Simulated dynamic cholesterol redistribution favors membrane fusion pore constriction

Abstract

Endo- and exocytosis proceed through a highly strained membrane fusion pore topology regardless of the aiding protein machinery. The membrane’s lipid components bias fusion pores toward expansion or closure, modifying the necessary work done by proteins. Cholesterol, a key component of plasma membranes, promotes both inverted lipid phases with concave leaflets (i.e., negative total curvature, which thins the leaflet) and flat bilayer phases with thick, ordered hydrophobic interiors. We demonstrate by theory and simulation that both leaflets of nascent catenoidal fusion pores have negative total curvature. Furthermore, the hydrophobic core of bilayers with strong negative Gaussian curvature is thinned. Therefore, it is an open question whether cholesterol will be enriched in these regions because of the negative total curvature or depleted because of the membrane thinning. Here, we compare all-atom molecular dynamics simulations (built using a procedure to create specific fusion pore geometries) and theory to understand the underlying reasons for lipid redistribution on fusion pores. Our all-atom molecular dynamics simulations resolve this question by showing that cholesterol is strongly excluded from the thinned neck of fusion and fission pores, revealing that thickness (and/or lipid order) influences cholesterol distributions more than curvature. The results imply that cholesterol exclusion can drive fusion pore closure by creating a small, cholesterol-depleted zone in the neck. This model agrees with literature evidence that membrane reshaping is connected to cholesterol-dependent lateral phase separation.

Significance

Using all-atom molecular dynamics simulations of complex membrane fusion pores, we demonstrate that cholesterol is depleted in the pore neck because of the membrane thinning inherent to such pores. This observation directly addresses a contradiction of cholesterol: that it thickens bilayers yet favors the curvature of a leaflet that is thinned. The ramifications for membrane elasticity directly impact the mechanism for how sterols change the membrane forces that resist reshaping.

Introduction

Membranes in both endo- and exocytosis proceed through a highly strained fusion pore (1,2,3) stage, regardless of the protein machinery that assists the specific process. These pore structures arise, for example, during vesicle fusion to release neurotransmitters (4,5) and preceding the fission step of clathrin-mediated endocytosis (6,7). It is predicted that metastable intermediate formation along the fusion/fission pathway has energetic barriers requiring tens to hundreds of $k_{\mathrm{B}}T$ to surpass (8,9,10,11). Therefore, nature uses a diverse set of proteins, performing specific tasks, to help overcome these large barriers during endo-/exocytosis processes: clathrin and adaptor proteins for forming buds with fusion pore necks (7); dynamin to complete fission of the budded vesicle (12); endosomal sorting complexes required for transport-III (ESCRT-III) for diverse outward membrane budding and fission (13); COPII for vesicular budding from the endoplasmic reticulum, allowing transport to the Golgi apparatus (14); and viral proteins that initiate enveloped virus fusion with the cell, such as SARS-CoV-2 spike proteins (15). Major questions persist regarding how protein and lipid components stabilize strained intermediates. Why does cholesterol, which stiffens membranes, support endocytosis (16)? What is the biophysical basis of sterol perturbations on viral entry and budding (17)?

Energetic models, supported by experimental validation, predict the pathway for pore formation and closure (2,18,19). The standard model for quantifying membrane shape energy is the Helfrich/Canham (HC) Hamiltonian (20,21), a spring-equivalent model of the deformation, which is quantified by the total curvature $\left(J=c_1+c_2\right)$ and Gaussian curvature $\left(K=c_1{c}_2\right)$. Here, $c_1$ and $c_2$ are the orthogonal principal curvatures of the surface, describing parabolic deviations away from the plane. Lipid chemical identity favors specific curvatures, quantified in the spontaneous curvature $\left(J_0\right)$ (22,23,24). The bending modulus $\left(\kappa \right)$ determines the stiffness of the membrane, setting the overall energy scale (25,26,27). The Gaussian curvature modulus $\left({\kappa }_{\mathrm{G}}\right)$ determines the coupling between principal curvature directions, differentiating, for example, spheres from tubes that might otherwise have the same $J$ (8,21,28,29,30). The model energy for a bilayer integrated at the midplane surface $\left(S\right)$ is

$$F_{\text{HC}}={\int }_S\mathrm{d}A\frac{1}{2}\kappa {\left(J_{\text{mid}}-J_0\right)}^2+{\kappa }_{\mathrm{G}}{K}_{\text{mid}}\text{.}$$ (1)

As an example, a small cylinder (e.g., a nascent fusion pore neck) with a height of 20 Å, a radius of 20 Å ($J={\left(20\mathrm{\AA }\right)}^{-1},K=0$), and a bending modulus of 20 kcal/mol, yields an $F_{\text{HC}}\approx$ 63 kcal/mol, which is attainable by dynamin and ESCRT-III. Moving from the cylindrical neck and into the pore’s hourglass shape, pore structure energetics are strongly determined by ${\kappa }_{\mathrm{G}}$. It is hypothesized that particular lipid mixtures can bring ${\kappa }_{\mathrm{G}}$ to zero (8,31), making the lipids potent “fusogens.” Thus, enrichment/depletion of lipids that influence $J_0$ and ${\kappa }_{\mathrm{G}}$ (e.g., possible fusogens), for example, can dramatically reduce endo-/exocytosis barriers.

This work focuses on cholesterol and the molecule’s unexpected redistribution given what we know about $F_{\text{HC}}$. A body of literature has demonstrated that cholesterol chemistry and concentration dramatically affect virus-endosome fusion (17,32) as well as plasma membrane fusion and fission (16,33). Understanding cholesterol’s role in mechanics is complicated by disparate observations. To discuss these observations and complications, we consider individual leaflet properties instead of solely bilayer properties. We continue to frame the discussion in terms of leaflets throughout the paper. Cholesterol induces negative leaflet $J_0$ in the fluid phase (34); however, cholesterol preferentially partitions into ordered, stiff regions in leaflets (35). As we show by geometry in materials and methods, both leaflets of an ideal catenoidal fusion pore have negative $J$. Therefore, should we anticipate that cholesterol is enriched or depleted in the fusion pore? It is possible that with its dual roles as ordering agent and negative $J_0$ lipid, cholesterol may serve different purposes at different stages of reshaping. Indeed, its ability to rapidly flip between the leaflets of membranes makes it uniquely suited to rapidly relieve stresses (36,37,38).

It is an open question as to what extent $J_0$ and ${\kappa }_{\mathrm{G}}$ are determined by individual or collective lipid properties—a stark case being lateral lipid phase separation. It is currently hypothesized that cellular outer leaflets exist near a liquid-ordered/liquid-disordered (L_o/L_d) transition because of the leaflets’ high saturated tail and cholesterol content (39,40,41). In an additive model (42), single lipid properties are compositionally weighted to determine an entire region’s mechanical properties. In reality, L_o and L_d phase mechanical properties are dramatically different (43), and individual lipids have “local extents” to which they influence their surroundings (42,44). In any system, an L_o/L_d phase separation creates a line tension because of hydrophobic mismatch between the thicker L_o and thinner L_d. Phase separation, line tension, and curvature are coupled (36,43,45,46,47), and it is hypothesized that this results in fusion pore closure (16,33,48,49); cholesterol is central to each of these elements. One might imagine a radial (from the pore center) enrichment/depletion of particular lipid species caused by strong curvature. If the redistribution creates two radial regions with different thicknesses, the pore will close (fission) to minimize the line tension (48,49). The simulations in this work link the geometric thinning of the hydrophobic interior to a driving force for phase separation (the difference in thickness between phases).

