Multicellular Systems - Molecular Theory of the Living Cell - page 538

where y i ,a i , and x i are, respectively, the metabolic rate, the proportionality constant
and the body size of the ith species with a distinct cell density, d i , and habitat
temperature, T i . Because of Assumption (3), a i is a function of both T i and d i :

a i = f(d i ; T i Þ

(15.13)

Taking the logarithm of both sides of Eq. (15.12) leads to:

log y i ¼ =

3 log x i þ

log a i (15.14)

3

Equation (15.14) predicts that, when the ith metabolic rate, y i , is plotted against
the ith body mass, x i , on a double logarithmic coordinate system, a straight line with
slope 3/4 would be obtained with different y -intercepts for different species, consis-
tent with Fig. 15.13 . Designating unicellular organisms as 1, poikilotherms as 2, and
homeotherms as 3, the data in Fig. 15.13 make it clear that a 1 < a 3, indicating

that the metabolic rates per unit mass increase from unicellular organisms to
poikilotherms to homeotherms. This observation is consistent with the FERFAC
(Free energy requirement for Active complexity) hypothesis, Statement 15.20,
which predicts that organisms with higher active complexities require higher energy
expenditures, since the active complexity (Sect. 5.2.3 ) of the groups of the organisms
considered here most likely increases in the same order as their intercept values, a i .
The life span (LS) of an organism can be viewed as the projection of the living
processes embodied in x on to the time dimension, which leads to:

a 2 <

a i x 1 = 4

LS i ¼

(15.15)

where LS i and a i are, respectively, the life span and the proportionality constant of
the ith species. Equation (15.15) is qualitatively consistent with the life span vs.
body-mass plots found in the literature (e.g., see http:/www.senescence.info/
comparartive.html ) and (West and Brown 2004).
The key difference between the West-Brown-Enquist (WME) approach to
developing a theory of allometric scaling and the one proposed in this topic is that
the former assumes that the 3/4 exponent can be derived mathematically from the
species-specific physical characteristics of organisms (e.g., vascular trees of
mammals, diffusion paths within cells in unicellular organisms), whereas the present
approach regards the exponent as resulting from the universal property of all living
systems, i.e., enzymic activity, regardless of differences in distribution networks
among different individuals and species. It is possible that both approaches are
relevant, since living systems embody two distinct processes - transport of matter
between micro-meso (e.g., cells) and macro-sites (e.g., lung) and transformation of
matter within cells. Based on the structural information of all living systems (e.g.,
role of mitochondrial membranes and lung alveoli membranes), it is likely that
transport processes scale as the body mass raised to the power of 2/3 as was first
suggested by Rubner in 1883 (White and Seymour 2005), and, based on the idea that