Spontaneous curvature generation by peptides in asymmetric bilayers

Abstract

Peptides and proteins play crucial roles in membrane remodeling by inducing spontaneous curvature. However, extracting spontaneous curvatures from simulations of asymmetric bilayers is challenging because differential stress (i.e., the difference of the leaflet surface tensions) arising from leaflet area strains can vary substantially among initial conditions. This study investigates peptide-induced spontaneous curvature ${\delta c}_0^{\mathrm{p}}$ in asymmetric bilayers consisting of a single lipid type and a peptide confined to one leaflet; the ${\delta c}_0^{\mathrm{p}}$ are calculated from the Helfrich equation using the first moment of the lateral pressure tensor and an alternative expression using the differential stress. It is shown that differential stress introduced during initial system generation is effectively relaxed by equilibrating using P2₁ periodic boundary conditions, which allows lipids to switch leaflets across cell boundaries and equalize their chemical potentials across leaflets. This procedure leads to robust estimates of ${\delta c}_0^{\mathrm{p}}$ for the systems simulated, and is recommended when equality of chemical potentials between the leaflets is a primary consideration.

GRAPHICAL ABSTRACT

The tendency of a membrane to bend, its spontaneous curvature, is modulated by peptides. This important property is difficult to calculate for asymmetric membranes using computer simulations because the leaflet surface tensions are not necessarily the same, and their difference (differential stress) varies. This paper shows how equilibration with P2₁ periodic boundary conditions, where lipids can switch leaflets across cell boundaries and equalize their chemical potentials, leads to robust estimates of peptide-induced spontaneous curvatures.

I. Introduction

Experimental methods for determining the asymmetry of cell membranes¹ and for making asymmetric liposomes^2–5 are now readily available. Simulations of asymmetric bilayers have also become common,^4,6–11 though these studies face technical and theoretical problems not encountered for symmetric ones. The first concerns the initial conditions. For an unstressed symmetric bilayer, both the overall and leaflet surface tensions are zero. Hence, it is straightforward to equilibrate most reasonably assembled starting structures. In contrast, leaflet surface tensions are not necessarily zero for asymmetric bilayers, and their values are not available a priori.¹² A related problem involves estimating the leaflet spontaneous curvature, $c_0^{\mathrm{m}}$.

For a symmetric bilayer, $c_0^{\mathrm{m}}$ can be obtained from the pressure tensor and leaflet bending modulus using the Helfrich equation.^13,14 Generation of spontaneous curvature by peptides and proteins is a critical step in membrane remodeling from pore formation to fusion,¹⁵ and simulations are routinely used to estimate this quantity for symmetric bilayers.^13,14,16,17 However, as elaborated by Hossein and Deserno,¹² extracting spontaneous curvatures for the leaflets of asymmetric bilayers requires numerous assumptions that are difficult to test. Fortunately, a Helfrich-based expression⁷ can be used for the entire bilayer due to the invariance of the bilayer bending moment with respect to the position of the bilayer midplane.

This paper focuses on asymmetric bilayers containing peptides in one leaflet and a single lipid species in both leaflets; this arrangement is termed here a “peptide-asymmetric bilayer”. Following assembly of the bilayers using an area per lipid (APL)-based method,^18–21 the lipids were equilibrated using P2₁ periodic boundary conditions (PBC),²² which equalizes their chemical potentials in both leaflets (Figure 1, left). The peptides, by construction, are confined to one leaflet, so asymmetry is guaranteed. While equal chemical potential is not the only plausible criterion for optimal lipid distribution in an asymmetric bilayer, it is a reasonable starting point for examining the ramifications of membrane asymmetry. To this end, after P2₁ PBC simulations, each system was then simulated in the standard P1 PBC (Figure 1, right), and relevant mechanical quantities were evaluated and compared with those calculated before equilibration with P2₁. The calculated mechanical quantities include the difference in the surface tensions of each leaflet, defined as the differential stress $\mathrm{\Delta }$, and the first moment of the pressure tensor over the bilayer. Substantial changes in these quantitates were shown previously for asymmetric lipid-only bilayers before and after equilibration with P2₁,⁹ but the present peptide-asymmetric systems are conceptually more straightforward and may be of more practical interest.

Figure 1.

Schematic of P2₁ (left) and P1 (right) periodic boundary conditions (PBC) for a peptide-asymmetric bilayer with a single peptide (yellow disc, representing a gramicidin A monomer). Lipids are represented as spheres, with light or dark blue for those in the peptide containing upper (cis) leaflet and pink or red for those in the lower (trans) leaflet. In P2₁ PBC, a lipid exiting the cis leaflet enters the trans, while it remains in cis in P1. Equilibration in P2₁ equalizes the chemical potentials of the lipids in both leaflets, similar to lipid flip-flop, which is not accessible on simulation timescales.

Two different peptide types are examined: gramicidin A (gA) monomers (individual and fused to model larger assemblies) and the fusion peptide (FP) from influenza hemagglutinin (see Figure S1 for their structures). gA is a β^6.3-helical peptide spanning vertically along the membrane normal due to anchoring Trp residues.^23,24 FP is an amphiphilic α-helical peptide adopting a tight α-helical hairpin arrangement;²⁵ it is surface associated at low concentrations, but aggregates and forms pores at higher concentrations.¹⁰ Curvature generation by proteins is a critical component of membrane remodeling,¹⁵ and the preceding two peptides are natural targets for quantitative studies. Specifically, upon dimerization of gA monomers in opposing leaflets, a monovalent cation channel is formed. This inclusion can alter the membrane spontaneous curvature and enrich hydrophobically matched lipids at the channel-lipid boundary²⁶ as discussed extensively by Sodt et al.²⁷ Hence, it becomes essential to quantify both bending and compression energetics. An asymmetric system of interest for FP involves aggregates of FP in pure 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), cholesterol (Chol):POPC, and 1,2-dioleoyl-sn-glycerol (DOG):POPC, where pore formation is highly correlated with the spontaneous curvatures of the pure lipid mixtures.¹⁰ Estimating the spontaneous curvature generation by the peptides in these mixtures is critical for understanding the experimental result that pore formation in the Chol:POPC and DOG:POPC mixtures is nearly identical, despite the significant differences in Chol and DOG.

