Theory of the asymmetric ripple phase in achiral lipid membranes
aa r X i v : . [ c ond - m a t . s o f t ] N ov Theory of the asymmetric ripple phase in achiral lipid membranes
Md. Arif Kamal, Antara Pal, V. A. Raghunathan, and Madan Rao
1, 2 Raman Research Institute, C V Raman Avenue, Bangalore 560 080, India National Centre for Biological Sciences (TIFR), GKVK Campus, Bangalore 560 065, India
We present a phenomenological theory of phase transitions in achiral lipid membranes in terms oftwo coupled order parameters – a scalar order parameter describing lipid chain melting , and a vectororder parameter describing the tilt of the hydrocarbon chains below the chain-melting transition.Existing theoretical models fail to account for all the observed features of the phase diagram, inparticular the detailed microstructure of the asymmetric ripple phase lying between the fluid andthe tilted gel phase. In contrast, our two-component theory reproduces all the salient structuralfeatures of the ripple phase, providing a unified description of the phase diagram and microstructure.
PACS numbers: 87.16.D-,61.30.Dk
Phospholipids self-assemble in water to form a richvariety of spatially modulated phases [1]. The simplestof these is the 1-dimensionally modulated fluid lamellarphase ( L α ) consisting of periodic stacks of lipid bilayermembranes separated by water, where the hydrocar-bon chains are floppy with liquid-like in-plane order.Changing the temperature or water content induces asequence of symmetry breaking transitions character-ized by unique microstructures.On reducing the temperature below the chain melting(main) transition ( T m ), the L α phase of phosphatidyl-cholines (PCs) transforms to a gel phase ( L β ′ ), charac-terized by fully-stretched all − trans chains which aretilted with respect to the bilayer normal [2–4]. In ad-dition, an asymmetric ripple phase ( P β ′ ) is found tooccur in between the L α and L β ′ phases in many PCsat high water content [1, 2, 5].Extensive studies using a variety of experimentaltechniques [1, 6–18], reveal that the P β ′ phase is char-acterized by a periodic saw-tooth height modulationof the bilayers having an amplitude of ∼ ∼
15 nm, and a bilayer thickness that isdifferent in the two arms of the ripple (fig. 1) [9, 10].As a result, the rippled bilayers lack a mirror plane nor-mal to the rippling direction. While in principle, thisdiscrete symmetry breaking can arise from an asymme-try in either shape (unequal lengths of the two arms) or bilayer thickness (unequal bilayer thickness in the twoarms), in practice these asymmetries seem to appearsimultaneously.At first it was believed that the origins of the asym-metric ripple lay in the chirality of lipid molecules [20].However, subsequent experiments using racemic mix-tures showed this was not the case [8, 15]. More re-cently, all-atom molecular dynamics simulations of lipidbilayers have observed that the degree of chain order-ing is different in the two arms of the ripple [21]. Theoccurrence of the ripple phase only in those lipids thatexhibit a L β ′ phase at lower temperatures [22], and inisolated bilayers [23], suggests an intimate connectionbetween chain tilt and the ability of the bilayers to formripples. Several theoretical models have been proposed to de-scribe the sequence of phase transitions in such lipidbilayers and the microstructure of the ripple phase[20, 24–34]. None of them accounts for all the ob-servations. We list three key features that should beexplained by any theory of the ripple phase in achiralbilayers : (1) occurrence of P β ′ phase between L α and L β ′ phases, separated by two first-order transitions; (2)unequal bilayer thickness in the two arms of the ripple;and (3) unequal lengths of the two arms.In this paper we present a phenomenological Landautheory to describe the ripple phase in an isolated, achi-ral lipid bilayer. Our free energy expression is writtenin terms of two order parameters: a scalar order pa-rameter ψ and a 2-D vector order parameter m (fig.2). ψ describes the melting of the bilayer and is the dif-ference in the bilayer thickness [28, 29] between thefluid ( L α ) and ordered ( P β ′ and L β ′ ) phases. Since the FIG. 1: Electron density map of the ripple phase of dimyris-toylphosphatidylcholine