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15, 9085

Mechanical properties of lipid bilayers: a note onthe Poisson ratio

M. Mert Terzi, † Markus Deserno * and John F. Nagle *

We investigate the Poisson ratio n of fluid lipid bilayers, i.e., the question how area strains compare to

the changes in membrane thickness (or, equivalently, volume) that accompany them. We first examine

existing experimental results on the area- and volume compressibility of lipid membranes. Analyzing

them within the framework of linear elasticity theory for homogeneous thin fluid sheets leads us to

conclude that lipid membrane deformations are to a very good approximation volume-preserving, with

a Poisson ratio that is likely about 3% smaller than the common soft matter limit n ¼ 12. These results are

fully consistent with atomistic simulations of a DOPC membrane at varying amount of applied lateral

stress, for which we instead deduce n by directly comparing area- and volume strains. To assess the

problematic assumption of transverse homogeneity, we also define a depth-resolved Poisson ratio n(z)

and determine it through a refined analysis of the same set of simulations. We find that throughout the

membrane’s thickness, n(z) is close to the value derived assuming homogeneity, with only minor

variations of borderline statistical significance.

I. Introduction

Biomembranes and lipid bilayers exhibit many interestingmechanical properties. Much emphasis has been placed onthe bending modulus KC, which is important for describing theflexural rigidity of membranes, and the area modulus KA, whichpertains to in-plane compression or stretching. These twoare in fact conceptually related, but the details depend onadditional microscopic assumptions, leaving the precise connectionopen to debate.1–5 Since local surface geometry is described by twoindependent curvatures, the bending modulus KC, which penalizesmean curvature, has a partner KG, which quantifies the cost ofGaussian curvature. However, due to the Gauss–Bonnet theorem theGaussian modulus usually only matters when the topology changes(say, pore opening or fission/fusion events), and for the same reasonit is also very difficult to measure.6–10

A mechanical property of lipid membranes that has seenconsiderably less attention is their Poisson ratio, n. This is thequantity that allows us to address the question: what relativearea change DA/A results if we impose a relative thicknesschange DD/D? Within linear elasticity, the ratio between thesetwo quantities, multiplied by �1

2, is called the Poisson ratio.

It is frequently assumed in biophysics, often without notingit explicitly, that volume remains constant upon membrane

deformation,4,11–21 but a few studies have let open the possibilitythat this is not the case for lipids near protein inclusions andlooked into it,22,23 although not in the way we do here. Since thevolume strain uV = DV/V can be expressed, to lowest order, asthe sum of the area strain uA = DA/A and the thickness strainuD = DD/D, one has

uV = uA + uD. (1)

Incompressibility (i.e., DV = 0) enforces the ratio of uA to uD tobe �1, which then implies a Poisson ratio of n ¼ 1

2.

Any sizable deviation from the constant volume assumptioncould, of course, have consequences for biophysical studies thatin one way or another rely on a membrane’s elastic properties.For instance, a lipid volume change would also occur inmembranes containing mechanosensitive channels and couldaffect their gating when they are activated by a mechanicalsurface tension.15,16,18,19 The passive permeability of suchmembranes would also increase due to both thinning of themembrane and lower mass density, but not necessarily by thesame amount due to the two effects.24

Experimentally, it is difficult to obtain the Poisson ratio bydirectly measuring both DA and DD. The continuum mechanicsrelations developed in Section II show us how to obtain n usingother quantities from experiment and simulation, namely, theisothermal area compression modulus KA already mentioned,and the isothermal bulk modulus KV along with the membranethickness D. However, there is also ambiguity in the parameters thatoccur in the continuum mechanics model. Fortunately, simulations

Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

E-mail: deserno@andrew.cmu.edu, nagle@cmu.edu

† Current address: LPTMS, CNRS, Univ. Paris-Sud, Universite Paris-Saclay,91405 Orsay, France.

Received 27th June 2019,Accepted 11th October 2019

DOI: 10.1039/c9sm01290g

rsc.li/soft-matter-journal

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obtain data that remove this ambiguity. The data are reviewed inSection III along with results for n.

This continuum mechanics approach assumes that bilayersare homogeneous. However, lipid membranes exhibit structurethat rapidly varies in the transverse z-direction. We know of noexperimental data that can test whether this heterogeneityeffects a conceivably depth-dependent Poisson ratio n(z); but,as is often the case, simulations can address properties notaccessible to experiment. In Section IV we therefore define n(z) anddetermine it from a refined analysis of the same set of simulations.

