A general basis for quarter-power scaling in animals

Abstract

It has been known for decades that the metabolic rate of animals scales with body mass with an exponent that is almost always <1, >2/3, and often very close to 3/4. The 3/4 exponent emerges naturally from two models of resource distribution networks, radial explosion and hierarchically branched, which incorporate a minimum of specific details. Both models show that the exponent is 2/3 if velocity of flow remains constant, but can attain a maximum value of 3/4 if velocity scales with its maximum exponent, 1/12. Quarter-power scaling can arise even when there is no underlying fractality. The canonical "fourth dimension" in biological scaling relations can result from matching the velocity of flow through the network to the linear dimension of the terminal "service volume" where resources are consumed. These models have broad applicability for the optimal design of biological and engineered systems where energy, materials, or information are distributed from a single source.

Fig. 1.

Sketches of 2D supply networks. Similar considerations apply to 3D animals. All three networks depict the supply routes from a single source to the service regions. (A) Radial explosion network. An individual route directly connects the central source to each service volume. The average length of a route is proportional to the length (L) of the animal, where L ∼ M^1/3. The shortest routes (solid lines) are those to service volumes adjacent to the source, so their length is l_s ∼ (V/B)^1/3 ∼ (M/B)^1/3. The scale of the velocity is set by these routes and yields Eq. 3. (B) Hierarchical branching network, similar to that described by West et al. (10). The shortest pipes (capillaries, solid lines) are proportional to the radius of the service volume (M/B)^1/3. There is backtracking through the pipes, so that the distance from the heart through the arteries to each service volume is the same and is equal to the length (L) of the animal, where L ∼ M^1/3. (C) Hierarchical branching network without backtracking. The network is similar to A in that there is no backtracking as blood flows from the central source to the service volumes and it is similar to B in that nearby supply routes are aggregated and the lengths of the shortest pipes (capillaries, solid lines) are proportional to the radius of the service volume (M/B)^1/3.

Fig. 2.

Schematic demonstration of how length scales change with animal mass (M) in the radial explosion network. In each animal, the length scale (L) and average distance between the central source and the service regions (d) are proportional to M^1/3, the volume of the service region is ∼M/B, and from Eq. 1, the length of the service volumes (l_s) is proportional to (M/B)^1/3. The length of the shortest pipe is ∼l_s, and thus the maximum separation between particles, s, is ∼l_s. From Eqs. 1 and 5, l_s ∼ M^1/12. Thus, in A, s is slightly larger (∼M^1/12) and d is much larger (∼M^1/3) than in B. Because velocity is proportional to s (Eq. 2), we have the following relationships: ,

Fig. 3.

Red blood cells in a branching network with no backtracking, as in Fig. 1C. Blood cells are packed into arteries at constant density, whereas blood cells destined for a particular service region are separated by a distance, s ∼ l_s ∼ M^1/12. Each cell is labeled with the service region it will be delivered to. In this case, l_s is approximately the length of two red blood cells. In a larger organism, l_s would be larger, and there would be more red blood cells between cells labeled with the same service region. Here there are eight service regions, each serviced by one capillary, so B ∼ 8 ∼ M^3/4, and eight red blood cells are released per unit time from the central heart. Those eight red blood cells are squeezed into an aorta with cross-sectional area ∼M^2/3, which causes the separation distance(s) and the velocity of blood cells, v, both to be proportional to M^1/12.