Structure of symmetric and asymmetric “ripple” phases in lipid bilayers

Olaf Lenz, Friederike SchmidFakultät für Physik, Universität Bielefeld, D – 33615 Bielefeld, Germany

We reproduce the symmetric and asymmetric “rippled” Pβ′ states of lipid membranes by MonteCarlo simulations of a coarse-grained molecular model for lipid-solvent mixtures. The structure andproperties compare favorably with experiments. The asymmetric ripple state is characterized by aperiodic array of fully interdigitated “defect” lines. The symmetric ripple state maintains a bilayerstructure, but is otherwise structurally similar. The formation of both ripple states is driven by thepropensity of lipid molecules with large head groups to exhibit splay.

PACS numbers: 87.16.Dg, 87.16.Ac, 82.70.Uv, 87.14.Cc

Membranes are ubiquitous in all living organisms [1].Their central structural element is a lipid bilayer, whichis kept together by the amphiphilic character of lipidmolecules – they self-assemble such that their hydrophilichead groups shield the hydrophobic tails from the waterenvironment. Pure lipid bilayers or monolayers (at theair-water interface) have been studied for a long time asmodel systems that can provide insight into the struc-tural properties of biomembranes. Already these seem-ingly simple systems exhibit a rich spectrum of structuresand phase transitions [2–5]. The most common state innature is the so-called “fluid” state (Lα), which is char-acterized by a large number of chain defects and highlipid mobility. If one decreases the temperature, one en-counters a phase transition (the “main” transition) intoa “gel” state where the lipid molecules are more orderedand less mobile. The structure of this low temperaturephase depends on the interactions between the lipid headgroups. Loosely speaking, lipids with small head groupssuch as phosphatidylethanolamines [4] assume a statewhere the long axes of the chains remain perpendicularto the bilayer normal (Lβ phase). Lipids with large headgroups and relatively strong head-head attraction such asphosphatidylcholines [5] exhibit tilt (Lβ′ phase). Finally,lipids with large head groups and weak head-head at-traction such as as ether linked phospatidylcholines [5, 6]form a phase Lint

β where both opposing lipid layers arefully interdigitated [4, 5].

The main transition has attracted considerable in-terest, since it occurs at temperatures that are simi-lar to typical temperatures on earth (between −200Cand 600C). The mechanism that governs the transitionLα ↔ Lβ to the untilted gel is comparatively straight-forward. The transition is driven by the competition ofthe entropy of chain disorder and the free energy of chainalignment [7, 8] (i.e., chain packing) and is thus in somesense related to the isotropic-nematic transition of liquidcrystals. At the transition Lα ↔ Lβ′ to the tilted gel,the situation is much more complicated. Here, the maintransition is preempted by a “pretransition”, and one ob-serves an intermediate state with a periodic, wave-likesurface structure: The “ripple” phase Pβ′ , first reported

by Tardieu et al. [9]. The microscopic structure of thismysterious phase has been debated for a long time.

In fact, at least two different rippled states have beenreported, which often coexist [10]. Electron density maps(EDMs) have recently been derived for both of them fromX-ray scattering data [11, 12]. One of the structures isasymmetric and has a sawtooth profile with alternatingthin and thick arms and a periodicity of 13-15 nms, whichcorresponds to roughly 20 lipid diameters. The other oneis symmetric and has a wavy sinusoidal structure withtwice the period of the asymmetric structure [13]. Theformation of the ripples depends strongly on the thermalhistory [10, 13, 14]. If the membrane is heated up fromthe gel state, asymmetric ripples are obtained. If onecools down from the fluid state, both types of ripples areformed – predominantly asymmetric ones if the coolingwas fast, and predominantly symmetric ones if the cool-ing was slow and if a long time was spent at the transitiontemperature. Dynamical X-ray [13] and AFM [15] stud-ies suggest that the symmetric ripple state is metastableand very slowly transforms into the asymmetric ripplestate; however, this does not yet seem to be fully settled.The degree of ordering in the ripple states largely resem-bles that in the gel state, except for a certain amount ofdisorder in the structure [2] – calorimetric studies sug-gest that approximately 10 % of all chains are melted.Most strikingly, the self-diffusion of lipids in the ripplestates is a few orders of magnitude higher than that inthe gel state, and highly anisotropic [16]. This has leadto the assumption that the ripple states might contain“coexisting” gel-state and fluid-state lipids.

