Phase Separation Model Research Articles

Critical behavior in active lattice models of motility-induced phase separation

Lattice models allow for a computationally efficient investigation of motility-induced phase separation (MIPS) compared to off-lattice systems. Simulations are less demanding, and thus, bigger systems can be accessed with higher accuracy and better statistics. In equilibrium, lattice and off-lattice models with comparable interactions belong to the same universality class. Whether concepts of universality also hold for active particles is still a controversial and open question. Here, we examine two recently proposed active lattice systems that undergo MIPS and investigate numerically their critical behavior. In particular, we examine the claim that these systems and MIPS in general belong to the Ising universality class. We also take a more detailed look on the influence and role of rotational diffusion and active velocity in these systems.Graphic

Temperature-Responsive Peptide-Nucleotide Coacervates.

Coacervates are a type of liquid–liquid phase separated (LLPS) droplets that can serve as models of membraneless organelles (MLOs) in living cells. Peptide–nucleotide coacervates have been widely used to mimic properties of ribonucleoprotein (RNP) granules, but the thermal stability and the role of base stacking is still poorly understood. Here, we report a systematic investigation of coacervates formed by five different nucleoside triphosphates (NTPs) with poly-l-lysine and poly-l-arginine as a function of temperature. All studied combinations exhibit an upper critical solution temperature (UCST), and a temperature-dependent critical salt concentration, originating from a significant nonelectrostatic contribution to the mixing free energy. Both the enthalpic and entropic parts of this nonelectrostatic interaction decrease in the order G/A/U/C/T, in accordance with nucleobase stacking free energies. Partitioning of two dyes proves that the local hydrophobicity inside the peptide–nucleotide coacervates is different for every nucleoside triphosphate. We derive a simple relation between the temperature and salt concentration at the critical point based on a mean-field model of phase separation. Finally, when different NTPs are mixed with one common oppositely charged peptide, hybrid coacervates were formed, characterized by a single intermediate UCST and critical salt concentration. NTPs with lower critical salt concentrations can remain condensed in mixed coacervates far beyond their original critical salt concentration. Our results show that NTP-based coacervates have a strong temperature sensitivity due to base stacking interactions and that mixing NTPs can significantly influence the stability of condensates and, by extension, their bioavailability.

Compressibility of Multicomponent, Charged Model Biomembranes Tunes Permeation of Cationic Nanoparticles.

Cells respond to external stress by altering their membrane lipid composition to maintain fluidity, integrity and net charge. However, in interactions with charged nanoparticles (NPs), altering membrane charge could adversely affect its ability to transport ions across the cell membrane. Hence, it is important to understand possible pathways by which cells could alter zwitterionic lipid composition to respond to NPs without compromising membrane integrity and charge. Here, we report in situ synchrotron X-ray reflectivity (XR) measurements to monitor the interaction of cationic NPs in the form of quantum dots, with phase-separated supported lipid bilayers of different compositions containing an anionic lipid and zwitterionic lipids having variable degrees of stiffness. We observe that the extent of NP penetration into the respective membranes, as estimated from XR data analysis, is inversely related to membrane compression moduli, which was tuned by altering the stiffness of the zwitterionic lipid component. For a particular membrane composition with a discernible height difference between ordered and disordered phases, we were able to observe subtle correlations between the extent of charge on the NPs and the specificity to bind to the charged and ordered phase, contrary to that observed earlier for phase-separated model biomembranes containing no charged lipids. Our results provide microscopic insight into the role of membrane rigidity and electrostatics in determining membrane permeation. This can lead to great potential benefits in rational designing of NPs for bioimaging and drug delivery applications as well as in assessing and alleviating cytotoxicity of NPs.

Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams.

The Edmond–Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B11, B12, B22) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B11, B12, B22). Since the Edmond–Ogston model is equivalent to the Flory–Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory–Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B12.

Viscoelastic phase separation model for ternary polymer solutions.

When a polymer solution undergoes viscoelastic phase separation, the polymer-rich phase forms a network-like structure even if it is a minor phase. This unique feature is induced by polymer dynamics, which are constrained by the temporal entanglement of polymer chains. The fundamental mechanisms of viscoelastic phase separation have already been elucidated by theory and experiments over the past few decades; however, it is not yet well understood how viscoelastic phase separation occurs in multicomponent polymer solutions. Here, we construct a new viscoelastic phase separation model for ternary polymer solutions that consist of a polymer, solvent, and nonsolvent. Our simulation results reveal that a network-like structure is formed in the ternary bulk system through a phase separation mechanism similar to that observed in binary polymer solutions. A difference in dynamics is also found in that the solvent, whose affinity to the polymer is similar to that of the nonsolvent, moves freely between the polymer-rich and water-rich phases during phase separation. These findings are considered important for understanding the phase separation mechanism of ternary mixtures often used in the manufacture of polymeric separation membranes.

