Undulations Drive Domain Registration from the Two Membrane Leaflets

Abstract

Phase separation in biological membranes plays an important role in protein targeting and transmembrane signaling. Its occurrence in both membrane leaflets commonly gives rise to matching liquid or liquid-ordered domains in the opposing monolayers. The underlying mechanism of such co-localization is not fully understood. The decrease of the line tension around the thicker ordered domain constitutes an important driving force. Yet, robust domain coupling requires an additional energy source, which we have now identified as thermal undulations. Our theoretical analysis of elastic deformations in a lipid bilayer shows that stiffer lipid domains tend to distribute into areas with lower fluctuations of monolayer curvature. These areas naturally align in the opposing monolayers. Thus, coupling requires both membrane leafs to display a heterogeneity in splay rigidities. The heterogeneity may either originate from intrinsic lipid properties or be acquired by adsorption of peripheral molecules. Undulations and line tension act synergistically: the gain in energy due a minimized line tension is proportional to domain radius and thus primarily fuels the registration of smaller domains; whereas the energetic contribution of undulations increases with membrane area and thus primarily acts to coalesce larger domains.

Introduction

Cell membranes are highly heterogeneous. The cytoplasmic and exofacial leaflets differ in their composition. Both of them contain physically distinguished regions. Small liquid-ordered (L_o) domains (<200 nm in diameter) that are enriched in cholesterol and sphingomyelin have been identified and termed “rafts” (1). They are thought to play an essential role in cell signaling (2), apoptosis (3), endocytosis (4), etc. This requires the L_o domains from opposing leaflets to be in register. It would otherwise be difficult to imagine how, for example, protein targeting to rafts occurs.

Experiments on much larger liquid-ordered domains in model membranes (5, 6, 7) have shown strong coupling of L_o domains that co-localize with L_o domains in the opposing leaflet. It has never been observed that two such L_o domains spontaneously decouple. On the contrary, it took considerable hydrodynamic shear stress to move domains in one leaflet of a supported bilayer out of register with domains in the other leaflet (8).

Thus far, no consensus has been reached about the mechanism of domain registration. It is only clear that it requires neither cholesterol nor any other specific lipid or proteinaceous component (7, 9, 10, 11, 12). Most prevalent is the idea that the two leaflets interact at the membrane midplane via overhang. However, measurements of interleaflet friction revealed the same mobility for long and short lipid molecules suggesting that there might not be permanent overhang (13). Nevertheless, the hypothesis about membrane midplane interaction is still en vogue, although, it is unclear 1) whether the liquid-disordered (L_d) domains and L_o domains repel each other, or 2) whether the phenomenon is caused by attraction of the two L_d domains and/or the two L_o domains, or 3) what the underlying attractive or repulsive forces may generate (14, 15). An alternative hypothesis with more mechanistic insight is that domain registration minimizes the line tension at the perimeter of the thicker L_o phases (11): a small shift in the boundary between L_d and L_o domains in the two leaflets suffices to minimize the elastic deformations that arise from the hydrophobic mismatch between the two phases. As a result, the energy of the registered domains is smaller than the energy of two isolated domains in the opposing monolayers. The hypothesis is supported by molecular dynamics simulations that have visualized the boundary mismatch between both membrane leaflets (16, 17, 18). Furthermore, it has been questioned if line tension alone may drive registration (19). That is, special conditions have been identified (the area S_R of all L_o domains per monolayer covers between 47% and 53% of the total membrane area S₀) in which minimization of the line tension fails to ensure domain registration (12).

Our previous experimental observation of mutual attraction between stiffed regions in opposing membrane monolayers (20) may well serve to close the gap between the current theoretical description and the experimental observation that registration always occurs independent of the ratio between L_o and L_d phases. We make use of the experimental observation that the splay rigidity of the L_o phase is two to five times larger than that of the L_d phase (21, 22). We are proposing a new coupling theory, to our knowledge, which is based on the difference in splay moduli. Accordingly, thermal undulations are suppressed in the L_o phase. Recruitment of weakly fluctuating monolayer L_o domains from opposing leaflets into the same membrane patch maximizes the area in which the membrane is free to undulate. Thus domain registration maximizes the entropy of the system. The proposed model builds on simple physical principles and does not impose any limitations on membrane composition.