Here, we focus on the spontaneous curvature of the entire bilayer and thereby circumvent the difficulties in estimating leaflet spontaneous curvatures. While details regarding the contributions of each leaflet and their coupling are lost, this approach directly yields a quantity of great interest to cell biology, the propensity of bending for the entire bilayer. The effects of equilibration with P2₁ and P1 PBC on differential stress and the first moment of the pressure tensor over the entire bilayer for these peptides, and two different ways of calculating the bilayer spontaneous curvature are presented in the Results. The Discussion covers the following topics: Sensitivity of calculated spontaneous curvature to deformation and system size; Estimates of the requirement for P2₁; Helfrich model with P2₁; General applicability of P2₁ periodic boundary conditions.

II. Methods

Spontaneous curvature generation by peptides

This section briefly describes estimation of the spontaneous curvature generated by a peptide in asymmetric bilayers from the bilayer bending moment or differential stress. We limit consideration to asymmetric bilayers with a peptide included in one leaflet (defined cis) and a single lipid type. For simplicity, the Gaussian curvature of the bilayer is not considered, which is valid for weakly curved membranes.

We start from the deformation energy density (${\overline{F}}_{\mathrm{b}}$) of a bilayer defined from its pivotal surface (where surface area does not change with bending), which is known as the Helfrich Hamiltonian,²⁸

$${\overline{F}}_{\mathrm{b}}=\frac{k_{AC}}{2}{\left(c-c_0\right)}^2$$ (1)

where $k_{AC}$ is the bilayer bending modulus, $c$ is the bilayer curvature, and $c_0$ is the spontaneous curvature at which the bending moment ($\partial {\overline{F}}_{\mathrm{b}}/\partial c$) vanishes. $k_{AC}$ defined for the bilayer midplane can be expressed using leaflet elastic moduli defined for two neutral surfaces^12,29,30

$$\begin{array}{l}{k}_{AC}=k_C+k_A;\hfill \\ k_A=K_A{z}_0^2\hfill \end{array}$$ (2)

where $k_C$ is the sum of the leaflet bending moduli, $k_A$ is the bilayer bending modulus associated with the area strain, $K_A$ is the sum of the leaflet area compressibility moduli, and $z_0$ is the mean distance of the leaflet’s neutral surfaces from the bilayer midplane. The second term, $k_A=K_A{z}_0^2$, can be understood as the area stretch contribution, appearing in the re-expressed leaflet bending moduli for the bilayer midplane, ²⁹ or that in the renormalized bilayer bending modulus.³⁰

In typical membrane simulations, the bilayer is planar ($c=0$) due to the applied P1 PBC and the net tension vanishes. The bending moment creates tensions ${\gamma }_1$ and ${\gamma }_2$ in leaflets 1 and 2, respectively, whose difference is defined as the differential stress $\mathrm{\Delta }\equiv {\gamma }_1-{\gamma }_2$. Asymmetry in leaflet bending moduli and associated spontaneous curvatures, as well as lipid packing (area strain) between leaflets contribute to $\mathrm{\Delta }$.⁹ The tensions and bending moment can be expressed as the zeroth and first moments of the lateral pressure profile, $p\left(z\right)\equiv p_{\mathrm{T}}\left(z\right)-p_N\left(z\right)$^31,32

$${\gamma }_1=-{\int }_0^{L_z/2}dz\left[p_{\text{T}}\left(z\right)-p_{\text{N}}\left(z\right)\right]$$ (3a)

$${\gamma }_2=-{\int }_{-L_z/2}^{0}dz \left[p_{\text{T}}\left(z\right)-p_{\text{N}}\left(z\right)\right]$$ (3b)

$$\begin{gathered} {\overline{F}}_{\text{b}}^{\text{'}}\left(0\right)=-k_{AC}{c}_0 \\ =-{\int }_{-L_z/2}^{L_z/2}dz z \left[p_{\text{T}}\left(z\right)-p_{\text{N}}\left(z\right)\right] \end{gathered}$$ (3c)

where ${\gamma }_i$ is the tension of the leaflet i, $p_{\mathrm{T}}\left(z\right)$ and $p_{\mathrm{N}}\left(z\right)=p_{\mathrm{N}}$ are the tangential and normal components of the pressure tensor, $z$ is the coordinate normal to the membrane surface, and $L_z$ is the simulation box dimension along the z-direction.

From the bending moments of the leaflets defined for the bilayer midplane, ${\overline{F}}^{\prime }\left(0\right)$ can also be written as²⁹

$$\begin{gathered} {\overline{F}}_{\text{b}}^{\prime }\left(0\right)={\overline{F}}_{1,\text{s}}^{\prime }\left(0\right)-{\overline{F}}_{2,\text{s}}^{\prime }\left(0\right)+z_0\Delta \\ \equiv {\overline{F}}_{\text{b},\Delta }^{\prime } \end{gathered}$$ (3d)

where ${\overline{F}}_{i,\mathrm{s}}^{\prime }\left(0\right)$ is the leaflet free energy derivative with respect to curvature from the planar cognate symmetric bilayer of the leaflet i. The symbol ${\overline{F}}_{\mathrm{b},\mathrm{\Delta }}^{\prime }$ is introduced to facilitate comparison with ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ obtained from Equation (3c). An analogous relation was obtained by Park et. al⁹ from the destressing approximation to $p\left(z\right)$ (Equation 3c) presented by Hossein and Deserno.¹²

Now let us suppose that a peptide is included in the cis leaflet of the bilayer and assume that it does not alter the elastic moduli. The change in the spontaneous curvature of the bilayer, $\delta c_0$, can then be readily expressed from the surface coverage weighted average of the spontaneous curvatures of the symmetric bilayer ($c_{0,\mathrm{s}}=0$) and the peptide ($c_0^{\mathrm{p}}$),^33,34

$$\delta c_0=\varphi \left(c_0^{\text{p}}-c_{0,\text{s}}\right)=\varphi \delta c_0^{\text{p}}$$ (4)

where $\varphi$ is the area fraction of peptide and ${\delta c}_0^{\mathrm{p}}$ is the spontaneous curvature generated by the peptide when it replaces the same surface area of the membrane. The second equality in Equation (4) will not hold in general for bilayers with compositionally asymmetric lipid distributions.