Numerous theoretical models for the ripple phase havebeen proposed, which explain the ripple formation by dif-ferent mechanisms: Chain packing effects [17, 18], dipolarinteractions [19], a coupling of monolayer curvature withlocal chain melting [20–22] or with tilt [17, 23, 24] in com-bination with chirality [24]. This list is far from complete.In contrast, molecular simulations of rippled membranestates are still scarce. Kranenburg et al. [25] were thefirst to reproduce a periodically modulated membranestate in a dissipative-particle dynamics (DPD) simula-tion of a coarse-grained lipid model. They observe a pe-

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riodic sequence of stripes with alternating gel and liquidorder, similar to a structure proposed theoretically byFalkovitz et al. [20]. Unfortunately, the distribution ofhead groups in that structure is not consistent with theexperimental EDMs – the structure is neither asymmet-ric, nor does it feature the waviness which characterizesthe symmetric ripple. Moreover, the relative fraction ofmolten molecules – 50 % – seems too high, compared toexperiments. A second, very interesting simulation hasrecently been carried out by de Vries et al. [26]. In anatomistic model of a lecithin bilayer, these authors founda structure containing a stretch of interdigitated mem-brane and a stretch of gel membrane. The interdigitatedpatch connects the neighboring gel membrane such thatthe upper leaflet of the bilayer on one side crosses overinto the lower leaflet on the other side. The authors as-sume that this structure will repeat itself periodically inlarger systems and identify it with an asymmetric ripple.It is worth noting that the lipids are not arranged in acontinuous bilayer – as had been assumed in all previousmodels for the ripple state.

In this letter, we present Monte Carlo simulations ofa simplified coarse-grained lipid model, which reproduceasymmetric and symmetric ripple states with propertiesthat compare very favorably to experiments. The struc-ture of the asymmetric ripple is similar to that proposedby de Vries et al. [26]. Our simulations show that it isindeed a periodic structure, and that it is generic, i.e., itdoes not depend on molecular details of the lipids. More-over, they enable us to propose a structural model for thesymmetric ripple as well, and to identify the mechanismsthat stabilize the rippled structures.

We employ a lipid model which we have used earlierto investigate phase transitions in Langmuir monolay-ers [27, 28]: Lipid molecules are represented by chainsmade of one head bead and six tail beads (Fig. 1b), whichare connected by anharmonic springs and subject to anintramolecular bending potential. The tail beads attractone another with a truncated and shifted Lennard-Jonespotential (diameter σ, well depth ∼ ε). The head beadsare larger than the tail beads (1.1σ) and purely repulsive.The other parameters and the exact form of the poten-tials can be found in Ref. 28 (the model correspondingto Fig. 7). Self-assembly of the “lipids” is enforced with

a) d)b) c)

FIG. 1: Illustration of our lipid model and snapshot of twolipid states (at P = 2ε/σ3). (a) All-atom model of DPPC (b)Coarse-grained bead-spring model used in this work (c) Thefluid phase Lα at kBT = 1.3ε (d) The tilted gel phase Lβ′

at kBT = 1.1ε. For better visualization, only heads (reducedsize) and tail bonds are shown.

1 1.2 1.4Temperature T [ε/kB]

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FIG. 2: Phase diagram of the lipid model as a function oftemperature and “pressure”, i.e., effective head interaction P .The phases are: Lα (fluid), Lβ′ (tilted gel), Lint

β (interdig-itated gel), P ′

β (ripple). At high temperatures, the bilayerdisintegrates. Open squares indicate transition points fromsimulation runs of small systems initially set up as fully or-dered, untilted bilayers. No ripples were observed in that case.Closed squares show transition points determined by heatingup and cooling down the system (different system sizes), withthe error bars corresponding to the width of the hysteresis.Open circles denote points where the membrane undergoes atransition from Lβ′ to Lint

β upon heating.