Starting from fusion pores built using a method to target specific pore dimensions, we simulated two pores to infer the thermodynamic effect of lipid composition, in particular cholesterol, on pore stability. Previous simulations of fusion pores and intermediates have typically focused on structural dynamics and intermediate evolution, and therefore have necessarily used coarse-grained resolution and/or simple compositions (50,51,52,53,54,55,56,57). Here, we focus on the energetics of a metastable, nascent fusion pore structure, similar to the work by Cui and coworkers (58). In that work, the authors established the correspondence of molecular simulations and continuum models at very small length scales, and in one case observed substantial thinning at the fusion pore neck. Instead, here, simulations were performed at all-atom resolution with a multicomponent, asymmetric, plasma membrane mimetic. The pores simulated here change shape considerably slower than the relaxation timescale of composition in the neck (∼200 ns; see the results section on timescales), which allows the simulation to quantify enrichment of lipids on structures whose shape is out of equilibrium. Fusion pore shape relaxation requires relaxation of water across the bilayer (osmotic and hydrostatic stress) as well as relaxation of lipids in the inner and outer leaflet (differential stress (38)). While we report the slowly changing shape of the fusion pores in the supporting material (see Figs. S8–S10), this work focuses on the redistribution of lipids on nearly static pores.

Independent, continuous ∼5.5-μs simulations of multicomponent, asymmetric pores simulated with Anton 2 (59) and the CHARMM all-atom force field (60) demonstrate that cholesterol is excluded from both leaflets of the fusion pore’s neck—an unexpected result for a lipid with strong negative $J_0$. Simulation results were interpreted through the HC model, which we implemented into a Monte Carlo (MC) program to predict equilibrium lipid distributions on the fusion pore. Using theory and analysis techniques, we show that cholesterol exclusion is driven primarily by its preference for a bilayer with a thicker interior and that the neck bilayer is thinned relative to the bulk (reducing lipid chain order). We further demonstrate that rather than favoring reshaping broadly, cholesterol drives fusion pore collapse, that is, smaller fusion pores, consistent with early theory on ${\kappa }_{\mathrm{G}}$ (45,46,61). Thus, cholesterol accumulation should favor endocytosis (e.g., viral entry through the plasma membrane) and budding (16,33,34,45,46,47,62) but disfavor viral escape from the endosome that requires expansion to a wide fusion pore (17,63,64). However, unlike previous models (16,48,49) that proposed fission would occur because of laterally phase separated domains, our model only assumes that 1) cholesterol prefers thicker bilayers and 2) fusion pore necks are thinned according to a straightforward mathematical and mechanical analysis. Along with these assumptions and associated controls, we expect our results to be general for any membrane composition containing physiological amounts of cholesterol.

The evidence and explanation of anomalous cholesterol redistribution in fusion pores is presented as follows. First, we demonstrate that cholesterol redistribution in the neck occurs on timescales that allow robust statistics. Next, methodology is shown for how a literature-informed HC model is applied to treat expected redistribution. The lipid distributions in the fusion pore necks determined by molecular dynamics simulations are compared with the HC model, highlighting the unexpected observed cholesterol depletion in the pore neck. We then propose that hydrophobic thinning in the pore neck due to negative $K$ is the mechanical basis for cholesterol depletion. Theory is presented for how dynamic redistribution of cholesterol acts to shrink pore structures, favoring the late stages of vesicle budding. Finally, we discuss the possible biological ramifications in the discussion section.

Materials and methods

Build and simulation

The multicomponent, planar bilayer with asymmetric leaflets was built using scripts from CHARMM-GUI (65). We define the outer leaflet as the exoplasmic leaflet that contains cholesterol (30%), monosialoganglioside (18:0/sphingosine; GM3 = 10%), palmitoyl-sphingomyelin (16:0/sphingosine; PSM = 28%), palmitoyl-arachidonoyl-phosphatidylcholine (16:0/20:4; PAPC = 7%), and palmitoyl-linoleoyl-phosphatidylcholine (16:0/18:2; PLPC = 25%). The inner leaflet is defined as the cytoplasmic leaflet that contains cholesterol (30%), PLPC (16%), stearoyl-arachidonoyl-phosphatidylethanolamine (18:0/20:4; SAPE = 34%), and stearoyl-arachidonoyl-phosphatidylserine (18:0/20:4; SAPS = 20%). See the supporting material for details on the composition choices and Tables S2–S4 for lipid concentrations for each system and each leaflet. The system was neutralized and brought to ∼150 mM KCl concentration. Starting from two $F_{\text{HC}}$ minimized continuum mesh surfaces (one with a radial constraint), asymmetric fusion pores with different radii (P_small and P_large) were built using in-house code that bends and grafts pieces of the planar bilayer onto the curved fusion pore geometry. Using this methodology, the bilayer and two pores had approximately the same lipid compositions and solution salt concentration. The pores and planar bilayer were simulated for ∼5–20 ns locally before being converted to dms format. The bilayer, P_small, and P_large were simulated 2 μs, 5.6 μs, and 5.5 μs, respectively using the Anton 2 supercomputer (59) and the CHARMM all-atom lipid force field (60). See Fig. 1 for a molecular image of P_large. Further build (particularly regarding the fusion pore construction) and simulation details can be found in the supporting material. The last 4.5 μs of the pore simulations and the entirety of the bilayer simulation were used for analysis.

Figure 1.

Molecular image of a snapshot of the large pore (P_large) with regions and leaflets labeled. To see this figure in color, go online.

Analysis methods

The leaflet neutral surface

A key parameter of the analyses is the lipidic neutral surface $\left(\delta \right)$, which is the height in a leaflet where curvature and lateral area elasticity are independent. We assume that the area of each lipid on the fluctuating leaflet does not depend on its local curvature when measured at $\delta$. Note that this assumption is clearly broken for the hexagonal phase, where total curvature and leaflet area are energetically coupled, leading to the divergence of the neutral surface and so-called pivotal plane where area is constant (66). Curvature-coupled-redistribution (CCR) analysis (see (42,44) and supporting material) indicates that the asymmetric planar bilayer has leaflet ${\delta }_0$ (where naught indicates a value for a flat bilayer) of 13.9 Å for the outer leaflet and 13.0 Å for the inner leaflet. The CCR method works by the fact that the average curvature of an undulating leaflet is only zero when sampled at ${\delta }_0$. If a surface is chosen above ${\delta }_0$ (i.e., too close to the lipid headgroups), the surface will appear to have total negative curvature as the headgroup atoms concentrate around a negative (concave) surface. The opposite is true for surfaces defined too close to the midplane. This corresponds with the variation of the area and curvature away from ${\delta }_0$, as illustrated by Fig. 2. As computed by our methodology, $\delta$ is somewhat sensitive to the area of a cholesterol molecule (see supporting material for sensitivity analysis).

Figure 2.

Left: diagram of curvature on an ideal ($J_{\text{mid}}=$ 0) fusion pore. Lipids with red heads have negative curvature in the plane of the paper, and lipids with blue heads have positive curvature in this plane. Note that perpendicular to this plane, curvature is positive for red-headed lipids and negative for blue-headed lipids. At the midplane (small purple dot), the total curvature $\left(J_{\text{mid}}\right)$ is zero. At the neutral surface of each leaflet (approximately the larger purple dots), the total curvatures ($J_{\text{in}}$ and $J_{\text{out}}$) are negative. Note that P_large's $J_{\text{mid}}=$ 1. $7\times 10$⁻² Å⁻¹ (sign of curvature determined by the outer leaflet), indicating that we actually simulated non-ideal fusion pores. Right: cartoon showing the variation of thickness in a highly curved bilayer. In our treatment, the atom of a leaflet’s $\delta$ is independent of curvature. For example, the $\delta$ position is three carbon atoms below the trans double bond in PSM’s sphingosine tail irrespective of curvature. Instead, the hydrocarbon thickness (i.e., from the midplane to $\delta$) does change with curvature. It is known from lipid inverse hexagonal experiments that negative curvature thins the hydrocarbon region (below $\delta$) and thickens the polar groups (above $\delta$) (67). In the cartoon, negative curvature (inner leaflet; red) thins the leaflet (changing ${\delta }_0$ to ${\delta }_{\text{in}}$), and positive curvature (outer leaflet; blue) thickens the leaflet (changing ${\delta }_0$ to ${\delta }_{\text{out}}$). The inner leaflet is thinned more than the outer leaflet is thickened. The length of the heavy red and blue lines are the same, implying that the area is conserved at $\delta$. However, the lengths at the midplane are different (the negatively curved lipid has a larger midplane area $\left(A_{\text{mid},\text{in}}\right)$ than the positively curved lipid $\left(A_{\text{mid},\text{in}}\right)$) between the red and blue lipids because of volume conservation (68). Note that the areas of the red and blue shaded regions are the same. We emphasize that total curvatures $\left(J\right)$ cannot generally be discerned by looking at projection in a single plane. However, at the central point on the neck (shown in purple), curvature can be inferred from the sum of the red and blue projections. To see this figure in color, go online.

