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Timur R.
Galimzyanov
*^{ab},
Peter I.
Kuzmin
^{a},
Peter
Pohl
^{c} and
Sergey A.
Akimov
^{ab}
^{a}Laboratory of Bioelectrochemistry, A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31/4 Leninskiy Prospekt, Moscow 119071, Russia. E-mail: gal_timur@yahoo.com
^{b}Department of Theoretical Physics and Quantum Technologies, National University of Science and Technology “MISiS”, 4 Leninskiy Prospect, Moscow 119049, Russia
^{c}Institute of Biophysics, Johannes Kepler University Linz, Gruberstrasse 40-42, Linz, 4020, Austria

Received
23rd October 2015
, Accepted 11th January 2016

First published on 14th January 2016

Archaeal membranes have unique mechanical properties that enable these organisms to survive under extremely aggressive environmental conditions. The so-called bolalipids contribute to this exceptional stability. They have two polar heads joined by two hydrocarbon chains. The two headgroups can face different sides of the membrane (O-shape conformation) or the same side (U-shape conformation). We have developed an elasticity theory for bolalipid membranes and show that the energetic contributions of (i) tilt deformations, (ii) area compression/stretching deformations, (iii) as well as those of Gaussian splay from the two membrane surfaces are additive, while splay deformations yield a cross-term. The presence of a small fraction of U-shaped molecules resulted in spontaneous membrane curvature. We estimated the tilt modulus to be approximately equal to that of membranes in eukaryotic cells. In contrast to conventional lipids, the bolalipid membrane possesses two splay moduli, one of which is estimated to be an order of magnitude larger than that of conventional lipids. The projected values of elastic moduli act to hamper pore formation and to decelerate membrane fusion and fission.

Theoretical investigations of conventional lipids' mechanics have been carried out in the framework of microscopic and macroscopic models. Microscopic models are represented by various molecular dynamic models^{13} and analytical solutions of statistical mechanics equations.^{14} Macroscopic models use the elasticity theory to treat membranes as a continuum elastic medium. Here we have focused on the lipid membrane elasticity theories. The first elasticity theory for conventional lipid membranes was developed by Helfrich.^{15} Despite the simplicity of Helfrich's model, it was successfully utilized for theoretical investigations of membrane structures and membrane-associated phenomena.^{16–19} Another big advance towards complete elasticity theory was work accomplished by Hamm and Kozlov,^{20} in which the authors accounted for the bilayer's intrinsic structure within the framework of so-called tilt deformation. This theory is still widely used for the investigation of various membrane processes and phenomena, such as poration, fission, fusion, and domain formation.^{6,21–29} These theory-based models enable the systematization of available experimental data and possess substantial predictive power. However, the afore-mentioned elasticity theory still requires an adaptation for bolalipid membranes.

Bolalipids have been experimentally investigated for a long time.^{11} However, not much theoretical research has been carried out, and all of it was completed in the framework of microscopic models: by means of molecular dynamics^{13,30} and analytical solutions of equations of statistical mechanics.^{14,31} A macroscopic elasticity theory for bolalipid membranes has not yet been developed.

Bolalipids have two conformations: (1) the so-called, O-shapes, in which polar heads are located on different sides of the membrane (Fig. 1a); (2) the so-called U-shapes, in which both polar heads are located on the same side of the membrane (Fig. 1b and c).

Fig. 1 Possible bolalipid configuration in the membrane: (a) O-shape; (b) U-shape and the O-shape mixture; (c) U-shape forming bilayer structure. |

Nuclear magnetic resonance experiments revealed that bolalipid membranes contain about 10% U-shapes and 90% O-shapes.^{10} Numerical experiments^{13} predict that the U-shape content depends on the particular experimental setup and may reach up to 60%. A membrane that mainly consists of O-shapes is likely to differ in its mechanical properties from a membrane in which two monolayers interact at the membrane midplane. The elasticity formalism has not yet been developed for the bolalipid membranes. The main aim of the present work is to fill that gap.

Firstly, we derive a general expression for the energy surface density of bolalipid membranes that exclusively consist of O-shaped lipids. As a starting point, we use the general elasticity theory of lipid membranes.^{20} Secondly, we consider U-shapes' contribution to the elastic energy. Thirdly, we suggest possible experiments and theoretical estimations for defining elasticity moduli and others parameters of the model.

