Nutrigonometry I: using right-angle triangles to quantify nutritional trade-offs in performance landscapes

Animals regulate their food intake to maximise the expression of fitness traits but are forced to trade-off optimal expression of some fitness traits due to differences in nutrient requirements of each trait (‘nutritional trade-offs’). Nutritional trade-offs have been experimentally uncovered using the Geometric Framework for Nutrition (GF). However, current analytical methods to measure such responses rely on either visual inspection or complex models of vector calculations applied to multidimensional performance landscapes, making these approaches subjective, or conceptually difficult, computationally expensive, and in some cases inaccurate. Here, we present a simple trigonometric model to measure nutritional trade-offs in multidimensional landscapes (Nutrigonometry), which relies on the trigonometric relationships of right-angle triangles and thus, is both conceptually and computationally easier to understand and use than previous quantitative approaches. We apply Nutrigonometry to a landmark GF dataset for the comparison of several standard statistical models to assess model performance in finding regions in the performance landscapes. This revealed that polynomial (Bayesian) regressions can be used for precise and accurate predictions of peaks and valleys in performance landscapes, irrespective of the underlying structure of the data (i.e., individual food intakes vs fixed diet ratios). We then identified the known nutritional trade-off between lifespan and reproductive rate both in terms of nutrient balance and concentration for validation of the model. This shows Nutrigonometry enables a fast, reliable, and reproducible quantification of nutritional trade-offs in multidimensional performance landscapes, thereby broadening the potential for future developments in comparative research on the evolution of animal nutrition.

Methods
The data was collected by Lee et al., 2008 and is available here and in the Dryad entry for the previous manuscript of this series (Morimoto and Lihoreau, 2019 Am Nat). The data consists of measures of life-histories in Drosophila based on diets, using the Geometric Framework for Nutrition.