  Phase separation, crucial for spatially segregating biomolecules in cells, is
well-understood in the simple case of a few components with pairwise
interactions. Yet, biological cells challenge the simple picture in at least
two ways: First, biomolecules, like proteins and nucleic acids, exhibit
complex, higher-order interactions, where a single molecule may interact with
multiple others simultaneously. Second, cells comprise a myriad of different
components that form various droplets. Such multicomponent phase separation has
been studied in the simple case of pairwise interactions, but an analysis of
higher-order interactions is lacking. We propose such a theory and study the
corresponding phase diagrams numerically. We find that interactions between
three components are similar to pairwise interactions, whereas
composition-dependent higher-order interactions between two components can
oppose phase separation. This surprising result can only be revealed from the
equilibrium phase diagrams, implying that the often-used stability analysis of
homogeneous states is inadequate to study these systems. We thus show that
higher-order interactions could play a crucial role in forming droplets in
cells, and their manipulation could offer novel approaches to controlling
multicomponent phase separation.
