Developing initial conditions for simulations of asymmetric membranes: a practical recommendation

Abstract

It has been proposed that the surface tension difference between leaflets (or differential stress) in asymmetric bilayers is generally nonvanishing. This implies that there is no unique approach to generate initial conditions for simulations of asymmetric bilayers in the absence of experimentally derived constraints. Current generation methods include individual area per lipid (APL) based, leaflet surface area (SA) matching, and zero leaflet tension based (0-DS). This work adds a bilayer-based approach that aims for achieving partial chemical equilibrium by interleaflet switching of selected lipids via P2₁ periodic boundary conditions. Based on a recently proposed theoretical framework, we obtained expressions for tensions in asymmetric bilayers from both the bending and area strains. We also developed a quantitative measure for the energetic penalty from the differential stress. The impacts of APL-, SA-, and 0-DS-based approaches on mechanical properties are assessed for two different asymmetric bilayers. The lateral pressure profile and its moments differ significantly for each method, whereas the area compressibility modulus is relatively insensitive. Application of P2₁ periodic boundary conditions (APL/P2₁, SA/P2₁, and 0-DS/P2₁) results in better agreement in mechanical properties between asymmetric bilayers generated by APL-, SA-, and 0-DS-based approaches, in which changes are the smallest for bilayers from the SA-based method. The estimated differential stress from the theory shows good agreement with that from the simulations. These simulation results and the good agreement between the predicted and observed differential stress further support the theoretical framework in which bilayer mechanical properties are outcomes of the interplay between intrinsic bending and asymmetric lipid packing. Based on the simulation results and theoretical predictions, the SA/P2₁-based, or at least the SA-based (when the differential stress is small), approach is recommended as a practical method for developing initial conditions for asymmetric bilayer simulations.

Significance

Molecular dynamics simulations of symmetric bilayers are now a standard tool in membrane biophysics. However, simulations of asymmetric bilayers, essential for modeling most cellular membranes, are considerably less well developed. This is because assumptions appropriate for symmetric bilayers may not be applicable. In particular, there is a potential for different surface tensions in the two leaflets (called differential stress) arising from coupling of intrinsic bending and asymmetric lipid packing. This work considers the assembly and simulation of two different asymmetric bilayers using several different methods and analyzes the results within a theoretical framework.

Introduction

Cell membranes typically have compositionally asymmetric leaflets (1, 2, 3). For example, in the plasma membrane of red blood cells (3), phosphatidylcholine (PC) and sphingomyelin (SM) are major components in the outer (exoplasmic) leaflet, whereas phosphatidylethanolamine (PE), phosphatidylserine (PS), and PC are major components in the inner (cytoplasmic) leaflet with small amounts of phosphatidylinositol. In addition to the headgroup asymmetry, the relative abundance of saturated or unsaturated lipids is also asymmetric with unsaturated lipids accumulated in the inner leaflet (3). The compositional asymmetry results in a tightly packed outer leaflet and loosely packed negatively charged inner leaflet (3,4), which has immediate implications in cell biology and functions. The former is likely related to a fundamental barrier function of the plasma membrane (5,6), and the latter influences the insertion and orientation of transmembrane proteins, known as the so-called positive inside rule (7). In addition to these fundamental roles, lipid asymmetry in cell membranes has profound effects on their functions such as membrane fusion (8, 9, 10) and lipid droplet (LD) budding in endoplasmic reticulum (11, 12, 13). In membrane fusion, lipids with negative intrinsic curvature (such as cholesterol (Chol) and PE) are necessary to promote and stabilize highly curved membrane structures and transient fusion-pore intermediates (8, 9, 10). In LD budding, the directionality of budding is regulated by asymmetric distribution of phospholipids, in which an LD emerges toward a leaflet with enriched phospholipids because of lowered surface tension of oil-water interface (11,12). Loss of asymmetry can be fatal to cells (14,15), and thus, lipid composition is dynamically regulated (16, 17, 18). In addition to apoptotic loss of asymmetry, recent studies revealed that nonapoptotic (transient, reversible, and localized) changes in PS asymmetry play a variety of roles in biological processes including cell-cell communication and fusion (19, 20, 21, 22) and intracellular signaling (23, 24, 25).

Compared with the biological and functional roles of asymmetry, much less is known for mechanical properties of asymmetric membranes, such as leaflet tension and elastic moduli, not only because of difficulties in measurements of these properties but also owing to technical difficulties in preparation of asymmetric membranes. Recently, three experimental techniques—lipid exchange (26, 27, 28), phase transfer (29), and hemifusion (30) protocols—were developed for the systematic preparation of (synthetic) asymmetric membranes (31, 32, 33, 34, 35). One interesting and puzzling result from the recent studies is that the rigidity of the asymmetric membrane was significantly higher than the average of those from the cognate symmetric membranes (33, 34, 35). This nonadditive rigidity was subsequently proposed to arise from area strain due to possible mismatch in surface area (SA) of monolayers in preparation of the asymmetric membrane (36). Thus, there remains a question as to whether asymmetric membranes from these protocols faithfully represent characteristics of biological membranes.

It also has remained challenging to build realistic asymmetric membranes for simulation studies because assumptions appropriate for symmetric bilayers may not be applicable. For example, the surface tension γ is zero for an unstressed symmetric bilayer, so the surface tension γ_m of each leaflet is zero. Lipid packing is optimal, and there is no area strain, ɛ = (A − A₀)/A₀ = 0, where A is the surface area and A₀ is that in a stress-free bilayer. The leaflet (or monolayer) bending moduli $\left(k_{\text{m}}^{\text{b}}\right)$ are of course equal in a symmetric bilayer, but not necessarily in an asymmetric one. Likewise, the spontaneous curvature $\left(c_{0,\text{b}}^{\text{b}}\right)$ of a homogeneous bilayer associated with bending is zero, even though spontaneous curvature of leaflets $\left(c_{0,\text{m}}^{\text{b}}\right)$ may be negative or positive. Inequalities in $k_{\text{m}}^{\text{b}}$ and $c_{0,\text{m}}^{\text{b}}$ of an asymmetric bilayer lead to nonzero bilayer curvature, and nonoptimal lipid packing causes differential area strain. Coupling between leaflets adds further complexity. Hossein and Deserno (HD) (36) have rigorously shown how asymmetric bilayers can be described by families of the preceding properties. Specifically, there can be many metastable states in which the leaflet surface tensions from area strains are not equal (i.e., nonzero differential stress) as a result of interplay of asymmetric $k_{\text{m}}^{\text{b}}$ and $c_{0,\text{m}}^{\text{b}}$ and differential area strain due to nonoptimal lipid packing (in a timescale shorter than lipid flip-flop (37,38)). Lastly, asymmetric bilayers in living cells are not necessarily at equilibrium, so equilibrium constraints such as equality of chemical potentials are not necessarily applicable.

This work considers practical approaches for developing initial conditions for simulations of asymmetric bilayers in the absence of precise experimental guidance. Table 1 lists the methods that are broadly classified as lipid based, leaflet based, and bilayer based. They are described in detail and related to theory in Approaches for generating asymmetric bilayers in the Materials and methods. Readers interested in the detailed theory are referred to Theory in the same section. Two asymmetric bilayers were simulated. The first is composed of 1,2-dioleyl-sn-3-glycero-phosphoethanolamine (DOPE), dipalmitoyl-sn-3-glycero-phosphocholine (DPPC), and Chol, and the second consists of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE), palmitoylsphingomyelin (PSM), and Chol. In each bilayer, the saturated lipid (DPPC or PSM) was enriched in one leaflet and unsaturated PE was enriched in the other. The bilayers were assembled using the area per lipid (APL)-based and SA-based methods noted in Table 1 and equilibrated in P1 periodic boundary conditions (PBC). For the P2₁ approach, the APL- and SA-based systems were re-equilibrated in P2₁ PBC (39) (in which selected lipids effectively switch leaflets in accordance with transformations using P2₁ symmetry), and then the simulations with P1 PBC using P2₁-equilibrated systems were performed for mechanical property calculations. Zero leaflet tension (0-DS) bilayers were also prepared, whose lipid composition was chosen using the method proposed by Doktorova and Weinstein (40) from the bilayers generated by the APL-based method. Mechanical properties of asymmetric membranes (such as area compressibility modulus, differential stress, and torque densities) were calculated from the lateral pressure profiles and compared for the assessment. Significant differences are obtained for the methods before simulating in P2₁, and the subsequent P1 simulations after P2₁ equilibration establish a better approach.

Table 1.

Simulation methods for generating asymmetric bilayers with principal assumptions

Class	Example	Procedure	Assumptions	References
Lipid based	APL	combine APL from homogeneous bilayers of each component	ideal mixing, no interleaflet coupling	(57) (58) (59) (60)
Leaflet based	SA	equilibrate symmetric bilayers, each of a different composition; combine one leaflet from each bilayer, matching SAs	zero differential strain, no interleaflet coupling	(61) (62)
Bilayer based	0-DS	adjust lipid composition of leaflets such that surface tension is zero in both leaflets	leaflet surface tensions equal to zero	(40)
	equal chemical potential (APL/P21, SA/P21, and 0-DS/P21)	assemble asymmetric bilayer with APL, SA, or 0-DS; equilibrate with P21 boundary conditions by restraining location of selected components	equal chemical potentials of nonrestrained components	this work

Materials and methods

Theory

This subsection is based on the framework for describing asymmetric bilayers developed by HD (36), where the membrane is assumed to be in a curvature-relaxed state with a spontaneous curvature $c_{0,\text{b}}^{\ast }$. One of the main findings from the HD work is that leaflet tensions from differential area strains do not necessarily vanish in a curvature-relaxed asymmetric bilayer with net zero bilayer tension. Here, we consider the leaflet tensions from both the bending and area strain, which is an extension of the HD work.

We start with definitions of the bilayer and the differential stresses from the leaflet tensions (Eq. 1) and provide the relations between the lateral pressure profile and tension-torque densities (Eqs. 2, (3a), (3b), 4, and 5). Then, we proceed with a brief review of the HD framework ((6a), (6b), (7a), (7b), (7c), (8a), (8b), (8c), 9, and 10) and obtain the bilayer and the differential stresses ((11a), (11b), (12a), (12b), and (13a), (13b)), which include the contributions from both the bending and area strain. We discuss the (constant) intrinsic tensions for net zero tension at a reference bilayer curvature, c_r (Eqs. 14 and (15b), (15a)). Subsequently, we show that the bilayer and differential stresses are formally expressed as their residuals with respect to those at the reference point ((16a), (16b)). Finally, for a physically plausible reference point, c_r = $c_{0,\text{b}}^{\ast }$, we derive a general expression for the differential stress in planar bilayers with P1 PBC from simulations ((17a), (17b), (17c)) and connect the differential stress to torque densities (Eqs. 20 and 21). The theory below is applied to planar membranes with P1 PBC, though it may be applicable to other geometries (e.g., membranes in the inverse hexagonal phase). A list and description of symbols is provided in the Glossary.

Here, we restrict ourselves to asymmetric bilayers without applied surface tension. In addition, we assume that the monolayers share a common midplane, along which they can slide with respect to each other. The bilayer is composed of monolayers m1 (upper leaflet) and m2 (lower leaflet) with bending moduli ($k_{\text{m}1}^{\text{b}}$, $k_{\text{m}2}^{\text{b}}$) and spontaneous curvatures associated with bending ($c_{0,\text{m}1}^{\text{b}}$,$c_{0,\text{m}2}^{\text{b}}$). The area compressibility moduli of the monolayers are K_A,m1 and K_A,m2, and the area strains due to asymmetric lipid packing are ɛ_m1,0 and ɛ_m2,0. From these area strains, another set of spontaneous curvatures ($c_{0,\text{m}1}^{\text{s}}$, $c_{0,\text{m}2}^{\text{s}}$) arise. These spontaneous curvatures are generalization of the spontaneous curvature (c₀) from the bending energy in the Helfrich Hamiltonian (41,42). Following the usual convention, a monolayer with a positive curvature is convex to the headgroup plane and vice versa. The bilayer curvature follows the same convention as the upper leaflet. The leaflet tensions in the monolayers are denoted γ_m1 and γ_m2. Hereafter, we will use subscripts m (as well as m1 or m2) and b to indicate various properties of monolayers and bilayers, respectively. The superscripts b and s denote the various properties from bending and area strain, respectively.

The bilayer tension (Σ) and differential stress (Δ) can be defined by the sum and difference of the monolayer tensions:

$$\text{Σ}\equiv {\gamma }_{\text{m}1}+{\gamma }_{\text{m}2}\text{and}\text{Δ}\equiv {\gamma }_{\text{m}1}-{\gamma }_{\text{m}2},$$ (1)

where Σ reflects the symmetric component and Δ captures the effects of asymmetry in γ_m. When Σ = 0, γ_m1 = −γ_m2 = Δ/2. To quantify the asymmetry in leaflet tensions of the asymmetric bilayer, HD denoted γ_m as the “differential stress.” In this work, we define the differential stress as the tension difference between leaflets (Eq. 1), which is generalizable to cases of applied surface tension.

