Predicting phase behavior in multicomponent mixtures

Phase Separation in Complex Mixtures with Many Components: Analytical Expressions for Spinodal Manifolds

The phase behavior is investigated for systems composed of a large number of macromolecular components N, with N ≥ 2. Liquid-liquid phase separation is modeled using a virial expansion up to the second order of the concentrations of the components. Formal analytical expressions for the spinodal manifolds in N dimensions are derived, which simplify their calculation (by transforming the original problem into inequalities that can be evaluated numerically using linear programming techniques). In addition, a new expression is obtained to calculate the critical manifold and composition of the coexisting phases. The present analytical procedure complements previous attempts to handle spinodal decomposition for many components using a statistical approach based on random matrix theory. The results are relevant for predicting the effects of polydispersity on phase behavior in fields like polymer or food science and liquid-liquid phase separation in the cytosol of living cells.

Multi-scale molecular simulation of random peptide phase separation and its extended-to-compact structure transition driven by hydrophobic interactions

Intrinsically disordered proteins (IDPs) often undergo liquid-liquid phase separation (LLPS) and form membraneless organelles or protein condensates. One of the core problems is how do electrostatic repulsion and hydrophobic interactions in peptides regulate the phase separation process? To answer this question, this study uses random peptides composed of positively charged arginine (Arg, R) and hydrophobic isoleucine (Ile, I) as the model systems, and conduct large-scale simulations using all atom and coarse-grained model multi-scale simulation methods. In this article, we investigate the phase separation of different sequences using a coarse-grained model. It is found that the stronger the electrostatic repulsion in the system, the more extended the single-chain structure, and the more likely the system forms a low-density homogeneous phase. In contrast, the stronger the hydrophobic effect of the system, the more compact the single-chain structure, the easier phase separation, and the higher the critical temperature of phase separation. Overall, by taking the random polypeptides composed of two types of amino acid residues as model systems, this study discusses the relationship between the protein sequence and phase behaviour, and provides theoretical insights into the interactions within or between proteins. It is expected to provide essential physical information for the sequence design of functional IDPs, as well as data to support the diagnosis and treatment of the LLPS-associated diseases.

Composition Dependent Instabilities in Mixtures with Many Components

Understanding the phase behavior of mixtures with many components is important in many contexts, including as a key step toward a physics-based description of intracellular compartmentalization. Here, we study phase ordering instabilities in a paradigmatic model that represents the complexity of-e.g., biological-mixtures via random second virial coefficients. Using tools from free probability theory we obtain the exact spinodal curve and the nature of instabilities for a mixture with an arbitrary composition, thus lifting an important restriction in previous work. We show that, by controlling the concentration of only a few components, one can systematically change the nature of the spinodal instability and achieve demixing for realistic scenarios by a strong composition imbalance amplification. This results from a nontrivial interplay of interaction complexity and entropic effects due to the nonuniform composition. Our approach can be extended to include additional systematic interactions, leading to a competition between different forms of demixing as density is varied.

Theory and Simulation of Multiphase Coexistence in Biomolecular Mixtures

Biomolecular condensates constitute a newly recognized form of spatial organization in living cells. Although many condensates are believed to form as a result of phase separation, the physicochemical properties that determine the phase behavior of heterogeneous biomolecular mixtures are only beginning to be explored. Theory and simulation provide invaluable tools for probing the relationship between molecular determinants, such as protein and RNA sequences, and the emergence of phase-separated condensates in such complex environments. This review covers recent advances in the prediction and computational design of biomolecular mixtures that phase-separate into many coexisting phases. First, we review efforts to understand the phase behavior of mixtures with hundreds or thousands of species using theoretical models and statistical approaches. We then describe progress in developing analytical theories and coarse-grained simulation models to predict multiphase condensates with the molecular detail required to make contact with biophysical experiments. We conclude by summarizing the challenges ahead for modeling the inhomogeneous spatial organization of biomolecular mixtures in living cells.

