2.1 Prediction of A1-LCD coexistence lines using different sequence-dependent models

The determination of the phase diagram establishes the thermodynamic conditions to investigate the formation and stability of biomolecular condensates [80, 81]. Numerous experiments have consistently shown how aromatic residues represent key ‘stickers’ to sustain protein phase separation of prion-like domain proteins [23, 77, 78, 82, 83]. Therefore, to evaluate the performance of different models such as the HPS [52], HPS-cation–π [56], HPS-Urry [54], CALVADOS2 [55], Mpipi [53] and Mpipi-Recharged [61] in reproducing condensate behaviour (see Section 4.1 for technical details about the models), we consider the different mutants of A1-LCD reported in Ref. [78] that display multiple variations in the type of aromatic residues along their sequence (see S1 Table to visualize the studied sequences). These variants are the following: (1) WT+NLS which corresponds to the A1-LCD wild-type sequence, which features a nuclear localization signal [78] and contains 8 tyrosines (Y) and 12 phenylalanines (F); (2) allF with 19F; (3) allY with 19Y; (4) allW with 19 tryptophans (W); (5) YtoW with 7W and 12F; (6) FtoW with 7Y and 12W; and (7) W⁻ with 13W. Importantly, the mutations in all variants are performed on the aromatic residues conserving the sequence patterning and just changing the chemical identity of the residues [78].

We compute the temperature-vs-density phase diagrams of the aforementioned sequences by performing Direct Coexistence (DC) [84, 85] Molecular Dynamics (MD) simulations of protein solutions (see Fig 1A and Section 4.3 for further methodological details). In the DC method, the two coexisting phases are simulated by preparing periodically replicated slabs of the two phases (the condensed and the diluted phase) in the same simulation box. Once the system is equilibrated, a density profile along the long axis of the box can be extracted to compute the density (or concentration) of the two coexisting phases after a production run of ∼ 1μs (please see S1 Video for a representative trajectory of our DC simulations). Importantly, we note that none of the six employed models includes explicit solvent, hence, effectively in our DC simulations the protein diluted phase corresponds to a vapour phase. In Fig 1A, a typical snapshot of the simulation slab is shown together with the associated density profile below the critical temperature (Top panel), and above the critical point (Bottom panel), where a single phase is present.

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Fig 1.

(A) Snapshots of a Direct Coexistence (DC) simulation used to calculate the phase diagram and critical temperature of A1-LCD (WT+NLS) protein, where each protein replica has a different colour. Top panel exhibits phase separation with a distinguishable condensed and diluted phase as depicted by the associated density profile. Bottom panel represents a system above the critical solution temperature where no phase separation occurs and a single phase is detected across the density profile. B-G Temperature–density phase diagrams of A1-LCD variants calculated via DC simulations employing the HPS (B), HPS-cation–π (C), HPS-Urry (D), CALVADOS2 (E), Mpipi (F) and Mpipi-Recharged (G) models. Critical temperatures are represented by empty circles while filled circles depict coexistence densities obtained from DC simulations. The lines serve as a guide to the eye. The colour coding is preserved throughout all the panels for all the variants as indicated in the legend of panel B.

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The phase diagrams for the seven A1-LCD variants, computed using six different models, are shown in Fig 1B–1G. These diagrams highlight the varying sensitivity of the models in capturing the effects of specific amino acid mutations across the protein sequence. Among the models, Mpipi and Mpipi-Recharged (Fig 1F and 1G, respectively) display the largest variation in the critical temperature (T_c), with T_c spanning more than 100 K from the lowest to the highest value across the variants. In contrast, the T_c values predicted by the HPS and HPS-family models vary by only about 30 K between the lowest and highest values (Fig 1B–1D). Despite this, the HPS-Urry model (Fig 1D) demonstrates greater sensitivity to amino acid mutations, as evidenced by the larger spacing in T_c values among the variants.

The HPS model [76] represents amino acid pair interactions using a Ashbaugh-Hatch potential parametrized based on the Kapcha and Rossky (KR) hydrophobicity scale [86]. Specifically, the hydrophobicity values from the KR scale (λ_i, where i is the amino acid) are used to define the interaction strength parameter of the Lennard-Jones term in the Ashbaugh-Hatch potential for each amino acid type. To determine the Lennard-Jones parameters for interactions between different amino acid types, the model uses the Lorentz-Berthelot mixing rules. The HPS model also incorporates salt-screened charge–charge interactions modelled through a Debye-Hückel potential. The HPS-cation–π model [56] is a reparametrization of the HPS model that enhances the relative strength of all cation–π pair interactions to better capture the experimentally observed phase separation propensity of DDX4 IDR variants [87]. More recently, the HPS-Urry model [54] was developed to improve the predictive accuracy of the HPS family of models. This version replaces the KR hydrophobicity scale with the Urry hydrophobicity scale [88], enabling improved predictions of the effects of R-to-K and Y-to-F mutations on the phase behaviour of protein solutions.

The CALVADOS family of models [41, 55], including the CALVADOS2 model tested here, builds on the foundation of the HPS model. However, instead of using a hydrophobicity scale to define the λ_i values, and subsequently the Lennard-Jones parameters, the CALVADOS models employ a Bayesian learning approach. In this approach, the λ_i values for all amino acids are optimized to achieve agreement with experimental single-molecule radii of gyration and paramagnetic relaxation enhancement (PRE) nuclear magnetic resonance (NMR) data across a broad range of intrinsically disordered regions (IDRs).

