A gentle introduction to scaling relations in biological systems

Abstract

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 β of this powerlaw 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 β = 2/3. Secondly, the West-Brown-Enquist theory explains such scaling properties due to the hierarchical and fractal nutrient distribution network, deriving β = 3/4.

Theoretical approach to community structure: body size, density and energy

“…Integrating allometric relationships and biological processes, such as births, deaths, developmental stages (adults, juveniles), and extinctions, is proposed to capture demographic patterns accurately. Ribeiro and Pereira (2023) have reported on the different values of the allometric exponent in the Rubner and West-Brown-Enquist models [1]. Furthermore, this study posits that environmental conditions impact energy availability, potentially influencing the exponent value in relation to the metabolic adaptation of organisms.…”

Mathematical models to explain the origin of urban scaling laws

“…In the second part of his 2013 paper [11], Bettencourt proposes a more detailed model to explain the scaling laws of urban infrastructure. The model was inspired by West, Brown, and Enquist (WBE)'s theory [64,65,66,67], which explains the allometric scaling in biology based on the hierarchical network properties of the distribution and allocation of resources.…”