Improved coarse‐grained model for studying sequence dependent phase separation of disordered proteins

Abstract

We present improvements to the hydropathy scale (HPS) coarse‐grained (CG) model for simulating sequence‐specific behavior of intrinsically disordered proteins (IDPs), including their liquid–liquid phase separation (LLPS). The previous model based on an atomistic hydropathy scale by Kapcha and Rossky (KR scale) is not able to capture some well‐known LLPS trends such as reduced phase separation propensity upon mutations (R‐to‐K and Y‐to‐F). Here, we propose to use the Urry hydropathy scale instead, which was derived from the inverse temperature transitions in a model polypeptide with guest residues X. We introduce two free parameters to shift (Δ) and scale (µ) the overall interaction strengths for the new model (HPS‐Urry) and use the experimental radius of gyration for a diverse group of IDPs to find their optimal values. Interestingly, many possible (Δ, µ) combinations can be used for typical IDPs, but the phase behavior of a low‐complexity (LC) sequence FUS is only well described by one of these models, which highlights the need for a careful validation strategy based on multiple proteins. The CG HPS‐Urry model should enable accurate simulations of protein LLPS and provide a microscopically detailed view of molecular interactions.

1. INTRODUCTION

Liquid–liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) has been shown to be the fundamental mechanism driving the formation of membraneless organelles in the cell, which are crucial for organizing and maintaining spatiotemporal balances of protein concentrations. ¹ , ² , ³ , ⁴ , ⁵ Recent experimental ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ and computational studies ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ have indicated that nonspecific interactions between amino acids driven by their hydrophobic, aromatic, or electrostatic character play the leading role in driving the protein LLPS. To study how these various interactions come into play for a specific IDP sequence one must conduct extensive experimental studies and use computational methods for gaining an in‐depth understanding of the phase separation mechanism at the molecular level. ⁹ , ¹¹ , ²² All‐atom simulations can provide detailed information on molecular interactions ¹⁶ , ²³ but cannot be used currently to study the LLPS of IDPs.

Coarse‐grained (CG) modeling techniques have been quite successful in assisting and leading studies on drivers of phase separation and assembly of IDPs previously. ⁸ , ¹⁷ , ¹⁹ , ²⁰ , ²¹ , ²⁴ , ²⁵ , ²⁶ , ²⁷ , ²⁸ , ²⁹ The CG simulations can be conducted with off ²¹ , ³⁰ , ³¹ , ³² , ³³ or on lattice ¹⁹ models with varying resolutions ³³ , ³⁴ and the interactions between the CG amino acids can be informed by physics‐ or knowledge‐based potentials. ³⁵ We have previously proposed a one bead per amino acid CG model ³⁰ using the Kapcha‐Rossky (KR) hydropathy scale ³⁶ referred to as the HPS‐KR model hereafter which has been successfully used in numerous previous studies of IDP LLPS. ⁸ , ⁹ , ¹⁷ , ²⁶ , ²⁷ , ²⁸ , ³⁷ , ³⁸ Specifically, the transferable HPS‐KR model has been used to study the phase separation of several IDP sequences like the intrinsically disordered region (IDR) of DEAD‐box helicase LAF‐1 protein also referred to as LAF‐1 RGG, ⁹ the low complexity (LC) domain of the Fused in Sarcoma (FUS) protein ⁸ , ²⁸ , ³⁸ , ³⁹ , ⁴⁰ and disordered C‐terminal domain (CTD) of the TAR DNA binding protein (TDP‐43). ²⁶ We also recently extended the HPS‐KR model to capture thermoresponsive LLPS behavior of IDPs, ³¹ the role of post‐translational modifications in perturbing LLPS, ³⁷ and the multicomponent LLPS of IDP and polynucleotide mixtures ¹⁷ ; the simulations results were in very good agreement with the experimental observations. ³⁹ , ⁴¹ , ⁴²