Materials and Methods

The model assumes that the membrane represents an infinitely thin film with splay moduli 2B_R and 2B_S in the regions of the bilayer where both monolayers are in the L_o and in the L_d phases, respectively (Helfrich approximation) (23). The area S_int of the membrane where the two phases are not in register is characterized by the splay modulus 2B_int = B_S + B_R (Fig. 1). In that area, both leaflets interact with an effective energy surface density (energy per unit area) W_int that plays the role of an external “field.” When negative in sign, it forces the domains from opposing leaflets to be out of register. To be able to describe both freestanding and supported bilayers, we also account for the interaction of the membrane with the solid support. The latter is determined by three factors: 1) the van der Waals attraction; 2) hydration repulsion; and 3) electrostatic interactions. The resulting energy term has a minimum at a certain separation distance between membrane and support. Thus, it seems reasonable to expand the interaction energy in the vicinity of the minimum up to the second order of smallness to account for the effect of the substrate on the membrane.

Figure 1.

Schematic representation of the spatial distribution of liquid-ordered L_o (black) and liquid-disordered L_d (gray) phases in the membrane. S_S, S_R, and S_int denote the areas of the bilayer where the L_d phases from both monolayers are in register, where the L_o phases from both monolayers are in register, and the area out of register, respectively. The corresponding splay moduli are equal to 2B_S, 2B_R, and B_S + B_R.

The registration of lipid domains should also depend on lateral tension because of its effect on line tension at domain boundaries (11). This favors the merger of domains, i.e., minimization of boundary energy is achieved via the shortening of the boundary length (24, 25). Increased domain size, in turn, is likely to augment the contribution of thermal fluctuations to domain coupling energy. This effect is opposed by the suppression of vesicle shape fluctuations, as has been shown by micropipette aspiration of giant unilamellar vesicles (26). Thus, when analyzing fluctuations as a driving force for domain registration, it is mandatory to account for the lateral membrane tension, σ.

We introduce both a Cartesian coordinate system so that its xy plane coincides with the support and the function U(r) = U₀(r) + u(r) to describe the instant shape of the fluctuating membrane. Here r and U(r) are the radius-vector of a small membrane patch and the minimal distance that separates it from xy plane, respectively. u(r) represents an arbitrary deviation of the membrane shape from its shape in the ground state U₀(r). Thus, for the total energy W of the system, we find

$$W=\underset{{\text{S}}_R}{\int }\left(\frac{\alpha }{2}{U}^2+\frac{2B_R}{2}{\left(\Delta U\right)}^2+\frac{\sigma }{2}{\left(\nabla U\right)}^2\right)+\underset{{\text{S}}_S}{\int }\left(\frac{\alpha }{2}{U}^2+\frac{2B_S}{2}{\left(\Delta U\right)}^2+\frac{\sigma }{2}{\left(\nabla U\right)}^2\right)+\underset{{\text{S}}_{\text{int}}}{\int }\left(\frac{\alpha }{2}{U}^2+\frac{B_S+B_R}{2}{\left(\Delta U\right)}^2+\frac{\sigma }{2}{\left(\nabla U\right)}^2\right)+S_{\text{int}}{W}_{\text{int}},$$ (1)

where Δ, ∇, and S_S are the Laplace operator, the gradient operator, and the region of the bilayer in which the L_d/L_d domains are in register, respectively. Each of the three area integrals represents a sum of three terms: the first is proportional to the elastic constant α and accounts for the bilayer-substrate interaction; the second is determined by splay deformation; and the third depends on lateral membrane tension, σ.