We abide by the previously established algorithm for conventional lipid monolayers.^{20} For convenience, we reproduced the basic equations without excessive mathematical details. Eqn (1) is the general expression for the elastic energy F of a laterally liquid medium, written up to the second-order term:^{20}

(1) |

The final expression for F is written in terms of splay and tilt deformations.^{20} Tilt deformations are characterized by the tilt-vector t. It describes the deviation of the average direction n (also called “director”, the unit vector) of lipids from the normal N to the membrane surface: t = n/(nN) − N. Splay deformations are characterized by the mean curvature J and the Gaussian curvature k of the pivotal surface of lipid monolayers. By definition, a surface is called pivotal when it does not stretch upon splay. Experimental evidence locates the pivotal surface of “normal” phospholipids in the region of the carbonyl groups.^{18} The curvatures are also expressed through n: J = −div(n), .

Bolalipid membranes' splay deformations (Fig. 2e) would locate the pivotal surface in the vicinity of the membrane midplane. However, for symmetric barrel-like deformations (Fig. 2d), two pivotal surfaces are required as this deformation resembles the symmetric splay of conventional lipid bilayers. They should be located near the head-group regions. It thus appears more convenient to abandon the pivotal surface and to define all deformations with respect to the surface at the membrane midplane. The drawback of such an approach is that we can no longer consider splay and compressing/stretching to be independent of each other.

A vector field of unit normal N to the midplane defines the shape of the midplane. Characterizing membrane deformations requires a pair of unit field vectors n. Otherwise the membrane would be reduced to an infinitesimally thin film with some independent internal structure, merely defined by the bolalipid's tilt. Such a description would neither capture highly curved membranes nor the influence of local disturbances, like those displayed upon protein insertion. Moreover, it would fail to describe the case of asymmetric content of U-shape molecules (see Fig. 1b). With a pair of n, the average orientation of bolalipids in the upper and the bottom parts of the membrane can be described relative to the midplane (see Fig. 2). All parameters corresponding to the upper and lower membrane halves will be denoted by indices 1 and 2 (see Fig. 2). In the unstrained symmetric membrane, the midplane is flat and N, n_{1} and n_{2} are collinear. Bolalipid membranes are considered both laterally liquid and locally volumetrically incompressible^{6,20,24–29} which are similar to membranes made from conventional lipids.

Tilt can be described by the following dependence of u on z^{20}:

(2) |

(3) |

In contrast, deformations of bolalipid membranes are parameterized by two pairs of curvatures: J_{1}, k_{1} and J_{2}, k_{2}. Thus, stretching a volume element located at distance ζ from the midplane takes the following form:

ε = (1 + ε_{m} + α)(1 + ζJ_{1} + ζ^{2}k_{1})θ(ζ) + (1 + ε_{m} + α)(1 − ζJ_{2} + ζ^{2}k_{2})θ(−ζ) − 1, | (4) |

For further calculations, we switch from the Oxyζ coordinate system to the Oxyz coordinate system using the volumetric incompressibility condition for a membrane patch of area A_{0}: . z may be expressed via ζ as:

(5) |

(6) |

(7) |

(8) |

(9) |

We have disregarded mixed deformations, such as simultaneously occurring splay and tilt, since they are energetically decoupled as has been derived for conventional lipids. The reasoning is that a linear vector term cannot be part of an energy expression. However, the second-order cross-term of the scalar quantity splay (div(n)) and the vector quantity tilt (tilt-vector t) does not obey that requirement. Thus, tilt and splay must be considered independently.

In contrast to the expressions for F of conventional lipid bilayers, the cross-term for the curvatures of opposing membrane parts exists in the corresponding expressions for bolalipids. The cross-terms for the tilts of opposing membrane parts, for α and J_{1} or J_{2} are absent. That does not transform the midplane surface into a neutral one, since midplane stretching ε_{m} still depends on curvature (eqn (7)). It only means that the energy contributions from the deformations induced by the lateral tension σ and by the applied torques are independent of each other. The cross-term in J originates from the fact that the upper and lower halves share a common midplane. Model accuracy allows us to neglect the Gaussian curvature cross-terms.

(10) |

(11) |

(12) |

Eqn (12) ignores the entropic contribution of mixing U-shapes with O-shapes, which should be encountered when the deformational energy is comparable to or smaller than the thermal energy k_{B}T. Any lateral inhomogeneity of U-shapes may favor membrane deformations that are laterally non-uniform. For this reason we estimate the spontaneous curvature of U-shapes below.