Although the HPS‐KR model (in tandem with experimental studies) has advanced our understanding of the sequence‐dependent LLPS of IDPs, several potential shortcomings in the model have also been identified. ⁴³ , ⁴⁴ For example, the radius of gyration (R_g) of many IDPs in this model is significantly different from their values estimated from experimental observables. ⁴³ Also, this model is not able to account for qualitative differences in LLPS when mutating Arg and Tyr residues to Lys and Phe, respectively. ⁶ , ⁷ , ⁹ , ⁴⁵ It was proposed that the nonbonded interactions parameters in the HPS‐KR model can be refined using a combination of maximum entropy optimization and least‐squares regression to better capture experimental R_g values. ⁴³ Recently, Das et al. ⁴⁴ also proposed another parameter refinement strategy focusing on modifying interaction strengths of certain residue pairs based on the expected behavior from experimental studies and guided by bioinformatics‐based potentials such as the Kim‐Hummer (KH) model which uses the statistical contact potential derived by Miyazawa and Jernigan (hereafter referred to as MJ). ⁴⁶

In this paper, we use a different model refinement strategy to address the aforementioned issues, while preserving the transferability of our CG framework by avoiding overfitting of parameters based on limited experimental data. To accomplish this, we replace the hydropathy scale used to inform nonbonded interactions between the CG beads from KR to the Urry scale. ⁴⁷ As the Urry scale was derived based on the LLPS of elastin‐like model polypeptides, we believe it is more appropriate for parameterizing a CG model to study protein phase separation. We also use a modified approach to tune the free parameters of the CG model to obtain an excellent match between the simulated and experimental R_g values for 42 IDPs covering an extensive range of sequence properties. Furthermore, the new model termed as HPS‐Urry can provide near‐quantitative prediction of coexistence densities for well‐studied IDP sequences, FUS LC and DDX4 at room temperature.

2. RESULTS AND DISCUSSION

2.1. Comparison between the KR and Urry hydropathy scales

There are over 100 hydropathy scales of 20 naturally occurring amino acids reported in the literature. ⁴⁸ Although straightforward, it is time and resource consuming to examine all the hydropathy scales for building an IDP CG model. Instead, in this study, we focus only on two hydropathy scales, namely the KR scale, ³⁶ which was used in the previous models, ¹⁷ , ³⁰ , ³⁷ and the scale proposed by Urry et al. ⁴⁷ (hereafter denoted as the Urry scale). We also compared the ability of these hydropathy scales to estimate single‐chain IDP properties with the 210 parameters MJ statistical contact potential used in the KH model we used previously to capture IDP phase behavior. The KR hydropathy scale is an atomistic scale that assigns each atom as hydrophilic or hydrophobic depending on their partial charges based on an all‐atom forcefield and it has one advantage over other scales since it is straightforward to account for nonnatural amino acids or amino acids modified via post‐translational modifications. The HPS model with the KR scale has also been applied to study the effects of posttranslational modifications on IDP phase behavior. ³⁷ On the other hand, the Urry hydropathy scale is based on the inverse transition temperature of a polypeptide (VPGXG)_n with n = 11, where X stands for the guest residue to be evaluated. This polypeptide was derived from the repeat elastin‐like sequence VPGVG (where the second valine residue is replaced by a guest residue) that undergo the lower critical solution temperature (LCST) transition in which the proteins transition from water‐soluble state to self‐assembled state upon increasing the temperature. The inverse transition temperature is defined as the temperature at which the polypeptide undergoes the LCST transition. This temperature depends on the hydrophobicity of constituent amino acid residues, and thus is well correlated to the hydrophobicity of the guest residue X.

