Lipid bilayer membranes are fundamental biological barriers both at the cellular and sub-cellular level. They are both very stable and extremely deformable, characteristics that make membrane vesicles an efficient system for drug delivery applications. In most cases, due to the scale separation between the membrane thickness and the vesicle size, fluid lipid vesicles can be described as elastic sheets that deform as prescribed by a curvature dependent energy. At the same time, vesicle scale and membrane thickness may become simultaneously important in several key biological processes, such as vesicle fusion/fission, which are also pivotal steps for drug delivery. Recently, we provided a diffuse interface description of lipid vesicles that contains both the large scale of the vesicle and the small thickness of the membrane, allowing to account for multiscale effects in membrane fusion/fission (Bottacchiari et al. in PNAS Nexus 3:300, 2024). Here, after reviewing the main features of the approach and the related results, we analyze an additional term for the diffuse interface that takes into account the so-called area-difference elasticity, namely an energy term that considers the cost for the flip-flop motion of a lipid between the two monolayers constituting the bilayer membrane. Results are validated against those obtained with the (sharp-interface) area-difference elasticity model.
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1 Introduction
Biological membranes, both at the cellular and sub-cellular level, are formed by a fluid lipid bilayer that separates two distinct environments. The most important example of fluid lipid membrane is the plasma membrane, which defines the boundary between the inside and outside of cells. Lipid bilayer vesicles are very stable providing an effective physical barrier function, and, at the same time, they are also remarkably deformable, e.g. think of erythrocytes that undergo considerable deformations to flow in narrow capillaries. These features, combined with the fact that vesicles can be generated in a controlled manner by microfluidic devices [1‐3], make them a system of great interest for drug delivery applications [4‐6].
Given the scale separation between the membrane thickness (\(\sim 5 \, \text {nm}\)) and the vesicle size (e.g. the plasma membrane is micrometer-sized), the large deformations vesicles undergo are commonly described by the Canham–Helfrich elastic energy [7, 8]
where \(\Gamma \) is the two-dimensional elastic sheet that represents the (ideally infinitely thin) membrane. More precisely, \(\Gamma \) coincides with the bilayer mid-surface, which has local mean curvature M and local Gaussian curvature G. Here, k is the bending rigidity of the membrane, which is about \(20 \, k_BT\) [9], with \(k_B\) the Boltzmann constant and T the ambient temperature. The other two elastic constants are the Gaussian modulus \(k_{\textrm{G}}\), which is about \(-20 \, k_BT\) [10], and the spontaneous curvature m, namely the preferred curvature of the membrane, which models an asymmetry between the two sides of the bilayer. In order to find vesicle shapes, the Canham–Helfrich elastic energy must be minimized under the constraints of constant surface area and enclosed volume. Indeed, since the number of lipids in the membrane is fixed and the membrane rupture tension is very small, the vesicle area A is substantially conserved at fixed temperature. At the same time, the enclosed volume V is given by the osmotic conditions.
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In most cases, the second term on the right hand side of Eq. (1) (the Gaussian energy) does not enter the minimization process. In fact, the Gauss-Bonnet theorem of differential geometry states that
where g is the genus of the vesicle surface \(\Gamma \) (compact surface), which means that the Gaussian energy is a topological invariant. Therefore, for any given topology and spontaneous curvature m, the minimization of the Canham–Helfrich energy leads to one vesicle shape, which is also determined by the constraints on the surface area A and enclosed volume V. Remarkably, the Canham–Helfrich energy is scale invariant and therefore the solutions are only determined by the (dimensionless) reduced volume \(v = V/(\pi D_{ve}^3/6)\) and reduced spontaneous curvature \(m_0 = m D_{ve}\), where \(D_{ve} = \sqrt{A/\pi }\) is the characteristic size of the vesicle. This spontaneous curvature model of Canham–Helfrich has proven capable of describing numerous observations on membranes, so much so that it is to this day the reference model for the interpretation of various experiments [11].
