Theory and Simulation of Multiphase Coexistence in Biomolecular Mixtures

TIDGN: A Transfer Learning Framework for Predicting Interactions of Intrinsically Disordered Proteins with High Conformational Dynamics

Interactions between intrinsically disordered proteins (IDPs) are crucial for biological processes, such as intracellular liquid-liquid phase separation (LLPS). Experiments (e.g., NMR) and simulations used to study IDP interactions encounter a variety of difficulties, highlighting the necessity to develop relevant machine learning methods. However, reliable machine learning methods face the challenge resulting from the scarcity of available training data. In this work, we propose a transfer learning-based invariant geometric dynamic graph model, named TIDGN, for predicting IDP interactions. The model consists of a pretraining task module and a downstream task module. The pretraining task module learns the dynamic structural encoding of IDP monomers, which is then used by the downstream task module for interaction site prediction. The IDP monomer structure data set and the IDP interaction event data set are constructed using all-atom molecular dynamics (MD) simulations. The transfer learning strategy effectively enhances the model's performance. Both homotypic interactions and heterotypic interactions between two IDPs are considered in this work. Interestingly, TIDGN performs well for the heterotypic interaction prediction. Additionally, the feature ablation analysis emphasizes the importance of invariant geometric graph features. Taken together, our work demonstrates that the integration of transfer learning and the invariant geometric graph network offers a promising approach for addressing data scarcity challenges of IDP interaction prediction.

Active phase separation: new phenomenology from non-equilibrium physics

In active systems, whose constituents have non-equilibrium dynamics at local level, fluid-fluid phase separation is widely observed. Examples include the formation of membraneless organelles within cells; the clustering of self-propelled colloidal particles in the absence of attractive forces, and some types of ecological segregation. A schematic understanding of such active phase separation was initially borrowed from what is known for the equilibrium case, in which detailed balance holds at microscopic level. However it has recently become clear that in active systems the absence of detailed balance, although it leave phase separation qualitatively unchanged in some regimes (for example domain growth driven by interfacial tension via Ostwald ripening), can in other regimes radically alter its phenomenology at mechanistic level. For example, microphase separation can be caused by reverse Ostwald ripening, a process that is hard to imagine from an equilibrium perspective. This and other new phenomena arise because, instead of having a single, positive interfacial tension like their equilibrium counterparts, the fluid-fluid interfaces created by active phase separation can have several distinct interfacial tensions governing different properties, some of which can be negative. These phenomena can be broadly understood by studying continuum field theories for a single conserved scalar order parameter (the fluid density), supplemented with a velocity field in cases where momentum conservation is also present. More complex regimes arise in systems described by multiple scalar order parameters (especially with nonreciprocal interactions between these); or when an order parameter undergoes both conserved and non-conserved dynamics (such that the combination breaks detailed balance); or in systems that support orientational long-range order in one or more of the coexisting phases. In this Review, we survey recent progress in understanding the specific role of activity in phase separation, drawing attention to many open questions. We focus primarily on continuum theories, especially those with a single scalar order parameter, reviewing both analytical and numerical work. We compare their predictions with particle-based models, which have mostly been studied numerically although a few have been explicitly coarse-grained to continuum level. We also compare, where possible, with experimental results. In the latter case, qualitative comparisons are broadly encouraging whereas quantitative ones are hindered by the dynamical complexity of most experimental systems relative that of simplified (particle-level or continuum) models of active matter.

Disordered Regions of Condensate-promoting Proteins Have Distinct Molecular Signatures Associated with Cellular Function

Disordered regions of proteins play crucial roles in cellular functions through diverse mechanisms. Some disordered regions function by promoting the formation of biomolecular condensates through dynamic multivalent interactions. While many have assumed that interactions among these condensate-promoting disordered regions are non-specific, recent studies have shown that distinct sequence compositions and patterning lead to specific condensate compositions associated with cellular function. Despite in-depth characterization of several key examples, the full chemical diversity of condensate-promoting disordered regions has not been surveyed. Here, we define a list of disordered regions of condensate-promoting proteins to survey the relationship between sequence and function. We find that these disordered regions show amino acid biases associated with different cellular functions. These amino acid biases are evolutionarily conserved in the absence of positional sequence conservation. Overall, our analysis highlights the relationship between sequence features and function for condensate-promoting disordered regions. This analysis suggests that molecular signatures encoded within disordered regions could impart functional specificity.