Figure 1 presents the KR and Urry hydropathy scales rescaled from zero for the least hydrophobic residue to unity for the most hydrophobic residue. The biggest difference between the two scales occurs for Arg where it is the least hydrophobic (0) in the KR scale but is in the middle (0.56) for the Urry scale. It has been pointed out that the KR scale enhances the LLPS of IDPs upon replacing Arg by Lys (R2K mutation) in contrast to experimental observations. ⁴⁴ This is due to the increase in the interaction strength for the modified IDPs since the hydropathy scale of Lys (0.51) is larger than that of Arg. In fact, this short‐coming of the KR scale has led us to examine the Urry scale, since it can be expected that the Urry scale would predict the weakening of the LLPS for such modified IDPs in agreement with experiments as the hydropathy scale of Lys (0.38) is smaller than that of Arg. Other residues that exhibit significant differences (Table S2) between the two scales are Glu (0.46 for KR vs. 0 for Urry), Pro (1.0 for KR vs. 0.76 for Urry), Leu (0.97 for KR vs. 0.72 for Urry), and Ile (0.97 for KR vs. 0.71 for Urry).

FIGURE 1.

The normalized (0 to 1) Kapcha‐Rossky (KR) and Urry hydropathy scales used in the HPS CG modeling framework for IDPs. Aromatic residues (Trp, Tyr, Phe, and His) are colored red, Hydrophobic residues (Leu, Ile, Met, Val, and Ala) are colored blue, Polar residues (Asn, Ser, Thr, and Gln) are colored green, other residues (Pro, Cys, and Gly) are colored gray and charged residues (Arg, Lys, Asp, and Gln) are colored yellow

2.2. Optimization of interaction parameters

The LJ interaction model contains two free parameters, μ and Δ, that account for the strength of non‐electrostatic contributions and the affinity between amino acids relative to that of solvent, respectively. To obtain the optimal values for these parameters, we first determine the optimal Δ value for a given μ by simulating R _g values for 42 IDP sequences (compared with 12 sequences previously used ³⁰ ) listed in Table S1. The scoring function to be minimized is given by the standard chi‐square function

$${\chi }^2=\frac{1}{N}\sum _{i=1}^N\frac{{\left(R_g^{i,\mathit{exp}}-R_g^{i,\mathit{sim}}\right)}^2}{{\left(R_g^{i,\mathit{exp}}\right)}^2},$$ (1)

where $R_g^{i,\mathit{exp}}$ and $R_g^{i,\mathit{sim}}$ are experimental and simulated R _g values for an ith IDP sequence. Figures 2a and S1 show the behavior of χ ² for the Urry scale as a function of the offset parameter Δ for four different μ values. For a given μ, χ ² exhibits a minimum close to 0.02, which suggests that the simulated R _g values deviate overall less than ~14% from the experimental R _g values at this parameter value. Interestingly, the minimum χ ² values are almost independent of the scale parameter μ, remaining close to 0.02 for 0.5 ≤ μ ≤ 2. For the KR hydropathy scale and MJ contact potentials, similar trends are observed with minimum χ ² between 0.015 and 0.02. The set of optimal (μ, Δ) values, at which χ ² is minimum, is then obtained by fitting the scoring function via polynomial curve and is shown in Figure 2b. For all three scales considered (KR, Urry, and MJ), the optimal μ and Δ exhibit a linear relationship given by

$${\mathrm{\Delta }}^{\mathit{opt}}=a{\mu }^{\mathit{opt}}+b,$$ (2)

where a ^Urry = 0.58, a ^KR = 0.69, and a ^MJ = 0.39, while b =  − 0.5 independent of scales used.

FIGURE 2.

(a) Fits to chi‐square values (χ²) from comparison of experimental and simulated Rg data on 42 IDP sequences on a range of shift (Δ) for a selected scale (μ) allows for locating the offset providing minimum chi square value. (b) Scale and offset for optimum chi‐square change linearly with respect to each other for Urry, Kapcha Rossky (KR), and Miyazawa Jernigan (MJ) interaction parameters. (c) Scatter plot of experimental and simulated Rg for 42 IDPs using an optimal model, Black (dot‐dash) lines represent 20% deviation in Rg. (scale (μ) = 1.0, offset (Δ) = 0.08) (d) T _θ versus scale (μ) for FUS LC and DDX4 sequences shows a scenario where unique sequences might show drastic changes in behavior between different optimal models (μ, Δ)

