Kleiber's law
Kleiber's law, named after Max Kleiber for his biology work in the early 1930s, is the observation that, for the vast majority of animals, an animal's metabolic rate scales to the power of the animal's mass. Symbolically: if formula_1 is the animal's metabolic rate, and formula_2 the animal's mass, then Kleiber's law states that formula_3. Thus, over the same timespan, a cat having a mass 100 times that of a mouse will consume only about 32 times the energy the mouse uses. The exact value of the exponent in Kleiber's  law is unclear, in part because there is currently no completely satisfactory theoretical explanation for the law. Proposed explanations for the law. Kleiber's law, as many other biological allometric laws, is a consequence of the physics and/or geometry of animal circulatory systems. Max Kleiber first discovered the law when analyzing a large number of independent studies on respiration within individual species. Kleiber expected to find an exponent of (for reasons explained below), and was confounded by the exponent of he discovered. Heuristic explanation. One explanation for Kleiber's law lies in the difference between structural and growth mass. Structural  mass involves maintenance costs, reserve mass does not. Hence, small adults of one species respire more per unit of weight than large adults of another species because a larger fraction of their body mass consists of structure rather than reserve. Within each species, young (i.e., small) organisms respire more per unit of weight than old (large) ones of the same species because of the overhead costs of growth. Exponent. Explanations for -scaling tend to assume that metabolic rates scale to avoid heat exhaustion. Because bodies lose heat passively via their surface, but produce heat metabolically throughout their mass, the metabolic  rate must scale in such a way as to counteract the square–cube law. The precise exponent to do so is . Such an argument does not address the fact that different organisms exhibit different shapes (and hence have different surface-to-volume ratios, even when scaled to the same size). Reasonable estimates for organisms' surface area do appear to scale linearly with the metabolic rate. Exponent. A model due to West, Enquist, and Brown (hereafter WEB) suggests that -scaling arises because of efficiency in nutrient distribution and transport throughout an organism. In most organisms, metabolism is supported by a circulatory  system featuring branching tubules (i.e., plant vascular systems, insect tracheae, or the human cardiovascular system). WEB claim that (1) metabolism should scale proportionally to nutrient flow (or, equivalently, total fluid flow) in this circulatory system and (2) in order to minimize the energy dissipated in transport, the volume of fluid used to transport nutrients (i.e., blood volume) is a fixed fraction of body mass. They then proceed by analyzing the consequences of these two claims at the level of the smallest circulatory tubules (capillaries, alveoli, etc.). Experimentally, the volume contained in those smallest tubules is constant across a wide range  of masses. Because fluid flow through a tubule is determined by the volume thereof, the total fluid flow is proportional to the total number of smallest tubules. Thus, if denotes the basal metabolic rate, the total fluid flow, and the number of minimal tubules,formula_4.Circulatory systems do not grow by simply scaling proportionally larger; they become more deeply nested. The depth of nesting depends on the self-similarity exponents of the tubule dimensions, and the effects of that depth depend on how many "child" tubules each branching produces. Connecting these values to macroscopic quantities depends (very loosely) on a precise  model of tubules. WEB show that, if the tubules are well-approximated by rigid cylinders, then, in order to prevent the fluid from "getting clogged" in small cylinders, the total fluid volume satisfiesformula_5.Because blood volume is a fixed fraction of body mass, formula_6. Non-power-law scaling. Closer analysis suggests that Kleiber's law does not hold over a wide variety of scales. Metabolic rates for smaller animals (birds under , or insects) typically fit to much better than ; for larger animals, the reverse holds. As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward , and fit better to quadratic models. In all cases, local fits exhibit exponents in the range. Modified circulatory models. Adjustments to the WBE model that retain assumptions of network shape predict "larger" scaling exponents, worsening the discrepancy with observed data. But one can retain a similar theory by relaxing WBE's assumption of a nutrient transport network that is both fractal and circulatory. (WBE argued that fractal circulatory networks would necessarily evolve to minimize energy used for transport, but other researchers argue that their derivation contains subtle errors.) Different networks are less efficient, in that they exhibit a lower scaling  exponent, but a metabolic rate determined by nutrient transport will always exhibit scaling between and . If larger metabolic rates are evolutionarily favored, then low-mass organisms will prefer to arrange their networks to scale as , but large-mass organisms will prefer to arrange their networks as , which produces the observed curvature. Modified thermodynamic models. An alternative model notes that metabolic rate does not solely serve to generate heat. Metabolic rate contributing solely to useful work should scale with power 1 (linearly), whereas metabolic rate contributing to heat generation should be limited by surface area and scale with power . Basal metabolic  rate is then the convex combination of these two effects: if the proportion of useful work is , then the basal metabolic rate should scale asformula_7where and are constants of proportionality. in particular describes the surface area ratio of organisms and is approximately ; typical values for are 15-20%. The theoretical maximum value of is 21%, because the efficiency of glucose oxidation is only 42%, and half of the ATP so produced is wasted. Experimental support. Analyses of variance for a variety of physical variables suggest that although most variation in basal metabolic rate is determined by mass, additional variables with  significant effects include body temperature and taxonomic order. A 1932 work by Brody calculated that the scaling was approximately 0.73. A 2004 analysis of field metabolic rates for mammals conclude that they appear to scale with exponent 0.749. Criticism of the law. Kozłowski and Konarzewski have argued that attempts to explain Kleiber's law via any sort of limiting factor is flawed, because metabolic rates vary by factors of 4-5 between rest and activity. Hence any limits that affect the scaling of "basal" metabolic rate would in fact make elevated metabolism — and hence all animal activity — impossible. WEB conversely  argue that animals may well optimize for minimal transport energy dissipation during rest, without abandoning the ability for less efficient function at other times. Other researchers have also noted that Kozłowski and Konarzewski's criticism of the law tends to focus on precise structural details of the WEB circulatory networks, but that the latter are not essential to the model. Kleiber's law only appears when studying animals as a whole; scaling exponents within taxonomic subgroupings differ substantially. Generalizations. Kleiber's law only applies to interspecific comparisons; it (usually) does not apply to intraspecific ones. In other kingdoms. A 1999  analysis concluded that biomass production in a given plant scaled with the power of the plant's mass during the plant's growth, but a 2001 paper that included various types of unicellular photosynthetic organisms found scaling exponents intermediate between 0.75 and 1.00. A 2006 paper in "Nature" argued that the exponent of mass is close to 1 for plant seedlings, but that variation between species, phyla, and growth conditions overwhelm any "Kleiber's law"-like effects. Intra-organismal results. Because cell protoplasm appears to have constant density across a range of organism masses, a consequence of Kleiber's law is  that, in larger species, less energy is available to each cell volume. Cells appear to cope with this difficulty via choosing one of the following two strategies: a slower cellular metabolic rate, or smaller cells. The latter strategy is exhibited by neurons and adipocytes; the former by every other type of cell. As a result, different organs exhibit different allometric scalings (see table).