a recently proposed “phantom solvent” environment [29]:We add “solvent” particles which interact only with lipidbeads (repulsively), and not with one another. Two ex-amples of self-assembled membranes in the Lα and theLβ′ state are shown in Fig. 1c,d). The phantom solventhas the simple physical interpretation that it probes theaccessible free volume for solvent particles in the pres-ence of lipids. It entropically penalizes lipid/solvent in-terfaces, and thus effectively creates an attractive “de-pletion” interaction between the lipid beads next to suchan interface, i.e., the head beads. Compared to otherexplicit solvent environments, it has the advantage thatit does not introduce an artificial solvent structure andartificial solvent-mediated interactions between periodicimages of bilayers. Moreover, it is computationally cheap- in Monte Carlo simulations, less than 10 % of the totalcomputer time is typically spent on the uninteresting sol-vent region. We have carried out Monte Carlo simulationat constant pressure P . The system sizes ranged from288 to 1800 lipids (corresponding to 2000-12600 beads),typical run lengths were 1-10 million Monte Carlo steps.We used periodic boundary conditions, and as simula-tion box a parallelepiped of fluctuating size and shape.This ensured that the overall pressure tensor remainedisotropic, and that the membranes had vanishing surfacetension. It is important to note that the “pressure” in oursystem should not be related to the physical pressure ina real experimental setup; it is just an additional modelparameter. Increasing P increases the phantom solventdensity in the lipid free regions according to the ideal gaslaw, ρ = P/kBT , which in turn enhances (linearly) theamplitude of the solvent-mediated interactions betweenthe lipids [30]. The other effects of P are comparativelyminor.

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The resulting phase diagram is shown in Fig. 2. Themodel reproduces the experimentally observed gel andfluid phases for lipids with large heads: The fluid phase(Lα), the interdigitated gel (Lint

β ) for low P , i.e., weakhead attraction, and the tilted gel (Lβ′) for higher P , i.e.,strong head attraction. The structures of these phasesand the phase transitions shall be discussed in detail else-where [31]. Here, we fix the “pressure” at P = 2ε/σ3,where the gel phase has the tilted Lβ′ structure. In thetransition region between Lβ′ and Lα, we observe modu-lated configurations which we identify with rippled states.

The “ripples” develop reproducibly when cooling a fluidmembrane or heating a tilted gel membrane in a temper-ature range close to the transition temperature. As inthe experiments, their exact structure depends on thethermal history. Fig. 3 (left) shows three examples ofripples that have formed spontaneously starting from dif-ferent initial configurations. The structure in Fig. 3a)emerged after cooling the system rapidly from the fluidphase down to kBT = 1.1ε. It exhibits two ripples ofwidth ∼ 15σ with a structure very similar to that foundby de Vries et al. [26] in their atomistic simulations: Ateach ripple, one of the membrane monolayers crosses fromthe lower side to the upper side of the membrane, pass-ing through a line with an interdigitated structure. Thesecond monolayer ends at this line with an edge of dis-ordered, melted chains. Our results strongly support thehypothesis that the structure reported by de Vries et

al. corresponds to the generic configuration of the asym-metric ripple state. There is only one difference: In oursimulations, the interdigitated stretch is very short, likea line defect. As we shall see below, this is consistentwith the structural information provided by the EDMs.

The second structure, Fig. 3b), resulted from heatingup a bilayer in the tilted gel state up to a temperatureclose to the main transition, kBT = 1.21ε. During thesimulation, the bilayer first fluctuated very strongly. Atsome stage, the tilt was so strong that the lipids in bothmonolayers slid along each other and connected with theother monolayer. The final structure (after 10 millionMonte Carlo steps) exhibits one asymmetric ripple andfluctuates much less. The formation of the second rippleis apparently prevented kinetically.

The third structure, Fig. 3c), corresponds to a mem-brane which has been cooled down from the fluid state toa temperature close to the main transition kBT = 1.18ε.In this case, a new type of structure emerges: The mem-brane maintains its bilayer structure, but the monolayersexhibit curved, ordered stripes with a width of roughly25 σ. These “gel” stripes on the upper and lower mono-layers are interlocked, such that the membrane assumesan overall sinusoidal shape. Each stripe ends on bothsides with conical regions of disordered chains, which arevery similar to the monolayer caps in the asymmetric rip-ple state. The total width of a ripple is ∼ 30σ, which istwice as much as the width of an asymmetric ripple. We

