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

Fundamental Aspects of Phase-Separated Biomolecular Condensates

Biomolecular condensates, formed through phase separation, are upending our understanding in much of molecular, cell, and developmental biology. There is an urgent need to elucidate the physicochemical foundations of the behaviors and properties of biomolecular condensates. Here we aim to fill this need by writing a comprehensive, critical, and accessible review on the fundamental aspects of phase-separated biomolecular condensates. We introduce the relevant theoretical background, present the theoretical basis for the computation and experimental measurement of condensate properties, and give mechanistic interpretations of condensate behaviors and properties in terms of interactions at the molecular and residue levels.

The effect of monomer polarizability on the stability and salt partitioning in model coacervates

Coacervation of charged polymer chains has been a topic of major interest in both polymer and biological sciences, as it is a subset of a phenomenon called liquid-liquid phase separation (LLPS). In this process the polymer-rich phase separates from the polymer-lean supernatant while still maintaining its liquid-like properties. LLPS has been shown to play a crucial role in cellular homeostasis by driving the formation of membraneless organelles. It also has the potential to be harnessed to aid in novel therapeutical applications. Recent studies have demonstrated that there is no one simple mechanism which drives LLPS, which is instead a result of the combined effect of electrostatic, dipolar, hydrophobic, and other weak interactions. Using coarse-grained polymer simulations we investigate the relatively unexplored effects of monomer polarizability and spatially varying dielectric constant on LLPS propensity, and these factors affect the properties of the resulting condensates. In order to produce spatial variations in the dielectric constant, all our simulations include explicit solvent and counterions. We demonstrate that polarizability has only a minor effect on the bulk behaviour of the condensates but plays a major role when ion partitioning and microstructure are considered. We observe that the major contribution comes from the nature of the neutral blocks as endowing them with an induced dipole changes their character from hydrophobic to hydrophilic. We hypothesize that the results of this work can aid in guiding future studies concerned with LLPS by providing a general framework and by highlighting important factors which influence LLPS.

OpenABC enables flexible, simplified, and efficient GPU accelerated simulations of biomolecular condensates

Biomolecular condensates are important structures in various cellular processes but are challenging to study using traditional experimental techniques. In silico simulations with residue-level coarse-grained models strike a balance between computational efficiency and chemical accuracy. They could offer valuable insights by connecting the emergent properties of these complex systems with molecular sequences. However, existing coarse-grained models often lack easy-to-follow tutorials and are implemented in software that is not optimal for condensate simulations. To address these issues, we introduce OpenABC, a software package that greatly simplifies the setup and execution of coarse-grained condensate simulations with multiple force fields using Python scripting. OpenABC seamlessly integrates with the OpenMM molecular dynamics engine, enabling efficient simulations with performance on a single GPU that rivals the speed achieved by hundreds of CPUs. We also provide tools that convert coarse-grained configurations to all-atom structures for atomistic simulations. We anticipate that OpenABC will significantly facilitate the adoption of in silico simulations by a broader community to investigate the structural and dynamical properties of condensates.

OpenABC Enables Flexible, Simplified, and Efficient GPU Accelerated Simulations of Biomolecular Condensates

Biomolecular condensates are important structures in various cellular processes but are challenging to study using traditional experimental techniques. In silico simulations with residue-level coarse-grained models strike a balance between computational efficiency and chemical accuracy. They could offer valuable insights by connecting the emergent properties of these complex systems with molecular sequences. However, existing coarse-grained models often lack easy-to-follow tutorials and are implemented in software that is not optimal for condensate simulations. To address these issues, we introduce OpenABC, a software package that greatly simplifies the setup and execution of coarse-grained condensate simulations with multiple force fields using Python scripting. OpenABC seamlessly integrates with the OpenMM molecular dynamics engine, enabling efficient simulations with performances on a single GPU that rival the speed achieved by hundreds of CPUs. We also provide tools that convert coarse-grained configurations to all-atom structures for atomistic simulations. We anticipate that Open-ABC will significantly facilitate the adoption of in silico simulations by a broader community to investigate the structural and dynamical properties of condensates. Open-ABC is available at https://github.com/ZhangGroup-MITChemistry/OpenABC.

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.

Calculating Binodals and Interfacial Tension of Phase-Separated Condensates from Molecular Simulations with Finite-Size Corrections

We illustrate three methods for calculating the binodals of phase-separated condensates from molecular simulations. Because molecular simulations can only be carried out for small system sizes, correction for finite sizes may be required for making direct comparison between calculated results and experimental data. We first summarize the three methods and then present detailed implementation of each method on a Lennard-Jones fluid. In the first method, chemical potentials are calculated over a range of particle densities in canonical-ensemble simulations; the densities of the dilute and dense phases at the given temperature are then found by a Maxwell equal-area construction. In Gibbs-ensemble Monte Carlo, the exchange between separated dilute and dense phases is simulated to obtain their densities. Lastly, slab-geometry molecular dynamics simulations model the dilute and dense phases in coexistence and yield not only their densities but also their interfacial tension. The three types of simulations are carried out for a range of system sizes, and the results are scaled to generate the binodals corrected for finite system sizes. Size-corrected interfacial tension is also produced from slab-geometry molecular dynamics simulations.

