Sizing Up Allometric Scaling Theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets

Sizing Up Allometric Scaling Theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.EJD acknowledges financial support from a National Institutes of Health/National Research Service Award (1F32 GM080123-01)

The Principle of Similitude in Biology: From Allometry to the Formulation of Dimensionally Homogenous `Laws'

Meaningful laws of nature must be independent of the units employed to measure the variables. The principle of similitude (Rayleigh 1915) or dimensional homogeneity, states that only commensurable quantities (ones having the same dimension) may be compared, therefore, meaningful laws of nature must be homogeneous equations in their various units of measurement, a result which was formalized in the $\rm \Pi$ theorem (Vaschy 1892; Buckingham 1914). However, most relations in allometry do not satisfy this basic requirement, including the `3/4 Law' (Kleiber 1932) that relates the basal metabolic rate and body mass, which it is sometimes claimed to be the most fundamental biological rate (Brown et al. 2004) and the closest to a law in life sciences (West \& Brown 2004). Using the $\rm \Pi$ theorem, here we show that it is possible to construct a unique homogeneous equation for the metabolic rates, in agreement with data in the literature. We find that the variations in the dependence of the metabolic rates on body mass are secondary, coming from variations in the allometric dependence of the heart frequencies. This includes not only different classes of animals (mammals, birds, invertebrates) but also different exercise conditions (basal and maximal). Our results demonstrate that most of the differences found in the allometric exponents (White et al. 2007) are due to compare incommensurable quantities and that our dimensionally homogenous formula, unify these differences into a single formulation. We discuss the ecological implications of this new formulation in the context of the Malthusian's, Fenchel's and the total energy consumed in a lifespan relations.Comment: A accepted for publication in Theoretical Ecology. Comments are welcome ([email protected]

Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods--two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths

Hierarchical ordering of reticular networks

The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.Comment: 9 pages, 5 figures, During preparation of this manuscript the authors became aware of a related work by Katifori and Magnasco, concurrently submitted for publicatio