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FIG. 3: Three examples of ripple configurations in a modelbilayer of 720 lipid molecules (left) with corresponding tiltprofiles θ(x) across the ripple (right). (a) Two asymmetricripples, formed after rapidly cooling down to the temperaturekBT = 1.1ε from the Lα phase (b) One asymmetric ripple,formed after heating up to kBT = 1.21ε from the Lβ′ phase(c) Symmetric ripple, formed after cooling down slowly tothe temperature kBT = 1.18ε from the Lα phase. The opencircles in the graphs correspond to the lower monolayer, theclosed squares to the upper monolayer. The hatched ellipsesindicate the regions with the interdigitated line defect, thethick solid lines in c) the slopes of θ on the ordered monolayerregions. Compared to the snapshots, the curves are shiftedand replicated periodically in the x direction.

identify this structure with the symmetric ripple state.To support this hypothesis, we superimpose the pro-

posed structures for the asymmetric and the symmetricripple with the EDMs of Sengupta et al. [12] in Fig. 4.They can be inserted very nicely. In the asymmetric case,they explain the sawtooth shape with the thin and thickarm (assuming that the interdigitated region is indeedsmall). In the symmetric case, they reproduce the sinu-soidal shape.

The simulations can be used to characterize the rip-

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FIG. 4: Sketch of the proposed microscopic structures of theripple states superimposed onto EDMs from Ref. 12. (a)asymmetric ripple (on an EDM for DMPC at 18.2 0 C) (b)symmetric ripple (on an EDM for DPPC at 39.2 0 C)

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ple states in more detail. We just summarize some ofthe results here, the data will be presented and discussedelsewhere [31]. The structure of the ripple states is inmany respect similar to that of a gel: Roughly ∼ 85%of the chains have chain lengths distributed as in thegel, only 15% have a reduced length. This is in roughagreement with the experimental findings on the amountof chain disorder in the ripple state. The head layer inthe ordered parts of the ripple state has the same thick-ness than in the gel state. The structure factor of theripple state indicates a large amount of positional order,and resembles that of an untilted gel. The most reveal-ing structural feature is the average tilt of the molecules.It points perpendicular to the ripple and is modulated.Fig. 3 (right) shows profiles of the average tilt angle θ forthe three ripples discussed above. The slope θ′(x) turnsout to be almost constant throughout the whole orderedpart of the monolayer. Moreover, the numerical valuesare comparable: θ′

∼ 2.6 on average for the two asym-metric ripples, and θ′

∼ 2.5 for the symmetric ripple.Even the single asymmetric ripple of Fig. 3b), which pre-sumably has an unfavorable period, still features a con-stant slope of θ′

∼ 2.3. This suggests strongly that theripple formation is driven by the propensity of lipids withlarge head groups to exhibit splay. In contrast, local cur-vature seems less important. In the system of Fig. 3b)the lipids always splay inwards, even though the localcurvature varies and even changes sign.

In sum, we have reproduced symmetric and asymmet-ric rippled states with a generic model for lipid mem-branes. The comparison with experiments is favorable:The structure is consistent with the available EDMs, theperiod length is of the same order as the experimental pe-riod length (∼ 15 lipid diameters), the amount of chaindisorder is comparable, and we observe the same depen-dence on thermal history. Therefore, we believe to havestrong evidence that our structures correspond to the realripple states observed in experiments. Factors that areimportant for the formation of these states are: (i) Thevicinity to the Lα phase, such that a small number ofchains can melt, (ii) a strong tendency of monolayersto splay inwards – caused by a mismatch between headgroup and tail size, and (iii) the possibility to interdigi-tate. Chirality is not necessary, in agreement with exper-iments [13]; the “lipids” do not even have to asymmetric.The factors (i) and (ii) have been pointed out before; oursimulations show how they work together to bring aboutthe rippled states.

The factor (iii) is only needed to stabilize the asym-metric ripple state. If it is absent, the system can stillform a symmetric ripple state. We note that the sym-metric and the asymmetric ripples are structurally quitesimilar. Both contain about the same amount of moltenchains, both have large ordered monolayer regions withcomparable splay. This explains why the two types ofripples coexist, and why it seems so hard to determine

which one is stable. Our results suggest that the answerto that question may depend on the type of lipid, e.g.,on the head interactions and other factors that promoteor prevent interdigitation. Coarse-grained lipid modelsmay help to study this systematically. Unfortunately, wehave not yet been able to develop an efficient strategyto determine the free energy difference between the twostates. This will be subject of future work.

We thank V.A. Raghunathan for providing us with theEDMs, and the NIC computer center Jülich for com-puter time. This work has been funded by the DeutscheForschungsgemeinschaft within the SFB 613.

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