Developing bonded potentials for a coarse-grained model of Intrinsically Disordered Proteins

Abstract

Recent advances in residue-level coarse-grained (CG) computational models have enabled molecular-level insights into biological condensates of intrinsically disordered proteins (IDPs), shedding light on the sequence determinants of their phase separation. The existing CG models that treat protein chains as flexible molecules connected via harmonic bonds cannot populate common secondary structure elements. Here, we present a CG dihedral angle potential between four neighboring beads centered at C_α atoms to faithfully capture the transient helical structures of IDPs. In order to parameterize and validate our new model, we propose C_α-based helix assignment rules based on dihedral angles that succeed in reproducing the atomistic helicity results of a polyalanine peptide and folded proteins. We then introduce sequence-dependent dihedral angle potential parameters (${\text{ε}}_{\text{d}}$) and use experimentally available helical propensities of naturally occurring twenty amino acids to find their optimal values. The single-chain helical propensities from the CG simulations for commonly studied, prion like IDPs are in excellent agreement with NMR-based α-helix fraction prediction, demonstrating that the new HPS-SS model can accurately produce structural features of IDPs. Furthermore, this model can be easily implemented for large-scale simulations due to its simplicity.

Graphical Abstract

Introduction

Intrinsically disordered proteins (IDPs) and regions (IDRs) play an important role in many biological functions in the human proteome, including DNA repair^{1, 2}, cellular signaling³, and stress response⁴, and have been related with a broad range of neurodegenerative diseases such as ALS¹, Alzheimer’s⁵, and dementia⁶. IDPs are also associated with the phenomenon called cellular liquid−liquid phase separation (LLPS) to form membrane-less organelles⁵ and may drive pathological aggregation¹. Interestingly, they are biologically functional as described above without adopting a well-defined, stable tertiary structure, but have many distinct residue-level conformations. Over the past three decades, several studies have focused on the determination of transient secondary structure in IDPs^7–11, and various methods have been introduced to predict those residual structures using the combination of both experimental and computational tools^12–21. Recent studies have shed light on the existence of residual helices^{22, 23}, and β-sheet structures^{24, 25} in IDPs, which are found to be important for phase separation, contributing to their intermolecular interactions. Accurately identifying these transient secondary structures of IDPs that promote LLPS is vital to understanding the driving forces of the membrane-less organelle formation and their role in disease pathology.

Numerous biophysical experimental methods have been used to structurally characterize IDP conformational properties, which can provide information on the average ensemble values rather than complete distributions of conformational and structural variables. Local secondary structure propensities can be obtained based on residues-specific chemical shifts and residual dipolar couplings from solution-based nuclear magnetic resonance (NMR) spectroscopy²⁶ and circular dichroism (CD) ellipticities²⁷, whereas small-angle X-ray scattering (SAXS) intensities can give low-resolution information on the shape, size and folding state of IDPs^{28, 29}. Recent studies have also suggested that the single-molecule Förster resonance energy transfer (smFRET) can provide transient populations of IDP structural conformations³⁰. However, this is a low-resolution tool that lacks atomistic details³¹, and thus molecular dynamics (MD) simulations will be essential as complementary tools in studying IDPs.

MD simulations based on the fully atomistic force fields and water models have been improved extensively in studying IDPs making them transferable, which do not need any experimental input³². Although the use of all-atom representation is a reliable tool to complement experimental methods, they are not computationally efficient to simulate longer proteins and LLPS, requiring very long timescales for convergence. In contrast, coarse-grained (CG) models that replace the atomistic details with a coarser protein description, often not more than 3 or 4 beads per residue, can allow the speed-up to reach the time and length scales needed to study phase separation of IDPs. The computational expense of conformational sampling is further reduced in CG simulations by representing solvent in an implicit manner rather than atomistic explicit solvent models.

Numerous CG models have been created to faithfully study the structural and dynamic conformation of IDPs. Commonly used coarse-grained force field, Martini³³, which uses a mapping of two to four non-hydrogen atoms to one bead, can accurately capture electrostatic interactions but relies on constraints from native structures to maintain secondary structures. Therefore, it is not able to explore the protein conformational fluctuations accurately and predict the transient secondary structure propensity of proteins. Other models like CAMELOT³⁴ parameterize the model for a specific system using machine-learning tools based on all-atom data, making it non-transferable. In recent CG models, higher levels of details have been added in the backbone to account for the change in structural conformations in IDPs. A hybrid resolution (HyRes) model developed by Chen and Liu³⁵ consists of an atomistic description of the backbone, providing the semi-quantitative secondary structure propensities, whereas the SOP-IDP force field³⁶ is a two-bead model parameterized to reproduce the SAXS scattering profile of IDPs. AWSEM-IDP developed by Wolynes, Papoian, and colleagues³⁷ include ${\text{C}}_{\text{α}}$, ${\text{C}}_{\text{β}}$, and O atoms and has been parameterized for IDPs relying on bioinformatics memory using NMR experimental data and atomistic simulations to structurally bias protein fragments. Despite the challenges to using single C_α-based beads to describe local structure propensities, Latham and Zhang developed MOFF-IDP³⁸ using the combinations of maximum entropy biasing, least-squares fitting, and basic principles of energy landscape theory that reproduces experimental radius of gyration for various IDPs, however, secondary structure potential of this model is protein-specific. Another attempt has been made by introducing a multibody pseudo-improper-dihedral potential term³⁹ to the single-bead model to reproduce the nature of backbone and side-chain interactions between amino acids, which were five times slower and harder to parallelize after the addition of the complex term. Even though there were many efforts to develop CG models with backbone constraints, there is still a need for a transferable CG model able to capture the transient secondary structure of IDPs.