calculated from x-ray diffractiondata [19]. Bands labeled h and c correspond to the head-group and hydrocarbon chain regions of the bilayer; w de-notes the water layer separating the bilayers. bilayer thickness is determined by the conformationsof the hydrocarbon chains, ψ can also be interpretedin terms of differences in chain conformations betweenthe fluid and ordered phases [30]. m is the projectionof the molecular axis on the bilayer plane [20]. A thirdorder parameter h , describing the height of the bilayer,can be integrated out of the expression for the total freeenergy density. This model is found to capture all thethree salient features of the ripple phase listed above. FIG. 2: The unit vector ˆ n represents the orientation of thelong axis of the lipid molecules relative to the bilayer normal ~ N . ~ m = ~ n − ( ~ N · ˆ n ) ˆ N is the projection of ˆ n on the bilayerplane. The total free energy per unit area is taken to be thesum of three terms; the stretching free energy density f s , the tilt free energy density f t , and the curvaturefree energy density f c . For an isolated lipid bilayer f s is given by [34], f s = 12 a ψ + 13 a ψ + 14 a ψ + 12 C ( ∇ ψ ) + 12 D (cid:0) ∇ ψ (cid:1) + 14 E ( ∇ ψ ) (1)where ψ ( x, y ) = δ ( x,y ) − δ δ , δ ( x, y ) being the mem-brane thickness at position ( x, y ) in the bilayer mea-sured with respect to a flat reference plane, and δ theconstant thickness of the membrane in the L α phase. ψ is taken to be positive for T < T m due to the stretch-ing of the chains. This is valid in general, even if thechains are tilted below T m . Explicit temperature de-pendence is assumed to reside solely in the coefficientof ψ : a = a ′ ( T − T ∗ ), T ∗ being a reference tempera-ture. a is taken to be negative, so that the continuoustransition at T ∗ is preempted by a first order meltingtransition at T m = T ∗ + a a ′ a . The coefficient Ccan either be positive or negative, but a , D and E arealways positive to ensure stability. With C >
0, theequilibrium phases are always homogeneous in space;either as L α or L β ( L β ′ ). However, with C < q . The ( ∇ ψ ) term is included, since in thecontext of a one dimensional model with a scalar orderparameter, it has been shown that such a term is nec-essary to stabilize a modulated phase with a non zeromean value of the order parameter [35]. The tilt free energy density can be written as, f t = 12 b | m | + 14 b | m | + ˜Γ ( ∇ · m ) + Γ (cid:0) ∇ m (cid:1) + Γ ( ∇ · m ) + Γ ψ | m | + Γ ( m · ∇ ψ ) + Γ ( m × ∇ ψ ) + Γ (cid:0) ∇ ψ (cid:1) ( ∇ · m ) + Γ ( ∇ ψ ) ( ∇ · m ) (2)The first four terms in eqn.(2) are the usual terms inthe expansion of the free energy in terms of a vectororder parameter. The ( ∇ · m ) term is included to beconsistent with the ( ∇ ψ ) term introduced in f s . Thenext term represents the coupling between ψ and m ,which is responsible for the appearance of tilted phasesin this model, as b is taken to be positive. If Γ > T m is L β with | m | =0. On theother hand, tilted phases can form if Γ <
0. The suc-ceeding two terms take into account the anisotropy ofthe tilted bilayer. The next two terms represent higherorder couplings between modulations in ψ and in m ,allowed by the symmetry of the system. −19 J ] T [ C ] L β′ L α P β′ FIG. 3: Phase diagram in the T − C plane calculated fromthe model. a ′ = 159.42 k B , T ∗ = 260.0 K. Values of theother coefficients in units of k B T ∗ are: a = -306.5, a =613.15, b = 0.2, b = 200.0, D = 557.41, E = 600.0, Γ =0.010, Γ = 1.80, Γ = 500.0, Γ = -3.0, Γ = -20.0, Γ =-20.0, Γ = -500.0, Γ = -750.0. Both the main-transition( L α → L β ′ ; L α → P β ′ ) and pre-transition ( P β ′ → L β ′ ) arefirst order. The curvature energy density of the bilayer can bewritten as [20, 31], f c = 12 κ (cid:0) ∇ h (cid:1) − γ (cid:0) ∇ h (cid:1) ( ∇ · m ) (3)where h ( x, y ) is the height of the bilayer relative to aflat reference plane, κ is the bending rigidity of themembrane, and γ couples the mean curvature to splayin m . ψ m x x [ 2 π /q ] h FIG. 4: Spatial variation of the different order parametersin the P β ′ phase at T = 310.5 K and C = -4.84 × − J.FIG. 5: Schematic of the bilayer profile obtained from themodel. The bilayer thickness is different in the two arms ofthe saw-tooth-like ripples.
