Nutrigonometry I: Using Right-Angle Triangles to Quantify Nutritional Trade-Offs in Performance Landscapes

Abstract

Animals regulate their food intake to maximize the expression of fitness traits but are forced to trade off the optimal expression of some fitness traits because of differences in the nutrient requirements of each trait (“nutritional trade-offs”). Nutritional trade-offs have been experimentally uncovered using the geometric framework for nutrition (GF). However, current analyt-ical methods to measure such responses rely on either visual in-spection 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) that 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 applied nutrigonometry to a landmark GF data set for comparison of several standard statistical models to assess model performance in finding regions in the performance land-scapes. 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 life span and reproductive rate in terms of both nutrient balance and concentration for validation of the model. This showed that nutrigonometry en-ables a fast, reliable, and reproducible quantification of nutritional trade-offs in multidimensional performance landscapes, thereby broad-ening the potential for future developments in comparative research on the evolution of animal nutrition.