Affinity and Valence Impact the Extent and Symmetry of Phase Separation of Multivalent Proteins

Biomolecular self-assembly spatially segregates proteins with a limited number of binding sites (valence) into condensates that coexist with a dilute phase. We develop a many-body lattice model for a three-component system of proteins with fixed valence in a solvent. We compare the predictions of the model to experimental phase diagrams that we measure in vivo, which allows us to vary specifically a binding site’s affinity and valency. We find that the extent of phase separation varies exponentially with affinity and increases with valency. Valency alone determines the symmetry of the phase diagram.

Figure 1

Schematic description of the lattice model for a solution of tetramers, dimers, and solvent. (a) Lattice model for the system on a square lattice. Regions R1 and R2 show the same particles in two different configurations: the configuration in R2 has higher enthalpy owing to satisfied interactions between $A$ and $B$ and a lower entropy as other configurations of $A$ and $B$ are restricted by the interaction. The configuration in R1 has lower enthalpy and higher entropy than R2. (b) Four possible complexes when the $B$ molecules are tetramers.

Figure 2

(a) Phase diagram for a solution of tetramers, dimers, and solvent. The dotted line denotes the binodal and the end points of the tie lines (in light blue) are the concentrations of the concentrated (dashed) and dilute (dotted) phases. The yellow dashed line is the spinodal that gives the limit of metastability. The two red stars denote the two critical points where the tie line length vanishes and the two coexisting phases become identical. (b) Binodal for solutions of tetramers, dimers, and solvent at two different values of the interaction strength, $J$, show stronger phase separation (larger area of the two-phase region and larger differences in concentrations of the two coexisting phases) at larger $J$. (c) The minimum distance $\mathrm{\Delta }$ of the phase boundary from the origin as a function of interaction strength $J$. The line is the theory and the symbols (with error bars) are experimental data. We note that the data point for the highest interaction strength (purple) overestimates $\mathrm{\Delta }$ because the corresponding concentrations of $A$ and $B$ in the dilute phase reach a value below the detection limit of the microscope. (d)–(f) Experimental phase diagrams for the dimer-tetramer system with different interaction strengths (affinities) ($J=-\log (IS)$ in units of $k_{B}T$).

Figure 3

(a) The symmetry axis of the phase diagrams for phase separation of tetramers and dimers lies along the zero of the abscissa for asymptotically large $J$. This shows that phase separation is strongest at the stoichiometric ratio (of the effective concentrations). (b) Comparison of the value of the minimal concentration, $\mathrm{\Delta }$, obtained from the full model (points) and simplified model (lines) along the symmetry axis. (c) Theoretical phase diagram for both tetramers and dimers and hexamers and dimers, both with $J=4.0$ shows stronger phase separation for the hexamers. (d)–(f) Experimental phase diagrams for the same interaction strength and different valence (d) Phase diagram for a dimer-tetramer system (e) Phase diagram for a dimer-hexamer system. (f) Experimental data from panels (d) and (e) are overlaid. Axes correspond to concentration of interaction sites and dotted lines are approximate phase boundaries.