Phase Transitions in Biological Systems with Many Components

Emergence of Multiphase Condensates from a Limited Set of Chemical Building Blocks

Biomolecules composed of a limited set of chemical building blocks can colocalize into distinct, spatially segregated compartments known as biomolecular condensates. While many condensates are known to form spontaneously via phase separation, it has been unclear how immiscible condensates with precisely controlled molecular compositions assemble from a small number of chemical building blocks. We address this question by establishing a connection between the specificity of biomolecular interactions and the thermodynamic stability of coexisting condensates. By computing the minimum interaction specificity required to assemble condensates with target molecular compositions, we show how to design heteropolymer mixtures that produce compositionally complex condensates by using only a small number of monomer types. Our results provide insight into how compositional specificity arises in naturally occurring multicomponent condensates and demonstrate a rational algorithm for engineering complex artificial condensates from simple chemical building blocks.

Exact analytical solution of the Flory-Huggins model and extensions to multicomponent systems

The Flory-Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory's ease of application. In this work, we derive an implicit analytical solution to the Flory-Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types.

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.

Preserving condensate structure and composition by lowering sequence complexity

Biomolecular condensates play a vital role in organizing cellular chemistry. They selectively partition biomolecules, preventing unwanted cross talk and buffering against chemical noise. Intrinsically disordered proteins (IDPs) serve as primary components of these condensates due to their flexibility and ability to engage in multivalent interactions, leading to spontaneous aggregation. Theoretical advancements are critical at connecting IDP sequences with condensate emergent properties to establish the so-called molecular grammar. We proposed an extension to the stickers and spacers model, incorporating heterogeneous, nonspecific pairwise interactions between spacers alongside specific interactions among stickers. Our investigation revealed that although spacer interactions contribute to phase separation and co-condensation, their nonspecific nature leads to disorganized condensates. Specific sticker-sticker interactions drive the formation of condensates with well-defined networked structures and molecular composition. We discussed how evolutionary pressures might emerge to affect these interactions, leading to the prevalence of low-complexity domains in IDP sequences. These domains suppress spurious interactions and facilitate the formation of biologically meaningful condensates.

Aromatic and arginine content drives multiphasic condensation of protein-RNA mixtures

Multiphasic architectures are found ubiquitously in biomolecular condensates and are thought to have important implications for the organization of multiple chemical reactions within the same compartment. Many of these multiphasic condensates contain RNA in addition to proteins. Here, we investigate the importance of different interactions in multiphasic condensates comprising two different proteins and RNA using computer simulations with a residue-resolution coarse-grained model of proteins and RNA. We find that in multilayered condensates containing RNA in both phases, protein-RNA interactions dominate, with aromatic residues and arginine forming the key stabilizing interactions. The total aromatic and arginine content of the two proteins must be appreciably different for distinct phases to form, and we show that this difference increases as the system is driven toward greater multiphasicity. Using the trends observed in the different interaction energies of this system, we demonstrate that we can also construct multilayered condensates with RNA preferentially concentrated in one phase. The "rules" identified can thus enable the design of synthetic multiphasic condensates to facilitate further study of their organization and function.

Phase Separation in Complex Mixtures with Many Components: Analytical Expressions for Spinodal Manifolds

The phase behavior is investigated for systems composed of a large number of macromolecular components N, with N ≥ 2. Liquid-liquid phase separation is modeled using a virial expansion up to the second order of the concentrations of the components. Formal analytical expressions for the spinodal manifolds in N dimensions are derived, which simplify their calculation (by transforming the original problem into inequalities that can be evaluated numerically using linear programming techniques). In addition, a new expression is obtained to calculate the critical manifold and composition of the coexisting phases. The present analytical procedure complements previous attempts to handle spinodal decomposition for many components using a statistical approach based on random matrix theory. The results are relevant for predicting the effects of polydispersity on phase behavior in fields like polymer or food science and liquid-liquid phase separation in the cytosol of living cells.

Patterning of multicomponent elastic shells by gaussian curvature

Recent findings suggest that shell protein distribution and the morphology of bacterial microcompartments regulate the chemical fluxes facilitating reactions which dictate their biological function. We explore how the morphology and component patterning are coupled through the competition of mean and gaussian bending energies in multicomponent elastic shells that form three-component irregular polyhedra. We observe two softer components with lower bending rigidities allocated on the edges and vertices while the harder component occupies the faces. When subjected to a nonzero interfacial line tension, the two softer components further separate and pattern into subdomains that are mediated by the gaussian curvature. We find that this degree of fractionation is maximized when there is a weaker line tension and when the ratio of bending rigidities between the two softer domains ≈2. Our results reveal a patterning mechanism in multicomponent shells that can capture the observed morphologies of bacterial microcompartments, and moreover, can be realized in synthetic vesicles.