The constraint on A does not prevent the variation of the surface area of the two constituent monolayers, which can be thought as two slightly displaced surfaces parallel to the bilayer mid-surface \(\Gamma \). Hence, the model implicitly assumes that there is at least one type of molecule within the bilayer that is able to flip-flop sufficiently fast between the two monolayers during the membrane dynamics. However, in many cases the exchange between the two monolayers is hindered such that not only the total number of lipids is fixed, but also the number of lipids in each monolayer must be conserved. Then, the Canham–Helfrich energy needs to be modified with an additional constraint that allows only deformations that preserve both the area of the bilayer mid-surface \(\Gamma \) and those of the two constituent monolayers, leading to the bilayer-coupling model [12]. Given the already considered conservation of A, it is sufficient to constrain the area difference \(\Delta A\) between the two parallel surfaces of the monolayers, which can be done by fixing the integral of the mean curvature of \(\Gamma \) [11]
where \(I_{M0}\) sets the preferred integrated mean curvature of the vesicle, namely the preferred area-difference between the two monolayers. Total surface area A and enclosed volume V must also be conserved. When the nonlocal bending rigidity\(k_{\Delta } \rightarrow 0\), the spontaneous curvature model of Canham–Helfrich is recovered, while, when \(k_{\Delta } \rightarrow +\infty \), the integral of the mean curvature \(I_{M}[\Gamma ]\) must be equal to \(I_{M0}\), that is the bilayer-coupling model is recovered. In fact, the nonlocal bending rigidity \(k_{\Delta }\) turns out to be of the same order of magnitude as the bending rigidity k, and often [14] it is assumed \(k = k_{\Delta }\). Following Hossein and Deserno [15], the area-difference elasticity leads to a differential area strain in the monolayers, that can be associated to a concomitant differential stress. Such calculations suggest to set \(k_{\Delta } = 2k\).
As anticipated, these elastic models work due to the scale separation between membrane thickness and vesicle size. Anyway, these two scales become simultaneously important in several key biological processes. For example, during the fusion of two vesicles, the two initially separated membranes merge together through a local rearrangement of their lipids. Therefore, such process involves not only a large scale deformation but also the scale of the membrane at the level of which the rearrangement takes place. The same is true for membrane fission, and both fusion and fission are at the basis of a plethora of biological processes, like endo- and exocytosis, neurotransmission, intercellular communications, and viral infections. The same merging mechanism is also involved to release the cargo of the lipid droplets used for drug delivery. Therefore, since in many important cases both scales must be considered, we used a diffuse interface version of the Canham–Helfrich energy, Eq. (1), also including the Gaussian energy term which is the leading term during topological transitions like membrane fusion/fission [16]. This approach is convenient since it is equivalent to the Canham–Helfrich elasticity, with the additional presence of the scale of the membrane thickness (the diffuse interface width), allowing the description of fusion/fission of large vesicles which is not possible with the sharp-interface models described above, and not accessible with molecular simulations due to their computational cost. Recently, we have shown that the diffuse interface inherits key features of the membrane structure, providing results in accordance with the hemifusion pathway for fusion [17].
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In this work, after briefly reviewing the diffuse interface description, we discuss an additional energy term to take into account the area-difference elasticity. We show that simulations with such a term reproduce the energy and shape of vesicles as obtained with the (sharp-interface) area-difference elasticity model, to which the diffuse interface approaches asymptotically in the limit of small membrane thickness.
2 The diffuse interface model
2.1 Phase-field and Ginzburg–Landau free energy
The phase-field approach replaces the sharp-interface with a small and smooth transition layer, identified as the diffuse interface [18‐22]. The approach is convenient as it allows to introduce the otherwise missing membrane scale by matching the diffuse interface width with the actual bilayer thickness. Moreover, the phase-field can naturally handle topological transitions, which are not possible with the sharp-interface model, where surgery is necessary to cut and paste patches of membrane surface.
The phase-field \(\phi (\textbf{x})\) is a smooth function that takes value in the range \((-1; \, 1)\) and is defined everywhere in the domain \(\Omega \subseteq \mathbb {R}^3\). The field is \(-1\) in the space region outside the membrane vesicle, \(+1\) inside, while it rapidly but smoothly varies between these two values in the small region referred to as the diffuse interface. The width of the interface region depends on the small parameter \(\epsilon \), which therefore needs to be related to the actual bilayer thickness \(\ell _{me} = 5 \, \text {nm}\). Often, we choose \(6 \epsilon = \ell _{me}\) since about the \(97 \%\) of the variation of the value assumed by \(\phi \) is within this (space) interval as will be clear in what follows. It is such a match that introduces the scale of the bilayer thickness in the diffuse interface approach and thus sets the scale of the considered vesicles.
