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Abstract
Biomolecular condensates are emerging as an important organizational principle within
living cells. These condensed states are formed by phase separation, yet little is
known about how material properties are encoded within the constituent molecules and
how the specificity for being in different phases is established. Here we use analytic
theory to explain the phase behavior of the cancer-related protein SPOP and its substrate
DAXX. Binary mixtures of these molecules have a phase diagram that contains dilute
liquid, dense liquid, and gel states. We show that these discrete phases appear due
to a competition between SPOP-DAXX and DAXX-DAXX interactions. The stronger SPOP-DAXX
interactions dominate at sub-stoichiometric DAXX concentrations leading to the formation
of crosslinked gels. The theory shows that the driving force for gel formation is
not the binding energy, but rather the entropy of distributing DAXX molecules on the
binding sites. At high DAXX concentrations the SPOP-DAXX interactions saturate, which
leads to the dissolution of the gel and the appearance of a liquid phase driven by
weaker DAXX-DAXX interactions. This competition between interactions allows multiple
dense phases to form in a narrow region of parameter space. We propose that the molecular
architecture of phase-separating proteins governs the internal structure of dense
phases, their material properties and their functions. Analytical theory can reveal
these properties on the long length and time scales relevant to biomolecular condensates.