Even Strong Energy Polydispersity Does Not Affect the Average Structure and Dynamics of Simple Liquids

Size-polydisperse liquids have become standard models for avoiding crystallization, thereby enabling studies of supercooled liquids and glasses formed, e.g., by colloidal systems. Purely energy-polydisperse liquids have been studied much less, but provide an interesting alternative. We here study numerically the difference in structure and dynamics obtained by introducing these two kinds of polydispersity into systems of particles interacting via the Lennard-Jones and EXP pair potentials. To a very good approximation, the average pair structure and dynamics are unchanged even for strong energy polydispersity, which is not the case for size-polydisperse systems. When the system at extreme energy polydispersity undergoes a continuous phase separation into lower and higher particle-energy regions whose structure and dynamics are different from the average, the average structure and dynamics are still virtually the same as for the monodisperse system. Our findings are consistent with the fact that the distribution of forces on the individual particles do not change when energy polydispersity is introduced, while they do change in the case of size polydispersity. A theoretical explanation remains to be found, however.

Thermodynamic origins of two-component multiphase condensates of proteins

Intracellular condensates are highly multi-component systems in which complex phase behaviour can ensue, including the formation of architectures comprising multiple immiscible condensed phases. Relying solely on physical intuition to manipulate such condensates is difficult because of the complexity of their composition, and systematically learning the underlying rules experimentally would be extremely costly. We address this challenge by developing a computational approach to design pairs of protein sequences that result in well-separated multilayered condensates and elucidate the molecular origins of these compartments. Our method couples a genetic algorithm to a residue-resolution coarse-grained protein model. We demonstrate that we can design protein partners to form multiphase condensates containing naturally occurring proteins, such as the low-complexity domain of hnRNPA1 and its mutants, and show how homo- and heterotypic interactions must differ between proteins to result in multiphasicity. We also show that in some cases the specific pattern of amino-acid residues plays an important role. Our findings have wide-ranging implications for understanding and controlling the organisation, functions and material properties of biomolecular condensates.

On kinetics and extreme values in systems with random interactions

Biological environments such as the cytoplasm are comprised of many different molecules, which makes explicit modeling intractable. In the spirit of Wigner, one may be tempted to assume interactions to derive from a random distribution. Via this approximation, the system can be efficiently treated in the mean-field, and general statements about expected behavior of such systems can be made. Here, I study systems of particles interacting via random potentials, outside of mean-field approximations. These systems exhibit a phase transition temperature, under which part of the components precipitate. The nature of this transition appears to be non-universal, and to depend intimately on the underlying distribution of interactions. Above the phase transition temperature, the system can be efficiently treated using a Bethe approximation, which shows a dependence on extreme value statistics. Relaxation timescales of this system tend to be slow, but can be made arbitrarily fast by increasing the number of neighbors of each particle.

Spontaneous phase separation of ternary fluid mixtures

We computationally study the spontaneous phase separation of ternary fluid mixtures using the lattice Boltzmann method both when all the surface tensions are equal and when they have different values. To rationalise the phase diagram of possible phase separation mechanisms, previous theoretical works typically rely on analysing the sign of the eigenvalues resulting from a simple linear stability analysis, but we find this does not explain the observed simulation results. Here, we classify the possible separation pathways into four basic mechanisms, and develop a phenomenological model that captures the composition regimes where each mechanism is prevalent. We further highlight that the dominant mechanism in ternary phase separation involves enrichment and instability of the minor component at the fluid-fluid interface, which is absent in the case of binary fluid mixtures.

Thermodynamic stability and critical points in multicomponent mixtures with structured interactions

Theoretical work has shed light on the phase behavior of idealized mixtures of many components with random interactions. However, typical mixtures interact through particular physical features, leading to a structured, nonrandom interaction matrix of lower rank. Here, we develop a theoretical framework for such mixtures and derive mean-field conditions for thermodynamic stability and critical behavior. Irrespective of the number of components and features, this framework allows for a generally lower-dimensional representation in the space of features and proposes a principled way to coarse-grain multicomponent mixtures as binary mixtures. Moreover, it suggests a way to systematically characterize different series of critical points and their codimensions in mean-field. Since every pairwise interaction matrix can be expressed in terms of features, our work is applicable to a broad class of mean-field models.