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.

Analytical Solution to the Flory-Huggins Model

A self-consistent analytical solution for binodal concentrations of the two-component Flory-Huggins phase separation model is derived. We show that this form extends the validity of the Ginzburg-Landau expansion away from the critical point to cover the whole phase space. Furthermore, this analytical solution reveals an exponential scaling law of the dilute phase binodal concentration as a function of the interaction strength and chain length. We demonstrate explicitly the power of this approach by fitting experimental protein liquid-liquid phase separation boundaries to determine the effective chain length and solute-solvent interaction energies. Moreover, we demonstrate that this strategy allows us to resolve differences in interaction energy contributions of individual amino acids. This analytical framework can serve as a new way to decode the protein sequence grammar for liquid-liquid phase separation.

Rules of Physical Mathematics Govern Intrinsically Disordered Proteins

In stark contrast to foldable proteins with a unique folded state, intrinsically disordered proteins and regions (IDPs) persist in perpetually disordered ensembles. Yet an IDP ensemble has conformational features-even when averaged-that are specific to its sequence. In fact, subtle changes in an IDP sequence can modulate its conformational features and its function. Recent advances in theoretical physics reveal a set of elegant mathematical expressions that describe the intricate relationships among IDP sequences, their ensemble conformations, and the regulation of their biological functions. These equations also describe the molecular properties of IDP sequences that predict similarities and dissimilarities in their functions and facilitate classification of sequences by function, an unmet challenge to traditional bioinformatics. These physical sequence-patterning metrics offer a promising new avenue for advancing synthetic biology at a time when multiple novel functional modes mediated by IDPs are emerging.

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. A field theory description of the system, that treats polyampholyte charge sequence dependent electrostatic interactions as well as excluded volume effects, is hereby given. Interestingly, analysis of the model using random phase approximation and field theoretic simulation consistently show 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 polyampholyte sequence charge pattern as well as ion-pair dissociation constant, volumetric effects on ion association, solvent quality, and temperature.

Limitations of field-theory simulation for exploring phase separation: The role of repulsion in a lattice protein model

Field-theory simulation by the complex Langevin method offers an alternative to conventional sampling techniques for exploring the forces driving biomolecular liquid-liquid phase separation. Such simulations have recently been used to study several polyampholyte systems. Here, we formulate a field theory corresponding to the hydrophobic/polar (HP) lattice protein model, with finite same-site repulsion and nearest-neighbor attraction between HH bead pairs. By direct comparison with particle-based Monte Carlo simulations, we show that complex Langevin sampling of the field theory reproduces the thermodynamic properties of the HP model only if the same-site repulsion is not too strong. Unfortunately, the repulsion has to be taken weaker than what is needed to prevent condensed droplets from assuming an artificially compact shape. Analysis of a minimal and analytically solvable toy model hints that the sampling problems caused by repulsive interaction may stem from loss of ergodicity.

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.

What are the distinguishing features and size requirements of biomolecular condensates and their implications for RNA-containing condensates?

Exciting recent work has highlighted that numerous cellular compartments lack encapsulating lipid bilayers (often called "membraneless organelles"), and that their structure and function are central to the regulation of key biological processes, including transcription, RNA splicing, translation, and more. These structures have been described as "biomolecular condensates" to underscore that biomolecules can be significantly concentrated in them. Many condensates, including RNA granules and processing bodies, are enriched in proteins and nucleic acids. Biomolecular condensates exhibit a range of material states from liquid- to gel-like, with the physical process of liquid-liquid phase separation implicated in driving or contributing to their formation. To date, in vitro studies of phase separation have provided mechanistic insights into the formation and function of condensates. However, the link between the often micron-sized in vitro condensates with nanometer-sized cellular correlates has not been well established. Consequently, questions have arisen as to whether cellular structures below the optical resolution limit can be considered biomolecular condensates. Similarly, the distinction between condensates and discrete dynamic hub complexes is debated. Here we discuss the key features that define biomolecular condensates to help understand behaviors of structures containing and generating RNA.

