Thermodynamics of Associative Polymer Blends

Theory of thermoreversible gelation and anomalous concentration fluctuations in polyzwitterion solutions

We present a theoretical framework to investigate thermoreversible phase transitions within polyzwitterion systems, encompassing macrophase separations (MPS) and gelation. In addition, we explore concentration fluctuations near critical points associated with MPS, as well as tricritical and bicritical points at the intersection of MPS and gelation. By utilizing mean-field percolation theory and field theory formalism, we derive the Landau free energy in terms of polyzwitterion concentration with fixed dipole strengths and other experimental variables, such as temperatures and salt concentrations. As the temperature decreases, the dipoles can form cross-links, resulting in polyzwitterion associations. The associations can grow to a gel network and enhance the propensity for MPS, including liquid-liquid, liquid-gel, and gel-gel phase separations. Remarkably, the associations also impact critical behaviors. Using the renormalization group technique, we find that the critical exponents of the polyzwitterion concentration correlation functions significantly deviate from those in the Ising universality class due to the presence of polyzwitterion associations, leading to crossover critical behaviors.

Preserving condensate structure and composition by lowering sequence complexity

Biomolecular condensates play a vital role in organizing cellular chemistry. They selectively partition biomolecules, preventing unwanted cross talk and buffering against chemical noise. Intrinsically disordered proteins (IDPs) serve as primary components of these condensates due to their flexibility and ability to engage in multivalent interactions, leading to spontaneous aggregation. Theoretical advancements are critical at connecting IDP sequences with condensate emergent properties to establish the so-called molecular grammar. We proposed an extension to the stickers and spacers model, incorporating heterogeneous, nonspecific pairwise interactions between spacers alongside specific interactions among stickers. Our investigation revealed that although spacer interactions contribute to phase separation and co-condensation, their nonspecific nature leads to disorganized condensates. Specific sticker-sticker interactions drive the formation of condensates with well-defined networked structures and molecular composition. We discussed how evolutionary pressures might emerge to affect these interactions, leading to the prevalence of low-complexity domains in IDP sequences. These domains suppress spurious interactions and facilitate the formation of biologically meaningful condensates.

Modeling the competition between phase separation and polymerization under explicit polydispersity

The dynamics of phase separation for polymer blends is important in determining the final morphology and properties of polymer materials; in practical applications, this phase separation can be controlled by coupling to polymerization reaction kinetics via a process called 'polymerization-induced phase separation'. We develop a phase-field model for a polymer melt blend using a polymerizing Cahn-Hilliard (pCH) formalism to understand the fundamental processes underlying phase separation behavior of a mixture of two species independently undergoing linear step-growth polymerization. In our method, we explicitly model polydispersity in these systems to consider different molecular-weight components that will diffuse at different rates. We first show that this pCH model predicts results consistent with the Carothers predictions for step-growth polymerization kinetics, the Flory-Huggins theory of polymer mixing, and the classical predictions of spinodal decomposition in symmetric polymer blends. The model is then used to characterize (i) the competition between phase separation dynamics and polymerization kinetics, and (ii) the effect of unequal reaction rates between species. For large incompatibility between the species (i.e. high χ), our pCH model demonstrates that the strength for phase separation directly corresponds to the kinetics of phase separation. We find that increasing the reaction rate k̃, first induces faster phase separation but this trend reverses as we further increase k̃ due to the competition between molecular diffusion and polymerization. In this case, phase separation is delayed for faster polymerization rates due to the rapid accumulation of slow-moving, high molecular weight components.

Determination of Pair Interaction Parameters of Multicomponent Polymer Systems

From the examples of three and four-component polymer-polymer systems characterized by amorphous separation, an original technique for determining the pair parameters of interaction between components based on the sorption isotherms of common solvent vapor, particularly water vapor, has been developed. The possibility of calculating thermodynamic characteristics of multicomponent polymer compositions with specific interactions of functional groups from experimentally obtained sorption isotherms is shown. An algorithm for calculating pair interaction parameters, estimating concentration dependences of chemical potential and Gibbs free energy of mixing, and predicting the phase state of polymer mixtures was presented for the first time for such systems. The technique was tested on the example of systems poly(N-vinylpyrrolidone) (PNVP)-polyethylene glycol (PEG), PNVP-PEG-Poly(acrylic acid) (PAA), poly(N-vinylcaprolactam) (PNVCL)-PEG, and polyvinyl alcohol (PVA)-PEG.

