Abstract Continuum mechanical tools are used to describethe deformation, energy density, and material symmetry of alipid bilayer with spontaneous curvature. In contrast to con-ventional approaches in which lipid bilayers are modeledby material surfaces, here we rely on a three-dimensionalapproach in which a lipid bilayer is modeling by a shell-like body with finite thickness. In this setting, the interfacebetween the leaflets of a lipid bilayer is assumed to coin-cide with the mid-surface of the corresponding shell-likebody. The three-dimensional deformation gradient is found toinvolve the curvature tensors of the mid-surface in the spon-taneous and the deformed states, the deformation gradientof the mid-surface, and the transverse deformation. Atten-tion is also given to the coherency of the leaflets and to thearea compatibility of the closed lipid bilayers (i.e., vesicles).A hyperelastic constitutive theory for lipid bilayers in theliquid phase is developed. In combination, the requirementsof frame indifference and material symmetry yield a repre-sentation for the energy density of a lipid bilayer. This repre-sentation shows that three scalar invariants suffice to describethe constitutive response of a lipid bilayer exhibiting in-planefluidity and transverse isotropy. In addition to exploring thegeometrical and physical properties of these invariants, fun-damental constitutively associated kinematical quantities areemphasized. On this basis, the effect on the energy den-sity of assuming that the lipid bilayer is incompressible isconsidered. Lastly, a dimension reduction argument is used
M. Maleki · E. Fried (B)Department of Mechanical Engineering, McGill University,Montréal, QC H3A 0C3, Canadae-mail: [email protected]
B. SeguinDepartment of Mathematics and Statistics, McGill University,Montréal, QC H3A 0B9, Canada
to extract an areal energy density per unit area from thethree-dimensional energy density. This step explains the ori-gin of spontaneous curvature in the areal energy density.Importantly, along with a standard contribution associatedwith the natural curvature of the lipid bilayer, our analysisindicates that constitutive asymmetry between the leaflets ofthe lipid bilayer gives rise to a secondary contribution to thespontaneous curvature.
Biomembranes are essential to the functions of cells, bac-teria, and viruses (Harrison and Lunt 1975; Yeagle 2001).Basic to all biological membranes are lipid bilayers, whichare thin, sheet-like structural elements composed of two adja-cent monomolecular leaflets joined by weak, noncovalentbonds (Bretscher 1973). In the liquid phase, lipid bilayers arevery flexible in bending but highly resistant to lateral stretch-ing (Bloom et al. 1991). The architecture of lipid bilayershinges on the amphiphatic chemical properties of the con-stituent phospholipid molecules (Tanford 1980). Such mole-cules consist of hydrophilic head groups and hydrophobictails. When suspended in aqueous solutions under suitabletemperature conditions and at appropriate concentrations,they form various self-assembled complexes with hydropho-bic tails facing one another and hydrophilic head groups incontact with the solution (Lasic 1988). These complexesinclude closed bilayers, known as vesicles or liposomes,which are typically a few nanometers thick and can range
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between fifty nanometers and tens of micrometers in diame-ter (Luisi and Walade 2000).
Biomembranes are generally heterogeneous multicom-ponent systems involving hundreds of lipid species alongwith various proteins and hydrocarbons. The inherent com-plexity of such systems has driven the development of bio-mimetic model systems (Chan and Boxer 2007). These modelsystems include giant unilaminar vesicles (GUVs), whichmay be composed of as few as two lipid species and a sin-gle type of cholesterol. Aside from providing platforms forfocused investigations into processes mediated by biomem-branes (Lipowsky and Sackmann 1995; Peetla et al. 2009),GUVs are of potential value in various pharmaceutical andtechnological applications, including biocompatible micro-capsules for targeted drug delivery and gene therapy (Allenand Cullis 2004; Attama 2011), adjuvants for immunization(Latif and Bachhawat 1984; Gregoriadis et al. 1996), signalcarrying and enhancement in medical diagnostics and ana-lytical biochemistry (Gómez-Hens and Fernández-Romero2005; Edwards and Baeumner 2006), and biochemical reac-tors (Tsumoto et al. 2001; Fischer et al. 2002; Michel et al.2004; Vriezema et al. 2005).