Evolved interactions stabilize many coexisting phases in multicomponent liquids

Phase separation has emerged as an essential concept for the spatial organization inside biological cells. However, despite the clear relevance to virtually all physiological functions, we understand surprisingly little about what phases form in a system of many interacting components, like in cells. Here we introduce a numerical method based on physical relaxation dynamics to study the coexisting phases in such systems. We use our approach to optimize interactions between components, similar to how evolution might have optimized the interactions of proteins. These evolved interactions robustly lead to a defined number of phases, despite substantial uncertainties in the initial composition, while random or designed interactions perform much worse. Moreover, the optimized interactions are robust to perturbations, and they allow fast adaption to new target phase counts. We thus show that genetically encoded interactions of proteins provide versatile control of phase behavior. The phases forming in our system are also a concrete example of a robust emergent property that does not rely on fine-tuning the parameters of individual constituents.

Instabilities of complex fluids with partially structured and partially random interactions

We develop a theory for thermodynamic instabilities of complex fluids composed of many interacting chemical species organised in families. This model includes partially structured and partially random interactions and can be solved exactly using tools from random matrix theory. The model exhibits three kinds of fluid instabilities: one in which the species form a condensate with a local density that depends on their family (family condensation); one in which species demix in two phases depending on their family (family demixing); and one in which species demix in a random manner irrespective of their family (random demixing). We determine the critical spinodal density of the three types of instabilities and find that the critical spinodal density is finite for both family condensation and family demixing, while for random demixing the critical spinodal density grows as the square root of the number of species. We use the developed framework to describe phase-separation instability of the cytoplasm induced by a change in pH.

Rich Phase Separation Behavior of Biomolecules

Phase separation is a thermodynamic process leading to the formation of compositionally distinct phases. For the past few years, numerous works have shown that biomolecular phase separation serves as biogenesis mechanisms of diverse intracellular condensates, and aberrant phase transitions are associated with disease states such as neurodegenerative diseases and cancers. Condensates exhibit rich phase behaviors including multiphase internal structuring, noise buffering, and compositional tunability. Recent studies have begun to uncover how a network of intermolecular interactions can give rise to various biophysical features of condensates. Here, we review phase behaviors of biomolecules, particularly with regard to regular solution models of binary and ternary mixtures. We discuss how these theoretical frameworks explain many aspects of the assembly, composition, and miscibility of diverse biomolecular phases, and highlight how a model-based approach can help elucidate the detailed thermodynamic principle for multicomponent intracellular phase separation.

Multiphase Coacervates Driven by Electrostatic Correlations

The liquid-liquid phase separation of a polyelectrolyte solution containing one type of negatively and two types of positively charged polymers with different charge densities is studied theoretically by random phase approximation (RPA). It is predicted that multicoacervate phases could coexist, driven purely by electrostatic correlations. The asymmetry of the linear charge density could induce an effective immiscibility between two positively charged polyelectrolytes, leading to the multiphase separation. Adding salt will induce the disappearance of the dilute phase, forming two coexisting complex phases, instead of fusion between coacervates. Raising temperature could either induce a two coexisting complex phase, or a dilute phase coexisting with a coacervate phase, depending on the bulk concentration. Our predictions are in good agreement with experiments and provide insights in the further designing of the multiphase coacervation system.

Dissecting the complexity of biomolecular condensates

Biomolecular condensates comprise a diverse and ubiquitous class of membraneless organelles. Condensate assembly is often described by liquid-liquid phase separation. While this process explains many key features, it cannot account for the compositional or architectural complexity that condensates display in cells. Recent work has begun to dissect the rich network of intermolecular interactions that give rise to biomolecular condensates. Here, we review the latest results from theory, simulations and experiments, and discuss what they reveal about the structure-function relationship of condensates.

