Calculating Transition Energy Barriers and Characterizing Activation States for Steps of Fusion

Abstract

We use continuum mechanics to calculate an entire least energy pathway of membrane fusion, from stalk formation, to pore creation, and through fusion pore enlargement. The model assumes that each structure in the pathway is axially symmetric. The static continuum stalk structure agrees quantitatively with experimental stalk architecture. Calculations show that in a stalk, the distal monolayer is stretched and the stored stretching energy is significantly less than the tilt energy of an unstretched distal monolayer. The string method is used to determine the energy of the transition barriers that separate intermediate states and the dynamics of two bilayers as they pass through them. Hemifusion requires a small amount of energy independently of lipid composition, while direct transition from a stalk to a fusion pore without a hemifusion intermediate is highly improbable. Hemifusion diaphragm expansion is spontaneous for distal monolayers containing at least two lipid components, given sufficiently negative diaphragm spontaneous curvature. Conversely, diaphragms formed from single-component distal monolayers do not expand without the continual injection of energy. We identify a diaphragm radius, below which central pore expansion is spontaneous. For larger diaphragms, prior studies have shown that pore expansion is not axisymmetric, and here our calculations supply an upper bound for the energy of the barrier against pore formation. The major energy-requiring deformations in the steps of fusion are: widening of a hydrophobic fissure in bilayers for stalk formation, splay within the expanding hemifusion diaphragm, and fissure widening initiating pore formation in a hemifusion diaphragm.

Introduction

Biological membranes can fuse because they are fluid; this fluidity is conferred by the lipids of the membranes. The movements and reconfigurations of membrane lipids are thus central to membrane fusion. Based on both experimental and theoretical studies, investigators in the field of membrane fusion presently have a consensus on key lipid intermediates of membrane fusion (1, 2, 3, 4, 5). In this model, the apposing, proximal, lipid monolayers of two separated membranes merge to create an intermediate structure known as a “stalk” (6, 7). The structure of stalks connecting stacks of membranes has been determined to ∼0.3 nm by x-ray diffraction experiments for several lipid compositions (8, 9, 10, 11). In the stalk structure, the distal monolayers of the two original membranes have not yet come into contact. When they do, a new intermediate arises: hemifusion, in which the two distal monolayers touch each other, creating a hemifusion diaphragm, which continues to separate aqueous compartments. Then, the formation of a pore, either directly from a stalk without formation of the diaphragm, or from within a diaphragm, connects the aqueous compartments.

Molecular dynamics (MD) (12, 13, 14, 15, 16), Monte Carlo (17, 18, 19), and continuum membrane mechanics studies have yielded the energies and the minimizations of these intermediates in some detail. But energy minimizations have not been used to determine an entire least-energy pathway. Obtaining this pathway would provide a means to evaluate the evolution of lipid orientations and monolayer shapes during the progression toward fusion. About 10 years ago, a practical and reliable method, the string method, was developed to calculate the pathway of least energy between two local energy minima. (These minima are referred to as basins (20, 21).) The string method is increasingly used for a number of problems, in part because it has been incorporated into standard MD software packages (16, 22).

Coarse-grain force-field calculations have used the string method to obtain the pathway of least energy from two parallel membranes to a stalk (13) and for a pore in a single bilayer (23). But these calculations do not provide the lipid movements nor any sense of where or how the energy is expended during migrations. Also, the forces that drive a topological change, as in the formation of a stalk or a pore, are central to fusion, but are not explicitly revealed in MD calculations. Continuum methods do not yield the importance of granularity to the transitions, but have the merit that the events needed to initiate topological changes are explicitly incorporated into calculations, and the deformations and their energies are obtained for both proximal and distal monolayers. In the case of membrane fusion, the continuum method also has the advantage that the parameters of a model make direct contact with experimentally controllable, adjustable, and measurable variables. Continuum mechanical calculations have been used to obtain the shape of a stalk (6, 7) and a fusion pore (FP) (24, 25, 26) of minimal energy. This calculated pore shape is the same as that obtained by MD using a Martini force field (25).

In this study, we use continuum mechanics to calculate the entire least-energy pathway of membrane fusion. These calculations provide a movie of lipid deformations as a function of time, as membrane geometry and topology evolve during the fusion process. At each point, the energies expended for each deformation within each monolayer are calculated. This allows us to identify the forces that dominate the transition energies for stalk, hemifusion diaphragm, and FP formation. To our knowledge, identifying the least energy path and calculating the activation energies for the transition between intermediate states in membrane fusion are each novel, never before obtained through the use of continuum membrane mechanics. Our approach provides a viable framework for a range of problems in membrane biophysics, in addition to membrane fusion.

Materials and Methods

See the Supporting Material.

Results

Energy is expelled or gained as monolayer shapes continuously change during the process of membrane fusion. Mathematically, the bilayer $\Sigma$ is assembled from proximal ${\Sigma }_{\text{P}}$ and distal ${\Sigma }_{\text{D}}$ monolayer neutral surfaces (Fig. 1 A). In hemifusion, ${\Sigma }_{\text{HD}}$ denotes the monolayers of the hemifusion diaphragm (HD) (Fig. 1 D). The energy of the bilayer is the sum of intrabilayer elastic contributions and long-range surface interactions between separate membranes:

$$\begin{array}{c}E=\underset{\Sigma }{\int }\frac{K_{\text{C}}}{2}\left[{\left(\text{div}\mathbf{d}+k_0\right)}^2-k_0^2\right]+\frac{K_{\theta }}{2}|\mathbf{d}\times \mathbf{n}{|}^2+\frac{K_{\text{A}}}{2}\frac{{\left(a-a_0\right)}^2}{a{a}_0}dS\\ +\underset{{\Sigma }_{\text{P}}}{\int }W\left(h\right){\text{cos}}^2\left(\theta \right)dS+V\left({\Sigma }_F\right).\end{array}$$ (1)

The elastic energy, defined by three fundamental, and independent, deformations—splay, tilt (27), and stretch (28)—results from the local variation in the mean orientational and extensional order of the lipids. The interactions between membranes are due to attractive van der Waals forces, repulsive hydration (and undulation) forces, and hydrophobic interactions that result when lipid configurations expose acyl chains to water.

Figure 1.

(A–F) Key lipid intermediates calculated by the string method and energy minimization: (A) PBs separated by equilibrium distance $h_0$ (the reference state), (B) widening of a fissure surface ${\Sigma }_{\text{F}}$ between bilayers (dashed lines), (C) stalk, (D) hemifusion diaphragm, (E) pore expansion, and (F) fusion pore. The gaps between the curves denoting the neutral surfaces have been exaggerated to emphasize that in our calculations, the central lipids form a free boundary between the bilayer continuum and the aqueous phases, and that this free boundary can open (creating a hydrophobic fissure) or close (creating a continuous monolayer surface).

The rotational diffusion of the polar headgroup and acyl chains of a lipid body are well described by rigid-body motion (29, 30). The director $\mathbf{d}$ is the average lipid orientation (longitudinal axis) directed from the lipid head to lipid tail; $\mathbf{n}$ is the surface normal directed outward from the bilayer core: $\left|\mathbf{d}\right|=\left|\mathbf{n}\right|=1$. We will assume that $k_0$ is uniform in the distal and proximal monolayers and use $k_{0,\text{D}}$ and $k_{0,\text{P}}$ to denote the distal and proximal values of spontaneous curvature, respectively. We will relax the assumption of uniformity in Hemifusion Diaphragm Formation and Expansion, where we consider the effect of compositional variation. There we use $k_{0,\text{HD}}$ to denote the value of spontaneous curvature along the rim of the HD. We call the bilayer symmetric if $k_{0,\text{D}}=k_{0,\text{P}}=k_{0,\text{HD}}=:k_0$. The subtraction of $k_0^2$ in Eq. 1 sets splay energy to zero for planar bilayers (PBs). Thus the energy of the reference state of the Hamiltonian E is the energy of two PBs with the same total surface area as $\Sigma$.