Continuum mesh fits to determine the fusion pore shape

The bilayer and pore shapes were characterized via a fit through their respective midplane surfaces. The midplane was fit as the midsurface between the $\delta$ surfaces (defined by the ${\delta }_0$ atoms determined by the CCR method), accounting for curvature. Using this method avoids uncertainties in midplane location, particularly in curved regions. These continuum surfaces contained the contribution from $F_{\text{HC}}$ as well as a term to favor the surface passing between the leaflet $\delta$ (Eq. S29).

The shape of the pore inner and outer leaflets, given the bilayer midplane surface

In Eq. 1, $J_{\text{mid}}$ and $K_{\text{mid}}$ are written in terms of the bilayer midplane. For strongly curved systems, like those considered here, midplane and leaflet curvature differ substantially. The following details how curvature, area, and thickness are determined for individual lipids in their respective leaflets, given a description of the bilayer midplane. Consistent with area preservation at ${\delta }_0$, we choose to write the lipid area in terms of ${\delta }_0$’s area, $A_0$. Fig. 2 shows a cartoon of the inner and outer lipid leaflets of a curved bilayer surface. At the bilayer midplane, the leaflet areas ($A_{\text{mid},\text{in}}$ and $A_{\text{mid},\text{out}}$) are (28)

$$\begin{array}{cc}{A}_{\text{mid},\text{in}}& =A_0\left(1+J_{\text{mid}}{\delta }_0-K_{\text{mid}}{\delta }_0^2+\mathcal{O}\left[{\delta }_0^3\right]\right)\\ A_{\text{mid},\text{out}}& =A_0\left(1-J_{\text{mid}}{\delta }_0-K_{\text{mid}}{\delta }_0^2+\mathcal{O}\left[{\delta }_0^3\right]\right)\end{array}$$ (2)

relative to $A_0$.

The values of $J_{\text{mid}}$ and $K_{\text{mid}}$ at the neck for each fusion pore are listed in Table 1. The lightly shaded areas of Fig. 2 represent lipid volumes, which are nearly constant with curvature (68). For the red leaflet, the midplane area $\left(A_{\text{mid},\text{in}}\right)$ is larger than $A_0$. To maintain constant volume in the red leaflet, the thickness between the midplane and $\delta$ decreases. The corresponding relations for leaflet interior thickness of the inner and outer leaflets are

$$\begin{array}{cc}{\delta }_{\text{in}}& ={\delta }_0\left(1-\frac{1}{2}{J}_{\text{mid}}{\delta }_0+\frac{2}{3}{K}_{\text{mid}}{\delta }_0^2+\mathcal{O}\left[{\delta }_0^3\right]\right)\\ {\delta }_{\text{out}}& ={\delta }_0\left(1+\frac{1}{2}{J}_{\text{mid}}{\delta }_0+\frac{2}{3}{K}_{\text{mid}}{\delta }_0^2+\mathcal{O}\left[{\delta }_0^3\right]\right)\text{.}\end{array}$$ (3)

Table 1.

Structural information for the two fusion pores

Property	Psmall	Plarge
Neck radius (Å)	15.9	34.4
Neck $J_{\text{mid}}$ (Å−1)	−0.01	0.013
Neck $K_{\text{mid}}$ (Å−2)	−0.0017	−0.0007
Bulk thickness (Å)	26.7	26.3
Neck thickness (Å)	16.2	23.0
Neck thinning (%)	39	13

The bilayer thickness $t_{\delta }$ between the neutral surfaces is the sum of the two:

$$t_{\delta }={\delta }_{\text{in}}+{\delta }_{\text{out}}=2{\delta }_0\left(1+\frac{2}{3}{K}_{\text{mid}}{\delta }_0^2\right)\text{.}$$ (4)

Note that separate values of ${\delta }_0$ can be used for the inner and outer leaflets. The neutral surfaces of the inner and outer leaflets here were computed to be within 1 Å (see “the leaflet neutral surface” above). For simplicity we thus use their average value here as a single ${\delta }_0$. See the supporting material (“interior thickness variation with $J_{\text{mid}}$ and $K_{\text{mid}}$”) for the derivations of Eq. 3, and refer to Fig. S6 for more information on the $K$-induced bilayer thinning. As discussed below, while Eq. 4 accurately estimates the bilayer thinning within P_large’s interior, it underestimates thinning for P_small.

Leaflet curvatures ($J_{\text{in}}$ and $J_{\text{out}}$), from which individual lipid curvature energetics are determined, are (8,69)

$$J_{\text{in}}=-\frac{J_{\text{mid}}+J_{\text{mid}}^2\delta -2K_{\text{mid}}\delta }{1-J_{\text{mid}}\delta +K_{\text{mid}}\delta }=-J_{\text{mid}}-J_{\text{mid}}^2{\delta }_0+2K_{\text{mid}}{\delta }_0+\mathcal{O}\left[{\delta }_0^2\right]$$ (5)

and

$$J_{\text{out}}=\frac{J_{\text{mid}}-J_{\text{mid}}^2\delta +2K_{\text{mid}}\delta }{1-J_{\text{mid}}\delta +K_{\text{mid}}\delta }=J_{\text{mid}}-J_{\text{mid}}^2{\delta }_0+2K_{\text{mid}}{\delta }_0+\mathcal{O}\left[{\delta }_0^2\right]\text{.}$$ (6)

According to Eqs. 5 and 6, leaflet $J$ ($J_{\text{in}}$ or $J_{\text{out}}$) changes from $J_{\text{mid}}$ as a function of $K_{\text{mid}}$. The change in leaflet curvature due to $K_{\text{mid}}$ can be accounted for in two ways: treating the effect in terms of leaflet curvature or as a change in ${\kappa }_{\mathrm{G}}$. Consider leaflet curvature models of Eq. 1 in which ${\kappa }_{\mathrm{G}}$, assumed to be a bilayer rather than leaflet property, is treated separately. Shifting from the midplane to the outer leaflet yields (the bar indicates a per-area quantity; analysis for ${\overline{F}}_{\text{HC},\text{in}}$ is equivalent)

$${\overline{F}}_{\text{HC},\text{out}}=\frac{1}{2}{\kappa }_{\mathrm{m}}{\left(J_{\text{mid}}\underset{\Delta J_{\text{out}}}{\underbrace{-J_{\text{mid}}^2\delta +2K_{\text{mid}}\delta }}-J_0\right)}^2\text{,}$$ (7)

where ${\kappa }_{\mathrm{m}}$ is the leaflet bending modulus $\left({\kappa }_{\mathrm{m}}=\frac{1}{2}\kappa \right)$, and $2K_{\text{mid}}\delta$ is the $K$-based contribution to the shift $\left(\Delta J_{\text{out}}\right)$. Moving $K_{\text{mid}}$ out of the squared quantity and dropping higher-order curvature terms yields:

$${\overline{F}}_{\text{HC},\text{out}}=\frac{1}{2}{\kappa }_{\mathrm{m}}{\left(J_{\text{mid}}-J_{\text{mid}}^2\delta -J_0\right)}^2\underset{\Delta {\kappa }_{\mathrm{G},\mathrm{c}}}{\underbrace{-2{\kappa }_{\mathrm{m}}{J}_0\delta }}{K}_{\text{mid}}+\mathcal{O}\left[K_{\text{mid}}^2,K_{\text{mid}}{J}_{\text{mid}}\right]\text{.}$$ (8)

