibel
(1981) undertook a study on
the allometry of the respiratory system with the hypothesis that
*V̇*_{O2max}∝*M*_{b}^{1},
on the grounds that the cost of running was proportional to
*M*_{b} and that the design of the O_{2} cascade is
matched to the maximal energy demands of the organism, thus introducing the
hypothesis of symmorphosis.

This latter argument preceded one of the central statements of the
universal model, namely that `*the economy of design...[is such that]...
structures and functions tend to just meet maximal demands*'
(Brown et al., 2000), the
essential postulate of symmorphosis
(Taylor and Weibel, 1981;
Weibel et al., 1991,
1992). This means that maximal
metabolic rate, i.e. the condition under which the resource distribution
network as well as the total effective surfaces must meet maximal demands,
must abide to the universal scaling law and scale as
*M*_{b}^{3/4}.

There is, however, no *a priori* reason why MMR should scale the
same way as BMR with body size. There are distinct differences in the
performance of the system under the two conditions. (1) At BMR, O_{2}
is consumed in all cells of the body mainly for maintenance of cell polarity,
protein synthesis etc.; at MMR, over 90% of O_{2} is consumed in a
single organ system, the locomotor muscles, for ATP resynthesis related to
work output (Fig. 2). (2) At
BMR, blood flow is distributed equitably through the (fractal) vascular tree
to all organs of the body; at MMR, over 90% of blood flow is directed to
contracting muscle tissue in response to increased energy demand. (3) Among
species with similar BMR, there are great differences in the capacity to
increase MR in support of exercise, e.g. in athletic *vs* sedentary
species (Taylor et al., 1987;
Jones et al., 1989).

In this paper we shall first critically review the existing data base for MMR scaling in mammals. From the systems physiology point of view, we then ask what factors may be determining MMR and how this will cause the scaling of MMR to be different from that of BMR. An important characteristic of this analysis is that body size is considered as only one of several factors determining MR.

We defined MMR as
*V̇*_{O2max} estimated
by the accepted standardized method, which is to run animals on a treadmill at
varying work intensity (speed and incline) measuring
*V̇*_{O2} and plasma
lactate concentration when steady state is achieved
(Seeherman et al., 1981); for
this, speed is kept constant during each run but varied between runs. Under
such experimental conditions
*V̇*_{O2max} is
achieved when a further increase in work output does not cause
*V̇*_{O2} to rise
further but the additional energy is provided by anaerobic metabolism, as
reflected by an increase in plasma lactate concentration.

## The scaling of MMR

In an extensive review of the literature, Weibel et al.
(2004) found 57 estimates of
*V̇*_{O2max} conforming
to the above-stated conditions, representing 34 mammalian species ranging in
body mass from the pigmy mouse at 7 g to the horse at over 500 kg. It includes
a representative range of mammalian species, wild and domesticated. The data
set covers five orders of magnitude and therefore encompasses the vast
majority of terrestrial mammalian species.

In Fig. 3,
*V̇*_{O2max} is plotted
against body mass on a double-logarithmic scale. The power law regression
calculated for the entire data set is:
(4)

This slope is significantly different (*F*=17.472, d.f.=1, 64,
*P*<0.01) from 3/4 that characterizes BMR and it is also different
from 1.

Fig. 3 also reveals the
great variability of
*V̇*_{O2max} in
relation to body mass. For example, in the size class of 25 kg the range of
*V̇*_{O2max} is almost
one order of magnitude between the goat on one hand and the pronghorn antelope
on the other. This is related to differences in the aerobic exercise capacity
of different species, which is higher in athletic mammals, such as horse, dog
and pronghorn, compared to the majority of `normal' or more sedentary species
(Taylor et al., 1987;
Jones et al., 1989;
Lindstedt et al., 1991). These
athletic species are marked by large, open triangles whereas small, filled
triangles mark the `normal' species; the separation is made on the grounds of
a high mass-specific
*V̇*_{O2max} of
athletic species.

Could the slope be steeper than that of BMR because of the presence of athletic species, which are prevalent in the larger size classes? In Fig. 3 the scaling of athletic and `normal' species is also plotted. We find the regression for the non-athletic species alone to be: (5)

with 95% confidence limits CL=0.799-0.899. The coefficient *a* is
lower by 20% but the slope *b* is not different from that of the
overall population (*F*=0.265, d.f.= 24, *P*=0.609). From this
we conclude that the difference between the power law slopes of BMR and
*V̇*_{O2max} is not due
to the inclusion of athletic species.

