Role of Strong Localized vs Weak Distributed Interactions in Disordered Protein Phase Separation

Interaction strength and localization are critical parameters controlling the single-chain and condensed-state properties of intrinsically disordered proteins (IDPs). Here, we decipher these relationships using coarse-grained heteropolymers comprised of hydrophobic (H) and polar (P) monomers as model IDPs. We systematically vary the fraction of P monomers XP and employ two distinct particle-based models that include either strong localized attractions between only H–H pairs (HP model) or weak distributed attractions between both H–H and H–P pairs (HP+ model). To compare different sequences and models, we first carefully tune the attraction strength for all sequences to match the single-chain radius of gyration. Interestingly, we find that this procedure produces similar conformational ensembles, nonbonded potential energies, and chain-level dynamics for single chains of almost all sequences in both models, with some deviations for the HP model at large XP. However, we observe a surprisingly rich phase behavior for the sequences in both models that deviates from the expectation that similarity at the single-chain level will translate to a similar phase-separation propensity. Coexistence between dilute and dense phases is only observed up to a model-dependent XP, despite the presence of favorable interchain interactions, which we quantify using the second virial coefficient. Instead, the limited number of attractive sites (H monomers) leads to the self-assembly of finite-sized clusters of different sizes depending on XP. Our findings strongly suggest that models with distributed interactions favor the formation of liquid-like condensates over a much larger range of sequence compositions compared to models with localized interactions.


■ INTRODUCTION
Concentrated assemblies of biomolecules that exist without a lipid membrane, termed membraneless organelles (MLOs) or biomolecular condensates, play a key role in several cellular processes that include cell organization, 1 signaling, 2 and stress response 3 in addition to their relevance in pathological studies. 4In general, MLOs have liquid-like characteristics and form via phase separation. 1,5,6It has been found that intrinsically disordered proteins (IDPs) or intrinsically disordered regions (IDRs) within proteins primarily drive phase separation. 7,8A common hypothesis for the prevalence of IDPs and IDRs in MLOs is their ability to form multivalent contacts 7,9 via hydrogen bonding, 10 electrostatic interactions, 11,12 cation−π interactions, 13,14 as well as sp 2 /π interactions. 15,16However, quantifying the relative contributions of these interaction types to the phase-separation process is nontrivial, thus impeding the accurate prediction of the phase behavior of proteins based on sequence. 9,17espite these challenges, significant progress has been made in elucidating the sequence determinants of phase separation through experimental and computational studies. 10−24 Among these studies, the work on the low-complexity domain (LCD) of hnRNPA1 21,25 and the FUS prion-like domain 22 have shown that the critical temperature T c and the saturation (dilute-phase) concentration c sat were altered when the number of aromatic residues (tyrosine) was varied.In addition, the mutation of arginine to lysine while maintaining the tyrosine content in FUS-like proteins led to a significant increase in c sat , thus implying a reduction in phase-separation propensity. 13,14urther work on the FUS-LCD, using a combination of nuclear Overhauser experiments and all-atom molecular simulations, demonstrated that residues such as serine, glycine, and glutamine also play an important role in the stabilization of liquid-like condensates through interactions with other residues in the sequence. 18,26Studies involving shuffling of charges in LAF-1 RGG 11 and DDX4 12 have shown that charge patterning, frequently quantified by the sequence charge decoration parameter, 27 influences the phase-separation propensity of a protein, highlighting the key role played by electrostatic interactions.These findings have led to different interpretations of the interaction scenarios governing the phase separation of proteins.
One class of models considers only a few residue types responsible for phase separation and introduces strong localized interactions between certain amino acid pairs such as tyrosine−tyrosine or tyrosine−arginine. 7,13,14,28Such models have been used to investigate single-component and multi-component phase separation.These models are in the spirit of the HP model popularized by Dill and co-workers to investigate protein folding, in which the protein is composed of hydrophobic (H) and polar (P) monomers, and there are energetically favorable interactions (attractions) between only the H monomers. 29−31 More recent simulations using an offlattice HP variant studied the different assemblies that formed when varying the fraction of H monomers at fixed chain length and fixed interaction strength. 32nother class of protein models considers the collective contribution of all amino acids in a protein sequence by employing weak interactions, comprised of various interaction modes, distributed along the protein. 10,18,33−36 An extension of the model applying an additional temperature-dependent parabolic scaling for the nonbonded interactions was shown to capture experimentally observed 37 upper critical or lower critical solution temperature transitions for a set of thermoresponsive model IDPs. 38n this work, we investigate the differences between these two classes of models by systematically studying the phase behavior of a coarse-grained heteropolymer consisting of H and P monomers.We consider an off-lattice variant of the HP model, which has strong localized attractions between only H monomers, and an extension we call the HP+ model, which has weak distributed attractions between both H−H and H−P monomer pairs.To make the different sequences and models comparable, we carefully scale the interactions between H monomers for each sequence to achieve the same single-chain dimensions as a homopolymer reference sequence.Despite similarities at the single-chain level, differences between sequences and models emerge in the condensed-state, emphasizing the importance of the nature of intra-and interchain interactions in the formation of a homogeneous liquid-like condensate.The implications of our findings are of particular interest to the experimental interpretation of distinct interaction scenarios driving the phase separation of a given disordered protein.