Self-Assembly of Biomolecular Condensates with Shared Components

Biomolecular condensates self-assemble when proteins and nucleic acids spontaneously demix to form droplets within the crowded intracellular milieu. This simple mechanism underlies the formation of a wide variety of membraneless compartments in living cells. To understand how multiple condensates with distinct compositions can self-assemble in such a heterogeneous system, which may not be at thermodynamic equilibrium, we study a minimal model in which we can "program" the pairwise interactions among hundreds of species. We show that the number of distinct condensates that can be reliably assembled grows superlinearly with the number of species in the mixture when the condensates share components. Furthermore, we show that we can predict the maximum number of distinct condensates in a mixture without knowing the details of the pairwise interactions. Simulations of condensate growth confirm these predictions and suggest that the physical rules governing the achievable complexity of condensate-mediated spatial organization are broadly applicable to biomolecular mixtures.

Biological Phase Separation and Biomolecular Condensates in Plants

A surge in research focused on understanding the physical principles governing the formation, properties, and function of membraneless compartments has occurred over the past decade. Compartments such as the nucleolus, stress granules, and nuclear speckles have been designated as biomolecular condensates to describe their shared property of spatially concentrating biomolecules. Although this research has historically been carried out in animal and fungal systems, recent work has begun to explore whether these same principles are relevant in plants. Effectively understanding and studying biomolecular condensates require interdisciplinary expertise that spans cell biology, biochemistry, and condensed matter physics and biophysics. As such, some involved concepts may be unfamiliar to any given individual. This review focuses on introducing concepts essential to the study of biomolecular condensates and phase separation for biologists seeking to carry out research in this area and further examines aspects of biomolecular condensates that are relevant to plant systems.

Higher-order organization of biomolecular condensates

A guiding principle of biology is that biochemical reactions must be organized in space and time. One way this spatio-temporal organization is achieved is through liquid–liquid phase separation (LLPS), which generates biomolecular condensates. These condensates are dynamic and reactive, and often contain a complex mixture of proteins and nucleic acids. In this review, we discuss how underlying physical and chemical processes generate internal condensate architectures. We then outline the diverse condensate architectures that are observed in biological systems. Finally, we discuss how specific condensate organization is critical for specific biological functions.

Systems with Size and Energy Polydispersity: From Glasses to Mosaic Crystals

We use Langevin dynamics simulations to study dense 2d systems of particles with both size and energy polydispersity. We compare two types of bidisperse systems which differ in the correlation between particle size and interaction parameters: in one system big particles have high interaction parameters and small particles have low interaction parameters, while in the other system the situation is reversed. We study the different phases of the two systems and compare them to those of a system with size but not energy bidispersity. We show that, depending on the strength of interaction between big and small particles, cooling to low temperatures yields either homogeneous glasses or mosaic crystals. We find that systems with low mixing interaction, undergo partial freezing of one of the components at intermediate temperatures, and that while this phenomenon is energy-driven in both size and energy bidisperse systems, it is controlled by entropic effects in systems with size bidispersity only.

Effect of Liquid State Organization on Nanostructure and Strength of Model Multicomponent Solids

When a multicomponent liquid composed of particles with random interactions is slowly cooled below the freezing temperature, the fluid reorganizes in order to increase (decrease) the number of strong (weak) attractive interactions and solidifies into a structure composed of domains of strongly and of weakly interacting particles. Using Langevin dynamics simulations of a model system we find that the tensile strength, mode of fracture, and thermal stability of such solids differ from those of one-component solids and that these properties can be controlled by the method of preparation.

