Membrane Curvature Sensing by Amphipathic Helices: Insights from Implicit Membrane Modeling

Abstract

Sensing and generation of lipid membrane curvature, mediated by the binding of specific proteins onto the membrane surface, play crucial roles in cell biology. A number of mechanisms have been proposed, but the molecular understanding of these processes is incomplete. All-atom molecular dynamics simulations have offered valuable insights but are extremely demanding computationally. Implicit membrane simulations could provide a viable alternative, but current models apply only to planar membranes. In this work, the implicit membrane model 1 is extended to spherical and tubular membranes. The geometric change from planar to curved shapes is straightforward but insufficient for capturing the full curvature effect, which includes changes in lipid packing. Here, these packing effects are taken into account via the lateral pressure profile. The extended implicit membrane model 1 is tested on the wild-types and mutants of the antimicrobial peptide magainin, the ALPS motif of arfgap1, α-synuclein, and an ENTH domain. In these systems, the model is in qualitative agreement with experiments. We confirm that favorable electrostatic interactions tend to weaken curvature sensitivity in the presence of strong hydrophobic interactions but may actually have a positive effect when those are weak. We also find that binding to vesicles is more favorable than binding to tubes of the same diameter and that the long helix of α-synuclein tends to orient along the axis of tubes, whereas shorter helices tend to orient perpendicular to it. Adoption of a specific orientation could provide a mechanism for coupling protein oligomerization to tubule formation.

Introduction

The binding of proteins to curved lipid membranes is an important event that controls many cellular functions. For example, the GTPase activating protein arfgap1 binds preferentially to positively curved lipid membranes via an amphipathic lipid-curvature-sensing domain called ALPS (1, 2). Endophilin, a protein involved in cellular processes such as synaptic vesicle endocytosis and receptor trafficking, functions by sensing as well as inducing lipid curvature (3, 4, 5). Synapsin (I), which regulates the release of neurotransmitters at the synapses, binds to synaptic vesicles in a curvature-dependent manner (6). Furthermore, membrane fusion is thought to proceed through highly curved intermediate states and is achieved by the binding of special types of proteins onto the lipid surface (7). C-reactive protein, a serum protein with an important role in apoptosis (8), and α-synuclein, a protein involved in Parkinson’s disease, bind to the membrane surface in a curvature-dependent manner (9, 10). Membrane curvature can also regulate the shape and function of various transmembrane proteins by imposing on them a substantial amount of deformation energy (11).

Membrane curvature sensing and generation by proteins has been proposed to occur by mainly three mechanisms. The first is insertion of an amphipathic helix acting as a “wedge” (12, 13, 14, 15). Certain amphipathic motifs are thought to sense the packing defects associated with positive curvature rather than the geometry itself (16). Proteins like arfgap1, α-synuclein, and ENTH domains fall under this category. The second mechanism of curvature sensing and generation is “scaffolding,” by which the intrinsic shape of monomers or oligomers of a protein forces the membrane into a certain shape. The best-studied example is the BAR (Bin, Amphysin, and Rvs) domains (17, 18, 19), whose dimers form crescent-shaped structures. The BAR domains are believed to bind to the lipid membrane because of electrostatic interaction between positively charged residues on their concave, membrane-facing surface and negatively charged lipid headgroups. Some BAR domains, such as endophilin, also have N-terminal amphipathic helices and are called N-BAR. This N-terminal helix is thought to function similarly to the ALPS motif and to assist in lipid curvature sensing and binding, although this notion has recently been questioned (20). Some family members (e.g., the IBAR domains) have a convex membrane-binding surface that allows preferential binding to negatively curved membranes (21). The last mechanism of curvature generation is protein crowding (22, 23), by which the lateral collisions between membrane-bound proteins generate steric pressure that forces the membrane to bend in the opposite direction (23, 24). The efficiency of this mechanism is found to correlate with membrane tension and generally requires high surface coverage (25). For example, epsin1 has been shown to generate membrane curvature even in the absence of its amphipathic helix, and its capacity to generate membrane curvature was directly correlated with its membrane coverage (26).

One question in regard to the “wedge” mechanism is whether curvature sensing is a generic property of any amphipathic helix or a property of specific sequences. One possible answer was provided by Antonny and co-workers, who showed that the electrostatic interactions between peptide cationic residues and the membrane weaken the curvature dependence of membrane binding (27). This hypothesis is consistent with most amphipathic α-helical curvature-sensing proteins. However, there are also reports of certain peptides, such as MARCKS-ED, for which the electrostatic effect appears to play a positive role in curvature sensing (28). One study showed that varying the charge density of the membrane between 20 and 67% does not significantly alter the curvature-sensing ability of α-synuclein (29), whereas another study reported curvature sensitivity only for anionic fractions <25% (30).

A number of atomistic and coarse-grained molecular dynamics (MD) simulations have been carried out for the mechanistic study of membrane curvature sensing and generation. Most of these simulations have supported the notion that membrane curvature sensing by amphipathic helices is due to the recognition of lipid-packing defects on the membrane surface. One atomistic simulation study quantified both the number and size distribution of lipid-packing defects as functions of membrane curvature (14). Others revealed that monounsaturated acyl chains and conical lipids act similarly to curvature in inducing membrane packing defects (31), and that binding of the ALPS motif takes place by stepwise insertions of bulky hydrophobic residues on the defect sites (32). All-atom and coarse-grained MD simulations of the N-BAR dimer on a DOPC/DOPS mixed membrane revealed that the helix 0 is not involved in membrane curvature generation (33), supporting the scaffolding mechanism for the N-BAR proteins. MD simulations of multiple N-BAR dimers on the membrane surface revealed that the arrangement of the N-BAR dimers greatly affects the curvature as well as the dynamics of membrane curvature generation (34). This result could explain why the radii of curvature of BAR-domain-induced tubes and vesicles fall in a wide range of values, even though the BAR domains themselves have a definite intrinsic curvature. A more analytical approach to curvature sensing has been offered by Kozlov et al., who showed that amphipathic helices recognize membrane curvature by sensing the intramembrane stresses (35). Stress sensing is based on the elastic energy of forming the void needed to accommodate protein insertion (15).

All-atom simulations of curved membranes are superbly informative but have significant computational cost. Useful insights into membrane binding could come from simplified models such as implicit membrane models (IMM1 (36), GBSAIM (37), GBSW (38), HDGB (39), etc.). However, all these models so far apply only to flat membranes. In this work, we extend IMM1 to spherical and cylindrical membranes. In this extension, two components have proven necessary for recapitulating experimental observations: membrane geometry and changes in lipid packing, captured via the lateral pressure profile. This extension will hopefully provide alternative lines of attack in the study of lipid curvature sensing and generating proteins.

