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.

Effects of Cosolvents and Crowding Agents on the Stability and Phase Transition Kinetics of the SynGAP/PSD-95 Condensate Model of Postsynaptic Densities.

The SynGAP/PSD-95 binary protein system serves as a simple mimicry of postsynaptic densities (PSDs), which are protein assemblies based largely on liquid-liquid phase separation (LLPS), that are located underneath the plasma membrane of excitatory synapses. Surprisingly, the LLPS of the SynGAP/PSD-95 system is much more pressure sensitive than typical folded states of proteins or nucleic acids. It was found that phase-separated SynGAP/PSD-95 droplets dissolve into a homogeneous solution at a pressure of tens to hundred bar. Since organisms in the deep sea are exposed to pressures of up to ∼1000 bar, this observation suggests that deep-sea organisms must counteract the high pressure sensitivity of PSDs to avoid neurological impairment. We demonstrate here that the compatible osmolyte trimethylamine-N-oxide (TMAO) as well as macromolecular crowding agents at moderate concentrations can mitigate the deleterious effect of pressure on SynGAP/PSD-95 droplet stability, extending stable LLPS up to almost a kbar level. Moreover, the formation of SynGAP/PSD-95 droplets is a very rapid process that can be switched on and off in seconds. In contrast with the marked effects of the cosolutes on droplet stability, at the cosolutes' respective biologically relevant concentrations, their impact on the phase transformation kinetics is rather small. Only a high TMAO concentration results in a significant kinetic retardation of LLPS. Taken together, these findings offer new biophysical insights into the neurological effects of hydrostatic pressure. In particular, our results indicate how pressure-induced neurological disorders might be alleviated by upregulating certain cosolutes in the cellular milieu.

Assembly of model postsynaptic densities involves interactions auxiliary to stoichiometric binding.

The assembly of functional biomolecular condensates often involves liquid-liquid phase separation (LLPS) of proteins with multiple modular domains, which can be folded or conformationally disordered to various degrees. To understand the LLPS-driving domain-domain interactions, a fundamental question is how readily the interactions in the condensed phase can be inferred from interdomain interactions in dilute solutions. In particular, are the interactions leading to LLPS exclusively those underlying the formation of discrete interdomain complexes in homogeneous solutions? We address this question by developing a mean-field LLPS theory of two stoichiometrically constrained solute species. The theory is applied to the neuronal proteins SynGAP and PSD-95, whose complex coacervate serves as a rudimentary model for neuronal postsynaptic densities (PSDs). The predicted phase behaviors are compared with experiments. Previously, a three SynGAP/two PSD-95 ratio was determined for SynGAP/PSD-95 complexes in dilute solutions. However, when this 3:2 stoichiometry is uniformly imposed in our theory encompassing both dilute and condensed phases, the tie-line pattern of the predicted SynGAP/PSD-95 phase diagram differs drastically from that obtained experimentally. In contrast, theories embodying alternate scenarios postulating auxiliary SynGAP-PSD-95 as well as SynGAP-SynGAP and PSD-95-PSD-95 interactions, in addition to those responsible for stoichiometric SynGAP/PSD-95 complexes, produce tie-line patterns consistent with experiment. Hence, our combined theoretical-experimental analysis indicates that weaker interactions or higher-order complexes beyond the 3:2 stoichiometry, but not yet documented, are involved in the formation of SynGAP/PSD-95 condensates, imploring future efforts to ascertain the nature of these auxiliary interactions in PSD-like LLPS and underscoring a likely general synergy between stoichiometric, structurally specific binding and stochastic, multivalent "fuzzy" interactions in the assembly of functional biomolecular condensates.

Small-Angle X-ray Scattering Signatures of Conformational Heterogeneity and Homogeneity of Disordered Protein Ensembles.

