Forming a Double-Helix Phase of Single Polymer Chains by the Cooperation between Local Structure and Nonlocal Attraction

Double-helix structures, such as DNA, are formed in nature to realize many unique functions. Inspired by this, researchers are pursuing strategies to design such structures from polymers. A key question is whether the double helix can be formed from the self-folding of a single polymer chain without specific interactions. Here, using Langevin dynamics simulation and theoretical analysis, we find that a stable double-helix phase can be achieved by the self-folding of single semiflexible polymers as a result of the cooperation between local structure and nonlocal attraction. The critical temperature of double-helix formation approximately follows $T^{\mathrm{cri}}\sim \ln (k_{\theta })$ and $T^{\mathrm{cri}}\sim \ln (k_{\tau })$, where $k_{\theta }$ and $k_{\tau }$ are the polymer bending and torsion stiffness, respectively. Furthermore, the double helix can exhibit major and minor grooves due to symmetric break for better packing. Our results provide a novel guide to the experimental design of the double helix.

Figure 1

Phase diagrams for the self-folding of single semiflexible polymers of length $N=30$ with reference angles ${\theta }_0=105°$ and ${\tau }_0=50°$. (a) Varying the torsion stiffness $k_{\tau }$ and the temperature $T$ at a fixed bending stiffness $k_{\theta }=10$. (b) Varying $k_{\theta }$ and $T$ at $k_{\tau }=5$. The dotted lines represent the transition temperatures. Colored regions stand for structural phases. (c) Representative structures in different phases: random coils ($R$), random coils with flickering helical ordering ($R^{*}$), globules ($G$), globules with local helical ordering ($G^{*}$), compact structures ($C_1$), compact structures with local helical ordering ($C_2$), two-helix bundles ($H_2$), and double-helix structures ($D{H}_1$). The linker in the polymer is marked in orange. (d) The minimum parameter sets of $k_{\theta }$ and $k_{\tau }$ for the formation of $D{H}_1$. (e) The dependence of entropy loss, $\mathrm{\Delta }S$, for the $G^{*}-D{H}_1$ transition on $k_{\theta }$ and $k_{\tau }$ at their respective transition temperatures. (f) The free energy landscape for the $G^{*}-D{H}_1$ equilibrium at $k_{\theta }=7$, $k_{\tau }=5$, and $T=0.7$. (g) The free energy difference $\mathrm{\Delta }F=F_{D{H}_1}-F_{G^{*}}$ at $k_{\tau }=5$. The transition temperatures corresponding to $\mathrm{\Delta }F=0$ are obtained from simulations and Eq. (9).

Figure 2

Influence of local structure on the formation of double helix. (a) The energy per bead $E_p({\theta }_0,{\tau }_0)/N$ of the ideal double-helix structures, constructed by the theoretical method, as functions of ${\theta }_0$ and ${\tau }_0$. Note that the region for $E_p({\theta }_0,{\tau }_0)/N>-0.5$, where the gaps in one helix cannot accommodate the other, is marked in white. (b) Typical simulated structures for a variety of values of ${\theta }_0$ and ${\tau }_0$ for the 30mer with $k_{\theta }=10$ and $k_{\tau }=10$ at $T=0.1$.

Figure 3

Local structure effect on the formation of major and minor grooves in the double-helix structures. (a) The dependence of the theoretical azimuthal angle $\omega$ between two helical strands on ${\theta }_0$ and ${\tau }_0$. (b) The representative simulated structures of the 30mer with $k_{\theta }=10$ and $k_{\tau }=10$ for a number of values of ${\theta }_0$ and ${\tau }_0$ at $T=0.1$, and the corresponding simulated values of $\omega$ shown at the bottom.

Figure 4

(a) Phase diagram in terms of $k_{\tau }$ and $T$ for the semiflexible polymers of $N=50$ with ${\theta }_0=105°$, ${\tau }_0=50°$ and $k_{\theta }=10$. The transition temperatures corresponding to the coil-globule or freezing transition are depicted by the black symbols connected by the dotted lines. (b) Typical structures as marked in the phase diagram. They are composed of $R$, $R^{*}$, $G$, $G^{*}$, and $C_1$, three-double-helix bundles ($D{H}_3$), opened two-double-helix bundles ($D{H}_2\mathrm{o}$), closed two-double-helix bundles ($D{H}_2\mathrm{c}$), and $D{H}_1$ structures.