Abstract
How a particular attribute of an organism changes or scales with its body size is known as an allometry. Biological allometries, such as metabolic scaling, have been hypothesized to result from selection to maximize how vascular networks fill space yet minimize internal transport distances and resistances. The West, Brown, Enquist (WBE) model argues that these two principles (space-filling and energy minimization) are (i) general principles underlying the evolution of the diversity of biological networks across plants and animals and (ii) can be used to predict how the resulting geometry of biological networks then governs their allometric scaling. Perhaps the most central biological allometry is how metabolic rate scales with body size. A core assumption of the WBE model is that networks are symmetric with respect to their geometric properties. That is, any two given branches within the same generation in the network are assumed to have identical lengths and radii. However, biological networks are rarely if ever symmetric. An open question is: Does incorporating asymmetric branching change or influence the predictions of the WBE model? We derive a general network model that relaxes the symmetric assumption and define two classes of asymmetrically bifurcating networks. We show that asymmetric branching can be incorporated into the WBE model. This asymmetric version of the WBE model results in several theoretical predictions for the structure, physiology, and metabolism of organisms, specifically in the case for the cardiovascular system. We show how network asymmetry can now be incorporated in the many allometric scaling relationships via total network volume. Most importantly, we show that the 3/4 metabolic scaling exponent from Kleiber’s Law can still be attained within many asymmetric networks.
Highlights
One of the pervasive characteristics of biology is that the metabolic rate, B, of an organism scales with its body mass, M
Our model shows how different scaling exponents, such as 2/3 and 1, can be associated with different levels of asymmetry, where 2/3 is the value associated with the idea that metabolic rate is limited by the ability of an organism to dissipate heat [37], and 1 is the value associated with isometric scaling
Of significant interest is examining the effect of asymmetric branching on the predicted values of the metabolic scaling exponent θ
Summary
One of the pervasive characteristics of biology is that the metabolic rate, B, of an organism scales with its body mass, M. B 1⁄4 B0My ð1Þ where B0 is a normalization constant, and θ, the allometric scaling exponent, tends to cluster around the value of 3/4 [1,2,3,4,5]. This metabolic scaling relationship spans more than twenty five orders of magnitude in mass, from respiratory complexes at 10-18 grams to the largest mammals at 107 grams [6]. The West, Brown, and Enquist (WBE) model offers an alternative hypothesis for the origin of allometric scaling exponents in biology, in particular the 3/4 exponent of metabolic scaling [9].
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