Scaling and optimal synergy: Two principles determining microbial growth in complex media

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.

💡 Research Summary

This paper presents a novel, systems-level phenomenological model designed to predict and provide mechanistic insight into microbial growth in complex media. The work addresses a key limitation of existing constraint-based methods like Flux Balance Analysis (FBA), which, while predictive, offer little understanding of how individual nutrients contribute to growth and are difficult to generalize.

Inspired by the virial expansion from statistical physics, the model formulates the biomass production rate (g) as a sum of contributions: independent terms for each nutrient (first-order) and pairwise synergy terms between nutrients (second-order). The first-order contribution of a nutrient i is its uptake flux (φ_i) multiplied by its biomass yield when used alone (ˆα_i). A significant finding is that ˆα_i is largely proportional to the nutrient’s “effective carbon number” (C_i)—the number of carbons actually catabolized—with a nearly constant proportionality factor across different nutrient classes (sugars, fatty acids, amino acids). This indicates a fundamental scaling of biomass yield based on carbon supply.

The central discovery is the “scaling” behavior of the pairwise synergy term β_ij(φ_i, φ_j). This synergy does not depend on the absolute uptake values but collapses onto a universal curve when expressed as a function of the rescaled variable (C_iφ_i)/(C_jφ_j). For example, synergy data for various sugar-fatty acid pairs all fall on a single master curve. This universal curve shows a linear regime when one nutrient is scarce and a saturation regime when it is in excess. The authors capture this behavior with a simple two-parameter phenomenological model using a hyperbolic tangent (tanh) function.

For synergies involving amino acids, the behavior is more complex. The authors classify amino acids into High-synergy (H) and Low-synergy (L) groups. Using logistic regression, they demonstrate that an amino acid’s group can be predicted by the set of core metabolic pathways it participates in (e.g., TCA cycle, glyoxylate shunt), linking the synergy phenomenon to specific network topology.

A critical challenge arises when more than two nutrients are present: how are the limited uptake fluxes allocated among multiple possible synergistic pairs? The authors compare an “Equitative Synergy” (ES) model, which divides fluxes equally, with an “Optimal Synergy” (OS) model designed to maximize total synergy. The OS model ranks potential synergies by size and allocates fluxes sequentially: in each pair, the flux of the “limiting nutrient” is fully consumed, while the remaining flux from the “excess nutrient” is passed on to the next-ranked synergy. Validation against FBA simulations for E. coli shows that the OS model predictions are significantly more accurate than those of the ES or a first-order “Idealized Metabolism” model, especially as the number of nutrients increases.

This result provides a profound reinterpretation: the microbial growth maximization principle, often assumed in models like FBA, can be understood phenomenologically as the outcome of optimally allocating nutrient uptake fluxes to maximize synergistic interactions between them. The proposed model successfully decouples growth prediction from the intricate details of any specific metabolic network, requiring only nutrient uptake data and a small set of class-dependent parameters that can be determined experimentally. It thus offers a powerful, interpretable, and generalizable framework for predicting growth and uncovering the systems-level principles governing nutrient utilization.