The bilayer tension Σ and torque density ${\overline{F}}_{\text{b}}^{\prime }$(0) are obtained from the lateral pressure profile, p(z) = p_T(z) − p_N(z), where p_T(z) = [p_xx(z) + p_yy(z)]/2 and p_N(z) = p_zz(z) are the tangential and normal components of the pressure tensor: Σ = −$\int$dz p(z) and ${\overline{F}}_{\text{b}}^{\prime }\left(0\right)=-\int dzzp\left(z\right)$. Because p(z) vanishes in the bulk water due to the isotropy (p_xx = p_yy = p_zz), the leaflet tension and torque density from a simulation can be written as

$${\gamma }_{\text{m}1}=-\underset{0}{\overset{L_z/2}{\int }}dzp\left(z\right)\text{and}{\gamma }_{\text{m}2}=-\underset{-L_z/2}{\overset{0}{\int }}dzp\left(z\right),$$ (2)

$${\overline{F}}_{\text{b}}^{\prime }\left(0\right)=-\underset{-L_z/2}{\overset{L_z/2}{\int }}dzzp\left(z\right)={\overline{F}}_{\text{m}1}^{\prime }\left(0\right)-{\overline{F}}_{\text{m}2}^{\prime }\left(0\right),$$ (3a)

$${\overline{F}}_{\text{m}1}^{\prime }\left(0\right)=-\underset{0}{\overset{L_z/2}{\int }}dzzp\left(z\right)\text{and}{\overline{F}}_{\text{m}2}^{\prime }\left(0\right)=\underset{-L_z/2}{\overset{0}{\int }}dzzp\left(z\right),$$ (3b)

where L_z is the system size along the z dimension, ${\overline{F}}_{\text{b}}^{\prime }$(0) is the bilayer torque density, and z = 0 is the bilayer center. A nonvanishing $\overline{F}$′(0) implies that a membrane would bend toward different directions depending on its sign in the absence of PBC.

For a flat membrane in the absence of area strain from asymmetric lipid packing (ɛ_m1,0 = ɛ_m2,0 = 0), the bilayer torque density ${\overline{F}}_{\text{b}}^{\prime }$(0) is connected to the bilayer bending modulus $\left(k_{\text{b}}^{\text{b}}\right)$ and its associated spontaneous curvature $\left(c_{0,\text{b}}^{\text{b}}\right)$ (43):

$$k_{\text{b}}^{\text{b}}{c}_{0,\text{b}}^{\text{b}}=-{\overline{F}}_{\text{b}}^{\prime }\left(0\right).$$ (4)

When there exist area strains from asymmetric lipid packing, ɛ_m1,0 ≠ 0 and/or ɛ_m2,0 ≠ 0, Eq. 4 does not hold because ${\overline{F}}_{\text{b}}^{\prime }$(0) includes the contribution from the area strain (see Eq. 18c below). For a symmetric membrane, ɛ_m1,0 = ɛ_m2,0 = 0; hence, an analogous connection holds for the monolayer spontaneous curvature (43)

$$k_{\text{m}}^{\text{b}}{c}_{0,\text{m}}^{\text{b}}=-{\overline{F}}_{\text{m}}^{\prime }\left(0\right).$$ (5)

Hereafter, we will use more compact notations for the torque densities, ${\overline{F}}_{\text{b}}^{\prime }$ and ${\overline{F}}_{\text{m}}^{\prime }$, for brevity.

Below, we define elastic energy for leaflet’s reference surfaces, whose distances from the bilayer midplane are z_m1 and z_m2 for monolayers m1 and m2, respectively. Typically, the reference surface is chosen to be either the neutral or pivotal plane of the monolayer. At the neutral surface, the bending and stretching modes are decoupled (44), whereas the area strain vanishes at the pivotal plane (45). These surfaces do not coincide, and the bending modulus defined for the pivotal plane can be significantly larger than that for the neutral surface, though the spontaneous curvature changes relatively little between them (46). The energy densities (for the monolayer reference surfaces) are then expressed for the bilayer midplane using the parallel surface theorem (47), from which HD considered the total energy densities to obtain a modified spontaneous curvature of the asymmetric bilayer.

The elastic energy densities (per unit area of each monolayer) from bending $\left({\text{e}}_{\text{m}}^{\text{b}}\right)$ and area strains $\left({\text{e}}_{\text{m}}^{\text{s}}\right)$ for an asymmetric bilayer are given by (up to a constant)

$${\text{e}}_{\text{m}1}^{\text{b}}=k_{\text{m}1}^{\text{b}}{\left(c_{\text{m}1}-c_{0,\text{m}1}^{\text{b}}\right)}^2/2{\text{e}}_{\text{m}2}^{\text{b}}=k_{\text{m}2}^{\text{b}}{\left(c_{\text{m}2}+c_{0,\text{m}2}^{\text{b}}\right)}^2/2$$ (6a)

and

$${\text{e}}_{\text{m}}^{\text{s}}=K_{A,\text{m}}{\epsilon }_{\text{m}}^2/2,$$ (6b)

where c_0,m and ɛ_m are the spontaneous curvature and the area strain in the monolayer m, respectively. In this work, the topological Gaussian curvature and the coupling between leaflets are not considered. Hereafter, the energy densities are shown up to a constant.

From the parallel surface theorem with an assumption of weak curvature (|cz_m| << 1, thus c_m1 = c = c_m2 up to the linear order in c), the bending and area strain energy densities given in (6a), (6b) can be rewritten for the bilayer midplane. The energy density (per unit bilayer area) from bending is

$${\text{e}}_{\text{b}}^{\text{b}}\approx \frac{k_{\text{m}1}^{\text{b}}}{2}{\left(c-c_{0,\text{m}1}^{\text{b}}\right)}^2+\frac{k_{\text{m}2}^{\text{b}}}{2}{\left(c+c_{0,\text{m}2}^{\text{b}}\right)}^2=\text{const}.+\frac{k_{\text{b}}^{\text{b}}}{2}{\left(c-c_{0,\text{b}}^{\text{b}}\right)}^2,$$ (7a)

$$k_{\text{b}}^{\text{b}}=k_{\text{m}1}^{\text{b}}+k_{\text{m}2}^{\text{b}},$$ (7b)

and

$$c_{0,\text{b}}^{\text{b}}=\frac{k_{\text{m}1}^{\text{b}}{c}_{0,\text{m}1}^{\text{b}}-k_{\text{m}2}^{\text{b}}{c}_{0,\text{m}2}^{\text{b}}}{k_{\text{b}}^{\text{b}}},$$ (7c)

which is minimized at a bilayer spontaneous curvature, c = $c_{0,\text{b}}^{\text{b}}$.

Because the monolayers share a common midplane, the stress from area strains would be relaxed by bending (analogous to a bimetallic strip under thermal stress) (36). With an additional assumption of weak area strain (|ɛ_m,0| << 1), one can obtain the nonlocal curvature elastic energy density from the area strain as (see Appendix A for details)

$${\text{e}}_{\text{nl}}^{\text{s}}\approx \frac{k_{\text{m}1}^{\text{s}}}{2}{\left(\overline{c}+c_{0,\text{m}1}^{\text{s}}\right)}^2+\frac{k_{\text{m}2}^{\text{s}}}{2}{\left(\overline{c}-c_{0,\text{m}2}^{\text{s}}\right)}^2=\text{const}.+\frac{k_{\text{b}}^{\text{s}}}{2}{\left(\overline{c}-c_{0,\text{b}}^{\text{s}}\right)}^2,$$ (8a)

$$k_{\text{b}}^{\text{s}}=k_{\text{m}1}^{\text{s}}+k_{\text{m}2}^{\text{s}},$$ (8b)

and

$$c_{0,\text{b}}^{\text{s}}=\frac{k_{\text{m}2}^{\text{s}}{c}_{0,\text{m}2}^{\text{s}}-k_{\text{m}1}^{\text{s}}{c}_{0,\text{m}1}^{\text{s}}}{k_{\text{b}}^{\text{s}}},$$ (8c)

where $\overline{c}=\underset{S}{\int }$dAc/A is the average curvature over a surface S and $\overline{c}=-c_{0,\text{m}1}^{\text{s}}$ and $\overline{c}=c_{0,\text{m}2}^{\text{s}}$ are the curvatures at which the area strain in the monolayers m1 and m2 vanishes, respectively (see (A3a), (A3b), (A3c)). Here, $k_{\text{m}}^{\text{s}}=K_{A,\text{m}}{z}_{\text{m}}^2$ is the nonlocal bending modulus of the monolayer, m. With an additional assumption z_m1 ≈ z_m2 ≈ z₀, the nonlocal bending modulus can be simplified to $k_{\text{b}}^{\text{s}}=K_A{z}_0^2$, where K_A = K_A,m1 + K_A,m2 is the bilayer area compressibility modulus (48). As one can see from Eqs. 8c and A3c, $c_{0,\text{b}}^{\text{s}}$ is characterized by lipid packing in the monolayers, unlike $c_{0,\text{b}}^{\text{b}}$, which is an intrinsic material property. HD obtained the right-hand side of Eq. 8a by assuming that the differential area strains vanish at a bilayer curvature c = $c_{0,\text{b}}^{\text{s}}$ (Eq. 15 in (36)); we relaxed this assumption in our derivation.

From the energy densities (7a and 8a), the total energy density can be expressed as the sum of those from bending and stretching by choosing reference surfaces of monolayers to be neutral planes,

$${\text{e}}_{\text{b}}\left(c,\overline{c}\right)\approx \frac{k_{\text{b}}^{\text{b}}}{2}{\left(c-c_{0,\text{b}}^{\text{b}}\right)}^2+\frac{k_{\text{b}}^{\text{s}}}{2}{\left(\overline{c}-c_{0,\text{b}}^{\text{s}}\right)}^2.$$ (9)

For a constant mean curvature surface (i.e., c = $\overline{c}$ over the whole membrane), e_b is minimized at a modified spontaneous curvature

$$c_{0,\text{b}}^{\ast }=\frac{k_{\text{b}}^{\text{b}}{c}_{0,\text{b}}^{\text{b}}+k_{\text{b}}^{\text{s}}{c}_{0,\text{b}}^{\text{s}}}{k_{\text{b}}^{\text{b}}+k_{\text{b}}^{\text{s}}},$$ (10)

which would not change upon bending and/or stretching (i.e., $c_{0,\text{b}}^{\text{b}}$ is a material property and $c_{0,\text{b}}^{\text{s}}$ remains constant for a given asymmetric bilayer; see Eq. 8c and A3c).

Below, we revisit the bilayer and differential tensions, Σ and Δ, for which we consider the contributions from both the bending and area strain. Then, we show that the bilayer and differential tensions (which are observable) can be formally expressed as their residuals with respect to tensions at a reference point for net zero tension. It follows the choice of $c_{0,\text{b}}^{\ast }$ as the reference point.

The bilayer tension and differential stress from the bending are obtained from Eq. 1 by noting that ${\gamma }_{\text{m}}^{\text{b}}={\text{e}}_{\text{m}}^{\text{b}}=\partial E_{\text{m}}^{\text{b}}$/∂A_m (49,50):

$${\text{Σ}}^{\text{b}}\left(c\right)\approx \frac{k_{\text{m}1}^{\text{b}}}{2}{\left(c-c_{0,\text{m}1}^{\text{b}}\right)}^2+\frac{k_{\text{m}2}^{\text{b}}}{2}{\left(c+c_{0,\text{m}2}^{\text{b}}\right)}^2$$ (11a)

and

$${\text{Δ}}^{\text{b}}\left(c\right)\approx \frac{k_{\text{m}1}^{\text{b}}}{2}{\left(c-c_{0,\text{m}1}^{\text{b}}\right)}^2-\frac{k_{\text{m}2}^{\text{b}}}{2}{\left(c+c_{0,\text{m}2}^{\text{b}}\right)}^2.$$ (11b)

Σ^b(c = 0) in Eq. 11a can be understood as the spontaneous (bilayer) tension arising from the bending at a weak curvature (|c| << 1) (49,50). The concept of the spontaneous tension has been successfully applied to describe the shape and stability of membranes, which was obtained as $k_{\text{b}}^{\text{b}}{c_{0,\text{b}}^{\text{b}}}^2$/2 from the right-hand side of Eq. 7a (by neglecting the constant term). Though useful, the spontaneous (bilayer) tension does not provide information on the differential stress from the bending in an asymmetric bilayer. The differential stress defined in Eq. 11b augments the spontaneous (bilayer) tension, which describes the effects of asymmetry in $k_{\text{m}}^{\text{b}}$ and $c_{0,\text{m}}^{\text{b}}$.

The contributions from area strain in Σ and Δ are obtained from ${\gamma }_{\text{m}}^{\text{s}}$ = K_A,mɛ_m, Eq. 1, and (A3a), (A3b), (A3c):

$${\text{Σ}}^{\text{s}}\left(c\right)\approx \frac{k_{\text{m}1}^{\text{s}}-k_{\text{m}2}^{\text{s}}}{z_0}c+\frac{k_{\text{m}1}^{\text{s}}{c}_{0,\text{m}1}^{\text{s}}+k_{\text{m}2}^{\text{s}}{c}_{0,\text{m}2}^{\text{s}}}{z_0}$$ (12a)

and

$${\text{Δ}}^{\text{s}}\left(c\right)\approx \frac{k_{\text{b}}^{\text{s}}\left(c-c_{0,\text{b}}^{\text{s}}\right)}{z_0},$$ (12b)

where the second term in the right-hand side of Eq. 12a corresponds to the bilayer tension from the area strain at c = 0. The area strains do not necessarily vanish, but the differential stress from area strains vanishes at c = $c_{0,\text{b}}^{\text{s}}$ (Eq. 12b).

