Superelasticity of Plasma‐ and Synthetic Membranes Resulting from Coupling of Membrane Asymmetry, Curvature, and Lipid Sorting

Abstract Biological cells are contained by a fluid lipid bilayer (plasma membrane, PM) that allows for large deformations, often exceeding 50% of the apparent initial PM area. Isolated lipids self‐organize into membranes, but are prone to rupture at small (<2–4%) area strains, which limits progress for synthetic reconstitution of cellular features. Here, it is shown that by preserving PM structure and composition during isolation from cells, vesicles with cell‐like elasticity can be obtained. It is found that these plasma membrane vesicles store significant area in the form of nanotubes in their lumen. These act as lipid reservoirs and are recruited by mechanical tension applied to the outer vesicle membrane. Both in experiment and theory, it is shown that a “superelastic” response emerges from the interplay of lipid domains and membrane curvature. This finding allows for bottom‐up engineering of synthetic biomaterials that appear one magnitude softer and with threefold larger deformability than conventional lipid vesicles. These results open a path toward designing superelastic synthetic cells possessing the inherent mechanics of biological cells.


Supplemental Figures
Fig. S1 -Control FLIM measurements. a) Fast-DiI fluorescent lifetime is sensitive to membrane order in synthetic POPC "liquid-discorded" (left) and liquid-ordered egg sphingomyelin:cholestetrol (right) 7:3 membranes. GUVs were prepared at the indicated lipid concentrations as described in the main text with 0.1 mol% Fast-DiI dye. Both signals show representative GUVs from separate experiments of each composition. Color code shows FLIM lifetime. b) In homogenous DOPC:DOPG GUVs (see main text method) which exhibit nanotubes of the same composition as the outer membrane segment, Fast-DiI does not have a significant shift in fluorescent lifetime, indicating that Fast-DiI is not sensitive to membrane curvature alone. Solid curves (data for outer membrane) and dashed curves (nanotubes) are 2component exponential decay fits. Traces in red/dark red correspond to signal from the same segment after aspiration with a small tension Σ. In this example, the tension-induced shift in outer membrane fluorescence lifetime corresponds to about Δ » 93 ps, while it was not resolvable for the nanotubes. c) Analysis of the shift in the average nanotube lifetime after application of a small tension difference 0.7 ± 0.3 mN/m for n=3 repeats.

Quenching Experiments
For each experiment, we first prepared a fresh 100 mM stock solution of sodium dithionite (Sigma) in 1M Tris-HCl buffer. 3/5/2 DOPG/eggSM/chol GUV with 1 mol% NBD-PG (1-oleoyl-2-[12-[(7-nitro-2-1,3-benzoxadiazol-4-yl)amino]dodecanoyl]-sn-glycero-3-[phospho-rac-(1glycerol)] (ammonium salt) (Avanti Polar Lipids)) (for other details see main text methods) were grown using electro-formation in 200 mM sucrose HEPES pH 7.4 + 0.1 mM EDTA buffer, similar to the protocol in Ref. (Steinkühler et al., 2018). Then, 1.5 µL of this quenching buffer was pipetted into 58.5 µL of vesicle suspension in 200 mM glucose buffer (40 µL of glucose + 18.5 µL of GUV). The solution was gently stirred to ensure homogenous distribution of the quenching agent. The vesicles were then incubated for 5 minutes and consecutively diluted in 200 mM glucose buffer (to a final volume of 300 µL). From this final mixture, 60 µL was used for observation. For each population (non-quenched and quenched), at least 18 GUVs were imaged at the equatorial plane. The membrane intensity was assessed from the average of 4 peak maxima from the line profiles, drawn in the 4 brightest regions of the membrane.

Es ma on of the chemically-induced super-elas c coefficient
Even though GPMV membranes are complex mixtures of several components, the essen al mechanism through which the aspira on-induced mixing of differently curved domains generates the super-elas c response can be captured by a simple binary model. Therefore, for simplicity, we treat the tGPMV membrane as a two-component fluid consis ng of two generic lipids species A and B. The membrane is stable and there is no significant lipid exchange with the solvent, so both individual lipid numbers N A and N B are conserved quan es. We define the global membrane composi on as which quan fies the rela ve number of A lipids in the membrane. The rela ve number of B lipids is given by 1 − Φ.
