Variational umbrella seeding for calculating nucleation barriers.

In this work, we introduce variational umbrella seeding, a novel technique for computing nucleation barriers. This new method, a refinement of the original seeding approach, is far less sensitive to the choice of order parameter for measuring the size of a nucleus. Consequently, it surpasses seeding in accuracy and umbrella sampling in computational speed. We test the method extensively and demonstrate excellent accuracy for crystal nucleation of nearly hard spheres and two distinct models of water: mW and TIP4P/ICE. This method can easily be extended to calculate nucleation barriers for homogeneous melting, condensation, and cavitation.

Principles of assembly and regulation of condensates of Polycomb repressive complex 1 through phase separation.

Polycomb repressive complex 1 (PRC1) undergoes phase separation to form Polycomb condensates that are multi-component hubs for silencing Polycomb target genes. In this study, we demonstrate that formation and regulation of PRC1 condensates are consistent with the scaffold-client model, where the Chromobox 2 (CBX2) protein behaves as the scaffold while the other PRC1 proteins are clients. Such clients induce a re-entrant phase transition of CBX2 condensates. The composition of the multi-component PRC1 condensates (1) determines the dynamic properties of the scaffold protein; (2) selectively promotes the formation of CBX4-PRC1 condensates while dissolving condensates of CBX6-, CBX7-, and CBX8-PRC1; and (3) controls the enrichment of CBX4-, CBX7-, and CBX8-PRC1 in CBX2-PRC1 condensates and the exclusion of CBX6-PRC1 from CBX2-PRC1 condensates. Our findings uncover how multi-component PRC1 condensates are assembled via an intricate scaffold-client mechanism whereby the properties of the PRC1 condensates are sensitively regulated by its composition and stoichiometry.

Mechano-dependent sorbitol accumulation supports biomolecular condensate.

Biomolecular condensates regulate a wide range of cellular functions from signaling to RNA metabolism 1, 2 , yet, the physiologic conditions regulating their formation remain largely unexplored. Biomolecular condensate assembly is tightly regulated by the intracellular environment. Changes in the chemical or physical conditions inside cells can stimulate or inhibit condensate formation 3-5 . However, whether and how the external environment of cells can also regulate biomolecular condensation remain poorly understood. Increasing our understanding of these mechanisms is paramount as failure to control condensate formation and dynamics can lead to many diseases 6, 7 . Here, we provide evidence that matrix stiffening promotes biomolecular condensation in vivo . We demonstrate that the extracellular matrix links mechanical cues with the control of glucose metabolism to sorbitol. In turn, sorbitol acts as a natural crowding agent to promote biomolecular condensation. Using in silico simulations and in vitro assays, we establish that variations in the physiological range of sorbitol, but not glucose, concentrations, are sufficient to regulate biomolecular condensates. Accordingly, pharmacologic and genetic manipulation of intracellular sorbitol concentration modulates biomolecular condensates in breast cancer - a mechano-dependent disease. We propose that sorbitol is a mechanosensitive metabolite enabling protein condensation to control mechano-regulated cellular functions. Altogether, we uncover molecular driving forces underlying protein phase transition and provide critical insights to understand the biological function and dysfunction of protein phase separation.

The liquid-to-solid transition of FUS is promoted by the condensate surface.

A wide range of macromolecules can undergo phase separation, forming biomolecular condensates in living cells. These membraneless organelles are typically highly dynamic, formed reversibly, and carry out essential functions in biological systems. Crucially, however, a further liquid-to-solid transition of the condensates can lead to irreversible pathological aggregation and cellular dysfunction associated with the onset and development of neurodegenerative diseases. Despite the importance of this liquid-to-solid transition of proteins, the mechanism by which it is initiated in normally functional condensates is unknown. Here we show, by measuring the changes in structure, dynamics, and mechanics in time and space, that single-component FUS condensates do not uniformly convert to a solid gel, but rather that liquid and gel phases coexist simultaneously within the same condensate, resulting in highly inhomogeneous structures. Furthermore, our results show that this transition originates at the interface between the condensate and the dilute continuous phase, and once initiated, the gelation process propagates toward the center of the condensate. To probe such spatially inhomogeneous rheology during condensate aging, we use a combination of established micropipette aspiration experiments together with two optical techniques, spatial dynamic mapping and reflective confocal dynamic speckle microscopy. These results reveal the importance of the spatiotemporal dimension of the liquid-to-solid transition and highlight the interface of biomolecular condensates as a critical element in driving pathological protein aggregation.

