Elasticity of $\alpha$-Helical Coiled Coils

Predicting large scale conformations of protein structures is computationally demanding. Here we compute the conformation and elasticity of double-stranded coiled coils using a simple coarse-grained elastic model. By maximizing the contact between hydrophobic residues and minimizing the elastic energy, we show that the minimum energy structure of a coiled coil is a supercoiled double helix of $\alpha$ helices. For realistic binding energies, the elastic energy of the $\alpha$ helices requires binding every 7th residue, which leads to a pitch and helix angle for the structure that is consistent with experimental measurements. Analysis of the model equations shows how the pitch varies with the helical repeat of the hydrophobic residues and with the ratio of the twisting modulus to the bending modulus and provides an estimate of the persistence length of around 150 nm, in agreement with previous experimental estimates.

Figure 1

(a) Diagram showing a single $\alpha$ helix. The ribbon follows the helix backbone. The dashed line shows the helix axis, and the cylinder represents the elastic rod defined by the $\alpha$ helix. The lollipop shapes represent the residue side chains. The helix axis arclength distance between residues is $h$. The circumferential angle between the bottommost residue and its nearest neighbor is defined as ${\theta }_1$. If every 4th residue (shaded black) binds to a neighboring residue on a second $\alpha$ helix, an effective torsion ${\tau }_4$ can be defined using the angle between those residues ${\theta }_4$, divided by the arclength between them ${\tau }_4={\theta }_4/4h$ (b). (c) The coiled-coil conformation when every 11th, 7th, and 4th residue bind. The pitch $P$ and the distance between the $\alpha$-helix center lines $2r_0$ are shown. (d) Schematic of the mathematical representation of the two $\alpha$ helices. Vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ define the center line of each helix.

Figure 2

(a) Pitch $P$ vs helix torsion ${\tau }_N$. $P$ is calculated using (11) with $\Gamma =0.05$ (×), 0.5 (dashed line), and 5 (solid line) and compared to the solution from (1) (○) with $r_0=0.5\mathrm{nm}$. The inset shows how the pitch varies with $r_0$. Taking into account the elasticity of the $\alpha$ helices, the pitch increases with $r_0$ (solid line), whereas (1) predicts a decrease in pitch (○). (b) The effective bend (solid line) and twist (dashed line) persistence lengths as a function of applied moment.

Figure 3

The elastic energy per length as a function of the binding energy ${ϵ}_R$ for $N=4$, 7, and 11. When there is less than $13k_{B}T$ of binding energy per residue, 11-residue periodicity is favored. For $13<{ϵ}_R<80$, 7th-residue periodicity is favorable.