Differential Effects of Sequence-Local versus Nonlocal Charge Patterns on Phase Separation and Conformational Dimensions of Polyampholytes as Model Intrinsically Disordered Proteins.

Conformational properties of intrinsically disordered proteins (IDPs) are governed by a sequence-ensemble relationship. To differentiate the impact of sequence-local versus sequence-nonlocal features of an IDP's charge pattern on its conformational dimensions and its phase-separation propensity, the charge "blockiness" κ and the nonlocality-weighted sequence charge decoration (SCD) parameters are compared for their correlations with isolated-chain radii of gyration (Rgs) and upper critical solution temperatures (UCSTs) of polyampholytes modeled by random phase approximation, field-theoretic simulation, and coarse-grained molecular dynamics. SCD is superior to κ in predicting Rg because SCD accounts for effects of contact order, i.e., nonlocality, on dimensions of isolated chains. In contrast, κ and SCD are comparably good, though nonideal, predictors of UCST because frequencies of interchain contacts in the multiple-chain condensed phase are less sensitive to sequence positions than frequencies of intrachain contacts of an isolated chain, as reflected by κ correlating better with condensed-phase interaction energy than SCD.

Biological condensates form percolated networks with molecular motion properties distinctly different from dilute solutions.

Formation of membraneless organelles or biological condensates via phase separation and related processes hugely expands the cellular organelle repertoire. Biological condensates are dense and viscoelastic soft matters instead of canonical dilute solutions. To date, numerous different biological condensates have been discovered, but mechanistic understanding of biological condensates remains scarce. In this study, we developed an adaptive single-molecule imaging method that allows simultaneous tracking of individual molecules and their motion trajectories in both condensed and dilute phases of various biological condensates. The method enables quantitative measurements of concentrations, phase boundary, motion behavior, and speed of molecules in both condensed and dilute phases, as well as the scale and speed of molecular exchanges between the two phases. Notably, molecules in the condensed phase do not undergo uniform Brownian motion, but instead constantly switch between a (class of) confined state(s) and a random diffusion-like motion state. Transient confinement is consistent with strong interactions associated with large molecular networks (i.e., percolation) in the condensed phase. In this way, molecules in biological condensates behave distinctly different from those in dilute solutions. The methods and findings described herein should be generally applicable for deciphering the molecular mechanisms underlying the assembly, dynamics, and consequently functional implications of biological condensates.

Numerical Techniques for Applications of Analytical Theories to Sequence-Dependent Phase Separations of Intrinsically Disordered Proteins.

Biomolecular condensates, physically underpinned to a significant extent by liquid-liquid phase separation (LLPS), are now widely recognized by numerous experimental studies to be of fundamental biological, biomedical, and biophysical importance. In the face of experimental discoveries, analytical formulations emerged as a powerful yet tractable tool in recent theoretical investigations of the role of LLPS in the assembly and dissociation of these condensates. The pertinent LLPS often involves, though not exclusively, intrinsically disordered proteins engaging in multivalent interactions that are governed by their amino acid sequences. For researchers interested in applying these theoretical methods, here we provide a practical guide to a set of computational techniques devised for extracting sequence-dependent LLPS properties from analytical formulations. The numerical procedures covered include those for the determination of spinodal and binodal phase boundaries from a general free energy function with examples based on the random phase approximation in polymer theory, construction of tie lines for multiple-component LLPS, and field-theoretic simulation of multiple-chain heteropolymeric systems using complex Langevin dynamics. Since a more accurate physical picture often requires comparing analytical theory against explicit-chain model predictions, a commonly utilized methodology for coarse-grained molecular dynamics simulations of sequence-specific LLPS is also briefly outlined.

Analytical Formulation and Field-Theoretic Simulation of Sequence-Specific Phase Separation of Protein-Like Heteropolymers with Short- and Long-Spatial-Range Interactions.

A theory for sequence-dependent liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) in the study of biomolecular condensates is formulated by extending the random phase approximation (RPA) and field-theoretic simulation (FTS) of heteropolymers with spatially long-range Coulomb interactions to include the fundamental effects of short-range, hydrophobic-like interactions between amino acid residues. To this end, short-range effects are modeled by Yukawa interactions between multiple nonelectrostatic charges derived from an eigenvalue decomposition of pairwise residue-residue contact energies. Chain excluded volume is afforded by incompressibility constraints. A mean-field approximation leads to an effective Flory-Huggins χ parameter, which, in conjunction with RPA, accounts for the contact-interaction effects of amino acid composition and the sequence-pattern effects of long-range electrostatics in IDP LLPS, whereas FTS based on the formulation provides full sequence dependence for both short- and long-range interactions. This general approach is illustrated here by applications to variants of a natural IDP in the context of several different amino-acid interaction schemes as well as a set of different model hydrophobic-polar sequences sharing the same composition. Effectiveness of the methodology is verified by coarse-grained explicit-chain molecular dynamics simulations.