In this form, $\Delta {\kappa }_{\mathrm{G},\mathrm{c}}$ emerges as a leaflet (${\kappa }_{\mathrm{m}},J_0$, and $\delta$) contribution to the bilayer treatment of Gaussian curvature. Note that the second term in Eq. 8 implies the result of ${\kappa }_{\mathrm{G}}=2\left({\kappa }_{\mathrm{G},\mathrm{m}}-2{\kappa }_{\mathrm{m}}{J}_0\delta \right)$ (8), where ${\kappa }_{\mathrm{G},\mathrm{m}}$ is the leaflet Gaussian curvature modulus. Later in this paper, we demonstrate that ${\kappa }_{\mathrm{G}}$ has a bilayer thickness component $\left(\Delta {\kappa }_{\mathrm{G},\mathrm{t}}\right)$ that is important for determining fusion pore lipid distributions.

Lipid properties

To model a lipid’s redistribution on the pore, we employ a local model of elasticity in the vicinity of the lipid. The elasticity energy density ($\overline{F}\left(J,t\right)$; where the bar indicates a density) is

$$\overline{F}\left(J,t\right)=\frac{{\kappa }_{\mathrm{m}}}{2}{\left(J_{\text{in}/\text{out}}-J_0\right)}^2+\frac{K_{\mathrm{A},\mathrm{m}}}{2}{\left(\frac{t-\left(t_0+\Delta t_0\right)}{t_0+\Delta t_0}\right)}^2\text{,}$$ (9)

where $J_{\text{in}/\text{out}}$ is the total curvature of the inner or outer leaflet (the signs of leaflet curvatures are determined by their leaflet normals, which point in opposite directions; e.g., $J_{\text{mid},\text{in}}=-J_{\text{mid},\text{out}}$), $t$ is the measured bilayer thickness, $t_0$ is the unperturbed, bulk bilayer thickness, $\Delta t_0$ is the difference in thickness preference of an individual lipid from the bulk, and $K_{\mathrm{A},\mathrm{m}}$ is half the bilayer area compressibility modulus (described in more detail in (70) and supporting material). In this local treatment of the HC energy density, $J_0$ refers to the spontaneous curvature of the specific lipid. Values of $J_0$ and $\Delta t_0$ are extracted using curvature-coupled relaxation and thickness-coupled relaxation (TCR) techniques, respectively, and determined by how lipids redistribute on a planar bilayer’s dynamically fluctuating modes (44). Cartoons describing the two modes of curvature-coupled relaxation (left) and TCR (right), as well as the models for cholesterol they imply, are shown in Fig. 3 (see also supporting material for detailed technical information). Note that bilayer thickness $t$ and $t_{\delta }$ quantities are not necessarily equivalent; $t_{\delta }$ refers specifically to the thickness between the neutral surfaces of the two leaflets. In our reference planar simulations, we use $t=t_{\delta }$ to compute bilayer thickness strains $\left(\frac{t-t_0}{t_0}\right)$ such that the same leaflet Fourier calculations are used for both curvature and peristaltic (thickness) modes.

Figure 3.

Illustration of the expected distribution of cholesterol according to its curvature (left) and thickness preference (right). To see this figure in color, go online.

Gaussian curvature modulus

The Gaussian curvature modulus $\left({\kappa }_{\mathrm{G}}\right)$ strongly impacts “topological stability:” the relative free energy of vesicles, sheets, tubes, and strange objects such as the cubic phase (8,9,30) and cubosomes (71). This is because unlike the curvature $J$, the integrated Gaussian curvature on a surface, ${\int }_{S}K$, is a constant that only depends on topology via the Gauss-Bonnet theorem. Although a full mechanistic understanding of ${\kappa }_{\mathrm{G}}$ is unknown, differences between lipids are primarily understood through ${\kappa }_{\mathrm{G}}$’s dependence on $J_0$ as in Eq. 8. In this work, we are able to extend the treatment of the lipid-specific dependence of ${\kappa }_{\mathrm{G}}$ by incorporating the thickness elasticity of the bilayer interior. As we argue below, the novel lipid-composition dependence we include in our model of ${\kappa }_{\mathrm{G}}$ extends its relevance beyond its effect on leaflet curvature and topological transformations. We propose that lipids that are sensitive to the thickness of the bilayer interior have predictable changes in ${\kappa }_{\mathrm{G}}$ and thus can support heterogeneous Gaussian curvature such as that present in fusion and fission intermediates.

According to Eq. 4, a bilayer with saddle curvature $\left(K<0\right)$ experiences net thinning within the thickness of the bilayer interior, $t_0=2{\delta }_0$. The strain is equal to

$${ϵ}_{\mathrm{K}}=\frac{2}{3}{\delta }_0^{2}K\text{,}$$ (10)

where ${\epsilon }_{\mathrm{K}}$ is the strain away from the total bilayer thickness. The individual lipid strain ${\epsilon }_{\mathrm{l}}$ is

$$\begin{array}{cc}{ϵ}_{\mathrm{l}}& =\frac{-\Delta t_0}{t_0+\Delta t_0}\\ & \approx -\frac{\Delta t_0}{t_0}\end{array}\text{,}$$ (11)

where $\Delta t_0$ is the difference in bilayer thickness preference of the lipid, compared with $t_0$. The numerator is the deviation from the preferred thickness $\left(t_0+\Delta t_0\right)$. To first order, strains are additive:

$${\overline{F}}_{\text{lateral}}\left(t\right)\approx \frac{K_{\mathrm{A}}}{2}{\left({ϵ}_{\mathrm{l}}+{ϵ}_{\mathrm{K}}\right)}^2\text{.}$$ (12)

Expanding the local elastic thickness energy in powers of $K$ yields

$${\overline{F}}_{\text{lateral}}\left(t\right)=\frac{K_{\mathrm{A}}}{2}{ϵ}_{\mathrm{l}}^2+\underset{\Delta {\kappa }_{\mathrm{G},\mathrm{t}}}{\underbrace{\frac{2}{3}{K}_{\mathrm{A}}{\delta }_0^2{ϵ}_{\mathrm{l}}}}K+\mathcal{O}\left[K^2\right]\text{.}$$ (13)

The first energy term in Eq. 13 is from the thickness strain (dependent on the compressibility modulus, $K_{\mathrm{A}}$), and the second term is from the Gaussian curvature strain. Written as a contribution to ${\kappa }_{\mathrm{G}}$, Combining Eq. 11 and the second term of Eq. 13 yields the energetic contribution in terms of a change in ${\kappa }_{\mathrm{G}}$:

$$\Delta {\kappa }_{\mathrm{G},\mathrm{t}}\approx -\frac{1}{6}{K}_{\mathrm{A}}{t}_0\Delta t_0\text{,}$$ (14)

with $t_0=2{\delta }_0$. Note that $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ is inherently a bilayer property, as it is determined by the interior thickness of a bilayer. Contrast this with the influence of lipid spontaneous curvature on ${\kappa }_{\mathrm{G}}$ (Eq. 8), which is solely an effect of leaflet spontaneous curvature.