Preserving condensate structure and composition by lowering sequence complexity

Biological condensates play a vital role in organizing cellular chemistry. They selectively partition biomolecules, preventing unwanted cross-talk and buffering against chemical noise. Intrinsically disordered proteins (IDPs) serve as primary components of these condensates due to their flexibility and ability to engage in multivalent, non-specific interactions, leading to spontaneous aggregation. Theoretical advancements are critical at connecting IDP sequences with condensate emergent properties to establish the so-called molecular grammar. We proposed an extension to the stickers and spacers model, incorporating non-specific pairwise interactions between spacers alongside specific interactions among stickers. Our investigation revealed that while spacer interactions contribute to phase separation and co-condensation, their non-specific nature leads to disorganized condensates. Specific sticker-sticker interactions drive the formation of condensates with well-defined structures and molecular composition. We discussed how evolutionary pressures might emerge to affect these interactions, leading to the prevalence of low complexity domains in IDP sequences. These domains suppress spurious interactions and facilitate the formation of biologically meaningful condensates.

Dynamical control enables the formation of demixed biomolecular condensates

Cellular matter can be organized into compositionally distinct biomolecular condensates. For example, in Ashbya gossypii, the RNA-binding protein Whi3 forms distinct condensates with different RNA molecules. Using criteria derived from a physical framework for explaining how compositionally distinct condensates can form spontaneously via thermodynamic considerations, we find that condensates in vitro form mainly via heterotypic interactions in binary mixtures of Whi3 and RNA. However, within these condensates, RNA molecules become dynamically arrested. As a result, in ternary systems, simultaneous additions of Whi3 and pairs of distinct RNA molecules lead to well-mixed condensates, whereas delayed addition of an RNA component results in compositional distinctness. Therefore, compositional identities of condensates can be achieved via dynamical control, being driven, at least partially, by the dynamical arrest of RNA molecules. Finally, we show that synchronizing the production of different RNAs leads to more well-mixed, as opposed to compositionally distinct condensates in vivo.

The Membrane Phase Transition Gives Rise to Responsive Plasma Membrane Structure and Function

Several groups have recently reported evidence for the emergence of domains in cell plasma membranes when membrane proteins are organized by ligand binding or assembly of membrane proximal scaffolds. These domains recruit and retain components that favor the liquid-ordered phase, adding to a decades-old literature interrogating the contribution of membrane phase separation in plasma membrane organization and function. Here we propose that both past and present observations are consistent with a model in which membranes have a high compositional susceptibility, arising from their thermodynamic state in a single phase that is close to a miscibility phase transition. This rigorous framework naturally allows for both transient structure in the form of composition fluctuations and long-lived structure in the form of induced domains. In this way, the biological tuning of plasma membrane composition enables a responsive compositional landscape that facilitates and augments cellular biochemistry vital to plasma membrane functions.

Collective catalysis under spatial constraints: Phase separation and size-scaling effects on mass action kinetics

Chemical reactions are usually studied under the assumption that both substrates and catalysts are well-mixed (WM) throughout the system. Although this is often applicable to test-tube experimental conditions, it is not realistic in cellular environments, where biomolecules can undergo liquid-liquid phase separation (LLPS) and form condensates, leading to important functional outcomes, including the modulation of catalytic action. Similar processes may also play a role in protocellular systems, like primitive coacervates, or in membrane-assisted prebiotic pathways. Here we explore whether the demixing of catalysts could lead to the formation of microenvironments that influence the kinetics of a linear (multistep) reaction pathway, as compared to a WM system. We implemented a general lattice model to simulate LLPS of a collection of different catalysts and extended it to include diffusion and a sequence of reactions of small substrates. We carried out a quantitative analysis of how the phase separation of the catalysts affects reaction times depending on the affinity between substrates and catalysts, the length of the reaction pathway, the system size, and the degree of homogeneity of the condensate. A key aspect underlying the differences reported between the two scenarios is that the scale invariance observed in the WM system is broken by condensation processes. The main theoretical implications of our results for mean-field chemistry are drawn, extending the mass action kinetics scheme to include substrate initial "hitting times" to reach the catalysts condensate. We finally test this approach by considering open nonlinear conditions, where we successfully predict, through microscopic simulations, that phase separation inhibits chemical oscillatory behavior, providing a possible explanation for the marginal role that this complex dynamic behavior plays in real metabolisms.

Viscoelasticity and advective flow of RNA underlies nucleolar form and function

The nucleolus is the largest biomolecular condensate and facilitates transcription, processing, and assembly of ribosomal RNA (rRNA). Although nucleolar function is thought to require multiphase liquid-like properties, nucleolar fluidity and its connection to the highly coordinated transport and biogenesis of ribosomal subunits are poorly understood. Here, we use quantitative imaging, mathematical modeling, and pulse-chase nucleotide labeling to examine nucleolar material properties and rRNA dynamics. The mobility of rRNA is several orders of magnitude slower than that of nucleolar proteins, with rRNA steadily moving away from the transcriptional sites in a slow (∼1 Å/s), radially directed fashion. This constrained but directional mobility, together with polymer physics-based calculations, suggests that nascent rRNA forms an entangled gel, whose constant production drives outward flow. We propose a model in which progressive maturation of nascent rRNA reduces its initial entanglement, fluidizing the nucleolar periphery to facilitate the release of assembled pre-ribosomal particles.

