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.

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.

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.

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.

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.

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.

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.

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.

Long-Time Phase Correlations Reveal Regulation of Beating Cardiomyocytes.

Spontaneous contractions of cardiomyocytes are driven by calcium oscillations due to the activity of ionic calcium channels and pumps. The beating phase is related to the time-dependent deviation of the oscillations from their average frequency, due to noise and the resulting cellular response. Here, we demonstrate experimentally that, in addition to the short-time (1-2 Hz), beat-to-beat variability, there are long-time correlations (tens of minutes) in the beating phase dynamics of isolated cardiomyocytes. Our theoretical model relates these long-time correlations to cellular regulation that restores the frequency to its average, homeostatic value in response to stochastic perturbations.

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.

Registry Kinetics of Myosin Motor Stacks Driven by Mechanical Force-Induced Actin Turnover.

Actin filaments associated with myosin motors constitute the cytoskeletal force-generating machinery for many types of adherent cells. These actomyosin units are structurally ordered in muscle cells and, in particular, may be spatially registered across neighboring actin bundles. Such registry or stacking of myosin filaments have been recently observed in ordered actin bundles of even fibroblasts with super-resolution microscopy techniques. We introduce here a model for the dynamics of stacking arising from long-range mechanical interactions between actomyosin units through mutual contractile deformations of the intervening cytoskeletal network. The dynamics of registry involve two key processes: 1) polymerization and depolymerization of actin filaments and 2) remodeling of cross-linker-rich actin adhesion zones, both of which are, in principle, mechanosensitive. By calculating the elastic forces that drive registry and their effect on actin polymerization rates, we estimate a characteristic timescale of tens of minutes for registry to be established, in agreement with experimentally observed timescales for individual kinetic processes involved in myosin stack formation, which we track and quantify. This model elucidates the role of actin turnover dynamics in myosin stacking and explains the loss of stacks seen when actin assembly or disassembly and cross-linking is experimentally disrupted in fibroblasts.

Physics of Spontaneous Calcium Oscillations in Cardiac Cells and Their Entrainment.

Mechanical contraction in muscle cells requires Ca to allow myosin binding to actin. Beating cardiomyocytes contain internal Ca stores whose cytoplasmic concentration oscillates. Our theory explains observed single channel dynamics as well as cellular oscillations in spontaneously beating cardiomyocytes. The Ca dependence of channel activity responsible for Ca release includes positive feedback with a delayed response. We use this to predict a dynamical equation for global calcium oscillations with only a few physically relevant parameters. The theory accounts for the observed entrainment of beating to an oscillatory electric or mechanical field.

Elastic interactions synchronize beating in cardiomyocytes.

Motivated by recent experimental results, we study theoretically the synchronization of the beating phase and frequency of two nearby cardiomyocyte cells. Each cell is represented as an oscillating force dipole in an infinite, viscoelastic medium and the propagation of the elastic signal within the medium is predicted. We examine the steady-state beating of two nearby cells, and show that elastic interactions result in forces that synchronize the phase and frequency of beating in a manner that depends on their mutual orientation. The theory predicts both in-phase and anti-phase steady-state beating depending on the relative cell orientations, as well as how synchronized beating varies with substrate elasticity and the inter-cell distance. These results suggest how mechanics plays a role in cardiac efficiency, and may be relevant for the design of cardiomyocyte based micro devices and other biomedical applications.

Response of adherent cells to mechanical perturbations of the surrounding matrix.

We present a generic and unified theory to explain how cells respond to perturbations of their mechanical environment such as the presence of neighboring cells, slowly applied stretch, or gradients of matrix rigidity. Motivated by experiments, we calculate the local balance of forces that give rise to a tendency for the cell to locally move or reorient, with a focus on the contribution of feedback and homeostasis to cell contractility (manifested by a fixed displacement, strain or stress) that acts on the adhesions at the cell boundary. These forces can be either reinforced or diminished by elastic stresses due to mechanical perturbations of the matrix. Our model predicts these changes and how their balance with local protrusive forces that act on the cell's leading edge either increase or decrease the tendency of the cell to locally move (toward neighboring cells or rigidity gradients) or reorient (in the direction of slowly applied stretch or rigidity gradients).

Prediction of the dependence of the line tension on the composition of linactants and the temperature in phase separated membranes.

