The Principle of Similitude in Biology: From Allometry to the Formulation of Dimensionally Homogenous `Laws'

Meaningful laws of nature must be independent of the units employed to measure the variables. The principle of similitude (Rayleigh 1915) or dimensional homogeneity, states that only commensurable quantities (ones having the same dimension) may be compared, therefore, meaningful laws of nature must be homogeneous equations in their various units of measurement, a result which was formalized in the $\rm \Pi$ theorem (Vaschy 1892; Buckingham 1914). However, most relations in allometry do not satisfy this basic requirement, including the `3/4 Law' (Kleiber 1932) that relates the basal metabolic rate and body mass, which it is sometimes claimed to be the most fundamental biological rate (Brown et al. 2004) and the closest to a law in life sciences (West \& Brown 2004). Using the $\rm \Pi$ theorem, here we show that it is possible to construct a unique homogeneous equation for the metabolic rates, in agreement with data in the literature. We find that the variations in the dependence of the metabolic rates on body mass are secondary, coming from variations in the allometric dependence of the heart frequencies. This includes not only different classes of animals (mammals, birds, invertebrates) but also different exercise conditions (basal and maximal). Our results demonstrate that most of the differences found in the allometric exponents (White et al. 2007) are due to compare incommensurable quantities and that our dimensionally homogenous formula, unify these differences into a single formulation. We discuss the ecological implications of this new formulation in the context of the Malthusian's, Fenchel's and the total energy consumed in a lifespan relations.Comment: A accepted for publication in Theoretical Ecology. Comments are welcome ([email protected]

Effects of body size and temperature on population growth

For at least 200 years, since the time of Malthus, population growth has been recognized as providing a critical link between the performance of individual organisms and the ecology and evolution of species.We present a theory that shows how the intrinsic rate of exponential population growth, , and the carrying capacity, rmax K, depend on individual metabolic rate and resource supply rate. To do this, we construct equations for the metabolic rates of entire populations by summing over individuals, and then we combine these population-level equations with Malthusian growth. Thus, the theory makes explicit the relationship between rates of resource supply in the environment and rates of production of new biomass and individuals. These individual-level and population-level processes are inextricably linked because metabolism sets both the demand for environmental resources and the resource allocation to survival, growth, and reproduction. We use the theory to make explicit how and why exhibits its characteristic dependence on body size and rmax temperature. Data for aerobic eukaryotes, including algae, protists, insects, zooplankton, fishes, and mammals, support these predicted scalings for. The metabolic flux of energy and materials also rmax dictates that the carrying capacity or equilibrium density of populations should decrease with increasing body size and increasing temperature. Finally, we argue that body mass and body temperature, through their effects on metabolic rate, can explain most of the variation in fecundity and mortality rates. Data for marine fishes in the field support these predictions for instantaneous rates of mortality. This theory links the rates of metabolism and resource use of individuals to life-history attributes and population dynamics for a broad assortment of organisms, from unicellular organisms to mammals

Metabolic Scaling of Stress Hormones in Vertebrates

Glucocorticoids (GCs) are stress hormones that can strongly influence physiology, behavior, and an organism’s ability to cope with environmental change. Despite their importance, and the wealth of studies that have sought to understand how and why GC concentrations vary within species, we do not have a clear understanding of how circulating GC levels vary within and across the major vertebrate clades. New research has proposed that much interspecific variation in GC concentrations can be explained by variation in metabolism and body mass. Specifically, GC concentrations should vary proportionally with mass-specific metabolic rates and, given known scaling relationships between body mass and metabolic rate, GC concentrations should scale to the -1/4 power of body mass and to the power of 1 with mass-specific metabolic rate. Here, we use HormoneBase, the newly compiled database that includes plasma GC concentrations from free-living and unmanipulated vertebrates, to evaluate this hypothesis. Specifically, we explored the relationships between body mass or mass-specific metabolic rate and either baseline or stress-induced GC (cortisol or corticosterone) concentrations in tetrapods. Our phylogenetically-informed models suggest that, whereas the relationship between GC concentrations and body mass across tetrapods and among mammals is close to -1/4 power, this relationship does not exist in amphibians, reptiles, and birds. Moreover, with the exception of a positive association between stress-induced GC concentrations and mass-specific metabolic rate in birds, we found little evidence that GC concentrations are linked to metabolic rate, although the number of species sampled was quite limited for amphibians and somewhat so for reptiles and mammals. Nevertheless, these results stand in contrast to the generally accepted association between the two and suggest that our observed positive association between body mass and GC concentrations may not be due to the well-established link between mass and metabolism. Large-scale comparative approaches can come with drawbacks, such as pooling and pairing observations from separate sources. However, these broad analyses provide an important counterbalance to the majority of studies examining variation in GC concentrations at the population or species level, and can be a powerful approach to testing both long-standing and new questions in biology

Allometric and Metabolic Scaling: Arguments for Design... and Clues to Explaining Pre-Flood Longevity?

