Probing the elastic response of lipid bilayers and nanovesicles to leaflet tensions via volume per lipid

Biological and biomimetic membranes are based on lipid bilayers, consisting of two monolayers or leaflets. One important but challenging physical parameter of these membranes is their tension. For a long time, this tension was explicitly or implicitly taken to be the bilayer tension, acting on the whole bilayer membrane. More recently, it has been realized that it is useful to decompose the bilayer tension into two leaflet tensions and that these tensions are accessible to molecular dynamics simulations. To divide the bilayer up into two leaflets, it is necessary to introduce a midsurface that defines the spatial extent of the two leaflets. In previous studies, this midsurface was obtained from the density profiles across the bilayer and was then used to compute the molecular area per lipid. Here, we develop an alternative approach based on three-dimensional Voronoi tessellation and molecular volume per lipid. Using this volume-based approach, we determine the reference states with tensionless leaflets as well as the optimal volumes and areas per lipid. The optimal lipid volumes have practically the same value in both leaflets, irrespective of the size and curvature of the nanovesicles, whereas the optimal lipid areas are different for the two leaflets and depend on the vesicle size. In addition, we introduce lateral volume compressibilities to describe the elastic response of the lipid volume to the leaflet tensions. We show that the outer leaflet of a nanovesicle is more densely packed and less compressible than the inner leaflet and that this difference becomes more pronounced for smaller vesicles.


S1 Radii of nanovesicles from molecular densities S1.1 CHAIN protocol for vesicle radii
For a spherical nanovesicle, the density and stress profiles depend only on the radial coordinate r, and the midsurface between the two bilayer leaflets can be characterized by its radius r = R mid .In order to compute the two leaflet tensions, we need to decompose the stress profile of the bilayer into two parts, corresponding to the inner leaflet with r < R mid and to the outer leaflet with r > R mid , see eqn (45) in the Methods section of the main text.
In the CHAIN protocol for nanovesicles, the radius R mid is taken to be the r-value, at which the density profile ρ C (r) of the hydrophobic C beads has an extremum, which is a maximum in the DPD approach used here.Furthermore, this protocol involves two additional computational steps.[1,2] First, the radii R iH and R oH of the inner and outer headgroup layers are located at the two maxima of the density profile ρ H (r) for the headgroup beads.Second, the midsurface radii R il and R ol of the inner and outer leaflets are computed in terms of the three radii R mid , R iH , and R oH according to and as in eqns ( 34) and ( 35) of the main text, which determine the areas of the spherical midsurfaces.
The areas per lipid, a il and a ol , in the inner and outer leaflets are then computed by dividing the areas of the midsurfaces by the number of lipids assembled within these leaflets, which leads to [1,2] When the CHAIN protocol is used to determine the areas per lipid for the smallest nanovesicle, assembled from N il + N ol = 1500, one obtains the data in Fig. S5.Inspection of this figure reveals that the reference state of the nanovesicle with tensionless leaflets is obtained for N ol = 1100 and N il = 400.For this reference state, the optimal areas per lipid, a 0 il for the inner and a 0 ol for the outer leaflet, are given by a 0 il = (1.682± 0.002) d 2 and a 0 ol = (1.096± 0.001) d 2 , (S4) corresponding to the vertical dotted lines in Fig. S5.In fact, the area per lipid, a il , in the inner leaflet is found to be always larger than the area per lipid, a ol , in the outer leaflet, as shown in Fig. S5a for the whole range of N ol values and by the blue data in Fig. 11 of the main text, which displays the optimal areas per lipid in the two leaflets for different vesicle sizes.