In fusion, the lipid directors generally deviate from their surface normal. The tilt deformation $\mathbf{d}\times \mathbf{n}$ with modulus $K_{\theta }=10\text{kT}{\text{nm}}^{-2}$ (31, 32) accounts for this deviation. If $\mathbf{d}\times \mathbf{n}\ne 0$, surface bending alone is inadequate for quantifying elastic energy; splay ($\text{div}\mathbf{d}$, surface divergence) generalizes bending to the case of nonzero tilt. In cylindrical coordinates (r, z),

$$\text{div}\mathbf{d}=\frac{\partial \mathbf{d}}{\partial s}\times \tau +r^{-1}\mathbf{d}\cdot {\mathbf{e}}_r,$$

where s and τ denote the arc length and unit tangent, respectively, of the monolayer surface cross section; and ${\mathbf{e}}_r$ is the r-basis vector. We use the modulus $K_{\text{C}}=10\text{kT}$ for the splay deformation, which corresponds to a bilayer bending modulus ∼0.8 × 10⁻¹² erg (28, 32).

Wherever $\mathbf{d}\times \mathbf{n}=0,$ $\mathbf{d}$ and $\mathbf{n}$ are parallel, and $\text{div}\mathbf{d}=-2H$ with H being the mean curvature, the average of the principal curvatures $k_1$ and $k_2$ of $\Sigma .$ If $\mathbf{d}$ and $\mathbf{n}$ were parallel everywhere, the first term in Eq. 1 would reduce to the familiar Canham-Helfrich mean curvature energy density (33):

$${\left(\text{div}\mathbf{d}+k_0\right)}^2={\left(-2H+k_0\right)}^2={\left(2H-k_0\right)}^2.$$

This explains why, in Eq. 1, $k_0$ is added (instead of subtracted): because the sign of splay is opposite to the sign of mean curvature for our (standard) choice of orientations of $\mathbf{d}$ and $\mathbf{n}.$ Assuming cylindrical symmetry, $k_1=-\left(\partial \mathbf{d}/\partial s\right)\cdot \tau$ for the meridian curvature, and $k_2=-r^{-1}\mathbf{d}\cdot {\mathbf{e}}_r$ for the parallel curvature. In this convention, as the area ratio of polar headgroup to acyl chain becomes smaller, the splay of the monolayers becomes more positive.

The third term in Eq. 1 accounts for the local change in area a per lipid upon extension (or compression) of a monolayer surface from its resting area $a_0.$ Here a measures the area of the lipid cross section perpendicular to $\mathbf{d},$ and corresponds to the surface area divided by the number of lipids when the lipids are normal to the monolayer surface. Thus $K_{\text{A}}=30\text{kT}{\text{nm}}^{-2}$ gives the two-dimensional modulus for increasing (decreasing) monolayer area (or ∼240 dyn cm⁻¹ for bilayer area (28)). Changes in lipid orientation do not alter lipid cross-sectional area, and hence stretch and tilt are independent deformations. Lipid bilayers are volumetrically incompressible, implying that $la=l_0{a}_0$, where l is the variable lipid length and $l_0$ is resting length. Then ${\left(a-a_0\right)}^2/a{a}_0={\left(l-l_0\right)}^2/l{l}_0$, permitting us to calculate the stretching energy density (the third term in Eq. 1) in terms of lipid length l in place of a.

The interaction between proximal neutral surfaces separated by a small distance h (Fig. 1 C) across water is repulsive and is given by the potential of hydration pressure:

$$W\left(h\right)=L{P}_0{e}^{-h/L}.$$ (2)

We use $P_0=0.25\times {10}^9\text{Pa}$ for pressure at zero separation and a decay length $L=0.35\text{nm}$, as experimentally determined from pressure-distance curves for the lamellar phase of DOPC/DOPE mixtures (11, 34). For distances $h\gg h_0$, the equilibrium distance between two PBs (Fig. 1 A), interactions are dominated by attractive van der Waals forces, where

$$W\left(h\right)=-\frac{H}{12\pi }\left[\frac{1}{h_{\text{w}}^2}-\frac{2}{{\left(h_{\text{w}}-2l_0\right)}^2}+\frac{1}{{\left(h_{\text{w}}+4l_0\right)}^2}\right]+W_0,$$ (3)

and $H=1\text{kT}$ is the Hamaker coefficient for hydrocarbon across water, $h_{\text{w}}=h-d_0$ is the water layer thickness, $d_0=8\AA$ is the lipid diameter, and $W_0$ is a constant that sets $W\left(h_0\right)=0$ (35). For intermediate distances, $h\approx h_0$, attractive and repulsive forces are comparable, and we define $W\left(h\right)$ by interpolating between Eqs. 2 and 3.

The apposing proximal monolayers are not flat in the fusion region, yielding a nonzero angle θ between the proximal surface and the horizontal plane (Fig. 1 C). To account for the interactions between nonparallel apposing bilayers, the energy density $W\left(h\right){\text{cos}}^2\theta$ in Eq. 1 smoothly interpolates between two extremes: the energy density equals that of two planar surfaces when θ = 0°, modeling two flat bilayers far from the fusion region. The energy density is minimal and zero when θ = 90°, modeling the absence of interactions between vertically aligned surfaces, e.g., the proximal monolayers of the stalk neck.

Transient hydrophobic fissures, exposing acyl chains to water, continually form and close in bilayers (36, 37). When hydrophobic fissures are directly opposite in apposing proximal monolayers (Fig. 1 B), hydrophobic attraction initiates stalk formation (3). We use a variational integral originally proposed by Marčelja (38) (see also Eriksson et al. (39) and Glaser et al. (40)) to calculate the energy of this attraction:

$$V\left({\Sigma }_{\text{F}}\right)=\sigma \underset{\eta }{\text{min}}\underset{\Omega }{\int }\rho |\nabla \eta {|}^2+{\rho }^{-1}|\eta {|}^{2}dx.$$ (4)

Here $\sigma =11\text{kT}{\text{nm}}^{-2}\approx 45\text{mJ}{\text{m}}^{-2}$ is the interfacial tension of a hydrocarbon-water interface (41, 42). The scalar field η satisfies $\eta =1$ on the water/acyl-chain interface (the fissure surface ${\Sigma }_{\text{F}}$) and $\eta =0$ on the monolayer surfaces $\Sigma ,$ which shields the bilayer core from water (40) (Fig. S1, B and C, in the Supporting Material). In the bulk region Ω, η measures water interaction and minimizers of Eq. 4 have the property that they decay exponentially from their value 1 on ${\Sigma }_{\text{F}}$ to the value 0 in the bulk. We use the attraction decay length $\rho =0.22\text{nm}$. The hydrophobic energy $V\left({\Sigma }_{\text{F}}\right)$ is thus zero when the fissure is closed (i.e., when opposing headgroup surfaces are either continuous or merged as in a stalk or PB), and approaches the surface energy $\sigma \left|{\Sigma }_F\right|$ when separation distances are large. The determination of ρ (Eq. S8) and the calculation of $V\left({\Sigma }_{\text{F}}\right)$ (Eqs. S9 and S10) are described in the Supporting Material. By treating elastic and hydrophobic energies additively (Eq. 1), we are implicitly assuming that a break in the continuity of the monolayer surface does not alter the value of the elastic moduli $K_{\text{C}}$, $K_{\theta }$, and $K_{\text{A}}$ in the vicinity of the fissure surface.

The Hamiltonian Eq. 1 thus utilizes the experimentally measured coefficients $K_{\text{C}}$, $K_{\theta }$, $K_{\text{A}}$, $k_{0,\text{P},\text{D},\text{HD}}$, $h_0$, $d_0$, $l_0$, L, $P_0$, H, σ, and ρ. We emphasize that in our notation, the moduli $K_{\text{C}}$, $K_{\theta }$, and $K_{\text{A}}$ are for a single monolayer, and their value is thus one-half of the bending, tilt, and area moduli for a bilayer (28, 32). All shapes and energies are calculated as outputs, rather than assumed. In particular, the structures shown in Figs. 1, 3 E, 5, 6 C, 8, and S3 and Movies S1, S2, S3, S4, and S5 are outputs from the calculations, and not diagrammatic illustrations. In the Results section, we investigate the dependence of energy on the parameters $K_{\text{A}}$, $k_{0,\text{P},\text{D},\text{HD}}$, $l_0$, and $h_0$ whose values have been determined accurately (Figs. 2 and 3). In Generalizing the Helfrich Hamiltonian to Predict Topological Changes, we investigate the sensitivity of energy to those parameters ($K_{\text{C}},$ $K_{\theta },$ $\rho ,$ $P_0,$ and L) that are not as experimentally well determined.