Given this mathematical structure, one has to introduce a proper free energy functional which depends on \(\phi (\textbf{x})\) and its derivatives (actually only first and second derivatives). The rational is that this Ginzburg–Landau type of free energy must converge to the sharp-interface energy of Canham–Helfrich in the limit of small membrane thickness. In order words, when \(\lambda = \epsilon /D_{ve}<< 1\), the Ginzburg–Landau free energy is asymptotically equivalent to the Canham–Helfrich energy, Eq. (1). The Ginzburg–Landau free energy is the sum of two contributions
models the Gaussian energy (second term on the right hand side of Eq. (1)). The phase-field version of the bending energy was initially introduced by Du et al. [23], and then used for other works by several groups [24‐31], while we introduced \(E_{\textrm{G}}[\phi ]\) in order to investigate topological transitions [16].
We showed that the entire functional \(E_{\textrm{CH}}[\phi ]\) approaches the Canham–Helfrich energy of Eq. (1) in the sharp-interface limit \(\lambda<< 1\). Indeed, denoting with \(d(\textbf{x})\) the signed distance between point \(\textbf{x} \in \Omega \) and \(\Gamma \), it is reasonable to assume that
where \(d^*(\textbf{x}) = d(\textbf{x})/\epsilon \). By a direct substitution of Eq. (10) in the Ginzburg–Landau free energy (Eq. (5)), one is left with
with the bar denoting the dimensionless lengths obtained by dividing by \(D_{\textrm{ve}}\), the prime denoting the derivative done with respect to \(d^*(\textbf{x})\), and \(\textbf{n} = \varvec{\nabla } d\). It is straightforward to recognize that, in the sharp-interface limit (\(\lambda<< 1\)),
in order to minimize the free energy. Hence, in this limit, the phase-field \(\phi (\textbf{x})\) approaches \(\tanh (d(\textbf{x})/(\epsilon \sqrt{2}))\), which is the solution of \(f'' - (f^2-1)f = 0\). Actually, it may be shown that
for which we refer to our former work [16]. This result shows that \(6 \epsilon \) actually contains about the \(97\%\) of the \(\phi \)-variation between \(\pm 1\), which are the values that label the inside and outside of the vesicle. Furthermore, the \( \phi = 0\) level set identifies the membrane mid-surface \(\Gamma \) (\(d(\textbf{x}) \equiv 0\)), whose inward-pointing unit normal is \(\textbf{n}(\textbf{x})\). Therefore, \(\varvec{\nabla } \cdot \textbf{n} = -2M\), and \((\varvec{\nabla } \cdot \textbf{n})^2 + \textbf{n} \cdot \varvec{\nabla } (\varvec{\nabla } \cdot \textbf{n}) = 2G\), as directly follows from differential geometry [32]. If we call \(f_0(\bar{d})(\textbf{x}) = \tanh \left( \bar{d}(\textbf{x})/(\lambda \sqrt{2})\right) \) the leading order of the phase-field \(\phi (\textbf{x})\), then one may notice that \(\sqrt{2} f_{0}' = (1 - f_{0}^2)\), and that \({f_{0}'^2(\bar{d}(\textbf{x})/\lambda )}/{\lambda } {\mathop {\longrightarrow }\limits ^{\mathcal {W}}} \; {2 \sqrt{2}}/3 \; \delta (\bar{d}(\textbf{x})),\)\({f_{0}'^4(\bar{d}(\textbf{x})/\lambda )}/{\lambda } {\mathop {\longrightarrow }\limits ^{\mathcal {W}}} \; {8 \sqrt{2}}/{35} \; \delta (\bar{d}(\textbf{x})),\) in the sharp-interface limit, where \(\delta (\bar{d}(\textbf{x}))\) is the Dirac’s delta function, which localizes an integral over the entire domain on the surface \(\bar{d}(\textbf{x}) = 0 \iff x \in \Gamma \). Thence, for \(\lambda<< 1\), the Ginzbug-Landau free energy, Eq. (5), reduces to an integral over \(\Gamma \):
which approaches the surface area of \(\Gamma \) in the sharp-interface limit. As regards the constraint on the enclosed volume, we prescribe the value of
which is indeed the measure of the inner environment since, for \(\lambda<< 1\), the phase-field approaches the Heaviside function with \(-1\) in the outer environment and \(+1\) in the inner environment.
