View source for Physics:Kleiber's law

[[File:Kleiber1947.svg|thumb|upright=1.5|Kleiber's plot comparing body size to metabolic rate for a variety of species.<ref>{{cite journal | vauthors = Kleiber M | title = Body size and metabolic rate | journal = Physiological Reviews | volume = 27 | issue = 4 | pages = 511–41 | date = October 1947 | pmid = 20267758 | doi = 10.1152/physrev.1947.27.4.511 }}</ref> ]] '''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 [[Basal metabolic rate|metabolic rate]] scales to the {{Frac|3|4}} power of the animal's mass.<ref name="hilgardia">{{cite journal | vauthors = Kleiber M | title = Body size and metabolism | journal = Hilgardia  | volume = 6 | issue = 11 | pages = 315–353 | date = January 1932 | doi = 10.3733/hilg.v06n11p315 | doi-access = free }}</ref> More recently, Kleiber's law has also been shown to apply in plants,<ref>{{cite journal | vauthors = Enquist BJ, Brown JH, West GB  | title = Allometric scaling of plant energetics and population density. | journal = Nature | volume = 395 | issue = 10 | pages = 163–165 | date = 1998 | doi = 10.1038/25977 }}</ref> suggesting that Kleiber's observation is much more general. Symbolically: if {{math|''B''}} is the animal's metabolic rate, and {{mvar|M}} is the animal's mass, then Kleiber's law states that {{math|''B''~''M''<sup>3/4</sup>}}. 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:Physics|physics]] and/or [[Geometry|geometry]] of [[Biology:Circulatory system|circulatory system]]s in biology.<ref>{{cite book|title=Scaling: Why is animal size so important?| vauthors = Schmidt-Nielsen K |publisher=Cambridge University Press|year=1984|isbn=978-0521266574|location=NY, NY|url-access=registration|url=https://archive.org/details/scalingwhyisanim0000schm}}</ref> Max Kleiber first discovered the law when analyzing a large number of independent studies on respiration within individual species.<ref name="hilgardia"/> Kleiber expected to find an exponent of {{Frac|2|3}} (for reasons explained below), and was confounded by the discovery of a {{Frac|3|4}} exponent.

=== Historical context and the {{Frac|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",<ref>{{cite journal| vauthors = Harris JA, Benedict, FG  |year=1919|title=A biometric study of basal metabolism in man | journal=Carnegie Inst. Of Wash|volume=6| issue = 279 |pages=31–266}}</ref> 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 {{Frac|2|3}}-scaling tend to assume that metabolic rates scale to avoid [[Medicine:Heat exhaustion|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|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.<ref name="surfaceLaw1">Thompson, D. W. (1917). On Growth and Form. Cambridge University Press.</ref> Rubner (1883) first demonstrated the law in accurate respiration trials on dogs.<ref>{{cite journal| vauthors = Rubner M |year=1883|title=Über den Einfluss der Körpergrosse auf Stoff- und Kraftwechsel| journal=Zeitschr. F. BioI.|volume=19|pages=535–562}}</ref>

=== 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.<ref name="hilgardia"/> 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 [[Biology:Surface-area-to-volume ratio|surface-area-to-volume ratio]]s, even when scaled to the same size). Reasonable estimates for organisms' surface area do appear to scale linearly with the metabolic rate.<ref name="History">{{cite journal| vauthors = Hulbert AJ |date=28 April 2014|title=A {{As written|Sceptics}} View: "Kleiber's Law" or the "3/4 Rule" is neither a Law nor a Rule but Rather an Empirical Approximation|journal=Systems | volume=2|issue=2|pages=186–202|doi=10.3390/systems2020186|doi-access=free}}</ref>

=== Exponent {{Frac|3|4}} ===
West, Brown, and Enquist, (hereafter WBE) proposed a general theory for the origin of many [[Biology:Allometry|allometric scaling laws]] in biology. According to the WBE theory, {{Frac|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.<ref name="WBEModel">{{cite journal | vauthors = West GB, Brown JH, Enquist BJ | title = A general model for the origin of allometric scaling laws in biology | journal = Science | volume = 276 | issue = 5309 | pages = 122–6 | date = April 1997 | pmid = 9082983 | doi = 10.1126/science.276.5309.122 | s2cid = 3140271 }}</ref>  

