Environmental Engineering Reference
In-Depth Information
lost through the body surface, it is logical to expect that
an organism will adjust its BMR accordingly and that the
rate will be proportional to M
2
=
3
. Rubner's surface law
of metabolism remained unchallenged for nearly half a
century.
Then Kleiber (1932), working with a set of 13 data
points (including two steers, a cow, and a sheep), showed
that BMR goes up as M
0
:
74
. Kleiber (1961) eventually
recommended a rounded expression of 70M
0
:
75
(in
kcal/day) or 3
:
4M
0
:
75
(in W). When plotted on double-
log axes, this exponential relation became one of the most
important generalizations in bioenergetics, the straight
mouse-to-elephant line of the 3
=
4 law (fig. 4.2). Unlike
the 2/3 law, the 3
=
4 law presented a challenge of causal
interpretation. Maynard Smith (1978) explained the ex-
ponent as a compromise between the surface-related
BMR (0.67) and the mass-related inputs needed to over-
come the gravitation (exponent 1.0). McMahon (1973)
based his explanation on the elastic criteria of limbs. The
weight of these loaded members is a fraction of M,so
their diameter will be proportional to M
3
=
8
. The power
output of muscles depends only on their cross-sectional
area (proportional to d
2
), and thus the maximum power
output is related to
ð
M
3
=
8
Þ
2
,orM
0
:
75
. If applicable to
any particular muscle, the scaling should rule the total or-
ganism, and BMR should be a function of M
0
:
75
.
West, Brown, and Enquist (1997) offered an explana-
tion based on the geometry and physics of a network of
tubes needed to distribute resources and remove wastes
in organisms. They argued that the rates and times of
life processes are ultimately limited by the rates at which
energy and material flows are distributed between the
surfaces where they are exchanged and the tissues where
they are used or produced. This means that distribution
networks must be able to deliver these flows to every
part of an organism; that their terminal branches must
have identical size because they have to reach individual
cells; and that the delivery process must be optimized
in order to minimize the total resistance and hence
the overall energy needed for the distribution. A complex
mathematical derivation shows that these properties re-
quire the metabolism of entire organisms to scale with
the 3
=
4 power of their mass.
The first conclusion demands that many structures and
functions of the delivery system (be they hearts or heart-
beats) must be scaled according to the size of organisms.
Because the obviously tightly interdependent compo-
nents cannot be optimized separately, they must be
balanced by an overall design that calls for a single under-
lying scaling. West, Brown, and Enquist (1997) showed
that the networks providing these fundamental services
have a fractal architecture that requires many structural
and functional attributes to scale as quarter powers of
body mass, and that this requirements applies equally
well to heterotrophs and plants. The second conclusion
regarding the invariant components of the delivery sys-
tem seems to be convincingly demonstrated by the iden-
tical radius of capillaries in mammals, whose sizes span 8
OM (from shrews to whales), or by the identical number
of heartbeats per lifetime, that is, by the identical total of
energy needed to support a unit mass of any organism
over its lifetime (Marquet et al. 2005). The third conclu-
sion rests on the economy of evolutionary design: organ-
isms develop structures and functions to meet but not
exceed maximal demand.
But Makarieva, Gorshkov, and Li (2005) concluded
that this model is logically inconsistent because its
assumptions of the size-invariance of the rate of en-
ergy supply in terminal units contradicts its central
prediction—namely that the mass-specific metabolic rate
