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Metabolic theory and ecological scaling Geoffrey West James Brown Brian Enquist The grand idea of scaling theory is that animal and plant branching systems and tissues are self similar. They are assumed to have a fractal structure. Vn VK V c1 r a rn rK a A c2 r b rn rK An AK b rK r K 2 L K Metabolic rate is proportional to tissue surface area LK M A Animal or plant body weight is proportional to body volume W V In the Euclidean case A V2/3 M B 2/3 M B 3/ 4 Kleiber’s rule Mn MK Wn WK x A textbook example for Kleiber’s rule Hemmingson classic plot of metabolic rate against body size. Each regression line has a slope of 3/4 How to derive Kleiber’s rule? The second grand idea of Brown, West and Eberhard is that life during evolution optimized the relation between tissue surface and volumen to get maximum energetic efficiency at a given body volume (size). 2 av 3 WBE proposition 3 M W 4 The optimization argument is still heavility disputed. Recent analysis rather point to rather complex relationships between tissue surface and volume. The ¾ power function is then only a rough approximation. Metabolism and temperature 0.14 kT 0.1 0.12 k k Ae Ea 0.08 0.06 Ea = Activation energy k = 8.314 Jmol-1K-1 T = absolute temperature A = Proportionality factor 0.04 0.02 0 273 293 313 333 353 373 393 413 Temperature [K] The Arrhenius equation holds approximatily for most enzymatic processes. Ea is often not a constant but related to the square root of T Ea c T 433 The basic equation of metabolic theory M M 0W 3/4 e Ea kT Andrew Allen James Gillooly M: metabolic rate Ea: Mean activation energy for biochemical reactions M0: basal metabolic rate k: Boltzmann factor: 8.314 Jmol-1K-1 = 0.0000862eVK-1 W: body weigth Adding the concentration of an assumed limiting resource gives M M 0 [ R ]W 3/4 e Ea kT The activation energy can be estimated from plots of M against W (at constant temperature) or from plots of M against 1/T for species of similar body size M M 0 [ R ]W f ( R ) f (W ) f (T ) Ea takes values from 0.6 to 0.7 eV with a mean of 0.65 eV = 62693Jmol-1 -Ea/k = 0.65/0.0000862 = 7541K 3/4 7541 e 1 T The basic assumption of metabolic theory • Living organsims are composed of fractal networks • Evolution operated as to optimize energetic efficiency • Limiting constraints operate in a multiplicative manner • Parameter values are whole organism means are fairly constant • Body weight is independent of temperature • Parameters are not functions of the amounts of resources available. • The theory cannot explain why animals of the same size can have strikingly different metabolic rates and lifespan. • Kleiber’s law holds even for organisms without fractal networks. Empirical support The temperature corrected metabolism – weight relation M e From Brown et al. (2004), Ecology 85: 1771-1789 E / kT M 0W 0 .7 6 It is now easy to derive other relationships that involve basic ecological variables: Abundance: Take the the expression MN as the total metabolism of of a population of size N. At equilibrium (dN/dt = 0) the product NM must be constant (NM = c) that is total resource use remains stable. Hence c N [ R ]M [ R ]M 3/4 e 3 / 4 E a / kT e E a / kT Hence at stable temperature metabolic theory predicts that abundance scales to body weight to a power of -3/4. From Brown et al. (2004), Ecology 851771-1789 The energy equivalence rule N M 0M 3 / 4 e M M 0W E a / kT N M M 0W 3 / 4 Ea e kT M 0W 3/4 e 3/4 e Ea kT Ea kT const The product NM is the total amount of energy use of the whole population of size N. In poikilothermes and plants NM is a proxi to population biomass. The equal biomass hypothesis Ontogenetic growth Assume multicellular animals (or plants) where energy is transported through a network of branches. The total energy can be expressed as the sum of the energy need to