Lipid bilayers readily change shape in response to shift-ing osmotic and thermal conditions and applied mechani-cal loads (Lipowsky and Bachhawat 1995). Efforts to modelsuch shape changes date back somewhat more than fourdecades. Canham (1970) emulated the methodology com-monly applied in the bending analysis of beams to yielda simplified model capable of predicting the shapes avail-able to a red blood cell. Treating a cell membrane as a sur-face, Canham (1970) showed that the shapes it manifestsin equilibrium emerge as a consequence of bending-energyminimization. Independently, Helfrich (1973) attributed asurface bending-energy to lipid bilayers. In the models ofCanham (1970) and Helfrich (1973), energy changes inducedby relative molecular misalignment are incorporated throughdeviations of the principal curvatures (or alternatively, themean and Gaussian curvatures) of the surface that servesas a proxy for the lipid bilayer. Specifically, according toCanham–Helfrich theory, the bending-energy density (thatis, the energy per unit surface area) is given by
ψ = 1
2κ(H − H◦)2 + κK , (1)
where H and K are the mean and Gaussian curvatures,respectively, and H◦ is the spontaneous mean curvature,which embodies the curvature of the bilayer in its naturalstate. In (1), changes of H and K are, respectively, penal-ized by bending moduli κ and κ , known, respectively, asthe splay (or ordinary) and saddle-splay (or Gaussian) mod-uli. As the brothers Cosserat and Cosserat (1909) notedin their work on elastic surfaces, Germain (1821) previ-ously derived an energy density quadratic in the principal
curvatures equivalent to (1), and the particular case cor-responding to zero spontaneous curvature (H◦ = 0) wasobtained by Poisson (1812). See also the historical remarksof Nitsche (1993).
The Canham–Helfrich energy is perhaps the simplestmodel believed suitable to situations where the shape of thelipid bilayer is dominated by bending and the radius of thecurvature of the lipid bilayer is much larger than its thickness.It can nevertheless be argued that (1) neglects energetic con-tributions associated with changes of local area or thicknessand merely considers the lipid bilayer as a two-dimensionalfluid surface that resists curvature deviations. Moreover, theCanham–Helfrich theory does not allow for changes in over-all area. This constraint is imposed by adding a term propor-tional to the area of the surface that models the lipid bilayerto the net bending-energy determined by integrating (1) overthat surface. However, as is clear from the discussion of theglobal and local area preservation provided by Steigmannet al. (2003), this approach does not rule out the local areachanges. Importantly, local area or thickness changes canoccur in the vicinity of phase interfaces in multi-componentlipid bilayers or heterogeneities such as protein molecules(Israelachvili 2011).
In contrast to an elastic shell, a lipid bilayer in the liquidphase does not have the ability to resist in-plane shear forces.This is because the lipid molecules may move freely withina lipid bilayer. Due to the absence of the preferred direc-tionality tangent to their surfaces in the liquid phase, lipidbilayers also exhibit in-plane isotropy. Bearing in mind thatthe general theory of elastic shells allows for a broad range ofpossible material symmetries, any shell-like model for a lipidbilayer should be consistent with the observed in-plane fluid-ity and isotropy. Working in the context of modern shell the-ory, Jenkins (1977) derived the general equations governingthe mechanical equilibrium for a shell with material symme-try consistent with that of a lipid bilayer. Steigmann (1999)subsequently reconsidered the mechanical modeling of fluidfilms with bending elasticity from a fundamental perspective.By treating the lipid bilayer as a two-dimensional (inviscid)fluid and choosing the full set of two-dimensional unimod-ular transformations as the appropriate material symmetrygroup, Steigmann (1999) obtained a general energy densitydepending not only on the mean and Gaussian curvaturesH and K but also on the areal stretch J . The areal stretchJ represents local expansion/contraction within the tangentplane of the bilayer. Since a lipid bilayer shows no resis-tance to in-plane shear forces, but rather only to the local areachanges, the areal stretch J is the sole kinematical ingredientneeded to reckon in-plane deformation of the lipid bilayer.A new free-energy density for biomembranes, based on treat-ing the lipid bilayer as a three-dimensional body rather than atwo-dimensional surface, was proposed by Zurlo (2006) andDeseri et al. (2008). An important feature of the formulation
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Kinematics, material symmetry, and energy densities 999
of these authors involves the introduction of a symmetrygroup that describes in-plane fluidity and isotropy at the levelof the bulk, three-dimensional, material. However, a proof ofthe corresponding representation theorem was not provided.Additionally, the analysis of Zurlo (2006) and Deseri et al.(2008) is based on considering a flat reference configuration,which means that spontaneous curvature is tacitly assumedto vanish. Following a dimension reduction from a three-dimensional shell-like structure to a two-dimensional mate-rial surface, Zurlo (2006) and Deseri et al. (2008) obtainedthe superficial energy density per unit area of the referencesurface.