Rethinking chromatin accessibility: from compaction to dynamic interactions

The genome is traditionally divided into condensed heterochromatin and open euchromatin. However, recent findings challenge this binary classification and the notion that chromatin condensation solely governs the accessibility of transcription factors (TFs) and, consequently, gene expression. Instead, chromatin accessibility is emerging as a factor-specific property that is influenced by multiple determinants. These include the mobility of the chromatin fiber, the capacity of TFs to engage repeatedly with it through multivalent interactions, and the four-dimensional organization of its surrounding diffusible space. Unraveling the molecular and biophysical principles that render a genomic target truly accessible remains a significant challenge, but innovative methods for locally perturbing chromatin, coupled with microscopy techniques that offer single-molecule sensitivity, provide an exciting experimental playground to test new hypotheses.

Sequence determinants of protein phase separation and recognition by protein phase-separated condensates through molecular dynamics and active learning

Elucidating how protein sequence determines the properties of disordered proteins and their phase-separated condensates is a great challenge in computational chemistry, biology, and biophysics. Quantitative molecular dynamics simulations and derived free energy values can in principle capture how a sequence encodes the chemical and biological properties of a protein. These calculations are, however, computationally demanding, even after reducing the representation by coarse-graining; exploring the large spaces of potentially relevant sequences remains a formidable task. We employ an "active learning" scheme introduced by Yang et al. (bioRxiv, 2022, https://doi.org/10.1101/2022.08.05.502972) to reduce the number of labelled examples needed from simulations, where a neural network-based model suggests the most useful examples for the next training cycle. Applying this Bayesian optimisation framework, we determine properties of protein sequences with coarse-grained molecular dynamics, which enables the network to establish sequence-property relationships for disordered proteins and their self-interactions and their interactions in phase-separated condensates. We show how iterative training with second virial coefficients derived from the simulations of disordered protein sequences leads to a rapid improvement in predicting peptide self-interactions. We employ this Bayesian approach to efficiently search for new sequences that bind to condensates of the disordered C-terminal domain (CTD) of RNA Polymerase II, by simulating molecular recognition of peptides to phase-separated condensates in coarse-grained molecular dynamics. By searching for protein sequences which prefer to self-interact rather than interact with another protein sequence we are able to shape the morphology of protein condensates and design multiphasic protein condensates.

The human RAD52 complex undergoes phase separation and facilitates bundling and end-to-end tethering of RAD51 presynaptic filaments

Human RAD52 is a prime target for synthetical lethality approaches to treat cancers with deficiency in homologous recombination. Among multiple cellular roles of RAD52, its functions in homologous recombination repair and protection of stalled replication forks appear to substitute those of the tumor suppressor protein BRCA2. However, the mechanistic details of how RAD52 can substitute BRCA2 functions are only beginning to emerge. RAD52 forms an undecameric ring that is enveloped by eleven ~200 residue-long disordered regions, making it a highly multivalent and branched protein complex that potentiates supramolecular assembly. Here, we show that RAD52 exhibits homotypic phase separation capacity, and its condensates recruit key players in homologous recombination such as single-stranded (ss)DNA, RPA, and the RAD51 recombinase. Moreover, we show that RAD52 phase separation is regulated by its interaction partners such as ssDNA and RPA. Using fluorescence microscopy, we show that RAD52 can induce the formation of RAD51-ssDNA fibrillar structures. To probe the fine structure of these fibrils, we utilized single-molecule super-resolution imaging via DNA-PAINT and atomic force microscopy and showed that RAD51 fibrils are bundles of individual RAD51 nucleoprotein filaments. We further show that RAD52 induces end-to-end tethering of RAD51 nucleoprotein filaments. Overall, we demonstrate unique macromolecular organizational features of RAD52 that may underlie its various functions in the cell.

Accelerated Simulations Reveal Physicochemical Factors Governing Stability and Composition of RNA Clusters