Lipid redistribution from Helfrich/Canham theory

In the simplest version of the HC model, lipids have a single spontaneous curvature $J_0$ that does not depend on the context of the surrounding leaflet (e.g., the composition). Further assuming the effect of a lipid $i$ is localized, its contribution to the bending free energy $\left(F_{\mathrm{i}}\right)$ is

$$F_{\mathrm{i}}=\frac{{\kappa }_{\mathrm{m}}{\tilde{A}}_{\mathrm{i}}}{2}{\left(J_{\mathrm{i}}-J_{0,\mathrm{i}}\right)}^2\text{,}$$ (15)

where ${\kappa }_{\mathrm{m}}$ is the leaflet bending modulus, ${\tilde{A}}_{\mathrm{i}}$ is lipid $i$’s lateral area, $J_{\mathrm{i}}$ is the curvature local to a lipid $i$, and $J_{0,\mathrm{i}}$ is lipid $i$’s intrinsic curvature. The localized effect of a single lipid is based on calculations of the spatial extent of single lipids in (44). While that study found that lipids were not perfectly localized (for the studied lipids, $J_0$ diminishes somewhat for short wavelength undulations), the localization is sufficient to discriminate between the bulk and neck regions of the pore. The likelihood $p_{\mathrm{i},\mathrm{r}}$ of observing the lipid in a particular region $r$ with curvature $J_{\mathrm{r}}$ is proportional to its Boltzmann factor,

$$p_{\mathrm{i},\mathrm{r}}\propto {\mathrm{e}}^{-\beta F_{\mathrm{i}}}\text{,}$$ (16)

but naturally will be influenced by similar energetics of the other lipids. Therefore, lipid populations in each region are predictable.

Here, we use a single-body MC model to predict equilibrium lipid distributions on the fusion pores. The MC is divided into two regions (bulk and neck) for both pore sizes and both leaflets. For both regions, $⟨J⟩$ and $⟨K⟩$ for each pore size and leaflet were calculated from the molecular dynamics (MD) simulations. Lipids were randomly added to these regions, and lipids were randomly selected to move to the new region (e.g., a lipid in the bulk changes places with a lipid in the neck). The move is accepted or rejected based on Eq. 16 and the common Metropolis criterion. First, the MC simulations are informed by $J_0,\stackrel{\sim }{A},\kappa$, and $K_{\mathrm{A}}$ that are determined either by previously published simulation or by analogy. Then, MC analysis strongly suggests that it is $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$, not related to $J_0$, that determines cholesterol distributions in small fusion pores (see below). The MC details, including Table S7 showing mechanical parameters, can be found in the supporting material.

Results

Cholesterol redistributes rapidly in the pore neck

Statistically meaningful estimates of enrichment of lipids in the pore neck require rapid relaxation of fluctuations. Lipids are expected to diffuse across the neck with timescale $q^{-2}{D}^{-1}$, where $D$ is the diffusion constant, $q=2\pi /\lambda$, and $\lambda$ is the length scale ($\sim$8 nm) of the region. With $D\approx$ 8 μm²/s, the composition is expected to relax in ∼200 ns. Statistically, the variance of cholesterol’s mol fraction in the region is ${\sigma }^2=\phi \left(1-\phi \right)/n$, where $\phi$ is the mol fraction of cholesterol and $n$ is the number of lipids in the region. For a ${\phi }_{\text{chol}}=$ 0.30 and ∼100 lipids in the neck of P_small, $\sigma =$ 0.046. In P_large, $n=$ 150 and $\sigma =$ 0.037. Fig. 4 shows the time series of ${\phi }_{\text{chol}}$ in P_small over the course of the simulation, with the autocorrelation function inset. Although the autocorrelation is not sufficiently converged to extract a diffusion constant, it is useful for a rough indication of the timescale. Both the magnitude of the fluctuations (which are damped by sampling over a 24-ns window) and timescale of relaxation are consistent with the simple estimate.

Figure 4.

Timescale of the cholesterol fraction $\left({\phi }_{\text{chol}}\right)$ in the neck of P_small. The mean over the last 4.5 μs is shown by a solid purple line. Dotted lines indicate one standard deviation expected from statistical fluctuations. Instantaneous fluctuations were reduced by computing the composition over 24-ns windows. The autocorrelation function (shown in the inset) is consistent with a $\sim$200-ns timescale for compositional relaxation. To see this figure in color, go online.

Phospholipids enrich in the pore according to their spontaneous curvature; cholesterol does not

Following $\sim$5.5-μs simulations, the lipid distributions for P_small and P_large were calculated (solid bars in Fig. 5). For P_small, the outer leaflet shows strong enrichment of PLPC and PAPC with depletion of GM3, PSM, and cholesterol. Relative phospholipid redistribution results are consistent with $J_0$: unsaturated PC lipids have a negative total curvature preference (72), and sphingolipids have a positive curvature preference (73). In the inner leaflet SAPE is enriched, SAPS is near the bulk value, and PLPC and cholesterol are depleted. Simulations indicate that SAPE outcompetes PLPC and SAPS, consistent with its strongly negative $J_0$. However, with its strong negative $J$, cholesterol should be enriched in the neck to a similar degree as SAPE. Given the small size of our bulk reservoir, we anticipate that the observed redistribution is a lower bound for physiological redistribution. The chemical potential could be affected by redistribution in our small system but would be essentially unchanged with an effectively infinite bulk region in the cell.

Figure 5.

Enrichment per region relative to the total mol % of a lipid species: direct observations from MD are solid bars and MC predictions are cross-hatched (enrichment = 100 $\times \left({\phi }_{\text{species},\text{observed}}-{\phi }_{\text{species},\text{total}}\right)/{\phi }_{\text{species},\text{total}}$). This MC prediction does not use $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$, and the disagreement between the MD observation and MC prediction (particularly for cholesterol) is large. Data for P_small’s outer (left) and inner (right) leaflets are on the top row. Data for P_large’s outer (left) and inner (right) leaflets are on the bottom row. See Table S2 for ${\phi }_{\text{species},\text{total}}$ values. To see this figure in color, go online.

To gain further energetic insight into these results, we utilized an MC model with HC energetic terms: $J_0$, κ, and area contributions to the free energy (i.e., no $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$; see Eq. S37). The results are shown as cross-hatched bars in Fig. 5. MC predicts depletion of GM3 and PSM based on their positive $J_0$ and a depletion of PLPC based on competition. Importantly, the MC model predicts strong cholesterol enrichment given its very strong negative $J_0$. In the inner leaflet, SAPE and cholesterol are expected to be enriched in the pore. However, cholesterol is depleted in the MD. Similar results are shown for P_large. Therefore, a simple model containing only information on $J_0,\kappa$, and area contributions does not adequately capture the complex energetics occurring in the MD. The stark mismatch between the MC and MD suggests that there are more complex curvature energies to be determined.

Cholesterol is excluded from the pore by its affinity for thick bilayers

The oily interior of a leaflet is thinned by negative total curvature and thickened by positive total curvature. To first order in curvature, the bilayer interior thickness $t_{\delta }$, which is the sum of the inner and outer leaflets, is constant. As shown in Eq. 4 and fully derived in materials and methods (Eqs. S30–S36), the bilayer thickness is proportional to $K$. The midplane Gaussian curvature $\left(K_{\text{mid}}\right)$ is negative for saddles, indicating that the bilayer interior thins. Applying Eq. 4 to the midplane saddle curvature in Table 1 yields the expected thinning in the bilayer interior $\left({\delta }_{\mathrm{b}}\right)$ at the most extreme curvature of the pore. For P_small, the expected thinning is $\frac{2}{3}{\delta }^2{K}_{\text{mid}}=\left(\frac{2}{3}\right)\left(-0.0017{\mathrm{\AA }}^{-2}\right)\left({13\mathrm{\AA }}^2\right)=$ 19%, while for P_large it is 8%. Simulations indicate that thinning for the small and larger is even more dramatic, at 39% and 13%, respectively (Fig. 6).