The spontaneous curvature generated by the peptide for the entire bilayer can be related to the bending moment⁷ or the differential stress. The following relation for $\delta c_0$ is obtained from Equations (3c) and (4)

$$\delta c_0=\varphi \delta c_0^{\text{p}}=-\frac{{\overline{F}}_{\text{b}}^{\prime }\left(0\right)}{k_{AC}}$$ (5a)

In addition, following Hossein and Deserno,¹² we assume that the intrinsic leaflet spontaneous curvatures (from their peptide-free cognate symmetric bilayers) are material properties, which are not altered by small perturbations (e.g., differential area strain^9,12). With this assumption, one can identify the first and second terms in the right hand side of the first equation in Equation (3d) as the leaflet bending moments from the corresponding peptide-free cognate symmetric bilayers. Therefore, the effects of peptide inclusion are solely manifested in $\mathrm{\Delta }$ (the last term in the right hand side of the first equation in Equation 3d).^9,12 Combining Equations (3c), (3d), and (4), one obtains an alternative expression for $\delta c_0$,

$$\delta c_0=\varphi \delta c_0^{\text{p}}=-\frac{d_{\text{h}}\Delta }{2k_{AC}}$$ (5b)

where we set $z_0=d_{\mathrm{h}}/2$, and $d_{\mathrm{h}}$ is the bilayer hydrophobic thickness.

Equation (5b) is slightly simpler than Equation (5a) (only the zeroth moment of the pressure tenor is needed), but an additional assumption is required. The predictions for each are compared in the Results.

Simulation details

We considered asymmetric bilayers made up of a single lipid type and a peptide included in the cis leaflet. Two lipid types were used: POPC and 1,2-dinervonoyl-sn-glycero-3-phosphocholine (DNPC). A series of gA-based peptides were simulated: a single monomer (termed gA), a fused tetramer (gA4), a fused nonamer (gA9), nine gA monomers (9gA), and a fused tetramer of a gA mutant whose Trp residues were replaced by Gln (gA4^W2Q). See Figure S1 for molecular images of the preceding compounds and sequences of the peptides. For the generation of fused gA assemblies, gA monomers were evenly spaced (on 2 × 2 or 3 × 3 grids with a grid spacing about the diameter of gA) followed by the alignment of Trp residues between the nearest gA monomers. These aligned Trp residues were then mutated to Cys residues to form a disulfide bond between the nearest gA pairs. A series of energy minimization and/or implicit membrane simulations were followed to remove bad contacts in the fused gA assemblies. These gA-based assemblies allowed an examination of the size dependence of chemically similar single leaflet spanning peptides. In addition, the influenza FP was simulated for a comparison with results from previous simulations of a symmetric POPC bilayer with one FP in each leaflet.¹⁰

Asymmetric bilayers were generated following the procedure described in our previous work.⁹ First, bilayers were generated using the area per lipid (APL) method, where leaflet areas were calculated using APLs of lipid types from homogenous bilayers and the two leaflets were area matched. This was followed by equilibration using P2₁ PBC, to reduce potential stresses generated by the initial step. Hereafter, the asymmetric bilayers generated by APL and APL/P2₁ methods are denoted 𝒜(APL) and 𝒜(APL/P2₁), respectively. For a chosen peptide-lipid bilayer system at a given peptide area fraction ($\varphi$), an initial asymmetric bilayer for 𝒜(APL) was generated by CHARMM-GUI Membrane Builder,^18,19 where the number of lipids in each leaflet was determined by matching the leaflet surface areas using the area per lipid from its homogeneous bilayer.^18–21 The generated initial 𝒜(APL) includes a lipid bilayer with a peptide in the middle of the upper leaflet, and 22.5 Å – 28.5 Å of bulk water with 0.15 M KCl both above and below the bilayer to ensure flat $p\left(z\right)$ in bulk regions (Figure S2).

The generated 𝒜(APL) were subjected to a series of initial constant volume and temperature (NVT) simulations followed by constant pressure and temperature (NPT) simulations. During the initial simulations, potentials to restrain the position and dihedral angles of the components were gradually relaxed to zero, after which a 500-ns NPT simulation with P1 PBC was carried out. The final snapshot was converted for a subsequent P2₁ equilibration, consisting of a series of short P2₁ simulations (a total of 0.3 ns with or without the same restraining potentials for initial NPT simulations of 𝒜(APL)), followed by a 50-ns P2₁ production run without a restraining potential, except for the 9gA system. In this case, following our previous work,⁹ two flat-bottomed harmonic potentials were applied to the center of mass of each peptide to keep them in one leaflet. A representative snapshot (in terms of the number of lipids in each leaflet from the distribution over the last 30-ns, see Figures S3 and S4) was converted back for P1 simulation, 𝒜(APL/P2₁). The same initial series of short NVT and NPT equilibrations were carried out, followed by a 500-ns production run. Modified input scripts from CHARMM-GUI Membrane Builder were used for conversion between P2₁ and P1 PBCs.⁹ Figure 2 illustrates the general procedure for gA9.

Figure 2.

Generation of 𝒜 (APL/P2₁) for gA9 in a DNPC bilayer (gA9-DNPC). The initial condition of 𝒜(APL) (left, at t = 0 ns) was generated by Membrane Builder, where water and ions are placed above and below the membrane. Subsequently, a series of short initial equilibration runs were carried out to prepare for production run, during which restraints to hold the positions and dihedral angles of components were gradually relaxed to vanish. Then, a 500-ns simulation of 𝒜(APL) was run, during which significant lipid condensation was developed in the trans leaflet (two middle panels at t = 1 ns and t = 500 ns). A P2₁ equilibration followed using the snapshot of 𝒜(APL) at t = 500 ns, during which the tight packing in the trans leaflet was relaxed. The snapshot with the representative lipid composition at the closest time to the final one (t = 500 ns) was chosen for a subsequent 500-ns P1 simulation, 𝒜(APL/P2₁) (at t = 500 ns, right). Each snapshot shows the heavy atoms of gA9 (ice blue), those of trans leaflet lipids in contact with gA9 (white), and phosphorous (orange). The other components are omitted for clarity.