Within the creationist community, there has been recent interest in applying engineering principles to the study of living things. Engineers routinely attempt to maximize certain qualities, such as efficiency and durability, while minimizing others, such as cost of construction, within certain design limits or constraints. Biologists have long suspected that living things are in some sense “optimized” and there is a long history of attempting to understand biological systems in terms of this optimization. This is strong prima facie evidence of intelligent design, yet evolutionists attribute this optimization to evolution and natural selection. Biological optimization is closely related to allometry, the study of the manner in which biological variables scale with body size. A large body of literature suggests that observable biological characteristics are often proportional to an organism’s mass raised to some power, often a simple multiple of 1/4. Some of these variables include lifespan, heart rate, and the radii of tree trunks and the mammalian aorta. One particularly well-known example was discovered by agricultural scientist Max Kleiber. He observed that for birds and many mammals, basal (resting) metabolic rate is proportional to body mass raised to the 3/4 power. This surprised biologists, as they were expecting, based on simple geometrical reasoning, an exponent of 2/3, rather than 3/4. In 1997 physicist Geoffrey West and biologists Brian Enquist and James Brown (WEB) published a theoretical explanation for Kleiber’s Law. WEB modeled a biological nutrient supply network as levels of interconnected, diverging pipes. The highest level of the network consisted of one single large pipe, which diverged into multiple pipes at the second and subsequent levels. The number of new branches (the branching ratio) at each junction was a constant, and pipes in subsequent levels decreased in size in a regular, predictable manner. The resulting pipe network was a self-similar fractal. WEB assumed that biological systems are designed to minimize the amount of energy needed to transport nutrients to the entire organism, given “volume filling” and other constraints. They used Lagrange multipliers to find optimal values (in terms of branching ratios) for pipe lengths and pipe cross-sectional areas in subsequent levels of the network. Their model did a reasonably good job of accounting for 16 features of the mammalian circulatory system and 17 characteristics of plant vascular systems. Nevertheless, the WEB theory has been the subject of controversy, with some questioning the empirical validity of Kleiber’s Law, as well as whether it is truly universal. Nevertheless, the WEB model reignited interest in using engineering principles to understand living things, inspiring others to propose other optimization models. Although some of these are quite impressive, none are universally successful, probably because some of the model constraints are too strong, whereas in many cases, additional constraints need to be added. This suggests that the optimization of living things is very sophisticated, involving the “balancing” of multiple different design considerations against one another. In fact, other researchers have already suggested design considerations that might perhaps need to be included in future models. Clearly, living things are designed according to optimization principles, and any creationist theory of biological design should make use of them. Yet surprisingly, these optimization/allometric theories seem to have received little attention in the creationist community. This paper is divided into four parts. Part 1 provides an overview of allometry and earlier attempts to apply optimization principles to biology. Part 2 provides an overview of the WEB theory, as it is perhaps the best known optimization theory currently in existence. I lay out the theory’s basics, filling in details which may not be obvious to newcomers, especially the “volume filling” requirement, which was not clearly explained by WEB. Part 3 discusses both successes and failures of the theory, as well as possible reasons for the failures. Part 4 discusses ways in which creationists could improve upon the WEB theory. I provide examples of how flowchart logic might be helpful in improving upon these models, as could observations and deductions from Scripture. This area is “wide open” for creation researchers. Although the WEB and other models are quite impressive, they are still relatively crude, as evidenced by their inability to explain all observations. Creationists have a real opportunity to improve on existing models to obtain a more comprehensive and successful theory of biological design

Novel morphological and physiological scaling relationships in the southern red wood ant

Red wood ants (Formica rufa) are visual navigators whose colonies contain workers that differ substantially in size. By investigating the allometry of the ants’ compound eyes, and the regions within them, I showed that facets in particular regions scaled differently: both grade and slope shifts occurred. Facets in some eye regions were absolutely larger than others, while other facet regions scaled at different rates with body size. I next compared eye scaling between nests from the same population. Nevertheless, the method by which ants increased their eye size differed between nests. I found that ants from some nests primarily increased eye size through facet number and others through facet diameter. This showed that scaling rules at the cellular levels can differ even within a single population. Comparisons among Formica species revealed that differential eye scaling was not restricted to just F. rufa. Differential scaling was found in F. sanguinea but not F. lugubris or F. fusca. Surprisingly, scaling between facet diameter and number was conserved across all four species, demonstrating that whole-organ scaling among species can be conservative whilst differing vastly between organ-regions. Moving beyond morphology, I next investigated whether physiological scaling was equally as variable among nests. Metabolic rate scaling was negatively allometric and the same among four nests. Respiratory water loss was found to be determined solely by metabolic rate. Metabolic rate co-varies with different ventilation types, however, switches in ventilation type are driven by movement. This demonstrates that increases in metabolic rate are not sufficient to explain changes in ventilation type but are sufficient to explain respiratory water loss