Figure 3.

(A–D) HD expansion is made energetically possible by the presence of a second lipid component in the distal monolayer. (Thick gray curves) Single distal component, $k_{0,\text{HD}}=k_{0,\text{D}}=0.01{\text{nm}}^{-1}$ and (thin black curves) two distal components, $k_{0,\text{HD}}=-0.13{\text{nm}}^{-1}$ and $k_{0,\text{D}}=0.01{\text{nm}}^{-1}$; $k_{0,\text{P}}=-0.20{\text{nm}}^{-1}$ for both curves. (Some of the single-component curves were essentially the same as for the two-component case. For clarity, these overlapping curves are plotted only once and in black.) For a single component, HD expansion continues to require energy (A, gray curve) because diaphragm splay increases with $r_{\text{HD}}$ (D, solid gray curve). With a second lipid component in the distal layer, hemifusion requires 20–30 kT in distal splay (C, solid curves), but the diaphragm thereafter continues to expand (D, solid black curve). For $k_{0,\text{HD}}\ne k_{0,\text{D}}$, the total energy E + F (Eqs. 1 and 5) decreases for $r_{\text{HD}}>1.5$ nm (A, black curve). A slight curvature at the rim of the HD (E) eliminated the tilt (D, dashed curves). Other combinations of spontaneous curvatures for a single distal component ($k_{0,\text{HD}}=k_{0,\text{D}}$, $k_{0,\text{P},\text{D}}=-0.2,-0.13{\text{nm}}^{-1}$, $k_{0,\text{P},\text{D}}=-0.2,+0.05{\text{nm}}^{-1}$, $k_{0,\text{P},\text{D}}=-0.22,+0.05{\text{nm}}^{-1}$, and $k_{0,\text{P},\text{D}}=-0.22,-0.13{\text{nm}}^{-1}$) yielded increasing total energy. (E) Sign of splay and its relationship to geometric curvature in the HD, distal, and proximal monolayers.

Figure 5.

Stalk shapes, using the definition of tilt in Eq. 1, derived from numerical energy minimization (Eq. 6, solid curves) are in agreement with the experimental neutral surface iso-contours (thick shaded curves), extracted from the Δρ = 0.3 levels in Fig. 3A of Aeffner et al. (11). If, however, the elastic terms in Eq. 1 are replaced with those of Eq. 8, the predicted stalk exhibits a pointed, cusp-shaped distal monolayer (dashed curve); this contrasts with the experimentally observed shapes. The numerical neutral surfaces are for our standard constants $l_0=1.5\text{nm}$, $K_{\text{C}}=10\text{kT}$, $K_{\theta }=10\text{kT}{\text{nm}}^{-2}$, $K_{\text{A}}=30\text{kT}{\text{nm}}^{-2}$, $P_0=0.25\times {10}^9\text{Pa}$, and $L=0.35\text{nm}$.

Figure 6.

(A–C) The energy of pore formation in a PB is sensitive to the hydrophobic parameters σ and ρ; total energy decreases with increasing ρ (B) and decreases with decreasing σ (C). Pore radius is the distance of separation between the axis of symmetry and the midplane surface (i.e., the widest point on the fissure). $k_0=+0.05{\text{nm}}^{-1}$ in all panels. Increasing the hydrophobic decay length to $\rho =0.44\text{nm}$ (B) reduces the hydrophobic interaction, along with the splay and stretch deformations. (C) With a lowered $\sigma =5.5\text{kT}{\text{nm}}^{-2}$ surface tension (hydrocarbon-air), the fissure surface expands radially, decreasing splay at the expense of increasing the now less costly fissure energy. (D) Fissure geometry is sensitive but neutral surface shape is insensitive to the hydrophobic parameters σ and ρ. The curves are the neutral and fissure surface cross sections for a pore in a PB at the activation energies in (A)–(C). Pore radii = (solid) 0.9 nm; (dashed) 1.1 nm; (dotted) 1.0 nm.

Figure 8.

Summary of the energy contribution of each deformation at the height of the activation barriers. The leftmost column tabulates, component-wise, the activation energy at the peaks of the profile in Fig. 4, relative to the left adjacent basin (in kT). The rows are for (top) stalk formation, (middle) HD expansion, and (bottom) pore formation. The images show the energy density at the peaks; the color bars are in units of kT nm⁻². In the splay column, the colors are for the splay energy density $\left(K_{\text{C}}/2\right){\left(\text{div}\mathbf{d}+k_0\right)}^2$, i.e., without normalization to the reference energy density. Splay energy decreases as the initially flat distal monolayer (purple) deforms into a curved monolayer (aqua), accommodating its spontaneous curvature ($k_{0,\text{P}}=-0.2{\text{nm}}^{-1}$). Splay is concentrated in regions of high parallel curvature—the nonvertical central lipids—and is independent of the break in the monolayer surface. Stretching is concentrated near the fissure surface, where lipids compress to minimize area, and on the diaphragm rim in hemifusion. Tilt is concentrated outside the regions where stretching is greatest.

Figure 2.

(A and D–F) Total energy E (A) Stalk energy and barrier height in symmetric bilayers decrease with spontaneous curvature $k_0.$ (B and C) Energy components for the thick curve in (A). (D) Stalk energy and barrier height are only slightly sensitive to bilayer separation $h_0$ at r_max = 100 nm. (E) The barrier height increases with resting lipid length $l_0$ but stalk energy is insensitive to $l_0.$ (F) Suppressing the extensional degree of freedom ($K_{\text{A}}=\infty ,l\equiv l_0$) significantly increases the barrier height and stalk energy.

We assume cylindrical and reflectional symmetry of the membrane bilayer (i.e., the two monolayers of each bilayer can have different compositions, but the apposing proximal monolayers are the same as are the apposing distal monolayers). The cross section of a monolayer surface is a curve in the (r, z) plane. We apply a far-field boundary condition that fixes the equilibrium separation $h_0$ between the proximal curves at a radius $r_{\text{max}}=100\text{nm}$ away from the fusion site. We assume a value for equilibrium separation $h_0$ in the range 3–5 nm. As will be seen, (e.g., Fig. 2 D), energy is relatively insensitive to $h_0,$ because the elastic deformations are localized to within 50 nm of the stalk, and van der Waals attractions keep bilayers nearly parallel outside this distance.

Near the fusion site, the fissure surface ${\Sigma }_{\text{F}}$ is a free boundary defining the interface between the continuous bilayer core and the aqueous phase. The endpoints of a curve for a continuous monolayer can either terminate along the axis of symmetry, or terminate on an apposing curve (e.g., Fig. 1, A and C–F). For an interrupted, noncontinuous monolayer, as must occur in fusion, the endpoints of the monolayer curves must end in a space that does not connect to another monolayer curve (representing a break in the monolayer surface, e.g., near the dashed portion of Fig. 1 B). (The gaps between the curves in Fig. 1 have been exaggerated to the width of one lipid to illustrate this degree of freedom. In the actual calculations, the small gaps representing monolayer continuity are only a few angstroms.) The elastic and long-range interactions are integrated along the continuous part of the bilayer (i.e., not across the discontinuity); the hydrophobic attraction $V\left({\Sigma }_{\text{F}}\right)$ assigns an energy to the free boundary surface. Cylindrical symmetry and a straight acyl-chain backbone requires that the fissure surface ${\Sigma }_{\text{F}}$ be conical. This can be assumed in the context of continuum mechanics because the average water/acyl-chain interface would be geometrically flat over any reasonable timescale (the frequency of lipid axial rotations is on the order of 10⁷ s⁻¹ (29, 30)).