II. Relations for the Poisson ratio n fora fluid membrane

The harmonic free energy density f of an elastic body cangenerally be written as

f ¼ 1

2lijkluijukl ; (2)

where uij is the strain tensor and repeated indices are summedover the three spatial dimensions. After exploiting a membrane’sin-plane translational and rotational symmetry, as well as itsfluidity, this reduces to25–27

f ¼ 1

2lAuA2 þ

1

2lzuz

2 þ lAzuAuz; (3)

where the notation has been simplified by introducing lA = lxxxx =lyyyy, lz = lzzzz, lAz = lxxzz, uA = uxx + uyy and uz = uzz. It haspreviously been assumed that lA = lz, because it simplifies somecalculations.25 In this paper we instead allow for an elasticasymmetry

a :¼ lz � lAlA

; (4)

whose value we then estimate.Eqn (3) neglects a term that accounts for lipid tilt; but such a

deformation involves non-diagonal strains, like uxz, that appearnot to be of concern for Poisson ratio considerations. Likewise,the elastic free energy of lipid membranes also contains a termproportional to a trans-bilayer lateral pre-stress, s0(z);11 butsince the zeroth moment of this pre-stress vanishes, and noneof our deformations are z-dependent, it also drops out of allsubsequent considerations.

To obtain the Poisson ratio n in terms of the elastic modulientering eqn (3), one fixes a value of uz and determines theensuing uA that minimizes f,

0 ¼ @f

@uA

� �uz

¼ lAuA þ lAzuz; (5)

which gives

n :¼ �uA

2uz¼ lAz

2lA: (6)

It proves convenient to re-express eqn (3) in terms of the Poissonratio n and the elastic asymmetry a; two equivalent forms,

which differ in the choice of independent strains, will beused:

f ðuA; uzÞlA

¼ 1

2uA

2 þ 1

2ð1þ aÞuz

2 þ 2nuzuA; (7a)

f ðuA; uVÞlA

¼ 1

2uV

2 þ a2ðuV � uAÞ2

þ ð2n � 1ÞuAðuV � uAÞ:(7b)

Next we obtain expressions for the experimentally determinedmoduli. The area modulus KA is defined by

minuz

Df ðuA; uzÞf g ¼ 1

2KAuA

2; (8)

where D is the thickness of the membrane. Enforcing the mini-mization condition via eqn (7a) leads to

(1 + a)uz + 2nuA = 0. (9)

Solving this equation for uz, inserting its value into eqn (7a),and combining with eqn (8), we find

KA

DlA¼ 1þ a� 4n2

1þ a: (10)

Similarly, the experimentally determined bulk modulus KV isdefined by

minuA

f ðuA; uVÞf g ¼ 1

2KVuV

2: (11)

Enforcing the minimization condition, this time via eqn (7b),leads to

(1 + a � 4n)uA � (1 + a � 2n)uV = 0. (12)

Solving this equation for uA, reinserting into eqn (7b), andcombining with eqn (11), we find

KV

lA¼ ð1þ aÞ � 4n2

aþ 2ð1� 2nÞ: (13)

For subsequent analysis of experimental and simulationdata, it is convenient to consider the ratio between a membra-ne’s area- and bulk modulus. Dividing eqn (10) by eqn (13)eliminates the less convenient modulus lA and gives

r :¼ KA

DKV¼ 2ð1� 2nÞ þ a

1þ a: (14)

Notice the occurrence of the membrane thickness D, whicharises for dimensional reasons.

Rearranging eqn (14), we can also expresses the Poissonratio n in terms of two in principle measurable elastic modulusratios, r and a:

n ¼ 1

2� r

4þ a

1� r4

: (15)

In passing, it is interesting to note that eqn (14) allows n tobe larger than 1/2 for small r when a is positive. Although thismay seem surprising in view of the well-known limit thatindeed holds for isotropic elastics, it should be appreciated

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that there are no bounds on n in the much larger universe ofanisotropic elastic materials.28 However, our system is constrainedto |2n| r 1 + a in order to satisfy the stability requirement thatboth KA and KV be non-negative in eqn (10) and (13).