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 {{Mvar|B}} denotes the basal metabolic rate, {{Mvar|Q}} the total fluid flow, and {{Mvar|N}} the number of minimal tubules,<math display="block">B\propto Q\propto N\text{.}</math>  Circulatory systems do not grow by simply scaling proportionally larger; they become [[Fractal|more deeply nested]]. The depth of nesting depends on the [[Fractal dimension|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 [[Physics:Incompressible flow|"getting clogged"]] in small cylinders, the total fluid volume {{Mvar|V}} satisfies<ref name="Reformulated" /><math display="block">N^4\propto V^3\text{.}</math>  (Despite conceptual similarities, this condition is inconsistent with [[Biology:Murray's law|Murray's law]].<ref>{{cite journal | vauthors = Painter PR, Edén P, Bengtsson HU | title = Pulsatile blood flow, shear force, energy dissipation and Murray's Law | journal = Theoretical Biology & Medical Modelling | volume = 3 | issue = 1 | pages = 31 | date = August 2006 | pmid = 16923189 | pmc = 1590016 | doi = 10.1186/1742-4682-3-31 | doi-access = free }}</ref>) Because blood volume is a fixed fraction of body mass,<ref name="WBEModel" /> <math display="block">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 {{Cvt|10|kg||disp=sqbr}}, or insects) typically fit to {{frac|2|3}} much better than {{frac|3|4}}; for larger animals, the reverse holds.<ref name="Empirics">{{cite journal | vauthors = Dodds PS, Rothman DH, Weitz JS | title = Re-examination of the "3/4-law" of metabolism | journal = Journal of Theoretical Biology | volume = 209 | issue = 1 | pages = 9–27 | date = March 2001 | pmid = 11237567 | doi = 10.1006/jtbi.2000.2238 | arxiv = physics/0007096 | bibcode = 2001JThBi.209....9D | s2cid = 9168199 }}</ref> As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models.<ref>{{cite journal | vauthors = Kolokotrones T, Deeds EJ, Fontana W | title = Curvature in metabolic scaling | journal = Nature | volume = 464 | issue = 7289 | pages = 753–6 | date = April 2010 | pmid = 20360740 | doi = 10.1038/nature08920 | bibcode = 2010Natur.464..753K | s2cid = 4374163 }}  <br />But note that a quadratic curve has undesirable theoretical implications; see {{cite journal | vauthors = MacKay NJ | title = Mass scale and curvature in metabolic scaling. Comment on: T. Kolokotrones et al., curvature in metabolic scaling, Nature 464 (2010) 753-756 | journal = Journal of Theoretical Biology | volume = 280 | issue = 1 | pages = 194–6 | date = July 2011 | pmid = 21335012 | doi = 10.1016/j.jtbi.2011.02.011 | bibcode = 2011JThBi.280..194M }}</ref> In all cases, local fits exhibit exponents in the {{Math|[{{frac|2|3}},{{frac|3|4}}]}} range.<ref name="ArbNetworks" />  

==== Modified circulatory models ====
Adjustments to the WBE model that retain assumptions of network shape predict ''larger'' scaling exponents, worsening the discrepancy with observed data.<ref>{{cite journal | vauthors = Savage VM, Deeds EJ, Fontana W | title = Sizing up allometric scaling theory | journal = PLOS Computational Biology | volume = 4 | issue = 9 | pages = e1000171 | date = September 2008 | pmid = 18787686 | pmc = 2518954 | doi = 10.1371/journal.pcbi.1000171 | bibcode = 2008PLSCB...4E0171S | doi-access = free }}</ref> But one can retain a similar theory by relaxing WBE's assumption of a nutrient transport network that is both [[Fractal|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 {{frac|2|3}} and {{frac|3|4}}.<ref name="ArbNetworks">{{cite journal | vauthors = Banavar JR, Moses ME, Brown JH, Damuth J, Rinaldo A, Sibly RM, Maritan A | title = A general basis for quarter-power scaling in animals | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 107 | issue = 36 | pages = 15816–20 | date = September 2010 | pmid = 20724663 | pmc = 2936637 | doi = 10.1073/pnas.1009974107 | bibcode = 2010PNAS..10715816B | doi-access = free }}</ref> (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.<ref name="Empirics" /><ref>{{cite journal| vauthors = Apol ME, Etienne RS, Olff H |date=2008|title=Revisiting the evolutionary origin of allometric metabolic scaling in biology |journal=Functional Ecology | volume=22|issue=6|pages=1070–1080|doi=10.1111/j.1365-2435.2008.01458.x |doi-access=free}}</ref>) If larger metabolic rates are evolutionarily favored, then low-mass organisms will prefer to arrange their networks to scale as {{frac|2|3}}, but large-mass organisms will prefer to arrange their networks as {{frac|3|4}}, which produces the observed curvature.<ref name = "Savage_2004">{{cite journal| vauthors = Savage VM, Gillooly JF, Woodruff WH, West GB, Allen AP, Enquist BJ, Brown JH |title=The predominance of quarter-power scaling in biology |journal=Functional Ecology | date = April 2004 |volume=18|issue=2|pages=257–282|doi=10.1111/j.0269-8463.2004.00856.x |quote=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 {{math|''M''{{sup|{{citefrac|3|4}}}}}} scaling. Thus, metabolic scaling relationships are predicted to show a slight curvilinearity  at  the  smallest  size  range.|doi-access=free }}</ref>