Alongside, numerous residue level CG models that mainly focuses on the LLPS of IDPs, namely Kim-Hummer⁴⁰, HPS-KR⁴¹, FB-HPS⁴², TSCL-M2⁴³, Mpipi⁴⁴ have been designed. We have also recently developed a C_α-based single-bead CG model that considers the physicochemical properties of all 20 naturally occurring amino acids, making it suitable for LLPS studying⁴⁵. The model uses short-range van der Waals (vdW) and long-range electrostatic to describe non-bonded interactions, while bonded interactions are only based on the harmonic bond potential. The short-range potential utilizes Ashbaugh-Hatch functional form⁴⁶, parameterized based on the hydrophobicity scale proposed by Urry et. al⁴⁷. Despite its success to study IDPs, predicting short sticky regions in IDPs that drive phase separation⁴⁵, it cannot capture the local secondary structure of IDPs since it lacks chain rigidity and depends solely on the harmonic bond potential between two consecutive beads.

Here, we introduce a transferable approach that doesn’t require further experimental input by developing new dihedral angle potentials with control parameters that modulate the population of conformations between the α-helix and extended configurations. The dihedral angle potential parameters are derived from the experimentally measured helical propensities of all twenty naturally occurring amino acids⁴⁸, using the Lifson & Roig model of helix-coil transition⁴⁹ with a model Ala-based host-guest peptide⁵⁰. We also introduce C_α-based helix definition rules based on dihedral angles to compute the helical propensity at the residue level. Sequence-dependent dihedral angle parameters are introduced to control the helicity and computed using various mixing rules combining contributions from neighboring residues. We show the ability of our model in predicting the transient α-helical structure of prion-like, low-complexity IDPs: TDP-43 CTD, hnRNPA2 LC, FUS LC and semi-randomly selected 9 test IDPs. Furthermore, we show how introducing the backbone potentials affect IDPs conformations compared to the fully flexible model and its implications for studying LLPS of IDPs.

Model development

A. Coarse-Grained IDP model

The previously developed IDP model, “HPS-Urry,” has been successfully applied for understanding the sequence-dependent LLPS of various IDPs⁴⁵. This model is based on a single-bead representation of each amino acid, where the beads are centered at the corresponding C_α positions and connected by harmonic springs. However, HPS-Urry model cannot capture the secondary structure elements of IDPs due to the absence of angle and dihedral angle potentials between successive beads. Here we develop novel, sequence-dependent dihedral angle potentials between four consecutive residues to properly capture the backbone rigidity and the secondary structure elements, especially the helical propensity. The total potential energy of the model is given by

$${\text{U}}_{\text{total}}={\text{U}}_{\text{vdw}}+{\text{U}}_{\text{elec}}+{\text{U}}_{\text{bond}}+{\text{U}}_{\text{angle}}+{\text{U}}_{\text{dihed}}$$ (1)

where U_vdw and U_elec are the non-bonded pairwise short-range van der Waals (vdW) and long-range electrostatic interactions, respectively, while U_bond, U_angle, and U_dihed are the bond, angle, and dihedral angle potentials between two, three, and four consecutive beads (Fig. 1A). Here, the first three terms, U_vdw, U_elec, and U_bond, in the potential energy are adopted from the HPS-Urry model (see Ref. ⁴⁵ for more details).

Figure 1.

(a) C_α based single-bead representation of amino acids. Schematic illustration of bond, angle, and dihedral angle between two, three and four consecutive beads in HPS-SS model. (b) Dihedral angle potential energy. Potential energy vs control parameter, ${\text{ε}}_{\text{d}}$, that accounts for the populations between extended and α-helical configurations. The dashed vertical lines denote the inflection points of the dihedral-angle free energy from the all-atom Ala₄₀ data (see Fig. S1). More negative ${\text{ε}}_{\text{d}}$ means a more stable α-helix configuration.

The non-bonded vdW interaction potential is modeled using the Ashbaugh and Hatch functional form given by⁴⁶

$${\text{U}}_{\text{vdw}}\left(\text{r}\right)=\left\{\begin{array}{cc}{\text{Φ}}_{\text{LJ}}\left(\text{r}\right)+\left(1-\text{λ}\right)ϵ,& \text{if r}\le 2^{1/6}\text{σ}\\ {\text{λΦ}}_{\text{LJ}}\left(\text{r}\right),& \text{otherwise }\end{array}\right.$$ (2)

where, ${\text{Φ}}_{\text{LJ}}\left(\text{r}\right)$ is the usual Lennard-Jones (LJ) potential:

$${\text{Φ}}_{\text{LJ}}\left(\text{r}\right)=4ϵ\left\{{\left(\frac{\text{σ}}{\text{r}}\right)}^{12}-{\left(\frac{\text{σ}}{\text{r}}\right)}^6\right\}.$$ (3)

As for previous studies⁴⁵, we set $ϵ=0.2$ kcal/mol and, for a pair of residue (i,j), ${\text{λ}}_{ij}=({\text{λ}}_i+{\text{λ}}_j)/2$, and ${\text{σ}}_{\text{ij}}=\left({\text{σ}}_{\text{i}}+{\text{σ}}_{\text{j}}\right)/2$, where ${\text{λ}}_i$ and ${\text{σ}}_{\text{i}}$ are the hydropathy value and vdW diameter of a residue i, respectively. The electrostatic interaction between two charged residues is given by the simple Debye-Hückel screened potential

$${\text{U}}_{\text{elec}}\left(\text{r}\right)=\frac{{\text{q}}_{\text{i}}{\text{q}}_{\text{j}}}{\text{Dr}}{\text{e}}^{-\text{κr}}$$ (4)

where q_i is the charge of residue i, D is the dielectric constant of water (set equal to 80), and $\text{κ}$ is the inverse screening length (= 0.1 Å^–1 at physiological salt concentration). Among twenty amino acids, we set ${\text{q}}_{\text{i}}\text{= +}{\text{q}}_{\text{0}}$ for ARG and LYS, -q₀ for ASP and GLU, where q₀ is the elementary charge of a proton.