We finally obtain the relative volume change uV resultingfrom a given area strain uA. Setting (qf/quV)uA

= 0 in eqn (7b)leads us to the associated strain ratio s,

s :¼ uV

uA¼ 1þ a� 2n

1þ a: (16)

In the next section we will determine values for the tworatios r and s, from which we then proceed to calculate theobservables of interest: Poisson ratio n and elastic asymmetry a.It is hence convenient to express the latter set directly in termsof the former:

n ¼ 1� s

2ð1þ r� 2sÞ; (17a)

a ¼ 2s� r1þ r� 2s

: (17b)

III. Data and results for homogeneousfluid membranes

The definition of the modulus ratio r in eqn (14) involves thearea expansion modulus KA, the bulk modulus KV, and themembrane thickness D. Experimental work has establishedthat KA depends remarkably weakly on the specific lipid understudy; values around (250 � 50) mN m�1 are typical.1,14,29,30

This range of values is encompassed in the first column ofTable 1 for the four experimental rows.

Because isotropic pressure P and its conjugate thermo-dynamic variable V are relatively less interesting than the morerelevant canonical surface pressure p and surface area A pair,fewer experimental results exist for a lipid bilayer’s bulk modulusKV = �V(qP/qV)T than for its area modulus KA = �A(qp/qA)T.However, the value 1.3 GPa has been measured for KV for DPPC(1,2-dipalmitoyl-sn-glycero-3-phosphocholine) bilayers in thefluid phase at T = 47.4 1C.31 A somewhat larger value of 2.2GPa has been reported for the shorter chain length lipid DMPC

(1,2-dimyristoyl-sn-glycero-3-phosphocholine) in its fluid phase atlower temperatures near T = 30 1C.32 Although another study ofDPPC under pressure did not quote a value for KV,33 from theirFig. 1 we estimate KV = 0.9 GPa in the fluid phase of DPPC atT = 45.13 1C and a smaller value of KV B 0.6 GPa closer to themain transition TM = 41.4 1C. It is well known that there isanomalous softening of the bending modulus KC in the fluidphase as the main transition is approached34,35 and an even morepronounced softening has recently been reported for the tiltmodulus Ky.

36 The few data for KV suggest a similar but weakertrend. A non-lipid bilayer (and non-anomalous) comparison isn-hexadecane, for which KV is about 0.6 GPa at T = 45 1C.37

Although hexadecane has similar hydrocarbon chains as DPPC, itis an isotropic liquid with more gauche–trans disorder thanDPPC,38 so its bulk modulus would be expected to be smaller.On the other hand, nearly 30% of the volume of lipids is in theheadgroup region, which, being surrounded by water, is not likelyto be compressible beyond that of the water. This would notchange DV, but it would reduce the effective compressible volumeV factor in KV, thereby reducing its relevant value by 30%, closerto that of hexadecane. Based on all these considerations, we willconsider the range of experimental estimates for KV shown inTable 1.

The D dependence in r, necessitated by dimensionality,causes some conceptual difficulties, because membrane thicknessis microscopic, and how to define a geometric reference surfacewithin a molecular-scale object is neither obvious nor in practiceunique.12 Two frequently encountered reference surfaces in thecontext of membrane bending are a leaflet’s pivotal plane,39 atwhich bending leads to no area strain, or its neutral surface,40 atwhich bending and stretching energies decouple. However, thesesurfaces are introduced to conceptually localize strain or simplifyelastic energies, not to serve as definitions of bilayer thickness—and hence they are not usually used for this purpose. A morecommon way to specify a membrane’s transverse dimension isthe Luzzati thickness DB, which refers to the surface that ariseswhen one imagines expelling all water from the headgroupregion and rearranging the lipids to form a gedanken sheet ofpure lipid. This definition approaches the problem thermo-dynamically, in the spirit of a Gibbs dividing surface. Typically,DB has values around 3.6nm.41 An alternative structure-baseddefinition is the hydrocarbon thickness, sometimes denoted2DC. It refers to the dividing surface for the methylene groupson the lipid tails. This choice identifies a bilayer with itshydrophobic lipid tail region; its value is about 2.7 nm fortypical bilayers.12 In our subsequent analysis we will examine

