Elastic ripening and inhibition of liquid-liquid phase separation

Condensate interfacial forces reposition DNA loci and probe chromatin viscoelasticity

Biomolecular condensates assemble in living cells through phase separation and related phase transitions. An underappreciated feature of these dynamic molecular assemblies is that they form interfaces with other cellular structures, including membranes, cytoskeleton, DNA and RNA, and other membraneless compartments. These interfaces are expected to give rise to capillary forces, but there are few ways of quantifying and harnessing these forces in living cells. Here, we introduce viscoelastic chromatin tethering and organization (VECTOR), which uses light-inducible biomolecular condensates to generate capillary forces at targeted DNA loci. VECTOR can be utilized to programmably reposition genomic loci on a timescale of seconds to minutes, quantitatively revealing local heterogeneity in the viscoelastic material properties of chromatin. These synthetic condensates are built from components that naturally form liquid-like structures in living cells, highlighting the potential role for native condensates to generate forces and do work to reorganize the genome and impact chromatin architecture.

Interplay of condensate material properties and chromatin heterogeneity governs nuclear condensate ripening

Nuclear condensates play many important roles in chromatin functions, but how cells regulate their nucleation and growth within the complex nuclear environment is not well understood. Here, we report how condensate properties and chromatin mechanics dictate condensate growth dynamics in the nucleus. We induced condensates with distinct properties using different proteins in human cell nuclei and monitored their growth. We revealed two key physical mechanisms that underlie droplet growth: diffusion-driven or ripening-dominated growth. To explain the experimental observations, we developed a quantitative theory that uncovers the mechanical role of chromatin and condensate material properties in regulating condensate growth in a heterogeneous environment. By fitting our theory to experimental data, we find that condensate surface tension is critical in determining whether condensates undergo elastic or Ostwald ripening. Our model also predicts that chromatin heterogeneity can influence condensate nucleation and growth, which we validated by experimentally perturbing the chromatin organization and controlling condensate nucleation. By combining quantitative experimentation with theoretical modeling, our work elucidates how condensate surface tension and chromatin heterogeneity govern nuclear condensate ripening, implying that cells can control both condensate properties and the chromatin organization to regulate condensate growth in the nucleus.

The condensation of HP1α/Swi6 imparts nuclear stiffness

Biomolecular condensates have emerged as major drivers of cellular organization. It remains largely unexplored, however, whether these condensates can impart mechanical function(s) to the cell. The heterochromatin protein HP1α (Swi6 in Schizosaccharomyces pombe) crosslinks histone H3K9 methylated nucleosomes and has been proposed to undergo condensation to drive the liquid-like clustering of heterochromatin domains. Here, we leverage the genetically tractable S. pombe model and a separation-of-function allele to elucidate a mechanical function imparted by Swi6 condensation. Using single-molecule imaging, force spectroscopy, and high-resolution live-cell imaging, we show that Swi6 is critical for nuclear resistance to external force. Strikingly, it is the condensed yet dynamic pool of Swi6, rather than the chromatin-bound molecules, that is essential to imparting mechanical stiffness. Our findings suggest that Swi6 condensates embedded in the chromatin meshwork establish the emergent mechanical behavior of the nucleus as a whole, revealing that biomolecular condensation can influence organelle and cell mechanics.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples of such assemblies include lipid rafts, and postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a two-dimensional (2D) diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to the mean-field approximation by applying the theory of diffusion in singularly perturbed domains. The latter is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains where small holes or perforations have been removed from the interior. Applications range from modeling intracellular diffusion, where interior holes could represent subcellular structures such as organelles or biological condensates, to tracking the spread of chemical pollutants or heat from localized sources. In this paper, we take the bounded domain to be the cell membrane and the holes to represent anchored clusters. The analysis proceeds by partitioning the membrane into a set of inner regions around each cluster, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Motivated by a recent study of light-activated protein oligomerization in cells, we also develop the analogous theory for cluster formation in a three-dimensional (3D) domain. The details of the asymptotic analysis differ from the 2D case due to the contrasting singularity structure of 2D and 3D Green's functions.

Ostwald ripening for designing time-dependent crystal hydrogels

Ostwald ripening (OR), a classic solution theory describing molecular transfer from metastable crystal to stable one, is applied to design time-dependent crystal hydrogels that can automatically change their mechanical properties. Using a system made from crosslinked polyacrylamide (PAM) and sodium acetate (NaAc), we demonstrate that metastable fibrous crystal networks of NaAc preferably form in PAM hydrogels via a polymer-involving mismatch nucleation. These fibrous crystals would undergo OR and evolve into isolated bulk crystals, leading to a significant reduction in material rigidity (179 folds) and interfacial adhesion (20 folds). This transformation can be applied to program time-dependent self-recovery in shape and self-delamination. Since OR is a ubiquitous, robust feature of various crystals, the approach reported here represents a new direction for designing advanced transient soft materials.

Formations of force network and softening of amorphous elastic materials from a coarsen-grained particle model

Amorphous materials, such as granular substances, glasses, emulsions, foams, and cells, play significant roles in various aspects of daily life, serving as building materials, plastics, food products, and agricultural items. Understanding the mechanical response of these materials to external forces is crucial for comprehending their deformation, toughness, and stiffness. Despite the recognition of the formation of force networks within amorphous materials, the mechanisms behind their formation and their impact on macroscopic physical properties remain elusive. In this study, we employ a coarse-grained particle model to investigate the mechanical response, wherein local physical properties are integrated into the softness of the particles. Our findings reveal the emergence of a chain-like force distribution, which correlates with the planar distribution of softness and heterogeneous density variations. Additionally, we observe that the amorphous material undergoes softening due to the heterogeneous distribution of softness, a phenomenon explicable through a simple theoretical framework. Moreover, we demonstrate that the ambiguity regarding the size ratio of the blob to the force network can be adjusted by the amplitude of planar fluctuations in softness, underscoring the robustness of the coarse-grained particle model.