The MMR of athletic species scales according to: (6)

which is significantly different from the non-athletic regression both with
respect to the coefficient *a* and the exponent *b* (95% CL for
*b*=0.889-0.995).

A steeper allometric slope of
*V̇*_{O2max} compared
to BMR suggests that larger mammals have a greater relative capacity to
increase metabolic rate above the resting state than small mammals. Weibel et
al. (2004) estimated the
so-called factorial aerobic scope (fAS), the ratio of
*V̇*_{O2max} to BMR, to
be on the order of 6-10 for normal species and 10-60 for athletic species; it
also increases with body mass, particularly in athletic species where it
scales in proportion to *M*_{b}^{0.18}. The fAS is
therefore related both to body mass and to aerobic capacity, and the range is
greater in large than small mammals, potentially giving large animals an
evolutionary advantage.

*V̇*_{O2max}
and muscle mitochondria

One of the reasons for the conjecture that MMR may scale differently from
BMR was the fact that the energetic cost of muscle activity increased linearly
with load (Fig. 1). Since MMR
as here defined reflects
*V̇*_{O2max} as
elicited by muscle work, we must first look for characteristics of the
locomotor muscle system that could cause the variation of
*V̇*_{O2max} to be
partly independent of BMR. Essential factors to consider are muscle mass and
aerobic capacity of muscle fibres.

Weibel et al. (2004) report
the results of a series of in-depth studies where correlated data were
obtained on
*V̇*_{O2max} and muscle
structure in a set of mammals ranging from the woodmouse of 20 g to the horse
and steer of 500 kg. The characteristic of these studies was that
*V̇*_{O2max}, whole
body muscle mass and the complete morphometry of locomotor muscle, as well as
lung morphometry, were estimated on the same animals
(Hoppeler et al., 1987). It
was first found that muscle mass, *M*_{m}, constituted 36% of
body mass *M*_{b} on average, ranging from 25% in the goat to
45% in the pronghorn, with athletic animals having a larger muscle mass, but
without any dependence on body size as *M*_{m} scales with
*M*_{b}^{1.01} (*r*^{2}=0.997).
Interestingly, the smallest animal, the (athletic) woodmouse, had one of the
highest relative muscle masses, 42%.

It has been noted for a long time that the higher aerobic capacity of
athletes is related to a higher mitochondrial content of the locomotor
muscles, both in humans (Hoppeler et al.,
1973) and in athletic mammals
(Hoppeler et al., 1987). This
suggests the hypothesis that the mitochondria of muscle could also determine
*V̇*_{O2max} with
respect to allometric variation. To test this hypothesis requires a study
design where the physiological and morphometric measurements are congruent,
i.e. relate to the same compartment and are obtained on the same animals. In
quadrupedal mammals running at
*V̇*_{O2max}, almost
the entire muscle mass is engaged and oxidative phosphorylation accordingly
occurs in all muscles, which may be different in humans using bipedal
locomotion (Hoppeler, 1990).
The estimate of muscle mitochondrial content must therefore reflect the
musculature of the whole body.

Estimates of whole body muscle mitochondrial volume
*V*_{(mt)} have been obtained in 11 mammalian species, for
which *V̇*_{O2max} was
also estimated on the same animals (Weibel
et al., 2004). Fig.
4 shows these data in a log-log plot against body mass. We note
that *V̇*_{O2max} and
*V*_{(mt)} are proportional in all instances, i.e. the overall
regression lines have identical slope of 0.96 (*F*=0.168, d.f.=1,
*P*=0.683), and the data points for athletic species are consistently
higher in both data sets.

*V̇*_{O2max}
and capillary blood supply

Mitochondria can only perform their high rate of oxidative phosphorylation
if they receive an adequate supply of O_{2} from capillary blood
(Fig. 5). This can be a
limiting factor for aerobic metabolism. We must therefore ask whether the
design of muscle microvasculature is matched to the varying demands observed
in large and small mammals. In previous studies comparing athletic with
sedentary mammals whose
*V̇*_{O2max} differs by
a factor of 2.5, it was found that the volume of the capillary network was
higher in the athletic species, but only by a factor of 1.7. However, athletic
species have a 1.8 times greater hematocrit
(Conley et al., 1987;
Kayar et al., 1994) so that,
as a consequence, the volume of capillary erythrocytes in the musculature is
2.5 times larger in athletic species, and hence matched to their higher
*V̇*_{O2max}
(Weibel et al., 1991). This is
also seen in Fig. 6, based on
the same animals as for the study of mitochondria
(Fig. 4): the data points for
capillary volume show the same distribution with respect to the regression
line as those of
*V̇*_{O2max}. We note
that the scaling exponent of capillary volume of 0.98 does not differ from
that of *V̇*_{O2max},
nor from that of the volume of muscle mitochondria.