Fig. 1
a A spherical vesicle, with its \(6 \epsilon = 5 \, \text {nm}\) membrane thickness. The right part of the sphere is replaced by the half cross section of the contour plot of the phase-field \(\phi \). Bottom, an enlargement of the diffuse interface, where some lipids are qualitatively sketched to show the interpretation of the phase-field. Next to the right of the enlargement is the lateral stress profile s(z) in function of the coordinate z normal to the mid-surface. The graph is for the case \(k_{\textrm{G}} = -0.7 k\), \(m = 0\), as reported in our previous work [17], and peaks are of the order of hundreds of bars for \(k \approx 20 \, k_BT\). b The contour plot of the phase-field in an intermediate step of the fusion between two spherical vesicles. The enlargement shows the hemifused structure of the merging region. Adapted from [16]. c A lipid tubule can be seveared by an external force field which mimics the action of the constrictase dynamin. Three regimes depending on the polymerization length of the dynamin are predicted by the diffuse interface: in regime I (short coats) no fission occurs; in regime II fission occurs at the center of the dynamin coat; in regime III (sufficiently long coats) fission occurs at the edges of the coat. Adapted from [33]
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2.2 Review of results on topological transitions
The diffuse interface approach is convenient since it naturally handles topological transitions. Moreover, as we discussed in our former work [16], the approach permits to evaluate the force fields associated with the Gaussian energy term \(E_{\textrm{G}}[\phi ]\), which are the leading forces during fusion/fission events – the force field associated with the Ginzburg–Landau free energy turns out to be \(\textbf{f} = (\delta E_{\textrm{CH}}/\delta \phi ) \varvec{\nabla } \phi \), where \(\delta E_{\textrm{CH}}/\delta \phi \) is the functional derivative of Eq. (5). This is remarkable since such forces are not accessible within the sharp-interface model of Canham–Helfrich, even if it predicts their leading role during a topology change due to the energy jump/drop prescribed by the Gauss-Bonnet theorem (Eq. 2). Thus, our Ginzburg–Landau free energy appears as the closest generalization of the Canham–Helfrich energy, which allows the computation of the Gaussian forces and the natural occurrence of topology changes.
More recently, we have also shown [17] that the diffuse interface has a lateral stress profile that is a coarse-grained version of those found with molecular models—the lateral stress profile is the distribution of tangential stresses across the bilayer thickness. The lateral stress profile is determined by the amphiphilic structure of lipids and is a key quantity used in molecular simulations to extract the elastic constants of the bilayer [19]. Conversely, the lateral stress of the diffuse interface is obtained as an explicit function of the elastic constants, providing an opposite point of view. This result is therefore a unique opportunity to bridge the scale between molecular and meso/macroscopic models, while, at the same time, shows clearly the multiscale capability of the diffuse interface approach, which has condensed into the elastic constants some of the fundamental characteristics of the smaller scale.
The upper part of Fig. 1a shows a spherical vesicle, with its \(6 \epsilon = 5 \, \text {nm}\) membrane thickness. The right part of the sphere is replaced by the half cross section of the contour plot of the phase-field \(\phi \), which identifies the inner (\(\phi = +1\)) and the outer environments (\(\phi = -1\)), and the bilayer mid-surface \(\Gamma \) of Canham–Helfrich (\(\phi = 0\)). The lower part of Fig. 1a shows an enlargement of the diffuse interface, where we also qualitatively sketched the lipids: one leaflet for \(\phi < 0\), and the other leaflet for \(\phi > 0\). Next to the right of the enlargement is the lateral stress profile s(z) of the diffuse interface [17], for the case \(k_{\textrm{G}} = -0.7 k\), \(m = 0\)—here, z is the coordinate normal to the bilayer, and the profile is for a sphere with \(\lambda \rightarrow 0\). Positive peaks represent attraction in the (hydrophilic) head group regions, which therefore tend to minimize the surface in contact with the aqueous environments, protecting the hydrophobic tails. The lipid tail region is repulsive, with a positive, small central stress bump at the bilayer mid-plane. The extent of this bump is controlled by the ratio \(k_{\textrm{G}} / k\), as we showed in our previous work [17]. For reasonable values of k, i.e. \(k = 20 \, k_BT\), peaks of s(z) are on the order of several hundreds of bars as found with molecular models [34].