Figure 2c shows the comparison between simulated and experimental R _g values for an optimal parameter set of μ = 1.0 and Δ = 0.08 for the Urry scale. The model shows excellent agreement between simulated and experimental R _g values for many IDP sequences. For most sequences the simulated R _g values lie within 20% of the experimental values (dot‐dashed lines). The largest discrepancy occurs for an16 (5.0 nm for experiment vs. 3.2 nm for simulation), where the model predicts more compact structures compared with the experimental observation. For protein‐L and hcyp, the trend is opposite where the simulated R _g values are about 35% higher than the experimental values (Figure S2). Comparing the amino acid compositions of these IDP sequences with the entire set, we do not observe any significant differences that could explain these discrepancies (Figure S3). However, in the absence of explicit backbone potentials, such as pseudo‐angle potential between three neighboring beads and pseudo‐dihedral angle potential between four neighboring beads, as well as considering possible differences in simulation and experimental conditions, the Urry scale captures the configurations of many IDP sequences very well. The KR and MJ scales after appropriate optimization based on Equation (8) also give the R _gvalues with similar accuracy to the experimental values (data not shown).

2.3. Effects of interaction parameters on IDP single‐chain properties

The optimal set of parameters μ and Δ shows high degeneracy in the (μ, Δ) parameter space where the scoring function, χ ², has minima of similar magnitude along a “critical” line. In order to narrow the range of the parameters for the model, we investigate the effects of the parameters along the critical line on the phase behavior of two IDPs, DDX4 IDR ⁴⁹ and FUS LC proteins. ⁶ , ⁸ , ¹⁶ They are quite different in their amino acid compositions (Figure 3a,c, sequences in Supporting Information) and have been well studied for their phase separation making them excellent candidates for testing the model's ability to predict the IDP phase behavior. It has been shown that the coil‐to‐globule transition temperature, T _θ, of a single‐chain IDP is highly correlated with its critical temperature of phase separation, T _c. ¹⁵ Since computing T _θ is computationally less expensive, we calculated T _θ as a first step for these IDPs using REMD simulations along the critical line in the parameter space. Figure 2d presents T _θ values for the DDX4 and FUS LC at different μ values along the critical line from the Urry scale. The coil‐to‐globule transition temperature of the DDX4 depends weakly on μ, while that of FUS LC increases sharply from ~330 K at μ = 0.5 to ~420 K at μ = 2.

FIGURE 3.

(a) Amino acid composition of DDX4 sequence used for this simulation. (b) Coexistence concentrations of DDX4 at 100 mM salt and room temperature from simulations (Red, circles) and experiments (Black, triangles). (c) Amino acid composition of FUS LC sequence used for this simulation. (d) Coexistence concentrations of FUS LC at 100 mM salt and room temperature from simulations (Green, circles), and experiments (Black, triangles)

The large difference in the behavior of these two sequences arises from their amino acid compositions. Using Equations (7) and (1), one can express λ _ij in terms of the optimal μ ^opt as

$${\lambda }_{\mathit{ij}}=\left({\lambda }_{\mathit{ij}}^0-a\right){\mu }^{\mathit{opt}}-b,$$ (3)

where, as μ ^opt increases, λ _ij increases for ${\lambda }_{\mathit{ij}}^0>a$, but decreases for ${\lambda }_{\mathit{ij}}^0<a$. For the Urry scale with a ^Urry = 0.58, the amino acids, Glu, Asp, and Lys, contribute to the decrease in the interaction energy when μ ^opt increases, while Trp, Tyr, Phe, Pro, His, Ile, Leu, Met, and Val contribute to the increase in the interaction energy. Since the DDX4 contains a significant number of Asp, Glu, Phe, Met, and Pro residues, the competition between opposing contributions from these residues leads to the fact that T _θ is almost independent of μ ^opt. On the other hand, FUS LC is a protein mostly consisting of Ser, Gln, Gly, and Pro residues. The normalized hydropathy scales of Ser, Gln, and Gly are about 0.57 resulting in λ _ij among themselves remaining close to 0.49 for 0.5 ≤ μ ^opt ≤ 2. On the other hand, λ _ij for Pro‐Pro varies from 0.58 at μ ^opt = 0.5 to 0.86 at μ ^opt = 2. Also λ _ij between Pro and other residues (Ser, Gln, and Gly) increases, albeit to a lesser degree, as μ ^opt increases. Consequently, the intramolecular interactions for FUS LC become more attractive as μ ^opt increases, resulting in the increase of T _θ values.