Several explanations for the existence of spontaneous cur-vature have been reported in the literature. Spontaneous cur-vature is believed to be a measure of the extent to which theupper and lower leaflets of the lipid bilayer are asymmet-ric (Seifert 1997). Asymmetry may arise due to differencesbetween the molecular compositions of the leaflets, differentproperties of the aqueous solutions adjacent to the sides ofthe lipid bilayer (Döbereiner et al. 1999), or interactions witha cytoskeleton (for a review see McMahon and Gallop 2005).For instance, the presence of molecules with different head-group or tailgroup conformations can lead to spontaneouscurvature (McMahon and Gallop 2005). Asymmetry betweenthe upper and lower leaflets of a bilayer can also be causedby helix insertion, scaffolding, transmembrane proteins, andclathrin coating (McMahon and Gallop 2005; Mashi and Bru-insma 1998; Agrawal and Steigmann 2009). Importantly, thepresence of different species on a lipid bilayer is not gener-ally sufficient to generate nonzero spontaneous curvature. Infact, spontaneous curvature appears to arise only when theflip-flop diffusion of unlike molecules between two leafletsis very slow compared to other time scales underlying shapechanges. Numerical models of vesicles that incorporate spon-taneous curvature reveal novel predictions of equilibriumshapes that appear to agree more closely with experimen-tal observations (Luisi and Walade 2000; Wintz et al. 1996;Mutz and Bensimon 1991; Michalet and Bensimon 1995)than otherwise.
In the present paper, continuum mechanical tools are usedto study the deformation, material symmetry, and energy den-sity of a lipid bilayer with spontaneous curvature. Attentionis restricted to lipid bilayers in the liquid phase. Inspired fromZurlo (2006) and Deseri et al. (2008), these thin structuralelements are treated as three-dimensional bodies rather thanmaterial surfaces. Specifically, the formulation encompassesbending, in-plane stretching, and thickening/thinning of thebilayer. It also enables characterizations of leaflet coherencyand area compatibility; whereas leaflet coherency concernsthe local coupling or sliding of upper and lower leaflets, areacompatibility concerns the integrity of closed lipid bilay-ers (i.e., vesicles). After discussing geometry and kinemat-ics, constitutive behavior is considered. Treating the lipid
bilayer as a three-dimensional body leads not only to a moreprecise understanding of the deformation of lipid bilayersbut also affords insight regarding the material symmetry ofthe lipid bilayers from a bulk material perspective. In par-ticular, stipulating that the lipid bilayer is hyperelastic andinvoking suitable material symmetry requirements lead to arepresentation for the energy density of a lipid bilayer. Inaddition, the impact of imposing the notion of incompress-ibility is considered. Finally, dimension reduction is used toderive an areal (two-dimensional) energy density from thethree-dimensional energy density. As such, it includes theCanham–Helfrich energy density as a particular case. Morebroadly, however, it incorporates possible asymmetry andincoherency of the leaflets.
