Kleiber's Law Research Articles

Biological scaling and physics.

Kleiber's law in biology states that the specific metabolic rate (metabolic rate per unit mass) scales as M- 1/4 in terms of the mass M of the organism. A long-standing puzzle is the (- 1/4) power in place of the usual expectation of (- 1/3) based on the surface to volume ratio in three-dimensions. While recent papers by physicists have focused exclusively on geometry in attempting to explain the puzzle, we consider here a specific law of physics that governs fluid flow to show how the (- 1/4) power arises under certain conditions. More generally, such a line of approach that identifies a specific physical law as involved and then examines the implications of a power law may illuminate better the role of physics in biology.

PHYLOGENY AFFECTS ESTIMATION OF METABOLIC SCALING IN MAMMALS

The relationship between body size and metabolic rate is a crucial issue in organismal biology and evolution. There has been considerable debate over whether the scaling exponent of the relationship is 0.75 (Kleiber's Law) or 0.67. Here we show that determination of this exponent for mammals depends on both the evolutionary tree and the regression model used in the comparative analysis. For example, more recent molecular-based phylogenies tend to support a 0.67 exponent, whereas older phylogenies, mostly based on morphological data, suggest a 0.75 exponent. However, molecular phylogenies yield more variable results than morphological phylogenies and thus are not currently helping to resolve the issue.

The Reconstruction of Kleiber's Law at the Organ-Tissue Level

The relationship between resting energy expenditure (REE) (kJ/d) and body mass (M) (kg) is a cornerstone in the study of energy physiology. By expressing REE as a function of body mass observed across mammals, Kleiber formulated the now classic equation: REE = 293M(0.75). The biological processes underlying Kleiber's law have been a topic of long-standing interest and speculation. In the present report we develop a new perspective of Kleiber's law by developing an organ-tissue level REE model consisting of five components: liver, brain, kidneys, heart and remaining tissues. The resting thermal output of each component is the product of the component's specific resting metabolic rate (K) and mass (T). With increasing body size, the K values for all five components had negative exponents and were directly proportional to M(-0.08--0.27), and all component T values were directly proportional to M(0.76-1.01). The resulting exponents of the product (K x T) were M(0.60-0.86) for the five components. Although the (K x T) values of individual components do not scale equally, their combined formula (286M(0.76)) is similar to that observed by Kleiber on the whole-body level. Modeling mammalian REE at the organ-tissue level provides new insights and pathways for future mechanistic explorations of REE-body composition relationships.

Allometric Slopes and Independent Contrasts: A Comparative Test of Kleiber's Law in Primate Ranging Patterns.

At the most fundamental level, the size of an animal's home range is determined by its energy needs. In the absence of confounding variables, home range size should therefore scale with body mass according to Kleiber's exponent for metabolic rate of 0.75. Comparative studies in a wide range of taxa have failed to confirm this prediction: home range size has commonly been found to scale with an exponent significantly >0.75. We develop a comparative measure of metabolic needs that incorporates both mass-specific metabolic rate and social-group size. We test the prediction that home range size in primates scales isometrically with this measure when an appropriate linear model is applied to data corrected for phylogenetic bias. Analyses using species values as data points indicate an exponent consistent with Kleiber's law. This result is misleading, however, because ecological factors confound the analysis, and the slopes within some ecologically homogeneous taxa are steeper. Accordingly, in analyses based on independent contrasts with reduced major axis, slopes are significantly greater than predicted by Kleiber's law. We examine the effects of other variables, and we find that systematic variation in substrate use, home range overlap, and diet account for the steeper than expected relationship between home range size and metabolic needs based on Kleiber's law. We therefore conclude that the scaling of home range size is subject to Kleiber's law but in combination with other factors. These results emphasize that the study of allometry requires detailed attention to statistical models and control of confounding variables.

Rivers, blood and transportation networks

The search for a theory to explain why the metabolic rate of mammals is proportional to the 3/4-power of body mass (Kleiber's law) has recently focused on the nutrient distribution network formed by arteries and capillaries. Banavar et al.1 argue that the law follows from the intrinsic properties of an outward-directed network. But careful analysis of their arguments reveals two implicit assumptions that may not be generally correct. Unless these assumptions are valid for mammalian circulation, these arguments cannot satisfactorily explain Kleiber's empirical relationship.

Allometric Slopes and Independent Contrasts: A Comparative Test of Kleiber's Law in Primate Ranging Patterns

At the most fundamental level, the size of an animal’s home range is determined by its energy needs. In the absence of confounding variables, home range size should therefore scale with body mass according to Kleiber's exponent for metabolic rate of 0.75. Comparative studies in a wide range of taxa have failed to confirm this prediction: home range size has commonly been found to scale with an exponent significantly >0.75. We develop a comparative measure of metabolic needs that incorporates both mass‐specific metabolic rate and social‐group size. We test the prediction that home range size in primates scales isometrically with this measure when an appropriate linear model is applied to data corrected for phylogenetic bias. Analyses using species values as data points indicate an exponent consistent with Kleiber’s law. This result is misleading, however, because ecological factors confound the analysis, and the slopes within some ecologically homogeneous taxa are steeper. Accordingly, in analyses based on independent contrasts with reduced major axis, slopes are significantly greater than predicted by Kleiber's law. We examine the effects of other variables, and we find that systematic variation in substrate use, home range overlap, and diet account for the steeper than expected relationship between home range size and metabolic needs based on Kleiber’s law. We therefore conclude that the scaling of home range size is subject to Kleiber's law but in combination with other factors. These results emphasize that the study of allometry requires detailed attention to statistical models and control of confounding variables.