A er tubula on, the tGPMV is at equilibrium with its surroundings and comprises primarily two environments: the nanotubes and the outer membrane domains. We label quan es rela ve to each of these domains respec vely with nt and om. The two domains have different sizes, which we quan fy via their areas A nt and A om and enclosed volumes V nt and V om . We assume that both the total area A tot and the total volume V tot of the tGPMV are fixed quan es at equilibrium. Therefore, the rela onships must hold. Note that because the nanotubes are inside the tGPMV, the volume V nt must be subtracted from the outer segment one, so that V tot < V om . We also assume, for simplicity, that the average area per lipid, a, is the same for both species and is not changed significantly during aspira on. Similarly to Eq. (S1) we define the rela ve area frac on occupied by the outer membrane domain so that the nanotubes occupy an area (1 − y)A nt . We also need to introduce the local composi on variables which give the rela ve amount of A lipids in the nanotubes and the outer membrane. These composi ons must match the global lipid frac on Eq. (S1) when weighted with the respec ve membrane area. By combining Eq. (S3) and Eq. (S4) together with N A = N A nt + N A om and N B = N B nt + N B om we obtain the further constraint Note that this expression entangles Φ om , Φ nt and the domain areas: a generic shape change of a domain affects its chemical nature.

The general form of the free energy
The free energy of the system is an extensive quan ty, so we can decompose it as where F nt,om are the local free energies of the respec ve domains. In principle, we should also include a line term quan fying the energy of the domain boundaries where the composi on smoothly interpolates between Φ om and Φ nt . Since the length of this interface-like por on is expected to be small and not to change significantly during aspira on, we will ignore such a contribu on in the following. In general, both terms in Eq. (S6) can be wri en as surface integrals of local densi es, which must depend on local intensive variables: where M nt and M om are the local mean curvatures of the membrane 1 . For the explicit dependence of f on the geometry, we choose a generalised Canham-Helfrich density where e(Φ) is the chemical free energy density per unit area, which, in general, will contain both internal and entropic (i.e. temperature dependent) contribu ons, while κ(Φ) and m(Φ) are respec vely the composi on-dependent bending modulus and spontaneous curvature. It is precisely the coupling terms in Eq. (S8) between Φ and M that allow, at equilibrium, the two membrane domains to sustain different composi ons, i.e. to have Φ om ̸ = Φ nt . Conversely, for homogeneous vesicles where Φ om = Φ nt = Φ (see Eq. (S5)), the free energy Eq. (S6) would reduce to the usual spontaneous curvature model of lipid membranes.
The equilibrium condi ons for the tGPMV can be obtained by minimizing Eq. (S7) with respect to composi ons and geometric variables. However, note that the variables entering Eq. (S6) are not all independent from each other: the global membrane composi on, the total area and the total volume are all fixed quan es (see Eq. (S2) and Eq. (S5)). Finding minima of Eq. (S6) is thus a constrained minimisa on problem which can be solved with the aid of Lagrange mul pliers. We thus must consider the generalised free energy with where λ is the Lagrange mul plier enforcing Eq. (S5), Σ is the Lagrange mul plier enforcing the total area to be A tot and ∆P = P in − P 2 is the pressure differen al between the inside and the outside of the tGPMV.
Equilibrium configura ons are those that sa sfy δG = 0, where varia ons can be carried out independently for variables rela ve to each domain. These deriva ves can be of two kinds: with respect to the chemical composi ons or with respect to membrane shape, and will produce two sets of equa ons, to which we refer respec vely as "chemical" and "mechanical" equilibrium condi ons.
Before aspira on, the solu on of these equa ons determines the equilibrium values Φ nt and Φ om . Because of the composi oncurvature couplings in Eq. (S8), we expect these to be different from each other. It is then useful to introduce the composi on difference which vanishes only for homogeneous vesicles.