Theory

IMM1 (36) is an extension of the EEF1 implicit solvation model for soluble proteins (40). The solvation free energy is considered to be the sum of group contributions

$$\mathit{\Delta }{\text{G}}^{\text{slv}}={\sum }_{\text{i}}\mathit{\Delta }{\text{G}}_{\text{i}}^{\text{slv}}={\sum }_{\text{i}}\mathit{\Delta }{\text{G}}_{\text{i}}^{\text{ref}}\mathrm{-}{\sum }_{\text{i}}{\sum }_{\text{j}\ne \text{i}}{\text{g}}_{\text{i}}\left({\text{r}}_{\text{ij}}\right){\text{V}}_{\text{j}},$$ (1)

where ΔG_i^slv is the solvation free energy of atom i, ΔG_i^ref is the solvation free energy of atom i in a small model compound, and the last term describes the loss of solvation because of surrounding groups. In the last term, g_i is the solvation free-energy density of atom i (a Gaussian function of r_ij), V_j is the volume of atom j, and r_ij is the distance between atoms i and j. The ionic side chains are neutralized, and a distant-dependent dielectric constant is used to account for the screening effect of the surrounding solvent. In IMM1, ΔG_i^ref is made a function of the vertical position,

$$\mathit{\Delta }{\text{G}}^{\text{ref}}\left({\text{z}}^{\prime }\right)=\text{f}\left({\text{z}}^{\prime }\right)\mathit{\Delta }{\text{G}}_{\text{i}}^{\text{ref,}\text{water}}+\left(1\mathrm{-}\text{f}\left({\text{z}}^{\prime }\right)\right){\sum }_{\text{i}}\mathit{\Delta }{\text{G}}_{\text{i}}^{\text{ref;}\text{chex}},$$ (2)

where ΔG_i^ref,water and ΔG_i^ref;chex are reference solvation free energies of atom i in water and cyclohexane, respectively. The membrane is oriented on the xy plane and z = 0 defines the midplane of the bilayer. The function f(z′) describes the transition from one phase to the other:

$$\text{f}\left({\text{z}}^{\prime }\right)={\text{z}}^{\prime \text{n}}/1+{\text{z}}^{\prime \text{n}},\text{where}{\text{z}}^{\prime }=\left|\text{z}\right|/\left(\text{T}/2\right),$$ (3)

where T is the thickness of the hydrophobic core of the membrane. The exponent n controls the steepness of the transition; a value of 10 gives an interface width consistent with experiments.

A lateral pressure term for flat membranes is already implemented in IMM1 (41) and is briefly described below. The bilayer is divided into slabs of equal width (h = 0.1 Å) along the membrane normal. The work done against the lateral pressure to insert a peptide into the bilayer is given by

$$\text{W}=\text{h}\left(1\mathrm{-}\mathit{\lambda }\right){\mathrm{\sum }}_{\text{k}}{\text{a}}_{\text{p,k}}{\text{p}}_{\text{k}},$$ (4)

where k runs over the slabs, a_p,k is the peptide cross-sectional area in slab k, p_k the lateral pressure in slab k, and λ is an expansion coefficient for the membrane, which can range from 0 (no expansion of the bilayer) to 1 (full expansion of the bilayer) upon peptide insertion. Because the appropriate value of this parameter under different experimental conditions is unknown, it is treated as adjustable.

At a finite protein to lipid ratio, the lateral pressure profile itself is affected by the presence of the peptide. The lateral pressure in each slab is a function of the cross-sectional area of the peptide. Therefore, Eq. 4 is modified as

$$\text{W}=\text{h}\left(1\mathrm{-}\mathit{\lambda }\right){\mathrm{\sum }}_{\text{k}}{\int }_{\text{0}}^{{\text{a}}_{\text{p},\text{k}}}{p}_{\text{k}}\left({\text{a}}_{\text{p,k}}\right){\text{da}}_{\text{p,k}}.$$ (5)

The explicit form of the term p_k(a_P,k) can be obtained based on thermodynamics by considering the compressibility modulus of the lipid (please refer to (41) for full derivation). The final form of the equation is

$$\text{W}=\text{h}\left(1\mathrm{-}\mathit{\lambda }\right){\mathrm{\sum }}_{\text{k}}{p}_k^0{\text{a}}_{\text{p,k}}+\frac{\text{h}{\left(1-\mathit{\lambda }\right)}^2}{2a_L^0}\left(n_p/n_l\right){\mathrm{\sum }}_{\text{k}}{\text{K}}_{\text{A,k}}{\text{a}}_{\text{p,k}}^2\text{,}$$ (6)

where K_A,k is the compressibility modulus of the slab, $a_L^0$ is the area per lipid in the pure bilayer, $n_{p/}$ $n_l$ is the peptide to lipid ratio, and $p_k^0$ is the lateral pressure in pure lipid. The first term on the right-hand side represents the lateral pressure energy in the pure bilayer (E^LAT). The second term is the lateral lipid compression energy upon peptide binding at finite peptide/lipid ratio based on lipid compressibility modulus (E^COM). The total energy of the peptide binding to the bilayer is thus given by

$$\text{E}={\text{E}}^{\text{IMM}1}+{\text{E}}^{\text{LAT}}+{\text{E}}^{\text{COM}},$$ (7)

where E^IMM1 is the standard IMM1 energy. The lateral pressure profile, together with the other components of the implicit membrane model (solvation, electrostatic potential), provide a static background on which the dynamics of the peptides is carried out.

In extending IMM1 to curved membranes, two issues need to be considered: 1) the different geometry (from a plane to a sphere or a cylinder) and 2) changes in lipid packing arising from membrane bending. The geometry change is straightforward to implement. If the center of the sphere is at the origin, and R defines the radius of the sphere mid-surface, then the equation for z above is modified to z′ = |r − R|/(T/2), where r is the distance of the atom from the origin, given by

$$\mathrm{r}=\sqrt{x^2+y^2+z^2}.$$ (8)

For a cylinder, the z component is omitted, assuming that the cylindrical axis is along the z axis.

The packing effect is a little more complex. We assume that lipid-packing changes can be captured by changes in the lateral pressure profile. Such profiles are not measurable experimentally but have been calculated by computer simulations (42, 43, 44, 45, 46, 47), analytical theory (48), and mean field theory (49). Most work so far has focused on flat membranes with few exceptions (50). Analytical and mean field theories have also been used to calculate the lateral pressure profile in curved membranes (51, 52).