S1.2 Alternative protocol based on headgroup layers
A previous simulation study based on the Martini force field applied another protocol to obtain the areas per lipid, a il and a ol , in the two leaflets of a vesicle bilayer.[3] In this study, the radii of the two leaflets were identified with the radii of the two head group layers, which were defined by the location of the phosphate groups.In the coarse-grained molecular model used here, the head groups are described by the H beads, see Fig. 13 in the main text.Therefore, the procedure in Ref [3] implies the alternative definitions for the radii of the inner and outer leaflets.Compared to eqns (S1) and (S2), the alternative definitions in eqn (S5) lead to a reduced value of the inner leaflet radius R il and to an increased value of the outer leaflet radius R ol .
Using the alternative definitions for the midsurfaces of the leaflets as given in eqn (S5), we again compute the areas per lipid in the two leaflets via eqn (S3).In order to obtain the leaflet tensions, we determine the location r = r mid of the midsurfaces by the CHAIN protocol and decompose the stress profiles into their leaflet contributions corresponding to r < R mid and r > R mid .The optimal lipid areas, a 0 il and a 0 ol , corresponding to the reference states with tensionless leaflets, are displayed by the red data symbols in Fig. S6; the numerical values of these data are in Table S13.Inspection of the red data in Fig. S6 reveals that the optimal area per lipid now increases with increasing mean curvature, where the mean curvatures of the inner and outer leaflets are taken to be negative and positive as in the main text.This functional dependence of the red data is opposite to the behavior of the blue data in Fig. 11 and Fig. S6.Thus, in contrast to the red data, the blue data for the optimal areas per lipid decrease with increasing mean curvature of the two leaflets, in agreement with the green data in Fig. 11 and Fig. S6 as obtained by the VORON protocol.S6.The corresponding data from the VORON protocol are displayed in Fig. 5 in the main text.The numerical values of the data are in Table S3.S7.S10.The corresponding data as obtained from the VORON protocol are displayed in Fig. 10 of the main text.

S2 Supplementary Figures
Figure S6: Optimal areas per lipid, a 0 le = a 0 il and a 0 ol , for the inner and outer leaflets versus the mean curvature M le = M il and M ol of the two leaflets, as determined by three different computational methods provided by the VORON protocol (green data symbols), CHAIN protocol (blue data symbols), and the procedure described in Section S1.2 [3] based on the headgroup layers.The green and blue data, which are identical with those in Fig. 11 of the main text, predict that the area per lipid, a 0 le , decreases monotonically with increasing mean curvature and that, for each nanovesicle, the optimal area per lipid, a 0 il , in the inner leaflet is always larger than the optimal area per lipid, a 0 ol , in the outer leaflet.This behavior is consistent with the intuitive view that lipids with two hydrocarbon chains prefer to reside in the more weakly curved surface, as provided by the outer leaflet, rather than in the more strongly curved inner leaflet.In contrast, the red data predict the opposite behavior, that is, that the areas per lipid in the outer leaflets are always larger than the areas per lipid in the inner leaflets.The numerical values of the green data are in Table S8, those of the blue and red data in Table S13.,

