Elektrotehniški vestnik 76(3): 92-96, 2009 Electrotechnical Review, Ljubljana, Slovenija Simulations of Inverted Hexagonal Lipid Structures Sarka Perutkova, Ales Iglic Laboratory of Physics, Faculty of Electrical Engineering, Trzaska 25, Sl-1000 Ljubljana, Slovenia E-mail: sarka.perutkova @fe. uni-lj.si, ales.iglic@fe. uni-lj.si Abstract. Phospholipid molecules are composed of multipolar headgroup and two electrically neutral hydrocarbon tails. The inverted hexagonal phase (Hjj) belongs to the biologically most significant non-lamellar phospholipid phases in biomembranes. Hence the geometric properties and conditions of transition to the Hu phase are nowadays widely studied. In our derivation of the free energy of lipid monolayers we assume that phospholipid molecules are in general anisotropic with respect to the axis perpendicular to the membrane plane. In our model the expression for the phospholipid monolayer free energy consists of two energy contributions: the bending energy which involves also deviatoric term, and the interstitial energy which describes the deformation energy due to stretching of the phospholipid molecule chains. On the basis of the derived expression for the phospholipid monolayer free energy we theoretically predict optimal geometry and physical conditions for the stability of the inverted hexagonal phase. Key words: inverted hexagonal phase, self-assembly of phospholipids, biomembranes, lamellar lipid phase, non-lamellar lipid phases Simulacije invertnih heksagonalnih lipidnih struktur Povzetek. Fosfolipidne molekule sestavljajo multipolna glava in dva nepolarna repa. Invertne heksagonalne faze (H//) spadajo med najpomembnejše fosfolipidne faze v bioloških membranah. Zato so geometrijske lastnosti ter pogoji faznega prehoda v H// fazo v zadnjem času ohranjajo široko zanimanje. V nasem teoretičnem modelu H// fosfolipidne faze upoštevamo, da sestavljata prosto energijo fosfolipidne enojne plasti dva prispevka: natezna energija repov fosfolipidnih molekul in upogibna energija fosfolipidne plasti. Pri tem upostevamo, da so fosfolipidne molekule na splošno anizotropne glede na njihovo vzdolzno glavno os. S pomočjo minimizacije proste energije fosfolipidne enojne plasti teoretično napovemo optimalno geometrijo ter pogoje stabilnosti invertne hek-sagonalne faze (H//) fosfolipidnih molekul. Ključne besede: invertne heksagonalne faze, samoz-druzevanje fosfolipidnih molekul, lamelarna lipidna faza, nelamelarne lipidne faze Received 3 February 2009 Accepted 18 March 2009 1 Introduction One of the main components of biological membranes are phospholipids. They have amphiphatic character, i.e. they comprise a polar head group as well as non-polar hydrocarbon chains in one molecule. Such molecules in aqueous solution undergo a self-assembling process and form various structures. Biologically important lipid/water systems are known for their rich polymorphism [1]. The driving force of this process is predominantly the hy-drophobic effect where the hydrophilic (polar) surfaces are in contact with water solution while the hydrophobic (non-polar) parts composed of lipid head-groups are hidden from water [2]. The most common and biologically the most relevant phase is the fluid lamellar lipid bilayer phase (La), see Fig. 1. The bilayer of lipid molecules represents the basic building block of the plasma membrane, which encloses the cell interior. Nevertheless, non-lamellar model membranes are subject of increasing interest [1, 3, 4, 5], due to their importance in living organisms and due to their promising technical applications such as in drug delivery [6, 7], gene transport and nanotechnol-ogy [8]. The bicontinuous cubic phase, inverse hexagonal phase and inverse micellar cubic phase belong to the bi- ologically most relevant non-lamellar mesophases. These mesophases resist excess of water and thus they can be stable under certain conditions in biological systems such as higher temperature and can be observed also in plasma membrane of some bacteria [5, 9]. 