Phase separation in fluids with many interacting components

Fluids in natural systems, like the cytoplasm of a cell, often contain thousands of molecular species that are organized into multiple coexisting phases that enable diverse and specific functions. How interactions between numerous molecular species encode for various emergent phases is not well understood. Here, we leverage approaches from random-matrix theory and statistical physics to describe the emergent phase behavior of fluid mixtures with many species whose interactions are drawn randomly from an underlying distribution. Through numerical simulation and stability analyses, we show that these mixtures exhibit staged phase-separation kinetics and are characterized by multiple coexisting phases at steady state with distinct compositions. Random-matrix theory predicts the number of coexisting phases, validated by simulations with diverse component numbers and interaction parameters. Surprisingly, this model predicts an upper bound on the number of phases, derived from dynamical considerations, that is much lower than the limit from the Gibbs phase rule, which is obtained from equilibrium thermodynamic constraints. We design ensembles that encode either linear or nonmonotonic scaling relationships between the number of components and coexisting phases, which we validate through simulation and theory. Finally, inspired by parallels in biological systems, we show that including nonequilibrium turnover of components through chemical reactions can tunably modulate the number of coexisting phases at steady state without changing overall fluid composition. Together, our study provides a model framework that describes the emergent dynamical and steady-state phase behavior of liquid-like mixtures with many interacting constituents.

Fluids composed of many components with multiple coexisting phases are widespread in living and soft matter systems. A striking example occurs in the eukaryotic cell, where distinct biochemical pathways are compartmentalized into membraneless organelles called biomolecular condensates, which often form through liquid–liquid phase separation (1–3). Unlike two-phase oil–water mixtures, the cellular milieu is organized into tens of coexisting phases, each of which is enriched in specific biomolecules (1, 2, 4–8). Other prominent examples include microbial ecosystems that organize into fluid-like communities (9–11), self-assembling colloidal systems (12, 13), and synthetic multiphase materials derived from biomolecules (14, 15). Despite their extensive prevalence, our understanding of how microscopic interaction networks between individual constituents encode emergent multiphase behavior remains limited.Delineating the coexisting phases of a heterogeneous mixture is a problem with a rich history (16)—determined by constraints of chemical, mechanical, and thermal equilibrium. In mixtures with few components (fewer than five), a combination of theory, simulation, and experiment has enabled extensive characterization of phase-separation kinetics and equilibrium coexistence (17–23) and the interplay between phase separation and chemical reactions (19, 24, 25). In the biological context, recent studies have begun to connect biomolecular features to their macroscopic phase behavior in binary or ternary mixtures (7, 26, 27). However, as the number of components increases, determining the emergent phase behavior from the underlying constraints becomes unwieldy and intractable—from both analytical and numerical standpoints, except for very particular systems such as polydisperse blends of a single species (28). An alternate approach, originally proposed by Sear and Cuesta (29), aims to characterize the phase behavior of mixtures that contain many components whose pairwise interactions are drawn from a random distribution. By building on results on properties of random matrices, originally identified by Wigner (30) and subsequently applied in various contexts (31–33), they relate the initial direction of phase separation to properties of the interaction distribution, subsequently confirmed independently by simulation (34). These results, however, are limited to describing only the initial direction of phase separation for marginally stable fluid mixtures (i.e., coinciding exactly at the spinodal). Consequently, little is known about the overall phase behavior of fluid mixtures that spontaneously demix (i.e., within the spinodal)—including kinetics beyond the initial direction of phase separation or the number and composition of coexisting phases at equilibrium. More generally, the emergent phase behavior of fluid mixtures with many randomly interacting components is not well understood. This lack of understanding, in turn, limits our ability to rationally program fluid mixtures with different macroscopic properties.Here, we develop a dynamic model of phase separation in fluid mixtures with many randomly interacting components. Through simulation of the model, we demonstrate that fluid mixtures with many components exhibit characteristic similarities in phase-separation kinetics and in the number and compositional features of coexisting phases at steady state, even when the underlying interactions are random. We propose a simple model, combining insights from random-matrix theory and dynamical systems analyses, that predicts dynamical and steady-state characteristics of the emergent phase behavior. We subsequently discuss two distinct ensembles (or component design strategies) that encode either linear or nonmonotic scaling (i.e., with an optima) between the number of coexisting phases and components. Finally, we extend our framework to incorporate chemical reactions and show that active turnover of components can tunably modulate the number of coexisting phases at steady state even without altering overall fluid composition. Overall, our model provides a framework to predict and design emergent multiphase kinetics, compositions, and steady-state properties in fluid mixtures with many interacting components.