As summarized in Tables 1 and 2, one or two additional peptide area fractions ($\varphi$) were examined for each system to obtain more reliable estimates of the spontaneous curvature generation by the peptide (Equation 5). The initial lipid numbers in two leaflets were determined based on those for the previously simulated 𝒜(APL/P2₁). Five independent replicas were simulated for each peptide-bilayer system at each $\varphi$ for better statistics.

Table 1.

Simulated asymmetric bilayers.

System[a]	gA-POPC		gA-DNPC		gA4-DNPC		gA4W2Q-DNPC		gA9-DNPC		9gA-DNPC		FP-POPC
Method[b]	N c [c]	N t [d]	N c	N t	N c	N t	N c	N t	N c	N t	N c	N t	N c	N t
APL	118	122	119	124	64	80	64	80	48	79	90	135	67	72
APL/P21	72	74	75	76	67	77	68	76	52	75	103	122	69	70
	120	120	121	122	105	115	105	115	110	132	148	165	109	110
									199	221	258	275

^[a]Peptide-lipid. gA: gramicidin A monomer; gA4: fused tetrameric gA assembly; gA9: fused nonameric gA assembly; 9gA: nine gA monomers; FP: fusion peptide of influenza hemagglutinin.

^[b]APL: individual area per lipid method; APL/P2₁: generation method using a P2₁ equilibration of (APL).

^[c]N_c: the number of lipids in the cis leaflet.

^[d]N_t: the number of lipids in the trans leaflet.

Table 2.

Peptide area fraction ($\varphi$) of simulated asymmetric bilayers.

System	gA-POPC	gA-DNPC	gA4-DNPC	gA4W2Q-DNPC	gA9-DNPC	9gA-DNPC	FP-POPC
Method
APL	0.04	0.04	0.20	0.19	0.40	0.31	0.08
APL/P21	0.06 (${\varphi }_1$)	0.06 (${\varphi }_1$)	0.19 (${\varphi }_1$)	0.18 (${\varphi }_1$)	0.38 (${\varphi }_1$)	0.28 (${\varphi }_1$)	0.08 (${\varphi }_1$)
	0.04 (${\varphi }_2$)	0.04 (${\varphi }_2$)	0.13 (${\varphi }_2$)	0.13 (${\varphi }_2$)	0.22 (${\varphi }_2$)	0.21 (${\varphi }_2$)	0.05 (${\varphi }_2$)
					0.14 (${\varphi }_3$)	0.13 (${\varphi }_3$)

Different peptide area fractions for (APL/P2₁) are denoted by ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$, where ${\varphi }_1>{\varphi }_2{>\varphi }_3$ for each peptide-bilayer system.

All P1 simulations were carried out using OpenMM³⁵ with the CHARMM36m protein³⁶ and lipid³⁷ force fields, and the TIP3P water model.^38,39 The integration time step was set to 2 fs with the SHAKE algorithm.⁴⁰ A shorter time step of 1 fs was used only for the initial short NVT and NPT runs. For P2₁ simulations, we used either the original implementation²² or a modified version⁴¹ of CHARMM⁴² with the same force fields and integration time steps for P1 simulations. The modified version of P2₁ in CHARMM shows excellent scalability comparable to P1.⁴¹

The Lennard-Jones interactions were calculated by a force-based switching function⁴³ which was switched off over 10–12 Å. The long-range electrostatic interactions were calculated by the particle mesh Ewald method⁴⁴ with a mesh size of ~ 1 Å. Temperature (T = 303.15 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,^45,46 with a pressure coupling frequency of 100 steps in OpenMM simulations. In CHARMM simulations, these were controlled by a Nosé-Hoover thermostat⁴⁷ and Langevin piston method⁴⁸ with a collision frequency of 20 ps⁻¹.

During the production runs, coordinates were saved every 0.1 ns and 10 ps for P1 and P2₁ simulations, respectively. For P1 simulations, velocity trajectories were also saved with the same interval, for which two in house python classes,⁹ VELFile and VELReporter, were used. The final 300-ns of the 𝒜(APL/P2₁) trajectories were used for analysis (Figure S5) and, for a chosen $\varphi$, the same length of the 𝒜(APL) trajectories were also analyzed.

Analysis

Prior to analysis, we aligned the bilayer midplane at $z=0$ along the membrane normal (z-direction). Although this could be done with more sophisticated methods,^9,14,49 due to the simplicity of the membranes considered here, we estimated the bilayer center as the average of the mean z-positions of phosphorous atoms in both leaflets. From the trajectories, the system size, $K_A$, bilayer hydrophobic thickness ($d_{\mathrm{h}}$), leaflet tensions and differential stress ($\mathrm{\Delta }$), and bilayer bending moment (${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$) were calculated. The standard errors in these properties were calculated over five independent simulations. For derived properties, such as the sum and product of these properties, the uncertainties were calculated by error propagation using standard errors.

The membrane area, $A$, was calculated from the lateral dimensions of the simulation box within the plane of the membrane (x and y). From its fluctuation, the bilayer $K_A$ was estimated using

$$K_A=\frac{k_{\text{B}}T{A}_0}{〈{\left(A-A_0\right)}^2〉}$$ (6)

where $k_{\mathrm{B}}$ is the Boltzmann’s constant and $A_0$ is the average area of the bilayer.

The hydrophobic thickness of the bilayer (d_h) was calculated as the difference of the average position of the C22 and C32 lipid tail atoms (i.e., the carbon atoms bonded to the lipid carbonyl group) in both leaflets. In addition, the radial hydrophobic thickness profiles, $z_{\mathrm{c}\mathrm{i}\mathrm{s}}\left(r\right)$ and $z_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}\left(r\right)$, were calculated, which are z-positions of the leaflet hydrophobic surfaces from the average z-position of the bilayer midplane (at $z=0$). To calculate tensions and the bilayer bending moment, $p\left(z\right)=p_{\mathrm{T}}\left(z\right)-p_{\mathrm{N}}\left(z\right)$, was calculated using a developmental version of CHARMM, where $p_{\mathrm{T}}\left(z\right)$ was calculated using the Harasima contour contour for slab geometry.⁵⁰ Because $p_{\mathrm{N}}\left(z\right)$ is not available from the Harasima contour,⁵¹ it was calculated as $p_{\mathrm{N}}=\int dz p_{\mathrm{T}}\left(z\right)/L_z$, where $L_z$ is the z-dimension of the simulation box, and the integral was carried over the entire simulation box along the z-direction. The tensions and bending moments were calculated from the pressure profile using Equation (3).