Location and Concentration of Aromatic-Rich Segments Dictates the Percolating Inter-Molecular Network and Viscoelastic Properties of Ageing Condensates.

Maturation of functional liquid-like biomolecular condensates into solid-like aggregates has been linked to the onset of several neurodegenerative disorders. Low-complexity aromatic-rich kinked segments (LARKS) contained in numerous RNA-binding proteins can promote aggregation by forming inter-protein ß-sheet fibrils that accumulate over time and ultimately drive the liquid-to-solid transition of the condensates. Here, atomistic molecular dynamics simulations are combined with sequence-dependent coarse-grained models of various resolutions to investigate the role of LARKS abundance and position within the amino acid sequence in the maturation of condensates. Remarkably, proteins with tail-located LARKS display much higher viscosity over time than those in which the LARKS are placed toward the center. Yet, at very long timescales, proteins with a single LARKS-independently of its location-can still relax and form high viscous liquid condensates. However, phase-separated condensates of proteins containing two or more LARKS become kinetically trapped due to the formation of percolated ß-sheet networks that display gel-like behavior. Furthermore, as a work case example, they demonstrate how shifting the location of the LARKS-containing low-complexity domain of FUS protein toward its center effectively precludes the accumulation of ß-sheet fibrils in FUS-RNA condensates, maintaining functional liquid-like behavior without ageing.

A Deep Potential model for liquid-vapor equilibrium and cavitation rates of water.

Computational studies of liquid water and its phase transition into vapor have traditionally been performed using classical water models. Here, we utilize the Deep Potential methodology-a machine learning approach-to study this ubiquitous phase transition, starting from the phase diagram in the liquid-vapor coexistence regime. The machine learning model is trained on ab initio energies and forces based on the SCAN density functional, which has been previously shown to reproduce solid phases and other properties of water. Here, we compute the surface tension, saturation pressure, and enthalpy of vaporization for a range of temperatures spanning from 300 to 600 K and evaluate the Deep Potential model performance against experimental results and the semiempirical TIP4P/2005 classical model. Moreover, by employing the seeding technique, we evaluate the free energy barrier and nucleation rate at negative pressures for the isotherm of 296.4 K. We find that the nucleation rates obtained from the Deep Potential model deviate from those computed for the TIP4P/2005 water model due to an underestimation in the surface tension from the Deep Potential model. From analysis of the seeding simulations, we also evaluate the Tolman length for the Deep Potential water model, which is (0.091 ± 0.008) nm at 296.4 K. Finally, we identify that water molecules display a preferential orientation in the liquid-vapor interface, in which H atoms tend to point toward the vapor phase to maximize the enthalpic gain of interfacial molecules. We find that this behavior is more pronounced for planar interfaces than for the curved interfaces in bubbles. This work represents the first application of Deep Potential models to the study of liquid-vapor coexistence and water cavitation.

On the possible locus of the liquid-liquid critical point in real water from studies of supercooled water using the TIP4P/Ice model.

