Core-shell model of the clusters of CPEB4 isoforms preceding liquid-liquid phase separation.

Protein solutions can undergo liquid-liquid phase separation (LLPS), where a dispersed phase with a low protein concentration coexists with coacervates with a high protein concentration. We focus on the low complexity N-terminal domain of cytoplasmic polyadenylation element binding-4 protein, CPEB4NTD, and its isoform depleted of the Exon4, CPEB4Δ4NTD. They both exhibit LLPS, but in contrast to most systems undergoing LLPS, the single-phase regime preceding LLPS consists mainly of soluble protein clusters. We combine experimental and theoretical approaches to resolve the internal structure of the clusters and the basis for their formation. Dynamic light scattering (DLS) and atomic force microscopy (AFM) show that both isoforms exhibit clusters with diameters ranging from 35-80 nm. Electron paramagnetic resonance (EPR) spectroscopy of spin-labeled CPEB4NTD and CPEB4Δ4NTD revealed that these proteins have two distinct dynamical properties in the clusters and coacervates. Based on the experimental results, we proposed a core-shell structure for the clusters, which is supported by the agreement of the DLS data on cluster size distribution with a statistical model developed to describe the structure of clusters. This model treats clusters as swollen micelles (microemulsions) where the core and the shell regions comprise different protein conformations, in agreement with the EPR detection of two protein populations. The effects of ionic strength and the addition of 1,6-hexanediol (HD) were used to probe the interactions responsible for cluster formation. While both CPEB4NTD and CPEB4Δ4NTD showed phase separation with increasing temperature and formed clusters, differences were found in the properties of the clusters and the coacervates. The data also suggested that the coacervates may consist of aggregates of clusters.

Erratum: Fluctuations and Shape Dependence of Microphase Separation in Systems with Long-Range Interactions [Phys. Rev. Lett. 131, 258401 (2023)].

This corrects the article DOI: 10.1103/PhysRevLett.131.258401.

Mechanosensitivity of phase separation in an elastic gel.

Liquid-liquid phase separation (LLPS) in binary or multi-component solutions is a well-studied subject in soft matter with extensive applications in biological systems. In recent years, several experimental studies focused on LLPS of solutes in hydrated gels, where the formation of coexisting domains induces elastic deformations within the gel. While the experimental studies report unique physical characteristics of these systems, such as sensitivity to mechanical forces and stabilization of multiple, periodic phase-separated domains, the theoretical understanding of such systems and the role of long-range interactions have not emphasized the nonlinear nature of the equilibrium binodal for strong segregation of the solute. In this paper, we formulate a generic, mean-field theory of a hydrated gel in the presence of an additional solute which changes the elastic properties of the gel. We derive equations for the equilibrium binodal of the phase separation of the solvent and solute and show that the deformations induced by the solute can result in effective long-range interactions between phase-separating solutes that can either enhance or, in the case of externally applied pressure, suppress phase separation of the solute relative to the case where there is no gel. This causes the coexisting concentrations at the binodal to depend on the system-wide average concentration, in contrast to the situation for phase separation in the absence of the gel.

Scaling perspectives of underscreening in concentrated electrolyte solutions.

We present a scaling view of underscreening observed in salt solutions in the range of concentrations greater than about 1 M, in which the screening length increases with concentration. The system consists of hydrated clusters of positive and negative ions with a single unpaired ion as suggested by recent simulations. The environment of this ion is more hydrated than average which leads to a self-similar situation in which the size of this environment scales with the screening length. The prefactor involves the local dielectric constant and the cluster density. The scaling arguments as well as the cluster model lead to scaling of the screening length with the ion concentration, in agreement with observations.

Affinity hierarchies and amphiphilic proteins underlie the co-assembly of nucleolar and heterochromatin condensates.