2 Geometry and kinematics
2.1 Basic considerations
Consider a lipid bilayer, either closed or open, representedby a three-dimensional body B (Fig. 1). Suppose that thelipid bilayer is in its spontaneous (natural) state. Assume thatthe thickness of the bilayer in that state is uniform. Allow,however, for the possibility that the upper and lower leafletsmay have different, but constant, thicknesses h+◦ and h−◦ ,respectively, in which case, the thickness of the bilayer in itsspontaneous state is simply h+◦ + h−◦ . It is useful to intro-duce a reference surface separating the leaflets. This surfaceis referred to as the mid-surface and is denoted by S◦. Theunit normal vector on S◦, directed outward from the regionenclosed by the vesicle, is denoted by m (Fig. 1). The curva-ture tensor of the mid-surface S◦ is denoted by L◦, as definedin (179),1 and is referred to as the spontaneous curvature ten-sor. Furthermore, H◦ and K◦ denote the corresponding spon-taneous mean and Gaussian curvatures, as defined in (180).It is assumed that S◦ does not intersect itself. Additionally,h+◦ and h−◦ are assumed to be sufficiently small relative to theradius of curvature of S◦ to ensure that the inner and outersurfaces of the lipid bilayer Si and So do not fold back onthemselves. A generic material point X located on a materialsurface Sξ◦ in B can then be uniquely described using its pro-jection X onto S◦ and a local coordinate ξ◦ ∈ [−h−◦ , h+◦ ] thatreckons the normal distance between X and X; specifically,
X = X + ξ◦m(X) = ˜X(X, ξ◦). (2)
Consider a deformation χ that maps the lipid bilayer in itsspontaneous state into the observed space. Under χ , the ref-erence placement of the body B, mid-surface S◦, and the nor-mal m to S◦ map to the observed placement B, mid-surface
1 To streamline the presentation, the precise definitions of L◦ and var-ious other useful geometrical objects are relegated to Appendix 9.1.
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Fig. 1 Two-dimensional schematic of a closed lipid bilayer in its spon-taneous state and its deformation to a generic spatial configuration
S◦, and unit normal n to S◦, respectively (Fig. 1). The cur-vature tensor of the surface S◦ is L, as defined in (179), withcorresponding mean and Gaussian curvatures H and K , asdefined in (181). In general, the thickness of the deformedbilayer may be nonuniform. As is customary, it is assumedthat the deformation χ is such that the images S◦, S i, and So
of S◦, Si, and So (see Fig. 1) do not fold back on themselves.Hence, there is, for each x in B, a unique X in B such that x =χ(X). In view of (2), the spatial point x can be described as
x = χ(X) = x(X, ξ◦), X ∈ S◦, ξ◦ ∈ [−h−◦ , h+◦ ]. (3)
The deformation gradient describing the local distortion ofB is given by
F = ∇χ(X), (4)
where ∇ indicates the gradient in the reference space.Consider a material point Xξ◦ on Sξ◦ along with a generic
point X located at a normal elevation z from Sξ◦ . Since thetangent planes T◦ and Tξ◦ at the points X and Xξ◦ are parallel(see Fig. 1), the unit normal m on S◦ is also normal to Sξ◦and, hence, can also be viewed as a function defined on Sξ◦ .Thus, bearing in mind that X coincides with Xξ◦ for z = 0,X can be expressed as X = Xξ◦ + zm(Xξ◦). The spatial pointx corresponding to X can thus also be described via
x = χ(X) = x(Xξ◦ , z), (5)
where Xξ◦ ∈ Sξ◦ and z∈ [(−h−◦ − ξ◦), (h+◦ − ξ◦)]. In view of(5), the deformation gradient F for a material point located
on Sξ◦ can be expressed as
F = ∇x|z=0 =(
∇Sξ◦ x + ∂ x∂z
⊗ m)∣
∣
∣
z=0, (6)
where ∇Sξ◦ indicates the surface gradient on Sξ◦ , defined as
∇Sξ◦ x|z=0 = (∇x|z=0) Pm, (7)
where Pm = 1 − m ⊗ m is the projection tensor onto thetangent plane T◦ (or, equivalently, Tξ◦ ).
Consider a material line element dXξ◦ tangent to the sur-face Sξ◦ and the corresponding line element dxξ◦ tangent tothe spatial image Sξ◦ of Sξ◦ . According to the description ofx in (3),
dxξ◦ = dx(X, ξ◦)|ξ◦=constant
= (∇x(X, ξ◦)|ξ◦=constant) dX
= (∇x(X, ξ◦)|ξ◦=constant)PmdX
= (∇S◦ x(X, ξ◦)|ξ◦=constant) dX, (8)
where ∇S◦ indicates the surface gradient on S◦, as defined by
Using (2) and elementary properties of the curvature tensorL◦ (see (179)), it is easy to arrive at the identity
dXξ◦ = dX + ξ◦dm
= dX − ξ◦L◦dX, (12)
which, on introducing
S◦ = Pm − ξ◦L◦, (13)
can alternatively be written as
dXξ◦ = S◦dX. (14)
Notice that dX is a tangent line element on S◦ at the point X.Since the tangent planes T◦ and Tξ◦ are parallel, dXξ◦ and dXcan be viewed as elements of the same tangent space, say T◦.
