Kleiber's law

Kleiber's law, named after Max Kleiber for his biological research in the early 1930s, states that, in the vast majority of cases, the basal metabolic rate scales to the 3⁄4 power of the animal's mass. More precisely: using w to denote the mass of the animal in kilograms, then BMR ≈ 70w3/4 kilocalories per day, or BMR ≈ 3.4w3/4 watts. Thus, over the same time span, a cat having a mass 100 times that of a mouse will consume only about 32 times the energy the mouse uses. It is presently unclear if the value of the exponent in Kleiber's law is correct, in part because the law currently lacks a single theoretical explanation that is entirely satisfactory. More recently, Kleiber's law has also been shown to apply in plants, suggesting that Kleiber's observation is much more general.

Proposed explanations for the law Kleiber's law, like many other biological allometric laws, is a consequence of the physics and/or geometry of circulatory systems in biology. 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 2⁄3 (for reasons explained below), and was confounded by the discovery of a 3⁄4 exponent.

Historical context and the 2⁄3 scaling surface law Before Kleiber's observation of the 3/4 power scaling, a 2/3 power scaling was largely anticipated based on the "surface law", 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. Rubner (1883) first demonstrated the law in accurate respiration trials on dogs.

Kleiber's contribution 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. 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.

Exponent 3⁄4 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). WBE 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. The model assumes that the energy dissipated is minimized and that the terminal tubes do not vary with body size. It provides a complete analysis of numerous anatomical and physiological scaling relations for circulatory systems in biology that generally agree with empirical data. More generally, the model predicts the structural and functional properties of vertebrate cardiovascular and respiratory systems, plant vascular systems, insect tracheal tubes, and other distribution networks. 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, B ∝ Q ∝ N . {\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 N 4 ∝ V 3 . {\displaystyle N^{4}\propto V^{3}{\text{.}}} (Despite conceptual similarities, this condition is inconsistent with Murray's law.) Because blood volume is a fixed fraction of body mass, B ∝ M 3 4 . {\displaystyle B\propto M^{\frac {3}{4}}{\text{.}}}

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