The energetic cost for forming voids (interstices) within a bilayer (43) is much greater than the combined tilt and splay energies required to fill a void with acyl chains (6). Electron density profiles of stalk architecture show that aqueous voids are not present in the bilayer core (8, 9, 10, 11). We prevent the introduction of voids by placing the terminal positions of opposing acyl chains at the same points along the midplane surface.

A nondimensional path coordinate s parameterizes the shape changes from one membrane configuration to another; s = 0 corresponds to an initial shape, s = 1 corresponds to a final shape, and $0<s<1$ tracks the transitional shapes. A least energy path minimizes the height of the energy barrier between energy basins. Numerical Methods in the Supporting Materials and Methods gives full implementation details for calculating bilayer movements (Eqs. S4 and S12) and describes how we used the simplified string method ((21) and Eqs. S14 and S15) to find least energy paths.

Stalk formation

To evaluate the energy required to form a stalk, we start with a path from two PBs (s = 0), each with a widening fissure surface, to a stalk (s = 1), consisting of a sequence of intermediate toroidally shaped bilayers. We minimize the energy of each of these shapes subject to fixed boundary conditions. This yields a reasonably close approximation to a least energy path. We then replace the fixed boundary condition with a free boundary condition for the fissure, and apply the string method to simultaneously adjust bilayer and fissure shape to energy minimality. A video comparing deformations for the initial, assumed path and for the least energy path is found in Movie S1.

The energy landscape of stalk formation has two energy basins: the PBs with ground state energy 0 kT, and the stalk whose minimal energy varies with experimental conditions. The energy barrier is overcome as the hydrophobic attraction draws opposing fissures together. For a symmetric bilayer with spontaneous curvature $k_0=-0.17{\text{nm}}^{-1}$, lipid length $l_0=1.5\text{nm}$, and bilayer separation $h_0=3\text{nm}$, the height of the energy barrier is 30 kT and the energy of the stalk is 4 kT (Fig. 2 A, thick curve).

Expansion of the fissure yields the dominant energetic term opposing stalk formation. The hydrophobic interaction between fissures reaches a maximum of 22 kT (Fig. 2 B, dash-dot-dotted curve). This energy term decreases rapidly as the distance between the fissures decreases and they merge. Splay energy also decreases significantly because the proximal monolayer favors a curved stalk shape over the planar shape (Fig. 2 B, solid curve, $k_{0,\text{P}}<0$). But the reduction in bilayer separation causes a slight increase (∼10 kT) in repulsion energy (Fig. 2 B, dotted curve). Hydrophobic interactions between lipids on a fissure surface cause compression, and consequently the proximal stretching energy also increases slightly, coincidentally ∼10 kT (Fig. 2 B, dash-dotted curve).

The dimpled distal surface of a stalk, consistent with experimental stalk architecture (see Fig. 5, thick gray curves), has a height considerably greater than twice the bilayer thickness (roughly 7 nm, Fig. 1 C). This thickness is achieved by stretching of the distal monolayers, accounting for most of the stored energy, and the lipids that intervene between the two distal monolayers in the stalk (Fig. 2 C, dotted curve). Because the distal tilt and splay deformations are localized near the neck of the stalk, they do not contribute much to the overall energy (Fig. 2 C, dashed and solid curves). Stalk formation is insensitive to distal spontaneous curvatures, even for asymmetric bilayers. In a stalk, the proximal splay is positive and meridian curvature $k_1$ is negative. A negative spontaneous curvature $k_{0,P}$ matches the negative meridian curvature. Consequently, after stalk formation, the remaining energetic contributions are entirely offset by the decrease in proximal splay energy.

To further illustrate how the energy barrier against stalk formation and the energy of a stalk itself depend on experimental conditions, we vary spontaneous curvature $k_0,$ resting lipid length $l_0,$ and equilibrium separation $h_0.$ The height of the energy barrier and, to a greater extent, the energy of the stalk decrease with decreasing (e.g., more negative) spontaneous curvature (Fig. 2 A). The energy of a stalk is sensitive to spontaneous curvature, but the shape of a stalk is insensitive (Fig. 5). For $k_0\ge 0,$ the energy profile nearly increases monotonically with s. Consequently, for lipids with positive spontaneous curvature, stalks will not form in practice because they require too much energy, and if they were to form, they would not be stable (evolving into an HD requires further energy). The minimal energy of a stalk is insensitive to lipid length (Fig. 2 E). But the height of the energy barrier increases with this length, mainly due to the proportionality of fissure area and $l_0.$ To evaluate the consequences of stretching, we fixed lipid length ($l\equiv l_0$) by assigning an effectively infinite stretching modulus $K_{\text{A}}.$ Suppressing the stretching deformation results in a 30 kT upward shift in the energy profile (Fig. 2 F). Stretching has generally been ignored in prior continuum mechanical studies of fusion (3, 4, 7), and so its contributions have not been appreciated. Perhaps surprisingly, we find that the stretching deformation must be included to not only yield agreement between numerically and experimentally derived minimal stalk shapes, but also to predict physically realizable transition energies. See Movie S2.

Formation of a fusion pore directly from a stalk

Large thermal fluctuations or external forces are needed for the stalk to evolve into a fusion pore (FP). Conceptually, this evolution could proceed either by the distal monolayers merging with each other directly from a stalk to form a pore or by moving toward each other until they abut to form another intermediate structure, a hemifusion diaphragm. To evaluate which possibility is energetically favored, we first determine the least energy path for forming a FP directly from a stalk. Here, the energy landscape starts from the basin of a stalk (s = 0) and terminates in a fusion pore (s = 1). Employing the string method as outlined in the previous section, for $k_0=-0.17{\text{nm}}^{-1}$ we find that the energy barrier is 120 kT to form a pore directly from a stalk, without going through hemifusion. This inordinately high value of energy is attributed to two dominating factors. The largest factor, ∼60 kT, stems from the splay energy of the distal monolayer. When the distal monolayers merge, the central distal lipids must deviate from their vertical orientation while at the same time maintaining continuity of the distal monolayer. Mathematically, the radial component of splay ($r^{-1}\mathbf{d}\cdot {\mathbf{e}}_r$, the parallel curvature) diverges because r remains small and $\mathbf{d}\cdot {\mathbf{e}}_r$ grows when the distal monolayers merge into a pore. (This is in contrast to stalk formation, where the headgroups of the central proximal lipids separate, so that r grows proportionally to $\mathbf{d}\cdot {\mathbf{e}}_r$.) In addition, the energy to form a fissure in the distal monolayer is also considerable, requiring ∼50 kT, further reducing the likelihood that a stalk will directly convert to a fusion pore.

Hemifusion diaphragm formation and expansion

We now analyze the alternate scenario: the distal monolayers move into the central region of the stalk until they abut and create a hemifusion diaphragm (HD). The HD joins the two distal monolayers in a Y-shape (Fig. 1 D), and is made up of two disklike monolayer surfaces with radius $r_{\text{HD}}$ that are not flat. In the following, “distal” refers to the portion of the distal monolayer that is outside the diaphragm.

After the initial expansion (up to ∼$r_{\text{HD}}=1.5\text{nm}$), the diaphragm directors are more or less vertical at the center and splay outwards, $\text{div}\mathbf{d}>0$ (Fig. 3 E, i). Outside the HD, the distal monolayer is positively curved and its directors splay inwards, $\text{div}\mathbf{d}<0$ and meridian curvature $k_1>0$ (Fig. 3 E, ii). The outward splay of the diaphragm and inward splay of the distal monolayer are in conflict and results in splay as the dominating energy opposing HD expansion. A positive spontaneous curvature in the distal monolayer would match the meridian curvature ($k_1>0$), promoting HD expansion, but this spontaneous curvature conflicts with the direction of splay in the diaphragm (Fig. 3 E). Conversely, negative spontaneous curvature in the distal monolayer conflicts with geometric curvature, again opposing HD expansion. To numerically illustrate, let $k_{0,\text{P}}=-0.20{\text{nm}}^{-1}$ and $k_{0,\text{D}}=+0.01{\text{nm}}^{-1}$, accommodating the geometric curvature of the proximal and distal monolayers. But, for $k_{0,\text{HD}}=k_{0,\text{D}}$, both $\text{div}\mathbf{d}$ and $k_0$ are both positive in the HD and their addition in Eq. 1 leads to a large splay energy in the diaphragm (Fig. 3 D, solid gray curve), preventing expansion. The effective line tension is significant, roughly 7 kT nm⁻¹ (Fig. 3 A, thick gray curve).

