A general model for metabolic scaling in self-similar asymmetric networks

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.

Rediscovering and Reviving Old Observations and Explanations of Metabolic Scaling in Living Systems

“…New RTN-based models (e.g., [84,[173][174][175][176]) or applications (e.g., [22,172,177]) continue to appear that have not sufficiently heeded the history of accumulating negative evidence against the assumptions, logic and predictions of previous RTN models.…”

Interspecific scaling of blood flow rates and arterial sizes in mammals

“…The empirical relationships between mammalian body size and heart mass, stroke volume, heart rate, cardiac output and blood pressure have each been shown to be related to metabolic rate, either directly or indirectly (Calder, 1996;Hillman and Hedrick, 2015;Seymour and Blaylock, 2000). The branching morphology of the arterial system has been measured and modelled to test the optimality theory of a space-filling fractal network that supplies oxygenated blood with the least energy cost (Brummer et al, 2017;Hunt and Savage, 2016;Kassab, 2012, 2016;Kassab, 2006;Newberry et al, 2015;Price et al, 2007;Tekin et al, 2016). However, most of these studies focused only on the morphology of the network, and so it is difficult to extract the relationships between arterial size and actual blood flow rate.…”

“…Correlations between the metabolic rates of animals and the structure of supply networks are well known, but the direction of dependence is confusing in the literature. West, Brown and Enquist began a revolution in thinking about physiological scaling by suggesting that quarter-power scaling of metabolic rate and other traits should arise if vascular networks are selected to fill space while minimizing the energy required to distribute resources (Brummer et al, 2017;West et al, 1997). The vascular system has therefore been hypothesized to determine rates of metabolism (Newberry et al, 2015).…”

Why allometric variation in mammalian metabolism is curvilinear on the logarithmic scale

“…The power equation in arithmetic domain consequently becomes The exponent in this complex equation appears to have two fitted parameters, b and c ; but the exponent seems also to vary with body size (in the form of log( x )). Although the function fails the test for parsimony (Anderson, ; Burnham & Anderson, ), several workers have embraced the equation and proposed changes to the underlying theoretical model for the MTE to bring its predictions into compliance with results of the statistical analyses (Brummer, Savage, & Enquist, ; Kolokotrones et al., ; Savage et al., ; Tekin, Hunt, Newberry, & Savage, ).…”