METHODOLOGY
We used a coarse-grained polymer model with a single bead per residue (monomer) and the solvent modeled implicitly in the effective monomer−monomer pair interactions.The sequences comprised of two types of monomers, namely hydrophobic (H) and polar (P) monomers having the same diameter σ = 0.5 nm and mass 100 g/mol.We fixed the chain length to N = 20 and randomly generated 21 sequences with the fraction of polar beads X P varying from purely hydrophobic, X P = 0, to purely hydrophilic, X P = 1 (Figure 1a).In addition to our primary sequence set, we also tested three other randomly generated sequence sets (RS1, RS2, and RS3 in Figure S1) as well as a highly patterned sequence set (RS4 in Figure S1).
Interactions between bonded beads were modeled using a harmonic potential, (1)   with distance r between monomers, spring constant k b = 2000 kcal/(mol nm 2 ) and equilibrium bond length r 0 = 0.38 nm.Nonbonded interactions were modeled using a modified Lennard−Jones potential, where the attractive contribution was scaled independently of the short-range repulsion by the pairwise hydropathy λ ij for monomers of type i and j, 19,39−41 (2) where U LJ is the standard Lennard−Jones potential, (3)   where ε is the interaction strength.We note that eq 2 reduces to the purely repulsive Weeks−Chandler−Andersen potential, 41 which describes excluded volume interactions between monomers under good solvent conditions, for λ ij = 0 and to eq 3, which also includes effective attraction between monomers as in a lower quality solvent, for λ ij = 1.
To determine λ ij , we accordingly assigned a nominal hydropathy λ P = 0 to the P monomers and a nonzero hydropathy λ H to the H monomers.(The specific value of λ H was varied as described later.)We set λ PP = λ P and λ HH = λ H so that H−H monomer pairs were attracted to each other but P− P monomer pairs were not.We then proposed two different models (HP and HP+) for the cross-interactions between H− P monomer pairs controlled by λ HP .The HP model has strong localized attractions between only a certain type of residue, e.g., self-interactions between aromatic residues like tyrosine, The Journal of Physical Chemistry B so we set λ HP = 0.The HP+ model also has weak distributed attractions with additional residues, e.g., cross interactions between aromatic residues and polar residues like glutamine, asparagine, serine, etc. (Figure 1b), so we set λ HP = (λ H + λ P )/2 = λ H /2 to the mean hydropathy of the H and P monomers.In both models, we set ε = 0.2 kcal/mol, which, in conjunction with other model parameters, was previously shown to accurately capture IDP properties like radius of gyration R g . 19,35e performed Langevin dynamics simulations in the lowfriction limit [friction coefficient 0.1 g/(mol fs)] at fixed temperature T = 300 K for a total duration of 1 μs using a time step of 10 fs.We performed single-chain simulations using LAMMPS (29 October 2020 version), 42 while condensedphase simulations were performed using HOOMD-blue (version 2.9.3) 43 with features extended using azplugins (version 0.10.1). 44We simulated phase coexistence in rectangular simulation boxes (10 nm × 10 nm × 75 nm) with 500 chains initially placed in a dense slab with surface normals aligned with the z-axis.We also performed simulations of the same number of chains in cubic boxes of edge length 40 nm to study the formation of finite-sized aggregates.Error bars on averaged quantities were estimated by subdividing the simulation trajectory into five blocks and computing the standard error of the mean between blocks.

