Research Article

Sizing Up Allometric Scaling Theory

  • Van M. Savage equal contributor,

    equal contributor Contributed equally to this work with: Van M. Savage, Eric J. Deeds

    Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America

  • Eric J. Deeds equal contributor,

    equal contributor Contributed equally to this work with: Van M. Savage, Eric J. Deeds

    Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America

  • Walter Fontana mail

    Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America

  • Published: September 12, 2008
  • DOI: 10.1371/journal.pcbi.1000171

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No paradigm shift in allometric scaling

Posted by Gregorio on 19 Jan 2009 at 06:07 GMT

We are met with the statements, "Over the past decade, the WBE model has initiated a paradigm shift in allometric scaling that has led to new applications, new measurements and the refinement of data analysis, and the recognition of connections between several variables that describe organismic physiology. However, WBE has also drawn intense criticism and sparked a heated debate." SDF's paper is in no way fatal to those criticisms that refute the relevance of the model to nature. There has been no paradigm shift that has led to new measurements. Instead new measurements have threatened the model and its archaic supporting data. The new applications of WBE allometric scaling are really new occurrences of it in the literature rather than real-world applications. There are no testable deductive inferences, just an attempt to rescue the model. "...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." The utility of WBE, it seems, concerns only 'reasoning about allometric scaling,' since the deviations from the data are so pronounced. The problem is eloquently expressed in the statement of SDF that "The challenge lies in pinpointing the physiological and evolutionary factors that constrain the shape of networks driving metabolic scaling." Rather than physiology and evolutionary factors that constrain shapes of networks, SDF should be intent upon physics, for it is the laws of the physical sciences that define metabolic scaling. To understand the point being made, one must know more of the historical context in which Kleiber's Law originated, and what was the prevailing account of metabolic energy from 1932 to the present.

In 1995 Berry and Grivell, in "An electrochemical description of metabolism" [Bioelectrochemistry of Cells and Tissues], decried the neglect into which the study of metabolism had slipped over the previous decades. Interest in the field began to wane in the 1950s, amongst medical researchers, with the 1953 paper by Eccles, Hodgkin, and Huxley. That paper codified the nature of nervous system electrical functioning in what became known as the ionic channel model of nerve impulse propagation. The paper was essentially a recapitulation of Bernstein's 1902 hypothesis that cellular electricity was driven by ion concentration gradients. Julius Bernstein appealed to the 1888, thermodynamic equation of Walther Nernst to account for this electrical nature. Nernst clearly stated his equation was not about electrical pressure, but was instead about entropic pressure.

The celebrated paper of Eccles et al. earned them a Nobel in 1963 for the model therein elaborated upon, a confabulation based upon an arrogation of the subject matter of the Nernst equation. Although thirty years later Eccles would abjure the model as inadequate for information traffic in the nervous system (and be completely ignored for it), in the meantime other avenues of study of the nature of bioelectricity fell into desuetude, despite promising findings. Bertil Hille [1991, Ionic Channels of Excitable Membranes] speaks of the 'epiphenomenalist school' that regarded the hypothesized voltages of Bernstein, first detected in the late 1930s, to be an indication of something propagating electrochemically along the nerve fiber. Clinical experiments in the 1940s [Sydney Licht, "The History of Electrotherapy", Therapeutic Electricity and Ultraviolet Radiation, 1969], re-enforced the suspicion that electrochemistry was the electricity of the body. The evidence was supplemented by later findings dealing with the repair of non-union fractures when exposed to DC currents, and, more recently, the destruction of tumors. The semi-conducting properties of liquid crystal proteins are inseparable from the amperage required for electrochemistry. In 2000 three chemists were awarded a Nobel for showing and reproducing semi-conduction on assembled, organic molecules. Instead, these things remain un-related artifacts of research without over-arching theoretical unification.

The Nobel to Eccles et al. was followed, in 1978, by a Nobel to Peter Mitchell for a reprise on the tune of Bernstein's, but that built on the authority of the 1963 award. The field of bioenergetics was fully invested in the idea that bioelectricity involved something called chemiosmosis, driven by what Mitchell called 'proticity.' Proticity involved a proton motive force that was in every way like electrochemistry, but did not involve electrons, or electrochemistry, something that could not be found in the physical sciences. It involved the equating of pH gradient with electrical voltage in a violation of sound scientific methodology. In the field that became known as biophysics, the study of metabolism was confounded. There was no way to simulate or manipulate metabolic proticity by resort to ion concentration gradient or pH manipulation. Serendipitously for the neuroscientists, apparently, was at least the ability to detect and control proticity using regular DC, as in the technique known as patch clamping.