Entropy Stress Based on Organ and Mitochondrial Metabolic Loading

he energy for sustaining life is released through the oxidation of glucose, fats, and proteins. A part of the energy released within each cell is stored as chemical energy of Adenosine Tri- Phosphate molecules, which is called work currency of the body, while the remainder is released as heat. Earlier literature introduced availability concepts from thermodynamics, related the specific irreversibility and entropy generation rate to metabolic efficiency, and energy release rate of each organ, and computed whole body specific entropy generation rate at any given age as a sum of entropy generation within four vital organs; Brain, Heart, Kidney, Liver and the rest of organs. The current work includes the effects of i) two additional organs: adipose tissue and skeletal muscles, for application to athletes, ii) proportions of nutrients oxidized which affects blood temperature and metabolic efficiencies, iii) converts the entropy stress from organ/cellular level to mitochondrial level, and iv) relates these parameters as biomarkers in biological aging process. Based on 7 organ model, considering a male of 84 kg steady mass, the lifetime energy expenditure is estimated to be 2726.46 MJ/kg body mass, with contributions of 86.4, 825.8, 274.8, 131.4, 316.4, 661.1, 430.4 MJ to each unit body mass by Adipose Tissue, Brain, Heart, Kidney, Liver, Rest of Mass, Skeletal Muscle, while lifetime entropy generated 6051 kJ/(K kg body mass) with contributions of 191.7, 1832.7, 610, 291.7, 702.3, 1467.2, 955.2 kJ/K to each unit body mass. Based on mitochondrial volume and 5 organ model, the lifetime energy expenditure is estimated to be 15529.6 MJ/ cm3 of mitochondrial volume of whole body, with contributions of 8250, 2435, 3040, 1805, 1.9E-05 MJ to each unit volume of mitochondria in organs, serving as biomarkers in the biological aging process of organs, while lifetime entropy generated is 34465 kJ/(K cm^3 of mitochondrial volume) with contributions of 18310, 5400, 6740, 4010.5, 4.3E-05 kJ/K respectively to each unit of mitochondrial volume. The organ entropy stress ranking based on unit volume of mitochondria within an organ {kJ/ (K cm^3 of mito of organ k) show brain being highest and liver lowest

The evolution of sperm morphometry in pheasants

Postcopulatory sexual selection is thought to be a potent evolutionary force driving the diversification of sperm shape and function across species. In birds, insemination and fertilisation are separated in time and sperm storage increases the duration of sperm female interaction and hence the opportunity for sperm competition and cryptic female choice. We performed a comparative study of 24 pheasant species (Phasianidae, Galliformes) to establish the relative importance of sperm competition and the duration of sperm storage for the evolution of sperm morphometry (i.e. size of different sperm traits). We found that sperm size traits were negatively associated with the duration of sperm storage but were independent of the risk of sperm competition estimated from relative testis mass. Our study emphasises the importance of female reproductive biology for the evolution of sperm morphometry particularly in sperm storing taxa

Allometric Scaling and Cell Ratios in Multi-Organ in vitro Models of Human Metabolism

Intelligent in vitro models able to recapitulate the physiological interactions between tissues in the body have enormous potential as they enable detailed studies on specific two-way or higher order tissue communication. These models are the first step toward building an integrated picture of systemic metabolism and signaling in physiological or pathological conditions. However, the rational design of in vitro models of cell–cell or cell–tissue interaction is difficult as quite often cell culture experiments are driven by the device used, rather than by design considerations. Indeed, very little research has been carried out on in vitro models of metabolism connecting different cell or tissue types in a physiologically and metabolically relevant manner. Here, we analyze the physiological relationship between cells, cell metabolism, and exchange in the human body using allometric rules, downscaling them to an organ-on-a-plate device. In particular, in order to establish appropriate cell ratios in the system in a rational manner, two different allometric scaling models (cell number scaling model and metabolic and surface scaling model) are proposed and applied to a two compartment model of hepatic-vascular metabolic cross-talk. The theoretical scaling studies illustrate that the design and hence relevance of multi-organ models is principally determined by experimental constraints. Two experimentally feasible model configurations are then implemented in a multi-compartment organ-on-a-plate device. An analysis of the metabolic response of the two configurations demonstrates that their glucose and lipid balance is quite different, with only one of the two models recapitulating physiological-like homeostasis. In conclusion, not only do cross-talk and physical stimuli play an important role in in vitro models, but the numeric relationship between cells is also crucial to recreate in vitro interactions, which can be extrapolated to the in vivo reality