From (11b), (11a) and (12b), (12a) with a consideration of the constant term in the energy densities, the bilayer and differential stresses can be written as

$$\text{Σ}={\text{Σ}}^{\text{b}}+{\text{Σ}}^{\text{s}}+{\text{Σ}}^{\text{I}}$$ (13a)

and

$$\text{Δ}={\text{Δ}}^{\text{b}}+{\text{Δ}}^{\text{s}}+{\text{Δ}}^{\text{I}},$$ (13b)

where Σ^I and Δ^I are the (constant) intrinsic tensions, whose values can be determined from the requirement of the zero tension at a reference point (bilayer curvature), c = c_r,

$$\text{Σ}\left(c=c_{\text{r}}\right)=0\text{and}\text{Δ}\left(c=c_{\text{r}}\right)=0.$$ (14)

Inserting (13a), (13b) into Eq. 14, Σ^I and Δ^I are obtained as

$${\text{Σ}}^{\text{I}}=-{\text{Σ}}^{\text{b}}\left(c=c_{\text{r}}\right)-{\text{Σ}}^{\text{s}}\left(c=c_{\text{r}}\right)$$ (15a)

and

$${\text{Δ}}^{\text{I}}=-{\text{Δ}}^{\text{b}}\left(c=c_{\text{r}}\right)-{\text{Δ}}^{\text{s}}\left(c=c_{\text{r}}\right).$$ (15b)

From (13b), (13a) and (15b), (15a), we find that the net tensions can be formally expressed as

$$\text{Σ}\left(c\right)={\text{Σ}}^{\text{res}}\left(c|c_{\text{r}}\right)\equiv \text{Σ}\left(c\right)-\text{Σ}\left(c=c_{\text{r}}\right)$$ (16a)

and

$$\text{Δ}\left(c\right)={\text{Δ}}^{\text{res}}\left(c|c_{\text{r}}\right)\equiv \text{Δ}\left(c\right)-\text{Δ}\left(c=c_{\text{r}}\right),$$ (16b)

where Σ^res(c|c_r) and Δ^res(c|c_r) are the residual bilayer and differential tensions with respect to those at the reference curvature, c_r. We note that Σ and Δ at a given bilayer curvature vary with different choices of c_r. For a symmetric bilayer, c_r = 0 is an obvious choice because of the symmetry. However, for an asymmetric bilayer, the choice of c_r requires more careful consideration.

Among possible choices, c_r = $c_{0,\text{b}}^{\ast }$ is physically appealing and is supported as follows. At the spontaneous curvature, the bilayer energy density is minimized (i.e., the membrane is stable). This implies that the net torque in the bilayer vanishes and so does the differential stress. Without applied surface tension, the bilayer tension in the membrane should vanish as well. Thus, Eq. 14 is naturally satisfied at c = $c_{0,\text{b}}^{\ast }$ (Fig. 1, column 2). This reference point was also chosen by HD in their consideration of an SA-matched asymmetric bilayer in P1 PBC (36), in which they noted that the additional tension to force the bilayer to be flat (required by P1 PBC) is the residual tension with respect to that at c = $c_{0,\text{b}}^{\text{∗}}$. In the presence of the applied tension, the net zero bilayer tension may not be satisfied, and the lateral packing in the monolayers may change as well. However, the new reference point still could be chosen to be the (new) spontaneous curvature of the bilayer for the same reason (net zero torque in the curvature-relaxed state).

Figure 1.

Schematic description of the tensions from simulations. From two symmetric bilayers $\mathcal{S}$m1 and $\mathcal{S}$m2 (first column), two patches of monolayers m1 and m2 are taken whose areas are A_m1,0 and A_m2,0, respectively. An asymmetric bilayer generated by the assembly of these patches would be in its curvature-relaxed state (c = $c_{0,\text{b}}^{\ast }$) in the absence of P1 PBC (second column). From the force and torque balance conditions at c = $c_{0,\text{b}}^{\ast }$, the net tensions (Σ and Δ) vanish; thus, Σ^I = −Σ^b$\left(c_{0,\text{b}}^{\ast }\right)-{\text{Σ}}^{\text{s}}\left(c_{0,\text{b}}^{\ast }\right)$ and Δ^I = −Δ^b$\left(c_{0,\text{b}}^{\ast }\right)-{\text{Δ}}^{\text{s}}\left(c_{0,\text{b}}^{\ast }\right)$ in this bilayer (see (15a), (15b)). When a P1 PBC is applied, the bilayer is forced to be planar (c = 0) (third column). The tension imposed to the monolayer via the P1 PBC is the residual differential leaflet tension, ${\gamma }_{\text{m}}^{\text{res}}\equiv {\gamma }_{\text{m}}\left(c=0\right)-{\gamma }_{\text{m}}\left(c=c_{0,\text{b}}^{\ast }\right)$. Because γ_m(c = $c_{0,\text{b}}^{\ast }$) = 0, ${\gamma }_{\text{m}}^{\text{res}}$ would be equal to γ_m from a simulation. In other words, Δ₀$\equiv$ Δ(c = 0) from a simulation can be estimated as a sum of the residual differential stresses, ${\text{Δ}}_0^{\text{b},\text{res}}\equiv {\text{Δ}}^{\text{b},\text{res}}\left(c=0|c_{\text{r}}=c_{0,\text{b}}^{\ast }\right)$ and ${\text{Δ}}_0^{\text{s},\text{res}}\equiv {\text{Δ}}^{\text{s},\text{res}}\left(c=0|c_{\text{r}}=c_{0,\text{b}}^{\ast }\right)$ ((17a), (17b), (17c)). To see this figure in color, go online.

Hereafter, we will focus on the differential stress in a planar membrane (c = 0) from simulations with P1 PBC, Δ₀ ≡ Δ^res(c = 0|c_r = $c_{\text{b}}^{\ast }$), where the effects of asymmetry on the surface tensions are reflected. From (13b), (13a) and (16b), (16a), the differential stress, Δ₀, can be written as

$${\text{Δ}}_0={\text{Δ}}_0^{\text{b},\text{res}}+{\text{Δ}}_0^{\text{s},\text{res}},$$ (17a)

where ${\text{Δ}}_0^{\text{b},\text{res}}\equiv {\text{Δ}}^{\text{b},\text{res}}\left(c=0|c_{\text{r}}=c_{0,\text{b}}^{\ast }\right)$ and ${\text{Δ}}_0^{\text{s},\text{res}}\equiv {\text{Δ}}^{\text{s},\text{res}}\left(c=0|c_{\text{r}}=c_{0,\text{b}}^{\ast }\right)$ are given by

$${\text{Δ}}_0^{\text{b},\text{res}}\approx -\frac{k_{\text{m}1}^{\text{b}}-k_{\text{m}2}^{\text{b}}}{2}{c_{0,\text{b}}^{\text{∗}}}^2+\left(k_{\text{m}1}^{\text{b}}{c}_{0,\text{m}1}^{\text{b}}+k_{\text{m}2}^{\text{b}}{c}_{0,\text{m}2}^{\text{b}}\right)c_{0,\text{b}}^{\text{∗}}$$ (17b)

and

$${\text{Δ}}_0^{\text{s},\text{res}}\approx -\frac{k_{\text{b}}^{\text{s}}{c}_{0,\text{b}}^{\ast }}{z_0}.$$ (17c)

Although (17a), (17b), (17c) is general, its direct application can be challenging (because of difficulties in accurate measurements of the elastic moduli and their associated spontaneous curvatures, especially for the bending). Hence, it is desired to have more practically applicable expression for Δ₀. We note that HD obtained such an expression for ${\text{Δ}}_0^{\text{s},\text{res}}$ (Eq. 26 in (36)) in a self-consistent way by introducing a destressing correction to the pressure profile. Below, we explain their self-consistent approach and obtain an expression for Δ₀. In addition, we obtain an alternative expression for Δ₀ directly from the destressing correction, which involves only torque densities, thicknesses, and the bilayer area compressibility modulus.

To obtain ${\text{Δ}}_0^{\text{s},\text{res}}$, HD introduced a destressing correction to the pressure profile, p(z), where the stress from the asymmetric lipid packing $\left({\gamma }_{\text{m}}^{\text{s},\text{res}}\right)$ is removed from p(z) with an assumption of evenly acting γ_m over the thickness (d_m) of the monolayer m:

$$p_0\left(z\right)\approx p\left(z\right)+\{\begin{array}{l}{\gamma }_{\text{m}1}^{\text{s},\text{res}}/d_{\text{m}1}(0<z<d_{\text{m}1})\\ {\gamma }_{\text{m}2}^{\text{s},\text{res}}/d_{\text{m}2}(-d_{\text{m}2}<z<0)\end{array}.$$ (18a)

Here, ${\gamma }_{\text{m}1}^{\text{s},\text{res}}=\left({\text{Σ}}^{\text{s},\text{res}}+{\text{Δ}}^{\text{s},\text{res}}\right)$/2 and ${\gamma }_{\text{m}2}^{\text{s},\text{res}}=\left({\text{Σ}}^{\text{s},\text{res}}-{\text{Δ}}^{\text{s},\text{res}}\right)$/2. The first moment of p₀(z) is given by

$$\int dzz{p}_0\left(z\right)\approx \int dzzp\left(z\right)+{\text{Δ}}_0^{\text{s},\text{res}}\frac{d_{\text{b}}}{4},$$ (18b)

where d_b = d_m1 + d_m2 is the bilayer thickness, and the contribution from the bilayer tension vanishes in the second term in the right-hand side because of the symmetry. Assuming that p₀(z) is a good approximation to the pressure profile from the bilayer without area strain ($c_{0,\text{b}}^{\text{s}}$ = 0), Eq. 18b can be re-expressed by combining with (3b), (3a) and 4 as

$$k_{\text{b}}^{\text{b}}{c}_{0,\text{b}}^{\text{b}}\approx -{\overline{F}}_{\text{b}}^{\prime }+{\text{Δ}}_0^{\text{s},\text{res}}\frac{d_{\text{b}}}{4}.$$ (18c)

From Eq. 18c, ${\text{Δ}}_0^{\text{s},\text{res}}$ is self-consistently obtained by combining with Eqs. 17c and 10

$${\text{Δ}}_0^{\text{s},\text{res}}\approx \frac{{\overline{F}}_{\text{b}}^{\prime }-k_{\text{b}}^{\text{s}}{c}_{0,\text{b}}^{\text{s}}}{\left(1+k_{\text{b}}^{\text{b}}/k_{\text{b}}^{\text{s}}\right)z_0+d_{\text{b}}/4}.$$ (19)

Noting that $c_{0,\text{b}}^{\ast }=-z_0{\text{Δ}}_0^{\text{s},\text{res}}/k_{\text{b}}^{\text{s}}$, Δ₀ is obtained from (17a), (17b), (17c), 19, and 5

$${\text{Δ}}_0\approx \left(\frac{{\overline{F}}_{\text{m}1,\text{s}}^{\prime }+{\overline{F}}_{\text{m}2,\text{s}}^{\prime }}{k_{\text{b}}^{\text{s}}/z_0}+1\right){\text{Δ}}_0^{\text{s},\text{res}}-\frac{k_{\text{m}1}^{\text{b}}-k_{\text{m}2}^{\text{b}}}{2}{\left(\frac{z_0{\text{Δ}}_0^{\text{s},\text{res}}}{k_{\text{b}}^{\text{s}}}\right)}^2,$$ (20)

where ${\overline{F}}_{\text{m}1,\text{s}}^{\prime }$ and ${\overline{F}}_{\text{m}2,\text{s}}^{\prime }$ are torque densities for monolayers, m1 and m2, in their stress-free state.

Alternatively, a different expression for Δ₀ can be obtained directly from the destressing correction

$${\text{Δ}}_0\approx \frac{4}{d_{\text{b}}}\left(\frac{{\overline{F}}_{\text{m}1,\text{s}}^{\prime }+{\overline{F}}_{\text{m}2,\text{s}}^{\prime }}{k_{\text{b}}^{\text{s}}/z_0}+1\right)\left[{\overline{F}}_{\text{b}}^{\prime }-\left({\overline{F}}_{\text{m}1,\text{s}}^{\prime }-{\overline{F}}_{\text{m}2,\text{s}}^{\prime }\right)\right],$$ (21)

which is practically parameter free because z₀ ≈ d_h/2, d_b, and $k_{\text{b}}^{\text{s}}\approx K_A{z}_0^2$ can be easily obtained and ${\overline{F}}_{\text{m}1,\text{s}}^{\prime }$ and ${\overline{F}}_{\text{m}2,\text{s}}^{\prime }$ can be obtained from cognate symmetric bilayers. Provided that the destressing correction is accurate, Eqs. 20 and 21 are expected to be good approximations for weakly coupled bilayers. The Results compare the differential stress from asymmetric bilayers generated by different approaches with the predictions from Eqs. 20 and 21.