The bond potential, U_bond, between two neighboring residues are represented by a harmonic potential

$${\text{U}}_{\text{bond}}\left(\text{r}\right)=\text{k}{\left(\text{r}-{\text{r}}_0\right)}^2$$ (5)

where r is the distance between two residues, k is the spring constant (set equal to 10 kcal/mol/Ų) and r₀ is the equilibrium distance (= 3.82 Å).

B. Angle potential

The angle potential between three consecutive C_α atoms is represented by a double-well Gaussian with the parameterized functional form taken from Kim and Hummer⁴⁰

$${\text{U}}_{\text{angle}}\left(\text{θ}\right)=\frac{-\text{ln}\left({\text{e}}^{-\text{γ}\left({\text{k}}_{\text{α}}{\left(\text{θ}-{\text{θ}}_{\text{α}}\right)}^2+{\text{ ε}}_{\text{α}}\right)}+{\text{ e}}^{-{\text{γk}}_{\text{β}}{\left(\text{θ}-{\text{θ}}_{\text{β}}\right)}^2}\right)}{\text{γ}}$$ (6)

where, $\text{θ}$ is the angle between three residues and the parameters are set as follows: $\text{γ}=0.1$ mol/kcal, ${\text{ε}}_{\text{α}}=4.3$ kcal/mol, ${\text{θ}}_{\text{α}}=1.60$ rad, ${\text{θ}}_{\text{β}}=2.27$ rad, ${\text{k}}_{\text{α}}=106.4$ kcal/(mol rad²), and ${\text{k}}_{\text{β}}=26.3$- kcal/(mol rad²). This double-well potential accounts for both helical (centered around ${\text{θ}}_{\text{α}}$) and extended ${\text{C}}_{\text{α}}\text{–}{\text{C}}_{\text{α}}\text{–}{\text{C}}_{\text{α}}$ angles (centered around ${\text{θ}}_{\text{β}}$).

C. Dihedral angle potential

Here our goal is to develop the dihedral angle potential that can accurately capture the transient α-helical structures of IDPs, that is prevalent in proteins and easily predictable. There have been several dihedral angle potentials introduced for protein CG models using periodic trigonometric functions⁵¹. A dihedral angle force field developed by Karanicolas and Brooks⁵² uses a four-term cosine-series for each pair of middle residues resulting in 1600 sets of parameters (four cosine terms for each pair) and is based on the statistical potentials derived from folded protein structures. In this article we instead derive the dihedral angle potential phenomenologically from experimentally available helical propensities of twenty amino acids⁴⁸. Specifically, the functional form of the dihedral angle potential is inspired by the dihedral angle distribution of the C_α-based coarse-grained trajectory of all-atom Ala₄₀ simulation data that shows a double-well shape (see Fig. S1). Therefore, we adopt the functional form similar to the double-well potential of the angle potential as:

$${\text{U}}_{\text{dihed}}\left(\text{ϕ},{\text{ε}}_{\text{d}}\right)\text{ = }-\text{ln}\left[{\text{U}}_{\text{dihed,α}}\left(\text{ϕ},{\text{ε}}_{\text{d}}\right)\text{ + }{\text{U}}_{\text{dihed, β}}\left(\text{ϕ},{\text{ε}}_{\text{d}}\right)\right],$$ (7)

where $\text{ϕ}$ is the dihedral angle between four consecutive ${\text{C}}_{\text{α}}$ atoms, ${\text{U}}_{\text{dihed,α}}$ and ${\text{U}}_{\text{dihed,β}}$ are the multi-Gaussian functions representing the helical and extended configurations, respectively, while ${\text{ε}}_{\text{d}}$ is the sequence-dependent parameter that controls the relative well-depth between two configurations. The helical part of the function, ${\text{U}}_{\text{dihed},\text{α}}\left(\text{ϕ},{\text{ε}}_{\text{d}}\right)$, is given by

$$\begin{array}{l}{\text{U}}_{\text{dihed,α}}\left(\text{ϕ},{\epsilon }_{\text{d}}\right)\text{ =}{\text{e}}^{-{\text{k}}_{\text{α,1}}{\left(\text{ϕ}-{\text{ϕ}}_{\text{α,1}}\right)}^{\text{2}}-{\text{ε}}_{\text{d}}}\text{+ }\\ {\text{e}}^{-{\text{k}}_{\text{α,2}}{\left(\text{ϕ}-{\text{ϕ}}_{\text{α,2}}\right)}^{\text{4}}\text{+ }{\text{e}}_{\text{0}}}\text{ +}{\text{e}}^{-{\text{k}}_{\text{α,2}}{\left(\text{ϕ}-{\text{ϕ}}_{\text{α,2}}\text{+2π}\right)}^{\text{4}}\text{ + }{\text{e}}_{\text{0}}},\end{array}$$ (8)

where ${\text{k}}_{\text{α},1}=11.4$, ${\text{k}}_{\text{α},2}=0.15$, ${\text{ϕ}}_{\text{α},1}=0.90$ , ${\text{ϕ}}_{\text{α},2}=1.02$, ${\text{e}}_{\text{0}}= 0\text{.27}$. The functional form for the extended configuration, ${\text{U}}_{\text{dihed},\text{β}}\left(\text{ϕ},{\text{ε}}_{\text{d}}\right)$, is given by

