Theory of the ripple phase in hydrated phospholipid bilayers

We begin with a one-dimensional discrete microscopic model for the ripple phase which is based on the packing competition between molecular head groups and hydrocarbon chains in the bilayer. This model leads to three distinct phases: (1) uniformly tilted chains, as in the low-temperature phase $L_{\beta \mathcal{’}}$; (2) uniform chain spacing with zero tilt, which we associate with the high-temperature phase $L_{\alpha }$; (3) long-wavelength periodic density modulation like the $P_{\beta \mathcal{’}}$ phase. The minimum-energy configuration in the modulated regime is composed of pinched regions of closely spaced chains separated by gaps. In the continuum limit, the bulk free energy of the pinched regions is described by a frustrated ${\phi }^4$ theory, and the gaps give rise to a defect term. The frustration may be thought of in two ways. It produces spontaneous splay of the hydrocarbon chains in the bulk, and it makes a negative contribution to the energy associated with the defect. Viewing the frustration the second way, we find that the modulated phase is preferred when the total energy of the defect becomes sufficiently negative.