High-throughput experimental techniques and bioinformatics tools make it possible to obtain reconstructions of the metabolism of microbial species. Combined with mathematical frameworks such as flux balance analysis, which assumes that nutrients are used so as to maximize growth, these reconstructions enable us to predict microbial growth. Although such predictions are generally accurate, these approaches do not give insights on how different nutrients are used to produce growth, and thus are difficult to generalize to new media or to different organisms. Here, we propose a systems-level phenomenological model of metabolism inspired by the virial expansion. Our model predicts biomass production given the nutrient uptakes and a reduced set of parameters, which can be easily determined experimentally. To validate our model, we test it against in silico simulations and experimental measurements of growth, and find good agreement. From a biological point of view, our model uncovers the impact that individual nutrients and the synergistic interaction between nutrient pairs have on growth, and suggests that we can understand the growth maximization principle as the optimization of nutrient synergies.