$$\begin{array}{l}{\text{U}}_{\text{dihed,}\beta }\left(\text{ϕ},{\epsilon }_{\text{d}}\right)\text{ =}{\text{e}}^{-{\text{k}}_{\beta \text{,1}}{\left(\text{ϕ}-{\text{ϕ}}_{\beta \text{,1}}\right)}^{\text{2}}\text{+}{\text{e}}_{\text{1}}\hspace{0.17em}\text{+ }{\text{ε}}_{\text{d}}}\text{+ }{\text{e}}^{-{\text{k}}_{\beta \text{,1}}{\left(\text{ϕ}-{\text{ϕ}}_{\beta \text{,1}}-2\pi \right)}^2\text{+ }{\text{e}}_{\text{1}}\hspace{0.17em}\text{+ }{\text{ε}}_{\text{d}}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{ +}{\text{e}}^{-{\text{k}}_{\beta \text{,2}}{\left(\text{ϕ}-{\text{ϕ}}_{\beta \text{,2}}\right)}^{\text{4}}\text{ + }{\text{e}}_{\text{2}}}\hspace{0.17em}+\hspace{0.17em}{\text{e}}^{-{\text{k}}_{\beta \text{,2}}{\left(\text{ϕ}-{\text{ϕ}}_{\beta \text{,2}}-2\pi \right)}^4\text{+ }{\text{e}}_2\hspace{0.17em}},\end{array}$$ (9)

Where ${\text{k}}_{\beta ,1}=1.80$, ${\text{k}}_{\beta ,2}=0.65$, ${\text{ϕ}}_{\beta ,1}\hspace{0.17em}=\hspace{0.17em}-1.55$, ${\text{ϕ}}_{\beta ,2}\hspace{0.17em}=\hspace{0.17em}-2.50$, ${\text{e}}_{\text{1}}= 0\text{.14}$, ${\text{e}}_{\text{2}}= 0\text{.40}$. Here ${\text{ε}}_{\text{d}}$ is the control parameter that depends on the sequence of the residues constituting the dihedral angle. Like in the angle potential, terms: e_0, e_1, and e₂ are responsible for the relative depths of the two basins as well as the energy barriers. The exponential functions with quartic terms are added to provide a reasonable energy barrier between two configurations (Fig. 1B). Without these quartic terms, the energy barriers will be too high for meaningful exchange. The terms with shifted exponents by $\pm 2\pi$ guarantee the periodic condition for U_dihed as a function of the dihedral angle. Width of the helix and extended basins are controlled by ${\text{k}}_{\text{α}}$ (large ${\text{k}}_{\text{α}}$ means narrow helix basin) and ${\text{k}}_{\beta }$ (smaller ${\text{k}}_{\beta }$ means wider extended basin) parameters whereas their minima are located by ${\text{ϕ}}_{\alpha }$ and ${\text{ϕ}}_{\beta }$ terms.

Figure 1B shows the dihedral angle potential function for different helix-control parameter, ${\text{ε}}_{\text{d}}$, values. As seen in the figure, the well depths of two configurations are controlled by ${\text{ε}}_{\text{d}}$, where negative ${\text{ε}}_{\text{d}}$ favors the helical conformation, while positive ε_d favors the extended conformation. The parameter ${\text{ε}}_{\text{d}}$ for a given dihedral angle will be determined by the identity of its constituent residue beads and, if necessary, neighboring residues as discussed below.

Simulation Details

CG simulations were performed using the LAMMPS molecular dynamics simulations package (Oct 2020 version)⁵³, in which both HPS-Urry and HPS-SS codes have been implemented. The simulations are done in the NVT ensemble using the Langevin thermostat with a damping coefficient of 1 ps^–1 and periodic boundary conditions. The duration of the production simulations was 5 μs with a 10ns equilibration time. Simulation data was saved every 1000th timestep.

All-atom simulations (AAMD) of Ala₄₀ were run using OpenMM 7.5.1⁵⁴ for 1.4 μs using Amber99SBws-STQ force field³² (with tip4p/2005 water model⁵⁵), solvated in a truncated octahedron box of 65 Å.

Results & Discussion

A. C_α-based helix assignment rules

Prediction of secondary structure elements (SSE) (helix, strand, coil) is an important step for the characterization of IDP conformational ensembles. The secondary structure of the IDPs can be predicted using automated tools such as DSSP^{56, 57} and Structural Identification (STRIDE)⁵⁸ at an atomistic level. DSSP is the most used predictor which assigns SSE based on the N-H and C=O hydrogen-bonding patterns. STRIDE utilizes both hydrogen bonding patterns with modified hydrogen bond energy function and backbone geometry for secondary structure analysis. Since our model is based on a single-bead representation for each amino acid, it becomes a challenge to compute the helix propensities from only ${\text{C}}_{\text{α}}$ coordinates. Several ${\text{C}}_{\text{α}}$-based assignment methods such as P-SEA⁵⁹, SABA⁶⁰, PCASSO⁶¹ have been developed. P-SEA uses ${\text{C}}_{\text{α}}-{\text{C}}_{\text{α}}$ distances (i to i + 2, i + 3, and i + 4) and two dihedral angle criteria for secondary structure assignment, whereas SABA utilizes pseudo-center (PC) position between neighboring ${\text{C}}_{\text{α}}$ beads and then assigns SSEs based on the PC-dependent geometric criteria. Recently developed PCASSO predicts SSEs based on the random forest (RF) approach, achieving high accuracy with respect to DSSP. Although these methods can be adopted in computing the helix propensities of ${\text{C}}_{\text{α}}$-based trajectories, here we first introduce new helix assignment rules solely based on the dihedral angles, mainly due to the fact that the helix conformation in our model is controlled by the dihedral angle potential only.