maintain the existing cells plus the energy needed to create new cells. M total dN NM E dt N w w cell N is the total number of cells at time t, M is energy demand. Mtotal is the total energy needed w is the ontogenetic mass and wcell the mean cell mass. M total M 0 w d 3/4 w w cell w W M 0 w cell w dt E W w W dw dt Cw w d w cell M E dt 3/4 M w E W 3/4 M 0 w cell 3 / 4 M dw w w dt E E dw dt M total e M 0 w cell W 1/ 4 w E E 3/4 M w E W dw kT dt Cw 3/4 e E kT dw Cw 3/4 E kT e dt dw w 3/4 t w Ce 1/ 4 E kT 4 Ce E dt kT 4 w 3 / 4 dw 4 w 1/ 4 E Ce kT E e dt C e t kT C 4 E E kT t 1/ 4 e kT C 33 33 w 1 /4 1 /4 y= y=eexxpp(a(a/T/T) ) y= y=ww 22 yy yy 22 11 11 00 00 00 55 1100 1155 2200 xx Body weight corrected developmental times should exponentially decrease with increasing environmental or body temperature 00 55 1100 1155 2200 xx Temperature corrected developmental times should scale to body weight to a power of ¼. The slope –E/k is predicted to be about -7500. 100 300 -E /k = -10000 1/4 60 -E /k = -12000 250 M a rine fish t/w t/w 1/4 80 40 200 A q ua tic e cto the rm s 150 100 20 50 0 0 270 280 290 300 310 B ird s 0 10 20 30 40 T /(1 + T /2 7 3 ) T /(1 + T /2 7 3 ) For a slope –E/k of about -11000 E should take a value of 0.95eV. Biological times should scale to body weight to the quarter power Examples: Generation time, lifespan, age of maturation, average lifetime of a species E t w 1/ 4 e kT The inverse of time are rates. Examples: Growth rates, mutation rates, species turnover rates, migration rates Hence biological rates should scale to body weight and temperature by r 1 t w 1 / 4 e E kT Criticisms Living organims are not or at least not in total fractals. The derivation of the ¾ scaling rule is mathematically flawed. The ¾ scaling rule has no empirical justification. The Arrhenius temperature term is much too simplified and the parameters are poorly defined. More important than universal scaling is the variability in metabolism. The theory predicts average scaling rates but does not account for the oberserved variance in scaling relationships. The variablility of living organisms prohibits any useful predictions from the theory. In other words, even if the thory is correct it does not tell much about the functioning of real world ecological systems. z 30 O b s e rv a tio n s 35 25 20 15 10 5 0 0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 1 .1 S lo p e Slopes of metabolic rate – body weight relationships of different taxonomic groups scatter around 0.75. There is no single slope for all taxa. Data from Peters 1983. 1 .2 Cell size theory (Kozłowski et al. 2003) Total body weight is the product of mean cell size Wcell and its number n Cell metabolic rate scales allometrically to cell size W nW cell M M 0 n W cell z z Upper boundary: If body size increases solely due to the increase in the number of cells we get z 1 M nM 0 W cell nM 0 W cell W z Lower boundary: If body size increases due to the increase in cell size we get M n M 0 W cell M 0 W z z z z<1 The theory predicts • Metabolic rate should scale allomtrically to body weight • The scaling exponent should be less than 1 • If cell metabolic rate is proportional to cell surface then z = 2/3 • 2/3 < z < 1 Major criticism • The theory uses species as as major driver in evolution. • Cells have a fractal nature. Hence the lower slope boundary should be 0.75. • The theory applies only to taxa with large differences in cell size like mammals. • The theory does not explain scaling intercepts. • The theory does not explain scaling in plants and invertebrates. Today’s reading Brown, J. H., Gillooly J. H., Allen A. P., Savage V. M., West G. B., 2004, Towards a metabolic theory of ecology. Ecology 85: 1771-1789. www.nceas.ucsb.edu/~drewa/pubs/brown_jh_2004_e85_1771.pdf Ecology’s big, hot idea: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=535575