The equilibrium height profile of the bilayer h ( x, y ) isrelated to the tilt m via the Euler-Lagrange equation, ∇ h = γκ ( ∇ · m ) (4)Eliminating h from the free energy density f leads tothe effective energy density f eff with a reduced splayelastic constant Γ = ˜Γ − γ / (2 κ ).To determine the mean field phase diagram we choosethe following ansatz for ψ and m , ψ = ψ + ψ sin ( qx ) m x = m x + m x cos ( qx ) + m x sin ( qx )+ m x cos (2 qx ) + m x sin (2 qx ) m y = m y (5)We do not consider two-dimensionally modulated rip-ples, since they do not appear to be generic; there be-ing, as far as we know, only one report of such a struc-ture [37]. Spatial modulation of m y is neglected as wedo not keep terms proportional to ( ∇ × m ) in eqn.(2)for reasons discussed below. Higher order Fourier com-ponents of m x are retained in order to account for theripple asymmetry. Three different ripple structures canbe described with this ansatz: (1) the P β with no mean tilt ( m x = m y = 0), (2) the P Tβ ′ with a mean tilt along y ( m x = 0, m y = 0) , and (3) the P Lβ ′ with a meantilt along x ( m x = 0, m y = 0). Of these only the P Lβ ′ structure has an asymmetric height profile [33].The phase diagram in the C-T plane obtained fromnumerical minimization of the effective free energy den-sity averaged over one spatial period, < f eff > =( q/ π ) R π/q f eff dx , is given in fig. 3. It is calcu-lated for a choice of parameter values, which reproduceclosely the main- and pre-transition temperatures ofdipalmitoylphosphatidylcholine at C = - 4.84 × − J .The values of T ∗ , a , a , a , and D are similar to thoseused in refs. [29] and [34]. As can be seen from fig.3, there are three distinct regions in the phase dia-gram corresponding to three different phases: L α ( ψ = ψ = m x = m y = m x = m x = m x = m x = 0), L β ′ ( ψ = 0 , m x = 0 , m y = 0; ψ = m x = m x = m x = m x = 0), and P Lβ ′ ( ψ = 0 , ψ = 0 , m x =0 , m x = 0 , m x = 0; m x = m x = m y = 0).It is interesting that of the three possible ripplestructures only the asymmetric P Lβ ′ , which is similarto the experimentally observed structure, is presentin this phase diagram [40]. The first-order transitionlines which separate these three phases meet at theLifshitz point located at C Lp = − . × − J and T = 38 . ◦ C . For C > C Lp , the first order L β ′ → L α transition line is parallel to the C axis and occurs at T m = 38 . ◦ C . But for C < C Lp , the intermediate P β ′ phase is found, separated from the other two phases byfirst-order transition lines. Further, the region occu-pied by the P β ′ phase expands at the expense of theother two as C becomes more negative.Typical spatial variation of the order parameters inthe ripple phase is shown in fig. 4. The height profileis asymmetric and resembles very closely those seen inexperiments (fig. 1) [9, 10, 18]. ψ is almost π /2 out ofphase with h , so that it is positive (negative) along thelonger (shorter) arm of the ripple, resulting in differentbilayer thicknesses in the two arms, again in agreementwith experimental observations (fig. 1) [9, 10]. Fig. 5shows a schematic of the structure of the bilayer in-ferred from these results. It is clear that the modelpresented here accounts for all the salient features ofthe ripple phase listed in the introduction.The essential term in the free energy expression re-sponsible for the asymmetric ripples is the one with thecoefficient Γ , since similar structures can be obtainedby setting Γ =Γ =Γ = 0, as long as Γ is <
0. Thusthe present model spontaneously picks out a non-zerovalue of the mean tilt along the rippling direction q ,even in the absence of any explicit in-plane anisotropyof the bending rigidity. This is in contrast to the modelpresented in refs. [20] and [31], where the mean tilt oc-curs in a direction normal to q , resulting in symmetricripples in the case of achiral bilayers; the bending rigid-ity has to be explicitly taken to be lower along the tiltdirection in order to obtain a non-zero mean tilt along q and to stabilize asymmetric ripples within this model[33]. If Γ is made sufficiently positive in the presentmodel, ψ and h become almost in phase, so that thebilayer thickness is modulated within each arm of theripple. It might be possible to tune this parameter bya suitable choice of an impurity which would prefer tosmoothen variations in ψ ; in such cases we predict theexistence of this new ripple phase.We have included only terms proportional to ( ∇ · m )in the expression for f t . In general there will also beterms proportional to ( ∇ × m ), which lead to a rip-ple structure with a non-zero winding number in themodel presented in refs. [20] and [31]. However, such astructure can be expected to be energetically very un-favorable in an achiral bilayer, since it is not consistentwith parallel close-packing of the chains demanded byvan der Waals interaction.A straightforward extension of this model would beto use a better description of the chain-melting tran-sition, instead of the reduced bilayer thickness ψ em-ployed here. 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