Figure 1

Correlation between the number of $\alpha$ amino acid residues computed using the DSSP ($n_{\mathrm{DSSP}}$) and dihedral-angle ($n_{\mathrm{dihedral}}$) definitions. The shading scale provides the probability of structures characterized by both $n_{\mathrm{DSSP}}$ and $n_{\mathrm{dihedral}}$, wherein the probabilities for values of $n_{\mathrm{dihedral}}$ for a given $n_{\mathrm{DSSP}}$ are normalized to that $n_{\mathrm{DSSP}}$. Three representative structures (ribbon diagrams) are shown for three values of ($n_{\mathrm{DSSP}},n_{\mathrm{dihedral}}$) with sticks representing main-chain H bonds.Reuse & Permissions

Figure 2

Correlation between the numbers of $\beta$ amino acid residues computed using the DSSP definition ($n_{\mathrm{DSSP}}$) and using dihedral angles ($n_{\mathrm{dihedral}}$). The shading-coded probabilities are normalized as in Fig. 1. Three representative structures are shown for three sets of ($n_{\mathrm{DSSP}},n_{\mathrm{dihedral}}$) values. In the conformations shown, red, blue, and gray sticks denote oxygen, nitrogen, and carbon atoms, respectively.Reuse & Permissions

Figure 3

Interplay between secondary structure geometry and hydrophobic energetics. (a) Distribution of C${}_{\beta }$-C${}_{\beta }$ distance $r_{ij}$ in $\alpha$ and $\beta$ structures in polyalanine for all amino acid residue pairs $I<J-1$. Here the conformations were simulated using the formulation in Ref. [47], which sets $M_{IJ}=0$ and thus $E_{\mathrm{HP}}=0$ for interactions between alanine sidechains. Both the dihedral-angle (dashed curves) and DSSP (solid curves) definitions were used to identify $\alpha$ and $\beta$ structures (as marked in the figure). Inset: the corresponding cumulative distributions using the DSSP definition. The corresponding cumulative distributions using the dihedral-angle definition (not shown) are essentially identical. (b) Potential of mean force of a pair of methane molecules (smooth line) in 396 TIP4P water molecules at 298.15 K computed using the test-particle insertion method [34]. Also shown is an example of our $E_{\mathrm{HP}}$ function in Eq. (5) (line with apparent rectilinear parts). The shaded box emphasizes that the first peak of the $\alpha$-helix distribution in (a) overlaps with the db region in (b).Reuse & Permissions

Figure 4

Effect of the attractive range of hydrophobic interactions on the ratio of amino acid residues in $\alpha$ versus $\beta$ conformations. $\langle N_{\alpha }\rangle /\langle N_{\beta }\rangle$ ratios were computed in accordance with the description in the text for $R_{\mathrm{cutoff}}$ $=4.50$, $5.25$, $6.25$, and $6.50$ Å, and plotted using different symbols. The arrows indicate increasing values of $R_{\mathrm{cutoff}}$. Secondary structures are determined using (a) the DSSP and (b) the dihedral-angle definitions. Here and in subsequent figures with $\epsilon /T$, $\epsilon =1$ is used to set the energy scale for the interaction strength $\epsilon /T$.Reuse & Permissions

Figure 5

Effect of the desolvation barrier (db) on the populations of $\alpha$ and $\beta$ conformations. Secondary structures were determined according to the dihedral-angle (a) and the DSSP (b) methods. Filled and open symbols are results from simulations without db (${\mathcal{E}}_{\mathrm{db}}=0$) and with db (${\mathcal{E}}_{\mathrm{db}}=5/9{\mathcal{E}}_{\mathrm{cm}}$), respectively, using Eqs. (4) and (5) with ${\mathcal{M}}_{IJ}=0.5$ and ${\mathcal{E}}_{\mathrm{cm}}=1$. Lines drawn through data points are merely guides for the eye.Reuse & Permissions