It is important to close this section by noting that any treatment of asymmetric elastic moduli and spontaneous curvatures requires subtle assumptions. Here, we assumed the additivity of the leaflet elastic moduli and chose the reference bilayer curvature, c_r = $c_{0,\text{b}}^{\text{∗}}$ for net zero tension from the force and torque balances for a curvature-relaxed state. When leaflet coupling becomes strong, the additivity assumption breaks down. Furthermore, the choice of c_r becomes more uncertain than for a bilayer of weakly coupled leaflets, and the tensions at c = 0 will vary with reference point (see (16a), (16b)). Thus, the theory is subject to different and reasonable interpretations, and future refinements are to be expected.

P2₁ PBC

To facilitate lipid exchange between leaflets, the P2₁ PBC was implemented in Chemistry at Harvard Macromolecular Mechanics (CHARMM) (39,51). As sketched in Fig. 2A, a lipid exiting one leaflet re-enters the opposite one via an orthogonal face. This is effected in P2₁(−x, y + 1/2, −z) with a primary cell (simulation box) rotated by 45° in the xy plane and translated by $\sqrt{2}$a/4, which leads to an up-down checkerboard arrangement of cells (Fig. 2B); a is the x length of the primary cell (before rotation in the xy plane). In the original implementation, the requirement of the entire unit cell in the particle mesh Ewald method (52) for electrostatic interactions adds 20–30% more computational time on a single node, and parallel scalability is poor. Recently, Prasad et al. have developed a modification of P2₁ PBC that is parallelizable (53). The modified P2₁ PBC represent the P2₁11 space group, i.e., the symmetry operation along the x axis is 2₁ with only translational symmetry along the y and z axes. The scalability of modified P2₁ PBC is comparable to that for P1 simulations.

Figure 2.

(A) Representation of P2₁ periodic boundary conditions (PBCs) for bilayer simulations. The spheres represent lipid headgroups, with those in the upper leaflet light blue and blue and in the bottom leaflet pink and red. The image of the blue particle is red, and the arrows show the direction of exit and entry for these particles. (B) The checkerboard arrangement of cells from P2₁(−x, y + 1/2, −z) transformation, looking down the bilayer normal (z axis) to the bilayer surface (xy plane), where the bilayer center is positioned at z = 0. The primary cell (asymmetric unit) is light blue and located in the center; the four particles (shown in red, green, blue, and orange circles) in the corners of the cell reside on the top leaflet. In the adjacent cells (pink), the same four particles are in the bottom leaflet and rotated with respect to the primary cell. The outer light blue cells are the result of the second transformation, and the four particles return to the top leaflet. The unit cell that generates the entire crystal by three dimensional translations is marked as a cyan rectangle, which contains cells equivalent to the primary cell and its flipped image (one of the pink cells from the P2₁ symmetry operation). To see this figure in color, go online.

The initial demonstrations of the P2₁ PBC included rebalancing the composition of a DPPC bilayer with 40 lipids on one side and 32 on the other and of an imbalanced DOPC bilayer with melittin in one leaflet (39). Here, the imposition of equal chemical potentials for the lipids (rapidly equalized by facilitated lipid flip-flop) is easily justifiable. Lipid flip-flop can also be accelerated by a transient pore in P1 simulations (54). Both P2₁ PBC and the transient pore method in P1 PBC rely on the lateral diffusion of lipids, and thus, the acceleration in flip-flop is expected to be comparable. However, mechanical properties of the bilayer are changed by the pore. Hence, even if the leaflets are equilibrated when the pore is in place, they may not be equilibrated when the pore is closed. On the other hand, P2₁ PBCs naturally allow equilibrated leaflets without perturbing the membrane by a transient pore.

The application of P2₁ to modeling asymmetric cell membranes requires additional considerations. In cell membranes, specific lipid types are restricted to one leaflet and others may be asymmetrically distributed in both leaflets (1, 2, 3). The protein-facilitated lipid asymmetry is dynamically regulated on a much shorter timescale (approximately microseconds to milliseconds) (55,56) than spontaneous flip-flop (approximately days) (37,38). Some lipid types (such as Chol) can readily flip-flop. Thus, the bilayer is likely in a steady state, and it is plausible that flip-flopping components stay in equilibrium between leaflets (partial chemical equilibrium). Assuming this, the regulation of lipid asymmetry is qualitatively modeled by imposing a restraining potential for specific lipid types to keep them in particular leaflet while facilitating interleaflet redistribution (i.e., flip-flop) of the other lipids by P2₁ PBC.

Approaches for generating asymmetric bilayers

This subsection describes each method in Table 1 and its connection to the theoretical framework described above. The lipid-based approach is the simplest one, in which SAs of m1 and m2 are matched using the APLs from homogeneous lipid bilayers. Referred to here as the APL approach, it contains the severe assumption of ideal mixing and ignores interleaflet coupling. Nevertheless, it is easy to use and available as a default option in Membrane Builder in CHARMM-GUI (57, 58, 59, 60), in which the modifications of component APLs to model mixtures are an option.

Other approaches aim to keep assumptions that hold for a symmetric bilayer. These are either optimal lipid packing or zero leaflet tension. In other words, these approaches try to minimize either differential area strain, ɛ_m ≡ A_m/A_m,0 − 1 (61,62), or differential stress, Δ (40), between leaflets. Here, A_m and A_m,0 are SAs of the monolayer m in its given and tensionless states, respectively.

Removing area strain is a leaflet-based approach. To satisfy ɛ_m = 0, the individual surface areas of components from cognate symmetric bilayers are used to match leaflet SAs without considering interleaflet interactions. Then, an asymmetric bilayer can be generated either by taking the SA-matched leaflets or de novo using the determined numbers of lipids in the leaflet. The latter can be an easier preparation protocol, for which Membrane Builder can be used. This approach is denoted SA and is illustrated in the second column of Fig. 3 for the two systems simulated here. Although the curvature due to area strain vanishes ($c_{0,\text{b}}^{\text{s}}$ = 0), there is still a potential contribution from the planar geometry constraint (imposed by the P1 PBC) as sketched in Fig. 1. The resulting Δ is given by (17a), (17b), (17c) with $c_{0,\text{b}}^{\text{s}}$ = 0.

Figure 3.

Snapshots from each stage of generation of asymmetric bilayers for (A) PE/PC/Chol and (B) PE/SM/Chol (see Table 2 for the system information). Component APLs from 1-μs simulations of symmetric bilayers ($\mathcal{S}$m1 and $\mathcal{S}$m2, first column) were used to match SAs of leaflets m1 and m2 to determine the numbers of lipids for generation of initial configuration of (SA). Initial configurations generated by CHARMM-GUI Membrane Builder (57, 58, 59, 60) for (APL) and (SA) (second column) were simulated over 1 μs, whose snapshots chosen between 500 ns and 1 μs (third column) were subject to subsequent P2₁ simulations. Once compositions in P2₁ simulations reached a steady state (within ∼100 ns), representative configurations were simulated over 0.5 μs with the P1 PBC (fourth column). Initial configurations for (0-DS) were prepared with the adjusted N_m1 from $\mathcal{A}$(APL) using Eq. 22, but not shown here. For clarity, only heavy atoms in bilayers (lines) and phosphorous atoms in lipids and oxygen atoms in Chols (spheres) were shown. Color code is given in each panel. To see this figure in color, go online.

Eliminating differential stress is a bilayer-based approach, in which the number of lipids in each leaflet is adjusted for zero differential stress (Δ = 0). This approach is denoted 0-DS here, where bilayer spontaneous curvature $c_{0,\text{b}}^{\text{∗}}$ = 0 (row 2 in column 2 of Fig. 1 and (17a), (17b), (17c)) as in a symmetric bilayer. This procedure makes use of both K_A and γ_m from simulations, in which the adjusted composition for 0-DS can be chosen by

$$\frac{N_{\text{m}1}}{N_{\text{m}2}}=\frac{N_{\text{m}1}^{\prime }\left({\gamma }_{\text{m}1}+K_A\right)}{N_{\text{m}2}^{\prime }\left({\gamma }_{\text{m}2}+K_A\right)}.$$ (22)

Here, $N_{\text{m}1}^{\prime }$/$N_{\text{m}2}^{\prime }$ and $N_{\text{m}1}/N_{\text{m}2}$ are the ratios of number of lipids in m1 and m2 before and after the adjustment, which requires the lateral pressure profile calculation. The 0-DS adjustment is consistent with (17a), (17b), (17c), where 0-DS (i.e., $c_{0,\text{b}}^{\text{∗}}$ = 0) can be realized by adjusting differential area strains (i.e., $c_{0,\text{b}}^{\text{s}}$; see Eqs. 8c and 10). This adjustment protocol may require iterative cycles to find optimal compositions.

An immediate implication of (17a), (17b), (17c) (see also Fig. 1) is that neither SA nor 0-DS is necessarily correct unless the underlying assumption (ɛ_m = 0 for SA and Δ = 0 for 0-DS) is verified for the asymmetric bilayer of interest. This motivated us to devise another bilayer-based approach that balances chemical potential between leaflets for some components using P2₁ PBC (39) while maintaining asymmetry by restraining other components (see also P21 periodic boundary conditions). Specifically, after assembly by a particular method (second column of Fig. 3 for APL and SA methods), an asymmetric bilayer is equilibrated in the P1 PBC (P1 simulations; Fig. 3, column 3). This is followed by a simulation with P2₁ PBCs (P2₁ simulation). Once a P2₁ simulation reaches its steady state, it is converted back to P1 for a subsequent simulation using a representative configuration (i.e., a snapshot with steady-state asymmetric compositions from P2₁ simulation) (Fig. 3, column 4).

The following approaches are assessed in this study: APL, SA, and the sequence described in the preceding paragraph, denoted APL/P2₁ and SA/P2₁, and 0-DS and 0-DS/P2₁. Hereafter, we denote a simulation system of an asymmetric bilayer as $\mathcal{A}$ with its preparation method given in the parentheses: $\mathcal{A}$(APL), $\mathcal{A}$(SA), $\mathcal{A}$(0-DS), $\mathcal{A}$(APL/P2₁), $\mathcal{A}$(SA/P2₁), and $\mathcal{A}$(0-DS/P2₁). Cognate symmetric bilayers for m1 and m2 in the asymmetric bilayer are denoted as $\mathcal{S}$m1 and $\mathcal{S}$m2, respectively.

Simulation details

Two asymmetric bilayers are simulated here. One is composed of DOPE, DPPC, and Chol (denoted PE/PC/Chol) but contains no DPPC in m2. The other is composed of DOPE, PSM, and Chol (PE/PSM/Chol), with no PSM in m2. Hence, m1 models the tightly packed outer leaflet and m2 the loosely packed inner leaflet of cell membranes. The molar ratios are listed in Table 2.

Table 2.

Compositions and numbers of lipids in the simulated systems

System	Composition
PE/PC/Chol	m1 (2:2:1)			m2 (4:0:1)
	DOPE	DPPC	Chol	DOPE	DPPC	Chol
$\mathcal{S}$m1	50	50	25	−	−	−
$\mathcal{S}$m2	−	−	−	100	0	25
$\mathcal{A}$(APL)	50	50	25	100	0	25
$\mathcal{A}$(APL/P21)	53	50	29	97	0	21
$\mathcal{A}$(SA)	56	56	28	104	0	26
$\mathcal{A}$(SA/P21)	55	56	31	105	0	23
$\mathcal{A}$(0-DS)	52	52	27	100	0	24
$\mathcal{A}$1(0-DS/P21)	53	52	28	99	0	23
$\mathcal{A}$2(0-DS/P21)	54	52	28	98	0	23
$\mathcal{A}$3(0-DS/P21)	55	52	28	97	0	23
PE/SM/Chol	m1 (1:1:1)			m2 (2:0:1)
	POPE	PSM	Chol	POPE	PSM	Chol
$\mathcal{S}$m1	42	42	42	−	−	−
$\mathcal{S}$m2	−	−	−	82	0	41
$\mathcal{A}$(APL)	42	42	42	82	0	41
$\mathcal{A}$(APL/P21)	45	42	40	79	0	43
$\mathcal{A}$(SA)	45	45	45	84	0	42
$\mathcal{A}$(SA/P21)	44	45	46	85	0	41
$\mathcal{A}$(0-DS)	43	43	43	82	0	41
$\mathcal{A}$(0-DS/P21)	45	43	43	80	0	41

$\mathcal{S}$m1 and $\mathcal{S}$m2 are symmetric bilayers corresponding to the outer (m1) and inner (m2) leaflets of an asymmetric bilayer. Asymmetric bilayers are denoted by $\mathcal{A}$ with the generation method in parentheses. For each monolayer, the molar ratio of components is given in parentheses. Subscript i in $\mathcal{A}$_i(0-DS/P2₁) for PE/PC/Chol represents that N_DOPE = 52 + i in m1.

The initial systems $\mathcal{A}$(APL) and $\mathcal{A}$(SA) were prepared for each asymmetric bilayer model composed of a bilayer of m1 and m2, and bulk water with 0.15 M KCl (column 2 in Fig. 3). The numbers of lipids in m1 and m2, N_m1 and N_m2, were determined by closely matching SAs of m1 and m2 using APLs either from a homogeneous bilayer for $\mathcal{A}$(APL) or from the corresponding $\mathcal{S}$m1 and $\mathcal{S}$m2 simulations (column 1 in Fig. 3) for $\mathcal{A}$(SA). For $\mathcal{A}$(0-DS), N_m1 and N_m2 are chosen using Eq. 22 with fixed N_m2, for which K_A and γ_m from (APL) were used. See Table 2 for compositional details. Five independent replicates for each system were generated by CHARMM-GUI Membrane Builder (57, 58, 59, 60).