Table 1 Values of area modulus KA, bulk modulus KV, membrane thick-ness D, modulus ratio r from eqn (14), strain ratio s from eqn (16), deviationof the Poisson ratio n from its usually assumed limit of 1/2 from eqn (17a),and the elastic asymmetry a from eqn (17b), for a select set of casesdiscussed in the text

CaseKA

[mN m�1]KV

[GPa]D[nm]

r[10�2]

s[10�2]

12� n½10�2�

a[10�2]

Exp 1a 250 1.3 3.6 5.3 3.0 1.2 0.7Exp 1b 250 1.3 2.7 7.1 4.0 1.6 0.9Exp 2 200 2.0 3.6 2.8 3.0 �0.1 3.3Exp 3 300 0.6 3.6 14.0 3.0 5.0 �7.3

Sim 1 138 1.3 3.9 2.7 3.0 0 3.4Sim 2a 277 1.3 3.8 5.6 3.0 1.3 0.4Sim 2b 277 1.3 2.8 7.6 4.0 1.8 0.4

Fig. 1 Geometry and coordinates in a stress-free and a laterally stressedpiece of flat material.

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both the Luzzati- and the hydrocarbon thickness, since theyhave different points in their favour. The Luzzati definitionbridges between continuum theory and molecular reality with-out having to make a potentially arbitrary structural choice. Thehydrocarbon thickness makes such a choice, but it is notarbitrary: simulations have shown that headgroup volumeremains essentially constant with area strain,42 consistent withit being immersed in water, and so one might not expect it tosupport anything other than n ¼ 1

2. Hence, the hydrocarbon

thickness focuses on that part of the bilayer for which we wouldexpect a ‘‘nontrivial’’ Poisson ratio.

Turning to simulations, row Sim 1 in Table 1 shows resultsfrom the only simulation43 we could find that reported resultsfor KA and D, as well as data from which KV can be extracted aswe now show. What was reported using the CHARMM27r forcefield was a bulk modulus for the entire system of Ksys = (1.5 �0.3) GPa. As nearly half (f = 0.42) of the system consisted ofwater, its contribution to the overall modulus must be takenout in order to arrive at the bulk modulus of the bilayer alone.Deviating slightly from the procedure proposed in ref. 43, wenote that the volume strains of the water- and membrane-phasemust add, which implies (in analogy to a springs-in-seriesargument)

1� fKV

¼ 1

Ksys� fKwater

: (18)

Taking the value Kwater = 2 GPa for the bulk modulus of TIP3Pwater from another CHARMM27r simulation,44 we then findKV = 1.3 GPa for the bilayer in very good agreement withexperiment.

Let us now turn to the strain ratio s. We are not aware of anyexperimental measurements of this variable, but we have beenable to extract s from the simulation data (and later simulationtrajectories) of Braun et al.,42 who used the united atom forcefield GROMOS 43A1-S345 to simulate flat bilayers (288 DOPC(1,2-dioleoyl-sn-glycero-3-phosphocholine) lipids, 9428 SPCEwater molecules) at T = 29.85 1C under fixed (projected) area A.Of the several areas simulated we have picked five, equally spacedfrom 0.64 nm2 to 0.72 nm2, whose surface tensions g had alsobeen determined. The volumes of the total lipid, VL, as well as ofthe hydrocarbon region alone, VC, were obtained using a simpleprocedure46 incorporated in the SIMtoEXP analysis software,47

and values were reported in the Supplementary Material of Braunet al.42 As the simulation conditions are identical to those foreqn (16), we obtained the value of s from a plot of ln V versus ln A.Using the entire lipid volume VL corresponds to choosingthe Luzzati thickness DB as the membrane thickness; it givesq(ln V)/q(ln A) = s = 0.0297 � 0.0041. Using only the hydrocarbonvolume corresponds to choosing the hydrocarbon thickness; it givess = 0.0399 � 0.0055. The area modulus KA and the thicknesses werealso reported;42 their values are shown in the Sim 2a and Sim 2brows of Table 1.