Figure 6.

Density of $\delta$ atoms for a 5-Å-wide slice. Left column: P_small. Right column: P_large. The top row is a 5-Å-wide slice in $xz$ slice through $y=0$. The bottom row is a 5-Å-wide slice in $xy$ through the pore at $z=0$. Each box is 200 Å by 200 Å. To see this figure in color, go online.

Thinning in the neck of the pore is a significant discrepancy between simulation and theory. The possibility that cholesterol depletion explains the thinness is excluded by supplemental simulations built with asymmetric compositions equivalent to those of the pore necks. Based on thickness analysis of these planar bilayers, the necks of P_small and P_large should have ${\delta }_{\mathrm{b}}$ similar to the bulk, far from the pore’s neck. See the supporting material for more information on these simulations and analyses.

Also, thinning is sensitive to the precise location of the neutral surface, going as the square of $\delta$. If $\delta$ decreased with cholesterol depletion, depletion would induce thinning and thus further depletion. Fig. S6 shows the variation of thinning as a function of $\delta$—large variations in thinning occur with small shifts of $\delta$. We used the CCR method (44) to determine $\delta$. The CCR method gives fluctuation-based (i.e., $q$-dependent) $J$ preferences by determining where lipids preferentially populate on a leaflet’s fluctuation modes. However, the result is sensitive to the area-per-lipid of cholesterol, an ambiguous quantity. Note also that throughout this paper we discuss cholesterol redistribution in terms of thickness. We do not fully deconvolve thickness from lipid order, to which it is strongly correlated.

Furthermore, thinning in the bilayer interior does not necessarily indicate headgroup-to-headgroup thinning. Indeed, the same considerations leading to thinning in the interior lead to thickening in the exterior (headgroup region) (67). This is the basis for the long-established theory for why lipids with negative $J_0$ favor cubic phases (8). Yet the dependence of ${\kappa }_{\mathrm{G},\mathrm{m}}$ on $J_0$ alone is inconsistent with cholesterol. Motivated by the fact that cholesterol resides mostly within the bilayer interior, we hypothesize that cholesterol is sensitive to interior thinning to the same degree it is sensitive to total bilayer thinning.

A change in ${\kappa }_{\mathrm{G}}$ can be related to a difference in thickness preference $\Delta t_0$ and the thickness/area elasticity $K_{\mathrm{A}}$ (Eq. 14). Given bilayer thinning from saddle Gaussian curvature is an important geometric constraint on these fusion pores, we introduced $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$ into the MC model (Fig. 7 and Eq. S37). The value of $\Delta t_0=$ 7.5 $\pm$ 2.2 Å for cholesterol (it prefers bilayers 7.5 Å thicker than our membrane mimic) was calculated using the TCR method on the planar bilayer simulation (see Fig. S4). Similar to the CCR method that obtains $q$-dependent $J$ preferences for a given lipid species, TCR obtains $q$-dependent thickness preferences. Through Eq. 14, this indicates that $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}=$ −20.1 $\pm$ 5.8 kcal/mol.

Figure 7.

Enrichment per region relative to the total mol % of a lipid species. Direct observations from MD are solid bars and MC predictions are cross-hatched (enrichment = 100 $\times \left({\phi }_{\text{species},\text{observed}}-{\phi }_{\text{species},\text{total}}\right)/{\phi }_{\text{species},\text{total}}$). This MC prediction includes $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$, and the cholesterol agreement is improved relative to Fig. 5. Data for P_small’s outer (left) and inner (right) leaflets are on the top row. Data for P_large’s outer (left) and inner (right) leaflets are on the bottom row. See Table S2 for ${\phi }_{\text{species},\text{total}}$ values. To see this figure in color, go online.

A lipid with negative $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ will be repelled from the pore where there is strong negative $K$ (Eq. 1). The hypothesis that only cholesterol is strongly influenced by $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ is justified because it is the only lipid that nearly fully resides in the bilayer hydrophobic interior, and therefore would be sensitive to interior thinning. We use the inner leaflet’s $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$, assuming that the lipids are in a disordered state. By introducing $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$ into the MC model (Fig. 7), all enrichments/depletions in both leaflets of the P_small and P_large are much better fit relative to the MC model without $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$ (Fig. 5). Still, the largest discrepancies occur in P_small’s outer leaflet, which has the strongest curvature considered in these simulations. An alternative hypothesis would be that polyunsaturated fatty acid (PUFA) lipids’ $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ determines the observed redistribution. To test this in the MC model, we used $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{PUFA}}=$ 6.0 kcal/mol while keeping $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}=$ 0 kcal/mol (Fig. S7). We demonstrate that adding $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{PUFA}}$ increases agreement between the MC model and MD, but not to the extent of adding $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$. Therefore, we conclude that $\Delta {\kappa }_{\mathrm{G},\mathrm{t},\text{chol}}$ is a critical energetic term for determining the lipid distributions of small fusion pores.

The fusion pore changes shape slowly

The shape of the fusion pore is determined by its internal elasticity as well as by external forces. Hydrostatic/osmotic pressures are caused by imbalances between the two distinct water compartments, and our method to assess and relieve hydrostatic pressure leads to relaxation times longer than a microsecond. Differential stress (38,74) is caused by an imbalance in lipid counts between the outer and inner leaflets. Cholesterol flip-flop rates that partially relax this stress are nearly one-tenth of a millisecond. We assess how these stresses affect the simulation results.

Artificial water channels reduce hydrostatic pressure

The periodically replicated fusion pore shape leads to two distinct aqueous regions that exchange water only by permeation through the oily interior (see also (58)). Water is nearly incompressible; therefore, the fixed water volumes serve as constraints on the membrane shape. These hydrostatic forces were a useful tool to restrain pore shape and examine lipid redistribution, but relaxing these restraints allows insight into the pores’ shape relaxation. To test the ramifications of our initial conditions, we added water-channeling carbon nanotubes (CNTs) into the membrane bulk and simulated for a further ∼1.5 μs using Amber (75,76) and the CHARMM all-atom force field (see supporting material for detailed simulation methodology). In P_small water exits the interior compartment, whereas in P_large water enters. The water flow in the systems enables the radius of P_small to expand and for the P_large to contract (see Figs. S8 and S10). The opposite water flow directions suggest that the internal forces that dictate the shape of the fusion pore are weak; that is, leaflets have been built properly to favor this approximate pore size.

Cholesterol flipping partially relaxes the leaflet tension imbalance

Differential stress in a bilayer occurs, for example, when there are too many or too few lipids in a leaflet (i.e., an area imbalance) (38,74). This stress can be relieved by changing shape and by flipping lipids between leaflets. Cholesterol is notable in that it has, by far, the shortest timescale for flipping. Therefore, we quantify cholesterol flips as a qualitative indicator of differential stress, with the caveat that lipid-lipid interactions and curvature energetics also influence the chemical potential difference of cholesterol between leaflets (38). Cholesterol flips were identified by comparing cholesterol orientation normal with the continuum fit surface. Identified flips lasting $<$40 ns were filtered to avoid including assignment errors. We found that in the pores, cholesterol flips between leaflets at a submillisecond rate ($\tau \approx$ 0.05–0.20 ms, ∼5–10 flips/chol/ms), a timescale that was similar across all four simulations (P_large: $\tau$ = 0.15 ms; P_small: $\tau$ = 0.15 ms; P_{large, CNT}: $\tau$ = 0.19 ms; P_{small, CNT}: $\tau$ = 0.07 ms). The standard error is less than 35%, assuming Poisson statistics. Flips tended to occur near the pore neck where the bilayer is thinner; however, P_small, with its substantially thinner neck, had a similar number of flipping events as P_large. The rates are similar to previous observations of flip-flop in heterogeneous bilayers, in which the ordered-phase timescale (1 ms) was slower than that of a completely fluid phase (0.1 ms) (77). A rough comparison suggests that the flipping events in the relatively small part of the neck, where a minority of cholesterol is located, are occurring at fluid-phase timescales while transitions in the bulk are much slower, comparable with the ordered-phase timescale. However, targeted methodology is necessary to statistically differentiate the rate of flip with pore size, composition, and localization. See the supporting material for a schematic showing locations of the cholesterol flips on the pores (Fig. S8) and a further discussion on differential stress.