Physics-driven coarse-grained model for biomolecular phase separation with near-quantitative accuracy

Various physics- and data-driven sequence-dependent protein coarse-grained models have been developed to study biomolecular phase separation and elucidate the dominant physicochemical driving forces. Here, we present Mpipi, a multiscale coarse-grained model that describes almost quantitatively the change in protein critical temperatures as a function of amino-acid sequence. The model is parameterised from both atomistic simulations and bioinformatics data and accounts for the dominant role of π-π and hybrid cation-π/π-π interactions and the much stronger attractive contacts established by arginines than lysines. We provide a comprehensive set of benchmarks for Mpipi and seven other residue-level coarse-grained models against experimental radii of gyration and quantitative in-vitro phase diagrams; Mpipi predictions agree well with experiment on both fronts. Moreover, it can account for protein-RNA interactions, correctly predicts the multiphase behaviour of a charge-matched poly-arginine/poly-lysine/RNA system, and recapitulates experimental LLPS trends for sequence mutations on FUS, DDX4 and LAF-1 proteins.

Current Understanding of Molecular Phase Separation in Chromosomes

Biomolecular phase separation denotes the demixing of a specific set of intracellular components without membrane encapsulation. Recent studies have found that biomolecular phase separation is involved in a wide range of cellular processes. In particular, phase separation is involved in the formation and regulation of chromosome structures at various levels. Here, we review the current understanding of biomolecular phase separation related to chromosomes. First, we discuss the fundamental principles of phase separation and introduce several examples of nuclear/chromosomal biomolecular assemblies formed by phase separation. We also briefly explain the experimental and computational methods used to study phase separation in chromosomes. Finally, we discuss a recent phase separation model, termed bridging-induced phase separation (BIPS), which can explain the formation of local chromosome structures.

Phase separation vs aggregation behavior for model disordered proteins

Liquid-liquid phase separation (LLPS) is widely utilized by the cell to organize and regulate various biochemical processes. Although the LLPS of proteins is known to occur in a sequence-dependent manner, it is unclear how sequence properties dictate the nature of the phase transition and thereby influence condensed phase morphology. In this work, we have utilized grand canonical Monte Carlo simulations for a simple coarse-grained model of disordered proteins to systematically investigate how sequence distribution, sticker fraction, and chain length impact the formation of finite-size aggregates, which can preempt macroscopic phase separation for some sequences. We demonstrate that a normalized sequence charge decoration (SCD) parameter establishes a "soft" predictive criterion for distinguishing when a model protein undergoes macroscopic phase separation vs finite aggregation. Additionally, we find that this order parameter is strongly correlated with the critical density for phase separation, highlighting an unambiguous connection between sequence distribution and condensed phase density. Results obtained from an analysis of the order parameter reveal that at sufficiently long chain lengths, the vast majority of sequences are likely to phase separate. Our results suggest that classical LLPS should be the primary phase transition for disordered proteins when short-ranged attractive interactions dominate and suggest a possible reason behind recent findings of widespread phase separation throughout living cells.

Salt dependent phase behavior of intrinsically disordered proteins from a coarse-grained model with explicit water and ions

Multivalent proteins and nucleic acids can self-assemble into biomolecular condensates that contribute to compartmentalize the cell interior. Computer simulations offer a unique view to elucidate the mechanisms and key intermolecular interactions behind the dynamic formation and dissolution of these condensates. In this work, we present a novel approach to include explicit water and salt in sequence-dependent coarse-grained (CG) models for proteins and RNA, enabling the study of biomolecular condensate formation in a salt-dependent manner. Our framework combines a reparameterized version of the HPS protein force field with the monoatomic mW water model and the mW-ion potential for NaCl. We show how our CG model qualitatively captures the experimental radius of the gyration trend of a subset of intrinsically disordered proteins and reproduces the experimental protein concentration and water percentage of the human fused in sarcoma (FUS) low-complexity-domain droplets at physiological salt concentration. Moreover, we perform seeding simulations as a function of salt concentration for two antagonist systems: the engineered peptide PR₂₅ and poly-uridine/poly-arginine mixtures, finding good agreement with their reported in vitro phase behavior with salt concentration in both cases. Taken together, our work represents a step forward towards extending sequence-dependent CG models to include water and salt, and to consider their key role in biomolecular condensate self-assembly.

Finite-size shifts in simulated protein droplet phase diagrams

Computer simulation can provide valuable insight into the forces driving biomolecular liquid-liquid phase separation. However, the simulated systems have a limited size, which makes it important to minimize and control finite-size effects. Here, using a phenomenological free-energy ansatz, we investigate how the single-phase densities observed in a canonical system under coexistence conditions depend on the system size and the total density. We compare the theoretical expectations with results from Monte Carlo simulations based on a simple hydrophobic/polar protein model. We consider both cubic systems with spherical droplets and elongated systems with slab-like droplets. The results presented suggest that the slab simulation method greatly facilitates the estimation of the coexistence densities in the large-system limit.