Nucleation of twisted and tubular states in chiral ribbons

Bilayers of chiral molecules can self-assemble into twisted and tubular structures, as was recently shown with chiral molecular constituents such as ssDNA-amphiphiles. I show that the dynamics of the transition between these topologies is driven by a nucleation mechanism that bears a striking formal similarity to that encountered in first-order wetting and dewetting transitions. Exploiting this analogy enables the critical nuclei of the transition to be calculated.

Figure 1

(a) Schematics of an ssDNA-decorated amphiphile and the placement of the amphiphiles in the bilayer in a cross-section view; (b) the resulting ribbon topologies, namely the twisted and the tubular ribbon.

Figure 2

Schematic ribbon free energy $E\left(w\right)$ as obtained from the WLC model in [11]. The free energy describes the case when the tubular ribbon is stable since $E(w=\infty )<E\left(w_0\right)$ with $w_0$ as the location of the minimum at small $w$.

Figure 3

Phase diagram of the ribbon morphologies, based on the minima of $E\left(w\right)-\mathrm{\Delta }\mu w$. For $\mathrm{\Delta }\mu \ge 0, T>T_t$, the tubular ribbons are the stable configurations. For $\mathrm{\Delta }\mu \le 0, T<T_t$, narrow twisted ribbons are stable; at the transition line for $\mathrm{\Delta }\mu <0, T>T_t$, twisted and tubular ribbons exchange stability if the chemical potential or temperature is changed to cross the line (from the shaded into the white area or reverse); a path at constant $T$ is indicated by an arrow. The transition line is bounded by a spinodal region, indicated by dashed lines, within which $F\left(w\right)$ has two minima, corresponding to a stable twisted ribbon and a metastable tubular ribbon state in the shaded area, and a metastable twisted ribbon and a stable tubular ribbon in the white area. Outside of the dashed lines, $F\left(w\right)$ only has one minimum, i.e., either a twisted ribbon or a tubular ribbon in the corresponding regions of the phase diagram.

Figure 4

Schematic drawing of nucleation at a ribbon, going from a small-width ribbon to a large-width ribbon in (a), representing the nucleation of the tubular state from the twisted ribbon, and the inverse case in (b). The initial state of width $w_0$ is drawn with broken lines. The bending of the ribbon in space, although ignored in the model, is tentatively indicated.

Figure 5

Ribbon nucleus (red) vs membrane at a step (blue). The energy $E\left(w\right)$ is approximated by a cubic polynomial, which is valid away from the transition, when the minimum at $w_0$ is not very pronounced (near the spinodal). $E\left(w\right)=w^2-0.3w^3$ served as the model potential for this case; $\tau =\gamma =1$.