Table 1 illustrates ranges for the values of the KA, KV, and Ddata discussed in this section, and these give the values in thecolumn of the modulus ratio r. For the D column, the smallervalues are the hydrocarbon thicknesses and the larger values

are the Luzzati thicknesses. The values in the s column comefrom the previous paragraph, and they correspond to the choiceof D. The Poisson ratio is close to 1/2, so Table 1 shows thedifference 1

2� n. That difference is largest for Case Exp 3 which

puts KA at its upper range and KV at it lower range, and thedifference even becomes slightly negative for the opposite endof the KA and KV ranges as seen for Case Exp 2. Using hydro-carbon thickness versus Luzzati thickness makes less differenceas seen by comparing Cases Exp 1a and 1b and between Sim 2aand 2b; that is due to a compensating effect of s. Based on thesefour cases we suggest that 1

2� n � 0:015, a 3% deviation from

the usual assumption that n ¼ 12. The strain ratio s directly gives

the relative volume change uV for a given area strain uA. Lipidbilayers typically rupture when the area strain uA exceeds 6%,48

so relative changes in volume would be less than 0.2% for CasesExp 1a and 1b and Sim 2a and 2b.

The final column in Table 1 provides, to the best of ourknowledge, the first estimate for the magnitude of the elasticasymmetry a from eqn (4), which quantifies the extent to whichthe diagonal elements lA = lxxxx = lyyyy and lz = lzzzz of thegeneral elastic from eqn (3) differ from one another because amembrane’s anisotropy breaks full rotational symmetry. Thevalues for Cases Exp 1a and 1b and Sim 2a and 2b suggest thatlz is about 0.5% larger than lA. However, uncertainties in theratios r and s are consistent with a = 0, so we simply concludethat any symmetry breaking is quite small.

It may also be noted that a coarse-grained simulation studyobtained elastic ratios,49 but it does not appear to relate thoseresults to the classical Poisson ratio defined in Section II.

IV. Inhomogeneous Poisson ratio

A fluid lipid bilayer is laterally uniform, but along the normaldirection it exhibits significant inhomogeneity, for instance interms of its structure,12 or lateral pressure profile,11,50,51 oreven its lateral area modulus profile.27 This need not imply thatresults using homogeneous elasticity theory are wrong; but itbehooves us to examine this issue, especially since any sizablevariation with position could have implications for other theorieswhich instead assume that n is uniform (and close to 1/2). It isfurthermore of interest whether it is possible to operationallyascribe a local Poisson ratio to characteristic subregions of thebilayer.

A first and relatively straightforward step in this direction isto recognize the elastic difference between a membrane’s headgroup region compared to the hydrocarbon tails. As noted above,a detailed analysis of molecular volumes indeed indicates thatlipid head group moieties essentially do not change their volumeupon bilayer stretching.42 If so, the volume change is confined tothe hydrocarbon tails, and so we expect the bilayer-averaged valueof 1

2� n to be larger when the focus is only on the hydrocarbon

region as shown in Table 1 by the cases Exp 1a and 1b and bySim 2a and Sim 2b.

Let us now strive to go beyond a binary division betweenheads and tails and define a truly local Poisson ratio n(z). As we

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know of no experiment that has the sensitivity to measureinhomogeneity in the Poisson ratio at such a small scale, weonce more turn to simulation and propose a refined way toanalyze the previous set of simulations.42

At first glance, it might seem promising to examine howmuch the mean positions zm of various molecular groups ‘m’move as the membrane strain varies. Such an approach wouldbe flawed, though, because the overall distribution of materialvaries as the molecular conformations perforce vary. This isespecially easy to understand with regard to the terminal methylgroups, because a surprisingly larger fraction of the chain endsturn back towards the headgroup region when chain packingbecomes more jumbled at larger area per lipid.52,53 Indeed, thesimulations42 readily show that the proportion of terminalmethyls in a central slab of thickness 1 nm decreases whenthe membrane area A is increased.

Our way to address this issue is by taking the focus awayfrom individual lipids, or specific locations within them, andinstead consider the total amount of material, no matter ofwhat provenance, within specified regions of space. Considertherefore material slabs of thickness 2L(M) centered on thebilayer midplane, where the argument M identifies how muchmaterial is contained within the slab. If we now considerdifferent areas Ai, the constraint of fixed M then yields theassociated values of Li. In the simulations different areas Ai

were enforced by imposing a biaxial strain, and hence thecondition required by eqn (9) which gives:

�2n12ðMÞ1� d

¼ uD

uA¼ DLðMÞhLðMÞi

�DAhAi (19a)

¼ L1ðMÞ � L2ðMÞL1ðMÞ þ L2ðMÞ

�A1 � A2

A1 þ A2: (19b)