Effect of Polymer Gel Elasticity on Complex Coacervate Phase Behavior

Gels are key materials in biological systems such as tissues and may control biocondensate formation and structure. To further understand the effects of elastic environments on biomacromolecular assembly, we have investigated the phase behavior and radii of complex coacervate droplets in polyacrylamide (PAM) networks as a function of gel modulus. Poly-l-lysine (PLL) and sodium hyaluronate (HA) complex coacervate phases were prepared in PAM gels with moduli varying from 0.035 to 15.0 kPa. The size of the complex coacervate droplets is reported from bright-field microscopy and confocal fluorescence microscopy. Overall, the complex coacervate droplet volume decreases inversely with the modulus. Fluorescence microscopy is also used to determine the phase behavior and concentration of fluorescently tagged HA in the complex coacervate phases as a function of ionic strength (100-270 mM). We find that the critical ionic strength and complex coacervate stability are nonmonotonic as a function of the network modulus and that the local gel concentration can be used to control phase behavior and complex coacervate droplet size scale. By understanding how elastic environments influence simple electrostatic assembly, we can further understand how biomacromolecules exist in complex, crowded, and elastic cellular environments.

Structure-Property Relationships of Granular Hybrid Hydrogels Formed through Polyelectrolyte Complexation

Hybrid hydrogels are hydrogels that exhibit heterogeneity in the network architecture by means of chemical composition and/or microstructure. The different types of interactions, together with structural heterogeneity, which can be created on different length scales, determine the mechanical properties of the final material to a large extent. In this work, the microstructure-mechanical property relationships for a hybrid hydrogel that contains both electrostatic and covalent interactions are investigated. The hybrid hydrogel is composed of a microphase-separated polyelectrolyte complex network (PEC) made of poly(4-styrenesulfonate) and poly(diallyldimethylammonium chloride) within a soft and elastic polyacrylamide hydrogel network. The system exhibits a granular structure, which is attributed to the liquid-liquid phase separation into complex coacervate droplets induced by the polymerization and the subsequent crowding effect of the polyacrylamide chains. The coacervate droplets are further hardened into PEC granules upon desalting the hydrogel. The structure formation is confirmed by a combination of electron microscopic imaging and molecular dynamics simulations. The interpenetration of both networks is shown to enhance the toughness of the resulting hydrogels due to the dissipative behavior of the PEC through the rupture of electrostatic interactions. Upon cyclic loading-unloading, the hydrogels show recovery of up to 80% of their original dissipative behavior in less than 300 s of rest with limited plasticity. The granular architecture and the tough and self-recoverable properties of the designed hybrid networks make them good candidates for applications, such as shape-memory materials, actuators, biological tissue mimics, and elastic substrates for soft sensors.

Heterogeneous elasticity drives ripening and controls bursting kinetics of transcriptional condensates

Many biomolecular condensates, including transcriptional condensates, are formed in elastic mediums. In this work, we study the nonequilibrium condensate dynamics in a chromatin-like environment modeled as a heterogeneous elastic medium. We demonstrate that the ripening process in such an elastic medium exhibits a temporal power-law scaling of the average condensate radius, depending on the local stiffness distribution and different from Ostwald ripening. Moreover, we incorporate an active process to model the dissolution of transcriptional condensates upon RNA accumulation. Intriguingly, three types of kinetics of condensate growth emerge, corresponding to constitutively expressed, transcriptional-bursting, and silenced genes. Furthermore, the simulated burst frequency decreases exponentially with the local stiffness, through which we infer a lognormal distribution of local stiffness in living cells using the transcriptome-wide distribution of burst frequency. Under the inferred stiffness distribution, the simulated distributions of bursting kinetic parameters agree reasonably well with the experimental data. Our findings reveal the interplay between biomolecular condensates and elastic mediums, yielding far-reaching implications for gene expression.

3D monitoring of the microphase separations inside the intraocular lens

Glistenings often occur after implanting the intraocular lens (IOL) due to the formation of numerous microvacuoles (MVs) and may lead to deterioration of vision quality. Previous studies showed the formation of MVs was associated with the hydrophobicity of IOL materials. Yet, the mechanism remains an open question due to the complexity of IOL polymer networks. In this study, two commercialized IOLs with similar hydrophobicity are found distinct in the formation of MVs. The 3D growth kinetics of MVs during cooling processes are captured for the first time by digital holographic microscopy (DHM) and the components of MVs are measured by DHM and Raman spectroscopy. The results reveal that the growth of MVs stems from the microphase separation of water and surrounding IOL polymers. A polymer swelling model is thus proposed to describe the microphase separation process which is found dependent on the elasticity of IOL polymer networks. The total volume of MVs is determined by the IOL hydrophobicity, while the elastic force of IOL polymer networks determines the number density and size of MVs. This study demonstrates an approach for characterizing the phase separation of crosslinked polymeric materials in biosystems and sheds lights on the refinement of IOL materials. STATEMENT OF SIGNIFICANCE: Glistenings due to the formation of numerous microvacuoles (MVs) in intraocular lens (IOL) can occur after IOL implantation, which may induce poor quality of vision. However, the underlying mechanism of MVs formation is still an open question. This study establishes an in-situ 3D imaging platform to monitor growth kinetics of the MVs in IOLs, which allows to uncover the mechanism of glistenings formation resulting from the microphase separation. The findings imply the material hydrophobicity influences the total volume of MVs, while the local elasticity of IOL polymer networks determines the number density and the size of MVs. This study offers a new approach for characterizing phase separation in crosslinking biosystems and sheds lights on the refinement of IOL materials.