Preserving condensate structure and composition by lowering sequence complexity

Biological condensates play a vital role in organizing cellular chemistry. They selectively partition biomolecules, preventing unwanted cross-talk and buffering against chemical noise. Intrinsically disordered proteins (IDPs) serve as primary components of these condensates due to their flexibility and ability to engage in multivalent, non-specific interactions, leading to spontaneous aggregation. Theoretical advancements are critical at connecting IDP sequences with condensate emergent properties to establish the so-called molecular grammar. We proposed an extension to the stickers and spacers model, incorporating non-specific pairwise interactions between spacers alongside specific interactions among stickers. Our investigation revealed that while spacer interactions contribute to phase separation and co-condensation, their non-specific nature leads to disorganized condensates. Specific sticker-sticker interactions drive the formation of condensates with well-defined structures and molecular composition. We discussed how evolutionary pressures might emerge to affect these interactions, leading to the prevalence of low complexity domains in IDP sequences. These domains suppress spurious interactions and facilitate the formation of biologically meaningful condensates.

Condensation Goes Viral: A Polymer Physics Perspective

The past decade has seen a revolution in our understanding of how the cellular environment is organized, where an incredible body of work has provided new insights into the role played by membraneless organelles. These rapid advancements have been made possible by an increasing awareness of the peculiar physical properties that give rise to such bodies and the complex biology that enables their function. Viral infections are not extraneous to this. Indeed, in host cells, viruses can harness existing membraneless compartments or, even, induce the formation of new ones. By hijacking the cellular machinery, these intracellular bodies can assist in the replication, assembly, and packaging of the viral genome as well as in the escape of the cellular immune response. Here, we provide a perspective on the fundamental polymer physics concepts that may help connect and interpret the different observed phenomena, ranging from the condensation of viral genomes to the phase separation of multicomponent solutions. We complement the discussion of the physical basis with a description of biophysical methods that can provide quantitative insights for testing and developing theoretical and computational models.

Hydrogen Bond Network Disruption by Hydration Layers in Water Solutions with Salt and Hydrogen-Bonding Polymers (PEO)

A mean field theory model describing the interaction of ion hydration layers with the network of hydrogen bonds of both water and the nonionic polymer poly(ethylene oxide) (PEO) is presented. The predictions of the model for types and statistics of hydrogen bonds, the number of water molecules bound to PEO, or their dependence on temperature are successfully verified from all-atom simulations at different NaCl and PEO concentrations. Furthermore, our simulations show that the binding of cations to PEO increases monotonically with salt concentration, in agreement with recent experimental results, through a mechanism in which the sum of the number of bound water and cations is independent of salt concentration. The model introduced is general and can describe any salt or hydrogen-bond-forming polymer.

Dynamics of protein condensates in weak-binding regime

Weak complementary interactions between proteins and nucleic acids are the main driving forces of intracellular liquid-liquid phase separation. The sticker-spacer model has emerged as a unifying principle for understanding the phase behavior of these multivalent molecules. It remains elusive how specific interactions mediated by stickers contribute to the rheological properties of the liquid condensates. Previous studies have revealed that for strong binding strength ɛ_{b}, the bulk diffusivity D depends on the effective bond lifetime τ, viz., D∝τ^{-1}. Consequently, equal concentrations of the complementary stickers induce a slow down in the dynamics of the condensates D∝e^{-1.5ɛ_{b}}. However, for weak-binding strength, it is expected that the resulting condensates are dynamic, loose network liquids rather than kinetically arrested, compact clusters. We develop a mean-field theory using the thermodynamics of the associative polymers and perform molecular-dynamics simulations based on the sticker-spacer model to study the controlling factors in the structure and dynamics of such condensates in the weak-binding regime. Through scaling analysis, we delineate how the free sticker fraction W_{f} and the bulk diffusivity D decrease with increasing binding energy and find that the internal dynamics of such network liquids are controlled by the free sticker fraction D∝W_{f}∝e^{-0.5ɛ_{b}} rather than the effective bond lifetime. Referred to as the free-sticker-dominated diffusivity, the microscopic slowdown due to a gradual loss of the free stickers affects the viscosity of the condensates as well, with the scaling of the zero-shear viscosity η∝e^{0.5ɛ_{b}}. Therefore, the way of controlling the structure, diffusivity, and viscosity of the condensates through the binding energy can be tested experimentally.