Lipid and area recruitment upon aspira on
We model the tGPMV aspira on process as shown in Fig. S4. As the outer membrane is exposed to two different external pressures (denoted P 1 and P 2 ), its shape deviates significantly from a sphere. We capture the essen al features of the tGPMV geometry as shown in Fig. S4b: the outer membrane segment consists of the union a spherical outer segment of radius R v with a cylindrical por on inside the pipe e of radius R p and length L p ending on a tongue-like spherical cap 2 of radius R t . At the ini al aspira on stage there is no cylindrical segment (i.e. L p = 0) and the tongue radius decreases from R t = R om (the original outer sphere radius) towards its minimum value R t = R p . In homogeneous tubulated vesicles it is precisely at this point that the droplet-like instability develops, as the suc on pressure crosses a cri cal threshold and the whole vesicle suddenly flows inside the pipe e [2]. In the present case, however, such instability is absent and the vesicle can sustain larger pressures so to have L p > 0 while maintaining R t = R p . To avoid confusion with quan es referring to the non-aspirated tGPMV we will denote any variable rela ve to the aspirated vesicle with a prime ′ , so that e.g. A ′ om refers to the area of the aspirated vesicle. Fig. 2a we see that the super-elas c response happens for rela ve area changes, ∆A/A om , in the range of 5 − 25%. Using the surface parametriza on sketched in Fig. S4b, we can calculate exactly every geometrical aspects of the aspira on process, including ∆A. At the ini al stage of the aspira on we have L p = 0 and the area increase is due only to the decrease of the tongue radius towards R p . Outer membrane (om) Figure S4: a) Cartoon showing the effect of aspira on on tGPMV shape and composi on. Top: before aspira on, the tGPMV comprises two domains: an outer sphere of radius R om and composi on Φ om (red), and the nanotubes of cross-sec onal radius R nt and composi on Φ nt (green). The tubes are inward poin ng, so V tot < V om . Bo om: during aspira on new lipids are recruited from nanotubes (dashed outline) so that the outer segment changes shape but also composi on, shi ing to a value Φ ′ om (orange), closer to Φ nt . b) The geometry of the outer segment during aspira on is well approximated by the union of two spherical caps and a cylindrical segment. The angle θ subtends a cone such that R p = sin θR v . Ini al aspira on has L p = 0 and R t > R p , un l when R t = R p and L p starts increasing. c) Rela ve area increase ∆A/A om for the shape shown in b as a func on of the aspirated length, from Eq. (S17). We model this aspira on either at constant volume (dashed lines) or at constant outer sphere radius R v = R om (solid lines). The colours refer to different rela ve pipe e sizes as shown in the legend. The do ed line is the maximal L p for constant volume aspira on, beyond which the tGPMV is en rely suc oned. d) Reduced volume v ′ om = 6 √ πV ′ om (A ′ om ) −3/2 as a func on of area increase, with the same colour coding as in c: constant R v = R om aspira ons (solid lines) discriminate between different R p , while constant volume aspira ons collapse on the single dashed curve (∆A/A om ) −3/2 . The solid black line shows the boundary between the ini al aspira on regime with R t > R p and the suc on with L p ̸ = 0. The red dot shows the maximum achievable ∆A/A om with L p = 0 for constant volume aspira ons. and Fig. S4d show how this ini al phase is not relevant for the experimentally measured area changes, as for a wide range of rela ve pipe e sizes R p /R om the maximal increases at this stage are always below ∼ 5%. Dashed and con nuous lines in these plots refer to two different assump ons on the outer membrane volume V ′ om during aspira on: the former refer to constant outer radius deforma ons, i.e. R v = R om is kept constant while V ′ om increases; the la er refers to constant volume aspira ons where V ′ om = V om is kept constant and the outer sphere radius decreases, R v < R om . Experimentally it is hard to dis nguish between these two limi ng cases: on the one hand, the outer membrane segment radius seems to be constant within op cal resolu on, while on the other one would expect no significant volume change due to smallness of V nt ∝ R 2 nt . Likely, the experimental setup lies somewhere in between these two limits. Regardless, Fig. S4c and Fig. S4d clearly show that we can ignore the ini al stage of aspira on and focus on the later stage with L p > 0 and R t = R p , as we shall do in the following.
