R. J. Molotkovsky

Energy of the interaction between membrane lipid domains calculated from splay and tilt deformations

Specific domains, called rafts, are formed in cell membranes. Similar lipid domains can be formed in model membranes as a result of phase separation with raft size may remaining small (∼10–100 nm) for a long time. The characteristic lifetime of a nanoraft ensemble strongly depends on the nature of mutual raft interactions. The interaction energy between the boundaries of two rafts has been calculated under the assumption that the thickness of the raft bilayer is greater than that of the surrounding membrane, and elastic deformations appear in order to smooth the thickness mismatch at the boundary. When rafts approach each other, deformations from their boundaries overlap, making interaction energy profile sophisticated. It has been shown that raft merger occurs in two stages: rafts first merge in one monolayer of the lipid bilayer and then in another monolayer. Each merger stage requires overcoming of an energy barrier of about 0.08–0.12 kBT per 1 nm of boundary length. These results allow us to explain the stability of the ensemble of finite sized rafts.

Stabilization of bilayer structure of raft due to elastic deformations of membrane

Line tension of the boundary of specific domains (rafts) rich in sphingomyelin was calculated. The line tension was calculated based on macroscopic theory of elasticity under assumption that the bilayer in raft is thicker than in the surrounding membrane. The calculations took into account the possibility of lateral shift of the domain boundaries located in different monolayers of the membrane. The line tension was associated with the energy of elastic deformations appearing in the vicinity of the boundary in order to compensate for the difference in the thickness of the monolayers. Spatial distribution of deformations and the line tension was calculated by minimization of elastic free energy of the system. Dependence of the line tension on the distance between the domains boundaries located in different monolayers was obtained. It was shown that the line tension is minimal if the distance is about 4 nm. Thus, membrane deformations stabilize the bilayer structure of rafts observed experimentally. The calculated value of line tension is about 0.6 pN for the difference between the monolayer thickness of raft and surrounding membrane of about 0.5 nm, which is in agreement with the experimental data available.

Model of membrane fusion: Continuous transition to fusion pore with regard of hydrophobic and hydration interactions

We consider the process of fusion of lipid membranes from the stage of stalk with minimal radius to the stage of fusion pore. We assume that stalk directly developed into the fusion pore, omitting the stage of hemifusion diaphragm. Energy of intermediate stages is calculated on the basis of the classical elasticity theory of liquid crystals adapted for lipid membranes. The trajectory of transition from stalk to pore is obtained with regard to hydrophobic and hydration interactions. Continuous change of orientation of lipids in distal monolayers occurs along the trajectory. The orientation changes from the direction along rotational axis of the system specific to stalk to the direction corresponding to the fusion pore. Dependence of energy of intermediate stages on the value of spontaneous curvature of distal monolayers of the fusing membranes is obtained. We demonstrate that the energy barrier of the stalk-to-pore transition decreases when distal monolayers have positive spontaneous curvature, which is in accordance with available experimental data.

Switching between Successful and Dead-End Intermediates in Membrane Fusion

Fusion of cellular membranes during normal biological processes, including proliferation, or synaptic transmission, is mediated and controlled by sophisticated protein machinery ensuring the preservation of the vital barrier function of the membrane throughout the process. Fusion of virus particles with host cell membranes is more sparingly arranged and often mediated by a single fusion protein, and the virus can afford to be less discriminative towards the possible different outcomes of fusion attempts. Formation of leaky intermediates was recently observed in some fusion processes, and an alternative trajectory of the process involving formation of π-shaped structures was suggested. In this study, we apply the methods of elasticity theory and Lagrangian formalism augmented by phenomenological and molecular geometry constraints and boundary conditions to investigate the traits of this trajectory and the drivers behind the choice of one of the possible scenarios depending on the properties of the system. The alternative pathway proved to be a dead end, and, depending on the parameters of the participating membranes and fusion proteins, the system can either reversibly enter the corresponding “leaky” configuration or be trapped in it. A parametric study in the biologically relevant range of variables emphasized the fusion protein properties crucial for the choice of the fusion scenario.

