# 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 34 power of the animal's mass. Symbolically: if q0 is the animal's metabolic rate, and M is the animal's mass, then Kleiber's law states that q0~M3/4. 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.

The exact value of the exponent in Kleiber's law is unclear, in part because the law currently lacks a single theoretical explanation that is entirely satisfactory.

Kleiber's plot comparing body size to metabolic rate for a variety of species.[1]

## 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.[2] Max Kleiber first discovered the law when analyzing a large number of independent studies on respiration within individual species.[3] Kleiber expected to find an exponent of 23 (for reasons explained below), and was confounded by the exponent of 34 he discovered.[4]

### 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.[5]

### Exponent 2⁄3

Explanations for 23-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 23.[6]

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.[5]

### Exponent 3⁄4

A model due to West, Enquist, and Brown (hereafter WEB) suggests that 34-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.[7]

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 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. 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 V satisfies[8]
${\displaystyle N^{4}\propto V^{3}{\text{.}}}$
(Despite conceptual similarities, this condition is inconsistent with Murray's law.)[9] Because blood volume is a fixed fraction of body mass,[7]
${\displaystyle B\propto M^{\frac {3}{4}}{\text{.}}}$

### 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 10 kg [22 lb], or insects) typically fit to 23 much better than 34; for larger animals, the reverse holds.[6] As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models.[10] In all cases, local fits exhibit exponents in the [23,34] range.[11]

#### Modified circulatory models

Adjustments to the WBE model that retain assumptions of network shape predict larger scaling exponents, worsening the discrepancy with observed data.[12] 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 23 and 34.[11] (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.[6][13]) If larger metabolic rates are evolutionarily favored, then low-mass organisms will prefer to arrange their networks to scale as 23, but large-mass organisms will prefer to arrange their networks as 34, which produces the observed curvature.[14]

#### 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 23. 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;[4] typical values for f are 15-20%.[15] 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.[4]

## Criticism of explanations

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.[16] 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.[17]

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.[8]

## 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.[18][19]

A 1932 work by Brody calculated that the scaling was approximately 0.73.[5][20]

A 2004 analysis of field metabolic rates for mammals conclude that they appear to scale with exponent 0.749.[14]

## Generalizations

Kleiber's law only applies to interspecific comparisons; it (usually) does not apply to intraspecific ones.[21] Indeed, Kleiber's law only appears when studying animals as a whole; scaling exponents within taxonomic subgroupings differ substantially.[22][23]

### In other kingdoms

A 1999 analysis concluded that biomass production in a given plant scaled with the 34 power of the plant's mass during the plant's growth,[24] but a 2001 paper that included various types of unicellular photosynthetic organisms found scaling exponents intermediate between 0.75 and 1.00.[25]

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.[26]

### 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: smaller cells or a slower cellular metabolic rate. Neurons and adipocytess exhibit the former; every other type of cell, the latter.[27] As a result, different organs exhibit different allometric scalings (see table).[5]