For each replica, a series of short constant volume and temperature (NVT) and constant pressure and temperature (NPT) simulations were carried out. In these initial runs, there were various restraint potentials to hold positions and dihedral angles of the components, which were gradually relaxed to zero. Then, a 1-μs NPT production run (with P1 PBC) was performed for each replica. The snapshots between 500 ns and 1 μs of each production run for $\mathcal{A}$(APL), $\mathcal{A}$(SA), and $\mathcal{A}$(0-DS) (column 3 in Fig. 3) were then converted for a P2₁ simulation using modified input scripts from CHARMM-GUI Membrane Builder, for which restraint potentials to DPPC in the PE/PC/Chol system and PSM in the PE/SM/Chol system were applied to prevent their interleaflet migration (to maintain asymmetry).

We applied two flat-bottomed planar harmonic restraint potentials to the center of mass of each DPPC and PSM molecule along the directions perpendicular to the xy boundaries of the leaflets in a P2₁ simulation: one along the vector (x, y) = (1, 1) and the other along the vector (x, y) = (1, −1), centered at (x, y) = (L/4, 0), where L = 2^1/2L_x,P1 and L_x,P1 is the lateral dimension of the primary unit cell along the x direction in P1 PBC (see Fig. 2 B for visualization). Each restraint potential is flat in the inner region of the leaflets and starts acting at 8 Å (about a lateral dimension of a lipid) from the associated boundaries of the simulation box. The force constant for the restraint potential was set to 2 kcal/(mol ⋅ Ų).

Analogous series of short P2₁ NVT and NPT simulations were followed by a P2₁ production run (120 ns, 110 ns, and 25 ns for $\mathcal{A}$(APL), $\mathcal{A}$(SA), and $\mathcal{A}$(0-DS), respectively) while restraint potentials to DPPC/PSM were kept. For each $\mathcal{A}$(APL) and $\mathcal{A}$(SA), steady-state compositions of m1 and m2 were calculated from five independent P2₁ simulations (Table 2). Five representative configurations with the steady-state compositions were then back converted for P1 simulations ($\mathcal{A}$(APL/P2₁) and $\mathcal{A}$(SA/P2₁)). For $\mathcal{A}$(0-DS), the same procedure was applied for PE/SM/Chol, whereas five configurations for each of three different compositions were converted for subsequent P1 simulations for PE/PC/Chol $\mathcal{A}$(0-DS/P2₁) (Table 2). These back-converted bilayers were subject to the same series of short NVT and NPT simulations, followed by a 500-ns NPT production run with P1 PBCs (column 4 in Fig. 3).

All P1 and P2₁ simulations were carried out using OpenMM (63) and CHARMM (51), respectively, with the CHARMM36 lipid force field (64,65) and TIP3P water model (66,67). The integration time step was set to 2 fs with the SHAKE algorithm (68), except for the initial short NVT and NPT runs with 1-fs integration time step. Lennard-Jones interactions were switched off over 10–12 Å by a force-based switching function (69), and the electrostatic interactions were calculated by the particle mesh Ewald method (52) with a mesh size of ∼1 Å. Temperature (T = 310.15 K) and pressure (p = 1 atm) were controlled by Langevin dynamics with a friction coefficient of 1 ps⁻¹ and a semi-isotropic Monte Carlo barostat (70,71), with a pressure coupling frequency of 100 steps in OpenMM P1 simulations (72). In CHARMM P2₁ simulations, these were controlled by a Nosé-Hoover thermostat (73) and Langevin piston method (74) with a collision frequency of 20 ps⁻¹. During a production run, coordinate trajectories were saved every 0.1 and 0.05 ns for P1 and P2₁ simulations, respectively. For P1 simulations, velocity trajectroies were also saved, for which two in-house python classes VELFile and VELReporter were introduced. These classes were modified from analogous classes, DCDFile and DCDReporter for generation of coordinate trajectories in OpenMM (63).

Analysis

Before analysis, for each frame in each trajectory, the simulation system was shifted along the membrane normal (z direction) with P1 PBC to align the bilayer center at z = 0. In this work, the bilayer center was defined as the crossing point between z-density profiles of monolayers. From the recentered trajectories, component APL, the area compressibility modulus (K_A), leaflet and differential tensions (γ_m and Δ), torque densities (${\overline{F}}_{\text{m}}^{\prime }$(0) and ${\overline{F}}_{\text{b}}^{\prime }$(0)), and bilayer (d_b) and hydrophobic thicknesses (d_h) were calculated for comparison between different approaches (Table 2). For each approach, averages and standard errors of these properties were calculated over five independent asymmetric bilayer simulations.

APL

For multicomponent bilayers (either symmetric or asymmetric), Voronoi tessellation has been a typical approach for APL calculations (75, 76, 77), in which an individual molecule is represented by a finite number of points. In this work, we represented Chol using a single atom and the other components using three atoms: O3 for Chol; C21 (first carbon on sn-2 acyl chain), C2 (central carbon in the glycerol backbone), and C31 (first carbon on sn-1 acyl chain) for PC and PE; and C1F (carbonyl carbon on β chain), C2S (central carbon branching β chain and γ chain), and C5S (unsaturated carbon on γ chain) for PSM. For a given leaflet in each frame, the representative atoms and their periodic images along the xy plane were projected onto the xy plane, from which each component’s APL was calculated as the average area of polygons representing individual molecules of the same type in the leaflet.

Area compressibility modulus

The K_A for a bilayer was estimated from fluctuations in SA,

$$K_A=k_{\text{B}}T\frac{A_0}{\langle {\left(A-A_0\right)}^2\rangle },$$ (23)

where k_B is the Boltzmann’s constant, T is the temperature, and A₀ is an average SA of the bilayer.

Leaflet and differential stresses and torque density

To calculate tensions (γ_m and Δ) and torque densities (${\overline{F}}_{\text{m}}^{\prime }$(0) and ${\overline{F}}_{\text{b}}^{\prime }$(0)), the lateral pressure profile (p(z)) was obtained from recentered trajectories using a developmental version of CHARMM, in which p_T(z) was calculated using the Harasima contour for slab geometry (78). Because p_N(z) cannot be correctly estimated by Harasima contour (79), p_N was calculated by p_N = $\int$dz p_T(z)/L_z. The resulting p(z) (= p_T(z) − p_N(z)) does not include the contribution from Σ, from which γ_m, Δ, ${\overline{F}}_{\text{m}}^{\prime }$(0), and ${\overline{F}}_{\text{b}}^{\prime }$(0) were calculated using Eqs. 2 and (3b), (3a).

Thicknesses

The bilayer thickness (d_b) from simulations were calculated using P atom positions. First, two-dimensional profiles of P atom positions for monolayers, z_P,m1(x, y) and z_P,m2(x, y), were calculated. Then, d_b was calculated as an average of d_b(x, y) = |z_P,m1(x, y) − z_P,m2(x, y)| over the grid space. The hydrophobic thicknesses (d_h) were calculated analogously. From the average positions of C22 and C32 atoms (for phospholipids), C2F and C6S atoms (for PSM), and O3 atoms (for Chol), two-dimensional profiles for each monolayer, z_h,m1(x, y) and z_h,m2(x, y), were calculated. Then, d_h was calculated as an average of d_h(x, y) = |z_h,m1(x, y) − z_h,m2(x, y)| over the grid space.

Results

APL, SA, APL/P2₁, and SA/P2₁

As APLs in monolayers of an asymmetric bilayer can be a metric for the extent of lipid packing (compared with those from the corresponding symmetric bilayer), we calculated component APLs in m1 and m2 from the simulations (Table 3). Expected from its assumption of ideal mixing, the component APLs from $\mathcal{A}$(APL) deviated from those from $\mathcal{S}$m1 and $\mathcal{S}$m2; component APLs from $\mathcal{S}$m1 (tightly packed leaflet) were smaller, and those from $\mathcal{S}$m2 (loosely packed leaflet) were larger. In other words, m1 and m2 in $\mathcal{A}$(APL) are under tensile and compressive stress compared to those in $\mathcal{S}$m1 and $\mathcal{S}$m2, respectively. Component APLs from $\mathcal{A}$(SA) remained close to those from $\mathcal{S}$m1 and $\mathcal{S}$m2, which is also expected from the underlying assumption of zero area strain (ɛ_m1 = ɛ_m2 = 0).

Table 3.

Component APL from simulations

System	Leaflet
PE/PC/Chol	m1 (2:2:1)			m2 (4:0:1)
	DOPE	DPPC	Chol	DOPE	DPPC	Chol
$\mathcal{S}$m1	56.4 (0.1)	55.2 (0.1)	28.1 (0.0)	N/A	N/A	N/A
$\mathcal{S}$m2	N/A	N/A	N/A	60.3 (0.0)	N/A	30.1 (0.0)
$\mathcal{A}$(APL)	59.1 (0.1)	57.5 (0.2)	29.1 (0.1)	58.4 (0.1)	N/A	29.2 (0.0)
diff.	2.7 (0.1)	2.4 (0.2)	1.0 (0.1)	−1.9 (0.1)	N/A	−0.9 (0.0)
$\mathcal{A}$(SA)	56.4 (0.1)	55.3 (0.1)	28.1 (0.0)	60.5 (0.1)	N/A	30.2 (0.0)
diff.	0.0 (0.1)	0.1 (0.1)	0.0 (0.1)	0.2 (0.1)	N/A	0.1 (0.0)
PE/SM/Chol	m1 (1:1:1)			m2 (2:0:1)
	POPE	PSM	Chol	POPE	PSM	Chol
$\mathcal{S}$m1	51.2 (0.1)	50.2 (0.1)	25.8 (0.1)	N/A	N/A	N/A
$\mathcal{S}$m2	N/A	N/A	N/A	54.0 (0.0)	N/A	28.1 (0.0)
$\mathcal{A}$(APL)	52.4 (0.1)	51.3 (0.2)	26.2 (0.2)	52.9 (0.1)	N/A	27.5 (0.0)
diff.	1.2 (0.1)	1.1 (0.3)	0.4 (0.2)	−1.1 (0.1)	N/A	−0.6 (0.1)
$\mathcal{A}$(SA)	51.4 (0.1)	50.5 (0.1)	25.4 (0.1)	54.1 (0.0)	N/A	28.1 (0.1)
diff.	0.1 (0.1)	0.3 (0.2)	−0.5 (0.1)	0.0 (0.0)	N/A	0.0 (0.1)

Component APLs (in Ų) were calculated using five 900-ns trajectories from independent simulations. Standard errors are shown in parenthesis. The uncertainties in area difference (diff.) were estimated by the error propagation. N/A, not applicable.

Table 4 summarizes the compressibility moduli, differential surface tensions, and torque densities from simulations. The values of K_A from $\mathcal{A}$(APL) and $\mathcal{A}$(SA) are similar for both systems, and statistically close to $K_A^{\text{add}}=\left(K_A^{\mathcal{S}\text{m}1}+K_A^{\mathcal{S}\text{m}2}\right)$/2, where $K_A^{\mathcal{S}\text{m}1}$ and $K_A^{\mathcal{S}\text{m}2}$ are K_A from $\mathcal{S}$m1 and $\mathcal{S}$m2, respectively. This result is consistent with the observation by HD (36) that nonoptimal lipid packing (i.e., differential area strain) below a certain threshold and/or compositional asymmetry does not stiffen an asymmetric bilayer. The results imply that interleaflet couplings in the stretching mode in our model bilayers are weak, if present at all.

Table 4.