==== 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 {{Frac|2|3}}. Basal metabolic rate is then the [[Convex combination|convex combination]] of these two effects: if the proportion of useful work is {{Mvar|f}}, then the basal metabolic rate should scale as <math display="block">B=f\cdot kM+(1-f)\cdot k'M^{\frac{2}{3}}</math> where {{Mvar|k}} and {{Math|''k''{{prime}}}} are constants of proportionality. {{Math|''k''{{prime}}}} in particular describes the surface area ratio of organisms and is approximately {{Math|0.1&nbsp;kJ·h{{sup|−1}}·g{{sup|−2/3}}}};<ref name="balles">{{cite journal | vauthors = Ballesteros FJ, Martinez VJ, Luque B, Lacasa L, Valor E, Moya A | title = On the thermodynamic origin of metabolic scaling | journal = Scientific Reports | volume = 8 | issue = 1 | pages = 1448 | date = January 2018 | pmid = 29362491 | pmc = 5780499 | doi = 10.1038/s41598-018-19853-6 | bibcode = 2018NatSR...8.1448B }}</ref> typical values for {{Mvar|f}} are 15-20%.<ref>{{cite book |title= Thermodynamic Bases of Biological Processes: Physiological Reactions and Adaptations | vauthors = Zotin AI |date=1990|publisher=Walter de Gruyter|isbn=9783110114010|language=en}}</ref> The theoretical maximum value of {{Mvar|f}} is 21%, because the efficiency of glucose oxidation is only 42%, and half of the [[Chemistry:Adenosine triphosphate|ATP]] so produced is wasted.<ref name="balles" /> 

== 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.<ref>{{cite journal|vauthors=Kozlowski J, Konarzewski M|year=2004|title=Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?|journal=Functional Ecology|volume=18|issue=2|pages=283–9|doi=10.1111/j.0269-8463.2004.00830.x|doi-access=free}}</ref> 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.<ref>{{cite journal| vauthors = Brown JH, West GB, Enquist BJ |date=2005|title=Yes, West, Brown and Enquist's model of allometric scaling is both mathematically correct and biologically relevant |journal=Functional Ecology | volume=19|issue=4|pages=735–738|doi=10.1111/j.1365-2435.2005.01022.x |doi-access=free}}</ref>

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.<ref name="Reformulated">{{cite journal| vauthors = Etienne RS, Apol ME, Olff HA |date=2006 |title=Demystifying the West, Brown & Enquist model of the allometry of metabolism |journal=Functional Ecology | volume=20 |issue=2 |pages=394–399 |doi=10.1111/j.1365-2435.2006.01136.x |doi-access=free }}</ref>

== Experimental support ==
[[Analysis of variance|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.<ref>{{cite journal | vauthors = Clarke A, Rothery P, Isaac NJ | title = Scaling of basal metabolic rate with body mass and temperature in mammals | journal = The Journal of Animal Ecology | volume = 79 | issue = 3 | pages = 610–9 | date = May 2010 | pmid = 20180875 | doi = 10.1111/j.1365-2656.2010.01672.x | doi-access = free }}</ref><ref>{{cite journal | vauthors = Hayssen V, Lacy RC | title = Basal metabolic rates in mammals: taxonomic differences in the allometry of BMR and body mass | journal = Comparative Biochemistry and Physiology. A, Comparative Physiology | volume = 81 | issue = 4 | pages = 741–54 | pmid = 2863065 | doi = 10.1016/0300-9629(85)90904-1 | year = 1985 }}</ref>