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.

Heterotypic interactions can drive selective co-condensation of prion-like low-complexity domains of FET proteins and mammalian SWI/SNF complex

Prion-like domains (PLDs) are low-complexity protein sequences enriched within nucleic acid-binding proteins including those involved in transcription and RNA processing. PLDs of FUS and EWSR1 play key roles in recruiting chromatin remodeler mammalian SWI/SNF (mSWI/SNF) complex to oncogenic FET fusion protein condensates. Here, we show that disordered low-complexity domains of multiple SWI/SNF subunits are prion-like with a strong propensity to undergo intracellular phase separation. These PLDs engage in sequence-specific heterotypic interactions with the PLD of FUS in the dilute phase at sub-saturation conditions, leading to the formation of PLD co-condensates. In the dense phase, homotypic and heterotypic PLD interactions are highly cooperative, resulting in the co-mixing of individual PLD phases and forming spatially homogeneous condensates. Heterotypic PLD-mediated positive cooperativity in protein-protein interaction networks is likely to play key roles in the co-phase separation of mSWI/SNF complex with transcription factors containing homologous low-complexity domains.

Biofunctional coacervate-based artificial protocells with membrane-like and cytoplasm-like structures for the treatment of persistent hyperuricemia

Coacervate droplets formed by liquid-liquid phase separation have attracted considerable attention due to their ability to enrich biomacromolecules while preserving their bioactivities. However, there are challenges to develop coacervate droplets as delivery vesicles for therapeutics resulting from the lack of physiological stability and inherent lack of membranes in coacervate droplets. Herein, polylysine-polynucleotide complex coacervate droplets with favorable physiological stability are formulated to efficiently and facilely concentrate small molecules, biomacromolecules and nanoparticles without organic solvents. To improve the biocompatibility, the PEGylated phospholipid membrane is further coated on the surface of the coacervate droplets to prepare coacervate-based artificial protocells (ArtPC) with membrane-like and cytoplasm-like structures. The ArtPC can confine the cyclic catalytic system of uricase and catalase inside to degrade uric acid and deplete the toxicity of H2O2. This biofunctional ArtPC effectively reduces blood uric acid levels and prevents renal injuries in mice with persistent hyperuricemia. The ArtPC-based therapy can bridge the disciplines of synthetic biology, pharmaceutics and therapeutics.

Elastic microphase separation produces robust bicontinuous materials

Bicontinuous microstructures are essential to the function of diverse natural and synthetic systems. Their synthesis has been based on two approaches: arrested phase separation or self-assembly of block copolymers. The former is attractive for its chemical simplicity and the latter, for its thermodynamic robustness. Here we introduce elastic microphase separation (EMPS) as an alternative approach to make bicontinuous microstructures. Conceptually, EMPS balances the molecular-scale forces that drive demixing with large-scale elasticity to encode a thermodynamic length scale. This process features a continuous phase transition, reversible without hysteresis. Practically, EMPS is triggered by simply supersaturating an elastomeric matrix with a liquid, resulting in uniform bicontinuous materials with a well-defined microscopic length scale tuned by the matrix stiffness. The versatility of EMPS is further demonstrated by fabricating bicontinuous materials with superior mechanical properties and controlled anisotropy and microstructural gradients. Overall, EMPS presents a robust alternative for the bulk fabrication of homogeneous bicontinuous materials.

Heterotypic interactions in the dilute phase can drive co-condensation of prion-like low-complexity domains of FET proteins and mammalian SWI/SNF complex

Prion-like domains (PLDs) are low-complexity protein sequences enriched within nucleic acid-binding proteins including those involved in transcription and RNA processing. PLDs of FUS and EWSR1 play key roles in recruiting chromatin remodeler mammalian SWI/SNF complex to oncogenic FET fusion protein condensates. Here, we show that disordered low-complexity domains of multiple SWI/SNF subunits are prion-like with a strong propensity to undergo intracellular phase separation. These PLDs engage in sequence-specific heterotypic interactions with the PLD of FUS in the dilute phase at sub-saturation conditions, leading to the formation of PLD co-condensates. In the dense phase, homotypic and heterotypic PLD interactions are highly cooperative, resulting in the co-mixing of individual PLD phases and forming spatially homogeneous co-condensates. Heterotypic PLD-mediated positive cooperativity in protein-protein interaction networks is likely to play key roles in the co-phase separation of mSWI/SNF complex with transcription factors containing homologous low-complexity domains.

Counterion-controlled phase equilibria in a charge-regulated polymer solution

We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deprotonation processes in the presence of a dissolved acid, whose anions serve as screening counterions. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni-modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: Re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions.

Condensate functionalization with microtubule motors directs their nucleation in space and allows manipulating RNA localization

The localization of RNAs in cells is critical for many cellular processes. Whereas motor-driven transport of ribonucleoprotein (RNP) condensates plays a prominent role in RNA localization in cells, their study remains limited by the scarcity of available tools allowing to manipulate condensates in a spatial manner. To fill this gap, we reconstitute in cellula a minimal RNP transport system based on bioengineered condensates, which were functionalized with kinesins and dynein-like motors, allowing for their positioning at either the cell periphery or centrosomes. This targeting mostly occurs through the active transport of the condensate scaffolds, which leads to localized nucleation of phase-separated condensates. Then, programming the condensates to recruit specific mRNAs is able to shift the localization of these mRNAs toward the cell periphery or the centrosomes. Our method opens novel perspectives for examining the role of RNA localization as a driver of cellular functions.