## Structures of aerobic capacity and
*V̇*_{O2max}

The large scatter of data points observed in Figs
4 and
6 results from the fact that
aerobic capacity is determined by two properties, body size and athletic
prowess. These affect
*V̇*_{O2max} and the
morphometric characteristics of muscle, in parallel. If
*V̇*_{O2max} is plotted
against *V*_{(mt)} it becomes evident that the two variables
are tightly associated (Fig.
7). The `noise' in the allometric relations due to the presence of
athletic and sedentary species in all size classes
(Fig. 4) disappears and all
data points lie tightly around the linear regression
(7)

*V̇*_{O2max} and
*V*_{(mt)} show a high positive association (Spearman
*R*=0.909, *t*=6.547, d.f.=9, *P*<0.001), independent
of their relationship to *M*_{b}. This result also means that,
in all mammalian species considered, whether athletic or sedentary, whether
small or large, 1 ml mitochondria consumes 4.9±0.43 ml O_{2}
min^{-1} at
*V̇*_{O2max},
confirming the observation of Hoppeler and Lindstedt
(1985).

A similar association of functional and structural variables is found for
capillaries. What counts here, however, is the volume of capillary
erythrocytes that deliver O_{2} to the muscle cells, which is the
volume of capillaries times the hematocrit or the cells' volume fraction of
blood. It is found that the hematocrit is invariant with body mass, averaging
0.42, but it is higher in the athletic species. We then find that the volume
of capillary erythrocytes and
*V̇*_{O2max} are
linearly related across the entire size range as shown in
Fig. 7.

We thus conclude that, in all mammalian species whether small or large,
athletic or sedentary, 1 ml of capillary erythrocytes delivers 45 ml
O_{2} min^{-1} at
*V̇*_{O2max}. We also
note that muscle tissue contains about 1 ml of erythrocytes in capillaries for
every 10 ml of mitochondria in the muscle fibres. From this regression
analysis we cannot decide whether it is capillaries or mitochondria that set
the limit for O_{2} flow through the respiratory system.

## MMR and the scaling of active surfaces

In the subset of correlated species we have shown that MMR scales with the
same exponent, 0.96, for both the mitochondrial and capillary (or capillary
erythrocyte) volumes that represent the active surfaces of musculature. Is it
justified to generalize these findings obtained on a subset of species to the
entire range of mammals for which MMR was found to scale with the 0.87 power
of body mass? We believe so, for several reasons. First, the scaling exponent
of the subset is not statistically different from that of the overall
population. Second, the subset is evenly distributed over nearly the entire
size range, but it excludes the smallest species of the overall set.
Furthermore the subset includes both athletic and non-athletic species, but
with an overweight of athletic species in the larger size classes. In
addition, we have obtained approximate estimates of muscle mitochondrial and
capillary volume in 10 additional mammalian species with a similar result. We
therefore conclude that the scaling of the aerobic capacity of locomotor
muscle is strictly proportional to the scaling of
*V̇*_{O2max} with a
scaling exponent close to 0.87 for the entire population.

This agrees in principle with the conjectures of the second model of West
et al. (1999) that the scaling
of metabolic rate is determined by the scaling of what they called the `active
surfaces', specifically mentioning mitochondria and capillaries. We found that
MMR scales in parallel with the mitochondrial and capillary (or capillary
erythrocyte) volumes. It must be noted that the active mitochondrial surface,
the inner mitochondrial membrane (Fig.
8), where oxidative phosphorylation takes place, shows invariant
density in the mitochondria both with respect to body mass and aerobic
capacity (Hoppeler and Lindstedt,
1985; Schwerzmann et al.,
1989) so that the active surface is directly proportional to
mitochondrial volume. We therefore conclude that MMR indeed scales with the
surface area of inner mitochondrial membranes such that for each ml
O_{2} consumed per minute at
*V̇*_{O2max} the muscle
contains 7 m^{2} of active membrane. The capillary volume is
essentially determined by the total length of the capillary network; since the
capillary diameter varies weakly with body mass
(∝*M*^{0.01}) the capillary surface across which
O_{2} is delivered also scales about in proportion to
*V̇*_{O2max}. This
therefore suggests that the conjecture of West et al.
(1999) of a direct
proportionality between `active surfaces' and MMR is correct, although the
scaling exponent is not 0.75 but 0.87.