The tensor S◦ defined in (13) is fully tangential (seeAppendix 9.1 for the definition of such a tensor) and, thus,can be viewed as a mapping from T◦ to T◦. By (13), thesecond principal invariant I2(S◦) of S◦ can be expressed as
I2(S◦) = 1
2[(I1(S◦))2 − I1(S
2◦)]= 1 − 2ξ◦ H◦ + ξ2◦ K◦. (15)
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Kinematics, material symmetry, and energy densities 1001
Let c◦1 and c◦2 denote the principal curvatures of S◦, so thatH◦ = 1
2 (c◦1 + c◦2) and K◦ = c◦1c◦2. Then, by (15),
I2(S◦) = (1 − ξ◦c◦1)(1 − ξ◦c◦2). (16)
Granted the assumption (imposed to ensure avoiding foldingback of outer and inner surfaces So and Si on themselves)that the thicknesses h+◦ and h−◦ of upper and lower leafletsare very small relative to the radius of curvature of S◦, ξ◦must satisfy
ξ◦c◦α �= 1, α = 1, 2, (17)
and (16) implies that I2(S◦) �= 0. Thus, as described inAppendix 9.2, S◦ has a pseudoinverse Q◦ given by
Q◦ = (I2(S◦))−1(I1(S◦)Pm − S◦), (18)
where
I1(S◦) = tr(Pm − ξ◦L◦)= 2(1 − ξ◦H◦) (19)
is the first principal invariant ofS◦, and, since, as noted above,dXξ◦ and dX can be viewed as elements of the same tangentspace, (14) is equivalent to
dX = Q◦dXξ◦ . (20)
As regards Q◦, it is convenient to introduce
γ◦(ξ◦) = I2(S◦), (21)
and thus, using (13), (15), and (19) in the representation (18)yields
Since (23) holds for an arbitrary line element dXξ◦ , we con-clude that
∇Sξ◦ x|z=0 = (∇S◦ x|ξ◦=constant)Q◦. (24)
The second term on the far right-hand side of (6) includesthe normal derivative (∇x)m, as is clear from (176). Noticethat, according to the descriptions x = x(X, ξ◦) andx = x(Xξ◦ , z), changes of x in the m direction are controlledby ξ◦ and z, respectively. Thus,
(∇x)m = ∂ x∂ξ◦
= ∂ x∂z, (25)
which, in combination with (6), implies that the deformationgradient F may be written as
F = (∇S◦ x)Q◦ + ∂ x∂ξ◦
⊗ m. (26)
2.2 Orientation of phospholipid molecules at themid-surface
Compatible with physical observations of amphiphilic fluidfilms, we assume that, due to interatomic interactions andpacking requirements, the phospholipid molecules compris-ing the bilayer tend to remain perpendicular to the mid-surface S◦. This constraint is embodied by the kinematicalrequirement that: during a deformation, straight line ele-ments perpendicular to S◦ remain straight and perpendicu-lar to S◦ (Fig. 1). This assumption resembles Kirchhoff’s(1850) hypothesis in theories of thin plates and shells. How-ever, at variance with that hypothesis, the kinematical con-straint imposed here does not restrict the through-thicknessdeformation of the lipid bilayer.
To provide an analytical characterization of our constraint,it is useful to represent a generic spatial point x in thedeformed body in the form x = x + ξn( x ), where ξ indi-cates normal distance of x from S◦. Since x is the image ofX in the observed space, and also X can be described by Xand ξ◦ through (2), the out-of-plane coordinate ξ in the bodycan be expressed in the form
ξ = ˜ξ(X, ξ◦). (27)
Moreover, as a result of the constraint, the projection x of xonto S◦ coincides with the placement of the spatial image ofthe material point X on S◦ (Fig. 1). Consequently, bearing inmind (26) and (27), it transpires that
F = (∇S◦ (x + ξn))Q◦ + ∂(x + ξn)
∂ξ◦⊗ m, (28)
where ∇S◦ (x + ξn) and ∂(x + ξn)/∂ξ◦ are given by
∇S◦ (x + ξn) = ∇S◦ x + ξ∇S◦n + n ⊗ ∇S◦ξ (29)
and
∂(x + ξn)
∂ξ◦= ∂ξ
∂ξ◦n, (30)
respectively.It is now convenient to introduce some shorthand notation.
