Free energy of the edge of an open lipid bilayer based on the interactions of its constituent molecules

Abstract

Lipid-bilayers are the fundamental constituents of the walls of most living cells and lipid vesicles, giving them shape and compartment. The formation and growing of pores in a lipid bilayer have attracted considerable attention from an energetic point of view in recent years. Such pores permit targeted delivery of drugs and genes to the cell, and regulate the concentration of various molecules within the cell. The formation of such pores is caused by various reasons such as changes in cell environment, mechanical stress or thermal fluctuations. Understanding the energy and elastic behaviour of a lipid-bilayer edge is crucial for controlling the formation and growth of such pores. In the present work, the interactions in the molecular level are used to obtain the free energy of the edge of an open lipid bilayer. The resulted free-energy density includes terms associated with flexural and torsional energies of the edge, in addition to a line-tension contribution. The line tension, elastic moduli, and spontaneous normal and geodesic curvatures of the edge are obtained as functions of molecular distribution, molecular dimensions, cutoff distance, and the interaction strength. These parameters are further analyzed by implementing a soft-core interaction potential in the microphysical model. The dependence of the elastic free-energy of the edge to the size of the pore is reinvestigated through an illustrative example, and the results are found to be in agreement with the previous observations.

Micro-mechanical, continuum-mechanical, and AFM-based descriptions of elasticity in open cylindrical micellar filaments

“…21 Each amphiphile is considered as a one-dimensional rigid rod with the length ξ. 21,47 Further, according to the observation that the hydrophobic tails of the molecules are oriented along the normal of the cylindrical interface, 16,48 the amphiphiles at any interior point of the centreline $\mathrm{scriptC}$ are assumed to be perpendicular to $\mathrm{scriptC}$ with a uniform angular distribution (see Figure 1). Hence, it is concluded that the spherical head-groups of amphiphiles lie on a cylindrical surface with a constant circular cross-section.…”

“…Following Keller and Merchant, 51 we assume that the intermolecular potential Ω( χ , χ ′ , d′ , d ′) between two molecules separated by a distance | χ − χ ′| more than a fixed cutoff distance δ vanishes. In the present setting, δ is required to be small relative to the minimum radius of curvature ℓ introduced in assumption iii, 21,47 so that δ ≪ ℓ . Also, since the potential between a molecule at a given point on the micellar body and other molecules does not exist for the whole length of the micelle, the cutoff distance δ is assumed to be much smaller than the length L of the filament.…”

“…Therefore, we define $\stackrel{\sim }{\mathrm{\Omega }}false({\epsilon }^{-2}|\mathrm{boldr}{|}^2,\mathrm{boldr}·\mathrm{boldd},\mathrm{boldr}·{\mathbf{d}}^{\prime },\mathrm{boldd}·{\mathbf{d}}^{\prime }false)=\mathrm{normal\Omega }false(\mathrm{bold\chi },{\mathbf{\chi }}^{\prime },\mathrm{boldd},{\mathbf{d}}^{\prime }false)/2$ as the new form of the interaction potential (for more details, see Refs. 21,47 ). The particular choice of the arguments of the intermolecular potential in its new form is in accord with those of the interaction potential between a pair of protein monomers forming a lipid bilayer, in which the pairwise intermolecular potential is a function of the relative position of the two protein particles and the orientation of each of the particles.…”

“…The particular choice of the arguments of the intermolecular potential in its new form is in accord with those of the interaction potential between a pair of protein monomers forming a lipid bilayer, in which the pairwise intermolecular potential is a function of the relative position of the two protein particles and the orientation of each of the particles. 47,53 As a consequence of the foregoing discussion, the net free-energy ψ b of the cylindrical body of the micelle can be expressed as $${\psi }_b=true{\int }_0^L\frac{1}{2}\left({\displaystyle {\int }_0^{2\pi }ℰfalse(s,\theta false)\mathrm{normald}\theta }\right)\mathrm{scriptD}false(sfalse)\mathrm{normald}s,$$ where the integrand of the inner integral, $ℰ$ ( s , θ ), is $$ℰfalse(s,\theta false)=true{\int }_0^L{\displaystyle {\int }_0^{2\pi }2\stackrel{\sim }{\mathrm{\Omega }}false({\mathscr{H}}_1,{\mathscr{H}}_2,{\mathscr{H}}_3,{\mathscr{H}}_{4}false)}\mathcal{D}(t)\mathrm{normald}\eta \mathrm{normald}t,$$ with ${\mathscr{H}}_1={\epsilon }^{-2}|\mathrm{bold\chi }false(sfalse)-\mathrm{bold\chi }false(tfalse){|}^2$, ${\mathscr{H}}_2=\mathrm{bold\chi }false(sfalse)-\mathrm{bold\chi }false(tfalse)false)·\mathrm{boldd}false(s,\theta false)$, ${\mathscr{H}}_3=\mathrm{bold\chi }false(sfalse)-\mathrm{bold\chi }false(tfalse)false)·\mathrm{boldd}false(t,\eta false)$, and ${\mathscr{H}}_4=\mathrm{boldd}false(s,\theta false)·\mathrm{boldd}false(t,\eta false)$. Eq.…”

“…(A.3) in Ref. 21 and the definition t 1 = t o + sε , the four arguments of the interaction potential in (7) are obtained in terms of the arc-length s and the dimensionless parameter ε as 21,47 $${\epsilon }^{-2}{\left|\mathbf{\chi }\left(t_o\right)-\mathbf{\chi }\left(t_1\right)\right|}^2=s^2+{\mathbb{B}}_1{\epsilon }^2{s}^4+o\left({\epsilon }^2\right),$$ $$\langle \mathbf{\chi }\left(t_o\right)-\mathbf{\chi }\left(t_1\right)\rangle \cdot \mathrm{boldd}\left(t_o,\theta \right)={\mathbb{B}}_2{\epsilon }^2{s}^2+o\left({\epsilon }^2\right),$$ $$\langle \mathbf{\chi }\left(t_o\right)-\mathbf{\chi }\left(t_1\right)\rangle \cdot \mathrm{boldd}\left(t_1,\eta \right)={\mathbb{B}}_3{\epsilon }^2{s}^2+o\left({\epsilon }^2\right),$$ $$\langle \mathbf{d}\left(t_o,\theta \right)\cdot \mathbf{d}\left(t_1,\eta \right)\rangle ={\mathbb{B}}_o+{\mathbb{B}}_4\epsilon s+{\mathbb{B}}_5{\epsilon }^2{s}^2+o\left({\epsilon }^2\right),$$ where ${\mathbb{B}}_o=\text{cos}\left(\theta -\eta \right)$, ${\mathbb{B}}_1=-{\kappa }^2/12$, ${\mathbb{B}}_2=-\kappa \text{cos}\theta /2$, …”