Under certain conditions, RNA repeat sequences phase separate, yielding protein-free biomolecular condensates. Importantly, RNA repeat sequences have also been implicated in neurological disorders, such as Huntington's disease. Thus, mapping repeat sequences to their phase behavior, functions, and dysfunctions is an active area of research. However, despite several advances, it remains challenging to characterize the RNA phase behavior at a submolecular resolution. Here, we have implemented a residue-resolution coarse-grained model in LAMMPS─that incorporates both the RNA sequence and structure─to study the clustering propensities of protein-free RNA systems. Importantly, we achieve a multifold speedup in the simulation time compared to previous work. Leveraging this efficiency, we study the clustering propensity of all 20 nonredundant trinucleotide repeat sequences. Our results align with findings from experiments, emphasizing that canonical base-pairing and G-U wobble pairs play dominant roles in regulating cluster formation of RNA repeat sequences. Strikingly, we find strong entropic contributions to the stability and composition of RNA clusters, which is demonstrated for single-component RNA systems as well as binary mixtures of trinucleotide repeats. Additionally, we investigate the clustering behaviors of trinucleotide (odd) repeats and their quadranucleotide (even) counterparts. We observe that odd repeats exhibit stronger clustering tendencies, attributed to the presence of consecutive base pairs in their sequences that are disrupted in even repeat sequences. Altogether, our work extends the set of computational tools for probing RNA cluster formation at submolecular resolution and uncovers physicochemical principles that govern the stability and composition of the resulting clusters.

Reentrant Condensation of Polyelectrolytes Induced by Diluted Multivalent Salts: The Role of Electrostatic Gluonic Effects

We explore the reentrant condensation of polyelectrolytes triggered by multivalent salts, whose phase-transition mechanism remains under debate. We propose a theory to study the reentrant condensation, which separates the electrostatic effect into two parts: a short-range electrostatic gluonic effect because of sharing of multivalent ions by ionic monomers and a long-range electrostatic correlation effect from all ions. The theory suggests that the electrostatic gluonic effect governs reentrant condensation, requiring a minimum coupling energy to initiate the phase transition. This explains why diluted salts with selective multivalency trigger a polyelectrolyte phase transition. The theory also uncovers that strong adsorption of multivalent ions onto ionic monomers causes low-salt concentrations to induce both collapse and reentry transitions. Additionally, we highlight how the incompatibility of uncharged polyelectrolyte moieties with water affects the polyelectrolyte phase behaviors. The obtained results will contribute to the understanding of biological phase separations if multivalent ions bound to biopolyelectrolytes play an essential role.

Coarse-Grained MD Simulations of the Capillary Interaction between a Sphere and a Binary Fluid with Truncated Lennard-Jones Potentials

In atomic force microscopy experiments on fluid samples, a capillary bridge forms between the tip and the fluid, causing an attractive capillary force. Here, we present a computational model of the capillary interaction between a solid sphere and a coarse-grained Lennard-Jones fluid containing 10% antifreeze particles with an enlarged van der Waals radius. The capillary force acting on the sphere is obtained from the displacement of the sphere in a trap potential as the sphere is incrementally approached and then retracted from the fluid. This yields force-distance data similar to that obtained in atomic force microscopy experiments. We use this methodology to study the influence of the cutoff radius of the truncated Lennard-Jones potentials on the capillary force and its temperature dependence. The latter is found to scale with the critical temperature of the system. With the presented approach, the tip-sample interaction can be studied for a wide range of complex fluids, particle shapes, and force-probing schemes.

Multiphase Coexistence in an Ionic Liquid: 1-Decyl-3-methylimidazolium Nitrate

Complicated phase transitions were observed in a single-component 1-decyl-3-methylimidazolium nitrate ([C10mim][NO3]) ionic liquid (IL) using Raman spectroscopy and synchrotron small- and wide-angle X-ray scattering (SWAXS). Time-resolved synchrotron SWAXS could distinguish the phase transitions depending upon the cooling rate. Low-Q peaks representing a few kinds of layered structures were decomposed. Multiphase coexistence was observed in [C10mim][NO3] at specific cooling rates (8-9 K/min). Ionic liquid crystals (ILCs), hybrid-layered crystals, and hexagonal close-packed structures coexisted simultaneously. At the cooling rate region, the reentrant phase transition of the ILC phase upon heating was observed.

Differential Effects of Sequence-Local versus Nonlocal Charge Patterns on Phase Separation and Conformational Dimensions of Polyampholytes as Model Intrinsically Disordered Proteins

Conformational properties of intrinsically disordered proteins (IDPs) are governed by a sequence-ensemble relationship. To differentiate the impact of sequence-local versus sequence-nonlocal features of an IDP's charge pattern on its conformational dimensions and its phase-separation propensity, the charge "blockiness" κ and the nonlocality-weighted sequence charge decoration (SCD) parameters are compared for their correlations with isolated-chain radii of gyration (Rgs) and upper critical solution temperatures (UCSTs) of polyampholytes modeled by random phase approximation, field-theoretic simulation, and coarse-grained molecular dynamics. SCD is superior to κ in predicting Rg because SCD accounts for effects of contact order, i.e., nonlocality, on dimensions of isolated chains. In contrast, κ and SCD are comparably good, though nonideal, predictors of UCST because frequencies of interchain contacts in the multiple-chain condensed phase are less sensitive to sequence positions than frequencies of intrachain contacts of an isolated chain, as reflected by κ correlating better with condensed-phase interaction energy than SCD.