To implement eqn (19b), we must define how much material isin a slab. Consider for instance the hydrocarbon region. Eventhough it is dominated by methylene groups, there are alsoterminal methyls and the methine groups at the double bondsin DOPC. We assume that all of these additively contribute toM, but in different proportions and with different weights. Forthe Poisson ratio, which focuses on volumes, the appropriateweight for each component is its volume. As already mentioned,component volumes have been previously reported for eachsimulated area.42 Since differences in the ratios of the compo-nent volumes, shown in Table S1 of that paper, have little areadependence, we used the same weights for all areas—relative toa weight of unity for methylenes: wCH3

= 1.969, wCH1= 0.890,

wPO4= 1.229, w2(COO) = 2.899, wGly = 2.425, wChol = 5.683 and

wwater = 1.118. We emphasize that we only fix the relativeweights; fixing the component volumes would perforce requiren ¼ 1

2.

Instead of just calculating Poisson ratios for central slabs, letus proceed now to a method that allows using simulations atmany areas to obtain a Poisson ratio as a function of z.

Therefore, we can read our defining equation in a localizedversion,

�uDðzÞuA¼ 2nðzÞ

1þ a; (20)

where n(z) is the depth-dependent value of the Poisson ratio,and where the z-coordinate must be defined via a ‘‘materialperspective’’ analogous to the one outlined above.

To be more specific, consider one of the simulated areas, Az,to be a reference area and focus on a thin slice dz within thebilayer. For a different simulated area, Az, we can write the localz-strain as

uDðzÞ ¼dz� dz

ðdzþ dzÞ=2 ¼ �21� dz=dz1þ dz=dz

; (21)

where z is the height-variable corresponding to the stretchedbilayer, in the manner described after eqn (19b) (see alsoFig. 1). Using eqn (20) to eliminate uD(z) yields

nðMÞ1þ a

¼ 1

uA� 1� dz=dz1þ dz=dz

(22)

for each area strain uA.We used simulation data for five different bilayer systems

featuring areas per lipid of (64, 66, 68, 70, 72) Å2, choosing 68 Å2

as the reference Az, and directly calculated the area strains forthe other four areas as

uA ¼Az � Az

ðAz þ AzÞ=2: (23)

Using the same relative volume weights for the componentgroups of lipids (terminal methyls, methylenes, methines, etc.)as above, we determine a set of z(z) values that corresponds tothe same amount of material in the central slab between z(z)and �z(z) (for each area). We numerically differentiated z(z) foreach area strain uA to obtain dz/dz, which together with eqn (22)yields four estimates of n(z)/(1 + a) from which we obtainaverages with standard deviations for the uncertainties. It maybe noted that the elastic asymmetry a could conceivably dependupon z, but we know of no way to investigate that. Rather, giventhe small value of a suggested by Table 1 and the errors of ourlocal strains, we will proceed by ignoring a altogether.

Our main result for the z-dependent Poisson ratio is shownin Fig. 2. In the region in which there is only water, z \ 2.3 nm,n(z) is statistically consistent with a value of 1/2. For smaller z,the respective averages of n(z) are consistent with the valueobtained from the homogeneous analysis in Section II. Moreprecisely, averaging n(z) over the hydrocarbon region yields12� hnðzÞihc ¼ 2:1� 10�2, quite close to the Sim 2b value, in

which we proposed a Poisson ratio within the hydrocarbonregion using simpler arguments. Looking beyond averages,there appear to be two small ‘‘dips’’ that are borderline statis-tically significant: one at z E 0.4 nm, and one at z E 1.6 nm(near the location of the glycerol groups). We explored whetherthese oscillations could be removed by varying the relativeweights of the component groups. Although this affected the

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values of n(z) somewhat, we found no set of weights thateliminated the small oscillations.

V. Discussion

Our observations, derived from both experiment and simula-tion, and summarized in Table 1 and Fig. 2, indicate that thePoisson ratio for lipid bilayers is close to 1/2, with relativedeviations about 3%. The predicted relative volume changesDV/V are very small, even for strains that would rupture mostbilayers.48 Therefore, we should be on safe ground whenmaking the usual approximation that volumes do not changesignificantly under anisotropic mechanical stresses of biophysicalinterest.