We calculated the energy required for HD expansion for other combinations of spontaneous curvature (Fig. 3, legend). In all cases, the total energy E was an increasing function of $r_{\text{HD}}$, when $k_{0,\text{D}}=k_{0,\text{HD}}$. We conclude that HD expansion for bilayers with identical lipid composition in the diaphragm and distal monolayers requires the injection of energy.

These energetic considerations suggest that, experimentally, hemifusion expansion is made possible by the presence of a second lipid component with negative spontaneous curvature in the distal monolayer. To appreciate this, let the distal monolayer be a mixture of two or more lipid types, maintaining positive net spontaneous curvature. (Expansion is inhibited if net curvature is negative.) During hemifusion expansion, lipids with negative spontaneous curvature will preferentially occupy the HD rim; the distal monolayer has an infinite reservoir of lipid so its composition is not altered. As a result, $k_{0,\text{P},\text{HD}}<0$ and $k_{0,\text{D}}>0$. The preferential partitioning of the negative curvature lipid in the HD is entropically disfavored, with a free energy penalty:

$$\begin{array}{c}F=\underset{{\Sigma }_{\text{HD}}}{\int }e\left(\varphi ,{\varphi }_0\right)dS,\\ e\left(\varphi ,{\varphi }_0\right)=\frac{\text{kT}}{A_0}\left(\varphi \text{ln}\frac{\varphi }{{\varphi }_0}+\left(1-\varphi \right)\text{ln}\frac{1-\varphi }{1-{\varphi }_0}\right),\end{array}$$

where ϕ is the mole fraction, ${\varphi }_0$ is the average mole fraction of the negative lipids (44, 45), and $A_0=60{\AA }^2$ is the area per headgroup (28).

To obtain a value for F, we note that only a few negative curvature lipids per circumference are required to off-set the positive HD rim splay. Splay decays with a length $\delta =\sqrt{K_{\text{C}}/K_{\theta }}=1\text{nm}$ (46) (Fig. 3 E). In the center of the HD, the value of $k_{0,\text{HD}}$ is inconsequential because the bilayer is essentially planar. In the HD rim, we assume that a larger fraction ${\varphi }_{\mathbf{r}}$ of the lipids have negative spontaneous curvature; we therefore set $\varphi ={\varphi }_0$ in the range $0<r<r_{\text{HD}}-\delta$ and $\varphi ={\varphi }_{\mathbf{r}}>{\varphi }_0$ in the range $r_{\text{HD}}-\delta <r<r_{\text{HD}}$. These considerations yield:

$$F\sim \gamma r_{\text{HD}},\gamma =4\pi \delta e\left({\varphi }_{\mathbf{r}},{\varphi }_0\right),$$ (5)

where γ is the contribution of the entropic energy of demixing to line tension.

To evaluate the height of the energy barrier against HD enlargement, consider the values $k_{0,\text{P}}=-0.2{\text{nm}}^{-1}$, the slightly positive $k_{0,\text{D}}=+0.01{\text{nm}}^{-1}$ in the distal monolayer (DOPE/DOPS 1:5, ${\varphi }_0=0.2$), and the HD rim with a negative value $k_{0,\text{HD}}=-0.13{\text{nm}}^{-1}$ (DOPE/DOPS 1:1, ${\varphi }_{\text{r}}=0.5$, $\gamma =4.0\text{kT}{\text{nm}}^{-1}$). In this case, the energy E + F is a decreasing function for $r_{\text{HD}}>1.5$ nm (Fig. 3 A, thin black curve), in contrast to a single-component bilayer where the total energy E was increasing (Fig. 3 A, thick gray curve). Hence a diaphragm will spontaneously expand once it overcomes the initial energy barrier of roughly 30 kT relative to the stalk. Decreasing the value of proximal spontaneous curvature to $k_{0,P}=-0.25{\text{nm}}^{-1}$, further reduces the energy barrier of hemifusion to 20 kT and steepens the slope of the decrease in energy as a function of $r_{\text{HD}}$ (Fig. S3 C, dashed gray curve). See Movie S3.

Pore formation and expansion in the hemifusion diaphragm

We have provided the conditions sufficient for a circular HD to spontaneously expand, once an initial barrier against stalk widening is overcome. Under general conditions, the external force that proteins or tension must supply for expansion can be estimated from the curves displaying energy versus $r_{\text{HD}}$ (Fig. 3 A, for example). At any time during HD expansion, a pore may form and there is an HD radius for which pore nucleation and expansion is more favorable than for all other radii. To obtain this optimal radius, the least energy path must take into account the location of pore nucleation and the energy required for subsequent dilation of the HD.

To leading order, a pore in a large HD (tens to hundreds of nanometers in diameter) behaves as if it were a hole in a two-dimensional sheet. For a given area, the hole minimizes its circumference by making contact with the edge of the sheet, i.e., forming a rim-pore. A rim-pore obviously violates axisymmetry and cannot be calculated by our present formalism. But, one-dimensional rim-pore profiles have been calculated variationally (47, 48). These studies show that the barrier against pore expansion grows proportionally to $r_{\text{HD}},$ implying that complete fusion becomes energetically more costly as the HD grows (48). It is thus unlikely that biological fusion occurs through this route.

When the diameter of the HD is on the same order as the thickness of the bilayer, the two-dimensional approximation of rim-pore formation is invalid and the bilayer must be treated three-dimensionally. For $r_{\text{HD}}$ comparable to the splay decay length δ = 1 nm, the lipid orientations at the diaphragm rim correlate with those of the pore (Fig. S3 A). A rim-pore is no longer advantageous because the bending energy expended in lining the pore is comparable to the energy gained in the correlations with the HD rim. Thus, the optimal pore forms in the center of the HD as $r_{\text{HD}}$ approaches zero.

Interestingly, a pore in the center of a small HD can fully expand without external forces or interfacial tensions because the correlation in lipid orientations favors pore expansion (Fig. S3 C, i and iii). When $r_{\text{HD}}<5.0\text{nm}$, elastic energy is lowered by the movement of lipids into the distal monolayers (i.e., outside the HD), independent of lipid composition. Conversely, when $r_{\text{HD}}>5.0\text{nm}$, neither the rim-pore nor a central pore (Fig. S3 B) will enlarge in the absence of external forces (48) (Fig. S3 C, ii and iv). Our calculations show that the energy of pore formation in HDs of moderate radii is 35–40 kT (Figs. S3 C, i–iv, and 4), slightly less than the energy of pore formation ∼45 kT in a PB (see Fig. 6 A and Glaser et al. (40)). These pores will also not close, because resealing requires reversibly surmounting the energy barrier of pore formation. See Movie S4.

Figure 4.

The least energy profile for the entire, axisymmetric fusion process for $k_{0,\text{P}}=-0.20{\text{nm}}^{-1}$, $k_{0,\text{D}}=+0.01{\text{nm}}^{-1}$, and $k_{0,\text{HD}}=-0.13{\text{nm}}^{-1}$. The activation energy between intermediates is indicated in the parentheses.

Some studies have suggested that a hemifusion diaphragm is a dead-end state in SNARE-mediated fusion (49). The above considerations give a plausible explanation for why this might be the case. A pore that forms while the expanding HD is still sufficiently small will spontaneously enlarge. But based on the stability calculations of Risselada et al. (48), once the HD radius has surpassed this window, complete fusion becomes increasingly unlikely and a dead-end state is reached. As a result, there is only a small of window of HD sizes that allows for pore formation and expansion; this is experimentally observed as complete fusion.

