Calibrated Langevin-dynamics simulations of intrinsically disordered proteins

We perform extensive coarse-grained (CG) Langevin dynamics simulations of intrinsically disordered proteins (IDPs), which possess fluctuating conformational statistics between that for excluded volume random walks and collapsed globules. Our CG model includes repulsive steric, attractive hydrophobic, and electrostatic interactions between residues and is calibrated to a large collection of single-molecule fluorescence resonance energy transfer data on the interresidue separations for 36 pairs of residues in five IDPs: $\alpha \text{−}$, $\beta \text{−}$, and $\gamma$-synuclein, the microtubule-associated protein $\tau$, and prothymosin $\alpha$. We find that our CG model is able to recapitulate the average interresidue separations regardless of the choice of the hydrophobicity scale, which shows that our calibrated model can robustly capture the conformational dynamics of IDPs. We then employ our model to study the scaling of the radius of gyration with chemical distance in 11 known IDPs. We identify a strong correlation between the distance to the dividing line between folded proteins and IDPs in the mean charge and hydrophobicity space and the scaling exponent of the radius of gyration with chemical distance along the protein.

Figure 1

Absolute value of the electric charge per residue $Q$ versus the hydrophobicity per residue $H$ (using the shifted and normalized Monera hydrophobicity scale) for known IDPs (small circles) and 221 folded proteins [32] (small open squares). The IDPs $\alpha \text{S}$ (large circle), $\beta \text{S}$ (large square), $\gamma \text{S}$ (upward triangle), $\text{ProT}\alpha$ (diamond), MAPT (star), ATN (pentagon), HMG-17 (hexagon), TOPO-1 (leftward triangle), SPRP (rightward triangle), and the folded protein lysozyme C (X) are highlighted. The line $Q=2.785H-1.151$ represents the dividing line between IDPs (above the line) and natively folded proteins (below the line) given in Ref. [32].

Figure 2

(a) Electric charge $Q_i$ (in units of the electron charge $q_e)$ and (b) hydrophobicity ${\epsilon }_i$ as a function of the residue index $i$ originating from the N-terminus for the IDPs $\alpha \text{S}$ (thick, solid red line), $\beta \text{S}$ (thick, dashed blue line), $\gamma \text{S}$ (thick, dotted green line), MAPT (thin, solid purple line), $\text{ProT}\alpha$ (thin, dashed orange line), and the folded protein lysozyme C (thin, dotted black line). We quote the normalized and shifted Monera hydrophobicity scale [34], where 0 is the least and 1 is the most hydrophobic [see Eq. (8)]. Data for each $i$ is averaged over 31 nearby residues, with data at the endpoints reflected beyond the endpoints to reduce edge effects. This averaging is employed to visualize the general differences between biologically important regions of the proteins. Note that the curves for $Q_i$ and ${\epsilon }_i$ are not strongly sensitive to the averaging length.

Figure 3

The backbone dihedral angle distribution $P^{\text{UA}}\left({\varphi }_{ijkl}\right)$ obtained from the UA description of $\alpha \text{S}$ (light green solid line) with only hard-sphere atomic interactions plus stereochemical constraints obtained from the Dunbrack database of high-resolution protein crystal structures. We fit $P^{\text{UA}}\left({\varphi }_{ijkl}\right)$ for the UA model using four coefficients (Table 2) in the Fourier series in Eq. (4) (blue dotted line). We show that $P^{\text{CG}}\left({\varphi }_{ijkl}\right)$ from Langevin dynamics simulations of the CG model for $\alpha \text{S}$ with only bond-length, bond-angle, and dihedral angle interactions in Eqs. (2), (3), and (4) (thick solid red line) matches that from the hard-sphere UA model for $\alpha$-synuclein. $P\left({\varphi }_{ijkl}\right)$ from stretches of $\alpha$ helices (orange horizontal lines) and $\beta$ sheets (purple vertical lines) that are longer than 10 residues in the Dunbrack database of high-resolution protein crystal structures are also shown for comparison. For ease of visual comparison, the dihedral angle distributions from $\alpha$-helical and $\beta$-sheet structures were not normalized.

Figure 4

Seven commonly used hydrophobicity scales (Kyte-Doolittle [47], Monera [34], augmented Wimley-White [48, 49], Eisenberg [50], Miyazawa [51], Sharp, and Sharp with solvent-solute size difference corrections [52]) for each amino acid type that have been shifted and normalized so that $0\le {\epsilon }_i\le 1$. The “average” value for each residue indicates the shifted and normalized average over the seven shifted and normalized hydrophobicity scales. The residues are ordered according to their average ${\epsilon }_i$.

