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22-Agric-A1 Applied Plant, Animal or Human Physiology · May 2018

Question 3 of 6: Metabolic Rate – Body Size Scaling

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 04-Agric-A1 Applied Plant, Animal or Human Physiology, National Exams May 2018 — a three-hour closed-book examination; one of two approved calculator models (Casio or Sharp) is permitted. The rubric states that five (5) questions constitute a complete exam paper and that the first five questions appearing in the answer book are marked (worth 20 marks each, 100 marks total); all six (6) printed questions are worked here as a complete study resource.

Reference texts. M.K. Yousef (ed.), Stress Physiology in Livestock, Vol. I — Basic Principles, CRC Press (thermoregulation, thermoneutral zone, piloerection, endotherm/ectotherm physiology); K. Schmidt-Nielsen, Animal Physiology: Adaptation and Environment, 5th ed. (metabolic body-size scaling, Kleiber's law, thermoconformers, calorimetry); P. McDonald et al., Animal Nutrition, 7th ed. (gross/digestible/metabolizable/net energy, feed-energy partition, growth efficiency); R.L. Curtis, Environmental Management in Animal Agriculture, Iowa State University Press (thermoneutral zone, animal housing microclimate); D.M. Lewis & T.R. Morris, Poultry Lighting: the Theory and Practice (photoperiodism); ASABE Standards (American Society of Agricultural and Biological Engineers), Livestock Energetics and Thermal Environmental Management (design sensible heat production, calorimetry).

Question 3: Metabolic Rate – Body Size Scaling (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

The equation. The standard relationship between an animal's resting (basal) metabolic rate and its body mass is the allometric power law $$Q = a\,M^{b}$$ where $Q$ is metabolic rate (e.g. W, or kcal/day), $M$ is body mass (kg), $a$ is a taxon-specific normalization constant (the metabolic rate of a notional 1 kg animal of that group), and $b$ is the scaling exponent. Compiled across an enormous span of body sizes — from small mammals through livestock to whales — the best-fit exponent is empirically close to $b \approx 0.75$; this relationship is known as Kleiber's law after Max Kleiber, who first compiled the cross-species data in 1932. Because $b < 1$, metabolic rate rises more slowly than body mass itself, so mass-specific metabolic rate ($Q/M \propto M^{b-1} = M^{-0.25}$) falls as an animal gets larger — a large animal produces less heat per kilogram of tissue than a small one.

Reasoning behind the exponent — the surface-area hypothesis and its limitation. The historically first explanation (Rubner's "surface law," 1883) was purely geometric: if body shape (density and proportions) is held constant across sizes, surface area scales with length squared while mass scales with length cubed, so surface area $\propto M^{2/3}$. Since an endotherm's heat loss to the environment occurs mainly through its external body surface, the surface law predicted $b = 2/3 \approx 0.667$. However, when Kleiber compiled actual metabolic-rate data across species, the best-fit exponent was consistently closer to 0.75 than to 0.667 — a systematic discrepancy the pure surface-area argument could not explain.

Reasoning behind the exponent — the vascular-network (WBE) hypothesis. The now-widely-accepted explanation for the 3/4 exponent (West, Brown & Enquist, 1997) locates the limiting factor not in the animal's external heat-loss surface but in the internal geometry of its resource-distribution network — the branching circulatory system that must deliver oxygen and nutrients to every cell in the body. Modelling this network as a fractal-like, space-filling hierarchy of branching tubes that minimizes the energy cost of pumping blood to the periphery shows mathematically that whole-organism metabolic rate must scale with body mass to the 3/4 power, essentially independent of the two-dimensional surface-area argument. This network-based derivation reproduces the empirically observed $b \approx 0.75$ far more closely than the surface law, and — importantly — it also explains why the 3/4-power relationship holds even for organisms (e.g. plants, whose vascular analogue is the xylem) that do not lose metabolic heat through an external surface at all, something the surface-area hypothesis alone cannot account for.