Generalizing localized versus distributed interactions in disordered-protein phase separation: a general sequence-to-threshold mapping beyond minimal two-letter models

Rekhi et al. (J. Phys. Chem. B 2023) compared coarse-grained heteropolymers of hydrophobic (H) and polar (P) monomers in which attractive interactions are either strong and localized (only H–H pairs, HP model) or weak and distributed (H–H and H–P pairs, HP+ model), with attraction strengths tuned so that every single chain matches the radius of gyration of a reference homopolymer. Surprisingly, single-chain similarity did not translate into similar condensed-phase behavior: macroscopic phase separation occurs only below a model-dependent polar-fraction threshold $X_\mathrm{P}^{*}$ (about 0.45 for HP and 0.70 for HP+ at chain length $N=20$ and $T=300\,\mathrm{K}$), and above it the chains assemble into finite clusters despite favorable pair attractions (negative $B_{22}$). These trends were established in a minimal two-letter, fixed-length, single-temperature setting with implicit solvent; the authors explicitly deferred to future work the effect of interaction localization on chain conformations in the condensed phase and the time-dependent material properties of the assemblies, and the metastability of the observed finite clusters was assumed rather than demonstrated. How these trends generalize to chemically diverse interaction models, chain lengths, and many-residue-type compositions, and whether a mapping between sequence composition/patterning, interaction localization, and the phase-separation threshold can be established, therefore remains open. This problem asks for that general, quantitative framework: a matched-single-chain comparison protocol valid across chemically diverse interaction models, chain lengths, and many-residue-type compositions, delivering the sequence-pattern to threshold-and-assembly-outcome mapping, and settling whether the finite clusters are metastable or equilibrium structures.

Background

Intrinsically disordered proteins (IDPs) lack a stable folded structure and often demix from solution by liquid–liquid phase separation (LLPS), forming membrane-free biomolecular condensates that concentrate specific molecules in cells. Because residue-level detail is expensive to simulate, a standard approach uses coarse-grained models with one bead per residue: the HP model represents each residue as either hydrophobic (H) or polar (P), with attractions derived from a hydropathy scale (Dignon et al., PLOS Comput. Biol. 2018), and related residue-level models assign interaction strengths from the full amino-acid alphabet. A complementary view, the sticker-and-spacer framework, treats only designated sticky residues (e.g., aromatic or charged residues) as contributing strong localized attractions while the rest are inert spacers; experiments on prion-like domains showed that both the number (valence) and the sequence positioning of aromatic stickers control phase behavior (Martin et al., Science 2020). Whether strong localized attractions among a few residue types or weak distributed attractions among many residue types better represent real IDPs — and whether the choice even matters once single-chain properties are matched — is a live question in coarse-grained modeling. Two standard observables connect single chains to condensed phases: the single-chain radius of gyration $R_g$ (a compactness measure) and the osmotic second virial coefficient $B_{22}$ (the coefficient of the two-body term in the expansion of solution free energy; negative values signal effective interchain attraction). Rekhi et al. (J. Phys. Chem. B 2023) constructed model IDPs of $N=20$ monomers with polar fraction $X_\mathrm{P}$ from 0 to 1, including patterned sequence sets, and simulated two interaction scenarios: the HP model, in which scaled H–H attractions are strong and localized, and the HP+ model, in which weaker attractions act between H–H and H–P pairs. For every sequence they rescaled the attraction strength so the single-chain $R_g$ matched that of the purely hydrophobic reference chain. This matching produced nearly identical single-chain ensembles, energies, and dynamics across sequences and models. Nevertheless, phase separation (quantified via slab-geometry coexistence simulations giving dilute-phase saturation concentrations) occurred only up to a threshold $X_\mathrm{P}^{*}$ — roughly 0.45 for HP and 0.70 for HP+ — beyond which the limited number of attractive sites (valence) forced self-assembly into finite, micelle-like clusters, even though the dilute-pair $B_{22}$ remained favorable and interactions stronger in the HP model. The authors concluded that distributed interactions stabilize liquid-like condensates over a much wider composition range than localized interactions, and they flagged as unresolved the effect of interaction localization on chain conformations inside the condensed phase and the time-dependent material properties of the assemblies below and above $X_\mathrm{P}^{*}$. Within the study, the sequence dependence of $X_\mathrm{P}^{*}$ was examined across several random and patterned sequence sets at fixed $N=20$; by stated construction, the work is limited to the two-monomer alphabet at fixed chain length and single temperature with implicit solvent, relies on dilute-pair virial coefficients, and leaves the metastability of the observed clusters unverified.

Problem Statement

Establish a general and quantitative account of how interaction localization and sequence encoding jointly determine the assembly behavior of disordered-protein phase separation, generalizing the HP/HP+ comparison of Rekhi et al. (2023) beyond the minimal setting. Concretely: (i) can a general protocol be formulated for comparing arbitrary interaction models — localized versus distributed, or of different chemical resolution — at matched single-chain conformational ensembles (not merely matched $R_g$), so that differences in condensed-phase behavior are attributable to interaction localization rather than to single-chain compactness; (ii) construct the mapping between sequence pattern (composition, valence and positioning of attractive sites, chain length) and interaction model on the one hand and the phase-separation threshold and assembly outcome (liquid condensate, finite clusters, or dilute phase) on the other, quantifying how far pairwise quantities such as $B_{22}$ and two-chain potentials of mean force determine multi-chain behavior at finite concentration, and establishing how the composition thresholds $X_\mathrm{P}^{*}$ and the condensate-versus-finite-cluster crossover observed at $N=20$, $T=300\,\mathrm{K}$ with a two-monomer alphabet generalize to chains with many chemically distinct residue types and to substantially longer chains; and (iii) determine whether the finite clusters formed above threshold are metastable transient aggregates or equilibrium microphase-like structures. All three components, and every demand bundled within component (ii), are jointly required to resolve the task (no proper subset suffices) and together serve one objective — replacing the minimal model's empirical thresholds with a predictive, general sequence-to-phase-behavior mapping — and answers may additionally address robustness axes the source paper set aside (explicit solvent and temperature dependence, including possible upper/lower critical solution temperature behavior, and all-atom resolution). Because the central deliverable is a claim of generality, it must carry an explicit acceptance standard: the protocol and mapping must be demonstrated at least for one interaction-model family beyond the two-letter HP/HP+ case and over a stated, justified representative range of chain lengths and residue-type alphabets, and the answer must state a criterion for when the mapping counts as general and the domain over which its validity is characterized; this fixes a minimum demonstration scope without bounding the generality sought.

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