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

Abstract

Animals regulate their diet in order to maximise the expression of fitness traits that often have different nutritional needs. These 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 applied to multidimensional performance landscapes, making these approaches subjective, or conceptually difficult, computationally expensive, and in some cases inaccurate. This limits our ability to understand how animal nutrition evolved to support life-histories within and between species. Here, we introduce a simple trigonometric model to measure nutritional trade-offs in multidimensional landscapes (‘Nutrigonometry’). Nutrigonometry is both conceptually and computationally easier than current approaches, as it harnesses the trigonometric relationships of right-angle triangles instead of vector calculations. Using landmark GF datasets, we first show how 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). Using trigonometric relationships, we then identified the known nutritional trade-off between lifespan and reproductive rate both in terms of nutrient balance and concentration. 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.

Nutrigonometry III: curvature, area and differences between performance landscapes

“…This is important because the equation ax 2 + by 2 + cx + dy + exy is differentiable and enables easy calculations of surface integrals and gradients for the estimates of curvatures. We have recently analysed the performance of several statistical (machine learning) models and their performance in identifying peak and valley regions in performance landscapes [ 29 ]. The good quadratic approximation using LM presented in this study agrees with our extensive comparison of model performance in GF datasets.…”

“…Other models have been proposed to find peaks and valleys in multidimensional performance landscapes, using either bootstrapping [ 28 ] or machine learning models [ 27 ]. More recently, we also proposed a novel way to define the peaks and valleys of multidimensional performance landscapes for comparison of strengths of nutritional trade-offs using the angle θ i , j which strictly represents performance landscapes as right-angle triangles and uses trigonometry for estimates of nutritional trade-offs [ 29 ]. However, these models focus on obtaining information on either peak or valley regions (or both) of the multidimensional performance landscapes, overlooking other properties of the landscapes with potential biological significance.…”

Nutrigonometry II: Experimental strategies to maximize nutritional information in multidimensional performance landscapes

“…This approach uses a machine learning model to identify the peak region. More recently, I developed a conceptually simpler and computationally cheaper model to estimate peak regions and nutritional trade‐offs in GF studies using trigonometric relationships (“Nutrigonometry”), which enabled the comparison of different statistical methods to estimate nutritional trade‐offs and opened up new ways in which properties of performance landscapes can be estimated (Morimoto et al, 2021 ). The dataset used here was fundamental for the validation of these methods and is therefore used here.…”

“…Two nutrients were investigated—protein and carbohydrate—such that performance landscapes have three dimensions. This dataset was previously used on my conceptualization of the Vector of Position approach and Nutrigonometry, having important benchmark status in the field (Morimoto & Lihoreau, 2019; Morimoto et al, 2021). Briefly, the Vector of Positions approach was developed to n‐ dimensional performance landscapes from GF experiments as vector from which the strength of nutritional trade‐offs between traits can be estimated via the angle $$$ \theta $$$ between vectors of two traits (Morimoto & Lihoreau, 2019).…”

“…Here, I investigated the performance of different sampling strategies when reconstructing performance landscapes, using a landmark dataset in the field of nutritional ecology (Lee et al, 2008 ). I used the pioneering Nutrigonometry framework to identify and compare the peaks in the reconstructed performance landscapes and how congruent these estimates are across sampling strategies (Morimoto et al, 2021 ). I tested four different sampling strategies: standard GF, hexagonal, squared, and random points sampling grids (Figure 1a ).…”