  Biomolecular condensates self-assemble when proteins and nucleic acids
spontaneously demix to form droplets within the crowded intracellular milieu.
This simple mechanism underlies the formation of a wide variety of membraneless
compartments in living cells. To understand how multiple condensates with
distinct compositions can self-assemble in such a heterogeneous system, we
study a minimal model in which we can "program" the pairwise interactions among
hundreds of species. We show that the number of distinct condensates that can
be reliably assembled grows superlinearly with the number of species in the
mixture when the condensates are allowed to share components. Furthermore, we
show that we can predict the maximum number of distinct condensates in a
mixture without knowing the details of the pairwise interactions. Simulations
of condensate growth confirm these predictions and suggest that the physical
rules governing the achievable complexity of condensate-mediated spatial
organization are broadly applicable to biomolecular mixtures.