Phase Separation and Ion Diffusion in Ionic Liquid, Organic Solvent, and Lithium Salt Electrolyte Mixtures

The highly desirable characteristics of ternary mixtures of ionic liquids, organic solvents, and metal salts make them a promising candidate for use in various electrothermal energy storage and conversion systems. In this study, using large-scale classical molecular dynamics simulations, we looked into 10 different ternary electrolyte mixtures using combinations of [EMIM]+, [BMIM]+, and [OMIM]+ cations with [NO3]-, [BF4]-, [PF6]-, [ClO4]-, [TFO]-, and [NTf2]- anions, tetraglyme, and Li salt to study the effect of ionic liquid composition on the phase behavior of ternary electrolyte mixtures. We uncovered that in these electrolytes, phase separation is mainly a function of pairwise binding energy of the constituents of the mixture. To corroborate this theory, several simulations are performed at various temperatures ranging from 260 to 500 K for each mixture, followed by calculating the binding energy of ionic liquid pairs using density functional theory. Our results verify that the transition temperature for the phase separation of each system is indeed a function of the pairwise binding energy of its ionic liquid pairs. It is also found that in some cases, the diffusion coefficient of the Li+ ions decreased even with the increase in the temperature, an effect that is attributed to the presence of condensed ionic domains in the electrolyte. This study provides a new insight for the design of multicomponent electrolyte mixtures for a wide range of energy applications.

Kinetic Growth of Multicomponent Microcompartment Shells

An important goal of systems and synthetic biology is to produce high value chemical species in large quantities. Microcompartments, which are protein nanoshells encapsulating catalytic enzyme cargo, could potentially function as tunable nanobioreactors inside and outside cells to generate these high value species. Modifying the morphology of microcompartments through genetic engineering of shell proteins is one viable strategy to tune cofactor and metabolite access to encapsulated enzymes. However, this is a difficult task without understanding how changing interactions between the many different types of shell proteins and enzymes affect microcompartment assembly and shape. Here, we use multiscale molecular dynamics and experimental data to describe assembly pathways available to microcompartments composed of multiple types of shell proteins with varied interactions. As the average interaction between the enzyme cargo and the multiple types of shell proteins is weakened, the shell assembly pathway transitions from (i) nucleating on the enzyme cargo to (ii) nucleating in the bulk and then binding the cargo as it grows to (iii) an empty shell. Atomistic simulations and experiments using the 1,2-propanediol utilization microcompartment system demonstrate that shell protein interactions are highly varied and consistent with our multicomponent, coarse-grained model. Furthermore, our results suggest that intrinsic bending angles control the size of these microcompartments. Overall, our simulations and experiments provide guidance to control microcomparmtent size and assembly by modulating the interactions between shell proteins.

Composition Dependent Instabilities in Mixtures with Many Components

Understanding the phase behavior of mixtures with many components is important in many contexts, including as a key step toward a physics-based description of intracellular compartmentalization. Here, we study phase ordering instabilities in a paradigmatic model that represents the complexity of-e.g., biological-mixtures via random second virial coefficients. Using tools from free probability theory we obtain the exact spinodal curve and the nature of instabilities for a mixture with an arbitrary composition, thus lifting an important restriction in previous work. We show that, by controlling the concentration of only a few components, one can systematically change the nature of the spinodal instability and achieve demixing for realistic scenarios by a strong composition imbalance amplification. This results from a nontrivial interplay of interaction complexity and entropic effects due to the nonuniform composition. Our approach can be extended to include additional systematic interactions, leading to a competition between different forms of demixing as density is varied.

Theoretical and Data-Driven Approaches for Biomolecular Condensates

Biomolecular condensation processes are increasingly recognized as a fundamental mechanism that living cells use to organize biomolecules in time and space. These processes can lead to the formation of membraneless organelles that enable cells to perform distinct biochemical processes in controlled local environments, thereby supplying them with an additional degree of spatial control relative to that achieved by membrane-bound organelles. This fundamental importance of biomolecular condensation has motivated a quest to discover and understand the molecular mechanisms and determinants that drive and control this process. Within this molecular viewpoint, computational methods can provide a unique angle to studying biomolecular condensation processes by contributing the resolution and scale that are challenging to reach with experimental techniques alone. In this Review, we focus on three types of dry-lab approaches: theoretical methods, physics-driven simulations and data-driven machine learning methods. We review recent progress in using these tools for probing biomolecular condensation across all three fields and outline the key advantages and limitations of each of the approaches. We further discuss some of the key outstanding challenges that we foresee the community addressing next in order to develop a more complete picture of the molecular driving forces behind biomolecular condensation processes and their biological roles in health and disease.

Phase Transitions of Associative Biomacromolecules

Multivalent proteins and nucleic acids, collectively referred to as multivalent associative biomacromolecules, provide the driving forces for the formation and compositional regulation of biomolecular condensates. Here, we review the key concepts of phase transitions of aqueous solutions of associative biomacromolecules, specifically proteins that include folded domains and intrinsically disordered regions. The phase transitions of these systems come under the rubric of coupled associative and segregative transitions. The concepts underlying these processes are presented, and their relevance to biomolecular condensates is discussed.

The pros and cons of ubiquitination on the formation of protein condensates

Ubiquitination, a post-translational modification that attaches one or more ubiquitin (Ub) molecules to another protein, plays a crucial role in the phase-separation processes. Ubiquitination can modulate the formation of membrane-less organelles in two ways. First, a scaffold protein drives phase separation, and Ub is recruited to the condensates. Second, Ub actively phase-separates through the interactions with other proteins. Thus, the role of ubiquitination and the resulting polyUb chains ranges from bystanders to active participants in phase separation. Moreover, long polyUb chains may be the primary driving force for phase separation. We further discuss that the different roles can be determined by the lengths and linkages of polyUb chains which provide preorganized and multivalent binding platforms for other client proteins. Together, ubiquitination adds a new layer of regulation for the flow of material and information upon cellular compartmentalization of proteins.

