Accurate model of liquid-liquid phase behavior of intrinsically disordered proteins from optimization of single-chain properties

Abstract

Many intrinsically disordered proteins (IDPs) may undergo liquid-liquid phase separation (LLPS) and participate in the formation of membraneless organelles in the cell, thereby contributing to the regulation and compartmentalization of intracellular biochemical reactions. The phase behavior of IDPs is sequence dependent, and its investigation through molecular simulations requires protein models that combine computational efficiency with an accurate description of intramolecular and intermolecular interactions. We developed a general coarse-grained model of IDPs, with residue-level detail, based on an extensive set of experimental data on single-chain properties. Ensemble-averaged experimental observables are predicted from molecular simulations, and a data-driven parameter-learning procedure is used to identify the residue-specific model parameters that minimize the discrepancy between predictions and experiments. The model accurately reproduces the experimentally observed conformational propensities of a set of IDPs. Through two-body as well as large-scale molecular simulations, we show that the optimization of the intramolecular interactions results in improved predictions of protein self-association and LLPS.

Keywords: biomolecular condensates; force field parameterization; intrinsically disordered proteins; liquid–liquid phase separation; protein interactions.

Fig. 1

Assessing the HPS, AVG, and HPS-Urry models using experimental data reporting on single-chain conformational properties. (A) Probability distributions of the λ parameters calculated from 87 min–max normalized hydrophobicity scales. Lines are the λ parameters of the HPS model (blue), the average over the hydrophobicity scales (orange) and the HPS-Urry model (green) (28). Intramolecular PRE intensity ratios for (B) the S42C mutant of α-Synuclein and (C) the S143C mutant of A2 LCD from simulations and experiments (22, 43) (black). (D) ${\chi }^2$ values quantifying the discrepancy between simulated and experimental intramolecular PRE data, scaled by the hyperparameter $\eta =0.1$ (Materials and Methods). Relative difference between simulated and experimental radii of gyration (E) for proteins that do not readily undergo phase separation alone and (F) for variants of A1 LCD, with negative values corresponding to the simulated ensembles being more compact than in experiments.

Fig. 2

Flowchart illustrating the Bayesian parameter-learning procedure (Materials and Methods).

Fig. 3

Selection and performance of the M1–3 models with respect to the training data. (A) Overview of the optimal λ sets with $\eta \langle {\chi }_{\mathit{PRE}}^2\rangle <21$ and $\langle {\chi }_{R_g}^2\rangle <3$ collected through the parameter learning procedures started from ${\lambda }_0=$ AVG (upward triangles), M1 (squares), and M2 (downward triangles). The gray gradient shows the Spearman’s correlation coefficient between experimental and simulated R_g values for the A1 LCD variants in the training set. Colored open symbols indicate the M1 (blue upward triangle), M2 (orange square), and M3 (green downward triangle) scales, whereas the adjacent values are the respective Spearman’s correlation coefficients. (B) Covariance matrix of the λ sets with $\eta \langle {\chi }_{\mathit{PRE}}^2\rangle <21$ and $\langle {\chi }_{R_g}^2\rangle <3$. (C) M1 (blue), M2 (orange), and M3 (green) scales. Solid lines are guides for the eye, whereas the gray shaded area shows the mean ±2 SD of the λ sets with $\eta \langle {\chi }_{\mathit{PRE}}^2\rangle <21$ and $\langle {\chi }_{R_g}^2\rangle <3$. Comparison between (D) $\eta {\chi }_{\mathit{PRE}}^2$ and (E) ${\chi }_{R_g}^2$ values for the HPS model (gray) and the optimized M1 (blue), M2 (orange), and M3 (green) models.

Fig. 4

(A) Comparison between experimental and predicted radii of gyration (SI Appendix, Table S1), R_g, for the HPS, HPS-Urry, and M1–3 models. (B) Zoom-in on the R_g values of the A1 LCD variants, with Pearson’s r coefficients for this subset of the training data reported in the legend.