Emergence of Multiphase Condensates from a Limited Set of Chemical Building Blocks

Biomolecules composed of a limited set of chemical building blocks can colocalize into distinct, spatially segregated compartments known as biomolecular condensates. While many condensates are known to form spontaneously via phase separation, it has been unclear how immiscible condensates with precisely controlled molecular compositions assemble from a small number of chemical building blocks. We address this question by establishing a connection between the specificity of biomolecular interactions and the thermodynamic stability of coexisting condensates. By computing the minimum interaction specificity required to assemble condensates with target molecular compositions, we show how to design heteropolymer mixtures that produce compositionally complex condensates by using only a small number of monomer types. Our results provide insight into how compositional specificity arises in naturally occurring multicomponent condensates and demonstrate a rational algorithm for engineering complex artificial condensates from simple chemical building blocks.

Exact analytical solution of the Flory-Huggins model and extensions to multicomponent systems

The Flory-Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory's ease of application. In this work, we derive an implicit analytical solution to the Flory-Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types.

Fundamental Aspects of Phase-Separated Biomolecular Condensates

Biomolecular condensates, formed through phase separation, are upending our understanding in much of molecular, cell, and developmental biology. There is an urgent need to elucidate the physicochemical foundations of the behaviors and properties of biomolecular condensates. Here we aim to fill this need by writing a comprehensive, critical, and accessible review on the fundamental aspects of phase-separated biomolecular condensates. We introduce the relevant theoretical background, present the theoretical basis for the computation and experimental measurement of condensate properties, and give mechanistic interpretations of condensate behaviors and properties in terms of interactions at the molecular and residue levels.

Reaction-Driven Diffusiophoresis of Liquid Condensates: Potential Mechanisms for Intracellular Organization

The cellular environment, characterized by its intricate composition and spatial organization, hosts a variety of organelles, ranging from membrane-bound ones to membraneless structures that are formed through liquid-liquid phase separation. Cells show precise control over the position of such condensates. We demonstrate that organelle movement in external concentration gradients, diffusiophoresis, is distinct from the one of colloids because fluxes can remain finite inside the liquid-phase droplets and movement of the latter arises from incompressibility. Within cellular domains diffusiophoresis naturally arises from biochemical reactions that are driven by a chemical fuel and produce waste. Simulations and analytical arguments within a minimal model of reaction-driven phase separation reveal that the directed movement stems from two contributions: Fuel and waste are refilled or extracted at the boundary, resulting in concentration gradients, which (i) induce product fluxes via incompressibility and (ii) result in an asymmetric forward reaction in the droplet's surroundings (as well as asymmetric backward reaction inside the droplet), thereby shifting the droplet's position. We show that the former contribution dominates and sets the direction of the movement, toward or away from fuel source and waste sink, depending on the product molecules' affinity toward fuel and waste, respectively. The mechanism thus provides a simple means to organize condensates with different composition. Particle-based simulations and systems with more complex reaction cycles corroborate the robustness and universality of this mechanism.

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.

Active learning of the thermodynamics-dynamics trade-off in protein condensates

Phase-separated biomolecular condensates exhibit a wide range of dynamic properties, which depend on the sequences of the constituent proteins and RNAs. However, it is unclear to what extent condensate dynamics can be tuned without also changing the thermodynamic properties that govern phase separation. Using coarse-grained simulations of intrinsically disordered proteins, we show that the dynamics and thermodynamics of homopolymer condensates are strongly correlated, with increased condensate stability being coincident with low mobilities and high viscosities. We then apply an "active learning" strategy to identify heteropolymer sequences that break this correlation. This data-driven approach and accompanying analysis reveal how heterogeneous amino acid compositions and nonuniform sequence patterning map to a range of independently tunable dynamic and thermodynamic properties of biomolecular condensates. Our results highlight key molecular determinants governing the physical properties of biomolecular condensates and establish design rules for the development of stimuli-responsive biomaterials.