Solitons in cell membranes

Using a two-dimensional smectic liquid crystal model, we have shown the plausibility of electrical solitary wave propagation along a bimolecular leaflet such as the cell membrane of a nerve axon which consists of chiral, lipid building blocks. Our model is a head-to-tail correlated ferroelectric, chiral Sm-${\mathit{C}}^{\mathrm{*}}$ liquid crystal, which is a unique class of substances that combines the electric polarization and anisotropy of ferroelectric crystals with the hydrodynamic properties of liquids. Polar Sm-A models can also be used with the same results. In addition to the usual transverse ferroelectricity, characteristic of the Sm-${\mathit{C}}^{\mathrm{*}}$ liquid crystal, the head-to-tail correlation ensures a longitudinal ferroelectricity component. The electric polarization due to the latter can couple to the transmembrane electric field resulting from the ionic imbalance between the two sides of the membrane—a mechanism detailed in the so-called Hodgkin-Huxley set of partial differential equations for the propagation of the action potential. We obtain a Landau–de Gennes–like free energy, which is the sum of elastic, fluctuation, and polarization terms, together with a ferroelectric term showing a direct coupling between the electric field and the mechanical deformation variable. Minimizing and equating to a viscous damping term leads to an equation similar to one equation of the Fitzhugh-Nagumo coupled set of partial differential equations, which is a simplified version of the Hodgkin-Huxley equations.

The other equation of the set resembles an equation derived from the Nernst-Planck equation, which describes transmembrane ion transport and hence provides a mechanism for transmembrane potential variation. A more complete calculation of the velocity of the asymptotic wave form shows a lower wave speed than the estimate of Nagumo et al. The piezoelectric properties of the phase compete with its curvature elasticity to produce the soliton lattice of the cell membrane, which consists of juxtaposed regions of opposite tilt orientations. The propagation of the solitary wave requires a switching electric field, which is the form for the action potential and which moves the polarized domains by ferroelectric switching.