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Allometric Scaling
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Allometric Scaling

A special issue of Systems (ISSN 2079-8954).

Deadline for manuscript submissions: closed (28 February 2014) | Viewed by 70483

Special Issue Editors

Dr. Paul S. Agutter

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Guest Editor
Theoretical Medicine and Biology Group, 26 Castle Hill, Glossop, Derbyshire, SK13 7RR, UK
Interests: the aetiology of deep venous thrombosis and chronic venous insufficiency; allometric scaling of metabolic rate; mechanisms of intracellular transport; history and philosophy of medicine and biology
Prof. Dr. Lloyd A. Demetrius

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Guest Editor
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
Interests: ergodic theory of dynamical systems and its applications to the analysis of biological processes at molecular, cellular and population levels; quantum statistics as a formalism to investigate the dynamics of electron transport and proton transduction in cellular metabolism
Prof. Dr. Jack A. Tuszynski

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Guest Editor
Department of Oncology, University of Alberta, Edmonton, AB T6G 1Z2, Canada
Interests: computational drug design; systems biology of cancer and neurodegenerative diseases; tubulin
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Conventionally, allometric scaling is described by a two-parameter equation linking one biological variable (such as basal metabolic rate, B) to another (such as body mass, M):

B= aMß

where a is the proportionality constant and ß is the scaling exponent. There have been debates for some eighty years about the reality of allometric scaling, the values of a and especially ß for different groups of organisms, and how the phenomenon (if real) is to be explained.

Investigations were initially focused on studies of plants and animals. In recent years, metabolism has been shown to play critical roles in the origin of age-related human diseases, such as cancer and neurological disorders. Accordingly, analyses of scaling relations between metabolic rate and body mass have now addressed processes at the cellular and molecular levels.

During the past decade and a half, there have been several efforts to clarify the empirical basis of the scaling rules and to furnish analytical models to explain these rules.

Recent discussion with colleagues, who are invited contributors to this special issue, indicate that both the empirical and theoretical aspects of allometric scaling relations remain highly controversial. The controversy revolves around three main issues:

(a)    The range of values of the scaling exponent.
Is the exponent, ß = 3/4, universal—modulo statistical fluctuations, or can ß assume values which range from 0 to 1?

(b)    The dependency of the proportionality constant on environmental parameters.
Does the dependency on temperature, for example, follow a universal law, or does it depend on the level of biological organization—uni-cells, plants, animals.

(c)    The relation between the scaling exponent and the proportionality constant.
Does there exist a robust empirical relation valid for different group of organisms, between the scaling exponent and the proportionality constant?

The present controversy in the field of allometric scaling indicates that these issues, particularly the problems of theoretical interest, will not easily be resolved.

We hope that the various contributors and the different perspectives they represent will shed some further light on a problem whose implications range from metabolic processes in human diseases, to the study of the structure and organization of ecological networks.

Paul S. Agutter
Lloyd A. Demetrius
Jack Tuszynski
Guest Editors

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Keywords

  • basal metabolic rate
  • metabolic theory of ecology
  • scaling exponent
  • two-factor power law relationship

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Published Papers (6 papers)