For the KR scale, the hydropathy scales of most residues lie below a ^KR = 0.69, resulting in T _θ decreasing for both DDX4 and FUS LC as μ ^opt increases (see Figure S4). For the MJ scale, the T _θ of both DDX4 and FUS LC also decreases with increasing μ ^opt. Since T _θ is highly correlated with the critical temperature of LLPS, these observations suggest that the KR and MJ scales predict undesirably low T _c values for these two proteins.

2.4. Urry scale captures the IDP phase behavior

The single‐chain properties of DDX4 and FUS LC (see Figure 2d) suggest that the Urry scale may be more appropriate to study the phase behavior of these IDPs. Recently, experimentally derived coexistence densities were used to scale the interaction strengths in the MARTINI CG forcefield ³⁴ for capturing the LLPS of the FUS LC sequence. ⁵⁰ We want to test if our optimized Urry scale‐based CG model above can capture the concentrations of coexisting phases of FUS LC and DDX4 without any further modifications. We simulated the coexistence densities of these two proteins using the Urry scale with the optimal parameter set of μ = 1, and Δ = 0.08. Figure 3b,d show the temperature‐concentration phase diagrams of DDX4 and FUS LC together with the experimentally measured coexisting concentrations at 300 K. We see excellent agreement between simulated and experimental coexistence densities of both DDX4 ⁴⁹ (Figure 3b) and FUS LC ⁸ (Figure 3d) with the differences at condensed phases less than 7% for both DDX4 and FUS LC. Upon varying model parameters, we observe large fluctuations in the coexistence concentrations for FUS LC at 300 K (Figure S5) which corresponds to the higher sensitivity of FUS LC phase behavior on model parameters observed earlier with the coil‐to‐globule transition temperature in the previous section. Hence, this warrants a careful validation strategy for model parameters in our CG framework as certain sequences based on their amino acid composition could be more prone to dramatic changes in phase behavior between various model parameters which were deemed optimal for capturing IDP phase behavior based solely on only comparing R _g values. Using the same parameter set (μ = 1, Δ = 0.8), we then investigated the model's ability to capture the effect of certain bulk mutations to the IDP sequence (sequences in SI) on its phase separation propensity. As mentioned earlier, the Urry scale would predict that the replacement of arginine residues with Lysine residues (R2K) on a protein sequence reduces phase separation propensity since the hydropathy value of Arg is larger than that of Lys. Figure 4a shows this anticipated behavior for the bulk R2K mutation of DDX4 showing the loss in phase separation propensity for two sequences DDX4 (Figure 4a, R2K, red lines joining ∆) and LAF‐1 RGG (Figure 4b, R2K, red lines joining ∆). We see the phase boundary shrinking and the critical temperature decreasing for the mutated sequences compared with the wild type (WT) (Figure 4a,b, WT, black lines joining O) in agreement with experimental observations. ⁴⁹ On the other hand, the KR scale yielded the opposite trend on phase separation upon this mutation, which led to the proposed modifications in the interaction strengths of Arg and Lys residues based on experimental and bioinformatics studies. ⁴⁴

FIGURE 4.