In our scheme, each residue is assigned a binary code, either 0 or 1, depending on dihedral angles. Specifically, when a dihedral angle, ${\text{ϕ}}_{\text{i,i+3}}$, of residues between i and i+3 falls in a specified range of the helical basin, we assign either 0 or 1 to each residue depending on assignment rules. For example, the (i,i+3) assignment rule means that, when the dihedral angle ${\text{ϕ}}_{\text{i,i+3}}$ is in the helical range, the outer two residues i and i+3 are assigned 1, while the others 0 (see Fig. S2). The middle two residues, i+1 and i+2, are assigned 1 for the (i+1,i+2) assignment rule (Fig. S2). On the other hand, when the dihedral angle lies outside the helical range, all the constituting residues are assigned 0. Once a residue is assigned 1 from any of dihedral angles that the residue participates in, it stays with 1 regardless of the other dihedral angles. The residue assigned with 1 is referred to being in a pseudo-helical (h’) state that can initiate the α-helix formation (see Fig. S2). A residue i is then defined as helical (h), if the residue and its neighboring 2n residues (i-n, i-n+1, ..., i+n-1, i+n; n=0,1, 2...) are both in pseudo-helical (h’) state (see Fig. S2). For n = 0, the pseudo-helical state (h’) becomes automatically helical (h), while for n > 1, the cooperativity of neighboring residues is required for a residue to be a part of the helical conformation.

To determine the optimal helix assignment rule among all possible cases, we carried out an all-atom MD simulation on polyalanine, Ala₄₀, and calculated the residue-level helical fractions (including α, 3₁₀, π helices) using the DSSP algorithm (Fig. 2A). As anticipated from the high helical propensity of alanine, the all-atom MD simulation yields relatively high helical fractions (> 0.4) for Ala₄₀. To calculate the helical fractions using our assignment rules from ${\text{C}}_{\text{α}}$ coordinates, it is then necessary to first identify the helical range of the dihedral angle. The free energy profile of the dihedral angle between successive ${\text{C}}_{\text{α}}$ coordinates from the all-atom Ala₄₀ data shows a steep potential well around 0.9 with inflection points at ~0.28 and ~1.48 (see Fig. S1). As a starting point, it is then natural to define the helical basin as the region between these two inflection points.

Figure 2. Helical assignment rule.

(a) C_α-based helical definitions based on all-atom MD (AAMD) C_α-trajectory worked well to match the helix fraction of Ala₄₀ computed via all-atom DSSP criteria. (b) The selection of the single dihedral angle-based helix assignment rule, comparing the chi-squared (χ²) errors between the helix fractions obtained from AAMD C_α-trajectory and DSSP criteria.

The C_α-based helical fractions are then calculated for different assignment rules using (0.28,1.48) as a helical range. Figure 2B shows ${\chi }^2$, the deviation of the CG-based helical fractions from the all-atom DSSP helical fractions, for all possible eleven assignment rules where at least two residues are assigned pseudo-helical state (h’) with varying n. The rules with two pseudo-helical states, i.e., (i,i+1), (i,i+2), …, show the comparable minima of ${\chi }^2$ at $\text{n = 1}$, while those with more pseudo-helical states require $\text{n > 1}$ to reach the minima. Closer examination of Fig. 2B shows that the (i+1,i+2) rule exhibits the least deviation from the DSSP helical fractions, although the differences from other rules are insignificant as shown in Fig. 2B. The (i+1,i+2) rule can be considered as the most natural rule from single dihedral angle criteria, since the dihedral angle is defined by the rotation angle of the two planes consisting of (i,i+1,i+2) and (i+1,i+2,i+3) around the middle bond between i+1 and i+2 residues. Similarly, the dihedral angle potentials derived by Karanicolas and Brooks⁵² depend only on the sequences of two middle residues. Therefore, the (i+1,i+2) assignment rule with n = 1 is used herein to calculate the helicity of MD trajectories.

Sensitivity of the helical range of the dihedra angle is further tested using the (i+1,i+2) assignment rule (with n = 1) and is found to be modest (see Fig. S3). For the values around the inflection points (0.28 and 1.48), the CG-based helical fractions match the all-atom DSSP helical fractions very well with ${\chi }^2$ being less than 0.06 (Fig. S3). However, the optimal dihedral angle range (0.25,1.3) is chosen based on the least ${\chi }^2$ value and thus adopted throughout this article.

B. Determination of residue-specific dihedral-angle potential parameters

Once the helical assignment rule is chosen, our next goal is to determine the sequence-dependent helix-control parameter, ${\text{ε}}_{\text{d}}$, of the dihedral angle potential in Eqs. (8) and (9). For a given dihedral angle, ${\text{ϕ}}_{\text{i,i+3}}$, ${\text{ε}}_{\text{d}}$ is determined by four constituting residues i, i+1, i+2, and i+3 with appropriate weights given by

$${\text{ε}}_{\text{d}}=\frac{{{\displaystyle \sum }}_{\text{i}=1}^4{\text{ε}}_{\text{d},\text{i}}{\text{η}}_{\text{i}}}{{{\displaystyle \sum }}_{\text{i}=1}^4{\text{η}}_{\text{i}}},$$ (10)

where ${\text{ε}}_{\text{d,i}}$ is the helical “energy” of a residue i that depends on the residue type, and 𝜂_i is the unnormalized weight. Here we derive ${\text{ε}}_{\text{d,i}}$ from the experimental context-independent helical propensities of Moreau et. al.⁴⁸ for naturally occurring 20 amino acids. For this purpose, the Lifson & Roig (LR) model of the helix-coil transition is adopted to estimate the helical propensities of twenty amino acids from coarse-grained MD simulations. In the LR analysis, a residue is in a helical state if it is helical (h) according to our assignment rule, otherwise it is in a coil state. A residue in a coil state is assigned a statistical weight factor,1.0. A residue of type i in a helical state but not within a helical segment is assigned a weight v_i, while a residue in a helical segment is assigned w_i except for the terminal residues in the segment which have weight v_i.⁴⁹ The helical propensities, w_i, are then fitted against the experimental data.⁴⁸