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Research

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497 KiB  
Article
Scaling of Metabolic Scaling within Physical Limits
by Douglas S. Glazier
Systems 2014, 2(4), 425-450; https://doi.org/10.3390/systems2040425 - 1 Oct 2014
Cited by 68 | Viewed by 10082
Abstract
Both the slope and elevation of scaling relationships between log metabolic rate and log body size vary taxonomically and in relation to physiological or developmental state, ecological lifestyle and environmental conditions. Here I discuss how the recently proposed metabolic-level boundaries hypothesis (MLBH) provides [...] Read more.
Both the slope and elevation of scaling relationships between log metabolic rate and log body size vary taxonomically and in relation to physiological or developmental state, ecological lifestyle and environmental conditions. Here I discuss how the recently proposed metabolic-level boundaries hypothesis (MLBH) provides a useful conceptual framework for explaining and predicting much, but not all of this variation. This hypothesis is based on three major assumptions: (1) various processes related to body volume and surface area exert state-dependent effects on the scaling slope for metabolic rate in relation to body mass; (2) the elevation and slope of metabolic scaling relationships are linked; and (3) both intrinsic (anatomical, biochemical and physiological) and extrinsic (ecological) factors can affect metabolic scaling. According to the MLBH, the diversity of metabolic scaling relationships occurs within physical boundary limits related to body volume and surface area. Within these limits, specific metabolic scaling slopes can be predicted from the metabolic level (or scaling elevation) of a species or group of species. In essence, metabolic scaling itself scales with metabolic level, which is in turn contingent on various intrinsic and extrinsic conditions operating in physiological or evolutionary time. The MLBH represents a “meta-mechanism” or collection of multiple, specific mechanisms that have contingent, state-dependent effects. As such, the MLBH is Darwinian in approach (the theory of natural selection is also meta-mechanistic), in contrast to currently influential metabolic scaling theory that is Newtonian in approach (i.e., based on unitary deterministic laws). Furthermore, the MLBH can be viewed as part of a more general theory that includes other mechanisms that may also affect metabolic scaling. Full article
(This article belongs to the Special Issue Allometric Scaling)
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505 KiB  
Article
Exploring and Explaining Complex Allometric Relationships: A Case Study on Amniote Testes Mass Allometry
by Colin D. MacLeod
Systems 2014, 2(3), 379-392; https://doi.org/10.3390/systems2030379 - 22 Sep 2014
Cited by 6 | Viewed by 6362
Abstract
While many allometric relationships are relatively simple and linear (when both variables are log transformed), others are much more complex. This paper explores an example of a complex allometric relationship, that of testes mass allometry in amniotes, by breaking it down into linear [...] Read more.
While many allometric relationships are relatively simple and linear (when both variables are log transformed), others are much more complex. This paper explores an example of a complex allometric relationship, that of testes mass allometry in amniotes, by breaking it down into linear components and using this exploration to help explain why this complexity exists. These linear components are two size-independent ones and a size-dependent one, and it is the variations in the interactions between them across different body mass ranges that create the complexity in the overall allometric relationship. While the size-independent limits do not vary between amniote groupings, the slope and the intercept of the size-dependent component does, and it is this that explains why some amniote groups conform to allometric relationships with apparently very different forms. Thus, breaking this complex allometric relationship down into linear components allows its complexity to be explored and explained, and similar processes may prove useful for investigating other complex allometric relationships. In addition, by identifying size-independent upper and lower limits to the proportional investment in specific structures, it allows the prediction of when allometric relationships will remain simple and linear; and when they are likely to develop higher levels of complexity. Full article
(This article belongs to the Special Issue Allometric Scaling)
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274 KiB  
Article
Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals
by Thomas H. Dawson
Systems 2014, 2(2), 168-185; https://doi.org/10.3390/systems2020168 - 22 Apr 2014
Cited by 29 | Viewed by 12559
Abstract
The modeling of the cardiovascular system of mammals is discussed within the framework of governing allometric relations and related scaling laws for mammals. An earlier theory of the writer for resting-state cardiovascular function is reviewed and standard solutions discussed for reciprocal quarter-power relations [...] Read more.
The modeling of the cardiovascular system of mammals is discussed within the framework of governing allometric relations and related scaling laws for mammals. An earlier theory of the writer for resting-state cardiovascular function is reviewed and standard solutions discussed for reciprocal quarter-power relations for heart rate and cardiac output per unit body mass. Variation in the basic cardiac process controlling heart beat is considered and shown to allow alternate governing relations. Results have potential application in explaining deviations from the noted quarter-power relations. The work thus indicates that the cardiovascular systems of all mammals are designed according to the same general theory and, accordingly, that it provides a quantitative means to extrapolate measurements of cardiovascular form and function from small mammals to the human. Various illustrations are included. Work described here also indicates that the basic scaling laws from the theory apply to children and adults, with important applications such as the extrapolation of therapeutic drug dosage requirements from adults to children. Full article
(This article belongs to the Special Issue Allometric Scaling)
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519 KiB  
Article
A Fractional Probability Calculus View of Allometry
by Bruce J. West
Systems 2014, 2(2), 89-118; https://doi.org/10.3390/systems2020089 - 14 Apr 2014
Cited by 6 | Viewed by 7530
Abstract
The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. An apparent corollary of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and [...] Read more.
The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. An apparent corollary of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and distinct from clock time. However, these are only two of the many allometry relations that emerge from empirical studies in the physical, social and life sciences. Herein, we present a theory of allometry that provides a foundation for the allometry relation between a network function and the size that is entailed by the hypothesis that the fluctuations in the two measures are described by a scaling of the joint probability density. The dynamics of such networks are described by the fractional calculus, whose scaling solutions entail the empirically observed allometry relations. Full article
(This article belongs to the Special Issue Allometric Scaling)
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Review