In contrast, the Mpipi model [53], developed by our group, takes a fundamentally different approach by abandoning the Lorentz-Berthelot mixing rules and instead defining amino acid pair interactions in a pair-specific manner. Additionally, the model uses the Wang-Frenkel potential, which offers greater flexibility and computational efficiency compared to the Lennard-Jones potential. For charge–charge interactions, Mpipi employs a Debye-Hückel potential, similar to the HPS and CALVADOS families. However, we recently introduced the Mpipi-Recharged model [61], which enhances the original Mpipi model by refining its description of electrostatic interactions. Specifically, Mpipi-Recharged replaces the pair-agnostic Debye-Hückel potential with a pair-specific, non-symmetric Yukawa potential. This parametrization is based on atomistic simulations of amino acid pairs in explicit water with ions, allowing for a more accurate depiction of charge effects. By abandoning mixing rules in the Wang-Frenkel potential (in both Mpipi and Mpipi-Recharged models) and incorporating amino acid-specific descriptions of charge interactions (in the Mpipi-Recharged), the Mpipi family of models addresses critical limitations of standard mixing rules. These rules may fail to capture nuanced biomolecular interactions, such as the subtle balance between aromatic stacking, cation–π interactions, and charge–charge interactions. Treating pair interactions explicitly enables the Mpipi models to achieve excellent agreement with experimental data [53, 61, 68, 89]. This pair-specific approach does not only improve accuracy but also provides a more detailed understanding of the molecular forces driving phase behaviour [61].

To provide additional context on the differences between the models and their resulting phase diagrams, S1 Fig presents the relative interaction strength maps for each model, encompassing both hydrophobic and electrostatic interactions between all amino acid pairs. These interaction maps reveal significant differences in how interactions are parametrized in the models for the aromatic residues Y, F, and W. In the HPS and HPS-cation–π models, the strength of interactions between aromatic residues and the rest of the amino acids are relatively homogeneous. In contrast, Mpipi, Mpipi-Recharged, CALVADOS2, and HPS-Urry display more pronounced differences in their energy scales, with W > Y > F in terms of interaction strength. The variations in the energy scale of the models and the relative strength of interactions among amino acid pairs strongly influence how sensitive each model is to specific (aromatic) sequence mutations and their impact on the encoded phase behaviour. Models with greater differences in interaction strengths for aromatic residues tend to show enhanced sensitivity to mutations, as reflected in their broader T_c ranges and the corresponding phase diagrams shown in Fig 1B–1G.

The temperature-vs-concentration phase diagram of three of these sequences (WT+NLS, allY, and allF) has been recently measured in vitro [77, 78]. By fitting critical solution temperatures from the experimental data, we now establish a direct comparison between modelling and experiments so we can assess the ability of the models to predict A1-LCD phase behaviour. Accordingly, the phase diagrams in Fig 2 are plotted as a function of the protein concentration (and in logarithmic scale) to evaluate their predictive capability against experiments in the diluted phase. Determining the protein concentration in the diluted phase—specifically, the saturation concentration—poses significant challenges for Direct Coexistence simulations, particularly at low temperatures, due to the limited sampling inherent to these simulations [68].

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Fig 2. Experimental (solid stars) vs. simulated (empty circles) phase diagrams for the WT+NLS, allF, and allY variants of A1-LCD using different models: HPS (A), HPS-cation–π (B), HPS-Urry (C), CALVADOS2 (D), Mpipi (E), and Mpipi-Recharged (F).

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As illustrated in Fig 2A, the HPS model [76] consistently underestimates the critical solution temperatures for the three A1-LCD variants tested. When comparing the predicted critical temperatures from simulations (orange symbols in Fig 3) with the corresponding experimental values (derived by us from coexistence concentrations reported in Ref. [77]), the trend line deviates significantly from the perfect-fit reference (black line with a slope of 1 and intercept of 0; Fig 3). This suggests that the HPS model does not effectively capture the impact of aromatic amino acid mutations on the critical solution temperatures of the A1-LCD system. The discrepancy is likely due to the parametrization of this early version of the HPS model, which relies on the KR hydrophobicity scale [76, 86]. In the KR scale, F is ranked as more hydrophobic than W, and W is ranked as more hydrophobic than Y, potentially misrepresenting the relative contributions that these aromatic residues have in the phase separation of these systems.

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Fig 3. Comparison of the critical temperatures of WT+NLS (circles), allF (stars), and allY (triangles) calculated in simulations and estimated from experiments (by applying the law of rectilinear diameters and critical exponents [90] to measurements of saturation concentrations at several temperatures [77, 78]).

The black solid line indicates perfect correlation between experimental and simulation results.

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In contrast, the Urry hydrophobicity scale [88] ranks the hydrophobicity of aromatic residues as W > Y > F. Consistently with this, the HPS-Urry model qualitatively captures the experimental trend in the relative phase separation propensities of the WT+NLS, allF, and allY variants, as indicated by the violet symbols in Fig 3. While the predicted order of phase separation propensities is correct, the absolute values of the critical solution temperatures predicted by the HPS-Urry model deviate significantly (by about ∼20% on average) remaining consistently lower than the fitted experimental data (Fig 2C).

The HPS-cation–π model [56] predicts higher-than-expected critical temperatures (approximately ∼30% higher on average) for the three variants (Figs 2 and 3). This discrepancy can be attributed to a substantial overestimation of cation–π interactions in the HPS-cation–π model (S1 Fig). Importantly, the high discrepancy in the predicted critical temperature values between the HPS-cation–π and HPS/HPS-Urry models highlights the prominence of cation–π interactions within A1-LCD condensates.

The CALVADOS2 model predicts coexistence lines that are, on average, very close to the experimental values (critical solution temperatures within approximately ∼5% of the values fitted by us from experiments). While the values are very close to the experimental ones, the relative ordering among the three variants deviates from the experimental trend (pink data in Fig 3). It is important to note that the CALVADOS2 model was not specifically designed to predict critical solution temperatures, and this evaluation goes beyond its intended scope. In this context, a recent study by our group [89], consistent with prior research [41, 55, 91], emphasizes that models developed to capture single-molecule properties of IDRs do not necessarily reproduce condensate coexistence lines. While diverse parametrizations can yield reasonable predictions for radii of gyration for diverse IDRs, the accurate prediction of phase diagrams represents a far more rigorous and demanding validation test [89].

