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. Graphical abstract

Significance For over a century, it has been known that the ratio of the nuclear and cytoplasm volumes (NC ratio), rather than the separate volumes, is constant among cells of many types of organisms. Changes of the NC ratio are associated with cancerous transformations, suggesting that the ratio has physiological importance. Notably, the dominant regulatory mechanism of the NC ratio has not been identified. Here, we use physical estimates of the forces implicated in nuclear volume determination and show that they are dominated by the osmotic pressure of actively transported proteins. Inspired by this, we formulate a minimal model for the cytoplasmic and nuclear volumes that predicts the NC ratio and the factors that modulate it, in agreement with published experiments.

In the past decade, a large body of research has highlighted the biological significance of liquid-liquid phase separation (LLPS) in the formation of cellular, membraneless organelles. The biological advantage of LLPS is commonly attributed to a characteristic of phase separation of binary solutions: the concentrations of molecules within the two phases are an intensive property determined by the interaction energies among the different types of molecules. This suggests a physical basis for the concept of “concentration buffering”: fluctuations in the overall concentration of the molecules (e.g. by fluctuations of the translation/degradation rates) are “buffered” by changes in the volumes of the phases while the concentrations of the molecules within them remain unchanged. Here, we show theoretically that in phase separation of a scaffold-client-solvent ternary system, the concentrations of molecules in the different phases are not fixed. Instead, the equilibrium concentrations of molecules within each of the two coexisting phases in general, depend on the overall concentrations of those molecules. However, our theory predicts that nevertheless, there is a combination of those concentrations for each phase that is unchanged by variations of the overall concentrations. This dictates the stoichiometric relations among the concentration in each of the coexisting phases as a function of the overall concentrations. We speculate that the overall concentrations may therefore have been evolutionary tuned to result in a stoichiometry that is required for initiation of specific biochemical reactions. Using a generalized, multi-component, Flory-Huggins free energy, we provide theoretical predictions for the form of these conserved relations in the case of phase separation driven by purely homotypic vs. heterotypic interactions, and the transition between these two extremes in a simplified scaffold-client-solvent ternary system. Our predictions agree with recent experimental measurements, both in-vivo and in-vitro [Riback et al., Nature 2020].

Significance The stochastic nature of transcription/translation implies that the concentrations of cellular proteins are “noisy” and not constant in time or across cell populations. Liquid–liquid phase separation (LLPS) can reduce or “buffer” this noise by maintaining well-defined concentrations, even in the presence of concentration distributions. However, this idea was recently challenged experimentally in multicomponent systems. Our physical theory of LLPS in ternary systems (solutes ϕ and ψ in a solvent) predicts their LLPS properties as a function of the ϕ – ϕ (homotypic) and ϕ – ψ (heterotypic) interaction strengths. We show how buffering can be effective if the noise distribution aligns with the tie-lines of the phase diagram and suggest that evolution may optimize concentration buffering by selecting appropriate mutations.

Recent experiments provide accumulating evidence that the physical phenomenon of liquid-liquid phase separation is commonly used by cellular system as an organizational principle. We formulate a coarse-grained and robust model for phase separation of proteins in aqueous solution. Independent of molecular details, we represent the system as a solution of hard spheres with a square well, attractive potential. The simplicity of this description allows us to use a virial expansion of the free energy that analytically predicts the phase separation as a function of the protein concentration, and the generic depth and width of the square-well attraction. We find that weak, and relatively long-range attractions (e.g. VDW interactions), lead to phase separation, with critical volume fraction of 13%, a prediction that is consistent with published experimental measurements. In contrast, strong, short-range interactions such as multivalent bonding, causes oligomerization of the proteins that may possibly lead to gel formation. Accounting for both types of interactions, we analyze the phase separation of multivalent molecules and find that non-valent, divalent, and multivalent (valency>2) are each in a different universality class with different physical behavior. Surprisingly, we predict that a change in the valency of multivalent molecules is equivalent to a change of the interaction energy, which is supported by recently published experimental results. This allows us to outline various mechanisms used by the cell to control phase separation and to provide quantitative predictions that can guide future experiments relating phase separation of proteins to their structure and sequence.

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.