III. Results

Figure 2 shows snapshots of gA9 (the fused assembly of 9 gA monomers) in a DNPC bilayer during the course of simulations, from initial condition of 𝒜(APL) (left) to the final snapshot 𝒜(APL/P2₁) (right). Lipids in the trans leaflet condensed during the P1 simulation (two middle panels). Subsequent P2₁ equilibration relaxed the tight lipid packing in the trans leaflet through inter-leaflet lipid exchange, and the number of DNPC lipids in two leaflets eventually fluctuated around well-defined mean values (see Figures S3 and S4). In addition to relaxation of the tight lipid packing in the trans leaflet, the 500 ns snapshot of 𝒜(APL/P2₁) shows that gA9 remains below the average position of the lipid head groups in the cis leaflet and induces curvature in both leaflets. These snapshots clearly illustrate the essential role of the P2₁ equilibration following the generation of 𝒜(APL).

Figure 3 plots mechanical properties $K_A$, $\mathrm{\Delta }$, and ${\overline{F}}^{\prime }\left(0\right)$ of the simulated peptide-bilayer systems (see Table S1 for numerical values). $K_A$ is relatively independent of composition and initial conditions (Figure 3, top panel), which supports our assumption that the elastic moduli are not affected by peptide inclusion. In contrast, $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ are sensitive to initial conditions and peptide area fraction $\varphi$; see Table 2 for the numerical values of ${\varphi }_i$).

Figure 3.

Mechanical properties of simulated bilayers: (top to bottom) the area compressibility modulus ($K_A$), differential stress ($\mathrm{\Delta }$), and the bilayer bending moment (${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$). In each panel, for each system, data from 𝒜(APL) are shown in the left (red), followed by data from 𝒜(APL/P2₁) at different peptide area fractions (${\varphi }_1$ (green) > ${\varphi }_2$ (blue) > ${\varphi }_3$ (cyan)). The error bars are standard errors from the last 300-ns of five independent simulations. Gray vertical dotted lines demarcate the data for different peptide-bilayer systems.

Large $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ are developed in 𝒜(APL) for the gA monomer and its fused assemblies due to poorly controlled leaflet surface areas in the generation of its initial condition, and tended to increase with peptide size. However, there is no significant difference between the single large 9-gA assembly peptide (gA9) and nine small gA monomers (9gA) at comparable $\varphi$. Likewise, the mutation of Trp to Gln in gA4 (gA4^W2Q) does not affect $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ in 𝒜(APL) significantly. These results indicate that $\varphi$ is a dominant contributor to $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ for 𝒜(APL) with gA and its fused assemblies.

The large $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ in 𝒜(APL) were relaxed during the subsequent P2₁ equilibration and decreased in magnitude with decreasing $\varphi$ (Figure 3, middle and bottom panels). Note the sign changes in $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ for the bilayers with gA and FP after P2₁ is applied; in these cases, the inferred sign of the spontaneous curvature obtained in the APL simulations would be incorrect. While the results for gA9-DNPC and 9gA-DNPC remain comparable, $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ for gA4^W2Q-DNPC are significantly smaller than those for gA4-DNPC, consistent with the reported reduced bilayer perturbation by the Gln mutant of gA compared with Trp in gA.⁵²

Calculated $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ (from Equation 3c) are highly correlated for all compositions and initial conditions (upper panel in Figure 4). This is reflected in the excellent agreement between the direct estimate of ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ from Equation (3c) and the ${\overline{F}}_{\mathrm{b},\mathrm{\Delta }}^{\prime }$ from Equation (3d) (lower panel in Figure 4), and validates our assumption for Equation (5b); i.e., that peptide inclusion is primarily manifested in $\mathrm{\Delta }$, at least for the peptides in POPC and DNPC bilayers in this study. In addition, the good agreement verifies that the current choice of the bilayer midplane (the average z-position of phosphorous in the cis and trans leaflets) is sufficiently accurate and so is the hydrophobic thickness. For 𝒜(APL), ${\overline{F}}_{\mathrm{b},\mathrm{\Delta }}^{\prime }$ does not fully reproduce ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ for large $\varphi$ (> 0.19). Considering that the differential area strain grows with increasing $\varphi$, the deviations indicate that the assumption becomes poorer for bilayers with large differential area strain when P2₁ equilibration is not carried out.

Figure 4.

Correlation between $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ (Equation 3c) (upper panel) and comparison of bending moments from direct estimate (Equation 3c) and ${\overline{F}}_{\mathrm{b},\mathrm{\Delta }}^{\prime }$ (Equation 3d) (lower panel) from simulations. For clarity, error bars are shown only for 𝒜(APL). The line in the upper panel is the linear fit of ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ to $\mathrm{\Delta }$ and the diagonal line in the lower panel is included for a visual guide.

The upper panel of Figure 5 shows the peptide-induced bilayer spontaneous curvature, $\delta c_0$ ($=\varphi \delta c_0^{\mathrm{p}}$, Equation 4) for each system. The estimates of $\delta c_0$ from ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ (Equation 5a) and from $\mathrm{\Delta }$ (Equation 5b) agree well with each other as expected from Figure 4. The estimated $\delta c_0$ is negative for gA and positive for FP in the POPC bilayer, which we attribute to the difference in modes of action with the membrane between these peptides. This is consistent with the notion that the spontaneous curvature induced by a peptide changes from positive to negative as it inserts deeper into the hydrophobic core of the membrane⁵³ (see snapshots in Figure S6). The negative $\delta c_0$ by gA and its fused assembly in the DPNC bilayer can be interpreted as the net result of leaflet deformations (see Figure 5): larger negative curvature is generated by the peptide in the cis leaflet than in the trans leaflet, in P2₁ equilibrated bilayers. As expected, the magnitude of $\delta c_0$ decreases with decreasing $\varphi$.