Figure 5

FRET efficiencies $F_{\text{eff}}$ for $\alpha \text{S}$ (upper left), $\text{ProT}\alpha$ (upper right), $\beta \text{S}$ (lower left), and $\gamma \text{S}$ (lower right) from experimental measurements (black circles) and CG Langevin dynamics simulations. We include data for three choices for the strength of the hydrophobic and electrostatic interactions ${\overline{\epsilon }}_a$ and ${\overline{\epsilon }}_{\mathrm{es}}$ for each IDP: (1) ${\overline{\epsilon }}_a=0$ and ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$ such that the chains behave as extended coils (blue diamonds), (2) the optimal ${\alpha }_{\text{CG}}$ for each protein with ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$, where the root-mean-square deviations between the experimental and simulation $F_{\text{eff}}$ are minimized (red squares), and (3) the optimal ${\alpha }_{\text{CG}}$ for each protein with no electrostatic interactions ${\overline{\epsilon }}_{\text{es}}=0$ (purple triangles). The error bars for $F_{\text{eff}}$ from the simulations give the error in the mean. Error bars that are not visible are smaller than the symbols.

Figure 6

Root-mean-square deviation $\Delta$ in $F_{\text{eff}}$ between experiments and simulations versus the ratio ${\alpha }_{\text{CG}}$ of the hydrophobicity and electrostatic interactions for $\alpha \text{S}$ (red circles), $\beta \text{S}$ (blue squares), and $\gamma \text{S}$ (green triangles), MAPT (orange stars), and $\text{ProT}\alpha$ (purple diamonds).

Figure 7

FRET efficiencies $F_{\text{eff}}$ for MAPT from smFRET experiments (black solid line with circles) and three CG simulations: (1) ${\overline{\epsilon }}_a=0$ and ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$ (blue diamonds), (2) the optimal ${\alpha }_{\text{CG}}=0.52$ with ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$, where the root-mean-square deviations between the experimental and simulation $F_{\text{eff}}$ are minimized (red squares), and (3) the optimal ${\alpha }_{\text{CG}}=0.52$ with no electrostatic interactions ${\overline{\epsilon }}_{\text{es}}=0$ (purple triangles).

Figure 8

RMS deviations ${\Delta }_{\min }$ in $F_{\text{eff}}$ (and its error) between the experiments and simulations for (from left to right) $\alpha \text{S}$ (red), $\beta \text{S}$ (blue), $\gamma \text{S}$ (green), $\text{ProT}\alpha$ (purple), and MAPT (orange) for nine hydrophobicity models. The labeling scheme for the nine hydrophobicity models is (hydrophobicity scale)-(mixing rule) as numbered in Sec. II. The RMS deviations are calculated from CG simulations with ${\alpha }_{\text{CG}}$ chosen such that the RMS deviations are minimized for each hydrophobicity scale and mixing rule. The black dotted line indicates the average error in $F_{\text{eff}}$ from smFRET experiments.

Figure 9

Radius of gyration $R_g\left(N_p\right)$ of seven IDPs from experiments [27, 42, 56, 57, 58] (black circles) and simulations of three CG models: (1) ${\epsilon }_a=0$ and ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$ such that the chains behave as extended coils (blue diamonds), (2) ${\alpha }_{\text{CG}}=0.50$ and ${\overline{\epsilon }}_{\text{es}}={\kappa }_{\text{es}}$ (green pentagons), and (3) ${\alpha }_{\text{CG}}=0.50$ and ${\overline{\epsilon }}_{\text{es}}=0$ (purple triangles). The IDPs are ordered from shortest to longest (left to right).

Figure 10

(Left) Radius of gyration $R_g\left(n\right)$ (thick lines) versus chemical distance $n$ along the chain for several IDPs with $N_p\ge 90$ so that $R_g\left(n\right)$ is in the power-law scaling regime. Power-law fits of the data to $R_g=R^0{n}^{\nu }$ for $n>20$ are shown as thin lines. The error in $R_g$ is comparable to the line thickness. (Right) Power-law scaling exponent $\nu$ as a function of the distance $d$ from the dividing line between folded and intrinsically disordered proteins (Fig. 1). The dotted line follows $\nu =0.47+0.85d$.