Discussion

Continuum theory describes well the steps of membrane fusion

It may be questioned whether continuum theory accurately describes steps in the fusion process, because the process may depend on the details of molecular interactions. We suggest that continuum theory is, a priori, reliable for describing monolayer and bilayer deformations because over any reasonable timescale, a large number of molecular collisions and interactions average out granularities; the relevant averages are described well by the experimentally determined elastic constants used in a Helfrich Hamiltonian (28, 29, 33, 50, 51). Explicit minimum energy shapes of fusion pores determined by coarse-grain (Martini) molecular simulations and by continuum theory are in excellent agreement (25, 26). Also, the experimentally and theoretically derived stalk shapes are in agreement, suggesting that continuum theory accurately describes the static intermediates of fusion. It remains to be seen whether continuum theory captures the physics of transition states, or whether it describes nanoscopic processes where only a few lipids are involved. When continuum theory is used in these cases, the Helfrich equations are an ansatz of the model.

Assumptions of this study

The assumption of axisymmetry significantly reduces computational overhead and does provide global minimizers for stalk and HD shapes because both experimental data and MD simulation show that these structures are essentially axisymmetric. Nonaxisymmetric deformations may yield lower energy, and the minimized pathway of maximal free energy of rim-pore formation is probably lower than our axisymmetric calculations. But our calculations explicitly reveal the effects of nonlocal elastic interactions, which occur independently of bilayer symmetry. Continuum finite element studies using a surface-director Hamiltonian similar to Eq. 1 have simulated three-dimensional vesicles (52, 53), but these studies considered neither changes in topology nor separate monolayers of a bilayer.

In our study of compositional fluctuations, spontaneous curvature was taken as a piecewise constant function in each monolayer. These calculations certainly demonstrate the principle that local compositional variations are required for spontaneous HD expansion, but the treatment of spontaneous curvatures can be improved. We suggest that in future studies, the $k_0$ in the Helfrich Hamiltonian Eq. 1 be defined self-consistently as a spatially varying field variable (54), rather than as a physical constant set to match experimental conditions. We also assumed that the elastic moduli $K_{\text{C}}$, $K_{\theta }$, and $K_{\text{A}}$ are insensitive to a break in the continuity of a monolayer surface. It may be, however, that a hydrophobic fissure or an inclusion in a protein locally alters the quadratic dependence on the director gradient; further experimental and theoretical efforts are required to resolve this issue.

Subject to these assumptions, we used the continuum formulation to not only characterize static stalk shape and formation, but also to evaluate the entire pathway of membrane fusion, and to obtain lipid orientations that occur throughout the transition from separate to merged membranes, information not presently possible to obtain experimentally. These calculations lead to three major, and previously unappreciated, conclusions: (1) It is highly unlikely that a fusion pore forms directly from a stalk; the calculated energetics show that a hemifusion intermediate is required. (2) Hemifusion requires a small amount of energy roughly the same for all lipid compositions. Subsequent expansion is spontaneous if the intrinsic curvatures of the diaphragm monolayers are sufficiently negative. At least two lipid components are required for spontaneous expansion. Single-component hemifusion bilayers do not expand without a continual injection of energy. (3) Pores that form in the center of an extended diaphragm will expand provided the diaphragm radius is comparable to the bilayer thickness.

Some prior studies, both experimental and theoretical, have suggested that a fusion pore might form directly from a stalk (3, 15, 55), whereas others have argued that a hemifusion diaphragm is required (2, 56, 57). But a firm conclusion was not previously possible, because for neither case was the energy barrier against pore formation explicitly measured or calculated.

MD and continuum theory yield similar values for energy barriers against transitions

Recent studies have used coarse-grained MD and self-consistent field theory to model individual steps of fusion and their energy barriers (13, 16). This allows us to compare transition energies calculated by independent methods that are based on conceptually different approaches. A coarse-grained MD simulation estimates a 17 kT barrier against stalk expansion between small vesicles (∼20 nm in diameter) composed of POPE ($k_0=-0.30{\text{nm}}^{-1}$) (14). In Fig. 3 A, we calculated a 35 kT barrier against stalk expansion between PBs for a single component HD ($k_0=-0.20{\text{nm}}^{-1}$). In both the continuum and MD calculations, energy increases monotonically with diaphragm radius when only a single component is involved. For hemifusion between PBs, the distal monolayers must distort to meet the far-field boundary condition, requiring energy. For hemifusion between vesicles, however, the local curvature of the distal monolayers matches the curvature of the sphere and hence the distortions required in the case of PBs are absent. The continuum and MD simulation results agree quantitatively once this energy of distortion is accounted for (i.e., subtracting the barrier in Fig. 3 C from the barrier in Fig. 3 A).

A coarse-grained MD simulation for PBs (e.g., the same reference state as used in this article), has calculated a 40 kT barrier against stalk formation and a 110 kT barrier against direct transition to a fusion pore, for DOPC/DOPE 1:1 ($k_0=-0.19{\text{nm}}^{-1}$) (15). Continuum theory predicts nearly identical barriers, 30 kT against stalks and 120 kT against stalks transitioning to a fusion pore without passing through hemifusion, for $k_0=-0.17{\text{nm}}^{-1}$ (Fig. 2 A and the Results). The energy we find using the string method to form a pore in a single bilayer (Fig. 6 A) agrees with the energy calculated according to nucleation theory (37). This value is, however, significantly less than the 90 kT barrier calculated by self-consistent field theory and the string method (23).

Comparison between theory and experiment

To make a side-by-side comparison against stalk shapes obtained experimentally, we match our single stalks to the repeated and periodic stalks determined from electron density contours of experimental 8-nm-wide crystallographic hexagonal cells. Experimental stalk shapes are nearly axisymmetric within a 6 nm diameter about the stalk axis, and are vertically periodic with a period of 5 nm (Fig. 4A of Aeffner et al. (11)). We model this by requiring that the bilayer meet the cylinder side walls perpendicularly, accomplished by imposing a free surface boundary condition, with unrestricted separation at r = 5 nm. To account for vertical periodicity, we add to the energy of Eq. 1 the repulsive hydration interaction between apposing distal monolayers in experimental stacks of bilayers. The revised energy for periodic confinement is

$$\hat{E}=E+\underset{{\Sigma }_{\text{D}}}{\int }W\left(h_{\text{D}}\right){\text{cos}}^2\left(\theta \right)dS.$$ (6)

The length $h_{\text{D}}$ is the vertical distance of separation between the distal monolayers of the stalk and the lamellar bilayer phases lying immediately above and below the stalk (Fig. 5, main panel). For a symmetric bilayer composed of DOPC/DOPE 1:1 with $k_0=-0.19{\text{nm}}^{-1}$, minimizing $\hat{E}$ yields a continuum stalk shape (Fig. 5, solid black curve) that is indistinguishable, to within 3 Å, from the experimental neutral surface contour (Fig. 5, thick gray curve). Experimentally, stalk shape is independent of lipid composition, a phenomenon accounted for by our theory; the solid curves in the main panel and lower-left inset of Fig. 5, if superimposed, virtually overlap.

Generalizing the Helfrich Hamiltonian to predict topological changes

The continuity of a monolayer surface must be transiently interrupted if a topological change is to occur. We modeled this interruption by introducing a hydrophobic fissure surface ${\Sigma }_{\text{F}},$ a free boundary in the bilayer initiating both stalk merger and pore formation. We enhanced the modeling capabilities of Helfrich’s theory (27) by adding to it the hydrophobic energy of fissures, Eq. 4, resulting from exposure of acyl chains to water (3). It is of interest that decreasing the value of interfacial tension σ by a few kT nm⁻² results in a reversal of a stalk back to separate membranes, as can occur experimentally (58). Perhaps a coarse-grained or all-atom simulation could provide a more refined functional form for the hydrophobic fissure energy.

In stalk and pore formation, the outer edges of the fissure surfaces are nearly in contact at the maximum height of the activation barrier (Figs. 1 B, 6 D, and 8, first and third rows). The gap between these edges is spanned by two polar headgroups, indicating that the space between the fissures is shielded from bulk water. If the space is filled by vacuum, then here the appropriate value for surface tension σ between the acyl chains and vacuum would be ∼$5.5\text{kT}{\text{nm}}^{-2}$ (22.5 erg cm⁻² (59)). In Fig. 6, A–C, we illustrate how the energy of pore formation varies with the hydrophobic parameters σ and ρ. The splay and stretching deformations, but not tilt, adjust to the lowered surface tension σ and the increased hydrophobic length scale ρ. The simultaneous decrease in splay and stretch is achieved by a slight radially expansion and a lengthening of the fissure surface (Fig. 6 D). The curvature of the neutral surfaces, however, is preserved.