Figure 5.

Spontaneous curvature induced by a peptide for simulated bilayers, $\delta c_0$ (upper panel) and $\delta c_0^{\mathrm{p}}$ (lower panel) calculated from Equation (5). The data for each peptide-bilayer system are arranged in the same order as in Figure 3. The bilayer bending modulus, $k_{AC}$ (Equation 2), was calculated using leaflet elastic moduli from the literature.¹⁶ Estimates from Equations (5a) and (5b) are plotted side by side for better visual comparison. Shown together are the induced leaflet spontaneous curvature, $\delta c_0^{\mathrm{m}}$ and $\delta c_0^{\mathrm{m},\mathrm{p}}$ (orange squares in top and bottom panels, respectively), for a symmetric FP-POPC bilayer (FP-POPC (sym)) where one FP was included in each leaflet. The $\delta c_0^{\mathrm{m}}$ and $\delta c_0^{\mathrm{m},\mathrm{p}}$ are defined analogously to Equation (4), and were calculated by an analogous equation to Equation (5a) using data from Ref. 9.

The peptide-induced leaflet curvature, $\delta c_0^{\mathrm{m}}$, for a symmetric FP-POPC bilayer,¹⁰ FP-POPC (sym), is close to that of a P2₁ equilibrated asymmetric FP-POPC at a nearly identical $\varphi$, albeit with large uncertainty accumulated from those in the raw data. Here, $\delta c_0^{\mathrm{m}}\equiv c_0^{\mathrm{m},\mathrm{p}}-c_0^{\mathrm{m}}$ is defined analogously to Equation (4), where $c_0^{\mathrm{m},\mathrm{p}}$ and $c_0^{\mathrm{m}}$ are leaflet spontaneous curvatures with and without a FP in the leaflet, respectively. The bilayer in FP-POPC (sym) is composed of two FPs, one in each POPC leaflet with $\varphi =0.08$ at T = 310 K; in this system, the leaflet area strains are relaxed except those generated by FP due to vanishing net tension.

One possible interpretation of the comparable $\delta c_0$ and $\delta c_0^{\mathrm{m}}$ is that the (natural) removal of differential area strain in the symmetric FP-POPC mimics the buffering effects of a larger membrane, thus $\delta c_0^{\mathrm{m}}$ is close to $\delta c_0$ from the P2₁ equilibrated asymmetric FP-POPC, which also mimics a large biological membrane.⁹ However, this particular result might not be general even at low peptide area fractions, e.g., when peptides in two leaflets interact (like the formation of a gA channel), the calculated $\delta c_0^{\mathrm{m}}$ from a symmetric bilayer would be different than $\delta c_0$ from the peptide-asymmetric bilayer. Hence, it would likely not be obtained for symmetric bilayers with large peptide area fractions as well.

The lower panel in Figure 5 plots the intrinsic spontaneous curvature generation, $\delta c_0^{\mathrm{p}}$ ($=\delta c_0/\varphi$), by a peptide from simulations. There is good agreement in $\delta c_0^{\mathrm{p}}$ for a given peptide-bilayer system generated by APL/P2₁ among different $\varphi$’s, except gA-POPC and gA-DNPC (whose $\varphi$ values are significantly smaller than the gA assembly-bilayer systems, see also Sensitivity of calculated spontaneous curvature to deformation and system size in the Discussion for details). The variations and uncertainties in $\delta c_0^{\mathrm{p}}$ for gA-POPC and gA-DNPC can be attributed to large uncertainties in $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ comparable to or larger than the magnitudes of mean values (the second and third entries in the first two sets in Figure 3). The results of the other peptide-bilayer systems support that Equation (4) is a valid assumption for 𝒜(APL/P2₁) at least for the bilayers considered here. In contrast, $\delta c_0^{\mathrm{p}}$ for 𝒜(APL) (the first entry for each set) is clearly different from those for 𝒜(APL/P2₁), which we attribute to excessive differential area strain introduced in the generation. Again, the intrinsic leaflet spontaneous curvature generation, $\delta c_0^{\mathrm{m},\mathrm{p}}$ ($=\delta c_0^{\mathrm{m}}/\varphi$), by FP from symmetric FP-POPC are slightly smaller than $\delta c_0^{\mathrm{p}}$ for asymmetric FP-POPC, albeit with larger uncertainty.

IV. Discussion

Sensitivity of calculated spontaneous curvature to deformation and system size

The preceding section demonstrated the large differences in $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ that might arise in P1 and P2₁ simulations of peptide-asymmetric bilayers containing gramicidin A monomers and the influenza fusion peptide. The $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ are calculated from the lateral pressure profile, $p\left(z\right)$, for the slab geometry, which are affected by membrane deformation and thus the system size. This subsection considers the sensitivity of calculated spontaneous curvature to the preceding quantites. We start with a brief discussion of undulatory and peristaltic modes of deformation, and proceed to the sensitivity of $\delta c_0^{\mathrm{p}}$ to the system size (i.e., $\varphi$).

Undulation can be important when $L_X\gg d_{\mathrm{h}}$ where $L_X$ is the lateral dimension of the system ($A_0=L_X^2$). Large undulations can blur the lateral pressure profile (thus affecting calculated $\delta c_0$ and $\delta c_0^{\mathrm{p}}$). Without external driving forces for the undulation, the maximum undulation height, $\delta z$, is proportional to the lateral dimension of the system, $L_X$, and limited by thermal fluctuation of the elastic energy of the membrane (interested readers are referred to S1. Threshold system size for a given maximum undulation height in SI). Hence, there exists a threshold system size, $L_X^{\delta z}$, for a given $\delta z$,

$$\frac{L_X^{\delta z}}{\delta z}=32\sqrt{\frac{k_{AC}}{2k_{\text{B}}T}}\approx 8d_{\text{h}}\sqrt{\frac{7K_A}{3k_{\text{B}}T}}$$ (7)

where the second equation is obtained with $k_{AC}=7K_A{d}_{\mathrm{h}}^2/24$ from the polymer brush model.⁵⁴ For $\delta z=1$ Å corresponding to a typical resolution of $p\left(z\right)$, the ratio, $L_X/L_X^{\delta z}$, remains well below unity (0.37–0.69, see also Figure S8), indicating that calculated $\delta c_0$ is not affected by undulations for the present systems, i.e., the bilayers are flat. The small ratios are consistent with the flat bulk membrane (Figure S7).