An accurate account of disordered protein conformations is of central importance to deciphering the physicochemical basis of biological functions of intrinsically disordered proteins and the folding-unfolding energetics of globular proteins. Physically, disordered ensembles of nonhomopolymeric polypeptides are expected to be heterogeneous, i.e., they should differ from those homogeneous ensembles of homopolymers that harbor an essentially unique relationship between average values of end-to-end distance REE and radius of gyration Rg. It was posited recently, however, that small-angle X-ray scattering (SAXS) data on conformational dimensions of disordered proteins can be rationalized almost exclusively by homopolymer ensembles. Assessing this perspective, chain-model simulations are used to evaluate the discriminatory power of SAXS-determined molecular form factors (MFFs) with regard to homogeneous versus heterogeneous ensembles. The general approach adopted here is not bound by any assumption about ensemble encodability, in that the postulated heterogeneous ensembles we evaluated are not restricted to those entailed by simple interaction schemes. Our analysis of MFFs for certain heterogeneous ensembles with more narrowly distributed REE and Rg indicates that while they deviate from MFFs of homogeneous ensembles, the differences can be rather small. Remarkably, some heterogeneous ensembles with asphericity and REE drastically different from those of homogeneous ensembles can nonetheless exhibit practically identical MFFs, demonstrating that SAXS MFFs do not afford unique characterizations of basic properties of conformational ensembles in general. In other words, the ensemble to MFF mapping is practically many-to-one and likely nonsmooth. Heteropolymeric variations of the REE-Rg relationship were further showcased using an analytical perturbation theory developed here for flexible heteropolymers. Ramifications of our findings for interpretation of experimental data are discussed.

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.

Comparative roles of charge, π, and hydrophobic interactions in sequence-dependent phase separation of intrinsically disordered proteins.

Endeavoring toward a transferable, predictive coarse-grained explicit-chain model for biomolecular condensates underlain by liquid-liquid phase separation (LLPS) of proteins, we conducted multiple-chain simulations of the N-terminal intrinsically disordered region (IDR) of DEAD-box helicase Ddx4, as a test case, to assess roles of electrostatic, hydrophobic, cation-π, and aromatic interactions in amino acid sequence-dependent LLPS. We evaluated three different residue-residue interaction schemes with a shared electrostatic potential. Neither a common hydrophobicity scheme nor one augmented with arginine/lysine-aromatic cation-π interactions consistently accounted for available experimental LLPS data on the wild-type, a charge-scrambled, a phenylalanine-to-alanine (FtoA), and an arginine-to-lysine (RtoK) mutant of Ddx4 IDR. In contrast, interactions based on contact statistics among folded globular protein structures reproduce the overall experimental trend, including that the RtoK mutant has a much diminished LLPS propensity. Consistency between simulation and experiment was also found for RtoK mutants of P-granule protein LAF-1, underscoring that, to a degree, important LLPS-driving π-related interactions are embodied in classical statistical potentials. Further elucidation is necessary, however, especially of phenylalanine's role in condensate assembly because experiments on FtoA and tyrosine-to-phenylalanine mutants suggest that LLPS-driving phenylalanine interactions are significantly weaker than posited by common statistical potentials. Protein-protein electrostatic interactions are modulated by relative permittivity, which in general depends on aqueous protein concentration. Analytical theory suggests that this dependence entails enhanced interprotein interactions in the condensed phase but more favorable protein-solvent interactions in the dilute phase. The opposing trends lead to only a modest overall impact on LLPS.

Analytical Theory for Sequence-Specific Binary Fuzzy Complexes of Charged Intrinsically Disordered Proteins.

Intrinsically disordered proteins (IDPs) are important for biological functions. In contrast to folded proteins, molecular recognition among certain IDPs is "fuzzy" in that their binding and/or phase separation are stochastically governed by the interacting IDPs' amino acid sequences, while their assembled conformations remain largely disordered. To help elucidate a basic aspect of this fascinating yet poorly understood phenomenon, the binding of a homo or heterodimeric pair of polyampholytic IDPs is modeled statistical mechanically using cluster expansion. We find that the binding affinities of binary fuzzy complexes in the model correlate strongly with a newly derived simple "joint sequence charge decoration" parameter readily calculable from the pair of IDPs' sequence charge patterns. Predictions by our analytical theory are in essential agreement with coarse-grained explicit-chain simulations. This computationally efficient theoretical framework is expected to be broadly applicable to rationalizing and predicting sequence-specific IDP-IDP polyelectrostatic interactions.

A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes.

The physical chemistry of liquid-liquid phase separation (LLPS) of polymer solutions bears directly on the assembly of biologically functional dropletlike bodies from proteins and nucleic acids. These biomolecular condensates include certain extracellular materials and intracellular compartments that are characterized as "membraneless organelles." Analytical theories are a valuable, computationally efficient tool for addressing general principles. LLPS of neutral homopolymers is quite well described by theory, but it has been a challenge to develop general theories for the LLPS of heteropolymers involving charge-charge interactions. Here, we present a theory that combines a random-phase-approximation treatment of polymer density fluctuations and an account of intrachain conformational heterogeneity based on renormalized Kuhn lengths to provide predictions of LLPS properties as a function of pH, salt, and charge patterning along the chain sequence. Advancing beyond more limited analytical approaches, our LLPS theory is applicable to a wide variety of charged sequences ranging from highly charged polyelectrolytes to neutral or nearly neutral polyampholytes. This theory should be useful in high-throughput screening of protein and other sequences for their LLPS propensities and can serve as a basis for more comprehensive theories that incorporate nonelectrostatic interactions. Experimental ramifications of our theory are discussed.