On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures

Most undergraduate students who have followed a thermodynamics course would have been asked to evaluate the volume occupied by one mole of air under standard conditions of pressure and temperature. However, what is this task exactly referring to? If air is to be regarded as a mixture, under what circumstances can this mixture be considered as comprising only one component called “air” in classical statistical mechanics? Furthermore, following the paradigmatic Gibbs’ mixing thought experiment, if one mixes air from a container with air from another container, all other things being equal, should there be a change in entropy? The present paper addresses these questions by developing a prior-based statistical mechanics framework to characterise binary mixtures’ composition realisations and their effect on thermodynamic free energies and entropies. It is found that (a) there exist circumstances for which an ideal binary mixture is thermodynamically equivalent to a single component ideal gas and (b) even when mixing two substances identical in their underlying composition, entropy increase does occur for finite size systems. The nature of the contributions to this increase is then discussed.

Identity ordering and metastable clusters in fluids with random interactions

We use Langevin dynamics simulations to study dense two-dimensional systems of particles where all binary interactions are different in the sense that each interaction parameter is characterized by a randomly chosen number. We compare two systems that differ by the probability distributions from which the interaction parameters are drawn: uniform (U) and exponential (E). Both systems undergo neighborhood identity ordering and form metastable clusters in the fluid phase near the liquid-solid transition, but the effects are much stronger in E than in U systems. Possible implications of our results for the control of the structure of multicomponent alloys are discussed.

Phase behavior and morphology of multicomponent liquid mixtures

Multicomponent systems are ubiquitous in nature and industry. While the physics of few-component liquid mixtures (i.e., binary and ternary ones) is well-understood and routinely taught in undergraduate courses, the thermodynamic and kinetic properties of N-component mixtures with N > 3 have remained relatively unexplored. An example of such a mixture is provided by the intracellular fluid, in which protein-rich droplets phase separate into distinct membraneless organelles. In this work, we investigate equilibrium phase behavior and morphology of N-component liquid mixtures within the Flory-Huggins theory of regular solutions. In order to determine the number of coexisting phases and their compositions, we developed a new algorithm for constructing complete phase diagrams, based on numerical convexification of the discretized free energy landscape. Together with a Cahn-Hilliard approach for kinetics, we employ this method to study mixtures with N = 4 and 5 components. We report on both the coarsening behavior of such systems, as well as the resulting morphologies in three spatial dimensions. We discuss how the number of coexisting phases and their compositions can be extracted with Principal Component Analysis (PCA) and K-means clustering algorithms. Finally, we discuss how one can reverse engineer the interaction parameters and volume fractions of components in order to achieve a range of desired packing structures, such as nested "Russian dolls" and encapsulated Janus droplets.

Physical principles of intracellular organization via active and passive phase transitions

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.

Composition, morphology, and growth of clusters in a gas of particles with random interactions

We use Langevin dynamics simulations to study the growth kinetics and the steady-state properties of condensed clusters in a dilute two-dimensional system of particles that are all different (APD) in the sense that each particle is characterized by a randomly chosen interaction parameter. The growth exponents, the transition temperatures, and the steady-state properties of the clusters and of the surrounding gas phase are obtained and compared with those of one-component systems. We investigate the fractionation phenomenon, i.e., how particles of different identities are distributed between the coexisting mother (gas) and daughter (clusters) phases. We study the local organization of particles inside clusters, according to their identity-neighbourhood identity ordering (NIO)-and compare the results with those of previous studies of NIO in dense APD systems.

Phase Transitions in Biological Systems with Many Components

Biological mixtures such as the cytosol may consist of thousands of distinct components. There is now a substantial body of evidence showing that, under physiological conditions, intracellular mixtures can phase separate into spatially distinct regions with differing compositions. In this article we present numerical evidence indicating that such spontaneous compartmentalization exploits general features of the phase diagram of a multicomponent biomolecular mixture. In particular, we show that demixed domains are likely to segregate when the variance in the intermolecular interaction strengths exceeds a well-defined threshold. Multiple distinct phases are likely to become stable under very similar conditions, which can then be tuned to achieve multiphase coexistence. As a result, only minor adjustments to the composition of the cytosol or the strengths of the intermolecular interactions are needed to regulate the formation of different domains with specific compositions, implying that phase separation is a robust mechanism for creating spatial organization. We further predict that this functionality is only weakly affected by increasing the number of components in the system. Our model therefore suggests that, for purely physico-chemical reasons, biological mixtures are naturally poised to undergo a small number of demixing phase transitions.