One of the most accepted hypothesis to explain the anomalous behavior of water is the presence of a critical point between two liquids, the liquid-liquid critical point (LLCP), buried within the deep supercooled regime. Unfortunately, such hypothesis is hard to be experimentally confirmed due to fast freezing. Here, we show that the TIP4P/Ice water potential shifted by 400 bar can reproduce with unprecedented accuracy the experimental isothermal compressibility of water and its liquid equation of state for a wide pressure and temperature range. We find, both by extrapolation of response function maxima and by a Maxwell construction, that the location of the model LLCP is consistent with previous calculations. According to the pressure shift needed to recover the experimental behavior of supercooled water, we estimate the experimental LLCP to be located around 1250 bar and 195 K. We use the model to estimate the ice nucleation rate (J) in the vicinity of the hypothesized LLCP experimental location and obtain J = 1024 m-3 s-1. Thereby, experiments where the ratio between the cooling rate and the sample volume is equal or larger than the estimated nucleation rate could probe liquid-liquid equilibrium before freezing. Such conditions are not accessible in common experiments with microdroplets cooled at a few kelvin per second, but they could be, for instance, using nanodroplets of around 50 nm radius observed in a millisecond timescale.

Time-Dependent Material Properties of Aging Biomolecular Condensates from Different Viscoelasticity Measurements in Molecular Dynamics Simulations.

Biomolecular condensates are important contributors to the internal organization of the cell material. While initially described as liquid-like droplets, the term biomolecular condensates is now used to describe a diversity of condensed phase assemblies with material properties extending from low to high viscous liquids, gels, and even glasses. Because the material properties of condensates are determined by the intrinsic behavior of their molecules, characterizing such properties is integral to rationalizing the molecular mechanisms that dictate their functions and roles in health and disease. Here, we apply and compare three distinct computational methods to measure the viscoelasticity of biomolecular condensates in molecular simulations. These methods are the Green-Kubo (GK) relation, the oscillatory shear (OS) technique, and the bead tracking (BT) method. We find that, although all of these methods provide consistent results for the viscosity of the condensates, the GK and OS techniques outperform the BT method in terms of computational efficiency and statistical uncertainty. We thus apply the GK and OS techniques for a set of 12 different protein/RNA systems using a sequence-dependent coarse-grained model. Our results reveal a strong correlation between condensate viscosity and density, as well as with protein/RNA length and the number of stickers vs spacers in the amino acid protein sequence. Moreover, we couple the GK and the OS technique to nonequilibrium molecular dynamics simulations that mimic the progressive liquid-to-gel transition of protein condensates due to the accumulation of interprotein ß-sheets. We compare the behavior of three different protein condensates, i.e., those formed by either hnRNPA1, FUS, or TDP-43 proteins, whose liquid-to-gel transitions are associated with the onset of amyotrophic lateral sclerosis and frontotemporal dementia. We find that both the GK and OS techniques successfully predict the transition from functional liquid-like behavior to kinetically arrested states once the network of interprotein ß-sheets has percolated through the condensates. Overall, our work provides a comparison of different modeling rheological techniques to assess the viscosity of biomolecular condensates, a critical magnitude that provides information on the behavior of biomolecules inside condensates.

Direct Calculation of the Interfacial Free Energy between NaCl Crystal and Its Aqueous Solution at the Solubility Limit.

Salty water is the most abundant electrolyte aqueous mixture on Earth, however, very little is known about the NaCl-saturated solution interfacial free energy (γ_{s}). Here, we provide the first direct estimation of γ_{s} for several NaCl crystallographic planes by means of the mold integration technique, a highly efficient computational method to evaluate interfacial free energies with anisotropic crystal resolution. Making use of the JC-SPC/E model, one of the most benchmarked force fields for NaCl water solutions, we measure γ_{s} of four different crystal planes, (100), (110), (111), and (112[over ¯]) with the saturated solution at normal conditions. We find high anisotropy between the different crystal orientations with values ranging from 100 to 150 mJ m^{-2}, and the average value of the distinct planes being γ[over ¯]_{s}=137(20) mJ m^{-2}. This value for the coexistence interfacial free energy is in reasonable agreement with previous extrapolations from nucleation studies. Our Letter represents a milestone in the computational calculation of interfacial free energies between ionic crystals and aqueous solutions.

Surfactants or scaffolds? RNAs of varying lengths control the thermodynamic stability of condensates differently.