Evidence for a Self-Thinning Rule in Animals

An animal equivalent of the «−3/2 power rule» for self-thinning in plants has been proposed: the «−4/3 power rule». It is based upon the relationship between animal body weights and their metabolic rate (Kleiber's law). Here, I consider some problems with using this relationship and suggest a more general form for an animal self-thinning law. Some experimental support for such a broader approach is provided

Functional Response During Winter of a Herbivore, The Moose, in Relation to Age and Size

This study was undertaken to provide an estimate of the functional response curve of the moose during winter. The functional response consists of two components: the intake rate while foraging and the time available for food searching. In generalist herbivores an interaction may exist between those two components, because the quality of food eaten (expressed as digestibility) may limit the period available for grazing. Thus, variation in intake rate while foraging in relation to food availability may give a biased estimate of functional response in such animals. We studied the relationship between the different components of functional response during winter for a generalist browser, the moose Alces alces, by comparing the intake rates of free—ranging individuals foraging on three Norwegian winter ranges of different quality. The intake rate was determined by collecting feces defecated during a 24—h period by radio—collared individuals. The amount of food eaten was found by multiplying the dry mass of feces with the inverse of the digestibility of the twigs eaten during that period. No relationship was found between intake rate and either snow depth or temperature. Body mass accounted for a large proportion of the variance in intake rates. The slope in the regression of intake rate on body mass (b = 0.73) was close to the one expected from Kleiber's law. However, when calves and adults were analyzed separately, a significant allometric relationship was found only for the calves. The intake rates of both age groups increased linearly with resource availability. This relationship also remained after accounting for the effects of body mass in a multiple—regression analysis. The intake rate also increased with the digestibility of the twigs eaten. However, twig digestibility was closely correlated with resource abundance in the presence data set, so the separate effects of quality and quantity were difficult to estimate. In the present data set no significant relationship appeared between resource availability and the time used for foraging. Thus, the functional response of winter—foraging moose was linearly related to the resource availability. However, we suggest that the quality of the food also should be included in the computation of functional response curves of large herbivores. Consequently, the functional response should be described in three dimensions as a function of quality and quantity of the food resources.

A comparative study of macroscopic and microscopic dimensions of the intensine in five macropods (Marsupialia: Macropodidae). I. Allometric relationships.

Macroscopic and microscopic dimensions of the intestines in five macropod species (the red kangaroo, Macropus rufus; the eastern grey kangaroo, M. giganteus; the agile wallaby, M. agilis; the swamp wallaby, Wallabia bicolor; and the red-necked pademelon, Thylogale thetis) were investigated allometrically in relation to body mass. In general, the length of the small intestine changed in an area : volume (A:V) compensating manner, but the circumference showed negative allometry such that the overall change in surface areas, both ground and mucosal (including the contribution of villi), were isometric but also included the coefficient derived from 'Kleiber's Law' (0.75) in their 95 and 99% confidence limits, respectively. Villous height and width generally showed no significant correlations with body size, but villous density was lower in large individuals. The allometry coefficient for the length of the large intestine was generally near the A:V compensating value (0.5) in most intraspecific analyses but much higher in the interspecific analysis, suggesting that some factor other than body size might be important (possibly dietary fibre). Caecal length was significantly correlated with body size only in two largest species (M. rufus and M. giganteus) and the interspecific analysis gave a value near A:V compensation).

Size and shape in biology.

Arguments based on elastic stability and flexure, as opposed to the more conventional ones based on yield strength, require that living organisms adopt forms whereby lengths increase as the (2/3) power of diameter. The somatic dimensions of several species of animals and of a wide variety of trees fit this rule well. It is a simple matter to show that energy metabolism during maximal sustained work depends on body cross-sectional area, not total body surface area as proposed by Rubner (1) and many after him. This result and the result requiring animal proportions to change with size amount to a derivation of Kleiber's law, a statement only empirical until now, correlating the metabolically related variables with body weight raised to the (3/4) power. In the present model, biological frequencies are predicted to go inversely as body weight to the (1/4) power, and total body surface areas should correlate with body weight to the (5/8) power. All predictions of the proposed model are tested by comparison with existing data, and the fit is considered satisfactory. In The Fire of Life, Kleiber (5) wrote "When the concepts concerned with the relation of body size and metabolic rate are clarified, . . . then compartive physiology of metabolism will be of great help in solving one of the most intricate and interesting problems in biology, namely the regulation of the rate of cell metabolism." Although Hill (23) realized that "the essential point about a large animal is that its structure should be capable of bearing its own weight and this leaves less play for other factors," he was forced to use an oversimplified "geometric similarity" hypothesis in his important work on animal locomotion and muscular dynamics. It is my hope that the model proposed here promises useful answers in comparisons of living things on both the microscopic and the gross scale, as part of the growing science of form, which asks precisely how organisms are diverse and yet again how they are alike.

The Relationship between Summated Tissue Respiration and Metabolic Rate in the Mouse and Dog

The Relationship between Summated Tissue Respiration and Metabolic Rate in the Mouse and Dog