Reduced Model for Properties of Multiscale Porous Media with Changing Geometry

In this paper, we consider an important problem for modeling complex coupled phenomena in porous media at multiple scales. In particular, we consider flow and transport in the void space between the pores when the pore space is altered by new solid obstructions formed by microbial growth or reactive transport, and we are mostly interested in pore-coating and pore-filling type obstructions, observed in applications to biofilm in porous media and hydrate crystal formation, respectively. We consider the impact of these obstructions on the macroscopic properties of the porous medium, such as porosity, permeability and tortuosity, for which we build an experimental probability distribution with reduced models, which involves three steps: (1) generation of independent realizations of obstructions, followed by, (2) flow and transport simulations at pore-scale, and (3) upscaling. For the first step, we consider three approaches: (1A) direct numerical simulations (DNS) of the PDE model of the actual physical process called BN which forms the obstructions, and two non-DNS methods, which we call (1B) CLPS and (1C) LP. LP is a lattice Ising-type model, and CLPS is a constrained version of an Allen–Cahn model for phase separation with a localization term. Both LP and CLPS are model approximations of BN, and they seek local minima of some nonconvex energy functional, which provide plausible realizations of the obstructed geometry and are tuned heuristically to deliver either pore-coating or pore-filling obstructions. Our methods work with rock-void geometries obtained by imaging, but bypass the need for imaging in real-time, are fairly inexpensive, and can be tailored to other applications. The reduced models LP and CLPS are less computationally expensive than DNS, and can be tuned to the desired fidelity of the probability distributions of upscaled quantities.

Physical mechanisms of chromatin spatial organization.

In higher eukaryotes, chromosomes have a complex three-dimensional (3D) conformation in the cell nucleus serving vital functional purposes, yet their folding principles remain poorly understood at the single-molecule level. Here, we summarize recent approaches from polymer physics to comprehend the physical mechanisms underlying chromatin architecture. In particular, we focus on two models that have been supported by recent, growing experimental evidence, the Loop Extrusion model and the Strings&Binders phase separation model. We discuss their key ingredients, how they compare to experimental data and some insight they provide on chromatin architecture and gene regulation. Progress in that research field are opening the possibility to predict how genomic mutations alter the network of contacts between genes and their regulators and how that is linked to genetic diseases, such as congenital disorders and cancer.

Evolution of microstructures in radiation fields using a coupled binary-collision Monte Carlo phase field approach

The simulation of radiation effects in materials broadly falls into two categories. In the limit of short time and length scales lies the modeling of primary radiation damage, such as point defect creation, energy deposition, and ballistic mixing. This is followed by the modeling at longer time scales of thermally activated microstructure evolution and defect reactions, such as recombination, clustering, and coarsening. The binary collision Monte Carlo method is an established, numerically efficient method for the computation of primary radiation damage. Conversely, the phase-field method is a state-of-the-art technique for modeling microstructure evolution on longer time and length scales. We present a concurrent coupling of these two methods, overcoming the difference between the discrete object Monte Carlo paradigm for primary radiation damage and the continuum field variable approach for microstructure evolution. The coupling is bidirectional, in which the microstructure evolution in the MOOSE finite-element framework provides the spatial scattering data set for the charged particle transport and receives point defect, mass transport, and heat source terms from the simulated collision cascades that contribute to the field variable evolution. The concurrent coupling scheme is implemented in the code Magpie and demonstrated by investigating patterning for an irradiated immiscible binary model alloy. The results from the coupled binary collision Monte Carlo/phase-field simulations reproduce the results of analytical models for phase separation, and phase mixing and patterning, supporting the approach and indicating its utility for modeling real-world materials systems.

Dynamic 3D Locus Organization and Its Drivers Underpin Immunoglobulin Recombination.