To adequately describe the membrane shape fluctuations, we expand U(r) in a Fourier series: $U\left(\mathbf{r}\right)=(a^2/S){\sum }_{n,m=-L_{x,y}/2a}^{L_{x,y}/2a}\left(U_{0,nm}+u_{nm}\right)e^{-i{\mathbf{q}}_{\mathbf{n}\mathbf{m}}\mathbf{r}}$, where q_mn is the wave-vector with the coordinates $\left(\left(2\pi n/L_x\right),\left(2\pi m/L_y\right)\right)$ that is given by the inverse wavelength of membrane shape fluctuations. L_x and L_y indicate the membrane dimensions in the x and y directions, respectively. The characteristic size a determines the ultraviolet cutoff $q_{UV}=\pi /a$ of the wave-vector. u_nm denotes the complex coefficients of the Fourier expansion, whereas U_0,nm stands for the Fourier components of the membrane shape in the ground state. The membrane energy functional in the Fourier representation can now be written as

$$W=\underset{S}{\int }\left[\frac{\alpha }{2}U{\left(\mathbf{r}\right)}^2+\frac{\sigma }{2}{\left(\nabla U\right)}^2+\frac{2B}{2}{\left(\Delta U\left(\mathbf{r}\right)\right)}^2\right]d^{2}r=a^2\sum _{\begin{array}{l}n=1\\ m=-L/2a\end{array}}^{L/2a}\left[\frac{\alpha }{2}+\frac{\sigma }{2}\left(q_n^2+q_m^2\right)+\frac{2B}{2}{\left(q_n^2+q_m^2\right)}^2\right]{\left|U_{0,nm}-u_{nm}\right|}^2,$$ (2)

where q_n = $2\pi n/L_x$ and q_m = $2\pi m/L_y$. To arrive at a compact notation, we introduced the placeholder B for the splay moduli in the three membrane regions: B = B_S, B_R, or B_int. The thermodynamic average of all possible membrane shapes allows us to calculate S_int as follows:

$$\langle S_{\mathrm{int}}\rangle =\frac{1}{Z}\underset{0}{\overset{\text{min}\left\{2s_R,2s_S\right\}}{\int }}{e}^{-W/k_{B}T}{S}_{\mathrm{int}}Du\left(\mathbf{r}\right)d{s}_{\mathrm{int}},$$ (3)

where $Z={\int }_0^{\text{min}\left\{2s_R,2s_S\right\}}{e}^{-W/k_{B}T}Du\left(\mathbf{r}\right)d{s}_{\mathrm{int}}$ is a partition function of the system, and k_B and T are the Boltzmann constant and the absolute temperature, respectively. The upper limit of the integral is found by assuming that either all L_o domains or all L_d domains (whichever occupies the larger area) are out of register. It is thus equal to the minimal value of either 2S_R or 2S_S. Defining the integral over the membrane shapes as $Z_{def}={\int }_0^{\text{min}\left\{2s_R,2s_S\right\}}{e}^{-W/k_{B}T}Du\left(\mathbf{r}\right)$ and solving it as an integral over u_nm yields

$$Z_{def}=\prod _{n,m}\underset{-\infty }{\overset{\infty }{\int }}{e}^{-\frac{W}{k_{B}T}}d\mathrm{Im}\left(\frac{u_{nm}}{\sqrt{\pi }a}\right)\mathrm{Re}\left(\frac{u_{nm}}{\sqrt{\pi }a}\right)=\prod _{n,m}\frac{k_{B}T}{a^2\left[\frac{\alpha }{2}+\frac{\sigma }{2}\left(q_n^2+q_m^2\right)+\frac{2B}{2}{\left(q_n^2+q_m^2\right)}^2\right]}.$$ (4)

Taking the logarithm of both sides of Eq. 4 helps to find the double product $\underset{n,m}{\Pi }$, by converting it into a sum of logarithms. Then, passing from summation to integration, and substituting the variables of integration: $\tilde{n}=a\left(n/L_x\right),\tilde{m}=a\left(m/L_y\right)$, we derive the following:

$$\text{ln}{Z}_{def}=-\frac{S}{2a^2}\text{ln}\left(\frac{2B{\left(2\pi \right)}^4}{2k_{B}T}\right)-\frac{S}{a^2}\underset{0}{\overset{1/2}{\int }}\underset{-1/2}{\overset{1/2}{\int }}\text{ln}\left(\frac{\alpha a^4}{2B{\left(2\pi \right)}^4}+\frac{\sigma a^2}{2B{\left(2\pi \right)}^2}\left({\tilde{n}}^2+{\tilde{m}}^2\right)+{\left({\tilde{n}}^2+{\tilde{m}}^2\right)}^2\right)d\tilde{n}d\tilde{m}.$$ (5)