■ RESULTS AND DISCUSSION
Single-Chain Heteropolymer Ensembles are Matched by Scaling Attractions.Since our model sequences have the same degree of polymerization (N = 20), their single-chain dimensions (i.e., at infinite dilution) are mainly dictated by the fraction of polar monomers X P and specific intramolecular interactions.Strong monomer attraction would lead to the collapse of the chain, mimicking poor solvent conditions, whereas pure repulsion would lead to chain expansion analogous to good solvent conditions. 9,45−48 Here, we quantified single-chain compactness using the polymer's radius of gyration R g , which we computed from the average of the trace of the gyration tensor. 40,49To facilitate comparison of our sequence-and model-dependent results, we used a purely hydrophobic polymer (X P = 0) with λ H = 1 and a purely hydrophilic polymer (X P = 1) as reference sequences.For the purely hydrophobic polymer, the maximum monomer−monomer attraction was ∼0.3k B T, so it behaved like a slightly attractive chain rather than a collapsed globule at infinite dilution, as demonstrated by the scaling exponent of 0.46 for the intrachain distance (Figure S2).
We first measured R g for the different sequences and models while fixing λ H = 1.As expected, we found that R g increased with increasing X P , indicating a reduced hydrophobic character (Figures 2a and S3a).Furthermore, the probability distribution of R g widened with increasing X P , highlighting that the chains exhibited greater conformational fluctuations for large X P (Figure S3b).Since our primary goal is to study the effect of interaction model and sequence patterning on phase behavior, we tuned the monomer interactions so that all investigated polymers had a comparable single-chain R g (and thus similar compaction at infinite dilution) as the purely hydrophobic reference sequence.To account for the decrease in the hydrophobicity of the chain with increasing X P , we initially attempted to rescale λ H = 1/(1 − X P ) to maintain the same average hydropathy per monomer ⟨λ⟩ = N −1 ∑ i=1 N λ i as the purely hydrophobic reference sequence.Scaling the interactions in this way did not, however, result in constant R g , highlighting the need for fine-tuning of the interaction strengths for both models (Figure S3a).
Accordingly, we introduced an additional scaling factor a to modulate λ H = a/(1 − X P ).We selected a for each polymer by running a series of simulations to determine R g as a function of a (Figure 2b), then found the value of a that matched R g to the hydrophobic reference sequence.We found a > 1 for the HP model, suggesting that the required monomer−monomer attractions are stronger compared to those needed to maintain the average hydropathy of the sequence (i.e., ⟨λ⟩ > 1).Conversely, we found a < 1 in the HP+ model and thus ⟨λ⟩ < 1.With these scaling factors applied to the HP model, we observed that for all values of X P , other than 0.90 and 0.95, R g remained within 3% of that of the purely hydrophobic reference sequence (Figure 2a).In the X P = 0.90 sequence, strong attraction existed between the only two H monomers present in the sequence, and therefore, their position within the sequence mainly dictated R g (Figures 2a and S4), while in the X P = 0.95 case, the chain behaved as a purely hydrophilic chain due to the presence of only one H monomer.For the HP+ model, we matched R g for all X P including the 0.90 and 0.95 cases due to the additional attractive interaction between H and P monomers present in the model.