First, a parameter, ${\text{ε}}_{\text{d,A}}$, for alanine, is determined from the CG Ala₄₀ simulations by computing the helical propensities for different ${\text{ε}}_{\text{d,A}}$ values (Fig. S4). The chain length of 40 is chosen to minimize the effects of chain length on the helical propensities. Using the LR analysis, the helical propensity of alanine, w_A, is calculated from the CG trajectories. The estimated helical propensity of alanine decreases monotonically with ${\text{ε}}_{\text{d,A}}$, as anticipated from the fact that the helical basin well-depth of the dihedral angle potential becomes shallower with increasing ${\text{ε}}_{\text{d}}$ parameter. The non-bonded interactions have little effect on w_A as three different hydrophobicity, ${\text{λ}}_{\text{A}}$_, values yield ${\text{ε}}_{\text{d,A}}$ within 2% difference from one another. Simulated helical propensity, w_A, is then compared to the experimental helical propensity for alanine, ${\text{w}}_{\text{A,exp}}= 1\text{.39}$ to determine the ${\text{ε}}_{\text{d,A}}$.⁴⁸ As shown in Fig. S4, ${\text{ε}}_{\text{d,A}}=\hspace{0.17em}-1.65$ yields the experimental helical propensity of alanine.

Next, the parameters, ${\text{ε}}_{\text{d,i}}$, of the remaining 19 amino acids are determined by simulating Ala-based guest-host systems similar to the one used by Moreau et. al.⁴⁸ Here we use two systems, A₂₀X₄A₂₀ and A₂₀XA₂₀, where X is the guest residue. Unlike the polyalanine A₄₀, the identity of the guest residue influences the total potential energy, as the nonbonded interactions depend on the hydrophobicity value of X and the dihedral angle potential parameter, ${\text{ε}}_{\text{d}}$, is affected not only by the parameter ${\text{ε}}_{\text{d,X}}$ value, but also by the weight factors, ${\text{η}}_{\text{i}}$. However, since the polyalanine simulations show little or no effects of the non-bonded pair interactions on the helical propensity, we expect this to be the case for the guest-host systems also. Thus, we set ${\text{λ}}_{\text{A}}= 0\text{.523}$ for alanine taken from the HPS-Urry model, while ${\text{λ}}_{\text{X}}= 0$ and 1 are used for the guest residue to ensure that the non-bonded pairwise interactions have a negligible effect on estimating ${\text{ε}}_{\text{d,X}}$. For dihedral angle potential, two weighting schemes, or mixing rules, are used to see their effects on ${\text{ε}}_{\text{d,X}}$. These are referred as 0110 and 1111 mixing rules, where the 0110 mixing rule means that the unnormalized weight factor, $\text{η}$, is set equal to 1 for i+1 and i+2 and 0 for the others, while for the 1111 mixing rule, the weight factors, ${\text{η}}_{\text{i}}$, for all four residues are set equal to 1.

Figure 3A shows the helical propensity of the guest residue estimated via the LR analysis for the 1111 mixing rule, where the guest-residue helical propensity is shown to decrease to zero as ${\text{ε}}_{\text{d,X}}$ increases. As expected, the hydrophobicity value of the guest residue has no effect on the helical propensity. Using the nineteen helical propensity data of Moreau et. al⁴⁸, the control parameters, ${\text{ε}}_{\text{d,X}}$, are then determined for the remaining residues (see Table S1). Figure 3B shows that the estimated control parameters for 1111 and 0110 mixing rules from two guest-host systems have a linear relationship with the relative helical free energies derived from the NMR chemical shifts for the 20 amino acids from Moreau et. al.⁴⁸

Figure 3. Parameterization of the dihedral angle potential.

(a) Parameterization of amino acids using A₂₀XA₂₀ guest-host system, via the (i+1,i+2) assignment rule with n = 1 and 1111 mixing rule (b) Control parameters (${\text{ε}}_{\text{d,X}}$) obtained using the (i+1,i+2) assignment rule with n = 1 for different guest-host models and mixing rules have a linear relationship with the relative free energies for the 20 amino acids derived from the NMR chemical shifts⁴⁸.

C. CG prediction of TDP-43 helical propensities

To test the predictability of the helix secondary structure for IDPs using our CG model with the parameters derived above, we simulated the CTD domain of the TDP-43 protein, which is an RNA binding protein whose aggregates have been involved in neurodegenerative diseases, such as amyotrophic lateral sclerosis (ALS)²³. The CTD domain of TDP-43 has been shown to exhibit a transient α-helical region in the 320–341 domain,²³ and thus is well-suited to evaluate the performance of our HPS-SS model. Furthermore, ALS mutations, as well as target mutations designed by both experimental and computational efforts, have identified the importance of the helical region in tuning the phase behavior of TDP-43.^{23, 62} Thus it is desired that the HPS-SS model with ability to predict the helix structure can be applicable in simulating the LLPS of TDP-43 with right helical conformations.