There is a moderate degree of compositional variation and experimental uncertainty in the value of the other parameters used by the Hamiltonian, Eq. 1 (11, 60, 61). Variation in the hydration parameters $P_0$ and L is significant for repulsion between PBs. But in the stalk and hemifusion structures of this study, the proximal monolayers are vertical at the stalk/hemifusion neck. At small distances away from the neck, they are separated by more than the decay length L, and reach a maximum separation that exceeds the equilibrium bilayer separation $h_0$ (Fig. 3 E). This tendency to overshoot equilibrium separation is a result of minimizing splay, and is independent of repulsion (26). Consequently, repulsion accounts for only a fraction of the energy that opposes stalk formation (Fig. 2 B), and is overcome by the steeply decreasing, negative proximal splay in HD/FP expansion (Fig. 3 B). The contribution of tilt energy was also uniformly small throughout all the calculations (Figs. 2, B and C, 3, B–D, and 6, A–C). For example, the maximum angle α formed between $\mathbf{d}$ and $\mathbf{n}$ was 24° in stalk formation, 17° in HD formation and expansion, and 20° in pore formation. Accordingly, a 10% uncertainty in the tilt modulus $K_{\mathbf{\theta }},$ for example, results in <1 kT variability in transition energies, which is small compared to the characteristic energies of fusion. Splay was the dominant deformation among all the elastic interactions. The total elastic energy was comparable to fissure energy, the dominant long-range interaction, in stalk and pore formation (Figs. 2 B and 6). Based on the divisions of energy in Figs. 2, 3, and 6, we interpret the sensitivity of transition energies to splay modulus $K_{\text{C}}$ as follows: Increasing $K_{\text{C}}$ results in a decrease in the energy of stalk formation and hemifusion expansion (even when assuming proportionality with $K_{\theta }$ and $K_{\text{A}}$), due to the increasingly negative proximal splay energy (Figs. 2 B and 3 B). Conversely, in pore formation, pore radius is small, unavoidably making the radial component of splay ($r^{-1}\mathbf{d}\cdot {\mathbf{e}}_r$, parallel curvature) large. Consequently, increases in $K_{\text{C}}$ raise the energy of pore formation.

In addition to the splay and tilt deformations, the energetics of fusion are influenced by the saddle-splay energy

$$\underset{\Sigma }{\int }{K}_{\text{G}}\text{det}\nabla \mathbf{d}dS,$$ (7)

where $K_{\text{G}}$ is the saddle-splay modulus and $\nabla \mathbf{d}$ is the 2 × 2 matrix of directional derivatives of $\mathbf{d}$ along Σ. In cylindrical coordinates, $\text{det}\nabla \mathbf{d}=\left(\left(\partial \mathbf{d}/\partial s\right)\cdot \tau \right)\left(r^{-1}\mathbf{d}\cdot {\mathbf{e}}_r\right)$, evaluated numerically by Eq. S2. When tilt is zero, $\text{det}\nabla \mathbf{d}=k_1{k}_2$ and Eq. 7 simplifies to the Gauss curvature energy previously assumed for the standard theoretical descriptions of saddle-splay (62).

The addition of saddle-splay Eq. 7 to the Hamiltonian Eq. 1 does not alter the trajectory of the least energy paths we have calculated because saddle-splay encodes changes in genus and is insensitive to changes in bilayer shape when the genus is fixed. Fig. 7 illustrates the explicit calculation of Eq. 7 along the entire least energy path. The two transitions, corresponding to the topological change in going from a planar bilayer to a stalk, and another in transitioning from a stalk to a fusion pore, are nearly multiples 4π. It may be surprising that Eq. 7 replicates the conclusion of the Gauss-Bonnett theorem (63), because tilt is not everywhere zero, and hence $\text{det}\nabla \mathbf{d}$ is not everywhere equal to the Gaussian curvature. It is likely that this occurs because tilt is negligible whenever Eq. 7 is constant.

Figure 7.

The graphs show the (unitless) integrals of Eq. 7 of saddle splay for the proximal and distal surfaces for the entire fusion process; ${\Sigma }_{\text{D}}$ denotes the union of the HD and distal monolayers outside the HD. The saddle-splay transitions are given in parentheses. In stalk formation, saddle splay decreases by 12.5 ∼ 4π with the increase in genus of the proximal monolayer (solid curve). The stalk and HD are topologically equivalent, and so saddle splay is constant during HD expansion (solid and dotted curve). In pore formation, saddle splay further decreases by 12.5 with the increase in genus of the HD/distal monolayer (dashed curve). Both curves are constant when the fissure surface is closed (the monolayers are continuous), and smoothly transition with the widening (monolayer continuity is interrupted) and closing of the fissure surface.

Although the transition pathways are the same with the addition of the saddle-splay energy equation (Eq. 7), stalk and FP energies would be much higher than that estimated solely on the basis of monolayer splay, tilt, and stretching energies. The increase in genus that occurs, in the transition from a PB to a stalk phase, for example, results in a decrease in total Gaussian curvature. Consequently, a negative value for $K_{\text{G}}$ means the energy for these transitions is greater than we calculate. To illustrate, each transition in Fig. 7 requires a stepwise increase in energy of ∼100 kT, assuming the modulus $K_{\text{G}}=-8\text{kT}$ (62, 64). Although the saddle-splay modulus cannot be directly measured, a negative value for $K_{\text{G}}$ is predicted from the stability and temperature dependence of the lamellar, inverted cubic, rhombohedral, and inverted hexagonal lipid phases (65). But a value of $K_{\text{G}}$ that is on the order of −8 kT renders, in any continuum description, the other elastic deformations and interactions irrelevant to fusion, and this runs counter to the repeated experimental demonstrations that fusion depends on spontaneous curvatures and elastic moduli (1). What might be the cause of this paradox?

Self-consistent field theory calculations, which are independent of the value of the elastic moduli, predict more moderate changes in free energy for stalk and pore formation (16, 18, 23). This leads us to suspect that some of the theoretical assumptions used to derive $K_{\text{G}}$ need to be refined. For example, in the absence of a practical numerical method to find true minimizers, the analytical calculations in Siegel (62) and Siegel and Kozlov (65) assumed the midplane of inverted cubic and rhombohedral phases have zero total curvature, and truncated energy to express $K_{\text{G}}$ in closed form. But the Helfrich energy obtained by assuming shapes is often drastically different from that of true minimizers (24, 26). This occurs because energy must be injected into the system to maintain the assumed shape, rather than allowing it to adjust to minimum energy. The injected energy can be consequential, and in the case of $K_{\text{G}},$ its magnitude may be significantly altered by, a priori, assuming shape. We suggest that if the bilayer geometries were not artificially fixed in calculating of $K_{\text{G}},$ the paradox would disappear.

