Concise theory of chiral lipid membranes

A theory of chiral lipid membranes is proposed on the basis of a concise free energy density which includes the contributions of the bending and the surface tension of membranes, as well as the chirality and orientational variation of tilting molecules. This theory is consistent with the previous experiments [J.M. Schnur \textit{et al.}, Science \textbf{264}, 945 (1994); M.S. Spector \textit{et al.}, Langmuir \textbf{14}, 3493 (1998); Y. Zhao, \textit{et al.}, Proc. Natl. Acad. Sci. USA \textbf{102}, 7438 (2005)] on self-assembled chiral lipid membranes of DC$_{8,9}$PC. A torus with the ratio between its two generated radii larger than $\sqrt{2}$ is predicted from the Euler-Lagrange equations. It is found that tubules with helically modulated tilting state are not admitted by the Euler-Lagrange equations, and that they are less energetically favorable than helical ripples in tubules. The pitch angles of helical ripples are theoretically estimated to be about 0$^\circ$ and 35$^\circ$, which are close to the most frequent values 5$^\circ$ and 28$^\circ$ observed in the experiment [N. Mahajan \textit{et al.}, Langmuir \textbf{22}, 1973 (2006)]. Additionally, the present theory can explain twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions. The ratio between the width and pitch of twisted ribbons is predicted to be proportional to the relative concentration difference of left- and right-handed enantiomers in the low relative concentration difference region, which is in good agreement with the experiment [R. Oda \textit{et al.}, Nature (London) \textbf{399}, 566 (1999)].