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

## References

1. ^ Kleiber M (October 1947). "Body size and metabolic rate". Physiological Reviews. 27 (4): 511–41. doi:10.1152/physrev.1947.27.4.511. PMID 20267758.
2. ^ Schmidt-Nielsen K (1984). Scaling: Why is animal size so important?. NY, NY: Cambridge University Press. ISBN 978-0521266574.
3. ^ Max K (1932). "Body size and metabolism". Hilgardia. 6 (11): 315–351. doi:10.3733/hilg.v06n11p315.
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5. ^ a b c d Hulbert AJ (28 April 2014). "A Sceptics View: "Kleiber's Law" or the "3/4 Rule" is neither a Law nor a Rule but Rather an Empirical Approximation". Systems. 2 (2): 186–202. doi:10.3390/systems2020186.
6. ^ a b c Dodds PS, Rothman DH, Weitz JS (March 2001). "Re-examination of the "3/4-law" of metabolism". Journal of Theoretical Biology. 209 (1): 9–27. arXiv:physics/0007096. Bibcode:2001JThBi.209....9D. doi:10.1006/jtbi.2000.2238. PMID 11237567. S2CID 9168199.
7. ^ a b West GB, Brown JH, Enquist BJ (April 1997). "A general model for the origin of allometric scaling laws in biology". Science. 276 (5309): 122–6. doi:10.1126/science.276.5309.122. PMID 9082983. S2CID 3140271.
8. ^ a b Etienne RS, Apol ME, Olff HA (2006). "Demystifying the West, Brown & Enquist model of the allometry of metabolism". Functional Ecology. 20 (2): 394–399. doi:10.1111/j.1365-2435.2006.01136.x.
9. ^ Painter PR, Edén P, Bengtsson HU (August 2006). "Pulsatile blood flow, shear force, energy dissipation and Murray's Law". Theoretical Biology & Medical Modelling. 3 (1): 31. doi:10.1186/1742-4682-3-31. PMC 1590016. PMID 16923189.
10. ^ Kolokotrones T, Deeds EJ, Fontana W (April 2010). "Curvature in metabolic scaling". Nature. 464 (7289): 753–6. Bibcode:2010Natur.464..753K. doi:10.1038/nature08920. PMID 20360740. S2CID 4374163.
But note that a quadratic curve has undesirable theoretical implications; see MacKay NJ (July 2011). "Mass scale and curvature in metabolic scaling. Comment on: T. Kolokotrones et al., curvature in metabolic scaling, Nature 464 (2010) 753-756". Journal of Theoretical Biology. 280 (1): 194–6. Bibcode:2011JThBi.280..194M. doi:10.1016/j.jtbi.2011.02.011. PMID 21335012.
11. ^ a b Banavar JR, Moses ME, Brown JH, Damuth J, Rinaldo A, Sibly RM, Maritan A (September 2010). "A general basis for quarter-power scaling in animals". Proceedings of the National Academy of Sciences of the United States of America. 107 (36): 15816–20. Bibcode:2010PNAS..10715816B. doi:10.1073/pnas.1009974107. PMC 2936637. PMID 20724663.
12. ^ Savage VM, Deeds EJ, Fontana W (September 2008). "Sizing up allometric scaling theory". PLOS Computational Biology. 4 (9): e1000171. Bibcode:2008PLSCB...4E0171S. doi:10.1371/journal.pcbi.1000171. PMC 2518954. PMID 18787686.
13. ^ Apol ME, Etienne RS, Olff H (2008). "Revisiting the evolutionary origin of allometric metabolic scaling in biology". Functional Ecology. 22 (6): 1070–1080. doi:10.1111/j.1365-2435.2008.01458.x.
14. ^ a b Savage VM, Gillooly JF, Woodruff WH, West GB, Allen AP, Enquist BJ, Brown JH (April 2004). "The predominance of quarter-power scaling in biology". Functional Ecology. 18 (2): 257–282. doi:10.1111/j.0269-8463.2004.00856.x. The original paper by West et al. (1997), which derives a model for the mammalian arterial system, predicts that smaller mammals should show consistent deviations in the direction of higher metabolic rates than expected from M34 scaling. Thus, metabolic scaling relationships are predicted to show a slight curvilinearity at the smallest size range.
15. ^ Zotin AI (1990). Thermodynamic Bases of Biological Processes: Physiological Reactions and Adaptations. Walter de Gruyter. ISBN 9783110114010.
16. ^ Kozlowski J, Konarzewski M (2004). "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?". Functional Ecology. 18 (2): 283–9. doi:10.1111/j.0269-8463.2004.00830.x.
17. ^ Brown JH, West GB, Enquist BJ (2005). "Yes, West, Brown and Enquist's model of allometric scaling is both mathematically correct and biologically relevant". Functional Ecology. 19 (4): 735–738. doi:10.1111/j.1365-2435.2005.01022.x.
18. ^ Clarke A, Rothery P, Isaac NJ (May 2010). "Scaling of basal metabolic rate with body mass and temperature in mammals". The Journal of Animal Ecology. 79 (3): 610–9. doi:10.1111/j.1365-2656.2010.01672.x. PMID 20180875.
19. ^ Hayssen V, Lacy RC (1985). "Basal metabolic rates in mammals: taxonomic differences in the allometry of BMR and body mass". Comparative Biochemistry and Physiology. A, Comparative Physiology. 81 (4): 741–54. doi:10.1016/0300-9629(85)90904-1. PMID 2863065.
20. ^ Brody S (1945). Bioenergetics and Growth. NY, NY: Reinhold.
21. ^ Heusner AA (1982-04-01). "Energy metabolism and body size I. Is the 0.75 mass exponent of Kleiber's equation a statistical artifact?". Respiration Physiology. 48 (1): 1–12. doi:10.1016/0034-5687(82)90046-9. ISSN 0034-5687. PMID 7111915.
22. ^ White CR, Blackburn TM, Seymour RS (October 2009). "Phylogenetically informed analysis of the allometry of Mammalian Basal metabolic rate supports neither geometric nor quarter-power scaling". Evolution; International Journal of Organic Evolution. 63 (10): 2658–67. doi:10.1111/j.1558-5646.2009.00747.x. PMID 19519636. S2CID 16889020.
23. ^ Sieg AE, O'Connor MP, McNair JN, Grant BW, Agosta SJ, Dunham AE (November 2009). "Mammalian metabolic allometry: do intraspecific variation, phylogeny, and regression models matter?". The American Naturalist. 174 (5): 720–33. doi:10.1086/606023. PMID 19799501. S2CID 36468932.
24. ^ Enquist BJ, West GB, Charnov EL, Brown JH (28 October 1999). "Allometric scaling of production and life-history variation in vascular plants". Nature. 401 (6756): 907–911. Bibcode:1999Natur.401..907E. doi:10.1038/44819. ISSN 1476-4687. S2CID 4397261.
Corrigendum published 7 December 2000.
25. ^ Niklas KJ (2006). "A phyletic perspective on the allometry of plant biomass-partitioning patterns and functionally equivalent organ-categories". The New Phytologist. 171 (1): 27–40. doi:10.1111/j.1469-8137.2006.01760.x. PMID 16771980.
26. ^ Reich PB, Tjoelker MG, Machado JL, Oleksyn J (January 2006). "Universal scaling of respiratory metabolism, size and nitrogen in plants". Nature. 439 (7075): 457–61. Bibcode:2006Natur.439..457R. doi:10.1038/nature04282. hdl:11299/176835. PMID 16437113. S2CID 1484450.
For a contrary view, see Enquist BJ, Allen AP, Brown JH, Gillooly JF, Kerkhoff AJ, Niklas KJ, Price CA, West GB (February 2007). "Biological scaling: does the exception prove the rule?". Nature. 445 (7127): E9–10, discussion E10–1. Bibcode:2007Natur.445....9E. doi:10.1038/nature05548. PMID 17268426. S2CID 43905935. and associated responses.
27. ^ Savage VM, Allen AP, Brown JH, Gillooly JF, Herman AB, Woodruff WH, West GB (March 2007). "Scaling of number, size, and metabolic rate of cells with body size in mammals". Proceedings of the National Academy of Sciences of the United States of America. 104 (11): 4718–23. Bibcode:2007PNAS..104.4718S. doi:10.1073/pnas.0611235104. PMC 1838666. PMID 17360590.