Nucleoli are surrounded by Pericentromeric Heterochromatin (PCH), reflecting a close spatial association between the two largest biomolecular condensates in eukaryotic nuclei. This nuclear organizational feature is highly conserved and is disrupted in diseased states like senescence, however, the mechanisms driving PCH-nucleolar association are unclear. High-resolution live imaging during early Drosophila development revealed a highly dynamic process in which PCH and nucleolar formation is coordinated and interdependent. When nucleolus assembly was eliminated by deleting the ribosomal RNA genes (rDNA), PCH showed increased compaction and subsequent reorganization to a shell-like structure. In addition, in embryos lacking rDNA, some nucleolar proteins were redistributed into new bodies or 'neocondensates,' including enrichment in the core of the PCH shell. These observations, combined with physical modeling and simulations, suggested that nucleolar-PCH associations are mediated by a hierarchy of affinities between PCH, nucleoli, and 'amphiphilic' protein(s) that interact with both nucleolar and PCH components. This result was validated by demonstrating that the depletion of one candidate amphiphile, the nucleolar protein Pitchoune, significantly reduced PCH-nucleolar associations. Together, these results unveil a dynamic program for establishing nucleolar-PCH associations during animal development, demonstrate that nucleoli are required for normal PCH organization, and identify Pitchoune as an amphiphilic molecular link that promotes PCH-nucleolar associations. Finally, we propose that disrupting affinity hierarchies between interacting condensates can liberate molecules to form neocondensates or other aberrant structures that could contribute to cellular disease phenotypes.

Affinity hierarchies and amphiphilic proteins underlie the co-assembly of nucleolar and heterochromatin condensates.

Nucleoli are surrounded by Pericentromeric Heterochromatin (PCH), reflecting a close spatial association between the two largest biomolecular condensates in eukaryotic nuclei. This nuclear organizational feature is highly conserved and is disrupted in diseased states like senescence, however, the mechanisms driving PCH-nucleolar association are unclear. High-resolution live imaging during early Drosophila development revealed a highly dynamic process in which PCH and nucleolar formation is coordinated and interdependent. When nucleolus assembly was eliminated by deleting the ribosomal RNA genes (rDNA), PCH showed increased compaction and subsequent reorganization to a shell-like structure. In addition, in embryos lacking rDNA, some nucleolar proteins were redistributed into new bodies or 'neocondensates,' including enrichment in the core of the PCH shell. These observations, combined with physical modeling and simulations, suggested that nucleolar-PCH associations are mediated by a hierarchy of affinities between PCH, nucleoli, and 'amphiphilic' protein(s) that interact with both nucleolar and PCH components. This result was validated by demonstrating that the depletion of one candidate amphiphile, the nucleolar protein Pitchoune, significantly reduced PCH-nucleolar associations. Together, these results unveil a dynamic program for establishing nucleolar-PCH associations during animal development, demonstrate that nucleoli are required for normal PCH organization, and identify Pitchoune as an amphiphilic molecular link that promotes PCH-nucleolar associations. Finally, we propose that disrupting affinity hierarchies between interacting condensates can liberate molecules to form neocondensates or other aberrant structures that could contribute to cellular disease phenotypes.

Regulation of chromatin microphase separation by binding of protein complexes.

We show evidence of the association of RNA polymerase II (RNAP) with chromatin in a core-shell organization, reminiscent of microphase separation where the cores comprise dense chromatin and the shell, RNAP and chromatin with low density. These observations motivate our physical model for the regulation of core-shell chromatin organization. Here, we model chromatin as a multiblock copolymer, comprising active and inactive regions (blocks) that are both in poor solvent and tend to be condensed in the absence of binding proteins. However, we show that the solvent quality for the active regions of chromatin can be regulated by the binding of protein complexes (e.g., RNAP and transcription factors). Using the theory of polymer brushes, we find that such binding leads to swelling of the active chromatin regions which in turn modifies the spatial organization of the inactive regions. In addition, we use simulations to study spherical chromatin micelles, whose cores comprise inactive regions and shells comprise active regions and bound protein complexes. In spherical micelles the swelling increases the number of inactive cores and controls their size. Thus, genetic modifications affecting the binding strength of chromatin-binding protein complexes may modulate the solvent quality experienced by chromatin and regulate the physical organization of the genome.

Mesoscale, long-time mixing of chromosomes and its connection to polymer dynamics.