Figure 6

Desolvation barriers can be avoided by $\beta$-like conformations. Shown are the top (a) and front (b) views of an example $\beta$-hairpin conformation with significantly different separations $d_1$ and $d_2$ between pairs of sidechains above and below the plane of the sheet; see text for details.Reuse & Permissions

Figure 7

Effects of db's on the Ramachandran plot. Contours and color coding show the distribution of $-\ln \left[\Omega (\phi ,\psi )\right]+\mathrm{const}$, where $\Omega (\phi ,\psi )$ is the probability of a residue with dihedral angles within a bin of $2^{\circ }\times 2^{\circ }$ centered at $(\phi ,\psi )$. Results were simulated with ${\mathcal{M}}_{IJ}=0.5$ in Eq. (4) and are shown for four different parametrizations of $E_{\mathrm{HP}}$ defined by Eq. (5): (a) $E_{\mathrm{HP}}=0$ (${\mathcal{E}}_{\mathrm{cm}}={\mathcal{E}}_{\mathrm{db}}=0$); (b) no contact attraction (${\mathcal{E}}_{\mathrm{cm}}=0$) but with db (${\mathcal{E}}_{\mathrm{db}}=10/9$); (c) with contact attraction (${\mathcal{E}}_{\mathrm{cm}}=1$) but no db (${\mathcal{E}}_{\mathrm{db}}=0$); and (d) with both contact attraction and db (${\mathcal{E}}_{\mathrm{cm}}=1$, ${\mathcal{E}}_{\mathrm{db}}=10/9$).Reuse & Permissions

Figure 8

Combined effects of db and hydrophobic interaction strength on secondary structure in model polyleucine (a),(b) and polyvaline (c),(d). The average numbers of $N_{\alpha }$ and $N_{\beta }$ residues are defined by dihedral angles on the left (a),(c) and by DSSP on the right (b),(d). Data for no-db ($E_{\mathrm{db}}=0$) and with-db models ($E_{\mathrm{db}}>0$) are plotted, respectively, with filled and open symbols. Lines are merely guides for the eye. Results were computed using Eqs. (2) and (5) with $M_{IJ}=0.9$ [54] and ${\mathcal{E}}_{\mathrm{cm}}=1$. Following Ref. [36], ${\mathcal{E}}_{\mathrm{db}}=(5/9){\mathcal{E}}_{\mathrm{cm}}$ is used for the with-db models.Reuse & Permissions

Figure 9

Combined effects of db and hydrophobic interaction strength on secondary structure in the alternate model. As Fig. 8 except the results here were computed using the modified potential in Eq. (4) with ${\mathcal{M}}_{IJ}=0.1$ for polyleucine and ${\mathcal{M}}_{IJ}=0.05$ for polyvaline. As in Fig. 8, here we apply Eq. (5) with ${\mathcal{E}}_{\mathrm{cm}}=1$, ${\mathcal{E}}_{\mathrm{db}}=0$ for the no-db models and ${\mathcal{E}}_{\mathrm{cm}}=1$, ${\mathcal{E}}_{\mathrm{db}}=5/9$ for the with-db models.Reuse & Permissions

Figure 10

Effect of $R_{\mathrm{cutoff}}$ on secondary structures in amyloid peptides, computed using variations of the Irbäck and Mohanty model, which does not consider desolvation barriers (see text). $\alpha$ and $\beta$ structures are defined by DSSP [(a)–(c), top panels] and by dihedral angles [(d)–(f), bottom panels]. Filled and open symbols are results for $R_{\mathrm{cutoff}}=6.5$ Å and $5.0$ Å, respectively. Results for the peptides from $\alpha$-synuclein, amyloid-$\beta$, and protein H1 are shown, respectively, in the left (a),(d), middle (b),(e), and right (c),(f) panels.Reuse & Permissions

Figure 11

Effect of db on secondary structure content in amyloid peptides. As Fig. 10 except here filled and open symbols are results from no-db (${\mathcal{E}}_{\mathrm{db}}=0$) and with-db (${\mathcal{E}}_{\mathrm{db}}>0$) models. See text for details.Reuse & Permissions