The Mpipi model has demonstrated excellent performance in describing the phase behaviour of prion-like domain proteins [53, 68]. Consistently, our tests show that Mpipi accurately predicts the order of the critical solution temperatures for the WT+NLS, allF, and allY variants (cyan symbols in Fig 3). This agreement arises from the Mpipi model’s parametrization, which ranks the interactions of the three aromatic residues as W > Y > F. Despite its qualitative accuracy, Fig 2E reveals that Mpipi slightly overestimates the critical solution temperatures for all three variants, with an average deviation of approximately 6%. Importantly, our analysis reveals that the improved Mpipi-Recharged model achieves the highest level of accuracy in reproducing experimental values, successfully capturing both the trend and the absolute critical solution temperatures for the three variants with deviations of less than ∼3% (green symbols in Fig 3).

Overall, this analysis demonstrates that CALVADOS2, Mpipi, and Mpipi-Recharged offer a robust and accurate description of the condensate phase behaviour for A1-LCD aromatic mutants. In contrast, while the HPS-Urry model qualitatively captures the experimental phase-separation tendencies, it exhibits significant deviations in its quantitative predictions of critical solution temperatures.

2.2 Comparison of the phase-separation saturation concentration between experiments and simulations

The concentration threshold above which biomolecular phase separation becomes thermodynamically favourable at a given temperature—C_sat(T)—is one of the critical quantities that can be controlled by cells to trigger condensate formation and dissolution on demand [7, 92]. In that sense, in vitro studies are extremely useful to characterise such threshold, determining the border between a single homogeneous phase and the emergence of phase separated condensates [77, 93–95]. Most of these studies have focused on measuring C_sat at physiological conditions of salt, pH, and temperature [22, 24, 93, 95–100]. However, recent studies have also reported how C_sat varies with temperature and measured the highest temperature values beyond which phase separation is no longer achievable [77, 78]. The saturation concentration quantifies the ability of a given protein to phase separate, thus, the higher the value, the lower propensity to form condensates [22, 24, 93, 97]. Similarly, for condensates that present upper critical solution temperatures (herein critical temperature), the critical temperature evaluated through DC simulations serves as a direct indicator of the ability of a protein to undergo phase separation [69]. Indeed, the in vitro experimental values of saturation concentrations of the A1-LCD variants [77] seem to correlate inversely with the critical solution temperatures that we can fit from the experimental coexistence densities [77]. A higher critical temperature is directly indicative of greater thermodynamic stability of the condensates. Thus, we can correlate the critical temperature obtained from simulations to the experimental saturation concentration at a given constant temperature (e.g., 298 K) to obtain a qualitative overview of the models sensitivity for describing protein phase behaviour as a function of specific sequence mutations [61, 69].

In Fig 4, we show the simulated critical temperatures (normalised by the critical temperature of WT+NLS) against the experimental saturation concentrations [78] measured at 298 K for the different A1-LCD mutants. The experimental C_sat vary significantly, changing by more than 2 orders of magnitude among the variants studied [78]. Thus, the comparison we perform in this section tests the sensitivity of the six different models to describe the impact of mutations on modulating the thermodynamic stability of A1-LCD condensates (Fig 4A–4F).

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Fig 4. Predicted critical temperature (normalised by the critical temperature of the WT+NLS sequence) for the different models HPS (A), HPS-cation–π (B), HPS-Urry (C), CALVADOS2 (D), Mpipi (E), and Mpipi-Recharged (F) vs. the experimental saturation concentration at 298 K reported in Ref. [78] for the different A1-LCD mutants.

The panels include the Pearson correlation coefficient (r) and the slope (m) from a linear fit to the data. The error bars show the uncertainty in the critical temperature associated to its calculation using the laws of critical exponents and rectilinear diameters (see further details in Section 4.3).

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To compare the predictions of the various models, we establish a linear relationship between the normalised critical solution temperatures and the experimental saturation concentrations, expressed as , where m is the slope and b is a constant. We use r (the Pearson correlation coefficient) to quantify the quality of the fit. Notably, the HPS and HPS-cation–π exhibit the smallest absolute values for the slope m in this relationship, ranging from −0.5 × 10⁻² to −0.8 × 10⁻²mM⁻¹. This indicates that the critical temperatures predicted by these two models are only weakly affected by the mutations studied here (Fig 4A and 4B, respectively). Furthermore, the quality of the fit is relatively low for the HPS (r = 0.3952) and HPS-cation–π models (r = 0.4179), suggesting a poor correlation between their predicted critical temperatures and the experimental trends for C_sat.

The Mpipi and Mpipi-Recharged models predict the most significant changes in critical temperature as a function of the variations in saturation concentration among the variants (Fig 4E and 4F, respectively). This is reflected in the steep values of their slopes, −4.6 × 10⁻² and −5.6 × 10⁻²mM⁻¹, respectively. Additionally, both models exhibit the strongest linear correlation with the experimental data, achieving Pearson correlation coefficients exceeding 0.98.

Finally, the HPS-Urry and CALVADOS2 models exhibit intermediate performance compared to the HPS/HPS-cation–π and Mpipi/Mpipi-Recharged models. The critical temperatures predicted by both HPS-Urry and CALVADOS2 display a strong linear correlation with the experimental C_sat values, achieving Pearson correlation coefficients above 0.91. However, both models predict only a moderate decrease in T_c as a function of changes in the saturation concentration of the variants studied (Fig 4C and 4D). Because the CALVADOS2 model was not specifically designed to predict critical solution temperatures, this evaluation extends beyond the model’s intended scope.

Building on these results, we now focus on the CALVADOS2, Mpipi, and Mpipi-Recharged models to further investigate the phase behaviour of A1-LCD mutants. Specifically, we use these models to compute the saturation concentration at 298 K (C_sat) by extensively sampling the equilibrium protein concentration in the diluted phase in coexistence with the condensates through Direct Coexistence simulations. Estimating C_sat using this approach is computationally demanding due to the inherent challenges of sampling the diluted phase, particularly at low temperatures [68]. The diluted phase contains a very low concentration of molecules, meaning there are intrinsically few particles present. This low number of particles can result in significant variability in the measured concentrations across different regions of the simulation box, introducing substantial noise into the measurement. Furthermore, exchanges of molecules between the diluted and condensed phases are rare events, which require long simulation timescales to be adequately captured. These challenges are exacerbated at lower temperatures, where the equilibrium concentration in the diluted phase becomes even smaller. While computationally demanding, extracting C_sat from the DC method allows us to perform a direct comparison between the same observable measured in both simulations and experiments.