The diffuse interface approach was used to evaluate the minimal energy pathway for the fusion/fission between two spherical large unilamellar vesicles and a dumbbell shaped one [16]. As shown in Fig. 1b, the fusion of the two spheres occurred via the formation of an intermediate configuration, where only the proximal leaflets were merged, while the distal ones were still separated. This is the so-called hemifusion intermediate, observed both in experiments [35, 36] and molecular simulations [37, 38] – see Aeffner et al. [39] for the electron density map of the stalk-hemifusion intermediate. This intermediate state can be stable, metastable, or unstable, depending on the size of the spheres [17], or more in general on the curvature of the vesicle considered. For example, we also investigated the minimal energy pathway for the piercing of a large oblate spherical vesicle into a toroidal one [17]. In this case, the hemifusion intermediate was metastable, with the grade of metastability depending on the value of \(k_{\textrm{G}}/k\), reflecting the well-known influence of lipids’ shape on the stability of the hemifusion state [40].
The minimal energy pathway for the transition between two large spherical unilamellar vesicles and a dumbbell shaped one showed that the energy barrier for fission is due to the large-scale deformation the vesicle must undergo, while the fusion barrier is substantially topological due to the Gauss-Bonnet theorem, Eq. (2). The force fields needed to drive the process along the minimal energy pathway that we computed [16] had an intensity in accordance with experimental data on typical fission protein systems, \(\sim 20 \, \text {pN}\). At the same time, the forces driving the fusion process were found to be extremely localized and intense, and associated with a large and steep energy barrier of about \(226 \, k_BT\), which prevents the fusion process from being thermally activated. While this result is clearly in agreement with the fact that cell membranes must be extremely stable against fusion in order to avoid unwanted unions, how proteins can carry out such a formidable work is still unclear – proteins should not just bring two membranes in close proximity [41]. For these reasons, we also used the diffuse-interface approach to investigate the effect of a local variation of the Gaussian modulus in the merging region [42], namely in the region of the enlargement of Fig. 1b. We showed that such a local modification drastically lowers the fusion energy barrier, allowing the process to be thermally activated, and stabilizes the hemifusion intermediate. This local modification is compatible with several biological processes, such as the influenza virus infection [42].
The diffuse interface approach was also used to investigate fission of lipid tubules under the constriction of an external force field which mimics the action of dynamin [33], the first protein shown to catalyze the fission of lipid membranes [43]. The study showed a criticality in the length scale of the dynamin coat. In particular, three different regimes were found increasing dyanmin polymerization length, see Fig. 1c. In regime I, fission does not occur, which means that the dynamin coat must have a minimum length H. In regime II, fission takes place at the center of the coat and the energy needed to drive the process is substantially independent of H (\(\approx 130 \, k_BT\)) and in agreement with the minimal effective amount of energy the GTP molecules must provide to the tubule [44, 45]. In regime III, for sufficiently long coats (\(H>40 \, \text {nm}\)), fission occurs in two sites, namely at the edges of the dynamin coat. Here, the required energy for the process increases with H, showing therefore that lengths in regime II are optimal for an energetically least costly fission. Both the behaviours in regimes II and III have been identified in molecular dynamics simulations [46] and experiments [44, 45], but never before fully understood.