2 Inverted Hexagonal Phase Lipids in the inverted hexagonal phase are self-assembled in long tubes arranged in a hexagonal lattice. Fig. 1 shows the geometry of the H// phase: two neighbouring tubes with diameter r are located at the distance a. The phos-pholipid chains point outwards from the cylinder surface defined as the pivotal plane while the headgroups form polar nanotubes filled with aqueous solution. The geometry of H// phase is described by the mean and Gaussian curvatures [1]: H c1 + C1 2 and K = C1C2, (1) where Ci and C2 are the principal curvatures of the plane. However, for highly curved structures it is more suitable to use curvature deviator D [10]: D = |Ci - C2I/2. (2) The invariants H, K and D are connected through the relation H2 = D2 + K. It can be seen in Fig. 1 that all lipid tails in hexagonal lattice do not have the same length. There are triangular regions (called voids) between neighboring tubes that are considerably energetically expensive, because the lipid chains in these regions need to stretch beyond the average length Co. In theoretical studies of stability of inverted hexagonal phases it is therefore necessary to take into account an energy term which accounts for the stretching of the hydrocarbon chains in void regions [11, 12, 13, 14]. 3 Energy In general, solving the stability conditions for different lipid phases is a problem of defining the free energy of the system and its minimization. Thus we consider for the total free energy of the hexagonal phase two energy contributions: the energy of local bending and the interstitial energy (voids filling energy). Starting from a single molecule energy and applying the methods of statistical physics, the free energy of a lipid monolayer (bilayer) was derived [10]. The local bending energy of laterally homogeneous monolayer (bilayer) [15] was recovered, however, an additional contribution due to orientational ordering of lipid molecules - i.e. the contribution of the devia- Figure 1. Geometry of lamellar and inverted hexagonal phases. One bilayer and one neighbouring monolayer are depicted for the lamellar phase. Lattice unit of lamellar phase (d-spacing) (d) and distance between the two bilayers (dpol) are denoted. For inverted hexagonal phase three cylinders of radius r at the distance a are depicted. Zo denotes the optimal length of hydrocarbon chains. Hu phase requires stretching or compressing some of the hydrocarbon chains with respect to their optimal length Z0 as shown schematically. Slika 2. Primerjava geometrije lamelarne in invertne hek-sagonalne fosfolipidne faze. V primeru lamelarne faze (leva slika) sta prikazani ena dvojna ter ena enojna fosfolipidna plast. Prikazani sta tudi mreZna konstanta (d) ter razdalja med površinama dveh sosednjih dvojnih plasti (dpol). V she-matskem prikazu invertne heksagonalne faze je prikazana razdalja med geometrijskima osema dveh sosednjih cilindrov (a). Polmer posameznega cilindra (priblizšno v nevtralni ravnini enojne plasti) je označen z r. Simbol Z0 označuje optimalno dolzino repov lipidnih molekul. V invertni heksagonalni fazi so repi lipidnih molekul v splošnem raztegnjeni ali skrceni glede na optimalno dolzno Z0. toric bending was obtained [10]. The orientational ordering of anisotropic phospholipids lowers the free energy of the system; the effect is more pronounced for larger anisotropy of lipid molecules and stronger membrane curvature anisotropy [10]. We proposed the interstitial energy dependent on the stretching of the lipid chains in order to fill the voids in the lattice. The stretching of chains can be expressed with the term of the harmonic spring model [16]: fd = T (Z - Zo)2 (3) where Z and Z0 are the actual and the reference lengths of lipid chains, respectively. Introducing the geometrical properties of H// phase into this term, we can define the interstitial energy of one lipid cylinder in H// phase. From summating the bending energy and interstitial energy contribution we can define the total free energy per lipid molecule in inverted hexagonal phase [17]: i 2 F = 2 ((H - Hm)2 + D2 + Di) -kT ln ^2cosh ( (liA^ V kT + 6 + -T a2%/3 — - a(r + Cc)l^V/3+ g(r + Zo)2 (4) where a and r are geometric parameters described in Fig. 1, T is thermodynamic temperature, k is Boltzmann constant, £ and k are constants described in next section and Hm and Dm are mean intrinsic curvature and curvature deviator, respectively [10]. 