Cholesterol favors pore collapse

All-atom MD and HC-based MC simulations indicate that cholesterol favors thicker membranes and is therefore excluded from thin fusion pores. Therefore, we ask how a simple model predicts the energetics of pore closure as a function of cholesterol exclusion. The model used herein is related to that of Chen, Higgs, and MacKintosh, who determined how a minority membrane component can initiate pore collapse leading to fission (61). Our model focuses on the entropic cost of cholesterol depletion in the fusion pore and the neck’s thickness strain (Eq. 4). First, entropically, it is easier to deplete a small region of all cholesterol than a large region; therefore, the cost of cholesterol depletion increases with fusion pore size. Second, the total thickness strain is constant in the absence of redistribution because the integral of $K$ is a constant that only varies with topology (i.e., area strain is proportional to ${\int }_{S}K$). This effect is borne out in the following model, which is treated in terms of cholesterol but is applicable to arbitrary lipids.

Cholesterol’s free energy contribution to the pore $\left(F_{\text{pore}}\right)$ is modeled with two terms, $F_{\text{pore}}=F_{\text{entropy}}+F_{\text{HC}}$. First is the entropy of the mol fraction $\left(\phi \right)$ of cholesterol in the vicinity of the fusion pore $\left(F_{\text{entropy}}\right)$:

$$F_{\text{entropy}}=k_{\mathrm{B}}TN\phi \log \left(\frac{\phi }{{\phi }_0}\right)+k_{\mathrm{B}}TN\left(1-\phi \right)\log \left(\frac{1-\phi }{1-{\phi }_0}\right)\text{,}$$ (17)

where $F_{\text{entropy}}$ arises simply from counting the ways of arranging cholesterol and the other lipids on a surface accommodating $N$ total lipids (i.e., $N\phi$ is the number of cholesterol in the region). The free energy is minimized when $\phi$ equals the total leaflet mol fraction ${\phi }_0$. Expanding $F_{\text{entropy}}$ around ${\phi }_0\left(\phi ={\phi }_0+\Delta \phi \right)$ yields

$$F_{\text{entropy}}\approx k_{\mathrm{B}}TN\frac{{\left(\phi -{\phi }_0\right)}^2}{2{\phi }_0\left(1-{\phi }_0\right)}\text{.}$$ (18)

That is, there is an effective force constant $\frac{k_{\mathrm{B}}TN}{2{\phi }_0\left(1-{\phi }_0\right)}$ restricting fluctuations around ${\phi }_0$. As $N$ grows (proportional to the size of the pore), the force constant becomes more restrictive.

Second is cholesterol’s HC energy in the neck relative to the bulk ($F_{\text{HC}}$; where $J_0$ is the spontaneous curvature of the cholesterol with the background zero, and considering the average $J$ and $K$ of the region, $⟨J⟩$ and $⟨K⟩$, respectively:

$$\begin{array}{c}{F}_{\text{HC}}=N{\kappa }_{\mathrm{m}}\tilde{A}\left[{\left(⟨J⟩-\phi J_0\right)}^2-{\left(\phi J_0\right)}^2\right]+N\phi \tilde{A}\Delta {\kappa }_{\mathrm{G},\text{chol}}⟨K⟩\\ =N{\kappa }_{\mathrm{m}}\tilde{A}{⟨J⟩}^2+-N\phi \tilde{A}{\kappa }_{\mathrm{m}}⟨J⟩J_0+N\phi \tilde{A}\Delta {\kappa }_{\mathrm{G},\text{chol}}⟨K⟩\text{.}\end{array}$$ (19)

The first term on the left-hand side does not couple $\phi$ to the shape of the pore (e.g., $⟨J⟩$ or $⟨K⟩$) and so is not relevant to the redistribution of cholesterol on a fixed shape. The $F_{\text{HC}}$ equation is similar to the MC method for determining lipid enrichment, described in detail in Eq. S37. The shape, and thus $⟨J⟩$, of a fusion pore will depend on the lipids and their spontaneous curvatures, as well as the protein system (e.g. clathrin and dynamin) that is forcing the constriction. For simplicity we only model the effect of $⟨K⟩$ on the free energy $\left(F_{\text{chol},\text{saddle}}\right)$, equivalent to setting $⟨J⟩=0$:

$$F_{\text{chol},\text{saddle}}=F_{\text{entropy}}+F_{\text{HC}}\left(⟨J⟩=0\right)\text{.}$$ (20)

With $N\tilde{A}⟨K⟩={\int }_{S}K=-4\pi$ (the topological invariant for the saddle simulated here, see Fig. 2 of (8)):

$$F_{\text{HC}}=-4\pi \phi \Delta {\kappa }_{\mathrm{G},\text{chol}}\text{.}$$ (21)

Minimizing $F_{\text{chol},\text{saddle}}$ with respect to $\phi$ to obtain ${\phi }^{\ast }$ (the optimal $\phi$) yields

$${\phi }^{\ast }=\frac{{\phi }_0\exp \left(\frac{-4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}}{k_{\mathrm{B}}TN}\right)}{{\phi }_0\exp \left(\frac{-4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}}{k_{\mathrm{B}}TN}\right)+\left(1-{\phi }_0\right)}\approx {\phi }_0-4\pi \frac{{\phi }_0\left(1-{\phi }_0\right)\Delta {\kappa }_{\mathrm{G},\text{chol}}}{k_{\mathrm{B}}TN}+\mathcal{O}\left[\Delta {\kappa }_{\mathrm{G},\text{chol}}^2\right]\text{.}$$ (22)

That is, deviations from ${\phi }_0$ are damped as the pore size grows and, conversely, a shrinking pore (i.e., small $N$) allows larger deviations from ${\phi }_0$.

Inserting ${\phi }^{\ast }$ from Eq. 22 into $F_{\text{HC}}+F_{\text{entropy}}$ yields

$$F_{\text{chol},\text{saddle}}=\frac{-{\phi }_0\left(-4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}-k_{\mathrm{B}}TN\log \left({\phi }_0+\omega \left(1-{\phi }_0\right)\right)\right)+\omega k_{\mathrm{B}}TN\left(1-{\phi }_0\right)\log \left(1+\left({\omega }^{-1}-1\right){\phi }_0\right)}{-\omega \left(1-{\phi }_0\right)-{\phi }_0}\text{,}$$ (23)

where for clarity we have defined $\omega =\exp \left(\frac{-4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}}{k_{\mathrm{B}}TN}\right)$. Its asymptote for large $N$ is

$$F_{\text{chol},\text{saddle}}\approx -4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}{\phi }_0-{\left(4\pi \Delta {\kappa }_{\mathrm{G},\text{chol}}\right)}^2\frac{{\phi }_0-{\phi }_0^2}{2k_{\mathrm{B}}TN}+\mathcal{O}\left[N^{-2}\right]\text{,}$$ (24)

and its asymptote for small $N$ is

$$F_{\text{chol},\text{saddle}}\approx -k_{\mathrm{B}}TN\log \left(1-{\phi }_0\right)\text{.}$$ (25)

Note that Eq. 24 requires $\Delta {\kappa }_{\mathrm{G}}$ to deviate sufficiently from zero such that redistribution of the target lipid into or out of the pore is nearly complete for small pores.

