Evolutionary emergent metabolic interactions in cell cultures: A Statistical Mechanics point of view

Cell cultures exhibit rich and complex behaviors driven by dynamic metabolic interactions among cells. In this work, we present a model that captures these interactions through a framework inspired by statistical mechanics. Using Monte Carlo simulations, we explore the equilibrium and dynamical properties of a population of cells arranged in a two-dimensional lattice, where each cell is characterized by fluxes of three reactions: glucose consumption ($g$), respiration ($r$), and waste production/absorption ($w$). The system minimizes an energy function influenced by competitive ($J_g > 0$) and cooperative ($J_w < 0$) couplings between cells. Our results reveal three distinct phases: a competitive phase dominated by glucose competition, a cooperative phase marked by ordered waste exchange, and a disordered phase with local-scale cooperation. By incorporating evolutionary dynamics, we demonstrate how initially non-interacting cells can develop effective metabolic interactions, leading to heterogeneous cultures sustained by cross-feeding. These findings are further supported by analytical solutions derived using mean-field approximations. The model provides insights into how environmental constraints and stochastic fluctuations shape community structures, offering a versatile approach to study several emergent phenomena in biological systems.