As shown in Figure S7, given that undulatory deformation is negligible, the deformation by a peptide is peristaltic, which is asymmetric between trans and cis leaflets leading to a non-vanishing $\delta c_0$ (Figure 5). The peptide-induced deformation varies with the system size (i.e., $\varphi$), in a manner that the membrane deforms more steeply around the peptide at higher $\varphi$ (smaller system size). The peptide-induced deformation is relaxed along the lateral distance (r) from the peptide and eventually vanishes when the system size is sufficiently large. However, at large $\varphi$, the deformation may not be fully relaxed due to the small system size (gA9 in the DNPC bilayer, see panel E in Figure S7). The residual deformation in the bulk region (with respect to the relaxed membrane) may impact the calculated $\delta c_0$ and $\delta c_0^{\mathrm{p}}$. On the other hand, at low $\varphi$, though the deformation can be fully relaxed in the bulk region, the effects of peptide inclusion are also small, which may lead to uncertainty issues. Thus, calculated $\delta c_0$ and $\delta c_0^{\mathrm{p}}$ may be sensitive to the membrane deformation as well as the system size.

In the present theory (Equation 5), $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ are linear in $\varphi$ (so is $\delta c_0$), which is verified by the insensitivity of $\delta c_0^{\mathrm{p}}$ to $\varphi$ for gA assemblies and FP in P2₁ equilibrated bilayers (lower panel in Figure 5). Considering that the hydrophobic thickness of gA9-DNPC at the highest $\varphi$ (~0.4) is larger than those at lower $\varphi$, the insensitivity of calculated $\delta c_0^{\mathrm{p}}$ suggests that P2₁ can evenly distribute the impact of the residual deformation to the two leaflets, so accurate estimates of $\delta c_0^{\mathrm{p}}$ from the peptide-inclusion are still possible even at high $\varphi$, at least for the simulated bilayers.

The variations and uncertainties in $\delta c_0^{\mathrm{p}}$ for gA-POPC and gA-DNPC (lower panel in Figure 5) show an intrinsic difficulty in accurate estimation of $\delta c_0$ and $\delta c_0^{\mathrm{p}}$ at low $\varphi$, which can be checked by calculating two ratios, $\mathrm{\Delta }\delta c_0/\varphi$ and $\mathrm{\Delta }\delta c_0/\left|\delta c_0\right|$, where $\mathrm{\Delta }\delta c_0$ denotes the uncertainty in the calculated $\delta c_0$. When $\mathrm{\Delta }\delta c_0/\varphi >1$ (magnification of the uncertainty in $\delta c_0^{\mathrm{p}}$) or $\mathrm{\Delta }\delta c_0/\left|\delta c_0\right|<1$ (undeterminability of the sign of $\delta c_0$), one needs to consider extending the trajectories or simulating with a bilayer at a larger $\varphi$ to avoid uncertainty issues.

As opposed to the clear ϕ-dependence of the deformation, P2₁ equilibration introduces minimal change in deformation (Figure S7), consistent with small net numbers of migrated lipids between leaflets (see Table 1 and Figures S3 and S4). Together with the significant relaxation of $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ (Figures 3 and 4), the results here support the necessity of P2₁ equilibration for accurate estimation of the peptide-induced spontaneous curvature ($\delta c_0^{\mathrm{p}}$), which relaxes the differential area strain in 𝒜(APL) with minimal influence on the bilayer deformation (see Figure 2).

Estimates of the requirement for P2₁

Although P2₁ equilibration has been shown to be advantageous for accurate estimation of $\delta c_0$ and $\delta c_0^{\mathrm{p}}$ (Figure 5), it is not presently accessible in most major simulation packages; i.e., the two versions of P2₁ PBC^22,41 are only implemented in CHARMM.⁴² Thus it is worthwhile to determine criteria for when P1 simulations would be expected to yield reasonable results for the peptide-asymmetric bilayers studied here.

The area strain ($\equiv ⟨\delta A⟩/A_0$) in a bilayer would depend on how the asymmetric configuration is generated and the size of the system, where $A_0$ is the area of the bilayer with zero differential stress. When the magnitude of the area change, $\left|⟨\delta A⟩\right|$, is much smaller than the area fluctuation, ${⟨\delta A^2⟩}^{1/2}$, the bilayer can absorb the area strain, i.e., it is sensible that P1 simulation might yield reasonable results when ${⟨\delta A⟩}^2/⟨\delta A^2⟩<1$. Recalling that $\left|⟨\delta A⟩\right|=A_0\left|\mathrm{\Delta }\right|/K_A$ in the framework of Hossein and Deserno¹² and combining the preceding condition with Equation (6), we obtain

$$\left|\Delta \right|<{\Delta }_{\text{stretch}}={\left(\frac{K_A{k}_{\text{B}}T}{A_0}\right)}^{1/2}$$ (8)

where ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}$ is the threshold associated with the bilayer stretching described by the area compressibility modulus. A second threshold, ${\mathrm{\Delta }}_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}$, can be determined from the bending energetics: ${\overline{F}}_{\mathrm{b}}{A}_0<k_{\mathrm{B}}T$ and Equations (1), (3c) and (3d) with $k_{AC}=7K_A{d}_{\mathrm{h}}^2/24$ from the polymer brush model^9,54

$$\left|\Delta \right|<{\Delta }_{\text{bend}}=\sqrt{7/3}{\Delta }_{\text{stretch}}$$ (9)

The results for these two criteria are examined in Figure 6 where $\mathrm{\Delta }/{\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}$ and $\mathrm{\Delta }/{\mathrm{\Delta }}_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}$ are compared for the systems simulated here.

Figure 6.