We calculate the line tension between domains in phase separated, ternary membranes that comprise line active molecules (linactants) that tend to increase the compatibility of the two phase separating species. The predicted line tension, which depends explicitly on the linactant composition and temperature, is shown to decrease significantly as the fraction of linactants in the membrane increases toward a Lifshitz point, above which the membrane phase separates into a modulated phase. We predict regimes of zero line tension at temperatures close to the mixing transition and clarify the two different ways in which the line tension can be reduced: (1) The linactants uniformly distribute in the system and reduce the compositional mismatch between the two bulk domains. (2) The linactants accumulate at the interface with a preferred orientation. Both of these mechanisms have been observed in recent experiments and simulations. The second one is unique to line active molecules, and our work shows that it is increasingly important at large fraction of linactants and is necessary for the emergence of a regime of zero line tension. The methodology is based on the ternary mixture model proposed by Palmieri and Safran [Palmieri, B.; Safran, S. A. Langmuir 2013, 29, 5246], and the line tension is calculated via variationally derived, self-consistent profiles for the local variation of composition and linactant orientation in the interface region.

Hybrid lipids increase the probability of fluctuating nanodomains in mixed membranes.

A ternary mixture model is proposed to describe composition fluctuations in mixed membranes composed of saturated, unsaturated, and hybrid lipids (with one saturated and one unsaturated hydrocarbon chain). The hybrids are line-active and can reduce the packing incompatibility between the saturated and unsaturated lipids. We introduce a lattice model that extends previous studies by taking into account the dependence of the interactions of the hybrid lipids on their orientations in a simple way. A methodology to recast the free energy of the lattice model in terms of a continuous, isotropic field theory is proposed and used to analyze composition fluctuations in the one-phase region (above the critical temperature). The effect of hybrid lipids on fluctuation domains rich in saturated/unsaturated lipids is predicted. The correlation length of such fluctuations decreases significantly with increasing amounts of hybrids; this implies that nanoscale fluctuation domains are more probable compared to the case with no hybrids. Smaller correlated fluctuation domains arise even when the temperature is close to a critical point, where very large correlation lengths are normally expected. This decrease in the correlation length is largest as the hybrid composition tends toward a crossover value above which stripelike fluctuations are predicted. This crossover value defines the Lifshitz line. The characteristic wavelength of the stripelike fluctuations is large close to the Lifshitz point but decreases toward a molecular size in a membrane that contains only hybrids. Micrometer size, stripelike domains have recently been observed experimentally in giant unilamelar vesicles (GUVs) made of saturated, unsaturated, and hybrid lipids. These results suggest that the line activity of hybrid lipids in such mixtures may be significant only at large hybrid fractions; in that regime, the interface between domains can be diffuse and several hybrid molecules with correlated orientations can separate saturated and unsaturated lipid regions.

Sarcomeric pattern formation by actin cluster coalescence.

Contractile function of striated muscle cells depends crucially on the almost crystalline order of actin and myosin filaments in myofibrils, but the physical mechanisms that lead to myofibril assembly remains ill-defined. Passive diffusive sorting of actin filaments into sarcomeric order is kinetically impossible, suggesting a pivotal role of active processes in sarcomeric pattern formation. Using a one-dimensional computational model of an initially unstriated actin bundle, we show that actin filament treadmilling in the presence of processive plus-end crosslinking provides a simple and robust mechanism for the polarity sorting of actin filaments as well as for the correct localization of myosin filaments. We propose that the coalescence of crosslinked actin clusters could be key for sarcomeric pattern formation. In our simulations, sarcomere spacing is set by filament length prompting tight length control already at early stages of pattern formation. The proposed mechanism could be generic and apply both to premyofibrils and nascent myofibrils in developing muscle cells as well as possibly to striated stress-fibers in non-muscle cells.

Metabolic remodeling of the human red blood cell membrane.

The remarkable deformability of the human red blood cell (RBC) results from the coupled dynamic response of the phospholipid bilayer and the spectrin molecular network. Here we present quantitative connections between spectrin morphology and membrane fluctuations of human RBCs by using dynamic full-field laser interferometry techniques. We present conclusive evidence that the presence of adenosine 5'-triphosphate (ATP) facilitates non-equilibrium dynamic fluctuations in the RBC membrane that are highly correlated with the biconcave shape of RBCs. Spatial analysis of the fluctuations reveals that these non-equilibrium membrane vibrations are enhanced at the scale of spectrin mesh size. Our results indicate that the dynamic remodeling of the coupled membranes powered by ATP results in non-equilibrium membrane fluctuations manifesting from both metabolic and thermal energies and also maintains the biconcave shape of RBCs.