Nonlinear elasticity of an $\alpha$-helical polypeptide: Monte Carlo studies

We report on Monte Carlo studies of the elastic properties of the helix-coil wormlike chain model of $\alpha$-helical polypeptides. In this model the secondary structure enters as a scalar (Ising-like) variable that controls the local chain bending modulus. We characterize the nonlinear elastic properties of these molecules including their response to applied tensile forces and bending torques both individually and in combination. We find a pronounced effect of applied torque on the extensional compliance of the molecule and a similar effect of tension on the bending compliance. Finally we speculate that the strongly nonlinear response of $\alpha$-helical polypeptides to combinations of torque and force plays a role in allosteric transitions in proteins.

Figure 1

Schematic figure of an $\alpha$-helical polypeptide and its representation in the HCWLC. The upper figure shows an $\alpha$-helical polypeptide with one denatured segment. The dashed line represents intramolecular hydrogen bonding. The lower figure shows the representation of this molecular configuration in terms of the Ising-like secondary structure variables (open circles for random coil segments and filled ones for $\alpha$-helical ones) and the tangent vectors to the segments of the chain (denoted by arrows). Each basic unit of the model (monomer) is shown between dotted lines.

Figure 2

Radius of gyration of the HCWLC as a function of the free energy cost per segment to transform to the random coil, non-native state: $h$ obtained from Monte Carlo simulations (plotted using black dots) as compared with the theory [22]. In this curve ${\kappa }_{>}=100$, ${\kappa }_{<}=1$, $N=10$, and ${ϵ}_{\mathrm{w}}=10$. All the energy scales are measured in $k_{B}T$ and the error bars are smaller than the symbols in the plot.

Figure 3

Numerical torque vs angle data (filled points) for ${ϵ}_{\mathrm{w}}=8.0$, $h=10.0$, ${\kappa }_{>}=100$, and ${\kappa }_{<}=1$ compared with previous exact calculations [13, 14] (solid line) for an $N=10$ chain. The torque is measured in units of $k_{B}T$. The error bars are not shown where they would be smaller than the size of the points.

Figure 4

Force vs extension Monte Carlo data (points) and mean-field theory (solid line). For the upper panel the parameters are ${ϵ}_{\mathrm{w}}=0.5$, $h=1.0$, ${\kappa }_{>}=4.0$, ${\kappa }_{<}=2.0$, and $N=20$. In the lower panel the parameters are ${ϵ}_{\mathrm{w}}=8$, and $h=1.5$, and ${\kappa }_{>}=100$, ${\kappa }_{<}=1$, and $N=10$. In each the mean length of the chain normalized by the maximum chain length $N{\gamma }_{>}$ is plotted as a function of the applied force normalized by the length of a helix segment ${\gamma }_{<}$. A comparison of the two figures demonstrates the disappearance of the pseudoplateau (intermediate-force flat region) for sufficiently long or uncooperative chains.

Figure 5

Force vs extension Monte Carlo data (points) shown in the low (top panel), intermediate (middle panel), and high (bottom panel) cooperativity limit. In each the mean length of the chain normalized by the maximum chain length $N{\gamma }_{>}$ is plotted as a function of the applied force normalized by the length of a helix segment ${\gamma }_{<}$. The parameters are $h=10.0$, ${\kappa }_{>}=100.0$, ${\kappa }_{<}=1.0$, and $N=20$, and ${ϵ}_{\mathrm{w}}=0.1$ (top panel), 5.0 (middle panel), and 10.0 (bottom panel). The solid lines correspond to the mean-field calculation [22].

Figure 6

A map of the ${ϵ}_{\mathrm{w}}\text{−}{h}_{\mathrm{eff}}$ parameter space of the helix-coil degrees of freedom showing the region of validity of the mean-field (MF) approximation as determined by the Ginzburg criterion. The insets show the energy fluctuations over Monte Carlo steps in numerical simulations of the two regimes.

Figure 7

Force vs extension curves for HCWLC chain with constrained end tangents. The force vs extension is shown for chains where the first tangent lies along the direction of the pulling force while the second tangent makes an angle of $\psi =0$ (triangles) or $\pi ∕2$ (circles) with respect to the initial chain tangent. The parameters of the chain are ${\kappa }_{>}=40$, ${\kappa }_{<}=2$, ${ϵ}_{\mathrm{w}}=8$, $h=1.5$, and $N=20$. Note that the extensional compliance depends on the state of the final chain tangent. The inset shows that the chain extension in the direction of applied force saturates at high forces to different values as a result of the constrained end tangent.

Figure 8

The extension vs force curves for the HCWLC chain under a fixed torque applied to the last chain segment. The model parameters are given by $h=10.0$, ${ϵ}_{\mathrm{w}}=8.0$, ${\kappa }_{>}=100.0$, ${\kappa }_{<}=1.0$, and the applied torques are $\tau ∕k_{B}T=1.0$ (squares) and 10.0 (circles). The dashed lines are guides to the eye.

Figure 9

The torque-induced denaturation of the $\alpha$ helix under applied tension. The secondary structure order parameter $m$ is plotted against the applied torque on the polymer for chains under various tensile stresses. The pretensioning of the chain shifts the free energy difference between the native and denatured states of the molecule thereby weakening its effective bending modulus. The chain parameters are $h=3.0$, ${ϵ}_{\mathrm{w}}=8.0$, ${\kappa }_{>}=100$, ${\kappa }_{<}=1$. The dashed lines are the guides to eye.