3 Results
3.1 Area-difference elasticity term
As discussed in the Introduction, the spontaneous curvature model of Canham–Helfrich describes fluid lipid bilayers in which there is at least one type of molecule that sufficiently fast flip-flops between the two leaflets. Otherwise, the additional area-difference elasticity term must be taken into account. The Ginzburg–Landau free energy in presence of area-difference elasticity is
where \(E_{CH}[\phi ] = E_B[\phi ] + E_G[\phi ]\) is the energy that asymptotically behaves as the Canham–Helfrich one in the sharp-interface limit, Eq. (5). \(E_{ADE}[\phi ]\) is the additional term introduced to model area-difference elasticity,
with \(\psi _B^2\) that appears in the bending energy term \(E_B[\phi ]\) [23], Eq. (5). The addition of \(E_{ADE}[\phi ]\) to the free energy does not alter the asymptotic analysis carried out in Sect. 2.1, and the phase-field that minimizes \(E[\phi ] + E_{ADE}[\phi ]\) is still \(\phi (\textbf{x}) = f_0(\bar{d}(\textbf{x})) + O(\lambda ^2) = \tanh \left( \bar{d}(\textbf{x})/(\lambda \sqrt{2})\right) + O(\lambda ^2)\). Therefore,
and, since \(\varvec{\nabla } \cdot \textbf{n} = - 2M\) and \(f_0'(\bar{d}(\textbf{x})/\lambda )/\lambda {\mathop {\longrightarrow }\limits ^{\mathcal {W}}} 2 \delta (\bar{d}(\textbf{x}))\) in the sense of distributions for \(\lambda<< 1\) (sharp-interface limit), eventually,
that is \(E_{ADE}[\phi ]\) approaches the area-difference elasticity term of Eq. (4) in the sharp-interface limit.
In what follows, we will calculate vesicle shapes of minimal energy by means of a gradient flow dynamics and compare results with those of the sharp-interface model. For this purpose, the Ginzburg–Landau free energy of Eq. (16) is augmented with the constraints for the surface area \(A[\phi ]\) (Eq. 14) and enclosed volume \(V[\phi ]\) (Eq. 15):
Constraints are imposed by means of an augmented Lagrangian method [47], which assumes that \(A_0\) and \(V_0\) are the target area and enclosed volume, \(M_1\) and \(M_2\) two penalty constants, and \(\gamma \) and \(\Delta p\) two estimates of the Lagrange multipliers, updated at each time step of the evolution as
where M is the mobility coefficient and \(\delta \bar{\mathcal {E}} / \delta \phi \) is the functional derivative of the augmented energy. Such a dynamics leads to find steady states that are of minimal energy \(\mathcal {E}[\phi ] = E[\phi ] + E_{ADE}[\phi ]\), for fixed surface area and enclosed volume.
3.2 Numerical simulations
In order to investigate the phase-field area-difference elasticity term, we calculate vesicle shapes of minimal energy solving the Allen-Cahn gradient flow, Eq. (25), and compare the obtained results with the sharp-interface model. In particular, we calculate shapes for three different genera: a vesicle with the topology of the sphere (\(g=0\)), one toroidal vesicle (\(g=1\)), and one with \(g=2\). For the purpose of finding minimal energy shapes, we set \(k_{\textrm{G}} = 0\), since topological transitions do not matter, and exploit the scale invariance of the asymptotic sharp-interface model. This allows to significantly lower the computational cost with respect to the study of fusion and fission, since it is no longer necessary to match the width of the interface with the bilayer thickness. At the same time, the setting is sufficient to numerically test the asymptotic convergence of the extended phase-field model with area-difference elasticity to the sharp-interface limit.
Fig. 2
a A prolate (genus 0 vesicle) that minimizes the Ginzburg–Landau free energy, Eq. (16), with \(k_\Delta = 4 k\). The vesicle has a reduced volume \(v = 0.8\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 15.07\), and reduced spontaneous curvature \(m_0 = 0\). Axisymmetric simulation has been carried out in the \(r-z\) plane with a \(20 \times 66\) domain (\(\epsilon = 1\)), and a \(40 \times 132\) grid. Computed energies are \(E_B / 8 \pi k = 1.7\) and \(E_{ADE} / 8 \pi k = 9.1 \cdot 10^{-4}\). b A torus (genus 1 vesicle) that minimizes the Ginzburg–Landau free energy, Eq. (16), with \(k_\Delta = k\). The vesicle has a reduced volume \(v = 0.72\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 12.56\), and reduced spontaneous curvature \(m_0 = 0\). Simulation has been carried out in the \(r-z\) plane with a \(40 \times 40\) domain (\(\epsilon = 1\)), and a \(80 \times 80\) grid. Computed energies are \(E_B / 8 \pi k = 1.58\) and \(E_{ADE} / 8 \pi k = 1.71 \cdot 10^{-5}\)
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Figure 2a depicts a prolate (\(g=0\) vesicle) that minimizes the Ginzburg–Landau free energy, Eq. (16), with \(k_\Delta = 4 k\). The vesicle has a reduced volume \(v = 0.8\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 15.07\), and reduced spontaneous curvature \(m_0 = 0\). Simulation has been carried out in the \(r-z\) plane with a \(20 \times 66\) domain (\(\epsilon = 1\)), and a \(40 \times 132\) grid. Computed energies are \(E_B / 8 \pi k = 1.7\) and \(E_{ADE} / 8 \pi k = 9.1 \cdot 10^{-4}\). The shape is in accordance with the phase-diagram of Miao et al. [13].