4 Model Constants The minimal value of the free energy of a unit patch of the lipid monolayer dependent on the mean principal curvature Hm was calculated by using Eq. (4) while the model constants are given in Table 1. The value of the inter- Table 1. Geometrical parameters of the La and the Hjj phases at T = 74° C. The structural parameters are defined in Fig. 1. The experimental values are taken from [19]. Tabela 1. Geometrijski parametr La in Hu faze pri temperaturi T = 74° C. Pomen strukturnih prametrov je podan na sl. 1 La(74oC) Hn (74o C) d, a[nm] dpoi, r[nm] Zo(Zmin ,Zmax)[nm] a0[nm2] —H [nm-1] 4.99 2.5 1.47 0.65 0 7.24 2.67 1.13 (0.95,1.51) 0.65 0.187 was taken to be from 0 to 0.4 nm-1, corresponding to curvature radii down to 1 nm. To study the effect of the deviatoric bending, also the hypothetical case where the molecules are isotropic (Dm = 0) was considered. 5 Results To show stability of the inverted hexagonal phase, we compare two lipid phases: planar and cylindrical. The systems were described as surfaces with constant principal curvatures. In the planar system (La), H = D = 0 and in the inverted cylindrical system (H//), H = —D = — 1/2r, where r is the radius of the cylinder. The minimal value of the free energy of a unit patch of the lipid monolayer with respect to the mean curvature H was calculated by using Eq. (4), where for La phase the interstitial energy term is omitted. Fig. 2 shows the free energy per lipid molecule F/n0A in dependence on the intrinsic mean curvature Hm for the La and the H// phase. We can A Dm=\H, B Dm = 0 action constant was estimated from monolayer bending constant £ = 2kca0, where for POPE kc = 11kT it is the bending constant [18] and a0 = 0.65 • 10-18m2 is the area per phospholipid molecule [19]. The reference length of the phospholipid tails Z0 (Fig. 1) was taken to be 1.30 nm [19]. In calculation of the interstitial energy, the lipid stretching modulus t was taken to be 0.95 kT nm-2 and 9.5 kT nm-2 (see [16]). For the sake of simplicity it was assumed that the molecules favor cylindrical geometry, i.e. |Hm| = Dm, which represents the wedge-shape model of the lipid molecule. The effect of the temperature was simulated by increasing the intrinsic curvatures |Hm| and Dm with increasing temperature which is consistent with increased spreading of the phospholipid tails while the headgroup extensions in POPE remain relatively unchanged. The range of the intrinsic curvatures Figure 2. Free energy per lipid molecule F/n0A consisting of the bending and the interstitial contributions in dependence on the intrinsic mean curvature of lipid molecules |Hm| in the La and Hu phase for various values of stiffness of hydrocarbon chains t for (A) |Hm| = Dm and (B) Dm = 0. See Eq. (4). fc/fcT = 1. Slika 2. Izračunana prosta energija na lipidno molekulo (F/noA) v odvisnosti spontane povprecne ukrivljenosti lipid-nih molekul |Hm| v La in Hii lipidni fazi za razlicne vrednosti konstante t ter fc/fcT = 1. compare the total free energy per molecule for anisotropic and isotropic phospholipid molecules in dependance on the mean intrinsic curvature Hm. It can be seen in Fig. 2 that there are two curves corresponding to the inverted hexagonal phase with different stiffness constants t and one curve corresponding to the lamellar phase. For stiff hydrocarbon chains (high values of t) the lamellar phase has lower energy than the inverted hexagonal phase, while for decreasing t , the inverted hexagonal phase is energetically more favorable than the lamellar phase for high enough |Hm|. Isotropic lipid molecules in the inverted hexagonal phase also exhibit the lowest energy for less stiff hydrocarbon chains. By comparing Figs. 2A and 2B, it is important to point out that the anisotropy of phospholipid molecules evokes a steeper increase in the absolute value of the energy difference between the lamellar and the inverted hexagonal phases with temperature and therefore promotes and stabilizes the H// phase profoundly. is obvious if the value of t is large enough. Large diameter of the lipid cylinder r produces larger voids that are energetically unfavorable. One can also see that there is a proportional dependence between r and a, (Fig. 4). The equation of the proportional line is similar to the rough estimation observed a = 2r + 2