Given the measured low value of the elas c modulus, we can safely assume that the average area per lipid a is also constant, which otherwise would lead to a significantly s ffer elas c response, as is the case for non-tubulated vesicles (see Figs. 2b and  3c). This means that the observed ∆A is en rely due to lipid recruitment from the nanotubes, and thus represents an area transfer between domains. From this, it follows that the the total tGPMV area is constant during aspira on, and to every area increase of the om domain there must be a corresponding area decrease of the nt domain, namely so that A tot remains unchanged. Due to the mul -component nature of the membrane, this area transfer implies that the composi on of outer membrane can change during aspira on, as shown in Fig. 2c. If lipid recruitment from the nanotubes proceeds uniformly, i.e. the amount of A and B lipids transferred from the nanotubes to the outer membrane is uniquely determined by Φ nt , the recruited area will contain 2∆AΦ nt /a type A lipids and 2∆A(1 − Φ nt )/a type B lipids. We can drop this assump on at the expense of introducing a further variable quan fying the rela ve mole frac on Φ ∆A in the transferred area ∆A. This could explain the shi in tubes composi on observed during aspira on (see the shi of the FAST-DiI fluorescence life me distribu on in Fig. 2c of the main text), but it is not necessary for the sake of the argument presented here. For the me being, we thus take Φ ∆A = Φ nt as a simplifying assump on. It then follows that composi ons change as where ∆Φ = Φ nt − Φ om is the pre-aspira on equilibrium difference defined in Eq. (S11). the value of Φ om is pushed towards Φ nt (see Fig. S4a), and Eq. (S13b) clearly shows how the aspira on process entangles a geometric varia on to a change in the chemical composi on of the membrane. The outer membrane local curvature M ′ om of the aspirated tGPMV is no longer constant, but rather a piece-wise constant func on. Because of the curvature-composi on couplings one would expect also Φ ′ om to be a piece-wise constant func on over the three por ons of the aspirated tGPMV membrane as well, so that Eq. (S13b) would quan fy only an average composi onal shi . In the experiments, however, no significant inhomogeneity in the outer segment has been observed. We can explain this by no ng that the nanotubes typical curvature M nt = 1/2R nt is at least two orders of magnitude larger than any characteris c curvature of the outer membrane (either 1/R om , 1/R v or 1/R p ). We infer that the coupling mechanism inducing ∆Φ ̸ = 0 is too weak to generate a measurable inhomogeneity in the different por ons of the outer membrane. We thus consider both the tongue and the external spherical cap being formed by a uniform membrane of composi ons Φ ′ om . In order to link the applied pressure to geometric and composi onal changes, we must minimise the free energy. Note that now, because we are considering Φ nt a fixed quan ty, we must rewrite the constraint Eq. (S5) accordingly. Using Eq. (S13) and Eq. (S12) we have where the last term in the right-hand side is a constant. At this point, we can ignore the nanotube domain and focus on the outer membrane segment. We thus consider only the second term in Eq. (S9): where and the sums span over the three constant curvature por ons of the outer membrane domain, with A ′ om,i and V ′ om,i being their respec ve areas and volumes such that i A ′ om,i = A ′ om and i V ′ om,i = V ′ om . P i are the external pressures to which each por on is subject to, while there is a single composi on Φ ′ om over the whole domain. In Eq. (S15), the Lagrange mul plier Σ enforces constraint Eq. (S12b) while λ enforces Eq. (S14).
Specifically, the areas and curvatures of these por ons are, past the ini al aspira on stage, respec vely 1/R v and 2πR v (1+ cos θ) on the outer spherical segment, 1/2R p and 2πL p R p on the cylinder and 1/R p and 2πR p on the tongue (θ = arcsin R p /R v is the subtended angle of the pipe e on the outer spherical segment). Using the surface parametriza on of the aspirated outer membrane shown in Fig. S4b, the expression for the whole outer membrane domain area is so that the rela ve area increase is (S18) While this expression formally depends on two geometric ra os, R om /R v and L p /R v , ∆A truly depends on a single degree of freedom (e.g. the length L p ), since we s ll need to fix whether V om or R om is kept constant during aspira on. These different choices lead to two dis nct expressions for ∆A as a func on of L p , as shown respec vely by the dashed and solid lines in Fig.  S4c.
With the explicit form Eq. (S15) we can now carry out varia ons with respect to composi on and shape on each por ons of the outer membrane domain.