Theory and Simulation of Multiphase Coexistence in Biomolecular Mixtures

Biomolecular condensates constitute a newly recognized form of spatial organization in living cells. Although many condensates are believed to form as a result of phase separation, the physicochemical properties that determine the phase behavior of heterogeneous biomolecular mixtures are only beginning to be explored. Theory and simulation provide invaluable tools for probing the relationship between molecular determinants, such as protein and RNA sequences, and the emergence of phase-separated condensates in such complex environments. This review covers recent advances in the prediction and computational design of biomolecular mixtures that phase-separate into many coexisting phases. First, we review efforts to understand the phase behavior of mixtures with hundreds or thousands of species using theoretical models and statistical approaches. We then describe progress in developing analytical theories and coarse-grained simulation models to predict multiphase condensates with the molecular detail required to make contact with biophysical experiments. We conclude by summarizing the challenges ahead for modeling the inhomogeneous spatial organization of biomolecular mixtures in living cells.

Self-assembly coupled to liquid-liquid phase separation

Liquid condensate droplets with distinct compositions of proteins and nucleic acids are widespread in biological cells. While it is known that such droplets, or compartments, can regulate irreversible protein aggregation, their effect on reversible self-assembly remains largely unexplored. In this article, we use kinetic theory and solution thermodynamics to investigate the effect of liquid-liquid phase separation on the reversible self-assembly of structures with well-defined sizes and architectures. We find that, when assembling subunits preferentially partition into liquid compartments, robustness against kinetic traps and maximum achievable assembly rates can be significantly increased. In particular, both the range of solution conditions leading to productive assembly and the corresponding assembly rates can increase by orders of magnitude. We analyze the rate equation predictions using simple scaling estimates to identify effects of liquid-liquid phase separation as a function of relevant control parameters. These results may elucidate self-assembly processes that underlie normal cellular functions or pathogenesis, and suggest strategies for designing efficient bottom-up assembly for nanomaterials applications.

RNA-mediated demixing transition of low-density condensates

Biomolecular condensates play a key role in organizing cellular reactions by concentrating a specific set of biomolecules. However, whether condensate formation is accompanied by an increase in the total mass concentration within condensates or by the demixing of already highly crowded intracellular components remains elusive. Here, using refractive index imaging, we quantify the mass density of several condensates, including nucleoli, heterochromatin, nuclear speckles, and stress granules. Surprisingly, the latter two condensates exhibit low densities with a total mass concentration similar to the surrounding cyto- or nucleoplasm. Low-density condensates display higher permeability to cellular protein probes. We find that RNA tunes the biomolecular density of condensates. Moreover, intracellular structures such as mitochondria heavily influence the way phase separation proceeds, impacting the localization, morphology, and growth of condensates. These findings favor a model where segregative phase separation driven by non-associative or repulsive molecular interactions together with RNA-mediated selective association of specific components can give rise to low-density condensates in the crowded cellular environment.

Developments in describing equilibrium phase transitions of multivalent associative macromolecules

Biomolecular condensates are distinct cellular bodies that form and dissolve reversibly to organize cellular matter and biochemical reactions in space and time. Condensates are thought to form and dissolve under the influence of spontaneous and driven phase transitions of multivalent associative macromolecules. These include phase separation, which is defined by segregation of macromolecules from the solvent or from one another, and percolation or gelation, which is an inclusive networking transition driven by reversible associations among multivalent macromolecules. Considerable progress has been made to model sequence-specific phase transitions, especially for intrinsically disordered proteins. Here, we summarize the state-of-the-art of theories and computations aimed at understanding and modeling sequence-specific, thermodynamically controlled, coupled associative and segregative phase transitions of archetypal multivalent macromolecules.

A lattice model of ternary mixtures of lipids and cholesterol with tunable domain sizes

Much of our understanding of the physical properties of raft domains in biological membranes, and some insight into the mechanisms underlying their formation stem from atomistic simulations of simple model systems, especially ternary mixtures consisting of saturated and unsaturated lipids, and cholesterol (Chol). To explore the properties of such systems at large spatial scales, we here present a simple ternary mixture lattice model, involving a small number of nearest neighbor interaction terms. Monte Carlo simulations of mixtures with different compositions show excellent agreement with experimental and atomistic simulation observations across multiple scales, ranging from the local distributions of lipids to the phase diagram of the system. The simplicity of the model allows us to identify the roles played by the different interactions between components, and the interplay between them. Importantly, by changing the value of one of the model parameters, we can tune the size of the liquid-ordered domains, thereby simulating both Type II mixtures exhibiting macroscopic phase separation and Type I mixtures with nanoscopic domains. The Type II mixture simulation results fit well to the experimentally determined phase diagram of mixtures containing saturated DPPC/unsaturated DOPC/Chol. When the tunable parameter is changed, we obtain the Type I version of DPPC/DOPC/Chol, i.e., a mixture not showing thermodynamic phase transitions but one that may be fitted to the same phase diagram if local measures are used to distinguish between the different states. Our model results suggest that short range packing is likely to be a key regulator of the stability and size distribution of biological rafts.