S3 Supplementary tables for planar bilayers
Table S1: Area compressibility modulus K ELT and optimal area per lipid, a 0 , as obtained for ELT deformations of planar and symmetric bilayers, for three different values of the total lipid number N lip = N ll + N ul .For N ll = N ul = 841, the dependence of Σ le on a le is plotted in Fig. 4c the dependence of Σ le on v le in Fig. 6.S3: OLT deformations of planar bilayers with three different values of the total lipid number N lip = N ul + N ll : Lipid number N ul and volume per lipid, v ul , in the upper leaflet; volume per lipid, v ll , in the lower leaflet; and leaflet tensions Σ ul and Σ ll in the upper and lower leaflets.The leaflet tensions were computed using the VORON protocol for the location z mid of the midplane as in eqns ( 3) and (4) of the main text.The dependence of the two leaflet tensions on the area per lipid is plotted in Fig. 5c for N lip = 1682, the dependence of the leaflet tensions on the volume per lipid in Fig. 7 and Fig. S3.S6: OLT deformations of planar bilayers for total lipid numbers N lip = N ll + N ul = 2000, 1682 and 1200: Lipid number N ul and area per lipid, a ul , in the upper leaflet; area per lipid, a ll , in the lower leaflet; and leaflet tensions Σ ul and Σ ll in the upper and lower leaflet.The leaflet tensions were computed using the CHAIN protocol, that is, by identifying the location z mid of the midplane with the maximum of the C bead density.The lipid number N ll in the lower leaflet is equal to N lip − N ul .For N lip = 1682, the areas per lipid are plotted in Fig. S2a, the leaflet tensions in Fig. S2b as functions of the lipid number N ul in the upper leaflet.
(a) N lip = 2000     34) in the main text; as well as the inner and outer leaflet tensions Σ il and Σ ol , which were obtained by computing the midsurface radii R mid of the vesicle bilayer in terms of the volumes, see eqns ( 29) and (30) in the main text.
3.57435 ± 0.00031 3.57538 ± 0.00015 3.5785 ± 0.0008 3.5790 ± 0.0004 3.56073 ± 0.00019 3.5598 ± 0.00021 3.56005 ± 0.00028 3.56116 ± 0.00027 Table S10: Simulation data for OLT deformations of vesicle bilayers with four different sizes, corresponding to the total lipid number N lip = 10100, 6312, 2525, and 1500.The 1st column provides the lipid number N ol within the outer leaflet.The 2nd and 3rd columns display the areas per lipid, a il and a ol , in the inner and outer leaflets as obtained from eqn (S3) using the leaflet radii R le = R ol and R le = R il as in eqns (S1) and (S2).The areas per lipid, a il and a ol , in the 4th and 5th column are again obtained from eqn (S3) but with the midsurfaces of the two leaflets computed by eqn (S5).The last two columns display the leaflet tensions Σ il and Σ ol in the inner and outer leaflet as computed using the midsurface radius R mid as determined by the CHAIN protocol.
(a) N lip = 10100 R le from eqns (S1) and (S2)    S12: Area compressibility moduli and optimal areas per lipid for inner and outer leaflets of nanovesicles with four different total lipid numbers total lipid numbers N lip = N il + N ol as obtained from the VORON protocol via eqns (33) and (34) of the main text.The rows display the mean curvatures M il and M ol ; the area compressibility moduli K il and K ol ; as well as the optimal areas per lipid, a 0 il and a 0 ol of the inner and outer leaflets, respectively.The optimal areas per lipid and the area compressibility moduli are plotted as green data points in Fig. 11 of the main text.Table S13: Area compressibility moduli and optimal areas per lipid for inner and outer leaflets of nanovesicles with four different total lipid numbers N lip = N il + N ol : (a) Leaflet properties obtained by computing the midsurface radius R mid of the vesicle bilayers via the CHAIN protocol, the leaflet radii via eqns (S1) and (S2), and the areas per lipid as in eqn (S3).This procedure leads to the blue data points in Fig. 11 and Fig. S6. and (b) Leaflet properties obtained from eqns (S5) and eqn (S3).This procedure leads to the red data points in Fig. S6.The rows of the table display the mean curvatures M il and M ol ; the area compressibility moduli K il and K ol ; as well as the optimal areas per lipid, a 0 il and a 0 ol of the inner and outer leaflets, respectively.

Figure S1 :
Figure S1: Leaflet tension space for planar bilayers that contain a total number of N ll + N ul = 1682 lipids.The two coordinates are the leaflet tensions Σ ll and Σ ul in the lower and upper leaflets.Negative and positive leaflet tensions describe compressed and stretched leaflets.The reference state with tensionless leaflets, corresponding to Σ ll = Σ ul = 0, is obtained for the symmetric bilayer with N ll = N ul = 841 lipids.The red data points describe elastic deformations with equal leaflet tensions (ELT), Σ ll = Σ ul , for which the midplane is equal to the plane of symmetry.The blue and black data represent bilayers with opposite leaflet tensions (OLT), Σ ll = −Σ ul .All OLT states can be obtained from the reference state by reshuffling lipids from one leaflet to the other and adjusting the base area of the simulation box to obtain tensionless bilayers.The blue data points are obtained from the CHAIN protocol, whereas the black data points are obtained from the VORON protocol.Adjacent blue and black data points are obtained for the same values of the lipid numbers N ll and N ul in the two leaflets.It then follows that the leaflet tensions Σ ll = −Σ ul as obtained from the CHAIN protocol are smaller than those from the VORON protocol.

Figure S2 :
Figure S2: Elastic OLT deformations for planar bilayer with total lipid number N ll +N ul = 1682 as computed via the CHAIN protocol: (a) Areas per lipid and (b) Leaflet tensions as functions of the lipid number N ul in the upper leaflet.Black up-triangles correspond to the upper leaflet and red down-triangles to the lower one.In (a) and (b), filled data points are the results of simulations, hollow data points are obtained by interchanging the lipid numbers and leaflet tensions between the two leaflets according to N ′ ll = N ul , N ′ ul = N ll , Σ ′ ll = Σ ul , and Σ ′ ul = Σ ll ; and (c) Leaflet tensions versus projected area per lipid, a le .The red line represents the least squares fit to all data points using the second-order expression in eqn (14) of the main text.The dashed black line provides the best fit to the filled data symbols in (c), which represent those data that fulfill the linearity condition |∆a le | < K OLT /10K ′ .The gray shaded band indicates the 95% prediction band for the linear fit.The numerical values of the data are displayed in TableS6.The corresponding data from the VORON protocol are displayed in Fig.5in the main text.

