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First published online April 26, 2005
Journal of Experimental Biology 208, 1575-1592 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01589

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The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization

Geoffrey B. West^1,2,* and James H. Brown^1,3

¹ The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
² Los Alamos National Laboratory, Los Alamos, NM 87545, USA
³ Department of Biology, University of New Mexico, Albuquerque, NM 87131, USA

View larger version (25K): [in a new window]   Fig. 1. Kleiber's original 1932 plot of the basal metabolic rate of mammals and birds (in kcal/day) plotted against mass (Mb in kg) on a log-log scale (Kleiber, 1975). The slope of the best straight-line fit is 0.74, illustrating the scaling of metabolic rate as Mb3/4. The diameters of the circles represent his estimated errors of 10% in the data.

View larger version (10K): [in a new window]   Fig. 2. Extension of Kleiber's 3/4-power law for the metabolic rate of mammals to over 27 orders of magnitude from individuals (blue circles) to uncoupled mammalian cells, mitochondria and terminal oxidase molecules, CcO of the respiratory complex, RC (red circles). Also shown are data for unicellular organisms (green circles). In the region below the smallest mammal (the shrew), scaling is predicted to extrapolate linearly to an isolated cell in vitro, as shown by the dotted line. The 3/4-power re-emerges at the cellular and intracellular levels. Figure taken from West et al. (2002b) with permission.

View larger version (12K): [in a new window]   Fig. 3. Plot of heart rates (fH) of mammals at rest vs body mass Mb (data taken from Brody, 1945). The regression lines are fitted to the average of the logarithms for every 0.1 log unit interval of mass, but both the average (squares) and raw data (bars) are shown in the plots. The slope is -0.251 (P<0.0001, N=17, 95% CI: -0.221, -0.281), which clearly includes -1/4 but excludes -1/3. Figure taken from Savage et al. (2004b) with permission.

View larger version (17K): [in a new window]   Fig. 4. Plot of genome length (number of base pairs) vs mass (in g) for a sequence of unicells on a log-log scale. The best straight-line fit has a slope very close to 1/4.

View larger version (29K): [in a new window]   Fig. 7. Plot of mass-corrected resting metabolic rate, ln(B Mb-3/4) vs inverse absolute temperature (1000/°K) for unicells (A), plants (B), multicellular invertebrates (C), fish (D), amphibians (E), reptiles (F), and birds and mammals (G). Birds (filled symbols) and mammals (open symbols) are shown at normal body temperature (triangles) and during hibernation or torpor (squares). Figure taken from Gillooly et al. (2001) with permission.

View larger version (13K): [in a new window]   Fig. 5. Metabolic rates (in W) of mammalian cells in vivo (blue line) and cultured in vitro (red line) plotted as a function of organism mass (Mb in g) on a log-log scale. While still in the body and constrained by vascular supply networks cellular metabolic rates scale as Mb-1/4. When cells are removed from the body and cultured in vitro, their metabolic rates converge to a constant value predicted by theory (West et al., 2002b). The two lines meet at the mass of the smallest mammal (the shrew with mass 1 g, as predicted). Figure taken from West et al. (2002b) with permission.

View larger version (21K): [in a new window]   Fig. 6. The universality of growth is illustrated by plotting the dimensionless mass ratio (m/M)1/4 against a dimensionless time variable, as shown. When data for mammals, birds, fish and crustacea are plotted this way, they are predicted to lie on a single universal curve; m is the mass of the organism at age t, m0 its birth mass, Mb its mature mass, and a a parameter determined by theory in terms of basic cellular properties that can be measured independently of growth data. Figure taken from West et al. (2001) with permission.

View larger version (10K): [in a new window]   Fig. 8. Plot of maximum reported xylem flux rates (liters of fluid transported vertically through a plant stem per day) for plants (Enquist and Niklas, 2001). The RMA regression line is fitted to the average of the logarithm for every 0.1 log unit interval of plant biomass, but both the average (circles) and raw data (bars) are shown in the plot. The slope is 0.736 (P<0.0001, N=31, 95%CI: 0.647, 0.825). The regression fitted to the entire unbinned data set gives a similar exponent of 0.735 (P<0.0001, N=69, 95% CI: 0.682, 0.788). Figure taken from Enquist and Niklas (2001) with permission.

View larger version (67K): [in a new window]   Fig. 9. (A) A typical tree. (B) A page from Leonardo's notebooks illustrating his discovery of area-preserving branching for trees. (C) A plot of the number of branches of a given size in an individual tree versus their diameter (in cm) showing the predicted inverse square law behaviour. (D) A plot of the number of trees of a given size vs their trunk diameter (in cm) showing the predicted inverse square law behaviour. The data are from a forest in Malaysia taken at times separated by 34 years, illustrating the robustness of the result. Even though the individual composition of the forest has changed over this period the inverse square law has persisted. Figure taken from West and Brown (2004), with permission.

View larger version (15K): [in a new window]   Fig. 10. Histograms of the exponents of (A) biological rates, (B) mass-specific biological rates (Peters, 1986) and (C) biological times (Calder, 1984). Note that the peak of the histogram for biological rates is very close to 0.75 (0.749±0.007) but not close to 0.67. Moreover, the histogram for mass-specific rates peaks near -0.25 (-0.247±0.011) but not -0.33, and the histogram for biological times peaks at 0.25 (0.250±0.011) and not 0.33. All errors quoted here are the standard error from the mean for the distribution. In all cases, the majority of biological rates and times exhibit quarter-power, not third-power, scaling. Figure taken from Savage et al. (2004b) with permission.