To avoid bulky notations we introduce the functional Q[α, σ, B] to abbreviate the right-hand side of Eq. 5 as follows:

$$\text{ln}{Z}_{def}=-\frac{1}{2}\frac{S}{a^2}\text{ln}\left(Q\left[\alpha ,\sigma ,B\right]\right).$$ (6)

Eq. 6 allows us to calculate Z

$$Z=\left(Q{\left[\alpha ,\sigma ,B_R\right]}^{-\frac{{\mathit{S}}_R}{2a^2}}Q{\left[\alpha ,\sigma ,B_{\mathit{S}}\right]}^{-\frac{{\mathit{S}}_{\mathit{S}}}{2a^2}}\right)\underset{0}{\overset{\text{min}\left\{2{\mathit{s}}_R,2{\mathit{s}}_{\mathit{S}}\right\}}{\int }}\left({\left(\frac{Q\left[\alpha ,\sigma ,B_R\right]Q\left[\alpha ,\sigma ,B_{\mathit{S}}\right]}{Q{\left[\alpha ,\sigma ,B_{\mathrm{int}}\right]}^2}\right)}^{\frac{{\text{S}}_{\mathrm{int}}/2}{2a^2}}\right)e^{-\frac{W_{\mathrm{int}}{\mathit{S}}_{\mathrm{int}}}{k_{B}T}}d{\text{S}}_{\mathrm{int}},$$ (7)

and S_int explicitly:

$$\begin{array}{c}\langle S_{\mathrm{int}}\rangle =\frac{\underset{0}{\overset{\text{min}\left\{2s_R,2s_S\right\}}{\int }}{\left(\frac{Q\left[\alpha ,\sigma ,B_R\right]Q\left[\alpha ,\sigma ,B_S\right]}{Q{\left[\alpha ,\sigma ,B_{\mathrm{int}}\right]}^2}{e}^{-\frac{W_{\mathrm{int}}}{4k_{B}T}}\right)}^{\frac{{\text{S}}_{\mathrm{int}}/2}{2a^2}}{\text{S}}_{\mathrm{int}}d{\text{S}}_{\mathrm{int}}}{\underset{0}{\overset{\text{min}\left\{2s_R,2s_S\right\}}{\int }}{\left(\frac{Q\left[\alpha ,\sigma ,B_R\right]Q\left[\alpha ,\sigma ,B_S\right]}{Q{\left[\alpha ,\sigma ,B_{\mathrm{int}}\right]}^2}{e}^{-\frac{W_{\mathrm{int}}}{4k_{B}T}}\right)}^{\frac{{\text{S}}_{\mathrm{int}}/2}{2a^2}}d{\text{S}}_{\mathrm{int}}}\\ =\frac{16+8{\beta }^{\text{min}\left\{s_R,s_S\right\}/2a^2}\left(\text{ln}\left(\beta \right)\text{min}\left\{s_R,s_S\right\}/a^2-2\right)}{4\left({\beta }^{\text{min}\left\{s_R,s_S\right\}/2a^2}-1\right)\text{ln}\left(\beta \right)},\end{array}$$ (8)

where $\beta =\left(\left(Q\left[\alpha ,\sigma ,B_R\right]Q\left[\alpha ,\sigma ,B_S\right]\right)/\left(Q{\left[\alpha ,\sigma ,B_{\mathrm{int}}\right]}^2\right)\right)e^{-\left(4W_{\mathrm{int}}{a}^2/k_{B}T\right)}$. For min{S_R, S_S}/a² ≫ 1, i.e., for macroscopic domains, Eq. 8 can be rewritten as

$$\langle {\text{S}}_{\mathrm{int}}\rangle =\{\begin{array}{l}-\frac{4a^2}{\text{ln}\left(\beta \right)},\beta <1\\ \text{min}\left\{2{\mathit{s}}_R,2{\mathit{s}}_{\mathit{S}}\right\},\beta >1\end{array}.$$ (9)

