A Nonlinear Dynamical View of Kleiber's Law on the Metabolism of Plants and Animals

Kleiber's empirical law, which describes that metabolism increases as the mass to the power 3/4, has arguably remained life sciences' enigma since its formal uncovering in 1930. Why is this behavior sustained over many orders of magnitude? There have been quantitative rationalizations put forward for both plants and animals based on realistic mechanisms. However, universality in scaling laws of this kind, like in critical phenomena, has not yet received substantiation. Here, we provide an account, with quantitative reproduction of the available data, of the metabolism for these two biology kingdoms by means of broad arguments based on statistical mechanics and nonlinear dynamics. We consider iterated renormalization group (RG) fixed-point maps that are associated with an extensive generalized (Tsallis) entropy. We find two unique universality classes that satisfy the 3/4 power law. One corresponds to preferential attachment processes-rich gets richer-and the other to critical processes that suppress the effort for motion. We discuss and generalize our findings to other empirical laws that exhibit similar situations, using data based on general but different concepts that form a conjugate pair that gives rise to the same power-law exponents.

Polyploidy in Xenopus lowers metabolic rate by decreasing total cell surface area

Although polyploidization is frequent in development, cancer, and evolution, impacts on animal metabolism are poorly understood. In Xenopus frogs, the number of genome copies (ploidy) varies across species and can be manipulated within a species. Here, we show that triploid tadpoles contain fewer, larger cells than diploids and consume oxygen at a lower rate. Drug treatments revealed that the major processes accounting for tadpole energy expenditure include cell proliferation, biosynthesis, and maintenance of plasma membrane potential. While inhibiting cell proliferation did not abolish the oxygen consumption difference between diploids and triploids, treatments that altered cellular biosynthesis or electrical potential did. Combining these results with a simple mathematical framework, we propose that the decrease in total cell surface area lowered production and activity of plasma membrane components including the Na+/K+ ATPase, reducing energy consumption in triploids. Comparison of Xenopus species that evolved through polyploidization revealed that metabolic differences emerged during development when cell size scaled with genome size. Thus, ploidy affects metabolism by altering the cell surface area to volume ratio in a multicellular organism.

Physiological essay on Gulliver's Travels: a correction after three centuries

Gulliver’s Travels by Jonathan Swift, published in 1726, was analyzed from the viewpoint of scaling in comparative physiology. According to the original text, the foods of 1724 Lilliputians, tiny human creatures, are needed for Gulliver, but the author found that those of 42 Lilliputians and of 1/42 Brobdingnagians (gigantic human creatures) are enough to support the energy of Gulliver. The author further estimated their heartbeats, respiration rates, life spans and blood pressure. These calculations were made by the use of three equations, i.e., body mass index (BMI = W/H²) and quarter-power laws (E∝W^3/4 and T∝W^1/4), where W, H, E, and T denote body weight, height, energy and time, respectively. Their blood pressures were estimated with reference to that of the giraffe and barosaurus, a long-neck dinosaur. Based on the above findings, the food requirement of Gulliver in the original text should be corrected after almost three centuries.

Influence of body size, metabolic rate and life history stage on the uptake and excretion of the lampricide 3-trifluoromethyl-4-nitrophenol (TFM) by invasive sea lampreys (Petromyzon marinus)

Invasive sea lamprey (Petromyzon marinus) are controlled in the Great Lakes using the lampricide 3-trifluoromethyl-4-nitrophenol (TFM), which is applied to streams infested with larval lamprey. However, lamprey that survive treatments (residuals) remain a challenge because they may subsequently undergo metamorphosis into parasitic juvenile animals that migrate downstream to the Great Lakes, where they feed on important sport and commercial fishes. The goal of this study was to determine if body size and life stage could potentially influence sea lamprey tolerance to TFM by influencing patterns of TFM uptake and elimination. Because mass specific rates of oxygen consumption (M˙O₂) are lower in larger compared to smaller lamprey, we predicted that TFM uptake would be negatively correlated to body size, suggesting that large larvae would be more tolerant to TFM exposure. Accordingly, TFM uptake and M˙O₂ were measured in larvae ranging in size from 0.2-4.2g using radio-labelled TFM (¹⁴C-TFM) and static respirometry. Both were inversely proportional to wet mass (M), and could be described usingthe allometric power relationship: Y=aM^b, in which M˙O₂=1.86M^0.53 and TFM Uptake=7.24M^0.34. We also predicted that body size would extend to rates of TFM elimination, which was measured following the administration of ¹⁴C-TFM (via intraperitoneal injection). However, there were no differences in the half-lives of elimination of TFM (T _1/2-TFM). There were also no differences in M˙O₂ or TFM uptake amongst size-matched larval, metamorphosing (stages 6-7), or post-metamorphic (juvenile) sea lamprey. However, the T_1/2-TFM was significantly lower in larval than post-metamorphic lamprey (juvenile), indicating the larval lamprey cleared TFM more efficiently than juvenile lamprey. We conclude that larger larval sea lamprey are more likely to survive TFM treatments suggesting that body size might be an important variable to consider when treating streams with TFM to control these invasive species.

Covariations in ecological scaling laws fostered by community dynamics

Scaling laws in ecology, intended both as functional relationships among ecologically relevant quantities and the probability distributions that characterize their occurrence, have long attracted the interest of empiricists and theoreticians. Empirical evidence exists of power laws associated with the number of species inhabiting an ecosystem, their abundances, and traits. Although their functional form appears to be ubiquitous, empirical scaling exponents vary with ecosystem type and resource supply rate. The idea that ecological scaling laws are linked has been entertained before, but the full extent of macroecological pattern covariations, the role of the constraints imposed by finite resource supply, and a comprehensive empirical verification are still unexplored. Here, we propose a theoretical scaling framework that predicts the linkages of several macroecological patterns related to species' abundances and body sizes. We show that such a framework is consistent with the stationary-state statistics of a broad class of resource-limited community dynamics models, regardless of parameterization and model assumptions. We verify predicted theoretical covariations by contrasting empirical data and provide testable hypotheses for yet unexplored patterns. We thus place the observed variability of ecological scaling exponents into a coherent statistical framework where patterns in ecology embed constrained fluctuations.