Thermodynamic origins of two-component multiphase condensates of proteins

Intracellular condensates are highly multi-component systems in which complex phase behaviour can ensue, including the formation of architectures comprising multiple immiscible condensed phases. Relying solely on physical intuition to manipulate such condensates is difficult because of the complexity of their composition, and systematically learning the underlying rules experimentally would be extremely costly. We address this challenge by developing a computational approach to design pairs of protein sequences that result in well-separated multilayered condensates and elucidate the molecular origins of these compartments. Our method couples a genetic algorithm to a residue-resolution coarse-grained protein model. We demonstrate that we can design protein partners to form multiphase condensates containing naturally occurring proteins, such as the low-complexity domain of hnRNPA1 and its mutants, and show how homo- and heterotypic interactions must differ between proteins to result in multiphasicity. We also show that in some cases the specific pattern of amino-acid residues plays an important role. Our findings have wide-ranging implications for understanding and controlling the organisation, functions and material properties of biomolecular condensates.

Dynamical control enables the formation of demixed biomolecular condensates

Macromolecular phase separation underlies the regulated formation and dissolution of biomolecular condensates. What is unclear is how condensates of distinct and shared macromolecular compositions form and coexist within cellular milieus. Here, we use theory and computation to establish thermodynamic criteria that must be satisfied to achieve compositionally distinct condensates. We applied these criteria to an archetypal ribonucleoprotein condensate and discovered that demixing into distinct protein-RNA condensates cannot be the result of purely thermodynamic considerations. Instead, demixed, compositionally distinct condensates arise due to asynchronies in timescales that emerge from differences in long-lived protein-RNA and RNA-RNA crosslinks. This type of dynamical control is also found to be active in live cells whereby asynchronous production of molecules is required for realizing demixed protein-RNA condensates. We find that interactions that exert dynamical control provide a versatile and generalizable way to influence the compositions of coexisting condensates in live cells.

Dynamical control enables the formation of demixed biomolecular condensates

Macromolecular phase separation underlies the regulated formation and dissolution of biomolecular condensates. What is unclear is how condensates of distinct and shared macromolecular compositions form and coexist within cellular milieus. Here, we use theory and computation to establish thermodynamic criteria that must be satisfied to achieve compositionally distinct condensates. We applied these criteria to an archetypal ribonucleoprotein condensate and discovered that demixing into distinct protein-RNA condensates cannot be the result of purely thermodynamic considerations. Instead, demixed, compositionally distinct condensates arise due to asynchronies in timescales that emerge from differences in long-lived protein-RNA and RNA-RNA crosslinks. This type of dynamical control is also found to be active in live cells whereby asynchronous production of molecules is required for realizing demixed protein-RNA condensates. We find that interactions that exert dynamical control provide a versatile and generalizable way to influence the compositions of coexisting condensates in live cells.

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.

Multiple polarity kinases inhibit phase separation of F-BAR protein Cdc15 and antagonize cytokinetic ring assembly in fission yeast

The F-BAR protein Cdc15 is essential for cytokinesis in Schizosaccharomyces pombe and plays a key role in attaching the cytokinetic ring (CR) to the plasma membrane (PM). Cdc15's abilities to bind to the membrane and oligomerize via its F-BAR domain are inhibited by phosphorylation of its intrinsically disordered region (IDR). Multiple cell polarity kinases regulate Cdc15 IDR phosphostate, and of these the DYRK kinase Pom1 phosphorylation sites on Cdc15 have been shown in vivo to prevent CR formation at cell tips. Here, we compared the ability of Pom1 to control Cdc15 phosphostate and cortical localization to that of other Cdc15 kinases: Kin1, Pck1, and Shk1. We identified distinct but overlapping cohorts of Cdc15 phosphorylation sites targeted by each kinase, and the number of sites correlated with each kinases' abilities to influence Cdc15 PM localization. Coarse-grained simulations predicted that cumulative IDR phosphorylation moves the IDRs of a dimer apart and toward the F-BAR tips. Further, simulations indicated that the overall negative charge of phosphorylation masks positively charged amino acids necessary for F-BAR oligomerization and membrane interaction. Finally, simulations suggested that dephosphorylated Cdc15 undergoes phase separation driven by IDR interactions. Indeed, dephosphorylated but not phosphorylated Cdc15 undergoes liquid-liquid phase separation to form droplets in vitro that recruit Cdc15 binding partners. In cells, Cdc15 phosphomutants also formed PM-bound condensates that recruit other CR components. Together, we propose that a threshold of Cdc15 phosphorylation by assorted kinases prevents Cdc15 condensation on the PM and antagonizes CR assembly.

On kinetics and extreme values in systems with random interactions

Biological environments such as the cytoplasm are comprised of many different molecules, which makes explicit modeling intractable. In the spirit of Wigner, one may be tempted to assume interactions to derive from a random distribution. Via this approximation, the system can be efficiently treated in the mean-field, and general statements about expected behavior of such systems can be made. Here, I study systems of particles interacting via random potentials, outside of mean-field approximations. These systems exhibit a phase transition temperature, under which part of the components precipitate. The nature of this transition appears to be non-universal, and to depend intimately on the underlying distribution of interactions. Above the phase transition temperature, the system can be efficiently treated using a Bethe approximation, which shows a dependence on extreme value statistics. Relaxation timescales of this system tend to be slow, but can be made arbitrarily fast by increasing the number of neighbors of each particle.