Ratio of $\mathrm{\Delta }$ from simulations and threshold value, ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}$, from Equation (8). The data for each peptide-bilayer system are arranged in the same order as in Figure 3. In the calculation of ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}$, the bilayer area compressibility modulus, $K_A$, was taken from the literature.¹⁶ As visual guides, $\mathrm{\Delta }/{\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}=\pm 1$ (dark green lines) and $\mathrm{\Delta }/{\mathrm{\Delta }}_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}=\pm 1$ (orange lines) are shown together.

The criteria, Equations (8) and (9), work well for gA assemblies in the DNPC bilayer generated by the APL method whose $\left|\mathrm{\Delta }\right|/{\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}$ are significantly larger than 1 (ranging from 3.9 to 7.4). However, we note that P2₁ equilibration may still be needed even if these criteria are satisfied. An example is FP in the POPC bilayer, whose sign of $\delta c_0$ (and thus $\delta c_0^{\mathrm{p}}$) is altered after P2₁ equilibration even though $\left|\mathrm{\Delta }\right|/{\mathrm{\Delta }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}<1(0.7)$; gA in POPC and DNPC also exhibits sign changes in $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ after simulation with P2₁, though the APL values are within the nominal thresholds. Hence the proposed thresholds can identify the need for P2₁ when there are large values of $\mathrm{\Delta }$, but not necessarily for small ones.

Helfrich model with P2₁

One of unique advantages of P2₁ equilibration is its ability to effectively relax differential stress which may be associated with system generation (see Figure 2 and examples in Ref. ⁹). This relaxation significantly reduces $\mathrm{\Delta }$ (and hence ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ as well) for the systems studied here, eliminating $\mathrm{\Delta }$ and ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$ not generated by the peptide inclusion (see Figure 4). Hence, the spontaneous curvature generation by peptides for the entire bilayer can be directly calculated from both Equations (5a) and (5b).

Although it is oversimplified, the leaflet spontaneous curvature generation can be estimated from the corresponding symmetric bilayers with peptides (e.g., FP-POPC (sym), see Figure 5) using the simple Helfrich equation for leaflets.^13,14,16,17 However, this approach is inadequate for compositionally asymmetric bilayers. This is because the differential stress in an asymmetric bilayer due to compositional asymmetry cannot be accounted for in cognate symmetric bilayers of each leaflet. Rather, peptide-induced leaflet spontaneous curvatures must be directly calculated from the asymmetric bilayer. Unfortunately the rigorous theory needed to calculate these leaflet spontaneous curvatures has not yet been developed, which will allow clear distinction between two modes of bilayer deformation (undulatory vs. peristaltic) by peptide inclusion.

In contrast to the aforementioned approach based on the simple Helfrich equation for leaflets, it is anticipated that Equation (5) can still be used for compositionally asymmetric bilayers in (quasi) steady-state with slight corrections. Specifically, the bending moment, ${\overline{F}}_{\mathrm{b}}^{\prime }$ in Equation (5a), and the differential stress, $\mathrm{\Delta }$ in Equation (5b), would be replaced with ${\overline{F}}_{\mathrm{b}}^{\prime }-{\overline{F}}_{\mathrm{b},\mathrm{m}}^{\prime }$ and $\mathrm{\Delta }-{\mathrm{\Delta }}_{\mathrm{m}}$, respectively, where ${\overline{F}}_{\mathrm{b},\mathrm{m}}^{\prime }$ and ${\mathrm{\Delta }}_{\mathrm{m}}$ are the bending moment and differential stress in the corresponding bilayer without peptide.^9,12 These modifications are based on the results from our previous study on compositionally asymmetric bilayers, where the first equation in Equation (3d) has been shown to be accurate.⁹ Hence, the applicability of modified Equation (5) can be first determined by the first equation in Equation (3d), and then be tested for the linearity of $\delta c_0$ (and the insensitivity of $\delta c_0^{\mathrm{p}}$) with respect to $\varphi$ given in Equation (5). Without P2₁ equilibration, even applying Equation (5) becomes challenging due to the difficulty in accurate decomposition of the differential stress from the peptide and the generation method. Therefore, modified Equation (5) and P2₁ equilibration for compositionally asymmetric bilayers would be practically important.⁹

General applicability of P2₁ periodic boundary conditions

Implicit in the application of P2₁ PBC is the assumption that the system (or subsystem) is at equilibrium, and hence the chemical potentials of the relevant species is equal in both leaflets. A related assumption is that bilayer spontaneous curvatures calculated under the condition of low differential stress are the most biologically relevant. These assumptions likely hold true for many cases of interest, such as membrane-bound proteins in cell membranes, experimentally-created vesicles, and slow, quasi-equilibrium processes. However, not all membrane systems of interest are well modeled as being in equilibrium. Unlike systems modeled using P2₁ PBC, lipids in biological membranes cannot readily change leaflets (with the exception of Chol), and the idealized case of relaxed differential stress may not be hold. In these cases, indiscriminate simulation with P2₁ PBC would allow all lipids to translocate and lead to an unphysical loss of asymmetry. In other cases, such as when nonequilibrium conditions are actively maintained by the cell, P2₁ should be applied with care to avoid disrupting steady-state compositions. This problem can be rectified to some degree by restraining selected lipids, as was done by Park et al.⁹ in simulations of asymmetric membranes with saturated lipids only in one leaflet.

V. Conclusions

This study investigated the effects of peptide inclusion on the mechanical properties and spontaneous curvature of model asymmetric bilayers consisting of a single lipid type and a peptide confined to one leaflet. The initial conditions of these bilayers were generated by a simple APL method followed by P2₁ equilibration. P2₁ equilibration relaxes the differential stress in bilayers that is introduced by the system generation method, thus allowing an accurate estimation of $\delta c_0$ (and $\delta c_0^{\mathrm{p}}$) for asymmetric bilayers from the bending moment ${\overline{F}}_{\mathrm{b}}^{\prime }\left(0\right)$, for the entire bilayer or differential stress $\mathrm{\Delta }$, in the framework of the Helfrich model. Additionally, this method is likely to be applicable to bilayers with lipid compositional asymmetry after slight corrections based on a recent theory of asymmetric bilayers. These results highlight the importance of P2₁ equilibration in order to accurately calculate spontaneous curvature generation by peptides and proteins in asymmetric biological membranes.