The West-Brown-Enquist Model for Allometric Scaling

Chapter 2 of the 4th edition of Intermediate Physics for Medicine and Biology ends with a section on “Food Consumption, Basal Metabolic Rate, and Scaling.” Here Russ Hobbie and I discuss the famous “3/4-power law” (also known as Kleiber’s law), which relates the metabolic rate R (in Watts) to the body mass M (in kg) by the equation R = 4.1 M^0.751 (Eq. 2.32c in our book). We conclude the section by writing

A number of models have been proposed to explain a 3/4-power dependence [McMahon (1973);  Peters (1983); West et al. (1999); Banavar et al. (1999)]. West et al. argue that the 3/4-power dependence is universal: they derive it from a model that supplies nutrients through a branching network that reaches all parts of the organism, minimizes the energy required for distribution, and ends in capillaries (or terminal xylem in plants) that are all the same size. Whether it is universal is still debated [Kozlowski and Konarzewski (2004)]. West and Brown (2004) review quarter-power scaling in a variety of circumstances.

When we wrote this paragraph, the origin of the 3/4th power law was still being hotly debated in the literature. Readers of Intermediate Physics for Medicine and Biology might like an update.

First, this work is highly cited. West, Brown, and Enquist’s first paper in Science (“A General Model for the Origin of Allometric Scaling Laws in Biology,” Volume 276, Pages 122–126, 1997; not cited in our book) now has over 1000 citations. Their second paper, which we list in the references at the end of Chapter 2, has nearly 400 citations. The paper by Banavar, Maritan and Rinaldo cited in Chapter 2 has over 200 citations. Clearly, these studies have had a major impact on the field.

Second, the work has generated quite a bit of discussion in the press. The December 2008 issue of The Scientist has an article by Bob Grant titled “The Powers That Might Be” about West and his colleagues and how they have coped with criticisms of their work. An interview with Geoffrey West can be found at physicsworld.com, and one with Brian Enquist at www.in-cities.com. In 2004, John Whitfield published a feature in the open access journal PLOS Biology reviewing the field (“open access” means that anyone can access the paper over the internet, without the need for a journal subscription).

Third, several recent papers in scientific journals have addressed this topic. Savage et al. have analyzed what they refer to as the WBE model in an article appearing in the open access journal PLOS Computational Biology (Volume 4, Article e1000171, 2008). The authors’ summary states

The rate at which an organism produces energy to live increases with body mass to the 3/4 power. Ten years ago West, Brown, and Enquist posited that this empirical relationship arises from the structure and dynamics of resource distribution networks such as the cardiovascular system. Using assumptions that capture physical and biological constraints, they defined a vascular network model that predicts a 3/4 scaling exponent. In our paper we clarify that this model generates the 3/4 exponent only in the limit of infinitely large organisms. Our calculations indicate that in the finite-size version of the model metabolic rate and body mass are not related by a pure power law, which we show is consistent with available data. We also show that this causes the model to produce scaling exponents significantly larger than the observed 3/4. We investigate how changes in certain assumptions about network structure affect the scaling exponent, leading us to identify discrepancies between available data and the predictions of the finite-size model. This suggests that the model, the data, or both, need reassessment. The challenge lies in pinpointing the physiological and evolutionary factors that constrain the shape of networks driving metabolic scaling.

In another paper, published in the December 2006 issue of Physics of Life Reviews (Volume 3, Pages 229–261), de Silva et al. write that

One of the most pervasive laws in biology is the allometric scaling, whereby a biological variable Y is related to the mass M of the organism by a power law, Y = Y0M^b, where b is the so-called allometric exponent. The origin of these power laws is still a matter of dispute mainly because biological laws, in general, do not follow from physical ones in a simple manner. In this work, we review the interspecific allometry of metabolic rates, where recent progress in the understanding of the interplay between geometrical, physical and biological constraints has been achieved.

For many years, it was a universal belief that the basal metabolic rate (BMR) of all organisms is described by Kleiber’s law (allometric exponent b = 3/4). A few years ago, a theoretical basis for this law was proposed, based on a resource distribution network common to all organisms. Nevertheless, the 3/4-law has been questioned recently. First, there is an ongoing debate as to whether the empirical value of b is 3/4 or 2/3, or even nonuniversal. Second, some mathematical and conceptual errors were found [in] these network models, weakening the proposed theoretical arguments. Another pertinent observation is that the maximal aerobically sustained metabolic rate of endotherms scales with an exponent larger than that of BMR. Here we present a critical discussion of the theoretical models proposed to explain the scaling of metabolic rates, and compare the predicted exponents with a review of the experimental literature. Our main conclusion is that although there is not a universal exponent, it should be possible to develop a unified theory for the common origin of the allometric scaling laws of metabolism.

Now, five years after we included the topic in Intermediate Physics for Medicine and Biology, the controversy continues. It makes for a wonderful example of how ideas from fundamental physics can elucidate biological laws, and a warning about how complicated and messy biology can be, limiting the application of simple models. I can't tell you how this debate will ultimately be resolved. But it provides a fascinating case study in the interaction of physics and biology.