Specifically, given a quantity g dependent either explicitly orimplicitly on ξ◦, let
g|◦ := g|ξ◦= 0 (31)
denote its value at the referential mid-surface S◦. With thisconvention in mind, and according to the definition of surfacegradient of a vector field provided in Appendix 9.1, we have
∇S◦ x = (∇x)|◦Pm
= F|◦Pm
= : F◦, (32)
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1002 M. Maleki et al.
where F◦, which designates the superficial deformation gra-dient on the mid-surface S◦, maps material line elements onS◦ to spatial line elements on S◦.
Having introduced F◦, the areal stretch
J := I2(F◦) (33)
represents any changes in the area of the mid-surface thatmay accompany deformation.
In general,n is a superficial field defined on S◦—that is,ncan be expressed as a function of points x on S◦. However, nmay be extended to all of B. In particular, consider a normallyconstant extension ne of n. Then, in view of developmentspresented in Appendix 9.1,
∇S◦n = (∇ne(χ(X)))|◦Pm
= (∇xne(x))|ξ=0∇x|◦Pm, (34)
where ∇x indicates the spatial gradient. Next, using proper-ties of the surface gradient (see (175)1), we have
∇xne(x) = ∇Sξ◦n
e(x)+ ∂ne(x)∂ξ
⊗ ne(x), (35)
and, since ne does not change in the ξ -direction, we arriveat the identification
(∇xne(x))|ξ=0 = (∇Sξ◦n
e(x))|ξ=0
= ∇S◦n
= −L. (36)
In view of (32) and (36), ∇S◦n as defined by (34) can beexpressed as
∇S◦n = −L(∇x)|◦Pm
= −LF◦, (37)
which, in combination with (28), yields a useful alternativerepresentation,
F = (F◦ − ξLF◦ + n ⊗ ∇S◦ξ)Q◦ + ∂ξ
∂ξ◦n ⊗ m, (38)
for the deformation gradient.Notice that, for a trivial deformation (that is, a deformation
for which F = 1 everywhere on B), n, L, F◦, and ξ aregiven by n=m, L=L◦, F◦ =Pm, and ξ=ξ◦. Under thesecircumstances, bearing in mind (22), (38) specializes to
F = (γ◦(ξ◦))−1(Pm − ξ◦L◦)(Pm − 2ξ◦ H◦Pm + ξ◦L◦)+m ⊗ m
= (1 − 2ξ◦ H◦ + ξ◦2 K◦)−1
×(Pm − 2ξ◦ H◦Pm + 2ξ2◦ H◦L◦ − ξ2◦L2◦)+ m ⊗ m.
(39)
Further, on applying the Cayley–Hamilton theorem (seeAppendix 9.2) to the spontaneous curvature tensor L◦, (39)becomes
F = Pm + m ⊗ m = 1, (40)
which is consistent with what must be true under a trivialdeformation. Moreover, it can be immediately checked that(38) reduces to a result of Zurlo (2006) and Deseri et al.(2008) when the spontaneous curvature vanishes (in whichcase, (18) reduces to Q◦ = Pm).
2.3 Transformation of normal vectors
If the thickness of the lipid bilayer in the spatial configurationis not uniform, the unit normal vectors of its outer and innersurfaces So and S i may differ, respectively, from the mid-surface unit normal n and its negative −n. In this case, letthe unit normal vectors of inner and outer surfaces of bilayerin the reference and the spatial configuration be denoted bymi = −m, mo = m, and ni, no, respectively. Whereas ni
and no must transform according to
n = F−�m|F−�m|
∣
∣
∣
∣
ξ◦=h±◦, (41)
wherein n and m take the values consistent with
(n,m) = (ni,mi) or (n,m) = (no,mo), (42)
n transforms according to
n = F◦e1◦ × F◦e2◦|F◦e1◦ × F◦e2◦|
, (43)
for any two linearly independent tangent vectors e1◦ and e2◦ onthe mid-surface S◦. Substituting (38) into (41) and invoking(43) show that n and n differ unless
∇S◦ξ∣
∣
ξ◦=h±◦ −→ 0, (44)
meaning that any change of bilayer thickness on the spa-tial mid-surface S◦ must be negligibly small. However, sincethis need not be the case, it is important to maintain a distinc-tion between n and n. Existing two-dimensional approachesbased on modeling the lipid bilayer as a material surface, asexemplified by the theory of Steigmann (1999), work solelywith n.