The Journal of Physical Chemistry B
To further quantify the similarity established at the singlechain level, we next investigated if the match in average R g also extended to the probability distributions P(R g ).Indeed, we found similar P(R g ) in the HP model for X P ≲ 0.5, but the agreement quickly deteriorated for larger X P (Figure 3a), possibly due to the reduction in the number of available interaction sites (i.e., H monomers).For the HP+ model, we found a near perfect overlap of P(R g ) (Figure 3b), implying that the single-chain conformations in the HP+ model were almost identical to those of a purely hydrophobic reference sequence for all investigated X P values.To further test this similarity, we considered the shape anisotropy ⟨κ 2 ⟩, computed using the eigenvalues of the gyration tensor; 25,40 the nonbonded potential energies U nb ; and the relaxation time τ e of the end-to-end vector autocorrelation function 40 (Figure 3c).Remarkably, we found for almost all X P values ⟨κ 2 ⟩ ≈ 0.39, similar to the numerically determined value of a threedimensional random walk, 50 despite the very different sequence compositions.Again, we found that the HP+ model reproduced the reference state up to significantly larger X P compared to the HP model.We observed similar trends in the nonbonded potential energy U nb : for the HP model, U nb remained almost the same as in the purely hydrophobic reference sequence up to X P ≈ 0.5, while in the HP+ model, it remained the same throughout the entire X P range.Surprisingly, we did not see as pronounced a difference between the two models when considering the end-to-end vector relaxation times τ e with varying X P .To check the sensitivity of the employed models and single-chain indicators, we considered three additional sequence sets in the range from X P = 0 to X P = 1 (RS1, RS2, and RS3 in Figure S1), but did not find any strong sequence dependence (Figures S4 and S5).Additionally, we found that the scaling factor a remained similar for all sequence sets considered in this work (Figure S6).
To summarize, tuning the attraction strength between H monomers gave similar average single-chain properties and P(R g ) as the purely hydrophobic reference sequence across for all sequences in the HP+ model, but only about half of the sequences in the HP model.The better match over the entire X P range is due to the additional attractive interaction between H and P monomers in the HP+ model.The HP+ model is less sensitive to changes in the number of attractive monomers as compared to the HP model at the single-chain level and hence might be expected to better capture the condensed-phase behavior of the purely attractive homopolymer for different values of X P .
Localized Interactions Reduce Phase-Separation Propensity.Having established similar single-chain properties for both models, we next investigated the implications for the condensed-state properties of the sequences studied. Panagiotopoulos and coworkers 53 showed via on-lattice Monte Carlo simulations that in the limit of infinitely long homopolymer chains, the critical temperature, the Boyle temperature, and the coil-to-globule transition temperature are equivalent.Recent simulations of heteropolymeric model IDPs and naturally occurring IDPs have revealed a similar correlation between the extent of collapse at the single-chain level and phase-separation propensity. 48Extending the ideas above, corrections were proposed to account for the correlation between single-chain and condensed-state properties for charged heteropolymers. 14hese prior works suggest that principles from homopolymer theory can be applied with reasonable accuracy when attractive monomers are distributed rather uniformly along the protein sequence. 25With these considerations, we investigated whether the similarity established at the single-chain level by scaling monomer interactions also leads to similarities in phase behavior for our model sequences having strong localized or weak distributed interactions.