Area compressibility modulus, differential stress, stress index, and torque density from simulations

System	KA (dyn/cm)	Δ (dyn/cm)		σ	$\overline{F}$′ (cal/mol/Å)
		Sim.	Eq. 21	Eq. 26	${\overline{F}}_{\text{m}1}^{\prime }$	${\overline{F}}_{\text{m}2}^{\prime }$	${\overline{F}}_{\text{b}}^{\prime }$
PE/PC/Chol					m1 (2:2:1)	m2 (4:0:1)
$\mathcal{S}$m1	356 (18)	−0.3 (0.7)	N/A	N/A	330 (8)	N/A	−7 (20)
$\mathcal{S}$m2	309 (16)	−0.2 (0.3)	N/A	N/A	N/A	390 (6)	−3 (14)
$K_A^{\text{add}}$	332 (12)
$\mathcal{A}$(APL)	351 (15)	13.0 (0.8)	10.6 (1.4)	2.4 (0.2)	425 (8)	327 (11)	98 (19)
$\mathcal{A}$(APL/P21)	343 (16)	−9.6 (1.4)	−7.6 (1.6)	0.8 (0.1)	290 (19)	464 (7)	−174 (21)
$\mathcal{A}$(SA)	341 (10)	−7.7 (1.0)	−8.5 (1.9)	0.6 (0.1)	273 (14)	459 (13)	−186 (27)
$\mathcal{A}$(SA/P21)	362 (35)	−8.5 (1.1)	−8.6 (2.6)	0.7 (0.2)	251 (17)	443 (32)	−192 (47)
PE/SM/Chol					m1 (1:1:1)	m2 (2:0:1)
$\mathcal{S}$m1	1016 (62)	1.2 (1.2)	N/A	N/A	514 (14)	N/A	21 (33)
$\mathcal{S}$m2	603 (23)	0.9 (0.5)	N/A	N/A	N/A	550 (8)	16 (20)
$K_A^{\text{add}}$	810 (33)
$\mathcal{A}$(APL)	824 (47)	15.3 (0.5)	16.3 (1.7)	1.9 (0.2)	648 (9)	432 (14)	216 (22)
$\mathcal{A}$(APL/P21)	712 (68)	1.2 (1.2)	−1.6 (1.9)	0.0 (0.0)	498 (16)	559 (23)	−60 (26)
$\mathcal{A}$(SA)	841 (65)	−5.3 (0.8)	−4.7 (1.8)	0.1 (0.0)	469 (10)	577 (14)	−108 (23)
$\mathcal{A}$(SA/P21)	773 (45)	−4.6 (1.7)	−5.8 (2.7)	0.0 (0.0)	480 (14)	615 (21)	−135 (33)

Data were calculated from five 500-ns trajectories for (SA/P2₁) and (APL/P2₁). For the other systems, five 900-ns trajectories were used. Standard errors over five trajectories are shown in parentheses. Average K_A from $\mathcal{S}$m1 and $\mathcal{S}$m2 for the additive model $K_A^{\text{add}}$ are shown together for easy comparison. The stress index σ (Eq. 26) is a measure of the stress due to the P1 PBC. Uncertainties in $K_A^{\text{add}}$, Δ from Eq. 21, and σ from Eq. 26 were estimated by the error propagation. N/A, not applicable; Sim., simulation.

In contrast to K_A, Δ and $\overline{F}$′ were sensitive to different approaches, as shown in Table 4. Δ from $\mathcal{A}$(APL) was positive, whereas that from $\mathcal{A}$(SA) was negative. Positive Δ from $\mathcal{A}$(APL) is consistent with the APLs (m1 and m2 under tensile and compressive stresses, respectively), and negative Δ from $\mathcal{A}$(SA) is consistent with ${\overline{F}}_{\text{b}}^{\prime }$(0) ($\mathcal{A}$(SA) would bend toward more positive curvature). Although Δ₀ from Eq. 20 with $k_{\text{b}}^{\text{b}}=k_{\text{b}}^{\text{s}}$/6 (the polymer brush model, PBM) (80) agree well with those from $\mathcal{A}$(APL) (Δ₀ = 10.6 ± 0.8 dyn/cm for PE/PC/Chol and Δ₀ = 15.2 ± 1.3 dyn/cm for PE/SM/Chol), those for $\mathcal{A}$(SA) are less accurate (Δ₀ = −5.1 ± 0.8 dyn/cm for PE/PC/Chol and Δ₀ = −3.0 ± 0.7 dyn/cm for PE/SM/Chol). The results indicate that the interleaflet coupling in the bending mode may be significant (see Discussion), which is not considered in our work. Estimated Δ₀ using Eq. 21 show good agreement with Δ from simulations for both $\mathcal{A}$(APL) and $\mathcal{A}$(SA) (Table 4). There is a slight shift in calculated ${\overline{F}}_{\text{m}}^{\prime }$(0) from the tensionless monolayer, which can be attributed to the contribution from Δ₀ to torque density (see (18a), (18b), (18c)). The ${\overline{F}}_{\text{b}}^{\prime }$(0) for $\mathcal{A}$(APL) and $\mathcal{A}$(SA) have opposite signs, indicating that the effect of asymmetric lipid packing in $\mathcal{A}$(APL) was strong enough to alter the sign of $c_{0,\text{b}}^{\ast }$ (see Eq. 10).

Fig. 4 plots the change in the number of lipids (ΔN) in m1 during P2₁ simulations (see also Table 2 for steady-state ΔN-values). There is a net migration of ∼3 to 4 Chol to m1 in both $\mathcal{A}$(APL) and $\mathcal{A}$(SA) for PE/PC/Chol. However, the net migration of DOPE is different between $\mathcal{A}$(APL) and $\mathcal{A}$(SA): ∼4 to m1 in $\mathcal{A}$(APL) and ∼1 to m2 in $\mathcal{A}$(SA). For PE/SM/Chol, fewer Chols migrated compared to those in PE/PC/Chol (∼2 to m2 in $\mathcal{A}$(APL) and ∼1 to m1 in $\mathcal{A}$(SA)). Whereas ∼3 POPEs migrated to m1 in $\mathcal{A}$(APL), only ∼1 migrated to m2 in $\mathcal{A}$(SA). The overall lipid migration was more pronounced in $\mathcal{A}$(APL) than in $\mathcal{A}$(SA), which was toward m1. We attribute this to a relaxation of lipids in m2 of $\mathcal{A}$(APL) under compressive stress during P2₁ simulations. Lower net lipid migration for PE/SM/Chol than that for PE/PC/Chol can be attributed to tighter lipid packing (as reflected in K_A; see Table 4).

Figure 4.

Changes in the number of lipids (ΔN) during P2₁ equilibration of (APL) (left panels) and (SA) (right panels) for PE/PC/Chol (A and B) and for PE/SM/Chol (C and D). Initial numbers of lipids and color code are given in each panel. Standard errors from five independent simulations are shown as gray areas. To see this figure in color, go online.

The restraints to hold DPPC and PSM in m1 generally worked as desired, which is shown as flat lines with much smaller fluctuations in ΔN and its standard errors compared with those for PE and Chol. The ΔN for DPPC and PSM from individual replicas remained at zero (albeit with transient deviations). The $\mathcal{A}$(APL) for PE/PC/Chol (Fig. 4 A) is an exception, in which a fractional amount of DPPC (∼0.4) migrated to m2 after ∼70 ns. More specifically, in two out of five replicas, one DPPC migrated to m2 by crossing the boundaries of the simulation box (not a flip-flop between leaflets). We attribute the observed DPPC migration in $\mathcal{A}$(APL) for PE/PC/Chol to an artifact arising from weak flat-bottomed restraints along the xy plane (see Discussion). The steady-state ΔN from the other three replicas did not change significantly, i.e., the integer values of the steady-state ΔN remained to be the same to those from five replicas.

After ΔN reached a steady state, the P2₁ systems were converted to P1 using representative configurations whose compositions match those from the steady state. Whereas K_A remained statistically close to $K_A^{\text{add}}$ (Table 4), the pressure profiles from $\mathcal{A}$(APL) and $\mathcal{A}$(SA) are substantially different before P2₁ simulations for PE/PC/Chol (Fig. 5, top). Though the difference is smaller for PE/SM/Chol (Fig. 5, bottom), their moments (Δ and $\overline{F}$′) are clearly different (Table 4). After simulation in P2₁ PBCs, the pressure profiles from the APL and SA systems show good agreement. Furthermore, the profiles from $\mathcal{A}$(APL/P2₁) become closer to those from $\mathcal{A}$(SA), and the profiles from $\mathcal{A}$(SA/P2₁) are not notably different from those of $\mathcal{A}$(SA). The better agreement of $\mathcal{A}$(APL/P2₁) and $\mathcal{A}$(SA/P2₁) is also clearly shown in Δ (Table 4). The changes in $\overline{F}$′(0) for $\mathcal{A}$(SA/P2₁) are also smaller than those for $\mathcal{A}$(APL/P2₁) after P2₁ simulations.

Figure 5.

Lateral pressure profiles, p(z) (left panels), and the residual pressure profiles, p_res(z) (right panels) for asymmetric bilayers of PE/PC/Chol (top) and PE/SM/Chol (bottom) for different simulation conditions: APL (red), SA (green), APL/P2₁ (blue), and SA/P2₁ (black). The residual pressure profiles are calculated with respect to the pressure profile from SA/P2₁. In pressure profiles (left panel), standard errors from five independent simulations for SA/P2₁ are shown as the error bars (gray). To see this figure in color, go online.

0-DS and 0-DS/P2₁

The results from our P2₁ simulations of $\mathcal{A}$(APL) and $\mathcal{A}$(SA) indicate that the number of lipids in monolayers can be adjusted via P2₁ PBCs, and the P2₁-equilibrated bilayers, $\mathcal{A}$(APL/P2₁) and $\mathcal{A}$(SA/P2₁), have similar mechanical properties except statistically different Δ₀ for PE/SM/Chol. Although different, the agreement of Δ₀ between $\mathcal{A}$(APL/P2₁) and $\mathcal{A}$(SA/P2₁) is much better compared to that between $\mathcal{A}$(APL) and $\mathcal{A}$(SA) (see Table 4), indicating that our P2₁ method drives asymmetric bilayers from different methods to similar states with nonzero differential stress. These results are therefore inconsistent with the assumptions of 0-DS. This subsection considers the 0-DS refinement (Eq. 22) to $\mathcal{A}$(APL) and assesses mechanical properties of the resulting $\mathcal{A}$(0-DS) and the influence of our P2₁ based approach to it, $\mathcal{A}$(0-DS/P2₁).

The mechanical properties of $\mathcal{A}$(0-DS) are summarized in Table 5. Similarly to $\mathcal{A}$(APL) and $\mathcal{A}$(SA), K_A are statistically close to $K_A^{\text{add}}$. The 0-DS adjustment (Eq. 22) of $\mathcal{A}$(APL) for PE/PC/Chol resulted in Δ₀ = 0.8 ± 0.9 dyn/cm and ${\overline{F}}_{\text{m}}^{\prime }$ that agree very well with those from tensionless leaflets (see Table 4 for ${\overline{F}}_{\text{m}}^{\prime }$ in their cognate symmetric bilayers, $\mathcal{S}$m1 and $\mathcal{S}$m2). However, Eq. 22 did not work well for PC/SM/Chol; Δ₀ is reduced but still significantly deviates from zero. Torque densities also significantly deviate from the theoretical predictions. These results may be attributed to poorly adjusted N_m1 due to larger uncertainties in K_A for PE/SM/Chol than that for PE/PC/Chol. This indicates that it is difficult to use Eq. 22 when there are large uncertainties in K_A.

Table 5.

Area compressibility modulus, leaflet tension, stress index, and torque density of$\mathcal{A}$(0-DS) and$\mathcal{A}$(0-DS/P2₁)

System	KA (dyn/cm)	Δ (dyn/cm)		σ	$\overline{F}$′ (cal/mol/Å)
		Sim.	Eq. 21	Eq. 26	${\overline{F}}_{\text{m}1}^{\prime }$	${\overline{F}}_{\text{m}2}^{\prime }$	${\overline{F}}_{\text{b}}^{\prime }$
PE/PC/Chol					m1 (2:2:1)	m2 (4:0:1)
$\mathcal{A}$(0-DS)	354 (17)	0.8 (0.9)	−0.3 (1.7)	0.0 (0.0)	327 (12)	391 (14)	−63 (23)
$\mathcal{A}$1(0-DS/P21)	358 (19)	−6.7 (0.9)	−7.4 (2.1)	0.4 (0.1)	281 (9)	451 (22)	−170 (30)
$\mathcal{A}$2(0-DS/P21)	353 (18)	−9.3 (1.2)	−8.6 (2.3)	0.9 (0.1)	266 (22)	454 (12)	−188 (33)
$\mathcal{A}$3(0-DS/P21)	346 (23)	−15.1 (0.9)	−13.3 (1.4)	2.4 (0.3)	244 (18)	502 (13)	−258 (19)
PE/SM/Chol					m1 (1:1:1)	m2 (2:0:1)
$\mathcal{A}$(0-DS)	822 (50)	6.1 (1.9)	7.5 (2.3)	0.6 (0.1)	583 (20)	503 (15)	80 (32)
$\mathcal{A}$(0-DS/P21)	763 (69)	−14.6 (1.7)	−14.6 (2.5)	0.7 (0.1)	394 (21)	654 (16)	−261 (35)

Data were calculated from five 900-ns trajectories for (0-DS) and five 500-ns trajectories for (0-DS/P2₁). Subscript i in $\mathcal{A}$_i(0-DS/P21) represents that N_DOPE = 52 + i in m1. Standard errors over five trajectories are shown in parentheses. The stress index σ (Eq. 26) is a measure of the stress due to the P1 PBC. Uncertainties in Δ from Eq. 21 and σ from Eq. 26 were estimated by the error propagation.

P2₁ simulations of $\mathcal{A}$(0-DS) introduced a net migration of lipids to m1 (Fig. 6), similar to $\mathcal{A}$(APL) and $\mathcal{A}$(SA) (Fig. 4). For PE/PC/Chol, we assessed the sensitivity of mechanical properties of $\mathcal{A}$(0-DS/P2₁) to the number of lipids in m1 with three different numbers of DOPE, N_DOPE (Table 2). As anticipated, Δ decreases to more negative values as N_DOPE in m1 increases, where Δ is comparable to that for (SA/P2₁) when N_DOPE = 54 (an average value over 5–25 ns in Fig. 6). For PE/SM/Chol, 0-DS/P2₁ resulted in a state whose Δ is more negative than that from (SA/P2₁). Hence, the application of 0-DS/P2₁ to both systems indicates that the leaflet surface tensions are not zero (36).