During a first-order phase transition, an interfacial layer is formed between the coexisting phases and kinetically limits homogeneous nucleation of the new phase in the original phase. This inhibition is commonly alleviated by the presence of impurities, often of unknown origin, that serve as heterogeneous nucleation sites for the transition. Living systems present a theoretical opportunity: the regulated structure of living systems allows modelling of the impurities, enabling quantitative analysis and comparison between homogeneous and heterogeneous nucleation mechanisms, usually a difficult task. Here, we formulate an analytical model of heterogeneous nucleation of holes in the nuclear lamina, a phenomenon with implications in cancer metastasis, ageing and other diseases. We then present measurements of hole nucleation in the lamina of nuclei migrating through controlled constrictions and fit the experimental data to our heterogeneous nucleation model as well as a homogeneous model. Surprisingly, we find that different mechanisms dominate depending on the density of filaments that comprise the nuclear lamina.

In the flow of information in biological systems, DNA molecules are transcribed to RNA molecules, which are then translated to functional proteins. Although transcription is well understood at the in-vitro level, the mechanisms that control transcription within the nucleus depend on DNA spatio-temporal organization, which is poorly understood. The organization of DNA (in the form of chromatin) in the nucleus presents many non-trivial physical phenomena, e.g. fractal organization, glassy dynamics, and out-of-equilibrium steady states. At the largest length scale of the nucleus itself, the genome is broadly organized into two distinct chromatin states: Heterochromatin, which is dense and transcriptionally repressed, and euchromatin, which is less compact and transcriptionally active. In this work, we review experimental evidences showing that small protein (HP1) phase-separation within the nucleus is critical for heterochromatin domain formation. We will present a physics-based, minimal, theoretical model of HP1 phase-separation that drives formation of HP1-dense heterochromatin. Motivated by experimental findings, we consider that HP1 protein can exist in two distinct configurations (open or closed), each having a different valency that determines its ability to interact with itself or with chromatin. We show that this minimal model is sufficient to recapitulate many aspects of heterochromatin formation such as the mobility of HP1, chromatin compaction, and the observed, inverse relation between concentration of DNA within globules and HP1 phosphorylation. We use a novel method to measure the in vivo response of heterochromatin to temperature changes, which provides an important test of the theory. In addition, by considering chromatin mechanics, we predict that external compressive forces applied to the nucleus can promote heterochromatin formation. This implies that the process that forms heterochromatin is inherently mechanosensitive, and may play a key role in the ability of a cell to respond to and resist mechanical stimuli.

Cell migration through dense tissues or small capillaries can elongate the nucleus and even damage it, and any impact on cell cycle has the potential to affect various processes including carcinogenesis. Here, nuclear rupture and DNA damage increase with constricted migration in different phases of cell cycle-which we show is partially repressed. We study several cancer lines that are contact inhibited or not and that exhibit diverse frequencies of nuclear lamina rupture after migration through small pores. DNA repair factors invariably mislocalize after migration, and an excess of DNA damage is evident as pan--nucleoplasmic foci of phosphoactivated ATM and γH2AX. Foci counts are suppressed in late cell cycle as expected of mitotic checkpoints, and migration of contact-inhibited cells through large pores into sparse microenvironments leads also as expected to cell-cycle reentry and no effect on a basal level of damage foci. Constricting pores delay such reentry while excess foci occur independent of cell-cycle phase. Knockdown of repair factors increases DNA damage independent of cell cycle, consistent with effects of constricted migration. Because such migration causes DNA damage and impedes proliferation, it illustrates a cancer cell fate choice of "go or grow."

During migration of cells in vivo, in both pathological processes such as cancer metastasis or physiological events such as immune cell migration through tissue, the cells must move through narrow interstitial spaces that can be smaller than the nucleus. This can induce deformation of the nucleus which, according to recent experiments, may result in rupture of the nuclear envelope that can lead to cell death, if not prevented or healed within an appropriate time. The nuclear envelope, which can be modeled as a double lipid bilayer attached to a viscoelastic gel (lamina) whose elasticity and viscosity primarily depend on the lamin composition, may utilize mechanically induced, self-healing mechanisms that allow the hole to be closed after the deformation-induced strains are reduced by leakage of the internal fluid. Here, we present a viscoelastic model of the evolution of a hole nucleated by deformations of the nuclear lamina and estimate the herniation of chromatin through the hole and its relation to the lamin expression levels in the nuclear envelope.