Spontaneous phase separation of ternary fluid mixtures

We computationally study the spontaneous phase separation of ternary fluid mixtures using the lattice Boltzmann method both when all the surface tensions are equal and when they have different values. To rationalise the phase diagram of possible phase separation mechanisms, previous theoretical works typically rely on analysing the sign of the eigenvalues resulting from a simple linear stability analysis, but we find this does not explain the observed simulation results. Here, we classify the possible separation pathways into four basic mechanisms, and develop a phenomenological model that captures the composition regimes where each mechanism is prevalent. We further highlight that the dominant mechanism in ternary phase separation involves enrichment and instability of the minor component at the fluid-fluid interface, which is absent in the case of binary fluid mixtures.

Thermodynamic stability and critical points in multicomponent mixtures with structured interactions

Theoretical work has shed light on the phase behavior of idealized mixtures of many components with random interactions. However, typical mixtures interact through particular physical features, leading to a structured, nonrandom interaction matrix of lower rank. Here, we develop a theoretical framework for such mixtures and derive mean-field conditions for thermodynamic stability and critical behavior. Irrespective of the number of components and features, this framework allows for a generally lower-dimensional representation in the space of features and proposes a principled way to coarse-grain multicomponent mixtures as binary mixtures. Moreover, it suggests a way to systematically characterize different series of critical points and their codimensions in mean-field. Since every pairwise interaction matrix can be expressed in terms of features, our work is applicable to a broad class of mean-field models.

Evolved interactions stabilize many coexisting phases in multicomponent liquids

Phase separation has emerged as an essential concept for the spatial organization inside biological cells. However, despite the clear relevance to virtually all physiological functions, we understand surprisingly little about what phases form in a system of many interacting components, like in cells. Here we introduce a numerical method based on physical relaxation dynamics to study the coexisting phases in such systems. We use our approach to optimize interactions between components, similar to how evolution might have optimized the interactions of proteins. These evolved interactions robustly lead to a defined number of phases, despite substantial uncertainties in the initial composition, while random or designed interactions perform much worse. Moreover, the optimized interactions are robust to perturbations, and they allow fast adaption to new target phase counts. We thus show that genetically encoded interactions of proteins provide versatile control of phase behavior. The phases forming in our system are also a concrete example of a robust emergent property that does not rely on fine-tuning the parameters of individual constituents.

Less is more when forming gels by dilution

Molecular self-assembly yields soft materials arising from the liquid state when diluted.

Dilution-induced gel-sol-gel-sol transitions by competitive supramolecular pathways in water

Fascinating properties are displayed by synthetic multicomponent supramolecular systems that comprise a manifold of competitive interactions, thereby mimicking natural processes. We present the integration of two reentrant phase transitions based on an unexpected dilution-induced assembly process using supramolecular polymers and surfactants. The co-assembly of the water-soluble benzene-1,3,5-tricarboxamide (BTA-EG 4 ) and a surfactant at a specific ratio yielded small-sized aggregates. These interactions were modeled using the competition between self-sorting and co-assembly of both components. The small-sized aggregates were transformed into supramolecular polymer networks by a twofold dilution in water without changing their ratio. Kinetic experiments show the in situ growth of micrometer-long fibers in the dilution process. We were able to create systems that undergo fully reversible hydrogel-solution-hydrogel-solution transitions upon dilution by introducing another orthogonal interaction.

Aging can transform single-component protein condensates into multiphase architectures

Phase-separated biomolecular condensates that contain multiple coexisting phases are widespread in vitro and in cells. Multiphase condensates emerge readily within multicomponent mixtures of biomolecules (e.g., proteins and nucleic acids) when the different components present sufficient physicochemical diversity (e.g., in intermolecular forces, structure, and chemical composition) to sustain separate coexisting phases. Because such diversity is highly coupled to the solution conditions (e.g., temperature, pH, salt, composition), it can manifest itself immediately from the nucleation and growth stages of condensate formation, develop spontaneously due to external stimuli or emerge progressively as the condensates age. Here, we investigate thermodynamic factors that can explain the progressive intrinsic transformation of single-component condensates into multiphase architectures during the nonequilibrium process of aging. We develop a multiscale model that integrates atomistic simulations of proteins, sequence-dependent coarse-grained simulations of condensates, and a minimal model of dynamically aging condensates with nonconservative intermolecular forces. Our nonequilibrium simulations of condensate aging predict that single-component condensates that are initially homogeneous and liquid like can transform into gel-core/liquid-shell or liquid-core/gel-shell multiphase condensates as they age due to gradual and irreversible enhancement of interprotein interactions. The type of multiphase architecture is determined by the aging mechanism, the molecular organization of the gel and liquid phases, and the chemical makeup of the protein. Notably, we predict that interprotein disorder to order transitions within the prion-like domains of intracellular proteins can lead to the required nonconservative enhancement of intermolecular interactions. Our study, therefore, predicts a potential mechanism by which the nonequilibrium process of aging results in single-component multiphase condensates.