A 1932 work by Brody calculated that the scaling was approximately 0.73.<ref name="History" /><ref>{{cite book|title=Bioenergetics and Growth| vauthors = Brody S |publisher=Reinhold|year=1945|location=NY, NY}}</ref>

A 2004 analysis of field metabolic rates for mammals conclude that they appear to scale with exponent 0.749.<ref name = "Savage_2004" /> 

== Generalizations ==
Kleiber's law has been reported to interspecific comparisons and has been claimed not to apply at the intraspecific level.<ref>{{cite journal| vauthors = Heusner AA |date=1982-04-01|title=Energy metabolism and body size I. Is the 0.75 mass exponent of Kleiber's equation a statistical artifact?|journal=Respiration Physiology|volume=48|issue=1|pages=1–12|doi=10.1016/0034-5687(82)90046-9|pmid=7111915|issn=0034-5687}}</ref> The taxonomic level that body mass metabolic allometry should be studied has been debated.<ref>{{cite journal | vauthors = White CR, Blackburn TM, Seymour RS | title = Phylogenetically informed analysis of the allometry of Mammalian Basal metabolic rate supports neither geometric nor quarter-power scaling | journal = Evolution; International Journal of Organic Evolution | volume = 63 | issue = 10 | pages = 2658–67 | date = October 2009 | pmid = 19519636 | doi = 10.1111/j.1558-5646.2009.00747.x | s2cid = 16889020 | doi-access = free }}</ref><ref>{{cite journal | vauthors = Sieg AE, O'Connor MP, McNair JN, Grant BW, Agosta SJ, Dunham AE | title = Mammalian metabolic allometry: do intraspecific variation, phylogeny, and regression models matter? | journal = The American Naturalist | volume = 174 | issue = 5 | pages = 720–33 | date = November 2009 | pmid = 19799501 | doi = 10.1086/606023 | s2cid = 36468932 }}</ref> 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.<ref>{{cite journal | vauthors = Moses ME, Hou C, Woodruff WH, West GB, Nekola JC, Zuo W, Brown JH| title = Revisiting a model of ontogenetic growth: estimating model parameters from theory and data | journal = The American Naturalist | volume = 171 | issue = 5 | pages = 632–645 | date = 2008 |  doi = 10.1086/587073 }}</ref>

=== In other kingdoms ===
A 1999 analysis concluded that biomass production in a given plant scaled with the {{Frac|3|4}} power of the plant's mass during the plant's growth,<ref>{{cite journal| vauthors = Enquist BJ, West GB, Charnov EL, Brown JH |date=28 October 1999|title=Allometric scaling of production and life-history variation in vascular plants|journal=Nature | volume=401|issue=6756|pages=907–911|doi=10.1038/44819|issn=1476-4687|bibcode=1999Natur.401..907E|s2cid=4397261|url=https://digitalrepository.unm.edu/cgi/viewcontent.cgi?article=1055&context=biol_fsp}}  <br />
[https://www.nature.com/articles/35047140 Corrigendum] published 7 December 2000.</ref> but a 2001 paper that included various types of unicellular photosynthetic organisms found scaling exponents intermediate between 0.75 and 1.00.<ref>{{cite journal | vauthors = Niklas KJ | title = A phyletic perspective on the allometry of plant biomass-partitioning patterns and functionally equivalent organ-categories | journal = The New Phytologist | volume = 171 | issue = 1 | pages = 27–40 | date = 2006 | pmid = 16771980 | doi = 10.1111/j.1469-8137.2006.01760.x | doi-access = free }}</ref>  