A functional adaptive immune system must generate enormously diverse antigen receptor (AgR) repertoires from a limited number of AgR genes, using a common mechanism, V(D)J recombination. The AgR loci are among the largest in the genome, and individual genes must overcome huge spatial and temporal challenges to co-localize with optimum variability. Our understanding of the complex mechanisms involved has increased enormously, due in part to new technologies for high resolution mapping of AgR structure and dynamic movement, underpinning mechanisms, and resulting repertoires. This review will examine these advances using the paradigm of the mouse immunoglobulin heavy chain (Igh) locus. We will discuss the key regulatory elements implicated in Igh locus structure. Recent next generation repertoire sequencing methods have shown that local chromatin state at V genes contribute to recombination efficiency. Next on the multidimensional scale, we will describe imaging studies that provided the first picture of the large-scale dynamic looping and contraction the Igh locus undergoes during recombination. We will discuss chromosome conformation capture (3C)-based technologies that have provided higher resolution pictures of Igh locus structure, including the different models that have evolved. We will consider the key transcription factors (PAX5, YY1, E2A, Ikaros), and architectural factors, CTCF and cohesin, that regulate these processes. Lastly, we will discuss a plethora of recent exciting mechanistic findings. These include Rag recombinase scanning for convergent RSS sequences within DNA loops; identification of Igh loop extrusion, and its putative role in Rag scanning; the roles of CTCF, cohesin and cohesin loading factor, WAPL therein; a new phase separation model for Igh locus compartmentalization. We will draw these together and conclude with some horizon-scanning and unresolved questions.

Determination of the critical micelle concentration and thermodynamic parameters of phenylcarbamic acid derivatives using a fluorescence method

Critical micelle concentration (CMC) values of 1-[3-(2-alkoxyphenylcarbamoyloxy)-2-hydroxypropyl]-4-(4-fluorophenyl)piperazin-1-ium-chlorides (alkoxy = methoxy to propoxy; 6a–c) were determined by a fluorescence method using pyrene as a probe over a temperature interval T = 293.15–313.15 K. The dependence of CMC on T turned out forming the U-shape. The curve was drawn by fitting of observed values using a second-degree polynomial function and so-called power-law equation, minimum of the curve was observed at T = 298.15 K. Thermodynamic parameters of micellization, i.e., molar Gibbs energy (ΔG°), enthalpy (ΔH°), and entropy (ΔS°), respectively, were calculated according to a phase separation model. Following the observations, the enthalpy (ΔH°) and entropy contributions (−TΔS°) to the molar Gibbs energy (ΔG°) were calculated. It was found that contribution of enthalpy increased with the increase in temperature, and entropy contribution decreased within chosen range of the temperature. Finally, the enthalpy–entropy compensation and the compensation temperature (Tc) were determined.

Effects of liquid–liquid phase separation and relative humidity on the heterogeneous OH oxidation of inorganic–organic aerosols: insights from methylglutaric acid and ammonium sulfate particles

Abstract. Organic compounds residing near the surface of atmospheric aerosol particles are exposed to chemical reactions initiated by gas-phase oxidants, such as hydroxyl (OH) radicals. Aqueous droplets composed of inorganic salts and organic compounds can undergo phase separation into two liquid phases, depending on aerosol composition and relative humidity (RH). Such phase behavior can govern the surface characteristics and morphology of the aerosols, which in turn affect the heterogeneous reactivity of organic compounds toward gas-phase oxidants. In this work, we used an aerosol flow tube reactor coupled with an atmospheric pressure ionization source (direct analysis in real time) and a high-resolution mass spectrometer to investigate how phase separation in model aqueous droplets containing an inorganic salt (ammonium sulfate, AS) and an organic acid (3-methylglutaric acid, 3-MGA) with an organic-to-inorganic dry mass ratio (OIR) of 1 alters the heterogeneous OH reactivity. At high RH, 3-MGA/AS aerosols were aqueous droplets with a single liquid phase. When the RH decreased, aqueous 3-MGA/AS droplets underwent phase separation at ∼75 % RH. Once the droplets were phase-separated, they exhibited either a core–shell, partially engulfed or a transition from core–shell to partially engulfed structure, with an organic-rich outer phase and an inorganic-rich inner phase. The kinetics, quantified by an effective heterogenous OH rate constant, was found to increase gradually from 1.01±0.02×10-12 to 1.73±0.02×10-12 cm3 molec.−1 s−1 when the RH decreased from 88 % to 55 %. The heterogeneous reactivity of phase-separated droplets is slightly higher than that of aqueous droplets with a single liquid phase. This could be explained by the finding that when the RH decreases, higher concentrations of organic molecules (i.e., 3-MGA) are present at or near the droplet surface, which are more readily exposed to OH oxidation, as demonstrated by phase separation measurements and model simulations. This could increase the reactive collision probability between 3-MGA molecules and OH radicals dissolved near the droplet surface and secondary chain reactions. Even for phase-separated droplets with a fully established core–shell structure, the diffusion rate of organic molecules across the organic-rich outer shell is predicted to be fast in this system. Thus, the overall rate of reactions is likely governed by the surface concentration of 3-MGA rather than a diffusion limitation. Overall, understanding the aerosol phase state (single liquid phase versus two separate liquid phases) is essential to better probe the heterogenous reactivity under different aerosol chemical composition and environmental conditions (e.g., RH).