Eq. 9 specifies that S_int is negligibly small (S_int ≪ min{S_R, S_S}, complete phase registration) for β < 1 and that there is no phase registration for β > 1. For the critical value β = 1 we find the fluctuation induced interaction energy surface density W^∗ between identical phases in opposing leaflets as

$$W^{\ast }=-W_{\mathrm{int}}\left(\beta =1\right)=\frac{k_{B}T}{4a^2}\text{ln}\left(\frac{Q{\left[\alpha ,\sigma ,B_{\mathrm{int}}\right]}^2}{Q\left[\alpha ,\sigma ,B_R\right]Q\left[\alpha ,\sigma ,B_S\right]}\right).$$ (10)

According to Eq. 5, both α and σ enter the final expression for Q in the following combinations: $\left(\alpha a^4/2B{\left(2\pi \right)}^4\right),\left(\sigma a^2/2B{\left(2\pi \right)}^2\right)$, i.e., Q strongly depends on a. Characteristic length a should lie in the range of ∼0.5–2 nm because 1) the diameter of a lipid head group is ∼0.5 nm and 2) the monolayer thickness amounts to ∼2 nm. The cutoff wave-vector $q_{UV}=\pi /a$ was taken from molecular simulations (27). These simulation data revealed that splay fluctuations are thermally activated up to wave-vectors q_max ∼1.5 nm^–1. Fluctuations with wave-vectors q > q_max do not include splay deformations, rather protrusions-like defects. The energy of these defects is determined by the surface tension at the interface between lipid tails and water. It is thus similar for ordered and disordered membrane regions. If so, the wave-vector range q > q_max does not contribute to the domain coupling energy, and q_max can be treated as the ultraviolet cutoff in our model, which leads to the estimate a = π/q_UV ∼2 nm. Because there is no sharp boundary between splay and protrusion fluctuations (27), we consider a ≈ 1 – 2 nm in our calculations.

To simplify Eq. 10, we estimate the substrate-bilayer interaction constant α. In case of neutral lipids this interaction is manifested by attractive van der Waals forces and hydration repulsion

$$W_{l-s}=P_0{e}^{-b/l_h}-\frac{H_a}{12\pi b^2},$$ (11)

where b is the distance between substrate and lipid bilayer; H_a is the Hamaker constant; P₀ is the disjoining pressure (= hydration pressure that leads the repulsion of two parallel hydrophilic surfaces); and l_h is the characteristic length of hydration repulsion. These parameters can be estimated as follows: H_a ≈ (3 ÷ 10) × 10⁻²¹ J, P₀ ≈ 4 × 10⁷ ÷ 10¹⁰ Pa, and l_h ≈ 0.1 – 0.3 nm (28). We approximate W_l-s by its Taylor series with respect to the distance around the equilibrium value b_eq, which yields $\alpha =d^2{W}_{l-s}/d{b}^2$∼6 k_BT/nm⁴ (here k_BT ∼4 × 10⁻²¹ J).

For B_R ∼ 60 k_BT, B_S ∼ 20 k_BT, and σ ∼ 1 mN/m ≈ 0.25 k_BT/nm² (26), the above introduced dimensionless parameters adopt the following values: $\left(\alpha a^4/2B{\left(2\pi \right)}^4\right)<3\times {10}^{-4},\left(\sigma a^2/2B{\left(2\pi \right)}^2\right)<{10}^{-3}$. Because 1) they are smaller than 10⁻³, and 2) W^∗ slightly depends on σ (compare Fig. 3), the function Q[0, 0, B] = B can be used for calculations in a wide range of parameters. Denoting W^∗ under these conditions as W₀^∗ we can use a significantly simplified form of Eq. 10

$$W_0^{\ast }=\frac{k_{B}T}{4a^2}\text{ln}\left[\frac{{\left(B_S+B_R\right)}^2}{4B_S{B}_R}\right].$$ (12)

Because the term in the square brackets is always greater than one, its logarithm is a positive value. Consequently, W₀^∗ stands for a repulsive interaction between L_o and L_d phases.

Figure 3.

The dependence of W^∗ on σ for freestanding membranes. B_S and B_R are assumed to be equal to 10 k_BT and 30 k_BT, respectively. For all other parameters and the curve code see Fig. 2.