In Fig 5, we present the predicted saturation concentration values from simulations, normalised by the saturation concentration of the WT+NLS sequence, plotted on a logarithmic scale. The predictions from the CALVADOS2, Mpipi, and Mpipi-Recharged models are compared against experimentally determined normalised saturation concentrations for the studied mutants ((; see Section 4.6 for further information) reported in Ref. [78]. To quantify the agreement between the simulations and experiments, we calculate D, a metric that measures the average deviation of the simulation predictions from the experimental values (details provided in Section 4.6). Lower D values indicate closer agreement with the experimental data. Additionally, we perform a linear fit to examine the relationship between the normalised simulation results and the experimental data (red dashed line in Fig 5), comparing it to the perfect fit (diagonal black line in Fig 5). To assess the strength of the linear correlation, we calculate the Pearson correlation coefficient (r). This metric evaluates whether the models qualitatively capture the experimental trend, even if quantitative deviations are present.

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Fig 5. Simulated vs. experimental saturation concentration (C_sat), normalised by the WT+NLS C_sat [78], for the different variants and for the models CALVADOS2 (A), Mpipi (B) and Mpipi-Recharged (C).

The Pearson correlation coefficient (r) and the root mean square deviation from the experimental values (D) are displayed for each set of simulation data. The black lines indicate a perfect match between experimental and computational values, while the red dashed lines depict a linear regression for each set of data.

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We find that the prediction of the Mpipi-Recharged model for the normalised C_sat presents the lowest average deviation, D = 0.63, with respect to the experimental values and the highest Pearson correlation coefficient among the set (r = 0.9947), indicating exceptional agreement with the experimental trends. The Mpipi and CALVADOS2 models also provide good descriptions with mean deviations of D = 0.97 and D = 1.26, respectively, and Pearson correlation coefficients of r = 0.8766 and r = 0.7549, respectively.

In addition, we compare the absolute experimental and simulation C_sat values (S2 Fig). In this comparison, the values of the Pearson correlation coefficients remain unchanged from those in Fig 5. That is, the predictions of the Mpipi-Recharged model exhibit the strongest correlation with the experimental values of C_sat, followed by Mpipi, and finally CALVADOS2. In contrast, the size of the mean errors, D, do change, revealing that CALVADOS2 provides C_sat with the smallest overall error (D = 0.38), followed by the Mpipi-Recharged (D = 0.78) and finally the Mpipi model (D = 0.93).

Overall, our comprehensive evaluation in this Section shows that all three models—CALVADOS2, Mpipi, and Mpipi-Recharged—perform exceptionally well in predicting the phase-separation propensity of A1-LCD variants, with Mpipi-Recharged standing out as the most accurate potential. Their excellent performance is particularly noteworthy given the substantial challenges coarse-grained models face in reproducing experimental coexistence lines.

2.3 Condensate viscoelastic behaviour of A1-LCD mutants by different models

Condensate viscoelastic properties have been linked to the roles of these systems in health and disease [34, 101, 102]. Whereas liquid-like states in condensates are associated to biological function, their progressive transition into solid and gel-like states has been linked to the onset of multiple neurological disorders [24]. Examples of protein condensates displaying hardening over time have been reported for RNA-binding proteins such as FUS [103, 104], TDP-43 [105], or hnRNPA1 [106] among others [30]. Hence, it is important to determine the ability of different residue-resolution CG models to probe the material properties of condensates: i.e., whether they behave as liquids or gels, and whether their viscosity remains constant or increases over time [31]. Moreover, determining the ability of residue-resolution CG models to characterise which precise interactions and domains control their time-dependent transport properties [103, 107–109] is central. Such information is essential to understand the molecular onset between functional and dysfunctional behaviour.

Computationally, the generalised Green-Kubo relation [110, 111] (see Section 4.5) is one of the most efficient methods for evaluating viscosity of protein condensates using residue-resolution CG models [31, 69, 112]. Calculating the shear stress relaxation function (G(t)) using the Green-Kubo approach provides information of both inter- and intra-molecular protein interactions, which may vary significantly with the chosen CG model [69, 108, 112]. Since the predictive accuracy of most residue-resolution CG models has been optimized and tested using experimental single-protein radius of gyration and phase diagrams (either saturation concentrations or critical solution temperatures) [41, 43, 53, 54, 59], evaluating their ability to predict condensate viscosity provides a demanding benchmark that extends beyond the scope of their original parametrizations.

As the experimental reference, we use the in vitro viscosity measurements at T ∼ 298 K from Ref. [78] for the seven variants of A1-LCD studied in the previous sections. We only test the performance of the CALVADOS2, Mpipi, and Mpipi-Recharged models because 1) HPS and HPS-cation–π failed to recapitulate the relative phase-separation propensity of the A1-LCD variants (Fig 4A and 4B, respectively); and 2) HPS and HPS-Urry do not phase-separate at 298 K, where condensates experimental viscosities have been reported.

To estimate the viscosity, we calculate the shear stress relaxation function, G(t), of the different A1-LCD variants under bulk conditions at 298 K (see Fig 6A and S2 Video for representative configurations of such simulations, and Section 4.2 for the simulation details). For these calculations, we perform bulk simulations (in the canonical ensemble) of ∼ 3 − 5μs depending on each specific system as reported in Section 4.2.

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Fig 6.