All other elasticity moduli depend on lipid structures and properties, thereby precluding this type of simple estimation. They should be experimentally measured. However, assessing K_{G} is very difficult even in the case of conventional lipids. At the same time, K_{G} only needs to be accounted for in a narrow and peculiar set of problems, in which membrane topology changes. Below we focus on how to estimate B_{s} and B_{d}.

1.
B
_{d}
.
K
_{A} and B_{d} measurements are commonly based on monitoring the increment in vesicular membrane area upon application of hydrostatic pressure. The change in surface area of a giant unilamellar vesicle (GUV) is associated with undulations and expansion of the area per lipid molecule.^{33–36} GUVs with a diameter of about 10 μm are well suited for this purpose because the average curvature is small. Since J_{1} and J_{2} have different signs, J_{1} + J_{2} is much smaller than J_{1} − J_{2}. This means that B_{d} and K_{A} can be determined in such experiments.^{33–36} The energetic contribution of the Gaussian curvature is constant because the system's topology does not change during the experiment (Gauss–Bonnet theorem).

2.
B
_{s}
.
Luminal conductivity measurements of lipid nanotubes that are pulled from the membrane represent an alternative method for the determination of elastic properties.^{25,37–39} The measured conductivity allows the determination of the inner nanotube radius R_{2} = 1/J_{2}. For conventional lipids, R_{2} depends both on splay modulus and membrane lateral tension.^{25,37,38}J_{1} + J_{2} cannot be assumed to be small because R_{2} is comparable with membrane thickness. Moreover, the U-shaped bolalipids are likely to laterally redistribute. Due to the cylindrical symmetry of the nanotube, tilt deformations do not appear. In addition, Gaussian curvature does not contribute to the energy associated with changes in nanotube radius R_{2}.

where J_{1} = (1/J + h)^{−1}, J_{2} = −(1/J − h)^{−1}, J_{ss} = 2J_{s0} + J_{su}(x_{1} + x_{2}), J_{sd} = J_{du}(x_{1} − x_{2}). The indices “1” and “2” correspond to external and internal parts of the membrane that form the nanotube, respectively. We define tube radius R = 1/J at the membrane midplane; h is equal to half of the membrane thickness, J_{ss} and J_{sd} are spontaneous curvatures (eqn (10)). The energy density is multiplied by the area of the non-deformed state,^{40} which with sufficient accuracy may be assumed to be equal to the area of the nanotube midplane. F given by eqn (13) should be minimized with respect to J and the concentration of U-shapes, x_{1} and x_{2}. As a result we find R as a function of σ. B_{s} is then obtained by varying σ via the application of transmembrane voltage.^{37}

R′ is the derivative of R with respect to σ (R' < 0 for real systems); R′ could be measured experimentally. The expression can be simplified for large R (h/R ≪ 1):

The nanotube radius subsequently relaxes due to the lateral redistribution of U-shapes in both monolayers. The relaxation is governed by the independent minimization of elastic energy in each monolayer. As a result, a local effective transversal asymmetry of U-shaped molecules may emerge. The characteristic time amounts to about 1 s for conventional lipids (dioleoylphosphatidylethanolamine, DOPE).^{39} The resulting equilibrium curvature of the nanotube as well as x_{1} and x_{2} can be obtained by minimizing F (eqn (13)) with respect to R, x_{1}, and x_{2}. Energy minimization demands the absence of U-shapes (possessing positive J_{s}) on the internal part of the nanotube having negative curvature. This yields the expression for B_{d} in the limit of large R:

Thus, B_{d} can be obtained from experiments with GUVs, while B_{s} and J_{ss} are attainable by measuring nanotube radii in and out of equilibrium.

Applying lateral tension σ to a cylindrical tube alters F as follows:

(13) |

For conventional lipids, the elastic moduli are much greater than the characteristic energy of thermal fluctuations, k_{B}T. For instance, B_{s} is about^{36} 10k_{B}T. Similarly, we may thus assume that the lateral distribution of U-shapes is only governed by F. The formation of nanotubes occurs much faster than the lateral redistribution of membrane components with non-zero spontaneous curvature^{39} (U-shapes). Consequently the U-shape concentrations in the internal and the external parts of the nanotube membrane immediately after formation are equal to the U-shape concentration in a flat membrane and J_{sd} = 0. Minimizing F given by eqn (13) with respect to nanotube curvature, we obtain B_{d}:

(14) |

(15) |

(16) |

Our theoretical considerations should help to ascertain the major differences in the roles archaeal and conventional lipids play during cellular processes that involve membrane reshaping. Archaea possess at least three distinct membrane remodelling systems.^{41} The first uses an archaeal actin-related protein, the “cell division A” CdvA protein. The second is comprised of the bacterial-type system FtsZ. The third alternative cell division apparatus is homologous to the eukaryotic ESCRT-III (endosomal sorting complex required for transport). Remarkably enough, membrane scission by the yeast ESCRT-III complex does not require a special type of lipid in addition to certain amounts of anionic lipids to preserve a negative net charge^{42}—a requirement that can easily be met by bolalipids. However, the energetics of scission should be fundamentally different when bolalipids are involved because their elastic moduli are different. In contrast to scission, early stages of fusion, i.e. hemifusion and fusion pore formation depend on lipid curvature,^{43} and it would be interesting to see how bolalipids may meet these requirements. To get a first impression about the energetics involved in membrane remodelling, we will estimate the basic elastic parameters of bolalipids from our theory.

Estimates of B_{s}, B_{d}, and J_{s} are attainable from simple considerations: symmetric splay of bolalipids is analogous to the symmetric splay of conventional lipid bilayers. Comparing the energetic costs for their splay with eqn (10) indicates that B_{s} is of the same order of magnitude as that of conventional lipids. In contrast, the case of antisymmetric splay, (J_{1} = −J_{2}), could not have been reduced to splay-like deformations of a conventional lipid bilayer, since in that case, the dividing surfaces are necessarily subjected to compression/stretching deformations. The stretching, ε_{J}, is approximately equal to ε_{J} ≈ h(J_{1} − J_{2})/2 = hJ_{1}. By assuming that curvature-like deformations dominate the energetic costs, and that their contribution is similar to that of compression/stretching deformations of conventional lipid membranes, the total deformational energy in the antisymmetric case adopts the form: , where K_{A} = 120 mN m^{−1} = 30k_{B}T nm^{−2} for most types of conventional lipids. Comparing E with F in eqn (9) enables the assessment of B_{d} as: B_{d} = K_{A}h^{2} + B ≈ 130k_{B}T, which is an order of magnitude larger than the splay modulus of conventional lipid membranes.

The spontaneous curvature of a monolayer composed of U-shapes can be estimated using a toy-model. Symmetrical insertion of U-shaped bolalipid molecules into a membrane that consists of O-shapes (Fig. 3a) enlarges the membrane surface, S_{h}, more than it increases the midplane area, S_{t}. In the limiting case of a pure U-shape monolayer, the spontaneous curvature is positive, since all polar headgroups are located at the same side of the membrane (Fig. 3a and b).

If a_{h} is the area per lipid headgroup of both O-shapes and U-shapes, and if a_{t} is the area of an O-shaped or U-shaped molecule at the membrane midplane, we attain the following expressions for the membrane surface areas:

(17) |

(18) |

(19) |

Based on the estimates for E_{ves}, we expect that the fusion of bolalipid membranes requires an asymmetrical U-shape distribution. Components with non-zero spontaneous curvature substantially alter the rate of membrane fusion even in the case of conventional lipid membranes.^{3,44} It decelerates if the contacting (proximal) leaflets have positive spontaneous curvature, and accelerates if the positive spontaneous curvature is acquired by the distal monolayers. Thus, enrichment of U-shapes in the distal halves of the membrane should facilitate membrane fusion. The asymmetry has to be locally restricted to the fusion zone. While lysolipids, which play that role in conventional membranes, may be produced at little cost by phospholipases and selectively enriched by protein imposed curvature, the corresponding mechanisms in bolalipid membranes are not known. Both translocation of a charged bolalipid headgroup and protein-induced bending of the bolalipid membrane are certainly energetically much more costly than in the case of conventional lipid bilayers.

We conclude that bolalipids' unique chemical structure sustains the unique stability of archaeal membranes. Their self-assembly into a monolayer, instead of into a bilayer as is the case with conventional membranes, should significantly hinder membrane reshaping by fusion and fission. From our estimations, bolalipids possess a splay modulus that is an order of magnitude larger than that of conventional mammalian lipids. It acts to further hamper fusion and inhibit pore formation, thus allowing archaea to maintain the membrane barrier to ions and other molecules even in extremely aggressive environments. However, the projected price for this stability is rate deceleration in cell division or endocytotic uptake.

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