Particle dynamics in fluids with random interactions

We study the dynamics of particles in a multi-component 2d Lennard-Jones (LJ) fluid in the limiting case where all the particles are different (APD). The equilibrium properties of this APD system were studied in our earlier work [L. S. Shagolsem et al., J. Chem. Phys. 142, 051104 (2015).]. We use molecular dynamics simulations to investigate the statistical properties of particle trajectories in a temperature range covering both the fluid and the solid-fluid coexistence region. We calculate the mean-square displacement as well as displacement, angle, and waiting time distributions, and compare the results with those for one-component LJ fluid. As temperature is lowered, the dynamics of the APD system becomes increasingly complex, as the intrinsic difference between the particles is amplified by neighborhood identity ordering and by the inhomogeneous character of the solid-fluid coexistence region. The ramifications of our results for the analysis of protein tracking experiments in living cells are discussed.

Multicomponent adsorption in mesoporous flexible materials with flat-histogram Monte Carlo methods

We demonstrate an extensible flat-histogram Monte Carlo simulation methodology for studying the adsorption of multicomponent fluids in flexible porous solids. This methodology allows us to easily obtain the complete free energy landscape for the confined fluid-solid system in equilibrium with a bulk fluid of any arbitrary composition. We use this approach to study the adsorption of a prototypical coarse-grained binary fluid in "Hookean" solids, where the free energy of the solid may be described as a simple spring. However, our approach is fully extensible to solids with arbitrarily complex free energy profiles. We demonstrate that by tuning the fluid-solid interaction ranges, the inhomogeneous fluid structure inside the pore can give rise to enhanced selective capture of a larger species through cooperative adsorption with a smaller one. The maximum enhancement in selectivity is observed at low to intermediate pressures and is especially pronounced when the larger species is very dilute in the bulk. This suggest a mechanism by which the selective capture of a minor component from a bulk fluid may be enhanced.

Effect of Energy Polydispersity on the Nature of Lennard-Jones Liquids

In the companion paper [ Ingebrigtsen , T. S. ; Tanaka , H. J. Phys. Chem. B 2015 , 119 , 11052 ] the effect of size polydispersity on the nature of Lennard-Jones (LJ) liquids, which represent most molecular liquids without hydrogen bonds, was studied. More specifically, it was shown that even highly size polydisperse LJ liquids are Roskilde-simple (RS) liquids. RS liquids are liquids with strong correlation between constant volume equilibrium fluctuations of virial and potential energy and are simpler than other types of liquids. Moreover, it was shown that size polydisperse LJ liquids have isomorphs to a good approximation. Isomorphs are curves in the phase diagram of RS liquids along which structure, dynamics, and some thermodynamic quantities are invariant in dimensionless (reduced) units. In this paper, we study the effect of energy polydispersity on the nature of LJ liquids. We show that energy polydisperse LJ liquids are RS liquids. However, a tendency of particle segregation, which increases with the degree of polydispersity, leads to a loss of strong virial-potential energy correlation but is mitigated by increasing temperature and/or density. Isomorphs are a good approximation also for energy polydisperse LJ liquids, although particle-resolved quantities display a somewhat poorer scaling compared to the mean quantities along the isomorph.