(A) Snapshot of a bulk NVT simulation with 200 A1-LCD (WT+NLS) protein replicas (each of them depicted by a different tone of colour) employed to compute the condensate viscosity. (B) Shear stress relaxation modulus (G(t)) for the WT+NLS sequence at 298 K evaluated by different models. (C) G(t) for different A1-LCD mutants using the Mpipi-Recharged model. In both B and C panels, G(t) raw data obtained during the simulations are represented by filled symbols, while Maxwell’s mode fits to the data (as described in Section 4.5) are plotted with solid lines. D, E, and F: Predicted versus in vitro viscosity (both reduced by the corresponding WT+NLS value) for all variants under study and the CALVADOS, Mpipi, and Mpipi-Recharged models respectively. The meaning of r, D, and the red dashed and black solid lines is the same as in Fig 5.

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The calculation of G(t) for the WT+NLS sequence using the three models is shown in Fig 6B. While the predictions of the three models for G(t) at short timescales are rather similar, indicating that bond, angle, and intramolecular conformational relaxation is similar for all of them, we observe large differences at long times. At long timescales, the slope and terminal decay of G(t), which strongly determines the condensate viscosity, differs significantly among the three models, clearly demonstrating the critical influence of the model parametrization on the resulting viscoelastic properties of the condensates. Such variability is even more pronounced when comparing the behaviour of G(t) predicted by the three models for all the different A1-LCD mutants. We show the full G(t) plots predicted by the Mpipi-Recharged model in Fig 6C, and the viscosity values for the three models in Fig 6D–6F. The calculation of η from G(t) consists in integrating the shear stress relaxation modulus (G) as a function of time (for further details on this calculation see Section 4.5).

Fig 6C reveals that the amino acid mutations tested lead to drastic variations in the condensate viscosity predicted by the Mpipi-Recharged (of approximately two orders of magnitude), in perfect agreement with the experimental observations [78]. This results highlights the strong influence that specific sequence mutations have on regulating the transport properties of condensates, as previously found for FUS [113], TDP-43 [105], and Tau [114, 115].

We next compare the viscosity (η) values for all A1-LCD sequences as predicted by simulations using the CALVADOS2 (Fig 6D), Mpipi (Fig 6E), and Mpipi-Recharged (Fig 6F) models against the corresponding experimental values [78]. Both simulation and experimental viscosity values are normalised by the WT+NLS η value to enable a relative comparison. This normalisation is necessary because the absolute values predicted by all the residue-resolution coarse-grained models systematically underestimate the experimental viscosities by several orders of magnitude [69]. This discrepancy arises naturally from the implicit treatment of solvents and ions, as well as the neglect of atomic vibrations and detailed inter-atomic interactions among amino acids, due to the amino acids being represented as spherical beads. While the choice of the A1-LCD WT+NLS variant used to normalise the viscosities influences the absolute deviation between simulations and experiments, it does not alter the overall trend.

We observe that the sensitivity exhibited by the different models in their predictions of critical temperatures (Fig 4D–4F) is similarly reflected in the variation of condensate viscosity as a function of sequence mutations (Fig 6D–6F). The Mpipi-Recharged model exhibits the greatest sensitivity, with viscosity values spanning nearly two orders of magnitude among the different A1-LCD variants, and emerges as the best fit for capturing the viscoelastic behaviour of A1-LCD condensates. Specifically, the Pearson coefficient (r) assessing the strength of the correlation between simulation and experimental η values is high for Mpipi-Recharged (r = 0.8819), and moderate for Mpipi(r = 0.6308) and CALVADOS2 (r = 0.5281). In addition, the root mean squared deviation (D) describing the mean error from the experimental data is the smallest for the Mpipi-Recharged (D = 0.7314), and moderate for both Mpipi (D = 1.4359) and CALVADOS2 (D = 1.7512) models.

We also investigate whether a correlation between the critical temperature and condensate viscosity can be established for the different studied variants. A correlation between these two quantities is expected since intermolecular interaction strength, which favours condensation (i.e., increases T_c), should also diminish molecular mobility (i.e., increasing η). In Fig 7, we plot the critical temperature (normalised by the T_c of the WT+NLS sequence) vs. condensate viscosity (also normalised by the η of the WT+NLS sequence) for the different studied A1-LCD mutants as predicted by CALVADOS2 (A), Mpipi (B), and Mpipi-Recharged (C) models. Moreover, experimental in vitro data from Ref. [78] for the WT+NLS, allF and allY sequences are included (empty squares). As can be seen, the Mpipi-Recharged predicts the correlation experimentally found for these three variants (depicted by a grey band linearly extrapolated). The full phase diagram (including T_c) for the rest of the variants was not reported in Ref. [78], nevertheless, it is expected that a similar correlation as that predicted by the Mpipi-Recharged model holds since it reasonably predicts the experimental critical temperature, saturation concentration, and condensate viscosity of all these variants, as shown in Figs 2, 5 and 6, respectively. The CALVADOS2 also shows a correlation between the critical temperature and viscosity (Fig 7A). However, the CALVADOS2 predictions do not span the same range of values for viscosity as the experimental measurements (Fig 6D). For the case of Mpipi, a weak correlation between the critical solution temperature and the viscosity is observed (Fig 7B).

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Fig 7. Correlation between the critical temperature (normalised by the T_c of the WT+NLS sequence) and the condensate viscosity (also normalised by η of the WT+NLS) for the different studied A1-LCD mutants as predicted by the CALVADOS2 (A), Mpipi (B), and Mpipi-Recharged (C) models.

Experimental data from Ref. [78] for the WT+NLS, allF and allY sequences have been also included as empty squares (see legend). A linear trend to the experimental correlation between these two quantities for the three sequences measured in vitro [78] is shown with a grey band.

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Future work exploring whether a correlation between the critical solution temperature and the viscosity arises in other condensates would be highly valuable, as both properties are challenging to measure and are rarely reported under identical conditions [77, 78]. Establishing such a relationship could enable the qualitative inference of one property from the other, offering complementary insights into the stability and transport properties of biomolecular condensates.