Chromosomes are arranged in distinct territories within the nucleus of animal cells. Recent experiments have shown that these territories overlap at their edges, suggesting partial mixing during interphase. Experiments that knock-down of condensin II proteins during interphase indicate increased chromosome mixing, which demonstrates control of the mixing. In this study, we use a generic polymer simulation to quantify the dynamics of chromosome mixing over time. We introduce the chromosome mixing index, which quantifies the mixing of distinct chromosomes in the nucleus. We find that the chromosome mixing index in a small confinement volume (as a model of the nucleus), increases as a power-law of the time, with the scaling exponent varying non-monotonically with self-interaction and volume fraction. By comparing the chromosome mixing index with both monomer subdiffusion due to (non-topological) intermingling of chromosomes as well as even slower reptation, we show that for relatively large volume fractions, the scaling exponent of the chromosome mixing index is related to Rouse dynamics for relatively weak chromosome attractions and to reptation for strong attractions. In addition, we extend our model to more realistically account for the situation of the Drosophila chromosome by including the heterogeneity of the polymers and their lengths to account for microphase separation of euchromatin and heterochromatin and their interactions with the nuclear lamina. We find that the interaction with the lamina further impedes chromosome mixing.

The LINC Complex Inhibits Excessive Chromatin Repression.

The Linker of Nucleoskeleton and Cytoskeleton (LINC) complex transduces nuclear mechanical inputs suggested to control chromatin organization and gene expression; however, the underlying mechanism is currently unclear. We show here that the LINC complex is needed to minimize chromatin repression in muscle tissue, where the nuclei are exposed to significant mechanical inputs during muscle contraction. To this end, the genomic binding profiles of Polycomb, Heterochromatin Protein1 (HP1a) repressors, and of RNA-Pol II were studied in Drosophila larval muscles lacking functional LINC complex. A significant increase in the binding of Polycomb and parallel reduction of RNA-Pol-II binding to a set of muscle genes was observed. Consistently, enhanced tri-methylated H3K9 and H3K27 repressive modifications and reduced chromatin activation by H3K9 acetylation were found. Furthermore, larger tri-methylated H3K27me3 repressive clusters, and chromatin redistribution from the nuclear periphery towards nuclear center, were detected in live LINC mutant larval muscles. Computer simulation indicated that the observed dissociation of the chromatin from the nuclear envelope promotes growth of tri-methylated H3K27 repressive clusters. Thus, we suggest that by promoting chromatin-nuclear envelope binding, the LINC complex restricts the size of repressive H3K27 tri-methylated clusters, thereby limiting the binding of Polycomb transcription repressor, directing robust transcription in muscle fibers.

Volume regulation in adhered cells: Roles of surface tension and cell swelling.

The volume of adhered cells has been shown experimentally to decrease during spreading. This effect can be understood from the pump-leak model, which we have extended to include mechano-sensitive ion transporters. We identify a novel effect that has important consequences on cellular volume loss: cells that are swollen due to a modulation of ion transport rates are more susceptible to volume loss in response to a tension increase. This effect explains in a plausible manner the discrepancies between three recent, independent experiments on adhered cells, between which both the magnitude of the volume change and its dynamics varied substantially. We suggest that starved and synchronized cells in two of the experiments were in a swollen state and, consequently, exhibited a large volume loss at steady state. Nonswollen cells, for which there is a very small steady-state volume decrease, are still predicted to transiently lose volume during spreading due to a relaxing viscoelastic tension that is large compared with the steady-state tension. We elucidate the roles of cell swelling and surface tension in cellular volume regulation and discuss their possible microscopic origins.

Fluctuations and Shape Dependence of Microphase Separation in Systems with Long-Range Interactions.

The combination of phase separation and long-ranged, effective, Coulomb interactions results in microphase separation. We predict the sizes and shapes of such microdomains and uniquely their dependence on the macroscopic sample shape which also affects the effective interfacial tension of fluctuations of the lamellar phase. These are applied to equilibrium salt solutions and block copolymers. Nonequilibrium phase separation in the presence of chemical reactions (e.g., cellular condensates) is mapped to the Coulomb theory to which our predictions apply. In some cases, the effective interfacial tension can be ultralow.

Affinity and Valence Impact the Extent and Symmetry of Phase Separation of Multivalent Proteins.

Biomolecular self-assembly spatially segregates proteins with a limited number of binding sites (valence) into condensates that coexist with a dilute phase. We develop a many-body lattice model for a three-component system of proteins with fixed valence in a solvent. We compare the predictions of the model to experimental phase diagrams that we measure in vivo, which allows us to vary specifically a binding site's affinity and valency. We find that the extent of phase separation varies exponentially with affinity and increases with valency. Valency alone determines the symmetry of the phase diagram.