Composite elastomers with on-demand convertible phase separations achieve large and healable electro-actuation

Phase separation has been widely exploited for fabricating structured functional materials. Generally, after being fabricated, the phase structure in a hybrid material system has been set at a specific length scale and remains unchanged during the lifespan of the material. Herein, we report a strategy to construct on-demand and reversible phase switches among homogenous, nano- and macro-phase separation states in a composite elastomer during its lifespan. We trigger the nanophase separation by super-saturating an elastomer matrix with a carefully selected small-molecule organic compound (SMOC). The nanoparticles of SMOC that precipitate out upon quenching will stretch the elastomer network, yet remain stably arrested in the elastomer matrix at low temperatures for a long time. However, at elevated temperatures, the nano-phase separation will transform into the macro-one. The elastic recovery will drive the SMOC onto the elastomer surface. The phase-separated structures can be reconfigured through the homogeneous solution state at a further elevated temperature. Taking advantage of the reversible phase switches leads to a novel strategy for designing high-performance dielectric elastomers. The in situ formed nanoparticles can boost the electro-actuation performance by eliminating electro-mechanical instability and lead to a very large actuation strain (∼146%). Once the actuator broke down, SMOC could on-demand be driven to the breakdown holes and heal the actuator.

Liquid-liquid phase separation within fibrillar networks

Complex fibrillar networks mediate liquid-liquid phase separation of biomolecular condensates within the cell. Mechanical interactions between these condensates and the surrounding networks are increasingly implicated in the physiology of the condensates and yet, the physical principles underlying phase separation within intracellular media remain poorly understood. Here, we elucidate the dynamics and mechanics of liquid-liquid phase separation within fibrillar networks by condensing oil droplets within biopolymer gels. We find that condensates constrained within the network pore space grow in abrupt temporal bursts. The subsequent restructuring of condensates and concomitant network deformation is contingent on the fracture of network fibrils, which is determined by a competition between condensate capillarity and network strength. As a synthetic analog to intracellular phase separation, these results further our understanding of the mechanical interactions between biomolecular condensates and fibrillar networks in the cell.

Mesoscale molecular assembly is favored by the active, crowded cytoplasm

The mesoscale organization of molecules into membraneless biomolecular condensates is emerging as a key mechanism of rapid spatiotemporal control in cells1. Principles of biomolecular condensation have been revealed through in vitro reconstitution2. However, intracellular environments are much more complex than test-tube environments: They are viscoelastic, highly crowded at the mesoscale, and are far from thermodynamic equilibrium due to the constant action of energy-consuming processes3. We developed synDrops, a synthetic phase separation system, to study how the cellular environment affects condensate formation. Three key features enable physical analysis: synDrops are inducible, bioorthogonal, and have well-defined geometry. This design allows kinetic analysis of synDrop assembly and facilitates computational simulation of the process. We compared experiments and simulations to determine that macromolecular crowding promotes condensate nucleation but inhibits droplet growth through coalescence. ATP-dependent cellular activities help overcome the frustration of growth. In particular, actomyosin dynamics potentiate droplet growth by reducing confinement and elasticity in the mammalian cytoplasm, thereby enabling synDrop coarsening. Our results demonstrate that mesoscale molecular assembly is favored by the combined effects of crowding and active matter in the cytoplasm. These results move toward a better predictive understanding of condensate formation in vivo.

Hierarchical bubble size distributions in coarsening wet liquid foams

Coarsening of two-phase systems is crucial for the stability of dense particle packings such as alloys, foams, emulsions, or supersaturated solutions. Mean field theories predict an asymptotic scaling state with a broad particle size distribution. Aqueous foams are good model systems for investigations of coarsening-induced structures, because the continuous liquid as well as the dispersed gas phases are uniform and isotropic. We present coarsening experiments on wet foams, with liquid fractions up to their unjamming point and beyond, that are performed under microgravity to avoid gravitational drainage. As time elapses, a self-similar regime is reached where the normalized bubble size distribution is invariant. Unexpectedly, the distribution features an excess of small roaming bubbles, mobile within the network of jammed larger bubbles. These roaming bubbles are reminiscent of rattlers in granular materials (grains not subjected to contact forces). We identify a critical liquid fraction [Formula: see text], above which the bubble assembly unjams and the two bubble populations merge into a single narrow distribution of bubbly liquids. Unexpectedly, [Formula: see text] is larger than the random close packing fraction of the foam [Formula: see text]. This is because, between [Formula: see text] and [Formula: see text], the large bubbles remain connected due to a weak adhesion between bubbles. We present models that identify the physical mechanisms explaining our observations. We propose a new comprehensive view of the coarsening phenomenon in wet foams. Our results should be applicable to other phase-separating systems and they may also help to control the elaboration of solid foams with hierarchical structures.