Pressure Sensitivity of SynGAP/PSD-95 Condensates as a Model for Postsynaptic Densities and Its Biophysical and Neurological Ramifications.

Biomolecular condensates consisting of proteins and nucleic acids can serve critical biological functions, so that some condensates are referred as membraneless organelles. They can also be disease-causing, if their assembly is misregulated. A major physicochemical basis of the formation of biomolecular condensates is liquid-liquid phase separation (LLPS). In general, LLPS depends on environmental variables, such as temperature and hydrostatic pressure. The effects of pressure on the LLPS of a binary SynGAP/PSD-95 protein system mimicking postsynaptic densities, which are protein assemblies underneath the plasma membrane of excitatory synapses, were investigated. Quite unexpectedly, the model system LLPS is much more sensitive to pressure than the folded states of typical globular proteins. Phase-separated droplets of SynGAP/PSD-95 were found to dissolve into a homogeneous solution already at ten-to-hundred bar levels. The pressure sensitivity of SynGAP/PSD-95 is seen here as a consequence of both pressure-dependent multivalent interaction strength and void volume effects. Considering that organisms in the deep sea are under pressures up to about 1 kbar, this implies that deep-sea organisms have to devise means to counteract this high pressure sensitivity of biomolecular condensates to avoid harm. Intriguingly, these findings may shed light on the biophysical underpinning of pressure-related neurological disorders in terrestrial vertebrates.

Temperature, Hydrostatic Pressure, and Osmolyte Effects on Liquid-Liquid Phase Separation in Protein Condensates: Physical Chemistry and Biological Implications.

Liquid-liquid phase separation (LLPS) of proteins and other biomolecules play a critical role in the organization of extracellular materials and membrane-less compartmentalization of intra-organismal spaces through the formation of condensates. Structural properties of such mesoscopic droplet-like states were studied by spectroscopy, microscopy, and other biophysical techniques. The temperature dependence of biomolecular LLPS has been studied extensively, indicating that phase-separated condensed states of proteins can be stabilized or destabilized by increasing temperature. In contrast, the physical and biological significance of hydrostatic pressure on LLPS is less appreciated. Summarized here are recent investigations of protein LLPS under pressures up to the kbar-regime. Strikingly, for the cases studied thus far, LLPSs of both globular proteins and intrinsically disordered proteins/regions are typically more sensitive to pressure than the folding of proteins, suggesting that organisms inhabiting the deep sea and sub-seafloor sediments, under pressures up to 1 kbar and beyond, have to mitigate this pressure-sensitivity to avoid unwanted destabilization of their functional biomolecular condensates. Interestingly, we found that trimethylamine-N-oxide (TMAO), an osmolyte upregulated in deep-sea fish, can significantly stabilize protein droplets under pressure, pointing to another adaptive advantage for increased TMAO concentrations in deep-sea organisms besides the osmolyte's stabilizing effect against protein unfolding. As life on Earth might have originated in the deep sea, pressure-dependent LLPS is pertinent to questions regarding prebiotic proto-cells. Herein, we offer a conceptual framework for rationalizing the recent experimental findings and present an outline of the basic thermodynamics of temperature-, pressure-, and osmolyte-dependent LLPS as well as a molecular-level statistical mechanics picture in terms of solvent-mediated interactions and void volumes.

Pressure-Sensitive and Osmolyte-Modulated Liquid-Liquid Phase Separation of Eye-Lens γ-Crystallins.