Inclusion/Exclusion Behaviors of Small Molecules during Crystallization of Polymers in Miscible PLLA/TAIC Blend

In this work, PLLA/TAIC has been taken as a model system to investigate the inclusion and exclusion of small molecules during the crystallization of polymers in their miscible blend. Our results indicate that it is the growth rate and diameter of PLLA spherulites that dominate the localization of TAIC. On the one hand, crystallization temperature plays an important role. Crystallization at higher temperature corresponds to higher growth rates and a greater diameter of PLLA spherulites. The former improves the ability of PLLA crystals to trap TAIC while the latter leads to a lower volume fraction of space among neighboring PLLA spherulites. The combination of the two contributes to the enhanced inclusion behaviors. On the other hand, when compared to melt crystallization, cold crystallization results in much smaller spherulites (from higher nucleation density) and sufficient space among spherulites, which accounts for the enrichment of TAIC in interspherulitic regions and for its enhanced exclusion. In the adopted polymer–small molecule blend, TAIC can enrich in interspherulitic regions even in its miscible blend with PLLA, which can be attributed to its stronger diffusion ability.

Role of Solvent Compatibility in the Phase Behavior of Binary Solutions of Weakly Associating Multivalent Polymers

Condensate formation of biopolymer solutions, prominently those of various intrinsically disordered proteins (IDPs), is often driven by "sticky" interactions between associating residues, multivalently present along the polymer backbone. Using a ternary mean-field "stickers-and-spacers" model, we demonstrate that if sticker association is of the order of a few times the thermal energy, a delicate balance between specific binding and nonspecific polymer-solvent interactions gives rise to a particularly rich ternary phase behavior under physiological circumstances. For a generic system represented by a solution comprising multiassociative scaffold and client polymers, the difference in solvent compatibility between the polymers modulates the nature of isothermal liquid-liquid phase separation (LLPS) between associative and segregative. The calculations reveal regimes of dualistic phase behavior, where both types of LLPS occur within the same phase diagram, either associated with the presence of multiple miscibility gaps or a flip in the slope of the tie-lines belonging to a single coexistence region.

Current Understanding of Molecular Phase Separation in Chromosomes

Biomolecular phase separation denotes the demixing of a specific set of intracellular components without membrane encapsulation. Recent studies have found that biomolecular phase separation is involved in a wide range of cellular processes. In particular, phase separation is involved in the formation and regulation of chromosome structures at various levels. Here, we review the current understanding of biomolecular phase separation related to chromosomes. First, we discuss the fundamental principles of phase separation and introduce several examples of nuclear/chromosomal biomolecular assemblies formed by phase separation. We also briefly explain the experimental and computational methods used to study phase separation in chromosomes. Finally, we discuss a recent phase separation model, termed bridging-induced phase separation (BIPS), which can explain the formation of local chromosome structures.

Polyphasic linkage and the impact of ligand binding on the regulation of biomolecular condensates

Cellular matter can be spatially and temporally organized into membraneless biomolecular condensates. The current thinking is that these condensates form and dissolve via phase transitions driven by one or more condensate-specific multivalent macromolecules known as scaffolds. Cells likely regulate condensate formation and dissolution by exerting control over the concentrations of regulatory molecules, which we refer to as ligands. Wyman and Gill introduced the framework of polyphasic linkage to explain how ligands can exert thermodynamic control over phase transitions. This review focuses on describing the concepts of polyphasic linkage and the relevance of such a mechanism for controlling condensate formation and dissolution. We describe how ligand-mediated control over scaffold phase behavior can be quantified experimentally. Further, we build on recent studies to highlight features of ligands that make them suppressors vs drivers of phase separation. Finally, we highlight areas where advances are needed to further understand ligand-mediated control of condensates in complex cellular environments. These advances include understanding the effects of networks of ligands on condensate behavior and how ligands modulate phase transitions controlled by different combinations of homotypic and heterotypic interactions among scaffold macromolecules. Insights gained from the application of polyphasic linkage concepts should be useful for designing novel pharmaceutical ligands to regulate condensates.