Here, we approximate the lateral pressure profile for a curved membrane by modifying that for a flat membrane based on the area change at different positions when the membrane is bent from flat to spherical or cylindrical. The number of lipids is generally higher in the outer leaflet than in the inner leaflet (see Discussion). In this model, we consider equal number of lipid molecules in the inner and the outer leaflet, which corresponds to the “blocked exchange” regime of Helfrich (53). Bending of a membrane causes a change in area ΔA(z), where z is the coordinate perpendicular to the membrane (z = 0 in the middle of the membrane). ΔA is positive in the outer leaflet and negative in the inner leaflet. As a result, the lateral pressure in the outer leaflet is reduced and that in the inner leaflet is increased. The relationship between tension (γ = −∫p(z)dz) and area is derived as follows:

$$\text{E}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\text{K}}_{\text{a}}{\left(\text{A}\mathrm{-}{\text{A}}_{\text{o}}\right)}^2/{\text{A}}_{\text{o}},$$ (9)

$$\mathit{\gamma }=\text{dE}/\text{dA}={\text{K}}_{\text{a}}\left(\text{A}\mathrm{-}{\text{A}}_{\text{o}}\right)/{\text{A}}_{\text{o}},$$ (10)

and

$$\partial \mathit{\gamma }/\partial \text{A}={\text{K}}_{\text{a}}/{\text{A}}_{\text{o}}=>\mathit{\delta \gamma }={\text{K}}_{\text{a}}/{\text{A}}_{\text{o}}\mathit{\delta }\mathrm{{\rm A},}$$ (11)

where K_a is the stretching modulus. Applying this to each membrane slab (K_a(z) = K_a/T, where T is now the thickness of the entire membrane, not just the hydrophobic core) gives us the following:

$$\mathit{\delta \gamma }\left(\text{z}\right)={\text{K}}_{\text{a}}/\text{T}\ast \mathit{\delta }\mathrm{{\rm A}}\left(\text{z}\right)/{\text{A}}_{\text{o}}=>\mathit{\delta }\text{p}\left(\text{z}\right)=\mathrm{-}{\text{K}}_{\text{a}}/\text{T}\ast \mathit{\delta }\mathrm{{\rm A}}\left(\text{z}\right)/{\text{A}}_{\text{o}}.$$ (12)

The change in area is obtained from simple geometric relationships. A_o is the area of the midplane. For a sphere of radius R at the midplane, we have the following:

$$\begin{array}{l}\text{A}\left(\text{z}\right)/{\text{A}}_{\text{o}}=\text{r}{\left(\text{z}\right)}^2/{\text{R}}^2={\left(\text{R}+\text{z}\right)}^2/{\text{R}}^2=1+2\text{z}/\text{R}+{\text{z}}^2/{\text{R}}^2\hfill \\ \mathit{\delta }\text{A}\left(\text{z}\right)/{\text{A}}_{\text{o}}=\text{A}\left(\text{z}\right)/{\text{A}}_{\text{o}}-1=2\text{z}/\text{R}+{\text{z}}^2/{\text{R}}^2\hfill \end{array}.$$ (13)

For a cylinder, we have the following:

$$\text{A}\left(\text{z}\right)/{\text{A}}_{\text{o}}=1+\text{z}/\text{R}\text{and}\mathit{\delta }\text{A}\left(\text{z}\right)/{\text{A}}_{\text{o}}=\text{z}/\text{R}.$$ (14)

In general, A(z)/A_o = 1 + (c₁ + c₂)z + c₁c₂ × z², where c₁,c₂ are the principal curvatures.

In Eq. 12, replacing the total area by the area per lipid, a(z) = A(z)/n and a_o = A_o/n, where n is the number of lipids in each leaflet, we obtain

$$\mathit{\delta }\text{p}\left(\text{z}\right)=\mathrm{-}{\text{K}}_{\text{a}}/\text{T}\ast \mathit{\delta }\text{a}\left(\text{z}\right)/{\text{a}}_{\text{o}}$$ (15)

and

$$\mathit{\delta }\text{a}\left(\text{z}\right)/{\text{a}}_{\text{o}}=2\text{z}/\text{R}+{\text{z}}^2/{\text{R}}^2.$$ (16)

Replacing the δa(z)/a_o term in Eq. 15 yields the change in the lateral pressure profile because of the bending of the membrane. We refer to the modified IMM1 model as IMM1-curv.

Methods

The EEF1 module of the CHARMM program (54) was modified to implement the equations described in the Theory. The ALPS domain of arfgap1, magainin, α-synuclein, an ENTH domain, and mutants thereof were selected for MD simulation (the sequences are shown in Table 1). The binding studies of the peptides were done on 30% anionic membrane with a hydrocarbon core thickness of 25.4 Å. The expansion coefficient term (λ) in lateral pressure calculations was adjusted to 0.30, based on comparison of the transfer free energies of magainin with available experimental binding free energies. The peptide to lipid ratio (P/L) was set to 0.001 because that corresponds to most experimental conditions (2, 27, 55, 56, 57). The electrostatic effect of anionic membranes was modeled by the Gouy-Chapman theory previously implemented in IMM1 (58). Although the Gouy-Chapman theory strictly applies to planar geometry, we assume here that it is also approximately valid for spherical or cylindrical systems if the curvature is not too high. The Supporting Material and Fig. S2 show that this is a good approximation at the curvatures considered here.

Table 1.

List and Sequences of the Studied Peptides

Arfgap1
Wild-Type HOOC-FLNSAMSSLYSGWSSFTTGASKFAS-NH2
2Ki HOOC-FLNSAMSKLYSGWSSFKTGASKFAS-NH2
2Kt HOOC-FLNSAMKSLYSGWSSFTKGASKFAS-NH2
4Ki HOOC-FLNSAMSKLKSGWSKFKTGASKFAS-NH2
4Ki/4Et HOOC-FLNSAMEKLKEGWEKFKEGASKFAS-NH2
Magainin 2
Wild-Type HOOC-GIGKFLHSAKKFGKAFVGEIMNS-NH2
Mutated (4K/4S) HOOC-GIGSFLHSASSFGSAFVGEIMNS-NH2
Mutated (1K/1E) HOOC-GIGEFLHSAKKFGKAFVGEIMNS-NH2
Mutated (2K/2E) HOOC-GIGEFLHSAEEFGEAFVGEIMNS-NH2
Mutated (3K/3E) HOOC-GIGEFLHSAEEFGEAFVGEIMNS-NH2
Mutated (4K/4E) HOOC-GIGEFLHSAEEFGEAFVGEIMNS-NH2
α-Synuclein
Whole HOOC-MDVFMKGLSK AKEGVVAAAE KTKQGVAEAA GKTKEGVLYV GSKTKEGVVH
GVATVAEKTK EQVTNVGGAV VTGVTAVAQK TVEGAGSIAA ATGFVKKDQL
GKNEEGAPQE GILEDMPVDP DNEAYEMPSE EGYQDYEPEA-NH2
Mutated (12-SER) HOOC-MDVFMKGLSK AKEGVVAAAE KTKQGVAEAA GKTKEGVLYV GSKTKEGVVH GVATVAEKTK EQVTNVGGAV VTGVTAVAQK TVEGAGSIAA ATGFVKKSQL
GKNSSGAPQS GILSSMPVSP SNSAYSMPSSSGYQDYEPEA-NH2
Truncated (95 residues) HOOC-MDVFMKGLSK AKEGVVAAAE KTKQGVAEAA GKTKEGVLYV GSKTKEGVVH
GVATVAEKTK EQVTNVGGAV VTGVTAVAQK TVEGAGSIAA ATGFV-NH2
ENTH Domain
Full ENTH domain (see PDB: 1H0A)
Truncated ENTH HOOC-MSTSSLRRQMKNIVHN-NH2