Results and Discussion

Eq. 10 allows a depiction of W^∗ between monolayer L_d and L_o domains (Fig. 2). The domain interaction energy W_fl = S_RW^∗ induced by fluctuations increases with 1) domain area S_R and 2) the difference between the B_S and B_R values. However, minimization of W_fl is not the only reason for domain registration. We previously demonstrated (11) that minimization of the line tension around the thicker ordered domain also serves to bring the domains from opposing monolayers in register. The resulting gain in energy is proportional to the domain perimeter. These observations suggest that the registration of small domains is governed by the energy of line tension W_lt, whereas that of large domains is governed by undulations. We found the characteristic radius R^∗ for the crossover from the line-tension-dominated registration to the fluctuation-dominated one by equating both contributions for B_R = 2B_S, a = 1.5 nm and the line tension γ = 0.25 k_BT/nm (11) to be equal to 38 nm. The corresponding coupling energy amounted to 60 k_BT. Thus, W_lt only governs the coupling of small L_o domains with radius R < R^∗ as it linearly depends on R. For larger L_o domains (R > R^∗) W_fl dominates, because it depends on domain area, increasing with the square of the radius. For the very large domains (R ≫ R^∗) that are observed by conventional light microscopy in model membranes, W_lt is much smaller than W_fl. The finite domain size R provide the infrared cutoff, such that fluctuations with wavelength λ > R do not affect domains and do not contribute to domain registering. However, the energetic contribution of undulations sharply declines with increasing wavelengths. Only curvature undulations with wavelengths in the vicinity of the ultraviolet cutoff (λ_UV ∼ π/q_UV) substantially contributes to the coupling energy density W^∗, so W^∗ almost does not depend on domain size.

Figure 2.

The dependence of the L_o-L_d repulsion energy density, W^∗, on the splay modulus of the L_o phase for freestanding membranes at zero external lateral tension σ. The splay modulus of the L_d phase in one leaflet was assumed to be equal to B_S = 10 k_BT. The solid, dashed, and dotted curves correspond to a = 1, 1.5, and 2 nm, respectively.

We may thus compare our theoretical W^∗ with experiments in which hydrodynamic shear forces were used to decouple the macroscopic domain in the upper leaflet from its counterpart in the lower leaflet. For the reasonable B_R/B_S ratio of 2 and a = 1.5 nm we find excellent agreement with the experimental value of 0.016 k_BT/nm² (8) (Fig. 2).

In line with the general Eq. 10, lateral tension reduces the amplitude of the thermal membrane fluctuations, and thereby diminishes W^∗ (Fig. 3). Nevertheless, even values of σ as high as ∼10 mN/m have only a modest effect on W^∗, although they are large enough to result in membrane rupture (29). This result seems to be counterintuitive: micropipette aspiration of large unilamellar vesicles revealed that even moderate tension removes the undulations and flattening out the bilayer (26). However, only fluctuations with larger wavelengths are really suppressed, leaving unaffected small wavelength undulations, which are the key contributors to the domain coupling energy. The evaluation of bending and tension energy densities of fluctuations with wavelength λ and amplitude A, which are proportional to ∼BA²/λ⁴ and ∼σA²/λ², respectively, confirms the conclusion. That is, at constant A, undulations with short wavelengths survive because strong membrane bending has little effect on membrane area. These highly curved membrane areas (small λ) act to attract L_d lipids while the stiff L_o lipids accumulate in the flat area. Since the main contribution to W^∗ comes from the small wave-length fluctuations, W^∗ only weakly depends on σ (Fig. 3).