Decoding the physical principles of two-component biomolecular phase separation

Cells possess a multiplicity of non-membrane-bound compartments, which form via liquid-liquid phase separation. These condensates assemble and dissolve as needed to enable central cellular functions. One important class of condensates is those composed of two associating polymer species that form one-to-one specific bonds. What are the physical principles that underlie phase separation in such systems? To address this question, we employed coarse-grained molecular dynamics simulations to examine how the phase boundaries depend on polymer valence, stoichiometry, and binding strength. We discovered a striking phenomenon – for sufficiently strong binding, phase separation is suppressed at rational polymer stoichiometries, which we termed the magic-ratio effect. We further developed an analytical dimer-gel theory that confirmed the magic-ratio effect and disentangled the individual roles of polymer properties in shaping the phase diagram. Our work provides new insights into the factors controlling the phase diagrams of biomolecular condensates, with implications for natural and synthetic systems.

Generalized models for bond percolation transitions of associative polymers

Polymers with stickers-and-spacers architectures can drive phase-separation-aided bond percolation transitions. Here, we present a generalized mean-field model to enable the calculation of bond percolation thresholds for polymers with multiple types of stickers. Further, using graph-based Monte Carlo simulations we demonstrate how cooperativity in bond formation can give rise to reentrant phase behavior. When combined with recent advances for modeling phase separation, our approaches for calculating percolation lines could be useful for modeling hardening transitions for multivalent proteins.

Physical Principles Underlying the Complex Biology of Intracellular Phase Transitions

Many biomolecular condensates appear to form via spontaneous or driven processes that have the hallmarks of intracellular phase transitions. This suggests that a common underlying physical framework might govern the formation of functionally and compositionally unrelated biomolecular condensates. In this review, we summarize recent work that leverages a stickers-and-spacers framework adapted from the field of associative polymers for understanding how multivalent protein and RNA molecules drive phase transitions that give rise to biomolecular condensates. We discuss how the valence of stickers impacts the driving forces for condensate formation and elaborate on how stickers can be distinguished from spacers in different contexts. We touch on the impact of sticker- and spacer-mediated interactions on the rheological properties of condensates and show how the model can be mapped to known drivers of different types of biomolecular condensates.

Simulation Study of Process-Controlled Supramolecular Block Copolymer Phase Separation with Reversible Reaction Algorithm

A supramolecular diblock copolymer formed by reversible bonds between the two blocks shows a rich microphase separation behavior and has great application potential in stimuli-responsive materials. We propose a novel method to describe supramolecular reactions in dissipative particle dynamics, which includes a reversible reaction to accurately reproduce the strength, saturation, and dynamic properties of the reversible bonds in the simulations. The thermodynamic properties and dynamic processes of the supramolecular diblock copolymer melts in both equilibrium and non-equilibrium states were studied using this method. The simulation results show that the method can faithfully characterize phase behaviors and dynamic properties of supramolecular diblock copolymer melts, especially in a non-equilibrium state, which provides a novel tool to unveil self-assembly mechanism and describe the properties of supramolecular block copolymers.

Amphiphilic Janus nanosheets by grafting reactive rubber brushes for reinforced rubber materials

An amphiphilic Janus nanosheet with different reactive rubber brushes on two opposite sides can simultaneously strengthen and toughen rubber blends.

Studying polymer solutions with particle-based models linked to classical density functionals: co-non-solvency

We demonstrate the potential of hybrid particle-based models, where interactions are introduced through functionals of local order parameters, in describing multicomponent polymer solutions. The link to a free-energy-like functional is advantageous for controlling the thermodynamics of the model. We focus on co-non-solvency - the collapse of polymer chains in dilute mixtures with two miscible good solvents, having different affinities towards the polymer. We employ a simple model where polymers and solvents are represented, respectively, by worm-like chains and single particles. Non-bonded interactions are captured by a polynomial which is third order in local densities and can, therefore, describe liquid-vapour coexistence. The parameterisation of the functional benefits from an elementary mean-field approximation to the statistical mechanics of the model. The model provides a framework for Monte Carlo simulations using a particle-to-mesh algorithm. Studies with conventional generic bead-spring and all-atom models have demonstrated that co-non-solvency is caused by preferential binding of the better solvent (termed cosolvent) with polymer. Hence, segmental loops bridged by cosolvent molecules are formed, initiating polymer collapse. The mesoscopic hybrid model differs conceptually from the conventional microscopic descriptions. Yet, it reproduces the same co-non-solvency mechanism supporting its universality. Films of adsorbed ternary solutions, showing co-non-solvency in the dilute regime, are considered at high concentrations. In this case, chains do not collapse. The properties of loops and tails of the adsorbed polymer agree with early theoretical predictions obtained for concentrated binary solutions.