2.4 Key intermolecular interactions in A1-LCD condensates

Computational approaches can contribute to a deeper understanding of biomolecular phase separation by evaluating the underlying molecular information that is hardly accessible in experiments [69, 74, 116]. A powerful example is the calculation of intra and inter-molecular contact maps [52, 69, 89] which provide microscopic insights about the key amino acids and protein domains enabling biomolecular self-assembly. However, the outcome of this type of analysis is intimately related to the specific model characteristics and its interaction parameters. Therefore, different models may provide notably different contact domain probabilities for the same protein condensate.

Analyzing the differences in the inter-molecular contact maps predicted by the different models we benchmark here, also allows us to understand the reasons why such models present different performance in these benchmarks. Specifically, we compute the number of times a pair of amino acids belonging to different proteins within the condensate are in contact, where ‘in contact’ is defined as being closer than a cut-off distance which depends on the identity of the amino acids in the pair (see Section 4.4 for technical details on this calculation). We then evaluate the energy contribution of each contact considering the sum of the potential energy terms in the model at the interacting distance. That is, the sum of the Wang-Frenkel and Yukawa potentials for the Mpipi-Recharged model, Wang-Frenkel and Debye-Hückel for Mpipi, and Lennard-Jones and Debye-Hückel for CALVADOS2 and the HPS family (see Section 4.4). The resulting contact energy maps reveal the key intermolecular interactions that sustain the liquid network within the different condensates.

When comparing the contact energy maps for WT+NLS condensates predicted by the different models at 0.95 T_c (S3 Fig), we find significant differences among the behaviour of the various models. The HPS and HPS-Urry models present homogeneous maps of intermolecular interactions, with most amino acid pairs presenting contact energies that differ by at most 0.002 kJ/mol. In contrast, the HPS-cation–π, Mpipi and Mpipi-Recharged models give rise to heterogeneous maps, where the contact energies among amino acid pairs can differ by more than 0.004 kJ/mol. CALVADOS2 presents an intermediate behaviour between the two extremes. Such difference can be explained by the HPS and HPS-Urry models considering electrostatic and π-π interactions as stronger contributors to biomolecular phase separation than cation–π contacts (see S4 Fig, and the relative interaction strength maps in S1 Fig). In the CALVADOS2 model, such a difference is present, but is much more moderate [55]. The HPS-cation–π, Mpipi, and Mpipi-Recharged models predict that phase separation is most strongly contributed by some specific prion-like sub-domains. Whilst all models show a strong interacting region from the 105th to 115th residue (a sub-domain enriched in Y, F, and G), the HPS-cation–π, Mpipi, and Mpipi-Recharged also present a highly interacting sub-domain from the 130th amino acid to the 135th residue, which contains 2 arginines and 1 phenylalanine. In addition, the HPS and HPS-cation–π, but more prominently the Mpipi, and Mpipi-Recharged models show two sub-domains between the 75th and 85th position (enriched in asparagine and aromatic residues) and between the 15th and 45th (rich in N, F, R and G) that significantly contribute to the stability of A1-LCD condensates through cation-π and π-π interactions (S3 Fig).

Since the Mpipi-Recharged model has consistently emerged as the most accurate model of those we have tested (Figs 1–6), we now employ it to compare the impact of amino acid mutations on the contact energy maps of A1-LCD condensates. In Fig 8, we present the contact energy maps for condensates of the WT+NLS, allF, and allW sequences at 0.95 T_c, where T_c represents the critical temperature specific to each variant (Fig 8A–8C). Notably, the contact energy maps for the allF and WT+NLS condensates exhibit striking similarities (Fig 8A and 8B). In contrast, the contact energy map for the allW condensates (Fig 8C) reveals significantly stronger contact interactions among residues near the protein terminal region, particularly from the 90th residue onward. This increase in intermolecular contacts at the protein terminal region enhances the interaction energies across the whole protein sequence, suggesting a cooperative mechanism that reinforces the condensate assembly (Fig 8C). This increase of intermolecular interactions in the allW mutant explains the large increase in viscosity observed both computationally and experimentally (Fig 6) for this variant with respect to the WT+NLS and allF sequences.

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Fig 8.

Contact energy maps of pairwise intermolecular interaction energy for allF (A), WT+NLS (B), and allW (C) condensates at T = 0.95 T_c as predicted by the Mpipi-Recharged model. The most frequent intermolecular amino acid pairwise interactions sustaining the condensates of allF (D), WT+NLS (E), and allW (F) are also displayed. Pairwise contacts have been both normalised by the highest value of residue-residue pairwise interactions and their abundance across their sequence (see Section 4.4 for further details on these calculations).

https://doi.org/10.1371/journal.pcbi.1012737.g008

We now extract from Fig 8A–8C the most frequent amino acid pairs that stabilize the condensed phase, as predicted by the Mpipi-Recharged model (Fig 8D–8F). The relative interaction strength of each pair has been normalised by the contact pair with the highest energetic contribution. As expected, the key intermolecular interactions in WT+NLS condensates are primarily cation–π (e.g., R–Y and R–F) and π–π interactions (e.g., F–F, Y–F, and Y–Y). In allF condensates, the dominant contacts include cation-π (R–F) and π–π interactions (Y–F and F–F), along with interactions involving asparagine (N) with arginine (R) and phenylalanine (F). For allW condensates, the most prevalent interactions are W–W and W–R, followed by contacts between tryptophan (W) and asparagine (N), glutamine (Q), and aspartic acid (D).

These findings align with the stickers-and-spacers framework for protein phase transitions, in which multivalent proteins are conceptualized as heteropolymers comprising ‘stickers’ (binding sites for associative interactions) and ‘spacers’ (regions between stickers) [117, 118]. Within the stickers-and-spacers framework, as proposed in Ref. [77] for A1-LCD condensates, tyrosine (Y), phenylalanine (F), and tryptophan (W) function as the primary stickers, arginine (R) acts as a context-dependent sticker, and other amino acids serve as spacers. Our simulations demonstrate that the predictions of the Mpipi-Recharged model are in excellent agreement with this framework (Fig 8D–8F).