Reply to Comment to “Elastic membrane deformations governinterleaflet coupling of lipid-ordered domains”

Our recent publication in this journal [1] challenges the concept that domains in opposing membrane leaflets are in register because of interactions at a membrane midplane. Compelled by the lack of direct experimental proof for (i) midplane interaction via an overhang [2] or (ii) LO and LD phases repelling each other [3] we propose that minimization of line tension γ drives registration (R) [1]. We dismiss antiregistration (AR) as an unlikely event because its twofold larger domain area translates into a 2-fold larger boundary length. Moreover, the line tensions at the LD/LD−LD/LO and LO/LO−LD/LO interfaces (Cartoon 1), γDD and γOO, respectively, exceed the line tension at the LD/LD−LO/LO interface, γR rendering the elastic energy WR of the registered state smaller than the elastic energy WAR of the antiregistered state. Consequently, registration is energetically favorable. Also, γR > γDD, γOO because an isolated LD/LO boundary in only one leaflet leads to membrane bending. As readily observed in the Cartoon, for the membrane to remain flat, a substantial torque must be applied or an LD/LO boundary must be created in the upper monolayer to oppose the LD/LO boundary in the lower monolayer. Cartoon 1 Calculated membrane shape at raft boundary for L = 100nm. The transitional LO/LD zone is tilted. A flat membrane is assured in [1] by boundary conditions (Eq. 6), which set the LO/LO and LD/LD bilayers to a flat horizontal (in Cartoon 1 at x→+∞ and x→−∞, respectively). A tilt was only allowed for the transitional L zone to yield minimal W. Accounting for the spontaneous curvatures of LO, and LD, JO = −0.07 nm−1 and JD = −0.1 nm−1, respectively, in a 1:1:1 mixture of dioleoylphosphatdiylcholine:dipalmitoylphatdiylcholine:cholesterol [4] and assuming hD = 1.3 nm (LD-phase) and hO = 1.6 nm (LO-phase) [5] yields the line tensions (in pN) of γDD=1.06, γOO=1.54, and γR=0.52. This is in stark contrast to Williamson’s and Olmsted’s erroneous assumption [6] that γR−AR = γDD = γOO = γ∞/2. There γ∞ was defined as γR(L→∞). For the specific lipid mixture γ∞ is equal to 0.83 pN. Thus, for the physiological relevant case of small LO domains (signaling platforms = rafts) surrounded by a large area of LD lipids, the ratio WR/WAR=γR/(2γDD)=0.5/1.5≈0.34<1, clearly favors registration. This is true for values of lateral tension σ ≤ 6mN/m per monolayer. Higher values of σ result in membrane rupture [7] and may thus be disregarded. Experimental data are available also for a second 1:1:1 mixture of palmitoyloleoylphosphatdiylcholine:sphingomyelin:cholesterol: For JO = −0.2 nm−1 and JD = −0.1 nm−1 [4] we find γDD=1.02, γOO=1.65, γR=0.6, and γ∞= 0.74. For small LO domains within a sea of LD lipids, WR/WAR ≈ 0.41 < 1, indicating that antiregistration does not occur. We conclude that our theory works well for all physiologically relevant cases. Williamson and Olmsted [6] raised the issue of large LO domains occupying an area fraction (1−ϕ) that is comparable to that of LD phases. Although such a configuration precludes the LO phase from functioning as a signaling platform (raft), their analysis may be helpful for a generalization of the theory. For 1/4<ϕ<1/2 we find: WR=γR2ϕπA,WAR=γOO2(1−2ϕ)πA where ϕ×A is the area of the LO domain. The ratio WR/WAR =1 for a critical ϕ value, ϕcrit: ϕcrit=γOO22γOO2+γR2 to yield ϕ = 0.47 for both lipid mixtures. Thus, if only γ causes domain registration, registration might not occur in the interval 0.47<ϕ<0.53. Therefore, our theory should be extended to account for these rare cases. In [1] we ignored the doubling of the area that is stiff if antiregistration occurs. Because stiff LO areas show reduced undulations, antiregistration violates the tendency of the system toward maximum entropy. In contrast, the mutual attraction of stiff membrane regions from both monolayers maximizes the membrane area in which the membrane is free to undulate, thereby providing a gain in free energy [8]. Since energy is required to prevent the membrane from undulating [9], we envision that accounting for it will rule out antiregistration for all ϕ values. A paper in preparation will provide a full quantitative analysis.