Oligomers of Heat-Shock Proteins: Structures That Don't Imply Function

Most proteins must remain soluble in the cytosol in order to perform their biological functions. To protect against undesired protein aggregation, living cells maintain a population of molecular chaperones that ensure the solubility of the proteome. Here we report simulations of a lattice model of interacting proteins to understand how low concentrations of passive molecular chaperones, such as small heat-shock proteins, suppress thermodynamic instabilities in protein solutions. Given fixed concentrations of chaperones and client proteins, the solubility of the proteome can be increased by tuning the chaperone-client binding strength. Surprisingly, we find that the binding strength that optimizes solubility while preventing irreversible chaperone binding also promotes the formation of weakly bound chaperone oligomers, although the presence of these oligomers does not significantly affect the thermodynamic stability of the solution. Such oligomers are commonly observed in experiments on small heat-shock proteins, but their connection to the biological function of these chaperones has remained unclear. Our simulations suggest that this clustering may not have any essential biological function, but rather emerges as a natural side-effect of optimizing the thermodynamic stability of the proteome.

Design strategies for self-assembly of discrete targets

Both biological and artificial self-assembly processes can take place by a range of different schemes, from the successive addition of identical building blocks to hierarchical sequences of intermediates, all the way to the fully addressable limit in which each component is unique. In this paper, we introduce an idealized model of cubic particles with patterned faces that allows self-assembly strategies to be compared and tested. We consider a simple octameric target, starting with the minimal requirements for successful self-assembly and comparing the benefits and limitations of more sophisticated hierarchical and addressable schemes. Simulations are performed using a hybrid dynamical Monte Carlo protocol that allows self-assembling clusters to rearrange internally while still providing Stokes-Einstein-like diffusion of aggregates of different sizes. Our simulations explicitly capture the thermodynamic, dynamic, and steric challenges typically faced by self-assembly processes, including competition between multiple partially completed structures. Self-assembly pathways are extracted from the simulation trajectories by a fully extendable scheme for identifying structural fragments, which are then assembled into history diagrams for successfully completed target structures. For the simple target, a one-component assembly scheme is most efficient and robust overall, but hierarchical and addressable strategies can have an advantage under some conditions if high yield is a priority.

Effect of Size Polydispersity on the Nature of Lennard-Jones Liquids

Polydisperse fluids are encountered everywhere in biological and industrial processes. These fluids naturally show a rich phenomenology exhibiting fractionation and shifts in critical point and freezing temperatures. We study here the effect of size polydispersity on the basic nature of Lennard-Jones (LJ) liquids, which represent most molecular liquids without hydrogen bonds, via two- and three-dimensional molecular dynamics computer simulations. A single-component liquid constituting spherical particles and interacting via the LJ potential is known to exhibit strong correlations between virial and potential energy equilibrium fluctuations at constant volume. This correlation significantly simplifies the physical description of the liquid, and these liquids are now known as Roskilde-simple (RS) liquids. We show that this simple nature of the single-component LJ liquid is preserved even for very high polydispersities (above 40% polydispersity for the studied uniform distribution). We also investigate isomorphs of moderately polydisperse LJ liquids. Isomorphs are curves in the phase diagram of RS liquids along which structure, dynamics, and some thermodynamic quantities are invariant in dimensionless units. We find that isomorphs are a good approximation even for polydisperse LJ liquids. The theory of isomorphs thus extends readily to size polydisperse fluids and can be used to improve even further the understanding of these intriguing systems.

Phase separation in solutions with specific and nonspecific interactions

Protein solutions, which tend to be thermodynamically stable under physiological conditions, can demix into protein-enriched and protein-depleted phases when stressed. Using a lattice-gas model of proteins with both isotropic and specific, directional interactions, we calculate the critical conditions for phase separation for model proteins with up to four patches via Monte Carlo simulations and statistical associating fluid theory. Given a fixed specific interaction strength, the critical value of the isotropic energy, which accounts for dispersion forces and nonspecific interactions, measures the stability of the solution with respect to nonspecific interactions. Phase separation is suppressed by the formation of protein complexes, which effectively passivate the strongly associating sites on the monomers. Nevertheless, we find that protein models with three or more patches can form extended aggregates that phase separate despite the assembly of passivated complexes, even in the absence of nonspecific interactions. We present a unified view of the critical behavior of model fluids with anisotropic interactions, and we discuss the implications of these results for the thermodynamic stability of protein solutions.