Figure 6.

Changes in the number of lipids (ΔN) during P2₁ equilibration of (0-DS) for (A) PE/PC/Chol and (B) PE/SM/Chol. Initial numbers of lipids and color code are given in each panel. Standard errors from five independent simulations are shown as gray areas. To see this figure in color, go online.

Discussion

The results of the bilayer systems studied here show that Δ₀ and $\overline{F}$′ from $\mathcal{A}$(APL) are significantly different from those from $\mathcal{A}$(SA), which is attributed to asymmetric lipid packing in $\mathcal{A}$(APL) as shown in component APLs (Table 2); m1 is not packed as tightly as it should be and vice versa. This is a systematic error that arises from the ideal-mixing assumption in the APL approach. The estimates of Δ^s(c = 0) from Eq. 12b (Δ^s(c = 0) = 13.4 ± 0.7 dyn/cm for PE/PC/Chol and Δ^s(c = 0) = 17.2 ± 1.2 dyn/cm for PE/SM/Chol) agree well with observed Δ_sim (Table 4). This indicates that the differential stress in $\mathcal{A}$(APL) is dominated by area strains. Considering that the bending is the softer degree of freedom ($k_{\text{b}}^{\text{b}}\approx k_{\text{b}}^{\text{s}}$/6 in the PBM), the results indicate that the differential stress in $\mathcal{A}$(SA) would be generally smaller than those in $\mathcal{A}$(APL).

The differential stress Δ from our theory includes the contributions from both the bending and area strain. To test our theory, we compared Δ₀ from (3b), (3a) and (3b), (3a) with Δ_sim from our simulations in Fig. 7, where Δ^s(c = 0) from Eq. 12b are shown together to demonstrate the contribution from the bending in Δ₀. Here, we set z₀ = d_h/2 and $k_{\text{b}}^{\text{b}}=k_{\text{b}}^{\text{s}}$/6.

Figure 7.

Comparison of predicted (Δ₀) and calculated differential stresses (Δ_sim) from the simulations. Δ₀ were calculated from Eq. 20 (green circle) with $k_{\text{b}}^{\text{b}}=k_{\text{b}}^{\text{s}}$/6 and from Eq. 21 (blue diamond). Shown together are Δ^s(c = 0) (Eq. 12b, red triangle) and its linear fit (red line). For clarity, error bars are omitted. To see this figure in color, go online.

The tensional cost from the bending appears to be more significant as the monolayer m1 (leaflet with smaller APL) becomes more densely packed, which is clearly shown with the larger deviation of Δ^s(c = 0) from Δ_sim as it becomes more negative. Δ₀ from Eq. 20 shows significantly improved agreement with Δ_sim. However, the deviation of Δ₀ persists for negative Δ_sim and grows with decreasing Δ_sim. The bilayer bending is likely to be sensitive to the differential stress from area strain, and the bilayer becomes stiffer as the monolayer m1 (the leaflet with smaller APL) is more compressed (36). Δ₀ from Eq. 21 agrees very well with Δ_sim (blue diamonds in Fig. 7), indicating that the destressing correction is a good approximation for our simulated bilayers.

We further tested (3b), (3a) and (3b), (3a) using the data from a recent simulation study of Miettinen and Lipowsky (81), in which ganglioside GM1 was asymmetrically distributed in the upper leaflet of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayers at various concentrations of GM1 (from 0 to 27 mol%) (see Fig. S1 for details). Two GM1 models, the lollipop-like (82) and cone-like GM1s (83), were considered in their study.

For the lollipop-like GM1 (Fig. S1A), Δ₀ from both Eq. 20 and Eq. 21 agree well with simulation data at low GM1 concentration (<10 mol%) and deviate oppositely from Δ_sim at higher GM1 concentrations. For the cone-like GM1 (Fig. S1B), Δ₀ from Eq. 21 agree well with the data up to 10 mol% GM1 concentration, whereas those from Eq. 20 become poor even at 5 mol% GM1 concentration. We attribute the discrepancy from Δ_sim to possibly altered bending moduli due to the softened bending rigidity of GM1-POPC bilayer (84). The test results indicate that Eq. 21 can be applicable to bilayers with mild asymmetry, and Eq. 20 can be challenging to apply when there is significant leaflet coupling.

In this work, the net differential stress in a bilayer is shown to be identical to the residual differential stress (${\text{Δ}}_0^{\text{res}}$ = Δ₀), which is an externally imposed differential stress via applied P1 PBC that is required to force an asymmetric bilayer to be flat (c = 0) (see Eq. 17a; Fig. 1). As ${\text{Δ}}_0^{\text{res}}$ does not vanish unless $c_{0,\text{b}}^{\ast }$ = 0, there exists an energetic penalty arising from Δ₀ in planar membranes with P1 PBC. Therefore, it is worth further discussing the size dependence of Δ₀ and the energetic cost from P1 PBC in a large but planar membrane.

Let us consider a patch of the membrane in a rectangular simulation box with P1 PBC, whose box size along the x or y dimensions is L (= L_x = L_y) and that along the z dimension is L_z. When L is smaller than 2|$c_{0,\text{b}}^{\ast }$|⁻¹ + d_b, the membrane is required to span the simulation box. When L << |$c_{0,\text{b}}^{\ast }$|⁻¹, the membrane would be planar (Fig. 8, column 1). When L ∼${\left|c_{0,\text{b}}^{\ast }\right|}^{-1}$, the membrane may locally be curved (Fig. 8, column 2). Here, we assume that the membrane spans along xy dimensions. In this case, the mean curvature of the membrane ($\overline{c}=\underset{S}{\int }$dAc/A) is zero from the periodicity.

Figure 8.

Schematic illustration of asymmetric membrane in a unit cell in P1 (columns 1 and 2) and P2₁ PBC (column 3). Here, we consider a membrane with nonzero spontaneous curvature ($c_{0,\text{b}}^{\ast }$ ≠ 0) in a simulation box whose smallest box size along the x or y dimension is L. When the membrane spans the unit cell, the mean curvature of the membrane is required to be zero ($\overline{c}$ = 0) because of the periodicity. The two monolayers are shown in blue and red. In the P1 PBC, the membrane is planar when L << |$c_{0,\text{b}}^{\ast }$|⁻¹ (column 1) and may be locally curved when L ∼|$c_{0,\text{b}}^{\ast }$|⁻¹ (column 2). In P2₁ PBC, the unit cell contains a laterally patched membrane B1 and its flipped image B2, whose $\overline{c}$ = 0 (column 3). Hence, both B1 and B2 are curvature relaxed. A vertical dotted line is located at the interface between the primary simulation box and its image. To see this figure in color, go online.

The Δ in an arbitrarily large but planar bilayer is given by (see (16a), (16b))

$$\text{Δ}\left(\overline{c}\right)=\underset{S}{\int }dA{\text{Δ}}^{\text{res}}\left(c|c_{\text{r}}=c_{0,\text{b}}^{\ast }\right)/\underset{S}{\int }dA,$$ (24)

where Δ(c) is the local differential stress given by (3b), (3a) and (3b), (3a). Eq. 24 reduces to Eq. 17a; thus, Δ₀ of the membrane remains constant independent of the system size. Indeed, it has been reported that Δ₀/2 ≈ −5.7 dyn/cm for an (SA) composed of 580 1,2-dilauroyl-sn-glycero-3-phosphocholine in m1 and 522 POPC in m2 (36), in which the number of lipids are about four times more than those in our bilayer systems. The comparable Δ₀ to that for significantly smaller bilayers (see Δ₀ for $\mathcal{A}$(SA) in Table 4) supports our argument.

As opposed to a symmetric bilayer, there would be energetic penalty in a planar asymmetric bilayer because of nonvanishing spontaneous curvature ($c_{0,\text{b}}^{\ast }$ ≠ 0). The energetic cost from the imposed P1 PBC in an asymmetric membrane is obtained from Eq. 9,

$$E_{\text{b}}=\frac{A}{2}\left(k_{\text{b}}^{\text{b}}+k_{\text{b}}^{\text{s}}\right){c_{0,\text{b}}^{\ast }}^2.$$ (25)

The immediate implication of Eq. 25 is that a large planar asymmetric bilayer in the P1 PBC is energetically unfavorable unless $c_{0,\text{b}}^{\ast }$ = 0, as E_b grows linearly with A and increases quadratically with ${\text{Δ}}_0^{\text{res}}$ (see (A3a), (A3b), (A3c)). Thus, care should be taken in generation of a large (planar) bilayer with nonvanishing spontaneous curvature.

The simulation study of Park et al. provides some guidance (85). A mismatch of a few lipids between leaflets could introduce Δ₀ = ∼6–8 dyn/cm in a moderately sized system. This Δ₀ is comparable to that from $\mathcal{A}$(SA) and smaller than the ∼13–15 dyn/cm from $\mathcal{A}$(APL) (Table 4). In the same work, the criterion of allowable mismatch was given by a thermally accessible energy change, 2k_BT (86). The estimate of energetic penalty $\left(E_{\text{b}}^{\text{s}}={\text{e}}_{\text{nl}}^{\text{s}}A\right)$ in $\mathcal{A}$(APL) can be calculated using Eqs. A2, (A3a), (A3b), (A3c), and (A4a), (A4b), which are $E_{\text{b}}^{\text{s}}$ = 4.1 ± 0.2 k_BT for PE/PC/Chol and $E_{\text{b}}^{\text{s}}$ = 2.1 ± 0.3 k_BT for PE/SM/Chol. It indicates that ${\text{Δ}}_0^{\text{s}}$ ≥ 13 dyn/cm in a moderately sized bilayer (∼125 lipids per leaflet in our work) is likely an indication that the system is under significant stress.

To be more quantitative, we define a stress index, σ, as a measure for the energetic cost from the asymmetry in a bilayer,

$$\sigma \equiv \frac{E_{\text{b}}/A}{k_{\text{B}}T/A}\approx \frac{A\left(k_{\text{b}}^{\text{b}}+k_{\text{b}}^{\text{s}}\right){\text{Δ}}_0^2}{2k_{\text{B}}T{\left({\overline{F}}_{\text{m}1,\text{s}}^{\prime }+{\overline{F}}_{\text{m}2,\text{s}}^{\prime }+k_{\text{b}}^{\text{s}}/z_0\right)}^2},$$ (26)

which can be understood as the tensional cost in the units of thermal energy (k_BT). The estimates from Eq. 26 with $k_{\text{b}}^{\text{b}}=k_{\text{b}}^{\text{s}}$/6 (from the PBM) for our simulations are given in Tables 4 and 5. As shown in Table 4, σ from $\mathcal{A}$(APL) are 2.4 ± 0.2 for PE/PC/Chol and 1.9 ± 0.2 for PE/SM/Chol, which is significantly higher compared with those from $\mathcal{A}$(SA) ≤ 0.6. This agrees with the simulation results that the mechanical properties of $\mathcal{A}$(APL) changed significantly after P2₁ simulations, during which σ from $\mathcal{A}$(APL/P2₁) became significantly lowered below 1. When σ > 1 (this indicates that energetic penalty is above k_BT), one would need to check for possible artifacts.

There is a unique and significant benefit in our P2₁ method over the other available approaches with P1 PBCs. In the P2₁ PBC, the unit cell contains the primary simulation box and its flipped image, i.e., two patches of membranes, B1 and B2 (Fig. 8, column 3). Using this unit cell, a P2₁ crystal can be generated by P1 PBC, which is equivalent to an up-down checkerboard arrangement of cells sketched in Fig. 2B. This arrangement of cells leads to a membrane that is infinitely differentiable. So, even though its mean curvature does not vanish, the membrane is valid under P2₁ PBC without necessitating kinks. When we consider the membrane patches B1 and B2 in the unit cell, their spontaneous curvatures ($c_{0,\text{B}1}^{\ast }$ and $c_{0,\text{B}2}^{\ast }$) are opposite ($\overline{c}$ = 0) and both B1 and B2 would be curvature relaxed, i.e., Δ₀ of membrane patches in the unit cell would vanish. Hence, the interleaflet lipid redistribution would take place in a curvature-relaxed bilayer over the whole simulation time. This implies that the steady-state number of lipids in each monolayer in the asymmetric bilayer likely represents those in its curvature-relaxed state, i.e., the natural compositional asymmetry in biological membranes would be achieved by P2₁ simulations.

Another important feature of P2₁ simulations is an accelerated interleaflet lipid redistribution. In our all-atom P2₁ simulations, the steady state was reached around 100–120 ns, whereas Chol flip-flop took around (nonscaled) 800 ns to reach a steady state in a coarse-grained (CG) model of $\mathcal{A}$(SA) composed of 79 DPPCs and 18 Chols in m1 and 56 1,2-dilinoleoyl-sn-glycero-3-phosphocholine lipids and 14 Chols in m2 (36). In another study of a CG model of $\mathcal{A}$(SA) composed of 66 POPCs and 24 lollipop-like GM1s in m1 and 87 POPCs in m2, the steady state was reached around (nonscaled) 250 ns (81). Considering a typical scaling factor of 4 for mapping CG time into the real time (87), P2₁ PBC accelerated the Chol redistribution about an order of magnitude. As discussed above, the P2₁ PBC accelerates the flip-flop of lipids, whereas the bilayer remains in its curvature-relaxed state. Though our P2₁ method does not rigorously represent the way that cells regulate the compositional asymmetry, it mimics the lipid asymmetry in biological membranes by imposing a restraining potential to specific lipids and allowing others to exchange leaflets.

