(Dis-)appearance of liquid-liquid phase transitions in a heterogeneous activated patchy particle model and experiment

TL;DR

The paper investigates LLPS in heterogeneous ion-activated patchy particle models for protein solutions with multivalent ions. It extends previous single-type patch frameworks by introducing multiple patch types and ion-binding energies $\epsilon_b^{i,\alpha}$, and solves the resulting thermodynamics with Wertheim theory to predict binodals and critical points. Experimental data on $\mathrm{BSA}$ with trivalent salts ($\mathrm{HoCl_3}$, $\mathrm{YCl_3}$) validate the theory and reveal ion-specific LLPS behavior that the two-patch model captures with physically reasonable parameters. The results show that the distribution of ion binding across patches and the competition between like-patch and unlike-patch attractions can cause LLPS to appear, broaden, shrink, or disappear, providing a mechanism to understand charge-driven LLPS in biology with potential links to atomistic insights.

Abstract

The ion-activated patchy particle model is an important theoretical framework to investigate the phase behaviour of globular proteins in the presence of multivalent ions. In this work, we study and highlight the influence of patch heterogeneity on the extension, appearance and disappearance of the liquid-liquid coexistence region of the phase diagram. We demonstrate that within this model the binding energy between salt ions and patches of different type is a key factor in determining the phase behavior. Specifically, we show under which conditions liquid-liquid phase separation (LLPS) in these systems can appear or disappear for varying binding energy and ion-mediated attraction energy between ion-occupied and unoccupied patches. In particular we address the influence of the patch type dependence of these energies on the (dis)appearance of LLPS. These results rationalize our new results on ion-dependent liquid-liquid phase separation in solutions of bovine serum albumine with trivalent cations. In comparison with models with non-activated patches, where the gas-liquid transition disappears when the number of patches approaches two, we find the complementary mechanism that ions may shift the attractions from stronger to weaker patches (with an accompanying disappearance of the transition), if their binding energy to the patches changes. The results have implications for the understanding of charge-driven LLPS in biological systems and its suppression.

Figures (9)

• Figure 1: (a) Metal ions are bound to patches of type $\alpha$, $\beta$ ... with probability $\theta_{\alpha}, \theta_{\beta}$.... (b) Occupied patches (of any type $\alpha$) interact attractively with unoccupied patches (of any type $\beta$). Averaging (see text) gives an effective patch-patch interaction energy $\varepsilon_\text{pp}$ between occupied and unoccupied patches.
• Figure 2: Change of binodal loop in the single-type-of-patch model upon variation of $\beta \varepsilon_\text{uo}$ ("onion-shell behavior"). $\beta\varepsilon_b= -5$, $c_0$ = 1M, $R =2.8$ nm, $R_s=R/18$. (a) $\eta$-$c^\text{res}$ plane (reservoir salt concentration). (b) $\eta$-$c_s$ plane (physical salt concentration).
• Figure 3: Change of binodal loop in the single-type-of-patch model upon variation of $\beta \epsilon_b$ ("squeeze/stretch behavior"). $\beta\epsilon_{uo}=-14$, $c_0$ = 1M, $R_p =2.8$ nm. (a) $\eta$-$c^\text{res}$ plane (reservoir salt concentration). (b) $\eta$-$c_s$ plane (physical salt concentration).
• Figure 4: Upon variation of the salt chemical potential, the occupation probabilities $\theta_\alpha$ trace a curve in the $\theta_1$-$\theta_2$ plane, here shown for ion binding energies being equal, $\epsilon_b^1=\epsilon_b^2$, and different with $\beta \Delta \epsilon_b^{1 2}= \pm 2$.
• Figure 5: The case of like-patch attraction dominance, interaction energy parameters are $\beta \varepsilon^{11} = -22$, $\beta \varepsilon^{22} = -12$, $\beta \varepsilon^{12} = -11$. The number of patches is $M=4$, and $m_1=m_2=2$. The difference in ion binding energy $\Delta \epsilon_b^{1 2}=\epsilon_b^2-\epsilon_b^1$ is varied ($\beta \varepsilon^{2}_{b}= -1.5$). (a) Effective patch energy $\beta \varepsilon_\text{pp}(c^\text{res})$ as function of reservoir salt concentration $c^\text{res}$. The dashed line is the critical strength for LLPS. (b) Binodal loops in the $\eta$-$c^\text{res}$ plane.