Could this be explained if the fractal model was corrected to account for
the fact that the containing volume has dimension 3 and the fractal surface is
not completely space-filling, so that *D*_{f}<3, as
mentioned above? A scaling exponent *b*=0.87= *D*_{f}/3
is obtained if *D*_{f} =2.6. Even though it is uncertain
whether this is a realistic value for the fractal dimension of the inner
mitochondrial membrane surface it may be interesting that Paumgartner et al.
(1981) estimated the fractal
dimension of inner mitochondrial membranes of liver cells at 2.54. This was a
microscopic study at the level of the liver cell and it is not really
justified to extend this to the fractal properties of the entire membrane
system of skeletal muscle.

However, it may be fundamentally problematic to invoke fractal properties
of real physical membranes in finding an explanation for the scaling of
metabolic rate. We have seen that the surface density of membranes within the
mitochondria is invariant with body size, so the observed allometric variation
of aerobic capacity with
*V̇*_{O2max} is fully
explained by the proportional variation of the volume of structurally and
functionally invariant mitochondria. It is thus not necessary to involve
fractal properties in this analysis. Nevertheless an in depth study of the
putative importance of fractal geometry in setting up efficient functional
systems at the organismic level may be warranted.

## The scaling of the O_{2} supply cascade

In our concept of the pathway for oxygen
(Taylor et al., 1981; Weibel
et al., 1991,
1992)
*V̇*_{O2max} is the
functional parameter that defines the limiting flow rate through all steps of
the respiratory system, from the lung to the muscle mitochondrial respiratory
chain (Fig. 2;
Table 1). The flow rates are
determined by the product of a stepwise driving force and a set of design
parameters that essentially determine the conductance. We found that design
parameters of the last two steps in this cascade, the mitochondrial and
capillary volumes and surfaces, scale strictly in proportion to MMR. What
about the other two steps, the heart and the lung?

We must first ask whether and how the functional and structural properties
of the heart are matched to the variation in O_{2} supply to muscle
microcirculation with variations in body size. The two key parameters of blood
flow, heart frequency *f*h and stroke volume
*V*s (Table
1, line 2), must be considered. Stroke volume is proportional to
heart size, which is known to be larger in athletes but invariant with body
mass (Prothero, 1979). In
contrast, heart frequency depends on body size as it is higher in small than
in large species. At BMR, heart frequency is found to scale with body mass to
the power -0.25 (Fig. 9).
Resting heart frequency therefore scales in parallel with mass-specific basal
metabolic rate. During exercise, heart frequency increases to reach a maximum
at *V̇*_{O2max}. We
find that maximal heart frequency scales with an exponent -0.15
(Bishop, 1999;
Weibel and Hoppeler, 2004),
which agrees with the scaling of mass-specific MMR, for which we obtain an
exponent -0.13 from Fig. 1. It
is thus evident that cardiac output is adjusted to the needs of working muscle
in animals of varying body mass.

Such a close match is not easily found in the lung. This organ at the
interface to the environment maintains a variable excess capacity for
O_{2} uptake so that the scaling of its diffusing capacity does not
follow the simple relationship observed in the internal parts of the
respiratory pathway: the pulmonary diffusing capacity scales linearly with
body mass (Weibel et al.,
1991; Weibel,
2000). However, considering also the structure of the pulmonary
acinar airways that distribute O_{2} to the alveolar surface, one
finds that the scaling of the membrane conductance is tightly associated with
that of *V̇*_{O2max},
both with exponents of 0.9. The effect of O_{2} screening in the
acinar airways reduces the effective conductance at rest to scale with 0.735,
similar to the scaling of basal metabolic rate
(Sapoval et al., 2002).

## Conclusions

Can we now attempt some conclusions on the mechanistic explanation of the scaling exponents observed for MMR as elicited by exercise? Two observations stand out.