Biomolecular condensates can be functional (e.g., as membrane-less organelles) or dysfunctional (e.g., as precursors to pathological protein aggregates). A major physical underpinning of biomolecular condensates is liquid-liquid phase separation (LLPS) of proteins and nucleic acids. Here we investigate the effects of temperature and pressure on the LLPS of the eye-lens protein γ-crystallin using UV/vis and IR absorption, fluorescence spectroscopy, and light microscopy to characterize the mesoscopic phase states. Quite unexpectedly, the LLPS of γ-crystallin is much more sensitive to pressure than folded states of globular proteins. At low temperatures, the phase-separated droplets of γ-crystallin dissolve into a homogeneous solution at as low as ∼0.1 kbar whereas proteins typically unfold above ∼3 kbar. This observation suggests, in general, that organisms thriving under high-pressure conditions in the deep sea, with pressure of up to 1 kbar, have to cope with this pressure sensitivity of biomolecular condensates to avoid detrimental impacts to their physiology. Interestingly, our experiments demonstrate that trimethylamine- N-oxide, an osmolyte upregulated in deep-sea fish, significantly enhances the stability of the condensed protein droplets, pointing to a previously unrecognized aspect of the adaptive advantage of increased concentrations of osmolytes in deep-sea organisms. As the birth place of life on earth could have been the deep sea, studies of pressure effects on LLPS as presented here are relevant to the possible formation of protocells under prebiotic conditions. A physical framework to conceptualize our observations and further ramifications of biomolecular LLPS at low temperatures and high hydrostatic pressures is discussed.

Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters.

Biomolecular condensates undergirded by phase separations of proteins and nucleic acids serve crucial biological functions. To gain physical insights into their genetic basis, we study how liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) depends on their sequence charge patterns using a continuum Langevin chain model wherein each amino acid residue is represented by a single bead. Charge patterns are characterized by the "blockiness" measure κ and the "sequence charge decoration" (SCD) parameter. Consistent with random phase approximation (RPA) theory and lattice simulations, LLPS propensity as characterized by critical temperature Tcr* increases with increasingly negative SCD for a set of sequences showing a positive correlation between κ and -SCD. Relative to RPA, the simulated sequence-dependent variation in Tcr* is often-though not always-smaller, whereas the simulated critical volume fractions are higher. However, for a set of sequences exhibiting an anti-correlation between κ and -SCD, the simulated Tcr*'s are quite insensitive to either parameter. Additionally, we find that blocky sequences that allow for strong electrostatic repulsion can lead to coexistence curves with upward concavity as stipulated by RPA, but the LLPS propensity of a strictly alternating charge sequence was likely overestimated by RPA and lattice models because interchain stabilization of this sequence requires spatial alignments that are difficult to achieve in real space. These results help delineate the utility and limitations of the charge pattern parameters and of RPA, pointing to further efforts necessary for rationalizing the newly observed subtleties.

Pressure-Induced Dissolution and Reentrant Formation of Condensed, Liquid-Liquid Phase-Separated Elastomeric α-Elastin.

We investigated the combined effects of temperature and pressure on liquid-liquid phase separation (LLPS) phenomena of α-elastin up to the multi-kbar regime. FT-IR spectroscopy, CD, UV/Vis absorption, phase-contrast light and fluorescence microscopy techniques were employed to reveal structural changes and mesoscopic phase states of the system. A novel pressure-induced reentrant LLPS was observed in the intermediate temperature range. A molecular-level picture, in particular on the role of hydrophobic interactions, hydration, and void volume in controlling LLPS phenomena is presented. The potential role of the LLPS phenomena in the development of early cellular compartmentalization is discussed, which might have started in the deep sea, where pressures up to the kbar level are encountered.

Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates.

Liquid-liquid phase separation and related condensation processes of intrinsically disordered proteins (IDPs), proteins with intrinsically disordered regions, and nucleic acids underpin various condensed-liquid droplets or gel-like assemblies in the cellular environment. Collectively referred to as condensates, these bodies provide spatial/temporal compartmentalization, often serving as hubs for regulated biomolecular interactions. Examples include certain extracellular materials, transcription complexes, and membraneless organelles such as germ and stress granules and the nucleolus. They are critically important to cellular function; thus misregulation of their assembly is implicated in many diseases. Biomolecular condensates are complex entities. Our understanding of their inner workings is only in its infancy. Nonetheless, insights into basic biophysical principles of their assembly can be gained by applying analytical theories to elucidate how IDP phase behaviors are governed by the properties of the multivalent, solvent-mediated interactions entailed by the proteins' amino acid sequences. Here we briefly review the background of the pertinent polymer theories and outline the approximations that enable a tractable theoretical account of the dependence of IDP phase behaviors on the charge pattern of the IDP sequence. Of relevance to the homeostatic assembly of compositionally and functionally distinct condensates in the cellular context, theory indicates that the propensity for populations of different IDP sequences to mix or demix upon phase separation is affected by the similarity or dissimilarity of the sequence charge patterns. We also explore prospects of extending analytical theories to account for dynamic aspects of biomolecular condensates and to incorporate effects of cation-π, π-π, and temperature-dependent hydrophobic interactions on IDP phase properties.