The Journal of Physical Chemistry B
We performed direct coexistence simulations of the dilute and dense phases, computed time-averaged concentration profiles (Figure 4), and extracted the coexistence concentrations of the dilute and dense phases (Figure 5).In the HP model, we found similar dilute-phase concentrations c sat for X P ≤ 0.10, while beyond X P = 0.15, we observed an increase in c sat and a corresponding decrease in the phase-separation propensity (Figures 4 and 5).We also found that the densephase concentration started to decrease beyond X P = 0.15.For X P ≥ 0.50, we found parabolic concentration profiles centered at the origin, indicative of the formation of clusters rather than a bulk condensed-phase (see representative simulation snapshots 54 for X P = 0.50 and X P = 0.75 in Figure 4).From these observations, we concluded that the threshold value above which the sequences did not phase separate is roughly X P * = 0.45 for the HP model.
In the HP+ model, phase separation persisted for higher X P compared to the HP model (Figures 4 and 5), highlighting the stabilizing role played by the cross interactions between H and P monomers.We found that the coexistence concentrations remained almost the same only for X P ≤ 0.15.Beyond X P = 0.20, c sat increased, indicative of lower phase-separation propensity as compared to the purely hydrophobic reference sequence under the conditions of this study.However, compared to the HP model, we observed cluster formation only for large X P values (X P ≥ 0.75; Figure 4).Based on this finding, we defined X P * = 0.70 for the HP+ model, which suggests that at least 30% of the sequence must consist of hydrophobic monomers to form a liquid-like condensate at the simulated temperature.We find that, for both models, matching the single-chain dimensions through scaling of interactions leads to a dramatic increase in the threshold values (0.1 to 0.45 for HP and 0.25 to 0.70 for HP+) for phase separation when compared to the unscaled interactions (Figure S7).This finding indicates that scaling the interactions between H monomers in order to match the single-chain dimensions did offset the effect of changes in composition in both models.However, the extent to which this similarity established at the single-chain level could compensate for changes in composition of the sequence was limited.Furthermore, the chains were slightly expanded in the condensed phase compared to the dilute phase (Figures S8  and S9).A more thorough investigation is required in the future to elucidate the effect of localization of interactions on chain conformations in the condensed-phase.
The sequences with X P around the threshold X P * value (X P = 0.35 and 0.40 in the HP model, and X P = 0.65 and 0.70 in the HP+ model) formed a wider dense phase slab with its concentration about 75% of that of the purely hydrophobic reference sequence (X P = 0).To investigate whether this widening is due to the decrease in concentration or the formation of void volume within the slab, we computed the radial distribution function g(r) between H monomers for these sequences (Figure S10).We found only a marginal  The Journal of Physical Chemistry B increase in the magnitude of the first peak in g(r) compared to the X P = 0 sequence, highlighting the absence of preferential patterning of H and P monomers in the dense phase of these sequences.Both H and P monomers were well mixed within the dense phase of all sequences up to X P *, further substantiating their ability to form homogeneous liquid-like condensates.
We next probed whether the system could form a condensed-phase above the threshold X P * values (0.45 for the HP model and 0.70 for the HP+ model) by further increasing the attraction between H monomers.To this end, we ran simulations for all cases that did not form stable slabs using the HP+ model with a increased to 2, 5, and 10 times that needed to match the single-chain R g .At such high interaction strengths, we observed the formation of highly patterned slabs as reflected in their concentration profiles (Figure S11).Additionally, the concentrations of the dense phase of these sequences showed a system size dependence, analogous to those classified as aggregates by Panagiotopoulos and coworkers. 55From these observations, we concluded that insufficient interaction strength was likely not the reason why the sequences above X P * did not form liquid-like condensates.Though the condensates formed by the sequences below X P * appeared to be liquid-like, while those above X P * aggregated into finite-sized micelles, characterizing the time-dependent material properties 56,57 in the future would facilitate a comparison between the nature of assemblies formed by these sequences.
To investigate the sequence dependence of our observations, we also characterized the phase behavior of three additional sequence sets (RS1, RS2, and RS3 in Figure S1).We found that the threshold value X P * showed a stronger sequence dependence than the single-chain behavior (Figure S12).However, X P * for the HP+ model was always higher than that observed for the HP model for all sequence sets, consistent with our primary sequence set.Taken together, we found that matching single-chain properties did not prevent the decrease in the phase-separation propensity with increasing localization of interactions within the sequences, indicating that interaction strength alone did not dictate the formation of a homogeneous condensed-phase but rather a combination of interaction strength and the number of attractive monomers.