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1363 KiB  
Review
Metabolic Scaling in Complex Living Systems
by Douglas S. Glazier
Systems 2014, 2(4), 451-540; https://doi.org/10.3390/systems2040451 - 1 Oct 2014
Cited by 129 | Viewed by 18047
Abstract
In this review I show that four major kinds of theoretical approaches have been used to explain the scaling of metabolic rate in cells, organisms and groups of organisms in relation to system size. They include models focusing on surface-area related fluxes of [...] Read more.
In this review I show that four major kinds of theoretical approaches have been used to explain the scaling of metabolic rate in cells, organisms and groups of organisms in relation to system size. They include models focusing on surface-area related fluxes of resources and wastes (including heat), internal resource transport, system composition, and various processes affecting resource demand, all of which have been discussed extensively for nearly a century or more. I argue that, although each of these theoretical approaches has been applied to multiple levels of biological organization, none of them alone can fully explain the rich diversity of metabolic scaling relationships, including scaling exponents (log-log slopes) that vary from ~0 to >1. Furthermore, I demonstrate how a synthetic theory of metabolic scaling can be constructed by including the context-dependent action of each of the above modal effects. This “contextual multimodal theory” (CMT) posits that various modulating factors (including metabolic level, surface permeability, body shape, modes of thermoregulation and resource-transport, and other internal and external influences) affect the mechanistic expression of each theoretical module. By involving the contingent operation of several mechanisms, the “meta-mechanistic” CMT differs from most metabolic scaling theories that are deterministically mechanistic. The CMT embraces a systems view of life, and as such recognizes the open, dynamic nature and complex hierarchical and interactive organization of biological systems, and the importance of multiple (upward, downward and reciprocal) causation, biological regulation of resource supply and demand and their interaction, and contingent internal (system) and external (environmental) influences on metabolic scaling, all of which are discussed. I hope that my heuristic attempt at building a unifying theory of metabolic scaling will not only stimulate further testing of all of the various subtheories composing it, but also foster an appreciation that many current models are, at least in part, complementary or even synergistic, rather than antagonistic. Further exploration about how the scaling of the rates of metabolism and other biological processes are interrelated should also provide the groundwork for formulating a general metabolic theory of biology. Full article
(This article belongs to the Special Issue Allometric Scaling)
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494 KiB  
Review
A Sceptics View: “Kleiber’s Law” or the “3/4 Rule” is neither a Law nor a Rule but Rather an Empirical Approximation
by A. J. Hulbert
Systems 2014, 2(2), 186-202; https://doi.org/10.3390/systems2020186 - 28 Apr 2014
Cited by 47 | Viewed by 14308
Abstract
Early studies showed the metabolic rate (MR) of different-sized animals was not directly related to body mass. The initial explanation of this difference, the “surface law”, was replaced by the suggestion that MR be expressed relative to massn, where the scaling [...] Read more.
Early studies showed the metabolic rate (MR) of different-sized animals was not directly related to body mass. The initial explanation of this difference, the “surface law”, was replaced by the suggestion that MR be expressed relative to massn, where the scaling exponent “n” be empirically determined. Basal metabolic rate (BMR) conditions were developed and BMR became important clinically, especially concerning thyroid diseases. Allometry, the technique previously used to empirically analyse relative growth, showed BMR of endotherms varied with 0.73–0.74 power of body mass. Kleiber suggested that mass3/4 be used, partly because of its easy calculation with a slide rule. Later studies have produced a range of BMR scaling exponents, depending on species measured. Measurement of maximal metabolism produced scaling exponents ranging from 0.80 to 0.97, while scaling of mammalian MR during growth display multi-phasic allometric relationships with scaling exponents >3/4 initially, followed by scaling exponents <3/4. There is no universal metabolic scaling exponent. The fact that “allometry” is an empirical technique to analyse relative change and not a biological law is discussed. Relative tissue size is an important determinant of MR. There is also a need to avoid simplistic assumptions regarding the allometry of surface area. Full article
(This article belongs to the Special Issue Allometric Scaling)
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