2.4 Coherency of leaflets
The extent to which the upper and lower leaflets are coher-ent across the mid-surface S◦ may influence the mechanicalresponse of a lipid bilayer (Fischer 1992; Božic et al. 1992;Wiese et al. 1992; Seifert 1997). When lipid molecules ofopposite leaflets are interdigitated, their connection is verynearly coherent (Elson et al. 2010). If this is not the case, theleaflets may slide relative to one another. To describe leafletcoherency, consider the Hadamard compatibility conditionat S◦. Let F+ and F− denote the respective limiting values ofthe deformation gradient at S◦ from the outer and inner sides
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Kinematics, material symmetry, and energy densities 1003
of the bilayer. To facilitate the calculation, we will introducesome notation. Given a quantity g with potentially differentlimits on either side of the referential mid-surface S◦, defineits jump [[g]] and average 〈g〉 at S◦ by
[[g]] := g+ − g−, 〈g〉 := 1
2(g+ + g−). (45)
In view of (38), the jump [[F]] of the deformation gradient atthe mid-surface (i.e., at ξ◦ = 0) is given by
[[F]] = [[F◦]] +[[
∂ξ
∂ξ◦
]] ∣
∣
∣
∣◦n ⊗ m. (46)
For [[F◦]] = 0, (46) represents the Hadamard conditionfor a coherent surface. This condition corresponds to thelocal interdigitation or coupling of the leaflets. However,for [[F◦]] �= 0, the mid-surface is an incoherent interfaceacross which the leaflets may slide relative to one another. Ingeneral, [[∂ξ/∂ξ◦]]|◦ need not vanish—as occurs if the lipidbilayer is not symmetric and its constitutive properties arediscontinuous across the mid-surface S◦. As mentioned ear-lier, a disparity between the molecular compositions of theleaflets is a potential reason for the existence of spontaneouscurvature. For lipid bilayers with such induced spontaneouscurvature, the jump [[∂ξ/∂ξ◦]]|◦, therefore, does not gener-ally vanish.
2.5 Area compatibility
For closed lipid bilayers, an additional global compatibil-ity condition becomes important. During the deformationof a vesicle, whether or not the leaflets are coherent, theirbounding surfaces at the common interface (what is calledhere mid-surface) must share the same area. Otherwise, theleaflets may loose their integrity. In view of the definition(33) of the areal stretch, this condition can be codified in theform∫
S◦
[[J ]] dA◦ = 0, (47)
where, bearing in mind (45)1, [[J ]] denotes the jump of theareal stretch at the mid-surface S◦, and dA◦ represents thereferential area element of S◦.
3 Energy densities
Any change of energy that accompanies a change in the shapeof a lipid bilayer must be due to the displacement of neigh-boring phospholipid molecules. Various types of lipid mole-cules with different physical properties can be present in alipid bilayer. As long as no phase separation occurs and, thus,the chemical energy remains fixed, any energy change can
be attributed to purely mechanical phenomena. In this set-ting, a multi-component biomembrane can be viewed as aneffectively homogenous body. For a single-component lipidbilayer, no such assumption is needed.
Molecular displacements are accompanied by changes inthe amount of elastic energy stored within the lipid bilayer.This is modeled by introducing an energy density W (per unitreferential volume) as a function W of the deformation gra-dient F, so that the lipid bilayer is modeled as a hyperelasticmaterial:
W = W (F). (48)
As in the case of a conventional elastic body, we require thatthe energy density of a lipid bilayer has a local minimum atthe spontaneous state:
W (F) has local minimum at F = 1. (49)
Also, since the energy density of each material point maybe additively scaled (Gurtin et al. 2010), we impose, withoutloss of generality, the normalization
W (1) = 0. (50)
Zurlo (2006) and Deseri et al. (2008) studied the chemo-mechanical coupling of a lipid bilayer undergoing phase sep-aration and elastic deformation. Under these circumstances,W would also be a function of species concentrations and,potentially, also their gradients, which would penalize theformation of phase interfaces on the bilayer.