Galimzyanovet al.Reply

Our recent publication in this journal [1] challenges the concept that domains in opposing membrane leaflets are in register because of interactions at a membrane midplane. Compelled by the lack of direct experimental proof for (i) midplane interaction via an overhang [2] or (ii) LO and LD phases repelling each other [3] we propose that minimization of line tension γ drives registration (R) [1]. We dismiss antiregistration (AR) as an unlikely event because its twofold larger domain area translates into a 2-fold larger boundary length. Moreover, the line tensions at the LD/LD−LD/LO and LO/LO−LD/LO interfaces (Cartoon 1), γDD and γOO, respectively, exceed the line tension at the LD/LD−LO/LO interface, γR rendering the elastic energy WR of the registered state smaller than the elastic energy WAR of the antiregistered state. Consequently, registration is energetically favorable. Also, γR > γDD, γOO because an isolated LD/LO boundary in only one leaflet leads to membrane bending. As readily observed in the Cartoon, for the membrane to remain flat, a substantial torque must be applied or an LD/LO boundary must be created in the upper monolayer to oppose the LD/LO boundary in the lower monolayer. Cartoon 1 Calculated membrane shape at raft boundary for L = 100nm. The transitional LO/LD zone is tilted. A flat membrane is assured in [1] by boundary conditions (Eq. 6), which set the LO/LO and LD/LD bilayers to a flat horizontal (in Cartoon 1 at x→+∞ and x→−∞, respectively). A tilt was only allowed for the transitional L zone to yield minimal W. Accounting for the spontaneous curvatures of LO, and LD, JO = −0.07 nm−1 and JD = −0.1 nm−1, respectively, in a 1:1:1 mixture of dioleoylphosphatdiylcholine:dipalmitoylphatdiylcholine:cholesterol [4] and assuming hD = 1.3 nm (LD-phase) and hO = 1.6 nm (LO-phase) [5] yields the line tensions (in pN) of γDD=1.06, γOO=1.54, and γR=0.52. This is in stark contrast to Williamson’s and Olmsted’s erroneous assumption [6] that γR−AR = γDD = γOO = γ∞/2. There γ∞ was defined as γR(L→∞). For the specific lipid mixture γ∞ is equal to 0.83 pN. Thus, for the physiological relevant case of small LO domains (signaling platforms = rafts) surrounded by a large area of LD lipids, the ratio WR/WAR=γR/(2γDD)=0.5/1.5≈0.34<1, clearly favors registration. This is true for values of lateral tension σ ≤ 6mN/m per monolayer. Higher values of σ result in membrane rupture [7] and may thus be disregarded. Experimental data are available also for a second 1:1:1 mixture of palmitoyloleoylphosphatdiylcholine:sphingomyelin:cholesterol: For JO = −0.2 nm−1 and JD = −0.1 nm−1 [4] we find γDD=1.02, γOO=1.65, γR=0.6, and γ∞= 0.74. For small LO domains within a sea of LD lipids, WR/WAR ≈ 0.41 < 1, indicating that antiregistration does not occur. We conclude that our theory works well for all physiologically relevant cases. Williamson and Olmsted [6] raised the issue of large LO domains occupying an area fraction (1−ϕ) that is comparable to that of LD phases. Although such a configuration precludes the LO phase from functioning as a signaling platform (raft), their analysis may be helpful for a generalization of the theory. For 1/4<ϕ<1/2 we find: WR=γR2ϕπA,WAR=γOO2(1−2ϕ)πA where ϕ×A is the area of the LO domain. The ratio WR/WAR =1 for a critical ϕ value, ϕcrit: ϕcrit=γOO22γOO2+γR2 to yield ϕ = 0.47 for both lipid mixtures. Thus, if only γ causes domain registration, registration might not occur in the interval 0.47<ϕ<0.53. Therefore, our theory should be extended to account for these rare cases. In [1] we ignored the doubling of the area that is stiff if antiregistration occurs. Because stiff LO areas show reduced undulations, antiregistration violates the tendency of the system toward maximum entropy. In contrast, the mutual attraction of stiff membrane regions from both monolayers maximizes the membrane area in which the membrane is free to undulate, thereby providing a gain in free energy [8]. Since energy is required to prevent the membrane from undulating [9], we envision that accounting for it will rule out antiregistration for all ϕ values. A paper in preparation will provide a full quantitative analysis.

