A gentle introduction to scaling relations in biological systems

In this paper it is presented a gentle review of empirical and theoretical advances in understanding the role of size in biological organisms. More specifically, it deals with how the energy demand, expressed by the metabolic rate, changes according to the mass of an organism. Empirical evidence suggests a power-law relation between mass and metabolic rate, namely the allometric equation. For vascular organisms, the exponent $\beta$ of this power-law is smaller than one, which implies scaling economy; that is, the greater the organism is, the lesser energy per cell it demands. However, the numerical value of this exponent is a theme of extensive debate and a central issue in comparative physiology. A historical perspective is shown, beginning with the first empirical insights in the sec. 19 about scaling properties in biology, passing through the two more important theories that explain the scaling properties quantitatively. Firstly, the Rubner model considers organism surface area and heat dissipation to derive $\beta=2/3$ . Secondly, the West-Brown-Enquist theory explains such scaling properties due to the hierarchical and fractal nutrient distribution network, deriving $\beta=3/4$ .

Keywords
Complex systems; scaling theory; modelling

Authorship

Fabiano L. Ribeiro ^** Correspondence email address: fribeiro@ufla.br

Universidade Federal de Lavras, Departamento de Física, Lavras, MG, BrasilUniversidade Federal de LavrasBrasilLavras, MG, BrasilUniversidade Federal de Lavras, Departamento de Física, Lavras, MG, Brasil

http://orcid.org/0000-0002-2719-6061

William R. L. S. Pereira

Independent researcher, São Paulo, SP, BrasilIndependent researcherBrasilSão Paulo, SP, BrasilIndependent researcher, São Paulo, SP, Brasil

* Correspondence email address: fribeiro@ufla.br

SCIMAGO INSTITUTIONS RANKINGS

Independent researcher, São Paulo, SP, BrasilIndependent researcherBrasilSão Paulo, SP, BrasilIndependent researcher, São Paulo, SP, Brasil

Figures | Tables | Formulas

Figures (10)
Tables (3)
Formulas (44)

$y$ (Circulatory system)	$\alpha$ (predicted by WBE theory)	$\alpha$ (empirical)
aortic radius ( $r_{0}$ )	$3/8=0.375$	0.36
aortic pressure	$0$	$0.032$
blood velocity in the aorta ( $u_{0}$ )	$0$	$0.07$
total blood volume ( $V_{b}$ )	$1$	$1.00$
circulation time	$1/4=0.25$	$0.25$
circulation distance ( $\sum_{k=0\vphantom{{}_{0}}}^{K\vphantom{{}^{0}}}l_{k}$ )	$1/4=0.25$	not available
cardiac injection fraction	$1$	$1.03$
heart rate	$-1/4=-0.25$	$-0.25$
cardiac output	$3/4=0.75$	$0.74$
number of capillaries ( $N_{c}$ )	$3/4=0.75$	not available
capillary density	$-1/12=-0.083$	$-0.095$
oxygen affinity in the blood	$-1/12=-0.083$	$-0.089$
service volume radius	$1/12=0.083$	not available
Krogh cylinder radius	$1/8=0.125$	not available
peripheral resistance	$-3/4=-0.75$	$-0.76$
Womersley number	$1/4=0.25$	$0.25$
metabolic rate	$3/4=0.75$	$0.74$

$y$ ( respiratory system)	$\alpha$ (predicted by WBE theory)	$\alpha$ (empirical)
lung volume	$1$	$1.05$
respiratory rate	$-1/4=-0.25$	$-0.26$
volume flow to lung	$3/4=0.75$	$0.80$
interpleural pressure	$0$	$0.004$
tracheal diameter	$3/8=0.375$	$0.39$
air velocity in the trachea	$0$	$0.02$
tidal volume	$1$	$1.041$
dissipated energy	$3/4=0.75$	$0.78$
number of alveoli	$3/4=0.75$	not available
alveolar radius	$1/12=0.083$	$0.13$
surface area of alveoli	$1/6=0.1666...$	not available
surface area of the lung	$11/12=0.92$	$0.95$
oxygen diffusing capacity	$1$	$0.99$
total airway resistance	$-3/4=-0.75$	$-0.70$
rate of oxygen consumption	$3/4=0.75$	$0.76$

	mices	elephants
$r$	$1$	$100$
$A$	$\sim$ 1	$\sim$ 10.000
$V$	$\sim$ 1	$\sim$ 1.000.000
$M$	$\sim$ 1	$\sim$ 1.000.000