MD simulations were carried out at 300 K. The simulations were run for 10 ns with a 2-fs time step. The membrane transfer free energies (henceforth referred to as transfer energies) of the peptides were calculated by averaging the differences in energy of the peptide in the membrane and in the aqueous phase for every conformation generated during the last 2 ns of MD simulation. The full structure of the ENTH domain, truncated α-synuclein, and mutated 12SER α-synuclein were simulated for 20 ns, and the average transfer energies were calculated from the last 5 ns, whereas the full-length wild-type (WT) α-synuclein was simulated for 30 ns and average transfer energies were calculated from the last 15 ns. Three trials with different random number seeds were carried out for each curvature to obtain an estimate of the statistical uncertainty. The quantity calculated is not a rigorous membrane binding free energy; it includes solvation effects but not peptide translational and rotational entropy and conformational reorganization (e.g., folding) free-energy contributions. It merely serves as an easily calculated proxy that is expected to correlate with the membrane binding free energy. Although rigorous calculations of membrane binding free energies for small molecules or rigid peptides and domains have been reported (59, 60, 61, 62), such calculations for systems that undergo substantial conformational rearrangement upon membrane binding are quite laborious, even for implicit membrane models (63), and would be impractical in this context.

Curvature sensitivity is assessed experimentally in different ways. Some researchers measure the change in fraction of protein bound as a function of vesicle or tubule diameter (2, 27, 55). Others measure partition coefficients, and from those binding affinity, for vesicles of different size (64). Stamou and co-workers fitted their data of surface density (ρ) versus vesicle radius (r) to a power law ρ ∼ r^−a and used the exponent a as a quantitative measure of curvature sensitivity (29, 65). Here, the plots of transfer energies versus radius of the curvature showed that they are best fitted as a logarithmic function, which is consistent with the power law of Stamou et al. Using ΔG^o = −RTlnK_c with K_c = ρ/c and c the protein concentration in the bulk, one obtains

$$\mathit{\Delta }{\text{G}}^{\text{o}}=\text{constant}+\mathit{\alpha }\times \text{RT}\times \ln \text{r}\text{.}$$ (17)

The magnitude of α was taken as the measure of curvature sensitivity.

Changes in binding free energy with curvature (1–2 kcal/mol or even less) are comparable to the statistical uncertainty in simulation-calculated affinities. It is thus important to ascertain whether the observed differences between two peptides or mutants are statistically significant. A statistical analysis of the results obtained at different levels of confidence is presented in Table S8.

Results

Lateral pressure profiles for spherical and cylindrical membranes

Fig. 1 shows the lateral pressure profiles for flat, spherical (R = 10 nm), and cylindrical (R = 10 nm) membranes calculated as described in the Theory. As expected, the symmetric profile used as input for the flat membrane becomes asymmetric in the case of curved membranes. Due to the change in available area, the headgroup peak of the inner leaflet is more pronounced than that of the outer leaflet, and the interfacial dip is less negative in the inner leaflet than in the outer leaflet. The model presented here predicts that the shift in the magnitude of the peaks in the spherical membrane is approximately double that in the cylindrical membrane of the same radius. This implies that for the curvatures considered here, the quadratic term in Eq. 13 is not yet important. The presence of peptides alters the lateral pressure profile of the lipid bilayer. A plot of the lateral pressure profile for a peptide adsorbed at the membrane interface at different P/L ratios is shown in Fig. S3.

Figure 1.

Lateral pressure profiles for spherical, cylindrical, and flat membranes. To see this figure in color, go online.

Previous theoretical and computational work on the lateral pressure profile of curved membranes is very limited. Two articles also predict an asymmetric lateral pressure profile (50, 52). Ollila et al. computed the three-dimensional pressure field in vesicles having different internal pressures using coarse-grained MD simulations. The peaks in the lateral pressure profile were broader than those of the flat membrane, indicating the lack of a sharp boundary between the hydrophilic and hydrophobic regions in the vesicle. The interfacial peak heights of both inner and outer leaflets were lower than those of the corresponding flat membrane. This differs from our model, which predicts that the interfacial peak height increases for the outer leaflet and decreases for the inner leaflet when the membrane is curved. The discrepancy may be due to the presence of internal pressure in the vesicles of Ollila et al. Drozdova and Mukhin (51) computed the lateral pressure profile in the hydrophobic region of the curved lipid bilayer analytically, considering lipid hydrocarbon chains as flexible strings of finite thickness. As in our result, they found that when a membrane is bent, the peak height decreases in the inner leaflet and increases in the outer leaflet. A similar result was obtained using mean field theory (52). Lateral pressure profiles in asymmetric bilayers (having different number of lipids in two leaflets) of POPC and DMPC show a reduction in pressure in the less dense leaflet and an increase in pressure in the denser leaflet (66). This is qualitatively similar to the changes in density and pressure produced by bending in this model.

ALPS motif of arfgap1

As noted above, one distinguishing feature of the ALPS domain compared to other amphipathic helices is the presence of a large number of polar residues such as serine and threonine on its hydrophilic face. Substitution of the polar residues with charged residues resulted in decrease in the curvature-sensing capacity of ALPS proteins (27). To test whether our model could reproduce these results, we selected the ALPS motif of arfgap1 and four mutants thereof that were studied experimentally (27). These mutants are the following: 2K_i, in which two polar residues (serine and threonine) at the interface between polar and nonpolar faces are mutated to lysine; 2K_t, in which one threonine and one serine at the center of the hydrophilic face are mutated to lysine; 4K_i, in which four interfacial polar residues are mutated to lysine; and 4K_i/4E_t, in which, in addition, four central polar residues are mutated to glutamate. The experimental data on percent protein bound to the vesicles were converted into free energies of binding as explained in (67) and the Supporting Material. The obtained free energies of binding for the WT and four mutants are presented in Table S1.

