Undulation instability in a bilayer lipid membrane due to electric field interaction with lipid dipoles

Bilayer lipid membranes (BLMs) are an essential component of all biological systems, forming a functional barrier for cells and organelles from the surrounding environment. The lipid molecules that form membranes contain both permanent and induced dipoles, and an electric field can induce the formation of pores when the transverse field is sufficiently strong (electroporation). Here, a phenomenological free energy is constructed to model the response of a BLM to a transverse static electric field. The model contains a continuum description of the membrane dipoles and a coupling between the headgroup dipoles and the membrane tilt. The membrane is found to become unstable through buckling modes, which are weakly coupled to thickness fluctuations in the membrane. The thickness fluctuations, along with the increase in interfacial area produced by membrane buckling, increase the probability of localized membrane breakdown, which may lead to pore formation. The instability is found to depend strongly on the strength of the coupling between the dipolar headgroups and the membrane tilt as well as the degree of dipolar ordering in the membrane.

Figure 1

(a) An unperturbed bilayer and (b) the bilayer following a deformation.

Figure 2

Symmetric and antisymmetric dipolar orientation between leaflets.

Figure 3

The dipole orientations in an applied field. The field applied to each membrane $\left(\varphi =0.03\right)$ is equivalent to a voltage drop of 0.07 V across the membrane. The parameters used are given in Table . The dipole orientations were generated using the equations for the minimums of $\Delta$ and $\Sigma$ found in Appendix . The functional form of the bilayer deformation modes $u$ and $\overline{h}$ are imposed in order to display pure modes.

Figure 4

(Color online) The eigenvalue ${\lambda }_q^b$ associated with the bilayer undulations ${\overline{h}}_q$ as a function of $\widehat{q}$. The shaded region is unstable.

Figure 5

(Color online) The locus of instability as a function of $\widehat{q}$ and $\varphi$ for various values of (a) the dipole-membrane coupling ${\sigma }_p$ and (b) the dipole alignment-membrane coupling ${\sigma }_c$. The shaded region is unstable.

Figure 6

(Color online) The eigenvalues for both the peristaltic and bilayer modes (upper panel) and the dipole modes (lower panel) as functions of $\widehat{q}$. As ${\sigma }_c$ is increased at constant $\varphi \left(=0.21\right)$, the bilayer and peristaltic modes react differently. The lower branch of the bilayer modes becomes stable for $\widehat{q}>1$. The dipolar modes show little difference for field strengths above or below the critical field strength $\left({\varphi }_c=0.16\right)$ but respond strongly to changes in the dipole alignment-membrane coupling ${\sigma }_p$. As $\widehat{q}$ is increased, the dipolar modes deviate from their initial value, the direction determined by their weak association with either the bilayer modes ${\overline{h}}_q$ or the peristaltic modes $u_q$.

Figure 7

(Color online) The ratio between the fluctuations of the bilayer $\left({\overline{h}}_q\right)$ modes at $\varphi =0.15$ to $\varphi =0$ as a function of $\widehat{q}$.

Figure 8

(Color online) The unstable eigenvalue and corresponding eigenvector ${\widehat{e}}_{\lambda }$ as a function of $\widehat{q}$. ${\widehat{e}}_{\lambda }\cdot {\widehat{e}}^{\overline{h}}$ and ${\widehat{e}}_{\lambda }\cdot {\widehat{e}}^u$ are the contributions to ${\widehat{e}}_{\lambda }$ of the bilayer and peristaltic modes, respectively.