Computational Models for Liquid-Liquid Phase Separation of Intrinsically Disordered Proteins

Intrinsically disordered proteins (IDPs), lack of a well-defined folded structure, have been recently shown to play key roles in forming membraneless organelles via liquid-liquid phase separation (LLPS). Due to the flexible structures of IDPs, it could be challenging to investigate IDPs with solely experimental techniques. Computational models can therefore provide complementary views on a number of interesting questions, including the fundamental physics underlying LLPS, and the sequence determinants and the molecular interactions contributing to LLPS.

Formulation of a low-protein rice drink fortified with caseinomacropeptide concentrate

A rice drink with low protein and fat contents was developed by response surface methodology. Effects of three factors, levels of white rice flour (3–8 % w/v) and xanthan gum (0.01−0.05% w/v) and process temperature (80−90 °C) on physical stability of the drink were evaluated. Viscosity, phase separation, sedimentation, color, particle size and zeta potential of the samples were measured as response variables. Models were developed from the data to predict response variables from the factors evaluated. Models for viscosity, particle size, color, zeta potential and phase separation of the drink were found significant while that of sedimentation was not significant. Significant models were then used in optimization to achieve selected response variables, minimum phase separation, particle size and viscosity and zeta potential in range of values measured in commercial samples of milk-like drinks. Optimum conditions were found to be 3% rice flour and 0.05% xanthan gum in formula and process temperature of 80 °C applied for 15 min. Sucrose addition at a level of (2.5% w/v) to the formula increased acceptability of the drink in sensory analysis but it had no significant effect on physical properties of the rice drink. The drink was fortified with a caseinomacropeptide (CMP) concentrate for development of a milk-like drink for consumers with phenylketonuria. Fortification with CMP concentrate resulted in partial neutralization of zeta potential, slight sedimentation and an increase in particle size of the drink. Color and zeta potential of the drink became closer to those of low fat milk by the addition of CMP concentrate.

On the sharp interface limit of a model for phase separation on biological membranes

Abstract We rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

Global existence of solutions to some equations modeling phase separation of self-propelled particles

In this paper, we provide mathematical results on some models of phase separation of self-propelled particles. To this end, we first introduce a general model presented in Tiribocchi et al. (Phys Rev Lett 115:188302, 2015). We then proceed to handle some particular cases where $$\lambda $$ , whose role is to violate time reversal symmetry, is zero (passive) or nonzero (active). We also cover some models coupled with incompressible fluids.

Client proximity enhancement inside cellular membrane-less compartments governed by client-compartment interactions

Membrane-less organelles or compartments are considered to be dynamic reaction centers for spatiotemporal control of diverse cellular processes in eukaryotic cells. Although their formation mechanisms have been steadily elucidated via the classical concept of liquid–liquid phase separation, biomolecular behaviors such as protein interactions inside these liquid compartments have been largely unexplored. Here we report quantitative measurements of changes in protein interactions for the proteins recruited into membrane-less compartments (termed client proteins) in living cells. Under a wide range of phase separation conditions, protein interaction signals were vastly increased only inside compartments, indicating greatly enhanced proximity between recruited client proteins. By employing an in vitro phase separation model, we discovered that the operational proximity of clients (measured from client–client interactions) could be over 16 times higher than the expected proximity from actual client concentrations inside compartments. We propose that two aspects should be considered when explaining client proximity enhancement by phase separation compartmentalization: (1) clients are selectively recruited into compartments, leading to concentration enrichment, and more importantly, (2) recruited clients are further localized around compartment-forming scaffold protein networks, which results in even higher client proximity.

