Stoichiometry Controls the Dynamics of Liquid Condensates of Associative Proteins

Multivalent associative proteins with strong complementary interactions play a crucial role in phase separation of intracellular liquid condensates. We study the internal dynamics of such “bond-network” condensates comprising two complementary proteins via scaling analysis and molecular dynamics. We find that when stoichiometry is balanced, relaxation slows down dramatically due to a scarcity of alternative binding partners following bond breakage. This microscopic slow-down strongly affects the bulk diffusivity, viscosity, and mixing, which provides a means to experimentally test this prediction.

Figure 1

Stoichiometry controls the bond relaxation time of multivalent associative proteins. (a) Sketch of associative multivalent proteins, with complementary domains separated by flexible linkers. (b) Strong yet reversible binding between proteins leads them to condense into a network with most possible bonds formed. (c)–(e) Schematic of the bond relaxation mechanism. When two initially bound domains (c) unbind, the two are caged in a small volume $v_{\mathrm{cage}}$ (d). Two events can then occur: the initially bound domains can rebind, or, if a free domain is within reach, a new bond may form (e), which is the system’s basic relaxation mechanism. (f) Concentration of unbound domains $c_{\mathrm{free}}$ of both types as a function of stoichiometry difference. (g) Relaxation time [Eq. (1)] corresponding to the process of unbinding and then rebinding with a new partner (c)–(e), as a function of stoichiometry difference. Here, $\epsilon =e^{-\mathrm{\Delta }F}$.

Figure 2

MD simulations reveal the importance of stoichiometry to the dynamical properties of the condensate. (a) MD model for the multivalent associative proteins. Colored spheres represent $A$ and $B$ domains. (b) Representative snapshot of the dense, network-forming liquid condensate. (c) Bond relaxation time (see text) as a function of stoichiometry for different binding strengths. Symbols indicate MD simulations; solid curves indicate theory [Eq. (1)] with $c_{AB}$ estimated assuming full binding of the minority domains and with fitted values $v_{\mathrm{cage}}=11.4$, $v_0=0.4$, and ${\tau }_0=0.37$. (d) Bond relaxation time ${\tau }_{\mathrm{rel}}$ as a function of binding strength is consistent with predicted scaling for both equal and unequal stoichiometries [Eq. (1), Fig. 1]. (e) Mean squared displacement (MSD) of individual domains as a function of time reveals diffusive scaling (dashed line) at long times (here $\delta =0$). (f) Diffusion coefficient of the minority species as a function of binding strength at equal and unequal stoichiometry. (g) Longtime diffusion coefficient plotted against bond relaxation time, for all values of $\delta$ and $\mathrm{\Delta }F$. The dotted black line indicates $D\propto {\tau }_{\mathrm{rel}}^{-1}$. Transparent circles correspond to systems where one component is in large excess, $|\delta |>0.2c_{\mathrm{tot}}$. (h) Viscosity, obtained using the Green-Kubo relation, as a function of binding strength, reflects the scaling of the bond relaxation time (d).

Figure 3

Composition controls mixing rate near equal stoichiometry. (a) Snapshot of an MD simulation with initially tagged particles on the left side of the box. (b) Normalized concentration profiles for tagged particles along the long axis at different times, for equal stoichiometry $\delta =0$, showing slow relaxation towards the homogeneous state. (c) Relaxation of the tagged concentration difference between the two half-boxes, for different binding free energies. (d) Equilibration time as a function of binding strength. The unbalanced case has $\delta =0.14$.