Chemical equilibrium Taking a deriva ve with respect to Φ ′ om of G om in Eq. (S15) leads to which for the Canham-Helfrich density Eq. (S8) becomes where A ′ om is given by Eq. (S17) and we defined the two dimensionless auxiliary func ons Mechanical equilibrium The cylindrical part of the outer membrane is in direct contact with the pipe e wall which, being a solid, can exert an arbitrary reac on pressure P wall on the membrane, counterbalancing any mechanical force. Therefore this por on will not provide any physically relevant informa on about the the suc on process and we can focus on the two other parts of the domain: the outer segment and the tongue. Since both are spherical caps, their shape equa ons are which can be combined into a single equa on where ∆P = P 2 − P 1 is the difference between the pressure inside the pipe e and the external environment. This equa on generalises the well-known Laplace rela on for homogeneous vesicles [3], as the Lagrange mul plier λ depends non-trivially on the geometric degrees of freedom of the vesicle via Eq. (S20). One recovers the standard result by se ng λ = 0 and removing any Φ ′ om dependence. Solving for λ in Eq. (S20) and subs tu ng back into Eq. (S23) leads to where the auxiliary func ons h κ and h m are defined in Eq. (S21) and we defined Chemically-generated elas c modulus Within the assump ons of our model, Eq. (S24) is an exact rela on that links the pressure difference ∆P to the measured area increase ∆A. Note that the dependence on ∆A enters this expression both through Φ ′ om (via Eq. (S13b)) and through the geometric variables L p and A ′ om (respec vely via Eq. (S18) and via Eq. (S17)) 3 . From Fig. 2a it is clear that the aspira on tension depends, in first approxima on, linearly on ∆A/A om . Given that the measured values of ∆A are always below ∼ 25%, we can expand Eq. (S24) for small area increases, so to collect informa on on the coefficient of this linear dependence. For simplicity, we focus now on aspira ons that keep the outer membrane segment radius constant, i.e. R v = R om . Then, L p is related to ∆A via Eq. (S18) as By plugging this expression and Eq. (S13b) back into Eq. (S24) and expanding to first order in the area increase, we obtain where Σ app is a tension-like term that collects all ∆A-independent quan es, while K app is the chemically-generated elas c modulus, with general structure where we defined the four auxiliary func ons which depend only on θ and are displayed in Fig. S5. As reported in the main text, the measured value for K app from tGPMV aspira on is 3.1 ± 0.2mN/m. Typical parameter values are m ∼ 1/(100nm) for the spontaneous curvature, κ ∼ 10 −19 J for the bending rigidity. The typical size of a tGPMV is R om ∼ 10µm. Roughly, the deriva ves of κ(Φ om ) or m(Φ om ) with respect to concentra ons quan fy differences of these material parameters between membrane consis ng of pure phases (i.e. bilayers made en rely of either A or B lipids). Hence, it is temp ng to es mate their values to be at most of the same order of magnitude as the values for pure membranes. Furthermore, from Fig. S5 we see that the magnitude of the func ons appearing in Eq. (S29) is, in modulus, consistently below 10 for R p R v /5. Since |∆Φ| < 1, we infer that all terms appearing in Eq. (S28) besides the one depending on e(Φ om ) will contribute to K app with magnitudes of the order at most of ∼ 10µN/m, i.e. about two orders of magnitude less than the measured value. We therefore conclude that the main contribu on to the chemically-generated elas c modulus must come from the internal energy, i.e.: It is not possible to find a more precise es mate for K app without choosing an explicit form for the free energy density Eq. (S8), in par cular of the exact dependence on Φ of the curvature coupling func ons κ(Φ om ) and m(Φ om ).

An explicit model
In our deriva on we did not have to provide any detail about the explicit composi on dependence of the free energy density f (Φ, M) appearing in Eq. (S8). In fact, this was not necessary in order to prove that, when the tGPMV is aspirated into the pipe e, the rela ve area increase induces a shi in the outer segment composi on Φ om → Φ om ′ (see Eq. (S13b)). In turn, this composi onal shi generates an elas c term which opposes any further area increase. Such term, quan fied in first approxima on by the "apparent" elas c modulus K app appearing in Eq. (S28), is the only one that sources a linear response (i.e. a term propor onal to ∆A/A om ) to the applied pressure differen al ∆P = P 2 −P 1 . We can argue that it is precisely this term that avoids the "droplet-like" instability observed in homogeneous vesicles [2], since now the elas c term can counterbalance arbitrarily high ∆P values, trying to push the outer tGPMV back into its spherical configura on prior to aspira on.