We now focus on the most frequent interactions in WT+NLS condensates, as predicted by the Mpipi-Recharged (Fig 8E) and the CALVADOS2 model (S4D Fig). While the Mpipi-Recharged suggests that cation–π interactions such as R–Y and R–F are stronger contributors than Y–Y, F–F, and Y–F to the stability of A1-LCD condensates, the CALVADOS2 proposes that instead π–π contacts are more energetically favourable than cation–π interactions for these systems. As in the Mpipi-Recharged model, according to the Mpipi model, the five most energetically favourable contacts in A1-LCD condensates are R–Y, R–F, Y–F, F–F, and Y–Y (S4E Fig). In contrast, the HPS and HPS-Urry models do not predict cation–π interactions as main contributors for A1-LCD phase-separation (S4A and S4C Fig). Instead, HPS and HPS-Urry consider that electrostatic R–D and K–D contacts alongside π–π interactions (such as Y-F, F-F, and Y-Y, which in the model parametrization have a considerable interaction strength [76], see S1 Fig) are the key contacts driving biomolecular phase separation. The underestimation of cation–π interactions for biomolecular phase transitions was addressed by the HPS-cation–π reparametrization [56], which added an extra potential term to the model (see Section 1 for further details). Nevertheless, as shown in S4B Fig, the HPS-cation–π reparametrization significantly overestimates cation–π contacts predicting that R–Y, R–F, K–Y and K–F contribute four times more than aromatic interactions such as F–F or Y–F. Such overestimation of cation–π interactions by the HPS-cation–π model leads to very high critical temperatures for A1-LCD condensates, as shown in Fig 2B; this has been previously discussed in Refs. [31, 53, 56, 69, 87] for other RNA-binding proteins such as FUS, TDP-43, or DDX4 among others.

Overall, this analysis reveals that residue-resolution CG models considering both cation–π and π–π interactions as primary contributors to the stability of A1-LCD condensates align most closely with experimental observations. Notably, models such as Mpipi and Mpipi-Recharged, which emphasize the stronger contribution of cation–π interactions over π–π interactions, show the highest level of agreement with experimental data.

4.1 Model and methods

In our simulations, we represent the intrinsically disordered regions (IDRs) as coarse-grained structures with one bead per amino acid. The potential energy of the coarse-grained force field is defined as follows: (1)

In Table 1 we provide a summary of the different potentials used by each model studied in this work. To better visualize the differences in interactions between the different models, S1 Fig shows the normalised values of interaction strength for all amino acid pairs. Higher values indicate stronger attractive interactions, and lower values indicate weaker or repulsive interactions.

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Table 1. Table of equations used by each model.

The details of the equations are given below.

https://doi.org/10.1371/journal.pcbi.1012737.t001

Bonded interactions are modeled by a harmonic potential (2) where k = 9.6kJ/mol⋅Ų for all the models except for the HPS-cation-π which is k = 2.4kJ/mol⋅Ų and for the HPS-Urry which is k = 4.8kJ/mol⋅Ų. The equilibrium bond length is r₀ = 3.81 Å between bonded amino acid beads.

The electrostatic interactions (E_{Electrostatic}) among charged amino acids are governed by a Debye-Hückel potential (3) where q_i and q_j represent the charges of the beads i and j, ϵ_r = 80 ϵ₀, is the relative dielectric constant of water (being ϵ₀ the electric constant), r is the distance between the ith and jth beads, and κ = 1 nm⁻¹ is the Debye screening length that mimics the implicit solvent (water and ions) at physiological salt concentration (c_s∼150mM of NaCl) for the models HPS, HPS-cation-π, and the HPS-Urry model, and for the Mpipi model the value is κ = 1.26 nm⁻¹. For the CALVADOS2 and Mpipi-Recharged model κ depends on the salt concentration as (4) where B(ϵ_r) = e²/4πk_BTϵ₀ϵ_r is the Bjerrum length. The relative dielectric permittivity depends on the temperature according to this empirical law (5)

For the Mpipi-Recharged model, electrostatic interactions are modelled through a Yukawa potential given by (6) where A_ij controls the interaction between a pair of amino acids and κ modulates the salt concentration in an explicit way. The dielectric constant varies with temperature according to Eqs (4) and (5).

For the HPS, HPS-cation-π, HPS-Urry and CALVADOS2 models, the hydrophobic interactions are established using a scale of hydrophobicity derived from statistical analysis of amino acid contacts in PDB structures. These interactions are incorporated using the Ashbaugh-Hatch potential [43, 86, 120, 121] functional form given by (7) where λ_i and λ_j are parameters that account for the hydrophobicity of the ith and jth interacting particles respectively, being λ_ij = (λ_i + λ_j)/2 following the Lorentz-Berthelot mixing rules [122, 123]. The excluded volume of the different residues is given by σ_i and σ_j, where σ_ij = (σ_i + σ_j)/2, and r is the distance between the ij particles. The parameter ϵ_ij is set to 0.2 kcal/mol to reproduce experimental single-IDR radius of gyration [43].

In the Mpipi and Mpipi-Recharged models, the hydrophobic interactions are parametrized by the Wang-Frenkel potential given by (8) where (9) where the excluded volume of the different residues is given by σ_ij, r is the distance between the ij particles, ϵ_ij is the energy interaction parameter, r_c = 3σ_ij is the cut-off of the potential between those amino acids and μ = 1 and ν_ij are terms that are involved in the shape of the potential. The values of σ_ij and ϵ_ij are precisely parametrized for each interaction [61, 79]

For the HPS-cation-π model another additional component is added. A reparametrisation of the interactions between the positively charged amino acids and the aromatic ones as a Lennard-Jones potential is modelled as (10)

We employed a cut-off of 3σ_ij for the hydrophobic interactions, and 3.5 nm for the electrostatic ones [43], except for the HPS-Urry that uses a cut-off of 2.0 nm and for the CALVADOS2 that uses a cut-off of 4.0 nm [55].