Microcompartment assembly around multicomponent fluid cargoes

This article describes dynamical simulations of the assembly of an icosahedral protein shell around a bicomponent fluid cargo. Our simulations are motivated by bacterial microcompartments, which are protein shells found in bacteria that assemble around a complex of enzymes and other components involved in certain metabolic processes. The simulations demonstrate that the relative interaction strengths among the different cargo species play a key role in determining the amount of each species that is encapsulated, their spatial organization, and the nature of the shell assembly pathways. However, the shell protein–shell protein and shell protein–cargo component interactions that help drive assembly and encapsulation also influence cargo composition within certain parameter regimes. These behaviors are governed by a combination of thermodynamic and kinetic effects. In addition to elucidating how natural microcompartments encapsulate multiple components involved within reaction cascades, these results have implications for efforts in synthetic biology to colocalize alternative sets of molecules within microcompartments to accelerate specific reactions. More broadly, the results suggest that coupling between self-assembly and multicomponent liquid–liquid phase separation may play a role in the organization of the cellular cytoplasm.

The mechanobiology of nuclear phase separation

The cell nucleus can be thought of as a complex, dynamic, living material, which functions to organize and protect the genome and coordinate gene expression. These functions are achieved via intricate mechanical and biochemical interactions among its myriad components, including the nuclear lamina, nuclear bodies, and the chromatin itself. While the biophysical organization of the nuclear lamina and chromatin have been thoroughly studied, the concept that liquid–liquid phase separation and related phase transitions play a role in establishing nuclear structure has emerged only recently. Phase transitions are likely to be intimately coupled to the mechanobiology of structural elements in the nucleus, but their interplay with one another is still not understood. Here, we review recent developments on the role of phase separation and mechanics in nuclear organization and discuss the functional implications in cell physiology and disease states.

On the stability and layered organization of protein-DNA condensates

Multi-component phase separation is emerging as a key mechanism for the formation of biological condensates that play essential roles in signal sensing and transcriptional regulation. The molecular factors that dictate these condensates' stability and spatial organization are not fully understood, and it remains challenging to predict their microstructures. Using a near-atomistic, chemically accurate force field, we studied the phase behavior of chromatin regulators that are crucial for heterochromatin organization and their interactions with DNA. Our computed phase diagrams recapitulated previous experimental findings on different proteins. They revealed a strong dependence of condensate stability on the protein-DNA mixing ratio as a result of balancing protein-protein interactions and charge neutralization. Notably, a layered organization was observed in condensates formed by mixing HP1, histone H1, and DNA. This layered organization may be of biological relevance, as it enables cooperative DNA packaging between the two chromatin regulators: histone H1 softens the DNA to facilitate the compaction induced by HP1 droplets. Our study supports near-atomistic models as a valuable tool for characterizing the structure and stability of biological condensates.

Pairing statistics and melting of random DNA oligomers: Finding your partner in superdiverse environments

Understanding of the pairing statistics in solutions populated by a large number of distinct solute species with mutual interactions is a challenging topic, relevant in modeling the complexity of real biological systems. Here we describe, both experimentally and theoretically, the formation of duplexes in a solution of random-sequence DNA (rsDNA) oligomers of length L = 8, 12, 20 nucleotides. rsDNA solutions are formed by 4^L distinct molecular species, leading to a variety of pairing motifs that depend on sequence complementarity and range from strongly bound, fully paired defectless helices to weakly interacting mismatched duplexes. Experiments and theory coherently combine revealing a hybridization statistics characterized by a prevalence of partially defected duplexes, with a distribution of type and number of pairing errors that depends on temperature. We find that despite the enormous multitude of inter-strand interactions, defectless duplexes are formed, involving a fraction up to 15% of the rsDNA chains at the lowest temperatures. Experiments and theory are limited here to equilibrium conditions.

Several biological processes require that specific partner molecules succeed in binding after negotiating their way through a huge number of interactions with other molecules. How such molecular recognition emerges among millions distinct molecular species is an open problem. We have studied, both experimentally and theoretically, such process of “molecular recognition” in pools of highly diverse random DNA oligomers, which binds preferentially, but not exclusively, to its perfect complementary sequence. We find a complex behavior, in which some perfect pairing takes place with a non-trivial temperature dependence that we understand thorough statistical mechanics modelling. The pairing pattern of short random DNA is relevant in the context of the origin of life since the so-called “RNA World” was most probably based on the mutual recognition of random chains.

Instabilities of complex fluids with partially structured and partially random interactions

We develop a theory for thermodynamic instabilities of complex fluids composed of many interacting chemical species organised in families. This model includes partially structured and partially random interactions and can be solved exactly using tools from random matrix theory. The model exhibits three kinds of fluid instabilities: one in which the species form a condensate with a local density that depends on their family (family condensation); one in which species demix in two phases depending on their family (family demixing); and one in which species demix in a random manner irrespective of their family (random demixing). We determine the critical spinodal density of the three types of instabilities and find that the critical spinodal density is finite for both family condensation and family demixing, while for random demixing the critical spinodal density grows as the square root of the number of species. We use the developed framework to describe phase-separation instability of the cytoplasm induced by a change in pH.