Biomolecular condensates, thought to form via liquid-liquid phase separation of intracellular mixtures, are multicomponent systems that can include diverse types of proteins and RNAs. RNA is a critical modulator of RNA-protein condensate stability, as it induces an RNA concentration-dependent reentrant phase transition-increasing stability at low RNA concentrations and decreasing it at high concentrations. Beyond concentration, RNAs inside condensates can be heterogeneous in length, sequence, and structure. Here, we use multiscale simulations to understand how different RNA parameters interact with one another to modulate the properties of RNA-protein condensates. To do so, we perform residue/nucleotide resolution coarse-grained molecular dynamics simulations of multicomponent RNA-protein condensates containing RNAs of different lengths and concentrations, and either FUS or PR25 proteins. Our simulations reveal that RNA length regulates the reentrant phase behavior of RNA-protein condensates: increasing RNA length sensitively rises the maximum value that the critical temperature of the mixture reaches, and the maximum concentration of RNA that the condensate can incorporate before beginning to become unstable. Strikingly, RNAs of different lengths are organized heterogeneously inside condensates, which allows them to enhance condensate stability via two distinct mechanisms: shorter RNA chains accumulate at the condensate's surface acting as natural biomolecular surfactants, while longer RNA chains concentrate inside the core to saturate their bonds and enhance the density of molecular connections in the condensate. Using a patchy particle model, we additionally demonstrate that the combined impact of RNA length and concentration on condensate properties is dictated by the valency, binding affinity, and polymer length of the various biomolecules involved. Our results postulate that diversity on RNA parameters within condensates allows RNAs to increase condensate stability by fulfilling two different criteria: maximizing enthalpic gain and minimizing interfacial free energy; hence, RNA diversity should be considered when assessing the impact of RNA on biomolecular condensates regulation.

The Chromatin Regulator HMGA1a Undergoes Phase Separation in the Nucleus.

The protein high mobility group A1 (HMGA1) is an important regulator of chromatin organization and function. However, the mechanisms by which it exerts its biological function are not fully understood. Here, we report that the HMGA isoform, HMGA1a, nucleates into foci that display liquid-like properties in the nucleus, and that the protein readily undergoes phase separation to form liquid condensates in vitro. By bringing together machine-leaning modelling, cellular and biophysical experiments and multiscale simulations, we demonstrate that phase separation of HMGA1a is promoted by protein-DNA interactions, and has the potential to be modulated by post-transcriptional effects such as phosphorylation. We further show that the intrinsically disordered C-terminal tail of HMGA1a significantly contributes to its phase separation through electrostatic interactions via AT hooks 2 and 3. Our work sheds light on HMGA1 phase separation as an emergent biophysical factor in regulating chromatin structure.

Alternating one-phase and two-phase crystallization mechanisms in octahedral patchy colloids.

Colloidal systems possess unique features to investigate the governing principles behind liquid-to-solid transitions. The phase diagram and crystallization landscape of colloidal particles can be finely tuned by the range, number, and angular distribution of attractive interactions between the constituent particles. In this work, we present a computational study of colloidal patchy particles with high-symmetry bonding-six patches displaying octahedral symmetry-that can crystallize into distinct competing ordered phases: a cubic simple (CS) lattice, a body-centered cubic phase, and two face-centered cubic solids (orientationally ordered and disordered). We investigate the underlying mechanisms by which these competing crystals emerge from a disordered fluid at different pressures. Strikingly, we identify instances where the structure of the crystalline embryo corresponds to the stable solid, while in others, it corresponds to a metastable crystal whose nucleation is enabled by its lower interfacial free energy with the liquid. Moreover, we find the exceptional phenomenon that, due to a subtle balance between volumetric enthalpy and interfacial free energy, the CS phase nucleates via crystalline cubic nuclei rather than through spherical clusters, as the majority of crystal solids in nature. Finally, by examining growth beyond the nucleation stage, we uncover a series of alternating one-phase and two-phase crystallization mechanisms depending on whether or not the same phase that nucleates keeps growing. Taken together, we show that an octahedral distribution of attractive sites in colloidal particles results in an extremely rich crystallization landscape where subtle differences in pressure crucially determine the crystallizing polymorph.