From the available NMR data²³, a single residue-specific secondary structure propensity (SSP) score⁶³ is calculated based on the deviations of NMR chemical shifts from Poulsen IDP/IUP random coil chemical shifts⁶⁴, known as secondary chemical shifts.⁶⁵ The SSP score in Fig. 4 shows a helical region between residue number 320 and 341 with the helical fractions exceeding 0.2. MD simulation was then performed on the TDP-43 CTD with GHM tag (TDP-43_267–414) with our HPS-SS model. The residue-level helix fraction was computed using the ${\text{C}}_{\text{α}}$-based helical assignment rule and compared with experimental data. Figure 4 shows the comparison between CGMD results and experimental SSP score. Overall, the HPS-SS model with three different parameter sets is able to capture the most helical region (aa:321–334) correctly with comparable helix fractions. On the other hand, the HPS-SS model with parameters derived from the A₂₀X₄A₂₀ system overestimates the helicity outside 321–334. This can be explained by the fact that the parameters, ${\text{ε}}_{\text{d,i}}$, for non-alanine residues are lower than those from the A₂₀XA₂₀ system, thus making the corresponding dihedral angles more helical-friendly. The model parameters derived from the A₂₀XA₂₀ guest-host system with 1111 mixing rule underestimates the helix fraction by ~10 % per residue in the region 321–334, while predicting the overall low helix fractions outside this region in agreement with experimental data.

Figure 4.

Comparison of the helix fraction of TDP-43 CTD obtained via parameterization of residues using the (i+1,i+2) assignment rule with n = 1 for different mixing rules with the NMR-based SSP-predicted helix fraction.

The largest discrepancy between simulations and experiments occurs at residue number 320 and 321 (with 320 being proline). The NMR chemical shifts show that the α-helical segment (aa: 320–340) of TDP-43_267–414 starts with the Pro-320. On the other hand, the parameter, ${\text{ε}}_{\text{d,i}}$, of Pro in the HPS-SS model is most positive, implying that it is least helix stabilizing. Within the current mixing rules, the existence of Pro in the sequence proves to be helix-breaking, leading to near zero helix fraction.

D. New helical assignment rule based on two dihedral angles

The main drawback of the simple, single-dihedral-angle based helical assignment rule is the failure to detect the helicity of proline that starts as the first residue of a helix segment. Although proline is known to act as a helix breaker, it can act as a hydrogen bond acceptor via the carbonyl group, forming the hydrogen bond with the N-H group of the amino acid located four residues later (i+4) along the protein sequence⁶⁶ (Fig. 5A). Therefore, a new helical assignment rule that accounts for the (i,i+4) hydrogen bonding is warranted.

Figure 5.

Schematic representation of (a) hydrogen bond formation of proline (residue i) via its carbonyl oxygen to the amino group of residue i+4 to stabilize $\text{α-helix}$. (b) (i,i+4) helical assignment rule based on the two consecutive dihedral angles between residue i and i+4. (c) Control parameters (${\text{ε}}_{\text{d,X}}$) were obtained using (i,i+4) helical assignment rule for 1–1001-1 mixing rule and A₂₀XA₂₀ guest-host system.

Here we consider two consecutive dihedral angles, ${\text{ϕ}}_{\text{i,i+3}}$ and ${\text{ϕ}}_{\text{i+1,i+4}}$, between residues i and i+4 to define helicity. When these two dihedral angles lie in the helical range, the residues i and i+4 are defined as pseudo-helical (h’) as shown in Fig. 5B, thereby referred as (i,i+4) assignment rule. As above, a residue i is defined as helical (h) if the residue and its neighboring residues (i-1 and i+1 for n = 1) are in pseudo-helical (h’) state (Fig. 5B). Note that our (i,i+4) rule is similar to P-SEA scheme (with n = 0)⁵⁹ except for the distance criteria. The dihedral angle range for the helical basin is optimized by fitting the helix fraction against an all-atom Ala₄₀ simulation data. It is found that the (i,i+4)-rule based helical fractions for the helical region match very well with the all-atom DSSP results that averaged over a trajectory using the helical range of (0.25,1.7) (see Fig S5). Again, these values are not far from the inflection points (0.28,1.48) of the dihedral angle free energy of the all-atom Ala₄₀ trajectory (Fig. S1). The new rule, (i,i+4) is also in good agreement with all-atom DSSP results (α-helix only) to capture individual conformations of the given residue over the trajectory as presented in Fig. S6. For instance, the new rule was 93% accurate compared with DSSP estimates for 10th residue (See Fig. S6A and S7A) whereas the minimum accuracy of 72% is determined for residue at position 30 (See Fig. S6B and S7B). Additionally, the new helical assignment rule is validated against folded proteins from Lindorff-Larsen et al.⁶⁷, comparing the helix fraction from DSSP and (i,i+4) rule based on C_α coordinates from Protein Data Bank⁶⁸ (See Fig. S8). The average chi-squared, ${\chi }^2$ value is 0.14, which shows that the assignment rule is faithfully located the helix positions for folded proteins.

The control parameter ${\text{ε}}_{\text{d}}$ for a given dihedral angle ${\text{ϕ}}_{\text{i,i+3}}$ is determined by considering the fact that two hydrogen bonds, (i−1,i+3) and (i,i+4), enforce ${\text{ϕ}}_{\text{i,i+3}}$ to be in the helical range. Therefore, the dihedral angle potential is affected mostly by its two outer residues, i and i+3, as well as its neighboring residues, i-1 and i+4. Hence, a new mixing rule, denoted by 1–1001-1 (where 1- and -1 refer to the preceding and succeeding residues, respectively), for a dihedral angle ${\text{ϕ}}_{\text{i,i+3}}$ is given by

$${\text{ε}}_{\text{d}}=\frac{{{\displaystyle \sum }}_{\text{j}=\text{i}-1}^{\text{i}+4}{\text{ε}}_{\text{d},\text{j}}{\text{η}}_{\text{j}}}{{{\displaystyle \sum }}_{\text{j}=\text{i}-1}^{\text{i}+4}{\text{η}}_{\text{j}}},$$ (11)

with ${\eta }_j=1$ for j = i-1, i, i+3, and i+4 and 0 otherwise. The residue-specific parameters, ${\text{ε}}_{\text{d,i}}$, for twenty amino acids are derived via the same procedure as (i+1,i+2) rule above (see Fig. S9) and range from −2.59 for Ala to 3.70 for Pro as shown in Fig. 5C.