Balance of osmotic pressures determines the nuclear-to-cytoplasmic volume ratio of the cell.

The volume of the cell nucleus varies across cell types and species and is commonly thought to be determined by the size of the genome and degree of chromatin compaction. However, this notion has been challenged over the years by much experimental evidence. Here, we consider the physical condition of mechanical force balance as a determining condition of the nuclear volume and use quantitative, order-of-magnitude analysis to estimate the forces from different sources of nuclear and cytoplasmic pressure. Our estimates suggest that the dominant pressure within the nucleus and cytoplasm of nonstriated muscle cells originates from the osmotic pressure of proteins and RNA molecules that are localized to the nucleus or cytoplasm by out-of-equilibrium, active nucleocytoplasmic transport rather than from chromatin or its associated ions. This motivates us to formulate a physical model for the ratio of the cell and nuclear volumes in which osmotic pressures of localized proteins determine the relative volumes. In accordance with unexplained observations that are a century old, our model predicts that the ratio of the cell and nuclear volumes is a constant, robust to a wide variety of biochemical and biophysical manipulations, and is changed only if gene expression or nucleocytoplasmic transport is modulated.

Live imaging of chromatin distribution reveals novel principles of nuclear architecture and chromatin compartmentalization.

The three-dimensional organization of chromatin contributes to transcriptional control, but information about native chromatin distribution is limited. Imaging chromatin in live Drosophila larvae, with preserved nuclear volume, revealed that active and repressed chromatin separates from the nuclear interior and forms a peripheral layer underneath the nuclear lamina. This is in contrast to the current view that chromatin distributes throughout the nucleus. Furthermore, peripheral chromatin organization was observed in distinct Drosophila tissues, as well as in live human effector T lymphocytes and neutrophils. Lamin A/C up-regulation resulted in chromatin collapse toward the nuclear center and correlated with a significant reduction in the levels of active chromatin. Physical modeling suggests that binding of lamina-associated domains combined with chromatin self-attractive interactions recapitulate the experimental chromatin distribution profiles. Together, our findings reveal a novel mode of mesoscale organization of peripheral chromatin sensitive to lamina composition, which is evolutionary conserved.

Physical theory of biological noise buffering by multicomponent phase separation.

Maintaining homeostasis is a fundamental characteristic of living systems. In cells, this is contributed to by the assembly of biochemically distinct organelles, many of which are not membrane bound but form by the physical process of liquid-liquid phase separation (LLPS). By analogy with LLPS in binary solutions, cellular LLPS was hypothesized to contribute to homeostasis by facilitating "concentration buffering," which renders the local protein concentration within the organelle robust to global variations in the average cellular concentration (e.g., due to expression noise). Interestingly, concentration buffering was experimentally measured in vivo in a simple organelle with a single solute, while it was observed not to be obeyed in one with several solutes. Here, we formulate theoretically and solve analytically a physical model of LLPS in a ternary solution of two solutes (ϕ and ψ) that interact both homotypically (ϕ-ϕ attractions) and heterotypically (ϕ-ψ attractions). Our physical theory predicts how the coexisting concentrations in LLPS are related to expression noise and thus, generalizes the concept of concentration buffering to multicomponent systems. This allows us to reconcile the seemingly contradictory experimental observations. Furthermore, we predict that incremental changes of the homotypic and heterotypic interactions among the molecules that undergo LLPS, such as those that are caused by mutations in the genes encoding the proteins, may increase the efficiency of concentration buffering of a given system. Thus, we hypothesize that evolution may optimize concentration buffering as an efficient mechanism to maintain LLPS homeostasis and suggest experimental approaches to test this in different systems.

Mesoscale phase separation of chromatin in the nucleus.