The pyrenoid: the eukaryotic CO2-concentrating organelle

The pyrenoid is a phase-separated organelle that enhances photosynthetic carbon assimilation in most eukaryotic algae and the land plant hornwort lineage. Pyrenoids mediate approximately one-third of global CO2 fixation, and engineering a pyrenoid into C3 crops is predicted to boost CO2 uptake and increase yields. Pyrenoids enhance the activity of the CO2-fixing enzyme Rubisco by supplying it with concentrated CO2. All pyrenoids have a dense matrix of Rubisco associated with photosynthetic thylakoid membranes that are thought to supply concentrated CO2. Many pyrenoids are also surrounded by polysaccharide structures that may slow CO2 leakage. Phylogenetic analysis and pyrenoid morphological diversity support a convergent evolutionary origin for pyrenoids. Most of the molecular understanding of pyrenoids comes from the model green alga Chlamydomonas (Chlamydomonas reinhardtii). The Chlamydomonas pyrenoid exhibits multiple liquid-like behaviors, including internal mixing, division by fission, and dissolution and condensation in response to environmental cues and during the cell cycle. Pyrenoid assembly and function are induced by CO2 availability and light, and although transcriptional regulators have been identified, posttranslational regulation remains to be characterized. Here, we summarize the current knowledge of pyrenoid function, structure, components, and dynamic regulation in Chlamydomonas and extrapolate to pyrenoids in other species.

Phase separation and ripening in a viscoelastic gel

The process of phase separation in elastic solids and viscous fluids is of fundamental importance to the stability and function of soft materials. We explore the dynamics of phase separation and domain growth in a viscoelastic material such as a polymer gel. Using analytical theory and Monte Carlo simulations, we report a domain growth regime in which the domain size increases algebraically with a ripening exponent [Formula: see text] that depends on the viscoelastic properties of the material. For a prototypical Maxwell material, we obtain [Formula: see text], which is markedly different from the well-known Ostwald ripening process with [Formula: see text]. We generalize our theory to systems with arbitrary power-law relaxation behavior and discuss our findings in the context of the long-term stability of materials as well as recent experimental results on phase separation in cross-linked networks and cytoskeleton.

Elastic heterogeneity governs asymmetric adsorption-desorption in a soft porous crystal

Metal-organic frameworks (MOFs), which possess a high degree of crystallinity and a large surface area with tunable inorganic nodes and organic linkers, exhibit high stimuli-responsiveness and molecular adsorption selectivity that enable various applications. The adsorption in MOFs changes the crystalline structure and elastic moduli. Thus, the coexistence of adsorbed/desorbed sites makes the host matrices elastically heterogeneous. However, the role of elastic heterogeneity in the adsorption-desorption transition has been overlooked. Here, we show the asymmetric role of elastic heterogeneity in the adsorption-desorption transition. We construct a minimal model incorporating adsorption-induced lattice expansion/contraction and an increase/decrease in the elastic moduli. We find that the transition is hindered by the entropic and energetic effects which become asymmetric in the adsorption process and desorption process, leading to the strong hysteretic nature of the transition. Furthermore, the adsorbed/desorbed sites exhibit spatially heterogeneous domain formation, implying that the domain morphology and interfacial area between adsorbed/desorbed sites can be controlled by elastic heterogeneity. Our results provide a theoretical guideline for designing soft porous crystals with tunable adsorption hysteresis and the dispersion and domain morphology of adsorbates using elastic heterogeneity.

Bio-enabled Engineering of Multifunctional "Living" Surfaces

Through the magic of "active matter"─matter that converts chemical energy into mechanical work to drive emergent properties─biology solves a myriad of seemingly enormous physical challenges. Using active matter surfaces, for example, our lungs clear an astronomically large number of particulate contaminants that accompany each of the 10,000 L of air we respire per day, thus ensuring that the lungs' gas exchange surfaces remain functional. In this Perspective, we describe our efforts to engineer artificial active surfaces that mimic active matter surfaces in biology. Specifically, we seek to assemble the basic active matter components─mechanical motor, driven constituent, and energy source─to design surfaces that support the continuous operation of molecular sensing, recognition, and exchange. The successful realization of this technology would generate multifunctional, "living" surfaces that combine the dynamic programmability of active matter and the molecular specificity of biological surfaces and apply them to applications in biosensors, chemical diagnostics, and other surface transport and catalytic processes. We describe our recent efforts in bio-enabled engineering of living surfaces through the design of molecular probes to understand and integrate native biological membranes into synthetic materials.

Viscoelastic coarsening of quasi-2D foam

Foams are unstable jammed materials. They evolve over timescales comparable to their "time of use", which makes the study of their destabilisation mechanisms crucial for applications. In practice, many foams are made from viscoelastic fluids, which are observed to prolong their lifetimes. Despite their importance, we lack understanding of the coarsening mechanism in such systems. We probe the effect of continuous phase viscoelasticity on foam coarsening with foamed emulsions. We show that bubble size evolution is strongly slowed down and foam structure hugely impacted. The main mechanisms responsible are the absence of continuous phase redistribution and a non-trivial link between foam structure and mechanical properties. These combine to give spatially heterogeneous coarsening. Beyond their importance in the design of foamy materials, the results give a macroscopic vision of phase separation in a viscoelastic medium.

Bottom-Up Synthetic Biology Using Cell-Free Protein Synthesis

Technical advances in biotechnology have greatly accelerated the development of bottom-up synthetic biology. Unlike top-down approaches, bottom-up synthetic biology focuses on the construction of a minimal cell from scratch and the application of these principles to solve challenges. Cell-free protein synthesis (CFPS) systems provide minimal machinery for transcription and translation, from either a fractionated cell lysate or individual purified protein elements, thus speeding up the development of synthetic cell projects. In this review, we trace the history of the cell-free technique back to the first in vitro fermentation experiment using yeast cell lysate. Furthermore, we summarized progresses of individual cell mimicry modules, such as compartmentalization, gene expression regulation, energy regeneration and metabolism, growth and division, communication, and motility. Finally, current challenges and future perspectives on the field are outlined.