Scaled Interactions Lead to Favorable Interchain Interactions for All Sequence Compositions.A possible reason for the reduction in phase-separation propensity with increasing X P could be the decreasing favorability of interchain interactions despite favorable intrachain interactions.To quantify the strength of interaction between a pair of chains, we determined the chain−chain second virial coefficient B 22 .Positive B 22 indicates effective repulsion, while negative B 22 indicates effective attraction.We used a combination of replica exchange Monte Carlo and umbrella sampling methods to compute the dilute chain−chain potential of mean force (PMF), w(r).Then, B 22 was computed as 34,48 (4) The above method is known to estimate B 22 with a high degree of statistical certainty. 34,48Figure 6a shows the PMF for select X P values (see Figures S13 and S14 for the entire set), while the resulting B 22 values for all X P values are shown in Figure 6b.For X P ≲ 0.40, both models showed highly similar PMF to that of the purely hydrophobic reference sequence in terms of both the location and the depth of the attractive well.At intermediate to high values of X P , we observed changes between the two models: PMFs of the HP+ model were shallower and closer to the purely hydrophobic reference sequence as compared to the HP model (Figure 6a).This behavior implies that the effective chain−chain interactions in the HP model are stronger than in the HP+ model.For X P = 0.90, the PMF obtained for the HP model is an order of magnitude more attractive than that of the HP+ model, possibly owing to the higher attraction strength needed to match the single chain dimensions using only two attractive H monomers.
This comparison shows that the similar behavior achieved at the single-chain level for all values of X P translates to similar The Journal of Physical Chemistry B interactions at the two-chain level, with slightly negative B 22 values up to roughly X P *.Beyond X P *, for both models, we see that the B 22 values fluctuate, with the HP model showing more negative values than the HP+ model, indicating that the effective attraction between chains is stronger in the HP model than in the HP+ model as the number of attractive sites decreases.In some cases (X P ≥ 0.50 for the HP model and X P = 0.80, 0.90, and 0.95 for the HP+ model), the chain−chain interactions are more favorable compared to the purely hydrophobic case, which implies that from a pairwise interaction standpoint, these cases would be more prone to chain−chain association than the purely hydrophobic reference sequence.However, we did not observe the formation of a homogeneous condensate in those cases.This discrepancy prompts the question: what limits the system from undergoing phase separation if interactions remain favorable?
Beyond the Threshold for Phase Separation, Finite-Sized Clusters Are Formed.Motivated by our finding that sequences with X P > X P * do not form a condensed homogeneous phase despite exhibiting strong chain−chain attraction, we next investigated the energetically favorable assemblies formed by such sequences using both models.To this end, we let the chains self-assemble within a cubic simulation box, and then performed a clustering analysis of the resulting aggregates. 58Specifically, we considered two chains to be part of the same cluster if the distance between monomers of two chains was less than 1.5σ, and then computed the probability of finding a chain in a cluster of size N c (Figure 7).The probability distributions P(N c ) were relatively insensitive to the distance criteria chosen for analysis (Figure S15).To rule out possible finite-size effects resulting from different box sizes and geometries (slab vs cubic), 48,59 we first performed the analysis on sequences that undergo phase separation (i.e., X P < X P *; Figure S16).Consistent with the results from slab simulations, we found that the chains predominantly resided in a single large cluster for these sequences.
Beyond X P *, the probability of finding chains within smaller clusters, which are expected to be metastable, was nonzero for both models (Figure 7), indicating that interchain interactions are still favored in the limit of multiple chains.Interchain contacts are more favored in the case of the HP model as compared to the HP+ model, reflected by the higher probability of observing clusters of chains as compared to single chains in solution.The attraction between H and P monomers in the HP+ model allowed isolated chains to form intrachain contacts for all but X P = 0.95, which formed micelles with their cores consisting of H monomers from multiple chains (Figure 7).We hypothesize that the different behavior exhibited for X P = 0.95 in the HP+ model could be due to the rather uniform patterning of H monomers.Indeed, in the case of a highly patterned sequence set (RS4 in Figure S1), we found that all systems beyond X P * preferably formed small micellar clusters, thus highlighting the role of the arrangement of H monomers in dictating the formation of interchain or intrachain contacts (Figure S17). 32Despite some differences between the specific cluster morphologies in the two models, it is clear that the general emergence of smaller clusters beyond X P * is model-independent.This finding further highlights that the formation of a single high concentration phase is governed by the interplay between favorability of interactions and the number of available interaction sites. 60−63 Beyond X P *, the system is limited by the number of available interaction sites (or valence), resulting in its inability to form a single continuous stable condensed-phase.