Figure 6 shows the residue-level helical fractions for TDP-43 from the CG model with the new (i,i+4) helical assignment rule and ${\text{ε}}_{\text{d,i}}$ parameters (see Table S1) compared to the experimental SSP score. The CG model captures both the local trend and the magnitude of the α-helical fraction of TDP-43_267–414. In particular, the helical fraction at Pro-320 is significantly higher (~ 0.2) than the single-dihedral-angle based model presented above. Helix fractions of the conserved region (aa:320–343) match quantitively with the NMR data. The HPS-SS model is also applied to two other RNA binding proteins, namely FUS LC⁶⁹ and hnRNPA2 LC⁷⁰, which have experimentally been shown to be fully disordered (i.e., low helical fraction). Our model also predicts low helix fractions for both proteins in agreement with the NMR chemical shifts (Fig. S10). In summary, the HPS-model with dihedral potential parameters was able to accurately predict the local helical propensities of the prion-like, low-complexity IDPs.

Figure 6.

Helix fractions were obtained using the newly developed HPS-SS model for TDP-43 CTD compared against the SSP-predicted helix fraction. Our model was able to faithfully capture both the trend and the magnitude of the α-helical fraction. Snapshot of TDP-43 CTD simulated using the HPS-SS model (aa:320–334 is shown as red).

E. Effects of bonded potentials on conformational ensemble of IDPs

To explore how the addition of the backbone constraints via angle and dihedral angle potentials affects the conformations of IDPs, MD simulations are carried out on the previously studied⁴⁵ 42 IDPs of various lengths and sequences using the HPS-SS model with (i,i+4) rule-based parameters (see Table S1).

Figure 7 shows the average radius of gyration for these IDPs in comparison with the previous model (HPS-Urry) without angle and dihedral angle potentials. As shown in Fig. 7, the R_g values from the HPS-SS are smaller than those from the HPS-Urry except for a few (notably tau) IDPs. The distributions of most IDPs from the HPS-SS are shifted towards smaller R_g values (see Fig. S11). The smaller R_g values for the HPS-SS may be due to cooperative non-bonded interactions between locally rigid chain segments resulting in stronger attractions. However, for all 42 IDPs, the addition of the angle and dihedral angle potentials results in overall less than ~9% deviations from the HPS-Urry model based on the chi-square scoring function⁴⁵. Among the 42 IDPs simulated here, r15, r17, protα-N, protα-C, protein-L, and erm show significant helix fractions (> 0.3, see Fig. S12). The difference between the R_g distributions as well as the helix fractions from both models is measured using the relative entropy (Kullback-Leibler divergence)⁷¹ and Jensen-Shannon divergence metric⁷², showing that the major deviations in R_g distribution from the HPS-Urry model come from the high helical content of the proteins (ProTα-C, R17d, IBB, hCyp, K44) captured by the HPS-SS model (Fig S11 & S12).

Figure 7.

Scatter plot of simulated R_g using the HPS-Urry and the HPS-SS model for 42 IDPs, Black (dot-dash) lines represent an ideal linear relationship between these two models. Simulated R_g from two models has a strong linear relationship (${\text{R}}^{\text{2}}=0\text{.94}$).

F. Limitations of the model and future work

We showed that our model works well with prion-like, low-complexity IDPs. Predictive capabilities of the model are tested against semi-randomly selected 9 IDPs with available NMR data, namely: ACTR⁷³, α-synuclein⁷⁴, p53⁷⁵, tau-k18⁷⁶, SIC1⁷⁷, paaA2⁷⁸, Ntail⁷⁹, NCBD⁸⁰, and N17-polyQ⁸¹. Overall, our model correctly predicts the helix positions in most proteins but fails to match the magnitude of the helicity for some of them (See Fig. 8). Like for low-complexity IDPs with no substantial helical propensity, the new model doesn’t overestimate the helicities for p53, tau-k18, and SIC1. It also works to determine the helical regions of IDPs with high residual helicity such as paaA2, NCBD, Ntail, and N17-polyQ. Among the test proteins, α-synuclein is predicted to have a quite high helicity based on our model even though it should have a minimal helical propensity in its free state⁷⁴. However, the locations of the helices predicted by the models match with the helix locations when α-synuclein is micelle-bound⁷⁴. This result is true for ACTR^{73, 82} as well, for which the model predicts two helical domains compared to one main residual helical structure in the free state⁷³. Charged residues: arginine and glutamic acid are found to have larger deviations against experimental helix fractions when the amino acid sequence of test proteins is analyzed. Helix formation by the charged residues may be more complex than we assumed in our model since it involves complex salt bridges⁸³ and side-chain interactions⁸⁴. Given the simplicity of the model (based on Cα positions only), the current model can still be considered a good approach with the limitations that can be fixed in future work.

Figure 8.

Helix fractions obtained using the HPS-SS model for 9 IDPs is tested against SSP-predicted helix fraction. For ACTR and α-synuclein, experimental helix fraction in free (red) and bound state (magenta) is given.

Conclusions

Here, we have introduced new dihedral angle potentials that allow us to compute the transient helical propensity of IDPs. The dihedral angle potentials do not require any further input for helical fraction calculations but are parameterized using experimental helical context-independent propensities of amino acids. We proposed new C_α-based helical assignment rules based solely on dihedral angles that agree well with DSSP results of all-atom MD simulations and folded proteins. The single-chain helical propensity of TDP-43 CTD, FUS LC, hnRNPA2 LC computed using our CG model were in good agreement with NMR data. The new dihedral angle potentials allow us to accurately predict the helical propensities of IDPs with some limitations and can be easily implemented in the large-scale assembly simulations of IDPs, thus enabling us to explore the role of structure in the function of IDPs and phase separation.