Rich Phase Separation Behavior of Biomolecules

Phase separation is a thermodynamic process leading to the formation of compositionally distinct phases. For the past few years, numerous works have shown that biomolecular phase separation serves as biogenesis mechanisms of diverse intracellular condensates, and aberrant phase transitions are associated with disease states such as neurodegenerative diseases and cancers. Condensates exhibit rich phase behaviors including multiphase internal structuring, noise buffering, and compositional tunability. Recent studies have begun to uncover how a network of intermolecular interactions can give rise to various biophysical features of condensates. Here, we review phase behaviors of biomolecules, particularly with regard to regular solution models of binary and ternary mixtures. We discuss how these theoretical frameworks explain many aspects of the assembly, composition, and miscibility of diverse biomolecular phases, and highlight how a model-based approach can help elucidate the detailed thermodynamic principle for multicomponent intracellular phase separation.

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.

Phase separation in fluids with many interacting components

Fluids in natural systems, like the cytoplasm of a cell, often contain thousands of molecular species that are organized into multiple coexisting phases that enable diverse and specific functions. How interactions between numerous molecular species encode for various emergent phases is not well understood. Here, we leverage approaches from random-matrix theory and statistical physics to describe the emergent phase behavior of fluid mixtures with many species whose interactions are drawn randomly from an underlying distribution. Through numerical simulation and stability analyses, we show that these mixtures exhibit staged phase-separation kinetics and are characterized by multiple coexisting phases at steady state with distinct compositions. Random-matrix theory predicts the number of coexisting phases, validated by simulations with diverse component numbers and interaction parameters. Surprisingly, this model predicts an upper bound on the number of phases, derived from dynamical considerations, that is much lower than the limit from the Gibbs phase rule, which is obtained from equilibrium thermodynamic constraints. We design ensembles that encode either linear or nonmonotonic scaling relationships between the number of components and coexisting phases, which we validate through simulation and theory. Finally, inspired by parallels in biological systems, we show that including nonequilibrium turnover of components through chemical reactions can tunably modulate the number of coexisting phases at steady state without changing overall fluid composition. Together, our study provides a model framework that describes the emergent dynamical and steady-state phase behavior of liquid-like mixtures with many interacting constituents.

Multivalent polymers can control phase boundary, dynamics, and organization of liquid-liquid phase separation

Multivalent polymers are a key structural component of many biocondensates. When interacting with their cognate binding proteins, multivalent polymers such as RNA and modular proteins have been shown to influence the liquid-liquid phase separation (LLPS) boundary to both control condensate formation and to influence condensate dynamics after phase separation. Much is still unknown about the function and formation of these condensed droplets, but changes in their dynamics or phase separation are associated with neurodegenerative diseases such as amyotrophic lateral sclerosis (ALS) and Alzheimer's Disease. Therefore, investigation into how the structure of multivalent polymers relates to changes in biocondensate formation and maturation is essential to understanding and treating these diseases. Here, we use a coarse-grain, Brownian Dynamics simulation with reactive binding that mimics specific interactions in order to investigate the difference between non-specific and specific multivalent binding polymers. We show that non-specific binding interactions can lead to much larger changes in droplet formation at lower protein-polymer interaction energies than their specific, valence-limited counterparts. We also demonstrate the effects of solvent conditions and polymer length on phase separation, and we present how modulating binding energy to the polymer can change the organization of a droplet in a three component system of polymer, binding protein, and solvent. Finally, we compare the effects of surface tension and polymer binding on the condensed phase dynamics, and show that both lower protein solubilities and higher attraction/affinity of the protein to the polymer result in slower droplet dynamics. This research will help to better understand experimental systems and provides additional insight into how multivalent polymers can control LLPS.

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.

Hierarchical microphase separation in non-conserved active mixtures

Non-equilibrium phase separating systems with reactions, such as biomolecular condensates and bacteria colonies, can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative and non-conservative sectors of the dynamics can be governed by incompatible free energies; when both sectors are present, this is the leading-order TRS violation, captured in its simplest form by 'Model AB'. Second, the diffusive dynamics can break TRS in its own right. This happens only at higher order in the gradient expansion (but is the leading behaviour without reactions present) and is captured by 'Active Model B+' (AMB+). Each of the two mechanisms can lead to microphase separation, by quite different routes. Here we introduce Model AB+, for which both mechanisms are simultaneously present, and show that for slow reaction rates the system can undergo a new type of hierarchical microphase separation, which we call 'bubbly microphase separation'. In this state, small droplets of one fluid are continuously created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and diffusive dynamics.

Multiphase Coacervates Driven by Electrostatic Correlations

The liquid-liquid phase separation of a polyelectrolyte solution containing one type of negatively and two types of positively charged polymers with different charge densities is studied theoretically by random phase approximation (RPA). It is predicted that multicoacervate phases could coexist, driven purely by electrostatic correlations. The asymmetry of the linear charge density could induce an effective immiscibility between two positively charged polyelectrolytes, leading to the multiphase separation. Adding salt will induce the disappearance of the dilute phase, forming two coexisting complex phases, instead of fusion between coacervates. Raising temperature could either induce a two coexisting complex phase, or a dilute phase coexisting with a coacervate phase, depending on the bulk concentration. Our predictions are in good agreement with experiments and provide insights in the further designing of the multiphase coacervation system.