■ CONCLUSIONS
In this computational study, we comprehensively investigated the interplay between the localization and strength of interactions in dictating the phase behavior of model disordered proteins consisting of hydrophobic (H) and polar (P) monomers.We systematically varied the fraction of P monomers in the sequences, X P , and considered two model classes, representing interaction scenarios with either strong localized or weak distributed attractions.To establish a common ground for the large variation in IDP sequences, we carefully scaled the attraction strength in each case to match the single-chain dimensions of an attractive homopolymer, thereby providing all of the model proteins an equal opportunity to phase separate, from a single-chain perspective.To our surprise, scaling the interactions this way also led to

The Journal of Physical Chemistry B
similar conformational ensembles, nonbonded potential energies, and chain-level dynamics in both models for almost all investigated sequences.Based on the expectation that intramolecular interactions will translate directly to the intermolecular level, we probed the propensity of the IDPs to phase separate.We found similar phase-separation propensity for IDPs with sufficiently low X P , where a significant scale up in the attraction strength counterbalanced the decreasing number of H monomers.As X P increased, however, the phase-separation propensity dramatically declined because of limited number of attractive monomers.Beyond a certain X P , the model IDPs did not form a continuous condensedphase anymore, but instead self-assembled into finite-sized aggregates.These deviations from the reference attractive homopolymer were much more pronounced for the model with strong localized interactions, suggesting that weak distributed interactions between multiple residue types may better stabilize the liquid-like condensed-phase of disordered proteins. 10,18,23,26,33We believe that our work, performed at multiple scales, will aid in the mechanistic understanding of biomolecular phase separation and in the development of theoretical models to accurately capture the interactions of disordered proteins.

Figure 1 .
Figure 1.(a) Simulated sequences of length N = 20 with the fraction of polar beads X P ranging from X P = 0 (purely hydrophobic) to X P = 1 (purely hydrophilic).(b) Schematic highlighting the different interactions used in the HP and HP+ models.

Figure 2 .
Figure 2. (a) Radius of gyration R g as a function of X P before scaling (λ H = 1, open symbols) and after scaling (λ H = a/(1−X P ), closed symbols) for the HP (red) and HP+ (blue) models.(b) R g as a function of scaling factor a using the HP and HP+ models.Results for purely hydrophobic (X P = 0) and purely hydrophilic (X P = 1) reference sequences are shown as dashed and dotted black lines, respectively.

Figure 3 .
Figure 3. Probability distribution of radius of gyration R g for the (a) HP and (b) HP+ models.(c) Shape anisotropy ⟨κ 2 ⟩, nonbonded potential energy U nb , and end-to-end vector relaxation time τ e as functions of X P for the HP and HP+ models.The last quantity in (c) is normalized by that obtained for the purely hydrophobic reference sequence (X P = 0).Results for purely hydrophobic and purely hydrophilic (X P = 1) reference sequences are shown as dashed and dotted black lines, respectively.

Figure 4 .
Figure 4. Concentration profiles determined using direct coexistence simulations for the HP and HP+ models.The black line represents the purely hydrophobic reference sequence (X P = 0).Representative simulation snapshots for X P = 0.25, 0.50, and 0.75 are shown on the left (HP) and right (HP+) sides.

Figure 5 .
Figure 5. Phase diagrams for the HP and HP+ models.Concentrations of dilute and dense phases for all X P values were extracted from the concentration profiles shown in Figure 4. Dashed line represents the total concentration of the coexistence simulation.

Figure 6 .
Figure 6.(a) Potential of mean force (PMF) as a function of center-of-mass distance r com between two chains for select X P values using the HP (filled symbols) and HP+ (open symbols) models.(b) Second viral coefficient B 22 obtained from the PMFs for the HP and HP+ models.

Figure 7 .
Figure7.Probability of finding a chain in a cluster of size N c for select X P values that did not undergo phase separation in the HP and HP+ models.Representative simulation snapshots for X P = 0.80 and 0.85 on the left side correspond to the HP model, while those for X P = 0.80 and 0.95 on the right side correspond to the HP+ model.