This is why Grivell and Berry were grousing in their 1995 "An electrochemical description of metabolism." Examination of their description reveals it too was seriously hobbled by credibility given to the ionic channel model, and suffers from a misunderstanding of the nature of chemical energy. But they did insist that thermodynamics was inadequate for understanding biological energy. This insistence was made almost sixty years earlier by F.S.C. Northrop in a 1936 Yalell Journal of Biology and Medicine article entitled "The History of Modern Physics in Its Bearing on Biology and Medicine." An earlier Northrop piece, with H.S. Burr, "The Electro-Dynamic Theory of Life" [Quarterly Review of Biology, 1935, no. 10], also had no impact on the model that would finally be accepted for bioelectricity. Hille [1991], as a matter of fact, describes the golden age of biophysics as extending from 1935 to 1953, during which time Bernstein's thermodynamic electricity was considered verified, and granted iconic status. It is taught today as a fundamental of physiology and neuroscience.

Two years after the work of Grivell and Berry, WBE's 1997 Science paper, purporting to justify the preference of 3/4 for the exponent of biomass over 2/3, appeared. In it WBE argued hydrodynamic fractality of vascular delivery of nutrients, had a dimension-contributing affect that boosted metabolic efficiency, and that is why big things lived longer. Fluid dynamics play an important part in biophysics. Hille [1991] describes how bioelecricity is akin to the flow of fluids through small tubes. Fluid dynamics combines the world of Newtonian physics, with the world of thermodynamics, in the modeling of electromagnetism and the flow of bioenergy. This is why the study of metabolism remains a backwater.

Along come SDF, in the present article. They acknowledge deviation of the data from the WBE model calling for the 3/4 exponent in Kleiber's Law, but insist the data is higher, not lower. They address the discrepancies by expanding upon and relaxing the assumptions that underly the WBE model rather than question the model itself. The way they hope to do this is by: 1. considering larger things, perhaps of an infinite biomass, the ideal case, since mostly small mammals have been studied; 2. introducing the idea that constant branching ratios of vascular ramification can vary from one level to another in finite-sized creatures, from area-preserving to area-increasing, where the latter involves 'complexities in the hydrodynamics of blood flow'. This does not remove the discrepancy, nor does it explain it. This is all speculation. No hint is given as to what hydrodynamic complexities could possibly increase the effective area of a capillary, or even what a hydrodynamic complexity might be. SDF admit this approach to ameliorate the discrepancies 'will eventually require detailed hydrodynamical calculations and extensive knowledge of the cardiovascular system,' something they apparently think is missing, and the nature of which they don't specify.

The idea the MR of a biomass is exclusively related to its size, and not to its organization, is something WBE taught. SDF are now suggesting the empirical deviations inherent to the WBE model could be diminished by considering some aspect of structure and organization dealing with flow that is size dependent and area creating. This is a step in the right direction. But their vagueness is an attempt to avoid outright dismissal, and on a par with the claim of WBE that another dimension is introduced by the fractality of vascular branching. They are still helpless before the criticism that blood flow and BMR cannot account for motor behavior, especially of larger creatures, given the universal limits on capillary size for all creatures, even if the discrepancies could be explained away. As noted in another comment, both WBE and SDF have no opinion about the role of thermogenesis in metabolism, surely something that would have a bearing on the value of the exponent.

The regard of metabolism as strictly an electrochemical phenomena involving the electrical properties of organic mass, with ME being a ratio of anabolism to catabolism, eliminates all vagueness. Effectiveness of biological organization to capture and expend energy, and the availability of energy to the biomass with that organization, is contained in the value for ME. This effectiveness is measured in the recharge rate (MR) of that biomass's covalent bonds, which is directly related to the longevity of the biomass. Difference in value for ME is why birds live longer lives than rodents of equal mass, why larger dogs live shorter lives while larger cats live longer lives, and why small mammals live longer on meager rations. Nothing has to be said about fluid dynamics. The numbers show that at less than 25% ME, the exponent for biomass is a negative number, not 3/4. This is a larger deviation than the strategies of SDF to save WBE can possibly account for.