Because the ALPS motif is in a disordered part of the protein (2), only an isolated peptide was considered here, constructed as an ideal helix. The experimental studies, and hence our calculations, were carried out in vesicles with radii of 40, 90, and 120 nm. The results for the transfer energies are presented in Table 2. All peptides bind strongly to a 15% anionic membrane. As expected, introduction of lysines enhances the binding because of increased electrostatic attraction, whereas introduction of glutamates attenuates it. At low curvature, lysines at the center of the polar face of the helix (mutant 2K_t) have a smaller effect than those at the hydrophobic-hydrophilic interface (mutants 2K_i and 4K_i). Comparing these results to the binding free energies deduced from experiment (Table 2), we see some discrepancies in the trends. For example, the strongest binder is found to be 4K_i in the calculations but 2K_i in experiment. Also, the introduction of 4E at central polar positions is found to have no effect in the experiment but lowers affinity significantly in the calculations. These discrepancies are likely due to the fact that the calculated transfer energies do not include entropic and conformational free-energy contributions. For example, if the 4K_i mutant experiences a higher free-energy cost of folding into a helix, that would lower its membrane binding affinity. Indeed, the 2K_i and 4K_i/4E_t mutants exhibited some helical character in solution (27).

Table 2.

Average Transfer Energies of the ALPS Motif of ArfGAP1 and Its Mutants to 15% Anionic Lipid Vesicles of Different Curvature

Radius, nm	Wild-Type		2Ki		2Kt		4Ki		4Ki/4Et
	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.
40	−9.3 (1.0)	−5.5	−9.4 (1.6)	−6.8	−9.8 (0.7)	−5.7	−11.6 (0.2)	−5.4	−8.7 (0.3)	−5.3
90	−8.2 (0.4)	−3.4	−9.4 (0.5)	−6.8	−9.0 (0.4)	−4.2	−10.9 (0.3)	−4.6	−8.5 (0.5)	−4.7
120	−8.0 (0.2)	−3.3	−9.6 (0.5)	−6.8	−9.1 (0.2)	−3.6	−10.8 (0.3)	−4.2	−7.9 (1.0)	−4.3

The numbers in parentheses indicate SD. Experimental binding free energies of arfgap1 were obtained from floatation assay (27). The given % protein bound information in the article was converted into binding free energy using the method described in the Supporting Material.

The calculated and experimental values of the curvature sensitivity parameter α (Table 3) exhibit similar trends except for the 2K_t mutant. In both columns, all mutants are less curvature sensitive; the least curvature-sensitive mutant is 2K_i, followed by 4K_i/4E_t. The calculated values, however, are systematically lower than those deduced from experiments. A possible source of this discrepancy is that the calculations involved an isolated ALPS helix instead of the full protein. As is known, the full arfgap1 protein contains two ALPS motifs, and both contribute to curvature sensing (55).

Table 3.

Curvature Sensitivity Parameter α of the ALPS Motif Determined from the Logarithmic Plot of Transfer Energy versus Radius in nm

	Calculated with IMM1-curv	Experiment
Wild-Type	2.06 ± 0.23	3.44
2Ki	−0.24 ± 0.24	–
2Kt	1.20 ± 0.47	3.07
4Ki	1.28 ± 0.18	1.74
4Ki/4Et	1.05 ± 0.64	1.42

To isolate the contribution of lipid packing to curvature sensing, the binding of the WT arfgap1 to vesicles of different radii was also modeled without the lateral pressure effect. The results (Table S2) show that the incorporation of the lateral pressure effect in IMM1-curv is essential for modeling curvature sensing proteins. Similarly, the average Gouy-Chapman energy term for the binding of WT arfgap1 on the spherical membrane of different radii is presented in Table S3. The data shows that the Gouy-Chapman energy term remains more or less constant with respect to the curvature change. The P/L ratio was set at 0.001 in all of these calculations. For the WT ALPS, the MD simulations were also carried out at different P/L ratios, and the results are presented in Table S7. The results show that the curvature sensitivity decreases as P/L ratio increases. This can be rationalized by the fact that as peptide concentration increases, lateral pressure increases, and membrane insertion is hindered, which decreases the curvature sensitivity.

Magainin 2

To reinforce the conclusions obtained with the ALPS peptide, we also studied magainin 2 (hereafter magainin), a classical antimicrobial peptide with an amphipathic α-helical structure (68). One major difference between the ALPS motif and magainin is the presence in the latter of four positively charged lysines on the hydrophilic side of the helix. If electrostatic interactions weaken the curvature-sensing ability, we expect magainin to be less sensitive to membrane curvature. In fact, an older experimental work has shown that magainin binds similarly to large unilamellar vesicles and small unilamellar vesicles (69). In addition, we tested whether the mutation of the lysine residues to serines or even to a negatively charged residue like glutamate makes magainin more curvature sensing. The sequences of the studied peptides are shown in Table 1.

The configurations of the WT magainin helix on spherical vesicles of radii 10 and 120 nm are shown in Fig. 2. The computed transfer energies to vesicles of different curvature are presented in Table 4 together with the curvature sensitivity parameters. WT magainin is the least curvature-sensing with α = 0.51; this value is much lower in comparison to the WT ALPS. Mutation of all lysines to serine (mutant 4K/4S) results in an increase in curvature sensitivity (α = 0.74). The mutation of one lysine to glutamate also increases the curvature sensitivity. However, additional mutations of lysine to glutamate do not further increase the curvature sensitivity. This is likely due to the fact that the increased repulsion between the glutamates and the membrane gradually reduces the insertion of the hydrophobic residues, which generates curvature sensitivity.

Figure 2.

Configuration of WT magainin helix on spherical vesicles having radii of 10 nm (top) and 120 nm (bottom) at the end of a 10-ns MD simulation. To see this figure in color, go online.

Table 4.

Average Transfer Energies of Magainin and Its Mutants to 30% Anionic Lipid Vesicles of Different Curvature

Radius, nm	Wild-Type	4K/4S	1K/1E	2K/2E	3K/3E	4K/4E
40	−12.1 (0.1)	−9.0 (0.2)	−10.6 (0.4)	−8.0 (0.0)	−6.4 (0.1)	−4.7 (0.0)
90	−11.9 (0.4)	−8.7 (0.1)	−10.3 (0.4)	−7.7 (0.1)	−6.1 (0.2)	−4.5 (1.0)
120	−11.8 (0.4)	−8.5 (0.1)	−10.1 (0.3)	−7.5 (0.1)	−6.0 (0.2)	−4.2 (0.0)
α	0.51 ± 0.00	0.74 ± 0.11	0.72 ± 0.19	0.74 ± 0.08	0.52 ± 0.05	0.71 ± 0.26

The numbers in parentheses indicate SD. The curvature sensitivity parameter is given in the last row. The transfer energies are given in kcal/mol.