Homogeneous ice nucleation rates for mW and TIP4P/ICE models through Lattice Mold calculations.

Freezing of water is the most common liquid-to-crystal phase transition on Earth; however, despite its critical implications on climate change and cryopreservation among other disciplines, its characterization through experimental and computational techniques remains elusive. In this work, we make use of computer simulations to measure the nucleation rate (J) of water at normal pressure under different supercooling conditions, ranging from 215 to 240 K. We employ two different water models: mW, a coarse-grained potential for water, and TIP4P/ICE, an atomistic nonpolarizable water model that provides one of the most accurate representations of the different ice phases. To evaluate J, we apply the Lattice Mold technique, a computational method based on the use of molds to induce the nucleus formation from the metastable liquid under conditions at which observing spontaneous nucleation would be unfeasible. With this method, we obtain estimates of the nucleation rate for ice Ih and Ic and a stacking mixture of ice Ih/Ic, reaching consensus with most of the previously reported rates, although differing with some others. Furthermore, we confirm that the predicted nucleation rates obtained by the TIP4P/ICE model are in better agreement with experimental data than those obtained through the mW potential. Taken together, our study provides a reliable methodology to measure nucleation rates in a simple and computationally efficient manner that contributes to benchmarking the freezing behavior of two popular water models.

Protein structural transitions critically transform the network connectivity and viscoelasticity of RNA-binding protein condensates but RNA can prevent it.

Biomolecular condensates, some of which are liquid-like during health, can age over time becoming gel-like pathological systems. One potential source of loss of liquid-like properties during ageing of RNA-binding protein condensates is the progressive formation of inter-protein ß-sheets. To bridge microscopic understanding between accumulation of inter-protein ß-sheets over time and the modulation of FUS and hnRNPA1 condensate viscoelasticity, we develop a multiscale simulation approach. Our method integrates atomistic simulations with sequence-dependent coarse-grained modelling of condensates that exhibit accumulation of inter-protein ß-sheets over time. We reveal that inter-protein ß-sheets notably increase condensate viscosity but does not transform the phase diagrams. Strikingly, the network of molecular connections within condensates is drastically altered, culminating in gelation when the network of strong ß-sheets fully percolates. However, high concentrations of RNA decelerate the emergence of inter-protein ß-sheets. Our study uncovers molecular and kinetic factors explaining how the accumulation of inter-protein ß-sheets can trigger liquid-to-solid transitions in condensates, and suggests a potential mechanism to slow such transitions down.

Aging can transform single-component protein condensates into multiphase architectures.

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

Kinetic interplay between droplet maturation and coalescence modulates shape of aged protein condensates.

Biomolecular condensates formed by the process of liquid-liquid phase separation (LLPS) play diverse roles inside cells, from spatiotemporal compartmentalisation to speeding up chemical reactions. Upon maturation, the liquid-like properties of condensates, which underpin their functions, are gradually lost, eventually giving rise to solid-like states with potential pathological implications. Enhancement of inter-protein interactions is one of the main mechanisms suggested to trigger the formation of solid-like condensates. To gain a molecular-level understanding of how the accumulation of stronger interactions among proteins inside condensates affect the kinetic and thermodynamic properties of biomolecular condensates, and their shapes over time, we develop a tailored coarse-grained model of proteins that transition from establishing weak to stronger inter-protein interactions inside condensates. Our simulations reveal that the fast accumulation of strongly binding proteins during the nucleation and growth stages of condensate formation results in aspherical solid-like condensates. In contrast, when strong inter-protein interactions appear only after the equilibrium condensate has been formed, or when they accumulate slowly over time with respect to the time needed for droplets to fuse and grow, spherical solid-like droplets emerge. By conducting atomistic potential-of-mean-force simulations of NUP-98 peptides-prone to forming inter-protein [Formula: see text]-sheets-we observe that formation of inter-peptide [Formula: see text]-sheets increases the strength of the interactions consistently with the loss of liquid-like condensate properties we observe at the coarse-grained level. Overall, our work aids in elucidating fundamental molecular, kinetic, and thermodynamic mechanisms linking the rate of change in protein interaction strength to condensate shape and maturation during ageing.