The first is that scaling of MMR and BMR differs in three respects: (1) the
scaling exponent *b* for MMR of 0.87 differs significantly from that of
0.75 for BMR; (2) the coefficient *a* is larger in MMR than BMR by a
factor of around 10 that characterises the aerobic scope of `normal' mammals;
and (3) this factor ranges from about 8 in small mammals to 50 in large
athletic mammals and thus causes the range of MMR to be much larger than that
for BMR. The factorial aerobic scope, fAS, is thus also a function of body
mass, varying with *M*_{b}^{0.1} in normal and
*M*_{b}^{0.18} in athletic species. This means that MMR
can be expressed as a product of BMR and fAS such that for normal species
MMR∝BMR·fAS∝11.3*M*_{b}^{0.75}*8.3*M*_{b}^{0.1}=93*M*_{b}^{0.85}.
For athletic species one will have to consider, in addition, an `athleticity'
factor that also depends partly on *M*_{b}, as larger species
show a greater potential for athletic trait than small species.

Is there an explanation for the scaling exponent of 0.87 for MMR in
contrast to 0.75 for BMR? The cascade model of Darveau et al.
(2002) and Hochachka et al.
(2003) conceives the scaling
exponent of metabolic rate as a resultant of the weighted partial scaling of
the sequential functions that determine metabolic rate and that this is
different for BMR and MMR. The problem is that a sum of power functions cannot
be converted into an overall power function. This is not so important,
however, as we have shown here that the relevant scaling exponents for all the
steps involved (Table 1),
except the lung, are identical and the same as the scaling exponent for
*V̇*_{O2max}. Most
significantly we have found that the scaling of
*V̇*_{O2max} is tightly
correlated with the aerobic capacity of the locomotor muscles determined by
the complex of mitochondria and capillary blood. These are stressed to the
limit at *V̇*_{O2max}.
The heart also adjusts to the needs for oxygen supply determined by this
aerobic capacity of muscle: it does so by increasing heart frequency in
proportion to the factorial aerobic scope of the animal, which is greater in
large than in small animals.

The mechanistic explanation of the scaling of MMR appears easy because it relates to conditions where metabolism occurs predominantly in one functionally homogenous compartment, the locomotor musculature, which is dominant in the sense that under these conditions this organ system receives 90% of the blood flow and consumes over 90% of the oxygen taken up in the lung. Further since the key elements of all the steps between the lung and the muscle cells show tightly related scaling relationship this may explain the observed scaling of the overall MMR.

Interestingly, we found that the scaling of MMR could be explained by the
fractal nature of the effective active surface according to the notions
proposed by West et al.
(1999), but only after
modifying their fractal model to account for realistic conditions. On the
other hand, we pointed out that it is questionable to invoke fractal
properties of real membranes, and that this is not necessary to explain the
association of
*V̇*_{O2max} and
mitochondrial surface. We also found the capillaries in muscle to scale the
same as MMR and mitochondria, but here it is particularly problematic to
invoke a fractal design of the capillary network as this is incompatible with
the structure of this network. Capillaries may be simply adjusted to the
demand of the mitochondria for O_{2} supply
(Vock et al., 1996), just as
the heart adjusts its frequency to meet the different demands in blood flow.
By all that, however, the fact that MMR, or the muscles capacity for aerobic
work, scale with 0.87 or 7/8 power of body mass rather than with
*M*_{b}^{1} is not explained.

We conclude that the scaling of maximal metabolic rate is determined by the energy needs of the cells that are active during maximal work, which determines the quantity of oxidative enzymes and mitochondria as well as the capillary volume and surface needed for energy supply. The terminal units of the vascular supply network, the capillaries, are thus not invariant structures but are rather quantitatively adapted to the needs of the muscle cells at their working limit. This leads us to suggest that the optimization of the supplying vascular network, judiciously designed as a fractal tree, occurs as an adaptation of the oxygen supply system to meet the demands of working cells. It would thus be the effect rather than the cause of the evolution of metabolic rate scaling. Nevertheless, the question remains as to why the energy needs of locomotion scale with the 0.872 or 7/8 power of body mass.

## ACKNOWLEDGEMENTS

This work was supported by grants of the Swiss National Science Foundation and the Maurice E. Müller Foundation.

- © The Company of Biologists Limited 2005