The crucial role of elasticity in regulating liquid-liquid phase separation in cells

Liquid-liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here, we bridge this gap by combining recent experimental data with theoretical predictions that capture the subtleties of nonlinear elasticity and fluid transport. We show that within a biologically accessible range of material parameters, phase separation is highly sensitive to elastic properties and can thus be used as a mechanical switch to rapidly transition between different states in cellular systems. Furthermore, we show that this active mechanically mediated mechanism can drive transport across cells at biologically relevant timescales and could play a crucial role in promoting spatial localization of condensates; whether cells exploit such mechanisms for transport of their constituents remains an open question.

Molecular Determinants for the Layering and Coarsening of Biological Condensates

Many membraneless organelles, or biological condensates, form through phase separation, and play key roles in signal sensing and transcriptional regulation. While the functional importance of these condensates has inspired many studies to characterize their stability and spatial organization, the underlying principles that dictate these emergent properties are still being uncovered. In this review, we examine recent work on biological condensates, especially multicomponent systems. We focus on connecting molecular factors such as binding energy, valency, and stoichiometry with the interfacial tension, explaining the nontrivial interior organization in many condensates. We further discuss mechanisms that arrest condensate coalescence by lowering the surface tension or introducing kinetic barriers to stabilize the multidroplet state.

Dynamic Assembly of Viscoelastic Networks by Aqueous Liquid-Liquid Phase Separation and Liquid-Solid Phase Separation (AqLL-LS PS2 )

Living cells comprise diverse subcellular structures, such as cytoskeletal networks, which can regulate essential cellular activities through dynamic assembly and synergistic interactions with biomolecular condensates. Despite extensive efforts, reproducing viscoelastic networks for modulating biomolecular condensates in synthetic systems remains challenging. Here, a new aqueous two-phase system (ATPS) is proposed, which consists of poly(N-isopropylacrylamide) (PNIPAM) and dextran (DEX), to construct viscoelastic networks capable of being assembled and dissociated dynamically to regulate the self-assembly of condensates on-demand. Viscoelastic networks are generated using liquid-liquid phase-separated DEX droplets as templates and the following liquid-to-solid transition of the PNIPAM-rich phase. The resulting networks can dissolve liquid fused in sarcoma (FUS) condensates within 5 min. This work demonstrates rich phase-separation behaviors in a single ATPS through incorporating stimuli-responsive polymers. The concept can potentially be applied to other macromolecules through other stimuli to develop materials with rich phase behaviors and hierarchical structures.

Analytical Formulation and Field-Theoretic Simulation of Sequence-Specific Phase Separation of Protein-Like Heteropolymers with Short- and Long-Spatial-Range Interactions

A theory for sequence-dependent liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) in the study of biomolecular condensates is formulated by extending the random phase approximation (RPA) and field-theoretic simulation (FTS) of heteropolymers with spatially long-range Coulomb interactions to include the fundamental effects of short-range, hydrophobic-like interactions between amino acid residues. To this end, short-range effects are modeled by Yukawa interactions between multiple nonelectrostatic charges derived from an eigenvalue decomposition of pairwise residue-residue contact energies. Chain excluded volume is afforded by incompressibility constraints. A mean-field approximation leads to an effective Flory-Huggins χ parameter, which, in conjunction with RPA, accounts for the contact-interaction effects of amino acid composition and the sequence-pattern effects of long-range electrostatics in IDP LLPS, whereas FTS based on the formulation provides full sequence dependence for both short- and long-range interactions. This general approach is illustrated here by applications to variants of a natural IDP in the context of several different amino-acid interaction schemes as well as a set of different model hydrophobic-polar sequences sharing the same composition. Effectiveness of the methodology is verified by coarse-grained explicit-chain molecular dynamics simulations.

Thermodynamics predicts a stable microdroplet phase in polymer-gel mixtures undergoing elastic phase separation

We study the thermodynamics of binary mixtures with the volume fraction of the minority component less than the amount required to form a flat interface and show that the surface tension dominated equilibrium phase of the mixture forms a single macroscopic droplet. Elastic interactions in gel-polymer mixtures stabilize a phase with multiple droplets. Using a mean-field free energy we compute the droplet size as a function of the interfacial tension, Flory parameter, and elastic moduli of the gel. Our results illustrate the role of elastic interactions in dictating the phase behavior of biopolymers undergoing liquid-liquid phase separation.

Stiffening or softening of elastic media: Anomalous elasticity near phase transitions

We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique, positional order logarithmically stronger than quasi-long-range order, or, as the inversion asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size L exceeds a finite threshold. In three dimensions, stronger inversion-asymmetric couplings induce instability to the long-range positional order for all L. Sufficiently strong order parameter-displacement couplings can also turn the phase transition first order at all dimensions, concomitant with finite jumps in the elastic modulii across the transition. Our theory establishes a one-to-one correspondence between the order of the phase transitions and anomalous elasticity near the transitions.

Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion

We consider a minimal spin model for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We set up the elastic theory for this system. We use this theory to identify and study the nature of the fluctuations in the system near the second order phase transitions at T_{c} in the ZTE limit given by dT_{c}/dV=0, where V is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple asymmetrically or selectively with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near T_{c}, and also on the nature of the phase transition itself. Near second-order phase transitions and with weak asymmetry in the order parameter-strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size L in a universal fashion as [ln(L/a_{0})]^{2/3}; a_{0} is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by r scale as [ln(r/a_{0})]^{2/3} for large r. For stronger selectivity above a finite threshold, this variance diverge as L exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short-range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is L-independent corresponding to long-range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold value, then again the positional order is lost with a concomitant transition to a phase with short-range order. Large values of the order parameter-strain couplings can turn the phase transition into a first order as well. Our theory establishes a one-to-one correspondence between the order of phase transitions and anomalous elasticity near the transitions. Our theory should be a useful guide to possible synthesis of appropriate ZTE materials.