Fig. 5

Testing the M1–3 models using experimental findings on protein–protein interactions. Comparison between experimental (black) intermolecular PRE rates (SI Appendix, Table S3) and predictions from the M1 (blue), M2 (orange) and M3 (green) models for (A–C) FUS LCD and (D and E) A2 LCD calculated using the best-fit correlation time, τ_c. (F and G) Discrepancy between calculated and experimental intermolecular PRE rates ${\chi }_{\mathit{PRE}}^2$ as a function of τ_c. (H) Second virial coefficients, B₂₂, of FUS LCD (circles) and A2 LCD (squares) calculated from two-chain simulations of the M1–3 models. Error bars are SEMs estimated by bootstrapping 1,000 times 40 B₂₂ values calculated from trajectory blocks of 875 ns. (I) Probability of the bound state estimated from protein-protein interaction energies in two-chain simulations of the M1–3 models. (J) Dissociation constants, K_d, of FUS LCD (circles) and A2 LCD (squares) calculated from two-chain simulations of the M1–3 models. For p_B and K_d, error bars are SDs of 10 simulation replicas. Lines in H and J are guides for the eye.

Fig. 6

Protein concentrations (A–C) in the condensate and (D–F) in the dilute phase from slab simulations of the M1–3, HPS, and HPS-Urry models performed at 50 ${}^{°}$C (closed symbols), 37 ${}^{°}$C (crosses in H), and 24 ${}^{°}$C (open symbols). Red open squares show experimental measurements at ∼ 24 ${}^{°}$C (A, C, D, and F) and ∼ 4 ${}^{°}$C (B and E). Correlation between ${\log }_{10}(c_{\mathit{sat}}/\text{M})$ from simulations and experiments for (G) diverse sequences and (H) A1 LCD variants. Solid lines show linear fits to the simulation data at 50 ${}^{°}$C. Dashed lines show linear fits to the HPS-Urry data at 24 ${}^{°}$C (G and H) and to the M1–3 data at 37 ${}^{°}$C (H). Values reported in the legends are Pearson’s correlation coefficients. Error bars are SEMs of averages over blocks of 0.3 µs. We note that the correlation coefficients reported in G are associated with a substantial uncertainty as they are calculated over only three (HPS), four (HPS-Urry), and five points (M1–3).

Fig. 7

Correlation between chain compaction and LLPS propensity for aromatic and charge variants of A1 LCD. ${\log }_{10}(c_{\mathit{sat}}/\text{M})$ vs. ν_sim for A1 LCD variants from simulations performed using the (A) M1, (B) M2, and (C) M3 models. Black and colored circles indicate aromatic and charge variants, respectively. Black lines are linear fits to the aromatic variants. (D–F) Residuals from the linear fits of A–C for the charge variants of A1 LCD as a function of the NCPR. Values reported in the legends are Pearson’s correlation coefficients. Error bars of ${\log }_{10}(c_{\mathit{sat}})$ values are SEMs of averages over blocks of 0.3 µs. Error bars of ν_sim are SDs from fits to $R_{ij}=R_0|i-j{|}^{{\nu }_{\mathit{sim}}}$ in the long-distance region, $|i-j|>10$. Solid lines are linear fits to the data. Dotted lines in D–F are lines of best fit to the experimental data by Bremer et al. (16).

Fig. 8

Comparing residue–residue interactions in dilute solution and in the condensate. Energy maps from simulations of the M1 model of (A–C) FUS LCD and (D–F) A2 LCD calculated using nonelectrostatic interaction energies. The 1D projections of the energy maps for (G) FUS LCD and (H) A2 LCD, normalized by the absolute average interaction energy $|\langle E\rangle |$ and shifted vertically for clarity. Colors indicate that the energies were calculated within a single chain at infinite dilution (blue), between two chains in the dilute regime (orange), and between a chain located at the center of a condensate and the surrounding chains (green).