Phase diagrams for DDX4 (a) and LAF‐1 RGG (b) sequence variants show shifts in phase separation propensity upon bulk mutations as expected from experimental data

In addition to studying the effects of the Arg to Lys (R2K) mutation, we also tested the model for the effects of mutations of aromatic residues on phase separation propensity, for example, the Phe to Ala (F2A) mutation, which is also supposed to reduce phase separation propensity according to experimental studies. ⁶ , ⁷ , ⁴⁵ , ⁴⁹ For the DDX4 sequence we see the F2A mutation reducing phase separation propensity in agreement with experimental studies (Figure 4a, F2A, orange lines joining ⊲). The model was also able to predict the decrease in phase separation propensity upon replacement of Tyr with Phe (Y2F) residues for the LAF‐1 RGG sequence (Figure 4b, Y2F, blue lines joining ∇) as seen in experiments, which is not captured in the KR and the MJ scales. ³² Other mutations such as Tyr to Trp (Y2W) in LAF‐1 RGG (Figure 4b, Y2W, green lines joining ⊲) showed increased phase separation propensity in agreement with the experimental data for the LAF‐1 RGG sequence. ⁹

3. CONCLUSION

We propose a new CG model for studying IDP phase behavior, called the HPS‐Urry model which attempts to improve the previous HPS‐KR model using the hydropathy scale proposed by Urry et al. ⁴⁷ We also apply a rigorous parameterizing and testing strategy to tune the parameters by comparing easily measurable quantities like the radius of gyration and coexistence densities with their known experimental values. The use of the Urry hydropathy scale for informing nonbonded interactions between amino acids to study liquid–liquid phase separation is supported by the fact that this scale was derived from the transitions of polypeptides from a water‐soluble state to an assembly of proteins into a condensed phase. We are able to capture experimentally observed phase behavior of IDPs and effects of bulk mutations without requiring any additional parameter refinements to the HPS‐Urry model. Future work will focus on extending this model to capture sequence‐dependent LCST and UCST transitions in IDPs, similar to what was proposed earlier for the HPS‐KR model. ³¹

4. METHODS

In the HPS modeling framework developed earlier, ³⁰ each amino acid of an IDP is represented by a spherical bead that is connected to neighboring beads via a harmonic spring. The total interaction energy is given by

$$U_{\mathit{tot}}=\sum _{i,j}{\varphi }_{\mathit{ij}}^{\mathit{vdw}}+\sum _{i,j}{\varphi }_{\mathit{ij}}^{\mathit{el}}+\sum _{i=1}^{N-1}{k}_b{\left(r_{i,\mathrm{i}+1}-r_0\right)}^2,$$ (4)

where ${\varphi }_{\mathit{ij}}^{\mathit{vdw}}$, and ${\varphi }_{\mathit{ij}}^{\mathit{el}}$ are the short‐range van der Waals (vdW) and long‐range electrostatic interactions between residues i and j, while r _i,i + 1 is the distance between two neighboring residues, i and i + 1, with the spring constant k _b = 20 kJ/Ų and an equilibrium bond length r ₀ = 3.82 Å. The vdW interaction between residues i and j is modeled using the Ashbaugh and Hatch functional form given by ⁵¹

$${\varphi }_{\mathit{ij}}\left(r\right)=\left\{\begin{array}{c}{\varphi }_{\mathit{ij}}^{\mathit{LJ}}\left(r\right)+\left(1-{\lambda }_{\mathit{ij}}\right)ϵ,r\le 2^{1/6}{\sigma }_{\mathit{ij}}\\ {\lambda }_{\mathit{ij}}{\varphi }_{\mathit{ij}}^{\mathit{LJ}}\left(r\right),r>2^{1/6}{\sigma }_{\mathit{ij}},\end{array}\right.$$ (5)

where ${\varphi }_{\mathit{ij}}^{\mathit{LJ}}\left(r\right)$ is the standard Lennard‐Jones (LJ) potential shown below:

$$\begin{array}{c}{\varphi }_{\mathit{ij}}^{\mathit{LJ}}\left(r\right)=4ϵ\left[{\left(\frac{{\sigma }_{\mathit{ij}}}{r}\right)}^{12}-{\left(\frac{{\sigma }_{\mathit{ij}}}{r}\right)}^6\right].\end{array}$$ (6)

Here ϵ is set to 0.2 kcal/mol and σ _ij = (σ _i + σ _j)/2 where σ _i is the van der Waals radius of residue i, while λ _ij controls the interaction strength between residues i and j and is adapted from hydropathy scales or contact potentials.