Mechanism of temperature-induced asymmetric swelling and shrinking kinetics in self-healing hydrogels

Understanding the physical principle that governs the stimuli-induced swelling and shrinking kinetics of hydrogels is indispensable for their applications. Here, we show that the shrinking and swelling kinetics of self-healing hydrogels could be intrinsically asymmetric. The structure frustration, formed by the large difference in the heat and solvent diffusions, remarkably slows down the shrinking kinetics. The plateau modulus of viscoelastic gels is found to be a key parameter governing the formation of structure frustration and, in turn, the asymmetric swelling and shrinking kinetics. This work provides fundamental understandings on the temperature-triggered transient structure formation in self-healing hydrogels. Our findings will find broad use in diverse applications of self-healing hydrogels, where cooperative diffusion of water and gel network is involved. Our findings should also give insight into the molecular diffusion in biological systems that possess macromolecular crowding environments similar to self-healing hydrogels.

Leveraging ultra-low interfacial tension and liquid-liquid phase separation in embedded 3D bioprinting

Many recently developed 3D bioprinting strategies operate by extruding aqueous biopolymer solutions directly into a variety of different support materials constituted from swollen, solvated, aqueous, polymer assemblies. In developing these 3D printing methods and materials, great care is often taken to tune the rheological behaviors of both inks and 3D support media. By contrast, much less attention has been given to the physics of the interfaces created when structuring one polymer phase into another in embedded 3D printing applications. For example, it is currently unclear whether a dynamic interfacial tension between miscible phases stabilizes embedded 3D bioprinted structures as they are shaped while in a liquid state. Interest in the physics of interfaces between complex fluids has grown dramatically since the discovery of liquid-liquid phase separation (LLPS) in living cells. We believe that many new insights coming from this burst of investigation into LLPS within biological contexts can be leveraged to develop new materials and methods for improved 3D bioprinting that leverage LLPS in mixtures of biopolymers, biocompatible synthetic polymers, and proteins. Thus, in this review article, we highlight work at the interface between recent LLPS research and embedded 3D bioprinting methods and materials, and we introduce a 3D bioprinting method that leverages LLPS to stabilize printed biopolymer inks embedded in a bioprinting support material.

Capillary forces generated by biomolecular condensates

Liquid-liquid phase separation and related phase transitions have emerged as generic mechanisms in living cells for the formation of membraneless compartments or biomolecular condensates. The surface between two immiscible phases has an interfacial tension, generating capillary forces that can perform work on the surrounding environment. Here we present the physical principles of capillarity, including examples of how capillary forces structure multiphase condensates and remodel biological substrates. As with other mechanisms of intracellular force generation, for example, molecular motors, capillary forces can influence biological processes. Identifying the biomolecular determinants of condensate capillarity represents an exciting frontier, bridging soft matter physics and cell biology.

Micellization: A new principle in the formation of biomolecular condensates

Phase separation is a fundamental mechanism for compartmentalization in cells and leads to the formation of biomolecular condensates, generally containing various RNA molecules. RNAs are biomolecules that can serve as suitable scaffolds for biomolecular condensates and determine their forms and functions. Many studies have focused on biomolecular condensates formed by liquid-liquid phase separation (LLPS), one type of intracellular phase separation mechanism. We recently identified that paraspeckle nuclear bodies use an intracellular phase separation mechanism called micellization of block copolymers in their formation. The paraspeckles are scaffolded by NEAT1_2 long non-coding RNAs (lncRNAs) and their partner RNA-binding proteins (NEAT1_2 RNA-protein complexes [RNPs]). The NEAT1_2 RNPs act as block copolymers and the paraspeckles assemble through micellization. In LLPS, condensates grow without bound as long as components are available and typically have spherical shapes to minimize surface tension. In contrast, the size, shape, and internal morphology of the condensates are more strictly controlled in micellization. Here, we discuss the potential importance and future perspectives of micellization of block copolymers of RNPs in cells, including the construction of designer condensates with optimal internal organization, shape, and size according to design guidelines of block copolymers.

Stress accumulation by confined ice in a temperature gradient

When materials freeze, they often undergo damage due to ice growth. Although this damage is commonly ascribed to the volumetric expansion of water upon freezing, it is usually driven by the flow of water toward growing ice crystals that feeds their growth. The freezing of this additional water can cause a large buildup of stress. Here, we demonstrate a technique for characterizing this stress buildup with unprecedented spatial resolution. We create a stable ice-water interface in a controlled temperature gradient and measure the deformation of the confining boundary. Analysis of the deformation field reveals stresses applied to the boundary with [Formula: see text](micrometers) spatial resolution. Globally, stresses increase steadily over time as liquid water is transported to more deeply undercooled regions. Locally, stresses increase until ice growth is stalled by the confining stresses. Importantly, we find a strong localization of stresses, which significantly increases the likelihood of damage caused by the presence of ice, even in apparently benign freezing situations. Ultimately, the limiting stress that the ice exerts is proportional to the local undercooling, in accordance with the Clapeyron equation, which describes the equilibrium between a stressed solid and its melt. Our results are closely connected to the condensation pressure during liquid-liquid phase separation and the crystallization pressure for growing crystals. Thus, they are highly relevant in fields ranging from cryopreservation and frost heave to food science, rock weathering, and art conservation.