Intact-organism imaging of Drosophila larvae reveals and quantifies chromatin-aqueous phase separation. The chromatin can be organized near the lamina layer of the nuclear envelope, conventionally fill the nucleus, be organized centrally, or as a wetting droplet. These transitions are controlled by changes in nuclear volume and the interaction of chromatin with the lamina (part of the nuclear envelope) at the nuclear periphery. Using a simple polymeric model that includes the key features of chromatin self-attraction and its binding to the lamina, we demonstrate theoretically that it is the competition of these two effects that determines the mode of chromatin distribution. The qualitative trends as well as the composition profiles obtained in our simulations compare well with the observed intact-organism imaging and quantification. Since the simulations contain only a small number of physical variables we can identify the generic mechanisms underlying the changes in the observed phase separations.

Designer protein assemblies with tunable phase diagrams in living cells.

Protein self-organization is a hallmark of biological systems. Although the physicochemical principles governing protein-protein interactions have long been known, the principles by which such nanoscale interactions generate diverse phenotypes of mesoscale assemblies, including phase-separated compartments, remain challenging to characterize. To illuminate such principles, we create a system of two proteins designed to interact and form mesh-like assemblies. We devise a new strategy to map high-resolution phase diagrams in living cells, which provide self-assembly signatures of this system. The structural modularity of the two protein components allows straightforward modification of their molecular properties, enabling us to characterize how interaction affinity impacts the phase diagram and material state of the assemblies in vivo. The phase diagrams and their dependence on interaction affinity were captured by theory and simulations, including out-of-equilibrium effects seen in growing cells. Finally, we find that cotranslational protein binding suffices to recruit a messenger RNA to the designed micron-scale structures.

Equilibrium size distribution and phase separation of multivalent, molecular assemblies in dilute solution.

Multivalent molecules can bind a limited number of multiple neighbors via specific interactions. In this paper, we investigate theoretically the self-assembly and phase separation of such molecules in dilute solution. We show that the equilibrium size (n) distributions of linear or branched assemblies qualitatively differ; the former decays exponentially with the relative size n/N[combining macron] (N[combining macron] = n), while the latter decays as a power law, with an exponential cutoff only for n ⪆ N[combining macron]2 ≫ N[combining macron]. In some cases, finite, branched assemblies are unstable and show a sol-gel transition at a critical concentration. In dilute solutions, non-specific interactions result in phase separation, whose critical point is described by an effective Flory Huggins theory that is sensitive to the nature of these distributions.

Active volume regulation in adhered cells.

Recent experiments reveal that the volume of adhered cells is reduced as their basal area is increased. During spreading, the cell volume decreases by several thousand cubic micrometers, corresponding to large pressure changes of the order of megapascals. We show theoretically that the volume regulation of adhered cells is determined by two concurrent conditions: mechanical equilibrium with the extracellular environment and a generalization of Donnan (electrostatic) equilibrium that accounts for active ion transport. Spreading affects the structure and hence activity of ion channels and pumps, and indirectly changes the ionic content in the cell. We predict that more ions are released from the cell with increasing basal area, resulting in the observed volume-area dependence. Our theory is based on a minimal model and describes the experimental findings in terms of measurable, mesoscale quantities. We demonstrate that two independent experiments on adhered cells of different types fall on the same master volume-area curve. Our theory also captures the measured osmotic pressure of adhered cells, which is shown to depend on the number of proteins confined to the cell, their charge, and their volume, as well as the ionic content. This result can be used to predict the osmotic pressure of cells in suspension.

Cardiomyocyte Calcium Ion Oscillations-Lessons From Physics.

We review a theoretical, coarse-grained description for cardiomyocytes calcium dynamics that is motivated by experiments on RyR channel dynamics and provides an analogy to other spontaneously oscillating systems. We show how a minimal model, that focuses on calcium channel and pump dynamics and kinetics, results in a single, easily understood equation for spontaneous calcium oscillations (the Van-der-Pol equation). We analyze experiments on isolated RyR channels to quantify how the channel dynamics depends both on the local calcium concentration, as well as its temporal behavior ("adaptation"). Our oscillator model analytically predicts the conditions for spontaneous oscillations, their frequency and amplitude, and how each of those scale with the small number of relevant parameters related to calcium channel and pump activity. The minimal model is easily extended to include the effects of noise and external pacing (electrical or mechanical). We show how our simple oscillator predicts and explains the experimental observations of synchronization, "bursting" and reduction of apparent noise in the beating dynamics of paced cells. Thus, our analogy and theoretical approach provides robust predictions for the beating dynamics, and their biochemical and mechanical modulation.