Assuming a rough scaling of the fusion pore’s radius $\left(r\right)$ with $N$,

$$N\approx \frac{A_{\text{half}-\text{torus}}}{\tilde{A}}=\frac{2\pi r^2\left(\pi -2\right)}{\tilde{A}}$$ (26)

yields a model in terms of fusion pore size, where $A_{\text{half}-\text{torus}}$ is the inner-half area of a torus with saddle curvature $K=-r^{-2}$ at the inner rim. Variations on Eq. 26 from a half-torus will change the details of the free-energetic force to collapse the pore but will not change the energetic scale, which asymptotes to $-4\pi \Delta {\kappa }_{\mathrm{G}}{\phi }_0$ regardless of how the area scales with $N$.

Fig. 8 shows the variation of the free energy due to cholesterol redistribution as a function of approximate fusion pore size. As $r$ decreases, the entropic penalty (Eq. 18) for depleting cholesterol from the fusion pore decreases, while the elastic benefit for depletion per unit $\phi$ (Eq. 21) is constant. The asymptote in Eq. 24 (Fig. 8, blue dashed line) describes the variation in free energy well for nearly the entire region of interest before pore collapse. Once cholesterol is nearly completely depleted from the pore, the free energy continues to drop only due to the lessened entropic penalty (Fig. 8, red dashed line; Eq. 25). At this point the fusion pore radius is so small that an elastic analysis is likely inappropriate.

Figure 8.

A plot of the modeled effect of cholesterol redistribution on the fusion pore energy (Eq. 23) with modest $\Delta {\kappa }_{\mathrm{G},\text{chol}}=-10$ kcal/mol. The large $N$ (Eq. 24) and small $N$ (Eq. 25) asymptotes are shown. To see this figure in color, go online.

Discussion

Endo- and exocytosis typically require protein machinery to force membranes to undergo fusion or fission. This paper aimed to determine how lipids themselves can bias these processes. The key observations are that fusion pore necks are thinned by negative Gaussian curvature $\left(K\right)$, and cholesterol is depleted in this region. Further simulation and analysis showed that cholesterol couples more strongly to bilayer thickness $\left(t\right)$ than leaflet curvature $\left(J\right)$, leading to cholesterol depletion in the fusion pore necks instead of enrichment. We interpret the results primarily through the Gaussian curvature modulus $\left({\kappa }_{\mathrm{G}}\right)$ that contains contributions of the spontaneous curvature $\left(J_0\right)$ and thickness coupling. Therefore, the important energetic factors can be compared directly (see Eqs. 14 and 8).

We speculate that these results have broad implications for biological fusion and fission. According to this simple model, a cholesterol-enriched bilayer (or any multicomponent system with varied ${\kappa }_{\mathrm{G}}$, as predicted in (61)) will favor the transition from a flat clathrin plaque to a bud (i.e., fission), with the bilayer appearing to be softer than its bending modulus $\left(\kappa \right)$ might imply. This effect belongs to a general class of softening mechanisms for multicomponent systems (42,78,79,80,81). We believe this to be the mechanical origin of the requirement for sterols in clathrin-mediated endocytosis observed in (16,82). Consistent with this interpretation is that local lipid membrane composition selects for a process. For example, a given local composition could plausibly bias the likelihood of nascent pore formation (83,84). Importantly, cholesterol concentration and local membrane thickness are important parameters of this theory as well. This work builds on previous observations that sterols and sterol derivatives can be pro- or antiviral depending on where the energetic barrier is along the cycle of viral replication (17). Simply depleting cellular cholesterol by statins, for example, has deleterious effects (85,86), so further research is required to understand how to target cellular sterol accumulation or depletion pharmacologically. Finally, we note that ESCRT-III machinery assembles onto and creates membranes with negative Gaussian curvature in vivo (13). The negative Gaussian curvature, and subsequent cholesterol redistribution, caused by ESCRT-III could be a mechanistic driver of fission. That said, making case-by-base predictions at a fine-grained level is impossible without the full picture. For example, we do not include important cellular players—most notably, proteins. However, we view this paper as a step toward understanding the fusion pore on a molecular level, and we do so by focusing on the influence of arguably the most important lipids (cholesterol) on mechanics.

Conclusions

All-atom MD simulations of two fusion pores indicate substantial lipid redistribution between the pore and bulk, depending on pore size. Continuum mesh surfaces were fit to the pores to determine the total $\left(J\right)$ and Gaussian $\left(K\right)$ curvature in the bulk and neck region of the pores. Comparing the lipid distributions from simulation with the literature HC model, we found that lipid redistribution cannot be adequately described using the spontaneous curvature $\left(J_0\right)$ and bending modulus $\left({\kappa }_{\mathrm{m}}\right)$ alone. We explain the anomalous depletion of cholesterol from the pore neck on the basis of its Gaussian curvature modulus, ${\kappa }_{\mathrm{G}}$. Strong saddle curvature thins and disorders the bilayer interior (i.e., not just a single leaflet), which disfavors cholesterol. Dynamic redistribution of cholesterol on planar simulations of similar composition yielded estimates for the thickness-driven change in Gaussian curvature modulus $\left(\Delta {\kappa }_{\mathrm{G},\mathrm{t}}\right)$ for each lipid in the bilayer simulation. Importantly, cholesterol in the fluid inner leaflet has a strong negative $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ (−20.1 $\pm$ 5.8 kcal/mol), indicating a lipid that is repelled by saddle curvature.

Using this information, we informed the HC model to include the $K$ and $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ observed in the MD simulations. This addition improved the theoretical model’s fit to the simulation data, indicating that $K$ and $\Delta {\kappa }_{\mathrm{G},\mathrm{t}}$ determine lipid distributions during fusion pore opening and closure. In the smaller pore, ganglioside GM3, PSM, and cholesterol were all excluded from the fusion pore’s neck, suggesting a structural connection to liquid-ordered (L_o) domain formation that would likely further alter lipid material parameters. Theoretical consideration of a two-component system (e.g., cholesterol and the surrounding lipid matrix) demonstrated that it is energetically easier to deplete cholesterol from smaller structures, such that cholesterol is depleted to a much greater extent in smaller fusion pores. This is observed in the all-atom MD simulations, and we expect these results to be general for any membrane composition containing physiological amounts of cholesterol. Also, given the small size of the bulk region, we anticipate that the observed redistribution is a lower bound to physiological redistribution with an effectively infinite bulk. We hypothesize that $K$-driven cholesterol redistribution in fusion and fission pores favors pore closure. Therefore, increased cholesterol levels should favor the late stages of endocytosis and budding (which require closure) but disfavor endosomal viral escape, which requires fusion pore expansion.

Note added in proof

A small number of force field parameters were incorrectly assigned for simulations run in Amber (the pore simulations with CNTs added and the POPC/cholesterol control simulations used to test cholesterol redistribution into thick bilayer modes). Note that the simulations that used Anton and the simulations to test pore interior thickness (NAMD) have the correctly assigned force field. Out of these affected Amber simulations, only a subset (the small pore simulation including the CNTs and two out of six of the POPC/cholesterol simulation replicas) likely contain practical effects. In these affected simulations, short-range repulsion between cholesterol reduces the number of cholesterol-cholesterol contacts. The Supplemental Material lists each Amber simulation affected, the NBFIX parameters inadvertantly used, and the qualitative effect on the simulation that is inferred from other test simulations.

Author contributions

A.H.B. and A.J.S. developed the approach, designed the system setup, and performed the simulations. All authors performed the analysis and wrote the manuscript.

Editor: Padmini Rangamani.