Kleiber's law

Before Kleiber's observation of the 3/4 power scaling, a 2/3 power scaling was largely anticipated based on the "surface law",[6] which states that the basal metabolism of animals differing in size is nearly proportional to their respective body surfaces. This surface law reasoning originated from simple geometrical considerations. As organisms increase in size, their volume (and thus mass) increases at a much faster rate than their surface area. Explanations for 2⁄3-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. Because many physiological processes, like heat loss and nutrient uptake, were believed to be dependent on the surface area of an organism, it was hypothesized that metabolic rate would scale with the 2/3 power of body mass.[7] Rubner (1883) first demonstrated the law in accurate respiration trials on dogs.[8]

Max Kleiber challenged this notion in the early 1930s. Through extensive research on various animals' metabolic rates, he found that a 3/4 power scaling provided a better fit to the empirical data than the 2/3 power.[9] His findings provided the groundwork for understanding allometric scaling laws in biology, leading to the formulation of the Metabolic Scaling Theory and the later work by West, Brown, and Enquist, among others.

Such an argument does not address the fact that different organisms exhibit different shapes (and hence have different surface-area-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.[10]

West, Brown, and Enquist, (hereafter WBE) proposed a general theory for the origin of many allometric scaling laws in biology. According to the WBE theory, 3⁄4-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.[11]

They then analyze 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 B denotes the basal metabolic rate, Q the total fluid flow, and N the number of minimal tubules,

$${\displaystyle B\propto Q\propto N{\text{.}}}$$

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. WBE show that if the tubules are well-approximated by rigid cylinders, then, to prevent the fluid from "getting clogged" in small cylinders, the total fluid volume V satisfies[12]

$${\displaystyle N^{4}\propto V^{3}{\text{.}}}$$

(Despite conceptual similarities, this condition is inconsistent with Murray's law.[13]) Because blood volume is a fixed fraction of body mass,[11]

$${\displaystyle B\propto M^{\frac {3}{4}}{\text{.}}}$$

Closer analysis suggests that Kleiber's law can vary within and between species. Metabolic rates for smaller animals (birds under 10 kg [22 lb], or insects) typically fit to 2⁄3 much better than 3⁄4; for larger animals, the reverse holds.[14] As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models.[15] In all cases, local fits exhibit exponents in the [2⁄3,3⁄4] range.[16]

Adjustments to the WBE model that retain assumptions of network shape predict larger scaling exponents, worsening the discrepancy with observed data.[17] But one can retain a similar theory by relaxing WBE's assumption of a nutrient transport network that is both fractal and circulatory. 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 2⁄3 and 3⁄4.[16] (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.[14][18]) If larger metabolic rates are evolutionarily favored, then low-mass organisms will prefer to arrange their networks to scale as 2⁄3, but large-mass organisms will prefer to arrange their networks as 3⁄4, which produces the observed curvature.[19]

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 2⁄3. Basal metabolic rate is then the convex combination of these two effects: if the proportion of useful work is f, then the basal metabolic rate should scale as

$${\displaystyle B=f\cdot kM+(1-f)\cdot k'M^{\frac {2}{3}}}$$

where k and k′ are constants of proportionality. k′ in particular describes the surface area ratio of organisms and is approximately 0.1 kJ·h⁻¹·g^−2/3;[20] typical values for f are 15-20%.[21] The theoretical maximum value of f is 21%, because the efficiency of glucose oxidation is only 42%, and half of the ATP so produced is wasted.[20].

A 1999 analysis concluded that biomass production in a given plant scaled with the 3⁄4 power of the plant's mass during the plant's growth,[31] but a 2001 paper that included various types of unicellular photosynthetic organisms found scaling exponents intermediate between 0.75 and 1.00.[32]

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.[33]

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: smaller cells or a slower cellular metabolic rate. Neurons and adipocytes exhibit the former; every other type of cell, the latter.[34] As a result, different organs exhibit different allometric scalings (see table).[10]

Allometric scalings for BMR-vs.-mass in human tissue

Organ	Scaling exponent
Brain	0.7
Kidney	0.85
Liver	0.87
Heart	0.98
Muscle	1.0
Skeleton	1.1