Continuum modelling of structure formation of biosilica patterns in diatoms

Formation of regularly structured silica valves of various diatom species is a particularly fascinating phenomenon in biomineralization. Intensive investigations have been devoted to elucidate the formation mechanisms of diatom valve structures. Phase-separation of species-specific organic molecules has been proposed to be involved in pattern formation, where the evolving organic molecule structures serve as template for silica formation. In the present work, using a continuum approach, we investigate the conditions under which silica structures of high regularity can develop within a phase separation model. In relation to previously reported in vitro experiments of silica formation, which revealed the important role of phosphate ions in the self-assembly of organic molecules, we propose a model where phase separation is coupled with a chemical reaction. We analyze the impact of the reaction of phosphate ions with organic molecules on the appearing morphology of the organic template. Two- and three-dimensional simulations of the development of regular stationary patterns are presented. The influence of a confined geometry and an interaction of organic molecules with the walls on pattern formation is also addressed. We expect that our approach will be relevant for experimental studies aiming at inducing structure formation under controlled in vitro conditions.

Phase-field modeling of non-solvent induced phase separation (NIPS) for PES/NMP/Water with comparison to experiments

We develop a phase-field model to simulate the formation of porous polymeric membranes via non-solvent induced phase separation. The material system of interest is PES/NMP/Water (Polyethersulfone/N-methyl-2-pyrrolidone/Water), however the approach is broadly applicable to other materials. The three-component system is represented with two field variables: one representing the volume fraction of polymer, and the other the fractional composition of non-solvent N (water) vs solvent S (NMP). The exchange of solvent and non-solvent is solved with a Fickian diffusion model, thus capturing the in-flux of the coagulation bath into the polymer solution. As a demonstration of the predictive capabilities of the model, the concentration of solvent (NMP) in the coagulation bath was varied to draw comparisons with experiments. Two- and three-dimensional simulations were carried out to evaluate the cross-sectional pore morphology and the top surface pore size for membranes formed by NIPS. Experiments involving handcast membranes of a similar system were performed for comparison with the simulations, and an agreement was found concerning the dependence of pore morphology on the composition of the coagulation bath.

Unraveling Molecular Interactions in Liquid-Liquid Phase Separation of Disordered Proteins by Atomistic Simulations.

Membraneless organelles are dynamical cellular condensates formed via biomolecular liquid-liquid phase separation of proteins and RNA molecules. Multiple evidence suggests that in several cases disordered proteins are structural scaffolds that drive the condensation by forming a dynamic network of inter- and intramolecular contacts. Despite the blooming research activity in this field, the structural characterization of these entities is very limited, and we still do not understand how the phase behavior is encoded in the amino acid sequences of the scaffolding proteins. Here we exploited explicit-solvent atomistic simulations to investigate the N-terminal disordered region of DEAD-box helicase 4 (NDDX4), which is a well-established model for phase separation. Notably, we determined NDDX4 conformational ensemble at the single-molecule level, and we relied on a "divide-and-conquer" strategy, based on simulations of various protein fragments at high concentration, to probe intermolecular interactions in conditions mimicking real condensates. Our results provide a high-resolution picture of the molecular mechanisms underlying phase separation in agreement with NMR and mutagenesis data and suggest that clusters of arginine and aromatic residues may stabilize the assembly of several condensates.

Organization of the Pluripotent Genome.

In the past several decades, the establishment of in vitro models of pluripotency has ushered in a golden era for developmental and stem cell biology. Research in this arena has led to profound insights into the regulatory features that shape early embryonic development. Nevertheless, an integrative theory of the epigenetic principles that govern the pluripotent nucleus remains elusive. Here, we summarize the epigenetic characteristics that define the pluripotent state. We cover what is currently known about the epigenome of pluripotent stem cells and reflect on the use of embryonic stem cells as an experimental system. In addition, we highlight insights from super-resolution microscopy, which have advanced our understanding of the form and function of chromatin, particularly its role in establishing the characteristically "open chromatin" of pluripotent nuclei. Further, we discuss the rapid improvements in 3C-based methods, which have given us a means to investigate the 3D spatial organization of the pluripotent genome. This has aided the adaptation of prior notions of a "pluripotent molecular circuitry" into a more holistic model, where hotspots of co-interacting domains correspond with the accumulation of pluripotency-associated factors. Finally, we relate these earlier hypotheses to an emerging model of phase separation, which posits that a biophysical mechanism may presuppose the formation of a pluripotent-state-defining transcriptional program.