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.<ref>{{cite journal | vauthors = Reich PB, Tjoelker MG, Machado JL, Oleksyn J | title = Universal scaling of respiratory metabolism, size and nitrogen in plants | journal = Nature | volume = 439 | issue = 7075 | pages = 457–61 | date = January 2006 | pmid = 16437113 | doi = 10.1038/nature04282 | bibcode = 2006Natur.439..457R | hdl = 11299/176835 | s2cid = 1484450 | hdl-access = free }}  <br />
For a contrary view, see {{cite journal | vauthors = Enquist BJ, Allen AP, Brown JH, Gillooly JF, Kerkhoff AJ, Niklas KJ, Price CA, West GB | title = Biological scaling: does the exception prove the rule? | journal = Nature | volume = 445 | issue = 7127 | pages = E9–10; discussion E10–1 | date = February 2007 | pmid = 17268426 | doi = 10.1038/nature05548 | first8 = Brian J. | first5 = James F. | first6 = James H. | first7 = Andrew P. | bibcode = 2007Natur.445....9E | s2cid = 43905935 }} and associated responses.</ref>

=== 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. [[Biology:Neuron|Neuron]]s and adipocytes exhibit the former; every other type of cell, the latter.<ref>{{cite journal | vauthors = Savage VM, Allen AP, Brown JH, Gillooly JF, Herman AB, Woodruff WH, West GB | title = Scaling of number, size, and metabolic rate of cells with body size in mammals | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 104 | issue = 11 | pages = 4718–23 | date = March 2007 | pmid = 17360590 | pmc = 1838666 | doi = 10.1073/pnas.0611235104 | bibcode = 2007PNAS..104.4718S | doi-access = free }}</ref> As a result, different organs exhibit different allometric scalings (see table).<ref name="History" />

:{| class="wikitable sortable"
|+
Allometric scalings for [[Basal metabolic rate|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
* [[Biology:Evolutionary physiology|Evolutionary physiology]]
* [[Earth:Metabolic theory of ecology|Metabolic theory of ecology]]
* Scaling law
*[[Biology:Rate-of-living theory|Rate-of-living theory]]

== References ==
{{Reflist}}

== Further reading ==
{{refbegin}}
* {{cite journal | vauthors = Rau AR | title = Biological scaling and physics | journal = Journal of Biosciences | volume = 27 | issue = 5 | pages = 475–8 | date = September 2002 | pmid = 12381870 | doi = 10.1007/BF02705043 | s2cid = 23900176 }}
* {{cite journal | vauthors = Wang Z, O'Connor TP, Heshka S, Heymsfield SB | title = The reconstruction of Kleiber's law at the organ-tissue level | journal = The Journal of Nutrition | volume = 131 | issue = 11 | pages = 2967–70 | date = November 2001 | pmid = 11694627 | doi = 10.1093/jn/131.11.2967 | doi-access = free }}
* {{cite book | vauthors = Whitfield J |title=In the Beat of a Heart |url=https://archive.org/details/inbeatofheart00whit |url-access=registration |year=2006|publisher=Joseph Henry Press |location=Washington, D.C. |isbn=9780309096812}}
* {{cite journal | vauthors = Glazier DS | title = A unifying explanation for diverse metabolic scaling in animals and plants | journal = Biological Reviews of the Cambridge Philosophical Society | volume = 85 | issue = 1 | pages = 111–38 | date = February 2010 | pmid = 19895606 | doi = 10.1111/j.1469-185X.2009.00095.x | s2cid = 28572410 }}
* {{cite journal|last=Glazier|first=Douglas S. | name-list-style = vanc |date=1 October 2014 |title=Metabolic Scaling in Complex Living Systems |journal=Systems | volume=2 |issue=4 |pages=451–540 |doi= 10.3390/systems2040451 |doi-access=free }}
*{{cite news |url=http://courses.missouristate.edu/mcb095f/bio121/lab/Respiration/of_mice_and_elephants.htm |title=Of mice and Elephants | vauthors = Johnson G |date=12 January 1999 |archive-url=https://web.archive.org/web/20081203225711/http://courses.missouristate.edu/mcb095f/bio121/lab/Respiration/of_mice_and_elephants.htm |url-status=dead |archive-date=3 December 2008 }}
* {{cite web | vauthors = Woolley T |title=3/4 and Kleiber's Law |url= http://www.numberphile.com/videos/kleibers.html |work=Numberphile |publisher=[[Biography:Brady Haran|Brady Haran]] |access-date=2013-04-01 |archive-date=2017-05-22 |archive-url=https://web.archive.org/web/20170522031206/http://www.numberphile.com/videos/kleibers.html |url-status=dead }}
{{refend}}

{{modelling ecosystems|expanded=other}}

[[Category:Power laws]]
[[Category:Ecological theories]]

{{Sourceattribution|Kleiber's law}}