α-Synuclein

α-Synuclein is a 140-residue protein abundant in presynaptic nerve terminals and the prime component of Lewy bodies in Parkinson’s disease (70, 71). Although its function is not yet entirely clear, it is thought to play a role in neurotransmission (72). α-Synuclein is unstructured in solution, but its N-terminal 91–95 residues adopt a helical form upon binding to membranes (73). Two types of membrane-bound helical structures have been observed: a continuous helix and a pair of curved antiparallel helices connected by a linker (74, 75, 76, 77, 78). This protein has been found to be curvature-sensing, binding more strongly to smaller vesicles (9, 29, 64, 79, 80). Intriguingly, acidic residues in the disordered C-terminal region have been found to play a role in curvature sensing despite their lack of contact with the membrane (10). To test whether IMM1-curv can reproduce this result, we considered full-length WT, a mutant with 15 negatively charged residues in the C-terminal region mutated to serine, and truncated α-synuclein with only the first 95 residues. The first 95 residues were constructed as an ideal helix, and the remaining coordinates were taken from the crystal structure (Protein Data Bank (PDB): 1XQ8). The mutation of 15 residues was carried out with Visual Molecular Dynamics (81).

The experimental free energies for α-synuclein binding to 30% anionic membrane (POPC/POPG) (10), converted from dissociation constants, are presented in Table S4. The logarithmic plot of r versus binding free energy gives α = 1.85 for WT and 1.20 for the 12S mutant, in which 12 negatively charged residues in the tail region were mutated to serine. Another experiment carried out with WT on 50% anionic lipid (POPS/POPC) showed that the binding free energy decreases from −7.21 to −5.23 kcal/mol when the radius of the vesicle is increased from 23 to 81 nm (64), corresponding to an α of 2.63. Finally, a single-liposome assay determined α = 1.23 (29). Thus, there is considerable variation in the experimental estimates of curvature sensitivity.

The binding configurations of full-length α-synuclein to 10 and 120 nm radius vesicles are shown in Fig. 3. The tail region tends to remain away from the membrane because of electrostatic repulsion, but a part of it can interact with the helical region of the protein. Its compactness is likely overestimated by the implicit solvation model (82). The helix is highly flexible and adjusts to the curvature. The transfer energies of the three versions of α-synuclein to vesicles of various radii and the curvature sensitivity parameter α are shown in Table 5. WT α-synuclein appears highly sensitive to curvature, with α = 4.27. The mutation of 12 negatively charged residues into serine reduces curvature sensitivity (α = 1.89), in agreement with the experiment (10). Truncation of the entire C-terminal tail also reduces curvature sensitivity but less than the E to S mutations. This implies that other elements in the C-terminal tail actually weaken curvature sensing. These could be three lysines that interact with the membrane or interactions between the C-terminal region and the extended helix, which could weaken the interaction of the latter with the membrane.

Figure 3.

Configurations of full α-synuclein on vesicles with radii 10 nm (top) and 120 nm (bottom), respectively, after 10-ns simulation. Residues 1–95 are shown as purple, and the tail region is shown as green. To see this figure in color, go online.

Table 5.

Average Transfer Energies of α-Synuclein and Its Mutants to 30% Anionic Lipid Vesicles of Different Curvature

Radius, nm	WT Full-Length on Vesicles	Mutated, 12SER on Vesicles	Truncated, 95 Residues on Vesicles	Truncated, 95 Residues on Tube
40	−26.3 (0.4)	−28.9 (2.4)	−27.9 (0.7)	−26.6 (0.9)
90	−24.3 (0.2)	−27.9 (1.2)	−26.5 (0.5)	−26.4 (0.5)
120	−23.5 (0.3)	−27.7 (2.3)	−26.3 (0.3)	−26.5 (0.1)
α	4.27 ± 0.11	1.89 ± 0.19	2.54 ± 0.36	0.20 ± 0.21

The numbers in parentheses indicate SD. The curvature sensitivity parameter is given in the last row. The transfer energies are given in kcal/mol.

The hydrophilic surface of α-synuclein contains a large number of charged residues, and its hydrophobic surface is poorly developed, unlike other curvature-sensing amphipathic helices (9). To further analyze the role of lysines in curvature sensitivity, we created mutants of truncated α-synuclein, designated as 3K/3A, 3K/3S, and 3K/3E, in which LYS₁₂, LYS₄₃, and LYS₈₀ were changed to ALA, SER, or GLU, respectively. The changes in curvature sensitivity (Table 6) are essentially within statistical uncertainty. The mutation of LYS to ALA appears to increase curvature sensitivity, as expected, but mutation to GLU essentially does not, and mutation to SER actually decreases curvature sensitivity somewhat. As in the case of magainin, we see that when the hydrophobic contribution is relatively weak, electrostatic interactions may not compromise or even slightly contribute to curvature sensitivity by promoting hydrophobic insertion. These mutated charges are at the membrane interface and are more likely to synergize with hydrophobic insertion than the remote charges in the C-terminal region.

Table 6.

Average Transfer Energies of Mutated α-Synuclein with LYS₁₂, LYS₄₃, and LYS₈₀ Residues Mutated to ALA, SER, or GLU, Respectively, to 30% Anionic Lipid Vesicles of Different Curvature

	3K/3A	3K/3S	3K/3E
40	−26.2 (0.5)	−24.6 (0.4)	−21.0 (0.7)
90	−25.2 (0.5)	−23.5 (0.5)	−20.2 (1.0)
120	−24.3 (0.7)	−23.4 (0.6)	−19.2 (0.8)
α	2.76 ± 0.58	2.02 ± 0.44	2.41 ± 0.89

The numbers in parentheses indicate SD. The transfer energies are given in kcal/mol.

One experimental study suggested that the anionic fraction of the membrane also has a large effect on curvature sensitivity (30), although another study found very little effect between 20 and 67% (29). Based on the idea that electrostatic effects weaken the curvature sensitivity, the charge on the membrane should also have a similar effect as the charge on the peptide. Therefore, we examined with IMM1-curv the curvature-sensing behavior of α-synuclein at anionic lipid fractions from 0 to 100%. The results in Table 7 show that, in general, increasing anionic fraction weakens curvature sensitivity. The variation in the midrange (30–50%) is very small, consistent with (29). However, in a broader range, there is a clear effect. The model predicts that even in a fully charged membrane, there is significant curvature sensitivity, in apparent disagreement with (30), which found no perceptible difference in binding to DOPG small and large vesicles. Perhaps under the conditions of those experiments, the protein is fully membrane-bound regardless of vesicle size, giving a false appearance of curvature insensitivity (see Discussion).

Table 7.

