Physics:Kleiber's law

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

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 ​3⁄4 power of the animal's mass.[2] More recently, Kleiber's law has also been shown to apply in plants,[3] suggesting that Kleiber's observation is much more general. Symbolically: if B is the animal's metabolic rate, and M is the animal's mass, then Kleiber's law states that B~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.

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.[4] Max Kleiber first discovered the law when analyzing a large number of independent studies on respiration within individual species.[2] 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",[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

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.[9]

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,[math]\displaystyle{ B\propto Q\propto N\text{.} }[/math] 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[10][math]\displaystyle{ N^4\propto V^3\text{.} }[/math] (Despite conceptual similarities, this condition is inconsistent with Murray's law.[11]) Because blood volume is a fixed fraction of body mass,[9] [math]\displaystyle{ B\propto M^{\frac{3}{4}}\text{.} }[/math]

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 [math]\displaystyle{ B=f\cdot kM+(1-f)\cdot k'M^{\frac{2}{3}} }[/math] 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;[18] typical values for f are 15-20%.[19] 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.[18]

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. Neurons and adipocytes exhibit the former; every other type of cell, the latter.[32] As a result, different organs exhibit different allometric scalings (see table).[8]

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

References

  1. ↑ "Body size and metabolic rate". Physiological Reviews 27 (4): 511–41. October 1947. doi:10.1152/physrev.1947.27.4.511. PMID 20267758. 
  2. ↑ 2.0 2.1 2.2 "Body size and metabolism". Hilgardia 6 (11): 315–353. January 1932. doi:10.3733/hilg.v06n11p315. 
  3. ↑ "Allometric scaling of plant energetics and population density.". Nature 395 (10): 163–165. 1998. doi:10.1038/25977. 
  4. ↑ Scaling: Why is animal size so important?. NY, NY: Cambridge University Press. 1984. ISBN 978-0521266574. https://archive.org/details/scalingwhyisanim0000schm. 
  5. ↑ "A biometric study of basal metabolism in man". Carnegie Inst. Of Wash 6 (279): 31–266. 1919. 
  6. ↑ Thompson, D. W. (1917). On Growth and Form. Cambridge University Press.
  7. ↑ "Über den Einfluss der Körpergrosse auf Stoff- und Kraftwechsel". Zeitschr. F. BioI. 19: 535–562. 1883. 
  8. ↑ 8.0 8.1 8.2 "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. 28 April 2014. doi:10.3390/systems2020186. 
  9. ↑ 9.0 9.1 "A general model for the origin of allometric scaling laws in biology". Science 276 (5309): 122–6. April 1997. doi:10.1126/science.276.5309.122. PMID 9082983. 
  10. ↑ 10.0 10.1 "Demystifying the West, Brown & Enquist model of the allometry of metabolism". Functional Ecology 20 (2): 394–399. 2006. doi:10.1111/j.1365-2435.2006.01136.x. 
  11. ↑ "Pulsatile blood flow, shear force, energy dissipation and Murray's Law". Theoretical Biology & Medical Modelling 3 (1): 31. August 2006. doi:10.1186/1742-4682-3-31. PMID 16923189. 
  12. ↑ 12.0 12.1 "Re-examination of the "3/4-law" of metabolism". Journal of Theoretical Biology 209 (1): 9–27. March 2001. doi:10.1006/jtbi.2000.2238. PMID 11237567. Bibcode: 2001JThBi.209....9D. 
  13. ↑ "Curvature in metabolic scaling". Nature 464 (7289): 753–6. April 2010. doi:10.1038/nature08920. PMID 20360740. Bibcode: 2010Natur.464..753K. 
    But note that a quadratic curve has undesirable theoretical implications; see "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. July 2011. doi:10.1016/j.jtbi.2011.02.011. PMID 21335012. Bibcode: 2011JThBi.280..194M. 