X-ray diffraction images of stalks (8, 9, 10, 11) show that bilayer thickness varies within a stalk. This freedom in thickness is accounted for by the stretching term ${\left(a-a_0\right)}^2/a{a}_0={\left(l-l_0\right)}^2/l{l}_0$ in our energy formulation equation, Eq. 1. The tilt deformation $\mathbf{d}\times \mathbf{n}$ was introduced by Helfrich (27) (where it is also called “twist”) and gives the freedom in lipid orientation with respect to the normal of the neutral surface. An alternate definition of tilt has been introduced and systematically investigated (31). Its elastic energies are:

$$\underset{\Sigma }{\int }\frac{K_{\text{C}}}{2}\left[{\left(\text{div}\mathbf{d}+k_0\right)}^2-k_0^2\right]+\frac{K_{\theta }}{2}|\frac{\mathbf{d}}{\mathbf{d}\cdot \mathbf{n}}-\mathbf{n}{|}^2+\frac{K_{\text{A}}}{2}\frac{{\left(\mu -l_0\right)}^2}{\mu l_0}dS.$$ (8)

The stretching term ${\left(\mu -l_0\right)}^2/\mu l_0$, which we have added, is independent of tilt and allows for the observed variability in monolayer thickness μ. However, the formulation used by Kuzmin et al. (3), Markin and Albanesi (4), Kozlovsky and Kozlov (6), Kozlovsky et al. (7), and Hamm and Kozlov (31) assumes that bilayer thickness is constant when lipids tilt; and this tacitly introduces the side condition μ = l₀. Minimal stalk energies using Eq. 6 (based on Eq. 1) or using Eq. 8 for the elastic terms are comparable, due to the fact that the elastic terms in Eqs. 1 and 8 agree to second order for moderate tilt deformations: $|\mathbf{d}\times \mathbf{n}{|}^2-|\left(\mathbf{d}/\mathbf{d}\cdot \mathbf{n}\right)-\mathbf{n}{|}^2={\text{sin}}^2\alpha -{\text{tan}}^2\alpha =O\left({\alpha }^2\right)$ and $l-\mu =l\left(1-\text{cos}\alpha \right)=O\left({\alpha }^2\right)$. For example, the largest calculated angle 24° (α = 0.42 radians) yields an absolute difference of ∼1 kT for the two tilt formulations. Although the minimal energies are similar, Eq. 8 yields a somewhat pointed cusp-shape distal monolayer (Fig. 5, dashed curve), a feature not shared by the experimental data (Fig. 5, main panel and inset, thick gray curves). In other words, the definition of tilt used in liquid crystal theory (53, 66, 67), and introduced into continuum membrane mechanics by Helfrich (27), leads to results that are in agreement with the experimentally observed stalk shape.

Summary of steps including and subsequent to stalk formation

See Movie S5. We have identified and analyzed energy barriers between the steps of fusion under various experimental conditions. To illustrate the process for one lipid composition, Fig. 4 depicts the entire least energy profile for $k_{0,\text{P}}=-0.20{\text{nm}}^{-1}$, $k_{0,\text{D}}=+0.01{\text{nm}}^{-1}$, and $k_{0,\text{HD}}=-0.13{\text{nm}}^{-1}$. All energy barriers are moderate. Fissure expansion is energetically the most expensive deformation in stalk formation (Fig. 8, top row). Formation of a hemifusion diaphragm is accompanied by an increase in distal splay energy because the Y-shaped rim enforces a nonnormal boundary condition for the directors in the diaphragm and distal monolayer (but not in the proximal monolayer). Consequently, distal splay and tilt energy increases with initial diaphragm widening independently of lipid composition (Fig. 8, middle row). Pore formation in a hemifusion diaphragm is similar to pore formation in PBs, with most of the energy accounted for by hydrophobic interactions (Fig. 8, bottom row). The main difference is that a pore in a diaphragm of moderate radius will spontaneously enlarge because the lipid orientations at the diaphragm rim correlate with lipid orientations of the pore.

Author Contributions

R.J.R. and F.S.C. wrote the article and performed the research; R.J.R., T.S.K., and L.Y. contributed analytic tools and analyzed data.

Editor: Klaus Gawrisch.

Supporting Citations

Reference (68) appears in the Supporting Material.

Supporting Material

Movie S1. Comparing Assumed and Least Energy Paths

We assume an initial path for stalk formation (left) and apply the string method to yield the least energy path (right). The initial path is generated from a sequence of curves that culminate in a toroidal stalk. Directors are assumed to be perpendicular to the surface representing the proximal monolayer. Directors are oriented vertically throughout the distal monolayer, and lipid length is maintained constant within the monolayers. In contrast, in calculating the least energy path, all deformations are allowed and orientations of directors are outputs of the calculation; the distal lipids now exhibit stretching, and the curves for each monolayer are mathematically smooth. The left and right frames have been synchronized so that proximal monolayers meet at the same time in the movie. For the initial path, the hydrophobic fissure is assumed to form only after the headgroups of opposing proximal monolayers meet. For the least energy path, the calculations show that the fissure forms and its surface widens before the opposing proximal monolayers meet. The fissure energies are comparable in the two cases, but the elastic energies and repulsive interactions are much less for the least energy path.

Movie S2. Stalk Formation in Planar Bilayers

The least energy pathway shown in Movie S1 is replicated. The majority of the energy opposing stalk formation at the height of the barrier is contributed by the exposure of acyl chains to water—the hydrophobic fissure surface. The fissure surface is widest at the transition state (highlighted in red). The geometric curvature of the proximal monolayer (yellow spheres) becomes progressively more negative throughout stalk formation; the distal monolayers (magenta spheres) remain almost flat. Consequently, stalk formation is insensitive to the composition of the distal monolayer.

Movie S3. Hemifusion Diaphragm Expansion

Hemifusion occurs when distal monolayers abut. Abutment requires that the distal monolayers bend into a negative geometric curvature (positive splay for our sign convention for d; see text following Eq. 1). The change in curvature is initiated before abutment (red arrows mark the initiation of the change in splay). The splay of the distal monolayer is energetically the dominant deformation in the creation of the hemifusion diaphragm (e.g., see Fig 3, C and D). Consequently, lipids move and deform in ways to minimize splay energy. Distal lipids with negative spontaneous curvature (small magenta spheres) are required near the diaphragm rim to match spontaneous and geometric curvature. These lipids therefore preferentially occupy the rim region: the entropic penalty of lipid demixing is small compared to the reduction in elastic energy. This is illustrated in the movie by the greater concentration of lipids with small headgroups in the diaphragm compared to the region outside the diaphragm. Diaphragm expansion requires that lipids in the distal monolayers with negative spontaneous curvature partition into the enlarging rim of the diaphragm. But lipids with positive spontaneous curvature (large magenta spheres) are required to accommodate the geometric curvature outside the diaphragm. Thus, the distal monolayers must contain at least two different types of lipids for a diaphragm to expand without the need for external forces.

Movie S4. Fusion Pore Formation from a Hemifusion Diaphragm

In a diaphragm of small radius, the lipids lining a central pore and the diaphragm rim are near each other, and their orientations are correlated. The lipids in the rim and pore are oriented in the same general direction and thus pore formation requires only moderate energy. This common orientation allows pores in the center of the diaphragm to enlarge spontaneously. For a large diaphragm, there are considerable numbers of lipids intervening between the pore and rim; an energetically favorable correlation between orientations is absent. Both pore formation and enlargement (Fig. S3) is less favored than for a small diaphragm. Creation of a fissure that exposes acyl chains to water (marked in red) dominates the height of the energy barrier. Pore formation is, consequently, relatively insensitive to lipid spontaneous curvature. Because the fissure has sealed once a pore forms in a diaphragm, its energy depends solely on elastic terms: it decreases for increases in lipid spontaneous curvature.

Movie S5. The Overall Process of Membrane Fusion

(Solid lines) Neutral surfaces of monolayers of two parallel bilayer membranes; (dotted lines) midplanes. Work is required to draw the membranes toward each other. An energy barrier must be overcome for the two proximal monolayers to merge into a stalk. Most of the expended energy is regained as the stalk forms. Another energy barrier must be surmounted for distal layers to abut each other in creating a hemifusion diaphragm. Diaphragm expansion can release or consume energy, depending on spontaneous curvatures. An energy barrier always opposes pore formation within the diaphragm. But for a small diaphragm, pore enlargement is energetically favorable. After sufficient enlargement, the lipids of the diaphragm have completely incorporated into the external bilayer and the fusion pore has become three-dimensional. Its expansion is spontaneous for sufficiently negative spontaneous curvature of the proximal monolayer. Throughout the fusion process, van der Waals attractions keep the two original bilayers close and parallel to each other at large lateral distances from the fusion site. Near the site, elastic energy is more significant than van der Waals interactions, causing the fusing membrane to bulge. Starting with the stalk, the membrane(s) consequently exhibit the shape of a teardrop that becomes progressively more pronounced as the fusion process continues.