Kleiber's law
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.
Proposed explanations for the law
Kleiber's law, like many other biological
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",[5] 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.[6] Rubner (1883) first demonstrated the law in accurate respiration trials on dogs.[7]
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.[2] 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.[8]
Exponent 3⁄4
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,
Non-power-law scaling
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.[12] As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models.[13] In all cases, local fits exhibit exponents in the [2⁄3,3⁄4] range.[14]
Modified circulatory models
Adjustments to the WBE model that retain assumptions of network shape predict larger scaling exponents, worsening the discrepancy with observed data.[15] 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.[14] (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.[12][16]) 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.[17]
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 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
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.[20] WBE 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.[21]
Other researchers have also noted that Kozłowski and Konarzewski's criticism of the law tends to focus on precise structural details of the WBE circulatory networks, but that the latter are not essential to the model.[10]
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.[22][23]
A 1932 work by Brody calculated that the scaling was approximately 0.73.[8][24]
A 2004 analysis of field metabolic rates for mammals conclude that they appear to scale with exponent 0.749.[17]
Generalizations
Kleiber's law has been reported to interspecific comparisons and has been claimed not to apply at the intraspecific level.[25] The taxonomic level that body mass metabolic allometry should be studied has been debated.[26][27] Nonetheless, several analyses suggest that while the exponents of the Kleiber's relationship between body size and metabolism can vary at the intraspecific level, statistically, intraspecific exponents in both plants and animals tend to cluster around 3/4.[28]
In other kingdoms
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,[29] but a 2001 paper that included various types of unicellular photosynthetic organisms found scaling exponents intermediate between 0.75 and 1.00.[30]
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.[31]
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.
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
See also
- Allometric law
- Evolutionary physiology
- Metabolic theory of ecology
- Scaling law
- Rate-of-living theory
References
- PMID 20267758.
- ^ .
- doi:10.1038/25977.
- ISBN 978-0521266574.
- ^ Harris JA, Benedict, FG (1919). "A biometric study of basal metabolism in man". Carnegie Inst. Of Wash. 6 (279): 31–266.
- ^ Thompson, D. W. (1917). On Growth and Form. Cambridge University Press.
- ^ Rubner M (1883). "Über den Einfluss der Körpergrosse auf Stoff- und Kraftwechsel". Zeitschr. F. BioI. 19: 535–562.
- ^ .
- ^ S2CID 3140271.
- ^ .
- PMID 16923189.
- ^ S2CID 9168199.
- PMID 21335012.
- ^ PMID 20724663.
- PMID 18787686.
- .
- ^ .
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 M3⁄4 scaling. Thus, metabolic scaling relationships are predicted to show a slight curvilinearity at the smallest size range.
- ^ PMID 29362491.
- ISBN 9783110114010.
- .
- .
- PMID 20180875.
- PMID 2863065.
- ^ Brody S (1945). Bioenergetics and Growth. NY, NY: Reinhold.
- PMID 7111915.
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- doi:10.1086/587073.
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Corrigendum published 7 December 2000. - PMID 16771980.
- 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.S2CID 43905935. and associated responses. - PMID 17360590.
Further reading
- Rau AR (September 2002). "Biological scaling and physics". Journal of Biosciences. 27 (5): 475–8. S2CID 23900176.
- Wang Z, O'Connor TP, Heshka S, Heymsfield SB (November 2001). "The reconstruction of Kleiber's law at the organ-tissue level". The Journal of Nutrition. 131 (11): 2967–70. PMID 11694627.
- Whitfield J (2006). In the Beat of a Heart. Washington, D.C.: Joseph Henry Press. ISBN 9780309096812.
- Glazier DS (February 2010). "A unifying explanation for diverse metabolic scaling in animals and plants". Biological Reviews of the Cambridge Philosophical Society. 85 (1): 111–38. S2CID 28572410.
- Glazier DS (1 October 2014). "Metabolic Scaling in Complex Living Systems". Systems. 2 (4): 451–540. .
- Johnson G (12 January 1999). "Of mice and Elephants". Archived from the original on 3 December 2008.
- Woolley T. "3/4 and Kleiber's Law". Numberphile. Brady Haran. Archived from the original on 2017-05-22. Retrieved 2013-04-01.