Lateral Membrane Heterogeneity Regulates Viral-Induced Membrane Fusion during HIV Entry

Sphingomyelin- and cholesterol- enriched membrane domains, commonly referred to as “rafts” play a crucial role in a large number of intra- and intercellular processes. Recent experiments suggest that not only the volumetric inhomogeneity of lipid distribution in rafts, but also the arrangement of the 1D boundary between the raft and the surrounding membrane is important for the membrane-associated processes. The reason is that the boundary preferentially recruits different peptides, such as HIV (human immunodeficiency virus) fusion peptide. In the present work, we report a theoretical investigation of mechanisms of influence of the raft boundary arrangement upon virus-induced membrane fusion. We theoretically predict that the raft boundary can act as an attractor for viral fusion peptides, which preferentially distribute into the vicinity of the boundary, playing the role of ‘line active components’ of the membrane (‘linactants’). We have calculated the height of the fusion energy barrier and demonstrated that, in the case of fusion between HIV membrane and the target cell, presence of the raft boundary in the vicinity of the fusion site facilitates fusion. The results we obtained can be further generalized to be applicable to other enveloped viruses.

Membrane fusion. Two possible mechanisms underlying a decrease in the fusion energy barrier in the presence of fusion proteins

Here we explored a contribution of fusion proteins to stalk formation, the first stage of membrane fusion, and considered two likely mechanisms, by which these proteins could influence the membrane transformation. One mechanism represents the induction of spontaneous membrane curvature, while another is membrane disturbance by a force generated by attached proteins. The energy barrier arising due to the deformation of approaching membranes and hydration repulsion between them was calculated. In addition, a dependence of an energy barrier height on certain protein features, such as spontaneous curvature, was analyzed. It was found that if fusion proteins do not produce a force directly applied to fusing membranes, they negligibly affect the barrier height irrespective of a value of spontaneous protein curvature. Thus, the overall results provide evidence that if fusion proteins are unable to exert force, they cannot provide monolayer fusion of the membranes.

Line tension and structure of raft boundary calculated from bending, tilt, and lateral compression/stretching

Bilayer thickness in membrane domains enriched with sphingomielin and cholesterol (known as “rafts”) is bigger than thickness of neighboring membrane. Monolayers need to deform to compensate the thicknesses difference in the vicinity of the raft boundary. Line tension of the boundary of rafts associated with elastic deformations originating from the compensation of the thickness mismatch is calculated in the frame-work of the elasticity theory. In the calculations deformations of splay, tilt and lateral stretching/compression are considered. It is assumed that raft consists of two monolayer domains situated in the different membrane monolayers; it is also assumed that the boundaries of domains can shift in the lateral direction with respect to relative to each other. Dependence of the boundary energy of raft on the value of the relative shift of the boundaries is calculated. It is shown that the boundary energy is minimal when shift is equal to 4.5 nm. Dependence of the optimal shift on the mismatch of the monolayer thicknesses of raft and surrounding membrane as well as membrane shape in the vicinity of boundary are calculated. The calculated values of line tension are in a good agreement with available experimental data. Taking into account deformation of stretching/compression increases the accuracy of calculations by 30%; this exceeds the uncertainty of the line tension measurements by modern techniques.

Two possible approaches to quantitative analysis of compression diagrams of lipid monolayers

Experimental curves of lateral pressure and Volta potential previously measured vs. area of dimyristoylphosphatidylserine (DMPS) monolayer (PA diagram) (Ermakov et al., 2010) are analyzed in the framework of the model introduced by Ruckenstein and Li (1996, 1998) for lipid clusters. The same curves are described by some empirical parameters as an alternative mechanistic approach. Both approaches agree with each other and describe the compression diagrams at low and high pressures (high and low areas). Analytical expressions for the asymptotic approximations in the cluster model are suggested. The product of pressure and area (equal to the energy of monolayer compression) as well as Volta potential shown in lateral pressure scale are proposed for experimental data presentation suitable for the quantitative analysis. It is also proposed to represent this function, as well as Volta potential vs. the lateral pressure. Monolayer compressibility module evaluated for the “expanded liquid” area was used to describe the shape of experimental curves. This parameter approximates well the shape of the compression diagrams and Volta potential changes if the interface potential is assumed to be proportional to the mechanical work of the monolayer compression.