Average Transfer Energies of Truncated α-Synuclein on Vesicles Composed of Different Fractions of Anionic Lipid

Radius, nm	Neutral	10%	30%	50%	100%
40	−22.4 (0.9)	−24.3 (0.1)	−27.9 (0.7)	−30.4 (0.6)	−34.9 (0.0)
90	−20.6 (0.4)	−22.6 (0.2)	−26.5 (0.5)	−29.4 (0.4)	−34.2 (0.2)
120	−20.3 (0.4)	−22.5 (0.1)	−26.3 (0.3)	−28.9 (1.0)	−33.6 (0.3)
α	3.34 ± 0.41	2.93 ± 0.61	2.54 ± 0.36	2.26 ± 0.18	1.87 ± 0.43

The curvature sensitivity parameter is given in the last row. The transfer energies are given in kcal/mol.

The extended helix of α-synuclein was also simulated on cylindrical membranes of varying curvature. The simulations showed the helix orients along the axis of the tube, presumably to avoid the energy cost of bending if it were to wrap around the tube. This energy penalty for bending cannot be avoided on a spherical membrane. The transfer-energy values in the last column of Table 5 show that the truncated α-synuclein on a cylindrical membrane exhibits the least curvature sensitivity (α = 0.20). The total average energies and internal (total − solvation) energies are provided in Table S5. Although, again, the error bars are large, some trends can be discerned. First, the internal energy is higher on spheres than on tubes, reflecting the bending penalty mentioned above. Despite that, the total energy appears slightly lower on highly curved spheres. To our knowledge, experimental binding free energies of α-synuclein to tubes are not available.

ENTH domain

The protein epsin plays a key role in clathrin-mediated endocytosis (83, 84) by binding PtdIns(4,5)P₂ through its N-terminal homology (ENTH) domain and then providing binding sites for clathrin. The ENTH domain was found to tubulate vesicles (85) and partition preferentially from giant unilamellar vesicles into cylindrical tethers (86). However, it does not discriminate liposome size, at least above 50 nm (4). It is thus worthwhile to examine the binding of the ENTH domain on spherical and cylindrical membranes of varying radii with our model. The first 15 residues of ENTH are unstructured in solution but form an amphipathic helix, referred to as helix zero, upon binding to the membrane. This helix appears to be responsible for both curvature sensing and generation (85, 87).

The initial structure for the simulation was taken from the crystal structure (PDB: 1H0A). The simulations were carried out with either the full crystal structure (without the PIP2 lipid headgroup) or with only helix zero (first 16 residues). The simulations of the full ENTH domain led to two membrane-bound orientations, in the first of which binding occurred through helix zero (Fig. S1). Only this orientation, which gave larger transfer energies, was considered in this study (the alternate orientation may have resulted from the lack of specific PIP2 lipid binding). On cylindrical membranes, especially those with high curvature, the truncated ENTH tends to orient roughly perpendicular to the tube axis. Similar orientations on tubes were observed for magainin and for the ALPS peptide. This may happen because in the perpendicular orientation, the hydrophilic termini are positioned further away from the hydrocarbon core of the membrane. When the diameter of the tube is very large or on a sphere, all orientations are essentially equivalent. To confirm the role of the termini in the orientation of the short helices, the terminal charges were removed by capping with acyl and amide groups. The results are presented in Table S6. With capped termini, the ALPS helix of arfgap1 changes its orientation from perpendicular to parallel. However, the capping of magainin and ENTH helix zero does not have much effect on orientation. In the case of ENTH, this appears to be due to the presence of a series of S and T residues near the N-terminus that face toward the membrane (see Fig. 4). The arrangement of the hydrophobic sidechains also affects orientation. When they are positioned near the termini, as in ALPS, they tend to favor a parallel orientation. When they are positioned closer to the middle of the helix, as in magainin, they immerse in the membrane in any orientation. This allows other factors, e.g., the solvent exposure of the termini, to determine the orientation.

Figure 4.

The binding configurations of the terminally capped short α-helical peptides on 10-nm tubular membrane. (a) The ALPS helix of arfgap1, (b) magainin, and (c) helix zero of ENTH domain are shown. To see this figure in color, go online.

The transfer energies to cylindrical and spherical membranes are presented in Table 8, together with the curvature-sensing parameters α. The transfer energies of full ENTH are slightly smaller than the truncated ENTH, likely due to the fact that the truncated helix has more flexibility to optimize its interactions with the membrane than the full ENTH domain. The model produces a larger enhancement of binding to spherical over cylindrical membranes of the same radius, but the differences in transfer energies are less than 1 kcal/mol. To our knowledge, no experimental data are available yet on the preference of the ENTH domain to spherical or cylindrical membranes of the same radius. Both full ENTH domain and helix zero are more curvature-sensing on spherical membranes (α = 1.01 and 1.23, respectively) than on cylindrical membranes (α = 0.38 and 0.71, respectively). Helix zero alone (truncated ENTH) is slightly less curvature-sensitive than the entire domain.

Table 8.

Average Transfer Energies of Full ENTH Domain and Its Helix Zero to Spherical and Cylindrical Lipid Vesicles of Different Curvature

Radius, nm	Helix Zero of ENTH		Full ENTH
	Spherical	Cylindrical	Spherical	Cylindrical
20	−8.2 (0.4)	−7.5 (0.4)	−8.0 (0.5)	−7.1 (0.4)
40	−7.5 (0.1)	−7.4 (0.2)	−7.0 (0.9)	−7.1 (0.7)
90	−7.2 (0.2)	−7.2 (0.1)	−6.6 (0.5)	−6.4 (0.4)
120	−7.1 (0.4)	−7.1 (0.3)	−6.7 (0.4)	−6.5 (0.3)
α	1.00 ± 0.19	0.38 ± 0.04	1.23 ± 0.36	0.71 ± 0.37

The numbers in parentheses indicate SD. The curvature sensitivity parameter is given in the last row. The transfer energies are given in kcal/mol.

Discussion

A considerable number of peptides and proteins have been found to bind preferentially to highly curved membranes (6, 88, 89, 90). The predominant rationalization of this has been the idea that highly curved membranes contain packing defects to which hydrophobic peptides can bind (12, 32, 91). Such defects have been analyzed computationally by several studies (14, 16, 31, 92). An alternative, more “mesoscopic” viewpoint is that bending of the membrane changes the area per lipid in a depth-dependent manner and thus alters the lateral pressure profile, or, equivalently, peptide insertion generates stresses which are relieved by bending (15). In this view, the atomic-level heterogeneity is averaged out, temporally and spatially. Previous theoretical studies have found that curvature effects can be adequately captured by changes in the lateral pressure profile (52, 93).

Our work adopts this more mesoscopic view and extends the implicit membrane model IMM1 to curved lipid membranes, including cylindrical and spherical shapes. As expected, to reproduce experimental observations, it has been necessary to consider changes not only in geometry but also in lipid packing. The latter was taken into account by modifying the lateral pressure profile, which was already incorporated into the model in previous work (41). The advantages of this approach include computational efficiency, atomic detail in the representation of the peptide, and rapid estimate of membrane binding affinity. One drawback of any continuum model is that it neglects specific interactions between the peptide and membrane lipids, which have been suggested to play an important role in membrane curvature induction (94). Compared to purely elastic continuum models (15), the current model accounts for solvation effects and peptide-membrane electrostatic interactions, although the latter are treated in an approximate, mean field way. The ability to perform molecular dynamics on an implicit membrane background allows for coupling between conformational dynamics and membrane positioning.