The long‐range electrostatic interactions between charged residues are described by a simple Debye‐Hückel potential,

$$\begin{array}{c}{\varphi }_{\mathit{ij}}^{\mathit{el}}\left(r\right)=\frac{q_i{q}_j}{4\mathrm{\pi }\mathit{Dr}}{e}^{-\mathit{\kappa r}}\end{array}$$ (7)

where q _i is the charge of residue i located at the center of the corresponding bead, D is the dielectric constant of the solvent medium, and κ is the inverse Debye screening length. Here we set D = 80, the dielectric constant of water and κ = 1 nm⁻¹, the inverse Debye length at physiological salt concentrations of ~100 mM. Residue charges are set according to pH 7, such that q _i =  + e for Arg and Lys, and −e for Asp and Glu, where e is the elementary charge. Here the charge of His is set to zero instead of +0.5e adapted by the KH model. ³²

In the previous model, λ _ij values were fixed from the KR hydropathy values scaled from 0 (Arg) to 1 (Phe, Pro) while ϵ was optimized. In this study, λ _ij values are optimized against the experimentally measured radius of gyration (R _g) of IDPs ⁵² from hydropathy scales or contact potentials. Specifically, we set

$$\begin{array}{c}{\mathrm{\lambda }}_{\mathit{ij}}=\mathrm{\mu }{\lambda }_{\mathit{ij}}^0-\mathrm{\Delta }\end{array}$$ (8)

where ${\lambda }_{\mathit{ij}}^0$ values are contact potentials or hydropathy scales scaled from 0 to 1, while μ and Δ are free parameters. For hydropathy scales of 20 amino acids, the arithmetic sum was used to obtain the cross‐interaction parameters, that is, ${\lambda }_{\mathit{ij}}^0=\frac{1}{2}\left({\lambda }_i+{\lambda }_j\right)$, where λ _i is the rescaled (from 0 to 1) hydropathy value of residue i. The parameter μ scales the strength of the vdW potentials compared with the physical electrostatic interactions, while Δ is the offset parameter that balances the preference of residue–residue interactions relative to residue‐solvent interactions. These two parameters are optimized using the experimental R _g values of 42 IDP sequences (see Table S1).

Replica Exchange Molecular Dynamics (REMD) simulations of single chain IDPs were performed to compute R _gvalues and coil‐to‐globule transition temperatures (T _θ) using the LAMMPS ⁵³ software package. A total of 16 replicas were used with the temperature ranging from 200 to 600 K for IDPs with chain length less than 200. For longer IDP sequences, a total of 36 replicas with the temperature ranging from 170 to 640 K were used. In addition, coexistence simulations of several IDPs were conducted to map their phase diagrams using the HOOMD‐Blue 2.9.3 software package ⁵⁴ , ⁵⁵ following the protocol proposed in our previous work. ³⁰ , ⁵⁶ A python code is provided for readers to setup and simulate a given IDP sequence with the optimized model using HOOMD‐Blue at the following location (https://bitbucket.org/jeetain/hoomd_slab_builder).

AUTHOR CONTRIBUTIONS

Roshan Regy: Data curation; formal analysis; investigation; validation; writing‐original draft. Jacob Thompson: Data curation; formal analysis; investigation. Youngchan C. Kim: Conceptualization; data curation; formal analysis; investigation; methodology; project administration; software; supervision; validation; writing‐review & editing. Jeetain Mittal: Conceptualization; data curation; formal analysis; investigation; methodology; project administration; supervision; writing‐review & editing.

Supporting information

Appendix S1: Supporting Information

Regy RM, Thompson J, Kim YC, Mittal J. Improved coarse‐grained model for studying sequence dependent phase separation of disordered proteins. Protein Science. 2021;30:1371–1379. 10.1002/pro.4094