Field theory description of ion association in re-entrant phase separation of polyampholytes.

Phase separation of several different overall neutral polyampholyte species (with zero net charge) is studied in solution with two oppositely charged ion species that can form ion pairs through an association reaction. Hereby, a field theory description of the system, which treats polyampholyte charge sequence dependent electrostatic interactions as well as excluded volume effects, is given. Interestingly, analysis of the model using random phase approximation and field theoretic simulation consistently shows evidence of a re-entrant polyampholyte phase separation at high ion concentrations when there is an overall decrease of volume upon ion association. As an illustration of the ramifications of our theoretical framework, several polyampholyte concentration vs ion concentration phase diagrams under constant temperature conditions are presented to elucidate the dependence of phase separation behavior on the polyampholyte sequence charge pattern as well as ion pair dissociation constant, volumetric effects on ion association, solvent quality, and temperature.

Subcompartmentalization of polyampholyte species in organelle-like condensates is promoted by charge-pattern mismatch and strong excluded-volume interaction.

Polyampholyte field theory and explicit-chain molecular dynamics models of sequence-specific phase separation of a system with two intrinsically disordered protein (IDP) species indicate consistently that a substantial polymer excluded volume and a significant mismatch of the IDP sequence charge patterns can act in concert, but not in isolation, to demix the two IDP species upon condensation. This finding reveals an energetic-geometric interplay in a stochastic, "fuzzy" molecular recognition mechanism that may facilitate subcompartmentalization of membraneless organelles.

A Simple Explicit-Solvent Model of Polyampholyte Phase Behaviors and Its Ramifications for Dielectric Effects in Biomolecular Condensates.

Biomolecular condensates such as membraneless organelles, underpinned by liquid-liquid phase separation (LLPS), are important for physiological function, with electrostatics, among other interaction types, being a prominent force in their assembly. Charge interactions of intrinsically disordered proteins (IDPs) and other biomolecules are sensitive to the aqueous dielectric environment. Because the relative permittivity of protein is significantly lower than that of water, the interior of an IDP condensate is expected to be a relatively low-dielectric regime, which aside from its possible functional effects on client molecules should facilitate stronger electrostatic interactions among the scaffold IDPs. To gain insight into this LLPS-induced dielectric heterogeneity, addressing in particular whether a low-dielectric condensed phase entails more favorable LLPS than that posited by assuming IDP electrostatic interactions are uniformly modulated by the higher dielectric constant of the pure solvent, we consider a simplified multiple-chain model of polyampholytes immersed in explicit solvents that are either polarizable or possess a permanent dipole. Notably, simulated phase behaviors of these systems exhibit only minor to moderate differences from those obtained using implicit-solvent models with a uniform relative permittivity equals to that of pure solvent. Buttressed by theoretical treatments developed here using random phase approximation and polymer field-theoretic simulations, these observations indicate a partial compensation of effects between favorable solvent-mediated interactions among the polyampholytes in the condensed phase and favorable polyampholyte-solvent interactions in the dilute phase, often netting only a minor enhancement of overall LLPS propensity from the very dielectric heterogeneity that arises from the LLPS itself. Further ramifications of this principle are discussed.

Thermodynamics of amyloid formation and the role of intersheet interactions.

The self-assembly of proteins into ß-sheet-rich amyloid fibrils has been observed to occur with sigmoidal kinetics, indicating that the system initially is trapped in a metastable state. Here, we use a minimal lattice-based model to explore the thermodynamic forces driving amyloid formation in a finite canonical (NVT) system. By means of generalized-ensemble Monte Carlo techniques and a semi-analytical method, the thermodynamic properties of this model are investigated for different sets of intersheet interaction parameters. When the interactions support lateral growth into multi-layered fibrillar structures, an evaporation/condensation transition is observed, between a supersaturated solution state and a thermodynamically distinct state where small and large fibril-like species exist in equilibrium. Intermediate-size aggregates are statistically suppressed. These properties do not hold if aggregate growth is one-dimensional.