RNA length has a non-trivial effect in the stability of biomolecular condensates formed by RNA-binding proteins.

Biomolecular condensates formed via liquid-liquid phase separation (LLPS) play a crucial role in the spatiotemporal organization of the cell material. Nucleic acids can act as critical modulators in the stability of these protein condensates. To unveil the role of RNA length in regulating the stability of RNA binding protein (RBP) condensates, we present a multiscale computational strategy that exploits the advantages of a sequence-dependent coarse-grained representation of proteins and a minimal coarse-grained model wherein proteins are described as patchy colloids. We find that for a constant nucleotide/protein ratio, the protein fused in sarcoma (FUS), which can phase separate on its own-i.e., via homotypic interactions-only exhibits a mild dependency on the RNA strand length. In contrast, the 25-repeat proline-arginine peptide (PR25), which does not undergo LLPS on its own at physiological conditions but instead exhibits complex coacervation with RNA-i.e., via heterotypic interactions-shows a strong dependence on the length of the RNA strands. Our minimal patchy particle simulations suggest that the strikingly different effect of RNA length on homotypic LLPS versus RBP-RNA complex coacervation is general. Phase separation is RNA-length dependent whenever the relative contribution of heterotypic interactions sustaining LLPS is comparable or higher than those stemming from protein homotypic interactions. Taken together, our results contribute to illuminate the intricate physicochemical mechanisms that influence the stability of RBP condensates through RNA inclusion.

Surface Electrostatics Govern the Emulsion Stability of Biomolecular Condensates.

Liquid-liquid phase separation underlies the formation of biological condensates. Physically, such systems are microemulsions that in general have a propensity to fuse and coalesce; however, many condensates persist as independent droplets in the test tube and inside cells. This stability is crucial for their function, but the physicochemical mechanisms that control the emulsion stability of condensates remain poorly understood. Here, by combining single-condensate zeta potential measurements, optical microscopy, tweezer experiments, and multiscale molecular modeling, we investigate how the nanoscale forces that sustain condensates impact their stability against fusion. By comparing peptide-RNA (PR25:PolyU) and proteinaceous (FUS) condensates, we show that a higher condensate surface charge correlates with a lower fusion propensity. Moreover, measurements of single condensate zeta potentials reveal that such systems can constitute classically stable emulsions. Taken together, these results highlight the role of passive stabilization mechanisms in protecting biomolecular condensates against coalescence.

'RNA modulation of transport properties and stability in phase-separated condensates.

One of the key mechanisms employed by cells to control their spatiotemporal organization is the formation and dissolution of phase-separated condensates. The balance between condensate assembly and disassembly can be critically regulated by the presence of RNA. In this work, we use a chemically-accurate sequence-dependent coarse-grained model for proteins and RNA to unravel the impact of RNA in modulating the transport properties and stability of biomolecular condensates. We explore the phase behavior of several RNA-binding proteins such as FUS, hnRNPA1, and TDP-43 proteins along with that of their corresponding prion-like domains and RNA recognition motifs from absence to moderately high RNA concentration. By characterizing the phase diagram, key molecular interactions, surface tension, and transport properties of the condensates, we report a dual RNA-induced behavior: on the one hand, RNA enhances phase separation at low concentration as long as the RNA radius of gyration is comparable to that of the proteins, whereas at high concentration, it inhibits the ability of proteins to self-assemble independently of its length. On the other hand, along with the stability modulation, the viscosity of the condensates can be considerably reduced at high RNA concentration as long as the length of the RNA chains is shorter than that of the proteins. Conversely, long RNA strands increase viscosity even at high concentration, but barely modify protein self-diffusion which mainly depends on RNA concentration and on the effect RNA has on droplet density. On the whole, our work rationalizes the different routes by which RNA can regulate phase separation and condensate dynamics, as well as the subsequent aberrant rigidification implicated in the emergence of various neuropathologies and age-related diseases.