Similar to σ, a solid support also suppresses the thermal membrane fluctuations. Analogously, W^∗ is only modestly affected so that it still governs domain registration (Fig. 4). Substrate proximity hampers the collective motion of lipids (30, 31) while reducing the mobility of individual lipids by only two- to fivefold (32, 33). Thus, domains in the substrate-adhering leaflet are completely immobile laterally. Domain movement in the opposing leaflet must also be hampered because of the strong friction between the monolayers (13). In consequence domain motion is completely missing in the nonadhering leaflet of supported planar bilayers that were created by the Langmuir-Blodgett technique (30). Such movement is only observed upon exposure to large hydrodynamic forces. However, ordered domains in the nonadhering leaflet that are forced out of register with domains in the adhering monolayer do not move back into register when no longer exposed to the hydrodynamic force (8). Thus, it is no surprise that atomic force experiments on supported bilayers reported the misalignment of domains in the individual leaflets (34). These results are not in contrast to our calculations, as we only consider mobile domains. That is, substrate proximity should not alter the undulation spectrum of the membrane. Its effect on the domains’ mobility results in an increase of the characteristic time of domain registration but has little effect on coupling energy.

Figure 4.

The dependence of W^∗ on the membrane-support interaction in the absence of σ. Parameters and the curve code are chosen as in Fig. 3.

In contrast to domain registration in opposing monolayers, molecular dynamics simulations proposed that antiregistration may also be an energetically stable state (16). That is, ordered and disordered domains from the two monolayers distributed in such a way that every ordered domain was facing a disordered domain and vice versa. Hydrophobic mismatch is missing at the interface between the domains, because such a bilayer would be homogeneous in thickness, thus completely canceling it in the very peculiar stoichiometry of a 1:1 ratio of L_o and L_d lipids that was subject to molecular dynamics simulations (16). Nevertheless, the simulated antiregistration has not been observed in an experiment so far because all membranes in the vitro systems undulate. In contrast, the in silico system is unable to capture membrane undulations because of the common restrictions in both size and simulated time span. Thus, such calculations cannot reproduce the resulting repulsive interaction between L_o and L_d phases.

The ideal system for testing our theory is a lipid bilayer that is unable to undulate while its individual lipids and its lipid domains retain full lateral mobility in at least one monolayer. Such a system is very difficult to realize since undulations and mobility are linked to each other. Thus, neither a decrease in temperature nor depositing the bilayer on a solid support seems to be a promising approach. Tethering the bilayer in limited contact areas to a support may represent a solution. However, adhesion-induced domain formation has previously been observed in such a scenario (35). Thus it requires suspended bilayers with a vast number of extremely small adhesion sites. Such bilayers have recently been described on top of streptavidin crystals, which in turn were placed on top of supported lipid bilayers (36). It remains to be clarified whether high-frequency undulations are really suppressed on this platform or, as in the case of membranes under tension, these undulations persist and thus drive domain registration.

An alternative way to test our theory would be to prevent thermal fluctuations and hydrophobic mismatch to contribute to domain coupling energy. This can be achieved by designing a phase-separated membrane where ordered and disordered phases have nearly the same splay moduli and bilayer thicknesses. This should be possible by carefully choosing long-tail unsaturated and short-tail saturated lipids (26).

Our theory is in line with the observed registration of liquid-ordered domains in multilayer lipid stacks (37, 38). That is, the narrow spacing between individual bilayers dampens undulations. Placing the stiffed regions in the different bilayers directly on top of each other minimizes the effect. Thus registration in a multibilayer stack is due to the same mechanism that ensures registration between the monolayers of a single bilayer.

In summary, we have shown that thermal membrane fluctuations act to register L_o domains from opposing membrane leaflets. To minimize W^∗, the stiffer L_o domains tend to distribute into areas with lower curvature, which naturally coincide in the opposing monolayers. Thus, registration is driven by the difference between the splay rigidities of different membrane domains. Because the mechanism also works for supported bilayers, it nicely serves to explain the formation of stacks of macroscopic L_o domains in multilayer membrane structures (37, 38). There are no constraints on the origin of these rigidities. That is, the current theory also explains the registration of domains that are induced by the adsorption of peripheral peptides (20) or scaffolding proteins (39). It avoids the artificial introduction of (thus far enigmatic) interactions at the membrane midplane (14, 19) to explain registration and to rule out antiregistration.

Author Contributions

T.R.G. designed the research and performed calculations; P.I.K. generalized and improved the theoretical model; P.P. designed the research; and S.A.A. analyzed the results and served as the scientific coordinator. All authors wrote and corrected the manuscript.

Editor: Georg Pabst