Figure 2b shows a minimal energy torus (\(g=1\) vesicle), with \(k_\Delta = k\). The vesicle has a reduced volume \(v = 0.72\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 12.56\), and reduced spontaneous curvature \(m_0 = 0\). In this case, simulation has been carried out in the \(r-z\) plane with a \(40 \times 40\) domain (\(\epsilon = 1\)), and a \(80 \times 80\) grid. Computed energies are \(E_B / 8 \pi k = 1.58\) and \(E_{ADE} / 8 \pi k = 1.71 \cdot 10^{-5}\). The obtained shape is in agreement with the phase-diagram of [48], and has got the bending energy of a Clifford torus, which indeed has approximately the same reduced volume.
Finally, Fig. 3 reports a genus 2 vesicle of minimal energy, with \(k_\Delta = 2 k\). The \(g=2\) vesicle has a reduced volume \(v = 0.6\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 14.44\), and reduced spontaneous curvature \(m_0 = 0\). In this case, the simulation has been carried out in a full 3D \(160 \times 90 \times 60\) domain (\(\epsilon = 1\)), with a \(160 \times 90 \times 60\) grid. Computed energies are \(E_B / 8 \pi k = 1.84\) and \(E_{ADE} / 8 \pi k = 3.01 \cdot 10^{-6}\). Similar shapes can be found in the phase-diagram for the bilayer-coupling model [49] in the same range of reduced volume and reduced integrated mean curvature.
Fig. 3
A genus 2 vesicle that minimizes the Ginzburg–Landau free energy, Eq. (16), with \(k_\Delta = 2 k\). The vesicle has a reduced volume \(v = 0.6\), reduced non local spontaneous curvature \(I_{M0}/(D_{ve}/2) = 14.44\), and reduced spontaneous curvature \(m_0 = 0\). The full 3D simulation has been carried out in a \(160 \times 90 \times 60\) domain (\(\epsilon = 1\)), with a \(160 \times 90 \times 60\) grid. Computed energies are \(E_B / 8 \pi k = 1.84\) and \(E_{ADE} / 8 \pi k = 3.01 \cdot 10^{-6}\)
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4 Conclusions
In this work, we have reviewed the diffuse-interface approach for lipid bilayer vesicles, which has recently been used to study fusion/fission events of large vesicles with multiscale resolution, allowing simulations not achievable with molecular models. The method provides the possibility of investigating the influence of the elastic constants on such processes, and their link to the smaller few nm-length scale of the bilayer thickness. We also analyzed an additional energy term in the diffuse-interface context to take into account area-difference elasticity. After showing the rational behind its introduction, we showed vesicle shapes of minimal energy obtained in presence of such additional term. Results were in accordance with shapes and energy of the sharp-interface model, thus corroborating the effectiveness of the term, which can then be used in the future for the study of topological transitions with area-difference elasticity.
Acknowledgements
The research has received financial support from ICSC-Italian Research Center on High Performance Computing, Big Data, and Quantum Computing, funded by European Union-NextGenerationEU. Support is acknowledged from the 2022 Sapienza Large Project: Plants and plant-inspired microfluidics. Concerning computational resources, we acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support (ISCRA-B D-RESIN, ISCRA-B CAMAGE3D, ISCRA-C GaVesFu).
Declarations
Conflict of interest
The authors declare no conflict of interest.
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