The flat-bottomed restraints in our P2₁ method have two potential issues. One is restraining a lipid in the wrong leaflet. When a restrained lipid in its original leaflet overcomes the restraints, it can migrate to the opposite leaflet. If this lipid diffuses to the inner region of the opposite leaflet, the lipid will be restrained to reside in the wrong leaflet by the same restraints. This issue worsens as the restraints become weaker (because of the higher probability of lipid diffusion inside the wrong leaflet followed by overcoming the restraints). Indeed, it was observed in two replicas of $\mathcal{A}$(APL) for PE/PC/Chol, for which a (restrained) DPPC in m1 was migrated to m2 during P2₁ simulations. The solution is to apply sufficiently strong restraints. The other is issue is an osmotic pressure-like effect arising from restraining an individual component of the bilayer to one leaflet when applying P2₁, which requires further study. This is a general effect of applying restraints and could in principle be reduced or eliminated by restraining an equal number of another component (see (88) for an application involving a mixture of classical and quantum mechanical waters). For bilayer simulations involving a specific number of peptides restrained to one leaflet, the effect could be reduced by increasing the number of lipids that can exchange between both leaflets.

There are some implications in the results from P2₁ simulations of $\mathcal{A}$(0-DS). Providing that $\mathcal{A}$(0-DS) is metastable, the interleaflet lipid migration would be small if any, i.e., ${\text{Δ}}_0^{\text{res}}$ ≈ 0 and $c_{0,\text{b}}^{\ast }$ ≈ 0 in $\mathcal{A}$(0-DS/P2₁). In our work, the lipids migrated to m1 during P2₁ simulations (Fig. 6), indicating that $\mathcal{A}$(0-DS) is not likely to be metastable (i.e., $c_{0,\text{b}}^{\ast }$ ≠ 0) when there exist interleaflet lipid exchanges. The net lipid migration to m1 indicates that $\mathcal{A}$(0-DS/P2₁) would have similar mechanical properties to those of $\mathcal{A}$(SA/P2₁), which was shown in $\mathcal{A}$(0-DS/P2₁) for PE/PC/Chol. Considering these results and the susceptibility of Eq. 22 to uncertainty in K_A (for the 0-DS refinement), we conclude that 0-DS/P2₁ is less practical than SA/P2₁ for generation of initial configurations of asymmetric bilayers.

Conclusions

Several approaches for asymmetric bilayer simulations, which can be broadly classified as lipid based (APL), leaflet based (SA), and bilayer based (0-DS), have been compared. These approaches aim to keep assumptions appropriate for symmetric bilayers, which may not be applicable to asymmetric bilayers. In this study, the underlying assumptions in these approaches are connected to a theoretical framework on the asymmetric membranes recently proposed by HD (36). We proposed another bilayer-based approach, in which specific lipid types are restrained to reside in their original leaflet and the other components are allowed to migrate between leaflets via a P2₁ PBC. For the APL, SA, and 0-DS approaches, we have assessed the impact of each approach on mechanical properties. In our bilayer systems, K_A is not significantly altered between different approaches, whereas the other mechanical properties are sensitive to the approaches. We found that the 0-DS refinement (Eq. 22) is susceptible to the uncertainties in K_A and thus difficult to use in these cases. Imposition of P2₁ PBC to $\mathcal{A}$(APL), $\mathcal{A}$(SA), and $\mathcal{A}$(0-DS) results in better agreement in mechanical properties between $\mathcal{A}$(APL/P2₁), $\mathcal{A}$(SA/P2₁), and $\mathcal{A}$(0-DS/P2₁), including nonzero differential stress. These results are inconsistent with the assumptions of 0-DS.

The simulations reported here are consistent with the recent theoretical framework of HD (36), in which the bilayer mechanical properties are outcomes of the interplay between bending and asymmetric lipid packing. Here, we extended their results by considering the bilayer and differential stresses (Σ and Δ) from both the bending and area strain. From the torque balance condition for a curvature-relaxed bilayer (where the net torque vanishes), we show that the net differential stress Δ₀ in the membrane with P1 PBC is identical to the residual differential stress $\left({\text{Δ}}_0^{\text{res}}\right)$, which is an externally imposed differential stress via P1 PBC that is required to force a membrane to be flat (c = 0). We derived an expression for the differential stress in a planar bilayer ((17a), (17b), (17c)) and practically applicable relations between Δ₀ and the torque densities (Eqs. 20 and 21). In addition, we defined a quantitative measure (stress index, σ) for the stress arising from P1 PBC (Eq. 26). The estimates from Eq. 20 are significantly improved compared with ${\text{Δ}}_0^{\text{s}}$ from the area strain (Eq. 12b), and those from Eq. 21 show excellent agreement with Δ_sim from our work. However, the agreement with the data from a simulation study of Miettinen and Lipowsky (81) becomes poor at higher asymmetry between leaflets, which we attribute to possibly softened bending rigidity arising from ganglioside GM1. The good agreement in Δ₀ between theory (Eqs. 20 and 21) and our simulations and consistent simulation results with the stress index (Eq. 26) further support the theoretical framework.

In our P2₁ simulations, the lipid asymmetry is modeled by specific lipid types restrained to reside in their original leaflet (m1 in this work), and by interleaflet redistribution of the other lipids via P2₁ PBC. Though it is not rigorous, our model qualitatively describes the lipid asymmetry in cells, in which specific lipid types are restricted to one leaflet and asymmetric distributions of the others are dynamically regulated in a much shorter timescale (by specific proteins) than that for the spontaneous flip-flop. Thus, our P2₁ method likely allows a steady state resembling the lipid asymmetry in biological membranes. In our asymmetric bilayer systems, mechanical properties from $\mathcal{A}$(SA) did not change significantly after P2₁ simulation, in which the stress index σ remained well below 1 (Eq. 26). Considering these, our practical recommendation for generation of initial conditions for asymmetric bilayer simulations is the SA-based simulation followed by a confirmatory simulation using the P2₁ PBC (i.e., the SA/P2₁) or at least the SA approach (when Δ₀ is small). In addition, we hope that the P2₁ PBC can be implemented in other molecular dynamics engines. Of note, a recently implemented modified P2₁ in CHARMM shows excellent scalability comparable to P1 (53). Concurrently, studies validating the approximation of P2₁ through P1 PBC in the checkerboard arrangement—resulting from a four-patch simulation box (see Fig. 2B) with the appropriate flat-bottomed potentials in the xy plane to retain the patch compositions—would be most welcome.

Glossary

A list of symbols and notations in our work is given in Table 6.

Table 6.

Glossary

Symbol	Description
m1, m2, b	subscripts denoting upper (m1) and lower leaflets (m2) and bilayer (b)
c	bilayer curvature, measured at midplane
$k_{\text{b}}^{\text{b}}$, $c_{0,\text{b}}^{\text{b}}$	bilayer bending modulus and spontaneous curvature associated with bending, (7a), (7b), (7c)
$k_{\text{b}}^{\text{s}}$, $c_{0,\text{b}}^{\text{s}}$	nonlocal bilayer bending modulus and area strain relaxed bilayer curvature, (8a), (8b), (8c)
$c_{0,\text{b}}^{\ast }$	bilayer spontaneous curvature, Eq. 10
γm, Σ, Δ	leaflet, bilayer, and differential stresses, Eqs. 1, (11a), (11b), and (12a), (12b)
Σ(c), Δ(c)	bilayer and differential stresses in a bilayer at a curvature, $c$
Σb, Δb	bilayer and differential stresses from the bending, (11a), (11b)
Σs, Δs	bilayer and differential stresses from the area strain, (12a), (12b)
ΣI, ΔI	(constant) intrinsic bilayer and differential stresses required for net zero bilayer and differential stresses at a reference bilayer curvature, c = cr in the absence of applied surface tension, (13a), (13b)
Σres, Δres	residual bilayer and differential stresses with respect to those at a reference point, c = cr: Σres ≡ Σ − Σ(c = cr) and Δres ≡ Δ − Δ(c = cr), (16a), (16b)
${\text{Σ}}_0^{\text{res}}$, ${\text{Δ}}_0^{\text{res}}$	residual bilayer and differential stresses in a planar bilayer (c = 0) with respect to those at cr = $c_{0,\text{b}}^{\ast }$, (17a), (17b), (17c)
${\overline{F}}_{\text{m}}^{\prime }$(0),${\overline{F}}_{\text{b}}^{\prime }$(0)	leaflet and bilayer torque densities, (3a), (3b)
σ	stress index arising from the energetic penalty in a planar bilayer with P1 PBC: tensional cost in the units of thermal energy (kBT), Eq. 26
$k_{\text{m}}^{\text{b}}$, $c_{\text{m}}^{\text{b}}$	monolayer bending modulus and spontaneous curvature
Am,0, A0	leaflet and bilayer SAs in their stress-free state
KA,m, KA	monolayer and bilayer area compressibility moduli
ɛm1,0, ɛm2,0	monolayer area strains from asymmetric lipid packing
ɛm1, ɛm2	monolayer area strains
zm1, zm2	distances of monolayer neutral surfaces from bilayer midplane
z0	value of zm1 and zm2 if it is assumed that zm1 = zm2
dm, db, dh	monolayer and bilayer thicknesses and bilayer hydrophobic thickness

Author contributions

All authors designed the project, discussed the results, and wrote the manuscript. S.P. carried out the simulations and analysis.

Editor: Siewert-Jan Marrink.

Appendix a: area strain and associated energy in an asymmetric bilayer

In this appendix, we describe energy densities from area strains in a differentially stressed bilayer and define the nonlocal curvature elastic energy density from the local energy density.

Let us start with considering a differentially stressed asymmetric bilayer from two tensionless monolayers, m1 and m2, of areas A_m1,0 and A_m2,0. Consider the area strains in the planar asymmetric bilayer. When there is an area mismatch (A_m1,0 ≠ A_m2,0) between monolayers m1 and m2, these would be laterally expanded or compressed to match their SAs to a bilayer SA, A. The area strain in monolayer m is then given by

$${\epsilon }_{\text{m},0}\equiv A/A_{\text{m},0}-1.$$ (A1)

The elastic energy density (per unit area) of the planar bilayer from area strains is given by

$${\text{e}}_{\text{b}}^{\text{s}}={\text{e}}_{\text{m}1}^{\text{s}}+{\text{e}}_{\text{m}2}^{\text{s}}\text{and}{\text{e}}_{\text{m}}^{\text{s}}=K_{A,\text{m}}{\epsilon }_{\text{m}}^2/2,$$ (A2)

where ${\text{e}}_{\text{m}}^{\text{s}}$ is the energy density from area strain in the monolayer m. The differentially stressed bilayer can bend to relax the area strains in monolayers (analogous to a bimetallic strip). From Eq. A1 and the parallel surface theorem (47), the area strains at a bilayer curvature c are obtained as

$${\epsilon }_{\text{m}1}\left(c\right)=\left(1+{\epsilon }_{\text{m}\mathrm{1,0}}\right)z_{\text{m}1}\left(c+\frac{{\epsilon }_{\text{m}\mathrm{1,0}}/z_{\text{m}1}}{1+{\epsilon }_{\text{m}\mathrm{1,0}}}\right)\approx z_{\text{m}1}\left(c+c_{0,\text{m}1}^{\text{s}}\right),$$ (A3a)

$${\epsilon }_{\text{m}2}\left(c\right)=-\left(1+{\epsilon }_{\text{m}\mathrm{2,0}}\right)z_{\text{m}2}\left(c-\frac{{\epsilon }_{\text{m}\mathrm{2,0}}/z_{\text{m}2}}{1+{\epsilon }_{\text{m}\mathrm{2,0}}}\right)\approx -z_{\text{m}2}\left(c-c_{0,\text{m}2}^{\text{s}}\right),$$ (A3b)

and

$$c_{0,\text{m}}^{\text{s}}={\epsilon }_{\text{m},0}/z_{\text{m}},$$ (A3c)

where, at approximation sign (≈), it is assumed that |ɛ_m,0| << 1.

HD noted that the meaningful area strain is not the local one but the one distributed over the whole membrane because the individual leaflets can slide (36). Based on this notion, one needs to express the elastic energy density as the nonlocal curvature elastic energy density ${\text{e}}_{\text{nl}}^{\text{s}}$,

$${\text{e}}_{\text{nl}}^{\text{s}}\equiv \frac{\underset{S}{\int }dA{\text{e}}_{\text{b}}^{\text{s}}}{\underset{S}{\int }dA},$$ (A4a)

where ${\text{e}}_{\text{b}}^{\text{s}}$ is the energy density (per unit bilayer area) at its curvature c,

$${\text{e}}_{\text{b}}^{\text{s}}\left(c\right)=\left(1+c{z}_{\text{m}1}\right){\text{e}}_{\text{m}1}^{\text{s}}+\left(1-c{z}_{\text{m}2}\right){\text{e}}_{\text{m}2}^{\text{s}}.$$ (A4b)

Combining (A4a), (A4b), A2, and (A3a), (A3b), (A3c) with an additional approximation, |c| << 1, one obtains (8a), (8b), (8c).