The proposed model was employed to study the curvature-sensing behavior of experimentally studied amphipathic helical peptides, one of the most important classes of curvature-sensing motifs found in a wide range of proteins. A key question in this field is whether curvature sensitivity is a generic property of all amphipathic helices or whether it is determined or influenced by amino acid composition and sequence. Some experimental results provided important insights into this issue, showing that curvature sensing is modulated by the relative strengths of the hydrophobic and electrostatic interactions (27, 29, 90). Because the electrostatic component is insensitive to curvature, stronger electrostatic interactions should lead to curvature-independent membrane binding. This has been confirmed by our model in a number of proteins. For example, mutation of the polar residues present on the hydrophilic side of the ALPS motif of arfgap1 to positively charged residues makes it less curvature-sensing than WT. In α-synuclein, the negatively charged tail region contributes to curvature sensing by weakening the electrostatic component of interaction. The electrostatic component can be tuned not only by altering the charge in the peptides but also by changing the fraction of anionic lipids. On the other hand, as shown by the magainin K to E mutants and similar α-synuclein mutants, reducing electrostatic interactions does not always result in enhancement of curvature sensitivity. In weakly hydrophobic proteins, membrane insertion of hydrophobic residues, and thus curvature sensitivity, may actually be enhanced by electrostatic interactions. This could reconcile certain reports of electrostatics playing a positive role in curvature sensitivity (28).

One important finding of this study is the different orientation of short and long helical peptides along a tubular membrane. Very little experimental or theoretical information is available on this issue. Our simulations show that long helical peptides like α-synuclein prefer to bind parallel to the tube axis, which helps to avoid significant bending of the helix. On the other hand, short helices such as those from the ENTH domain of epsin, magainin, and the ALPS motif prefer to bind perpendicular to the tube axis when the diameter of the tube is small. A significant contributor to this is the hydrophilicity of the termini, whose contact with the membrane can be minimized by a perpendicular orientation to the tube axis. A similar observation has been made in recent all-atom simulations of lipoprotein nanodiscs (95), which found that short helical fragments adopt a “picket fence” arrangement in contrast to the widely accepted “double belt” model. If a parallel arrangement of the peptides facilitates oligomerization (87, 96), adoption of such an arrangement on tubes could provide a mechanism for coupling between oligomerization and membrane tubulation. Other than epsin, amphiphysin with an N-BAR domain is known to oligomerize and to tubulate vesicles while binding the membrane via an amphipathic helix (97).

We studied the preference of curvature-sensing peptides for vesicles or tubes of the same diameter. Short helical peptides like the ENTH Nt helix clearly prefer to bind to vesicles over tubes. This result is consistent with a coarse-grained simulation that found a higher frequency of defects in spheres than in tubes of similar diameter (16). For long helical peptides such as α-synuclein, the results are less clear due to statistical uncertainty, but the trend seems to hold. This result may explain why α-synuclein tubulates liposomes at moderate concentrations and vesiculates them at higher concentrations. For a symmetric bilayer, the bending free-energy density of vesicles is four times higher than that of tubes of the same diameter (98). Thus, although α-synuclein stabilizes vesicles more than tubes, it takes a larger number of them to overcome the higher free-energy cost of forming a small vesicle. An alternative explanation for the tubulation activity of α-synuclein is that the long helix produces anisotropic curvature stress in one direction, which deforms a planar bilayer into a cylindrical tube (99). However, this does not explain why vesicles are formed at high protein concentrations.

Hatzakis et al. concluded from their experimental studies that curvature sensitivity is due to a larger number of defects rather than a higher affinity of the peptides for the higher curvature surface (65). This is in apparent contradiction with other experimental studies that find a clear enhancement of affinity at higher curvature (10, 28, 64, 100). MD simulations showed that not only the number but also the size of the defects increase with increase in the membrane curvature (14, 16), implying that binding affinity of the peptides should also increase with the increase in curvature. Our model, which does not incorporate any discrete defects, clearly shows that transfer energies vary with the curvature size. As discussed above, this may be a result of spatial and temporal averaging over a heterogeneous surface. On the other hand, the conclusion of Hatzakis et al. may have arisen from their using the ideal Langmuir isotherm model and not taking into account variations in lipid concentration among different experiments (contrast Eq. 1 of (99) and Eq. 2 of (10)).

Experimental investigation of curvature sensitivity has employed a variety of techniques and methods of analysis. Different groups use different criteria or measures of curvature sensitivity, and this can lead to discrepant conclusions. When experiments are done under conditions that strongly favor the membrane bound state, one may conclude incorrectly that binding is curvature insensitive. In our view, the most rigorous strategy is the determination of the binding free energy (or partition coefficient) as a function of vesicle radius. This strategy works well independent of binding affinity.

The IMM1-curv model has been able to recapitulate several experimental findings regarding curvature-sensing proteins. However, in many cases, the determined curvature sensitivity is not in full quantitative agreement with experiments. It is noted, of course, that the uncertainty in experimental results is substantial and there is a considerable range in curvature sensitivity determined by different assays. On the other hand, there are aspects of the model that could easily produce quantitative discrepancies. One of them is the use of transfer energies to approximate membrane binding energies. This approximation assumes that the effects of conformational reorganization upon membrane binding are independent of curvature, which may not always be valid. Another restricting feature of the model is the inability of the implicit membrane to change shape in response to peptide binding. Thus, we have to resort to the thermodynamic coupling between sensing and generation (101) to assess the curvature-generating capacity of peptides. One final limitation is our assumption of equal number of lipids in the two leaflets of the bilayer. Although this may correspond to some experimental situations (53), in most cases the lipids will distribute between the two leaflets to minimize the free energy of the system. The ratio of lipids in the outer and inner leaflets depends on the vesicle size and has in some cases been measured experimentally (102). One would expect that this assumption should systematically exaggerate curvature sensitivity, but comparison of the calculated curvature sensitivity parameters with experiments does not always indicate that. Clearly, more theoretical and computational work on this issue is needed. The model presented here can incorporate lateral pressure data from any source and can be accordingly modified when such data become available. Extension to asymmetric membranes should also be straightforward.

Author Contributions

T.L. and B.N. designed the research. J.L. did the initial code development and performed preliminary simulations. B.N. finalized the code, performed simulations, and analyzed the data. B.N. and T.L. wrote the article.

Editor: Markus Deserno.