What we cannot do without an explicit expression for f (Φ, M), however, is to provide an exact quan ta ve es mate for K app in terms of the membrane material parameters. It is thus instruc ve to choose a specific form for the free energy density -and specifically for the three func ons e(Φ), m(Φ) and κ(Φ) -, and show how all the physically relevant quan es, namely the equilibrium composi on difference ∆Φ, the total tensionΣ om and the elas c modulus K app can be computed explicitly.
We need first to fix an explicit form of the curvature-independent free energy density per unit area e(Φ): a typical choice for qualita ve descrip ons of binary mixtures is the mean-field theory of two-dimensional la ce gas theory. Then, following e.g. [4], we set with the internal energy density u(Φ) and entropy density s(Φ) where ω > 0 is the net A − B lipid interac on strength, which promotes demixing, and T is the vesicle temperature. A thermodynamic system described by Eq. (S32) can undergo phase separa on for subcri cal temperatures T < T c = w 2k B and composi on values within the binodal interval.
Furthermore, we choose to model the curvature-composi on interac on by allowing the spontaneous curvature to be a linear func on where ∆m = m A − m B , with m A (m B ) the spontaneous curvature of a membrane consis ng purely of A (B) lipids. Conversely, we take for simplicity the bending modulus to be the same for both species, i.e. we set κ(Φ) = κ. For a general discussion on possible alterna ves to Eq. (S33) see [4]. From Eq. (S32) and Eq. (S33) we get the explicit form of the free energy density Eq. (S8): The first deriva ve of f (Φ, M) with respect to composi on is then A curvature coupling of the form Eq. (S33) produces a non-homogeneous vesicle prior to aspira on. To see this, we take the deriva ve with respect to Φ om and Φ nt of Eq. (S9), which leads to the chemical equilibrium condi on where M om = 1/R om and M nt = −1/(2R nt ) are respec vely the curvatures of the spherical outer membrane and the cylindrical nanotubes. We can cast this equa on in a simpler form by using Eq. (S5), which implies where Φ is the global mole frac on defined in Eq. (S1) and y = A om /A tot is the outer segment area frac on. Plugging Eq. (S37) back into Eq. (S36) we get Although it is not possible to find an analy c solu on to this equa on, we can find an approximate value for weak curvaturecomposi on couplings. In this case we can expanding Eq. (S38) for small ∆m and get where M nt and M om are the local mean curvatures of the two domains. As expected, ∆Φ vanishes for ∆m = 0. It also vanishes when the geometry of the vesicle is homogeneous (M om = M nt ) or when the mole frac on is saturated to extremal values (Φ = 0, 1), so that the membrane consists of a single component. In general, Eq. (S39) shows how a curvature-dependent free energy necessarily enforces a difference of equilibrium composi ons between different membrane domains -without the need to invoke thermodynamic phase-separa on even at subcri cal temperatures T < T c [4].
We can now compute the chemically-generated modulus K app for this specific model. First, note that since m(Φ om ) is a linear func on and κ is a constant, the general expression Eq. (S28) simplifies considerably: Now, as shown in Fig. S5, |j (1) m (θ)| < 10 for R p R v /5. If we use the approximate values |∆m| ≃ 1/(100nm), κ ≃ 10 −19 J and R v ≃ 10µm, we get | κ∆m Rv j (1) m | ≃ 1µN/m. Similarly, the κ∆m 2 term will have a magnitude of at most ≃ 10µN/m.
We thus see that both these terms cannot produce significant contribu ons to K app , whose measured value is much larger, of the order of a few mN/m. What is le to consider in Eq. (S40) is the contribu on origina ng from the thermodynamic free energy e(Φ om ). For small ∆Φ this can be approximated as which, at leading order, depends on ∆m only through ∆Φ (see Eq. (S39)). We can take some puta ve values for the domain composi ons in order to obtain an es mate for K app : consider a situa on where the nanotubes are almost purely made by A molecules, i.e. Φ nt ≃ 1. From the sor ng diagram in Fig. 3b, we can postulate that the outer segment composi on is about Φ om ≃ 0.65, so that ∆Φ ≃ 0.35. Assuming both environments have roughly the same rela ve area, we can es mate the total mole frac on to lie somewhere in between, say at Φ ≃ 0.85. We then have that Eq. (S41) becomes which would require ω ≃ 2.1k B T (or equivalently T ∼ 0.96T c ) in order for this expression to reproduce the measured value K app ≃ 3.1mN/m, with a lipid cross sec onal area of a ≃ 0.6nm 2 .