Motor guidance by long-range communication on the microtubule highway

Coupling of motor proteins within arrays drives muscle contraction, flagellar beating, chromosome segregation, and other biological processes. Current models of motor coupling invoke either direct mechanical linkage or protein crowding, which rely on short-range motor-motor interactions. In contrast, coupling mechanisms that act at longer length scales remain largely unexplored. Here we report that microtubules can physically couple motor movement in the absence of detectable short-range interactions. The human kinesin-4 Kif4A changes the run length and velocity of other motors on the same microtubule in the dilute binding limit, when approximately 10-nm-sized motors are much farther apart than the motor size. This effect does not depend on specific motor-motor interactions because similar changes in Kif4A motility are induced by kinesin-1 motors. A micrometer-scale attractive interaction potential between motors is sufficient to recreate the experimental results in a biophysical model. Unexpectedly, our theory suggests that long-range microtubule-mediated coupling affects not only binding kinetics but also motor mechanochemistry. Therefore, the model predicts that motors can sense and respond to motors bound several micrometers away on a microtubule. Our results are consistent with a paradigm in which long-range motor interactions along the microtubule enable additional forms of collective motor behavior, possibly due to changes in the microtubule lattice.

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.

Non-specific adhesive forces between filaments and membraneless organelles

Many membraneless organelles are liquid-like domains that form inside the active, viscoelastic environment of living cells through phase separation. To investigate the potential coupling of phase separation with the cytoskeleton, we quantify the structural correlations of membraneless organelles (stress granules) and cytoskeletal filaments (microtubules) in a human-derived epithelial cell line. We find that microtubule networks are substantially denser in the vicinity of stress granules. When microtubules are depolymerized, the sub-units localize near the surface of the stress granules. We interpret these data using a thermodynamic model of partitioning of particles to the surface and bulk of the droplets. In this framework, our data are consistent with a weak (≲k B T) affinity of the microtubule sub-units for stress granule interfaces. As microtubules polymerize, their interfacial affinity increases, providing sufficient adhesion to deform droplets and/or the network. Our work suggests that proteins and other objects in the cell have a non-specific affinity for droplet interfaces that increases with the contact area and becomes most apparent when they have no preference for the interior of a droplet over the rest of the cytoplasm. We validate this basic physical phenomenon in vitro through the interaction of a simple protein-RNA condensate with microtubules.

Putting the Squeeze on Phase Separation

Phase separation is a ubiquitous process and finds applications in a variety of biological, organic, and inorganic systems. Nature has evolved the ability to control phase separation to both regulate cellular processes and make composite materials with outstanding mechanical and optical properties. Striking examples of the latter are the vibrant blue and green feathers of many bird species, which are thought to result from an exquisite control of the size and spatial correlations of their phase-separated microstructures. By contrast, it is much harder for material scientists to arrest and control phase separation in synthetic materials with such a high level of precision at these length scales. In this Perspective, we briefly review some established methods to control liquid-liquid phase separation processes and then highlight the emergence of a promising arrest method based on phase separation in an elastic polymer network. Finally, we discuss upcoming challenges and opportunities for fabricating microstructured materials via mechanically controlled phase separation.

Chromatin network retards nucleoli coalescence

Nuclear bodies are membraneless condensates that may form via liquid-liquid phase separation. The viscoelastic chromatin network could impact their stability and may hold the key for understanding experimental observations that defy predictions of classical theories. However, quantitative studies on the role of the chromatin network in phase separation have remained challenging. Using a diploid human genome model parameterized with chromosome conformation capture (Hi-C) data, we study the thermodynamics and kinetics of nucleoli formation. Dynamical simulations predict the formation of multiple droplets for nucleolar particles that experience specific interactions with nucleolus-associated domains (NADs). Coarsening dynamics, surface tension, and coalescence kinetics of the simulated droplets are all in quantitative agreement with experimental measurements for nucleoli. Free energy calculations further support that a two-droplet state, often observed for nucleoli in somatic cells, is metastable and separated from the single-droplet state with an entropic barrier. Our study suggests that nucleoli-chromatin interactions facilitate droplets' nucleation but hinder their coarsening due to the coupled motion between droplets and the chromatin network: as droplets coalesce, the chromatin network becomes increasingly constrained. Therefore, the chromatin network supports a nucleation and arrest mechanism to stabilize the multi-droplet state for nucleoli and possibly for other nuclear bodies.

Complex networks of interacting stochastic tipping elements: Cooperativity of phase separation in the large-system limit

Tipping elements in the Earth system have received increased scientific attention over recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are subdominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks of different size and with different coupling parameters.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Surface densities prewet a near-critical membrane

Recent work has highlighted roles for thermodynamic phase behavior in diverse cellular processes. Proteins and nucleic acids can phase separate into three-dimensional liquid droplets in the cytoplasm and nucleus and the plasma membrane of animal cells appears tuned close to a two-dimensional liquid-liquid critical point. In some examples, cytoplasmic proteins aggregate at plasma membrane domains, forming structures such as the postsynaptic density and diverse signaling clusters. Here we examine the physics of these surface densities, employing minimal simulations of polymers prone to phase separation coupled to an Ising membrane surface in conjunction with a complementary Landau theory. We argue that these surface densities are a phase reminiscent of prewetting, in which a molecularly thin three-dimensional liquid forms on a usually solid surface. However, in surface densities the solid surface is replaced by a membrane with an independent propensity to phase separate. We show that proximity to criticality in the membrane dramatically increases the parameter regime in which a prewetting-like transition occurs, leading to a broad region where coexisting surface phases can form even when a bulk phase is unstable. Our simulations naturally exhibit three-surface phase coexistence even though both the membrane and the polymer bulk only display two-phase coexistence on their own. We argue that the physics of these surface densities may be shared with diverse functional structures seen in eukaryotic cells.