  14. ↑ 14.0 14.1 "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. September 2010. doi:10.1073/pnas.1009974107. PMID 20724663. Bibcode: 2010PNAS..10715816B. 
  15. ↑ "Sizing up allometric scaling theory". PLOS Computational Biology 4 (9): e1000171. September 2008. doi:10.1371/journal.pcbi.1000171. PMID 18787686. Bibcode: 2008PLSCB...4E0171S. 
  16. ↑ "Revisiting the evolutionary origin of allometric metabolic scaling in biology". Functional Ecology 22 (6): 1070–1080. 2008. doi:10.1111/j.1365-2435.2008.01458.x. 
  17. ↑ 17.0 17.1 "The predominance of quarter-power scaling in biology". Functional Ecology 18 (2): 257–282. April 2004. 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 MTemplate:Citefrac scaling. Thus, metabolic scaling relationships are predicted to show a slight curvilinearity at the smallest size range.". 
  18. ↑ 18.0 18.1 "On the thermodynamic origin of metabolic scaling". Scientific Reports 8 (1): 1448. January 2018. doi:10.1038/s41598-018-19853-6. PMID 29362491. Bibcode: 2018NatSR...8.1448B. 
  19. ↑ (in en) Thermodynamic Bases of Biological Processes: Physiological Reactions and Adaptations. Walter de Gruyter. 1990. ISBN 9783110114010. 
  20. ↑ "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?". Functional Ecology 18 (2): 283–9. 2004. doi:10.1111/j.0269-8463.2004.00830.x. 
  21. ↑ "Yes, West, Brown and Enquist's model of allometric scaling is both mathematically correct and biologically relevant". Functional Ecology 19 (4): 735–738. 2005. doi:10.1111/j.1365-2435.2005.01022.x. 
  22. ↑ "Scaling of basal metabolic rate with body mass and temperature in mammals". The Journal of Animal Ecology 79 (3): 610–9. May 2010. doi:10.1111/j.1365-2656.2010.01672.x. PMID 20180875. 
  23. ↑ "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. 1985. doi:10.1016/0300-9629(85)90904-1. PMID 2863065. 
  24. ↑ Bioenergetics and Growth. NY, NY: Reinhold. 1945. 
  25. ↑ "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. 1982-04-01. doi:10.1016/0034-5687(82)90046-9. ISSN 0034-5687. PMID 7111915. 
  26. ↑ "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. October 2009. doi:10.1111/j.1558-5646.2009.00747.x. PMID 19519636. 
  27. ↑ "Mammalian metabolic allometry: do intraspecific variation, phylogeny, and regression models matter?". The American Naturalist 174 (5): 720–33. November 2009. doi:10.1086/606023. PMID 19799501. 
  28. ↑ "Revisiting a model of ontogenetic growth: estimating model parameters from theory and data". The American Naturalist 171 (5): 632–645. 2008. doi:10.1086/587073. 
  29. ↑ "Allometric scaling of production and life-history variation in vascular plants". Nature 401 (6756): 907–911. 28 October 1999. doi:10.1038/44819. ISSN 1476-4687. Bibcode: 1999Natur.401..907E. https://digitalrepository.unm.edu/cgi/viewcontent.cgi?article=1055&context=biol_fsp. 
    Corrigendum published 7 December 2000.
  30. ↑ "A phyletic perspective on the allometry of plant biomass-partitioning patterns and functionally equivalent organ-categories". The New Phytologist 171 (1): 27–40. 2006. doi:10.1111/j.1469-8137.2006.01760.x. PMID 16771980. 
  31. ↑ "Universal scaling of respiratory metabolism, size and nitrogen in plants". Nature 439 (7075): 457–61. January 2006. doi:10.1038/nature04282. PMID 16437113. Bibcode: 2006Natur.439..457R. 
    For a contrary view, see "Biological scaling: does the exception prove the rule?". Nature 445 (7127): E9–10; discussion E10–1. February 2007. doi:10.1038/nature05548. PMID 17268426. Bibcode: 2007Natur.445....9E.  and associated responses.
  32. ↑ "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. March 2007. doi:10.1073/pnas.0611235104. PMID 17360590. Bibcode: 2007PNAS..104.4718S. 

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