The values of n obtained by comparing experimental andsimulation results (Cases Exp 1a compared to Sim 2a andExp. 1b compared to Sim 2b), and the values determined inour simulation (Cases Sim 2a and 2b) are in remarkably goodagreement. This is to some extent fortuitous, though, since theexperimental values for KV and KA have neither been determinedfor the same lipid, nor at the same temperature. Likewise,values of KA, KV and s have not been obtained from the samesimulation. It would be valuable to do simulations on the samesystem, varying both lateral pressure and bulk pressure, toobtain the three quantities independently.

While the most immediate lesson of our work is confirma-tion that a lipid bilayer is nearly perfect soft matter, in the sensethat it is highly deformable relative to its bulk compressibility,there may be more to be learned from further considering thelocal Poisson ratio. Admittedly, the variations in n(z), asextracted from our data, are statistically not very compelling;but they would be highly interesting if true and might merit

following up on. Observe that the deviation in 12� n at

z = 1.6 nm in Fig. 2 is about three times as large as in regionsaway from the two ‘‘dips’’. This means that KA(z)/KV(z) is aboutthree times larger, or, relatively speaking, it is about three timesharder to stretch membranes at this particular depth thanelsewhere. This is intriguing, because the location where areastrain is found to be particularly expensive happens to coincidewith the location of the pivotal plane, where area strain indeedvanishes. Moreover, Fig. 2 also suggests that the ‘‘active region’’for deviations of n(z) from 1/2 is not limited to the hydrocarbonregion, but appears to extend into the headgroup region nearlyas far as the Luzzati thickness DB.

Our z-dependent findings are consistent with observationsby Campelo et al.,27 who have shown (on the basis of simulationsusing the MARTINI54,55 model) that the lateral stretching modulusprofile has indeed a maximum near the neutral surface (which isnot identical to, but quite close to,56 the pivotal plane). Todetermine this profile, they followed an entirely different techni-que, based on monitoring the tension-dependence of the lateralpressure profile. This, however, requires measuring stressesfairly precisely, which is computationally expensive. In contrast,our method for accessing KA(z)/KV(z) via the Poisson ratio reliesexclusively on keeping track of local bilayer material rearrange-ments, which can be done based on configurations alone,without calculating stresses or energies. It is hence conceptuallyeasier and can—potentially—be done with higher accuracy. Theidea is similar in spirit to a recent proposal to measure the ratioof tilt and bending modulus by counting the fraction of lipidswithin slices of buckled membranes—i.e., again by a purelygeometric procedure.57

VI. Conclusion

In this paper we have shown that lipid bilayers behave liketypical soft condensed matter, having a Poisson ratio thatdeviates about 3% from the common soft matter limit ofn ¼ 1

2. While not unexpected, this fills a niche in the elasticity

theory of membranes, and it supports the approximation usedin many theories that area and thickness deformation arestrongly coupled, with only a negligible correction due tovolume change. We have also seen that the elastic asymmetry adescribing the difference between the lateral and perpendiculardiagonal elements of the full elastic tensor deviate by lessthan 1%.

Beyond this homogeneous finding, any depth dependence ofthe Poisson ratio, or of a membrane’s elastic moduli, would surelyhave profound implications for a number of local membraneprocesses, such as protein insertion, gating of channels (especiallymechanosensitive ones), fission, and fusion. This simply mirrorssome of the interesting possibilities opened by the depthdependence of the lateral pressure profile. However, it seemssafe to expect in the foreseeable future that the study of locallyresolved constitutive relations must rely on simulation approaches,for lack of sufficient resolution in present experimental techniques.This paper is a step in that direction.

Fig. 2 Position-dependent Poisson ratio n(z) as a function of distance zfrom the bilayer midplane, obtained from an atomistically simulated DOPCbilayer.42 The first average is over all z and the second average is only forthe chain region. Overlaid are the simulated main structural regions withthe hydrocarbon thickness DC and the Luzzati thickness DB indicated byarrows.

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Conflicts of interest

There are no conflicts to declare.

Acknowledgements

JFN acknowledges Fred Sachs for conversations about thetheory of mechanosensitive channels that led him to this study.We acknowledge Alex Sodt for pointing out that we hadoriginally not properly taken into account in-plane fluidity.MD acknowledges funding from the National Science Founda-tion via grant CHE 1764257.

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