4.2 Preparation of the systems and simulation details

All simulations were conducted using the LAMMPS software [124, 125]. Both Direct coexistence (DC) simulations and viscosity calculations were performed in the NVT ensemble employing a Langevin thermostat [126] with a relaxation time of 5 ps and a time step of 10 fs for the Verlet integration. The configurations for DC simulations were generated by placing 200 protein replicas in a slab with ∼17x17nm² of section and 120nm of length for a resulting average density of ∼0.1 g/cm³. Simulations to calculate the coexistence densities run a total time of the order of ∼1μs after reaching equilibrirum. Calculation of the viscosity was carried out from NVT simulations by placing 200 protein replicas in a cubic box that was isotropically compressed to reach the desired bulk density. After equilibration, production run took around ∼3 − 5μs to fully capture the stress relaxation of each specific system.

We provide Supporting information and a GitHub Repository at https://github.com/Reshiiiii/hnRNPA1_Data_Scripts that contains the LAMMPS input scripts for every studied model as well as the configuration file for the considered A1-LCD mutants. We also provide sampling videos of representative trajectories concerning the direct coexistence method, viscosity and different conformations of a protein.

4.3 Direct Coexistence technique

Employing Direct Coexistence (DC) simulations [84, 127–129], we determined the phase diagram for each of the studied proteins [78]. This method involves simulating the two coexisting phases within the same simulation box. In our approach, we arranged a high-density protein liquid alongside a very low-density counterpart. To accommodate the diverse densities, we employed an elongated simulation box, therefore allowing for the formation of both coexisting phases forming an interface perpendicular to the long direction of the simulation box. Equilibrium was attained through NVT simulations, and subsequently, we measured the equilibrium coexisting densities of both phases along the elongated side of the box. This process was repeated across various temperatures until reaching the critical temperature. To mitigate finite system-size effects near the critical point, we calculated the critical temperature (T_c) and density (ρ_c) using the law of critical exponents and rectilinear diameters [130] (11) and (12) where the notation ρ_l and ρ_v are the densities of the condensed and diluted phases, respectively. Moreover, s₁ and s₂ are fitting parameters, while the critical exponent, α = 3.06 for the three-dimensional Ising model [130]. The critical temperature error has been determined using the law of critical exponents and rectilinear diameters to estimate the system’s critical temperature by considering the last two, three, and four temperatures.

4.4 Calculating contact maps and top contacts

The computation of intermolecular and intramolecular contact maps within protein condensates is calculated from DC trajectories. Contacts were determined across all systems at temperatures approximately 0.95T_c, where T_c corresponds to the critical temperature of each system. Typically, molecular contacts are identified based on a distance criterion, with the assumption that the relative frequency of contact map occurrences (rather than absolute frequency) remains generally unaffected by the selected cut-off distance used in calculations, provided the cut-off values are reasonable. However, to accurately capture the most relevant and common residue-residue contact pairs that facilitate LLPS, it is highly recommended to account for the specific parametrized of each amino acid in terms of excluded volume and minimum potential energy interaction distance. Hence, we adopted a sequence-dependent cut-off distance equivalent to 1.2σ_ij, where σ_ij represents the mean excluded volume of the respective ith and jth amino acids [69]. Given that the minimum of the potential used lies at approximately 2^1/6σ_ij ≈ 1.122σ_ij, we set the cut-off distance slightly beyond this point, at 1.2σ_ij, to ensure significant binding. By implementing this innovative sequence-dependent cut-off scheme for each amino acid pair interaction, we can effectively filter out adjacent contacts that may coincide with actual interacting amino acids along the sequence, thus enhancing our ability to accurately identify the amino acids that positively contribute to stabilising condensates [69].

To define the relative interaction strength for the most relevant contacts, this interaction frequency described above (f_i,j) is weighted with the hydrophobic and electrostatic contributions of each amino acid pair for the model (), normalised against the occurrence frequency of each pair of amino acids in the sequence, as described by (13)

4.5 Calculating viscosities

We can determine the viscosity of the LCD model’s condensate using the Green-Kubo (GK) relation. The time-dependent mechanical response of a viscoelastic material under small shear deformation is characterized by the shear stress relaxation modulus (G(t)) [131]. When subjected to zero deformation, G(t) can be computed by correlating any off-diagonal component of the pressure tensor at equilibrium [132, 133]

(14)

In Eq 14, σ_α,β represents an off-diagonal component of the stress tensor, V is the volume, T is the system temperature and k_B is the Boltzmann constant. For isotropic systems, a more precise expression of G(t) is obtained by considering the six independent components of the pressure tensor [110, 112, 134, 135], given by (15) where N_α,β = σ_α,α − σ_β,β represents the normal stress difference. Once the relaxation modulus is obtained, shear viscosity (η) is calculated by integrating the shear stress relaxation modulus over time using the last GK formula (16)

To mitigate noise in the relaxation modulus observed in protein condensate simulations [31, 69], we adopt a specific strategy for viscosity estimation. While G(t) is smooth at short timescales so that it can be numerically integrated, at longer timescales G(t) is fitted to a series of Maxwell modes (G_i exp(-t/τ_i)) equidistant in logarithmic time, and then analytically integrated. Thus, viscosity is calculated by combining two terms as (17) where η(t₀) represents the term computed at short times by trapezoidal integration, corresponds to the part evaluated via the Maxwell modes fit at long timescales, and t₀ denotes the time after which all intramolecular oscillations of G(t) have decayed and the function becomes strictly positive and decays monotonically.

4.6 Deviation from the ideal line in interpolations

For cases where simulated saturation concentration have been plotted against experimental values (Fig 5 in the main text and S2 Fig), the following formula has been used to calculate the deviation of simulated results from experimental ones (18) where log represents the decimal logarithm.

For cases where simulated viscosity has been plotted against experimental values (Fig 6) the following formula has been used to calculate the deviation of simulated results from experimental ones (19) where log represents the decimal logarithm.