Figure S3 :
Figure S3: OLT deformations of planar bilayer assembled from N ll + N ul = 1682 lipids.Leaflet tensions Σ ul and Σ ll as functions of the volume per lipid, v le .The data shown here are identical with those in Fig. 7 of the main text, with the midplanes of the asymmetric OLT states computed by the VORON protocol, but including a least squares fit (red line) of all data to the secondorder expression in eqn (20) of the main text.The dotted vertical line is located at the optimal lipid volume v = v 0 = 3.566 d 3 .Filled data symbols represent those v-values for which the linearity condition |∆v| < B OLT /(10B ′ ) is fulfilled.The dashed black line represents the best linear fit to the filled data symbols, the grey shaded band represents the 95% prediction band.The numerical values of the data are in TableS3.

Figure S4 :
Figure S4: Elastic OLT deformations of the smallest nanovesicles with a total number of N il + N ol = 1500 lipids: Leaflet tensions Σ le = Σ ol and Σ il of the outer (black data, up-pointing triangles) and inner leaflets (red data, down-pointing triangles) versus volumes per lipid, v le = v il and v le = v ol , in the two leaflets of the tensionless bilayers.The data shown here are identical with those in Fig. 8 of the main text, with the midplanes of the asymmetric OLT states computed by the VORON protocol.The black and red lines represent the least squares fits of the data to the second-order expression in eqn (20) of the main text, where the subscript 'le' now stands for ol and il, corresponding to the outer and inner leaflets.The two dotted vertical lines are located at the optimal lipid volumes v = v 0 ol ≃ 3.56 d 3 and v = v 0 il ≃ 3.58 d 3 of the outer and inner leaflets.The numerical values of the data are in TableS7.

Figure S5 :
Figure S5: Elastic OLT deformations of the smallest nanovesicles with a total number of N il + N ol = 1500 lipids in both leaflets as computed by the CHAIN protocol (Section S1.1): (a,b) Areas per lipid, a il and a ol , as well as leaflet tensions Σ il and Σ ol of the inner and outer leaflets as functions of the lipid number N ol in the outer leaflet.The midsurfaces of the vesicle bilayers are determined via the CHAIN protocol, and the areas per lipid are computed as described in ... The reference state with tensionless leaflets corresponds to the vertical dotted lines at N ol = N 0 ol = 1100; and (c) Leaflet tensions Σ il and Σ ol as functions of the lipid areas.The data for the inner leaflet are in red, those for the outer leaflet in black.The vertical dotted lines in (c) represent the optimal areas per lipid as given by a 0 ol = 1.096 d 2 and a 0 il = 1.682 d 2 for the outer and inner leaflet, respectively.The numerical values of the data are displayed in TableS10.The corresponding data as obtained from the VORON protocol are displayed in Fig.10of the main text.

Table S2 :
ELT deformations of planar and symmetric bilayers with total lipid numbers N lip = N ll + N ul = 2000, 1682 and 1200 as determined by the VORON protocol: Projected area per lipid, a le = a ll = a ul ; volume per lipid, v le = v ll = v ul ; and leaflet tensions Σ le = Σ ll = Σ ul .

Table S4 :
Area compressibility modulus K OLT and optimal area per lipid, a 0 , as obtained for OLT deformations of planar bilayers and computed by the VORON protocol, for three different values of the total lipid number N lip = N ll + N ul .

Table S5 :
Lateral volume compressibilities B ELT and B OLT for ELT and OLT deformations of planar bilayers as well as optimal volumes per lipid, v 0 , as computed for three different values of the total lipid number N lip = N ll + N ul by the VORON protocol.

Table S7 :
OLT deformations of vesicle bilayers with four different lipid numbers N lip = N il +N ol as obtained by the VORON protocol: The columns display the lipid number N ol in the outer leaflet for fixed N lip ; the volumes per lipid, v il and v ol in the inner and outer leaflets; the areas per lipid, a il and a ol , computed via eqns (33) and (