Requiring the energy density in (48) to be frame-indifferentleads, in conventional fashion, to the conclusion that it maydepend on the deformation gradient through at most the rightCauchy–Green tensor C = F�F, whereby (48) is replaced by
W = W (C). (51)
An important point concerning leaflet asymmetry shouldnow be clarified. An asymmetric distribution of lipid mole-cules with different molecular shapes requires the shape ofthe lipid bilayer in its natural state to be curved (McMahonand Gallop 2005). This effect might be modeled by allowingfor nontrivial spontaneous curvature L◦. There is also con-siderable evidence pointing to marked differences betweenthe chemical compositions of lipid molecules in the innerand outer leaflets of animal cells (Luckey 2008; Janmey andKinnunen 2006; Devaux and Morris 2004). Observed differ-ences in mechanical properties (Janmey and Kinnunen 2006)might, therefore, be attributed to differences in molecularpacking, chemical composition, or both. To encompass dif-ferences in the mechanical properties of the leaflets, it mightbe sufficient to allow the expression determining the energydensity function W to be distinct in each leaflet.
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1004 M. Maleki et al.
4 Material symmetry
The constitutive relation (51), which holds for all hypere-lastic materials, is very general. To incorporate the prop-erties of a lipid bilayer, the response function W mustobey certain requirements of material symmetry. Two dis-tinguishing features of biomembranes are in-plane fluidityand in-plane isotropy. Specifically, experimental observa-tions demonstrate unambiguously that phopholipid mole-cules on the surface of a lipid bilayer in the liquid phase canfreely migrate. Additionally, there is no preferred direction inthe tangent plane of its mid-surface; and therefore, the lipidbilayer exhibits in-plane isotropy. In view of these observa-tions, when modeled as three-dimensional, a lipid bilayer islike a transversely isotropic material with m being the axisof isotropy and Sξ◦ being the surface of isotropy, where therange of ξ◦ covers the thickness of the lipid bilayer. Bearingthis in mind, we next derive an appropriate representationtheorem for an energy density which correctly incorporatesboth in-plane fluidity and transverse isotropy. To achieve this,it is necessary to determine a proper unimodular symmetrytransformation H of the reference configuration which leavesthe response of the body to deformation unchanged. As iscustomary, the set of all such symmetry transformations isdesignated by G and is referred to as the symmetry group.
4.1 Symmetry transformations
Let H be a symmetry transformation. To encompass the trans-verse isotropy of the lipid bilayer, H should preserve thedirection of any material line element parallel to m. In addi-tion, it is necessary to require that H preserves the length ofany material line element parallel to m. If H does not do so,the phospholipid molecules deform along the direction ofthe tail groups, and consequently, their physical character-istics will generally change. These two requirements simplyimply that H should map any normal material line element toitself. Consistent with the in-plane fluidity of lipid bilayers,H should also map any material line element perpendicularto m to a material line element perpendicular to m. Thus,given a unit normal vector m, H should satisfy
Hm = m,
if e · m = 0, then He · m = 0,det H = 1.
⎫
⎬
⎭
(52)
Under a symmetry transformation, the deformation gra-dient F becomes FH, and hence, the right Cauchy–Greentensor C becomes H�CH, while the energy measured by theresponse function W must be invariant:
W (H�CH) = W (C). (53)
As a first step toward determining a representation forthe symmetry group G consistent with (52) and (53), choose
two arbitrary linearly independent tangent vectors e1 and e2
satisfying e1 · m = e2 · m = 0 and, therefore, belonging tothe tangent space Tξ◦ . Without loss of generality, H can berepresented as
H = e1 ⊗ f + e2 ⊗ g + m ⊗ h, (54)
with f, g, and h being linearly independent. Since (e1 ⊗ f +e2 ⊗ g + m ⊗ h)m = (f · m)e1 + (g · m)e2 + (h · m)mand e1 and e2 are linearly independent tangent vectors, (52)1
implies that f and g must obey
f · m = g · m = 0 (55)
and, hence, must be tangent vectors and that h must obey
h · m = 1. (56)
Next, since (e1 ⊗ f + e2 ⊗ g + m ⊗ h)e = (f · e)e1 +(g · e)e2 + (h · e)m for any tangent vector e, on definingα1 = f · e and α2 = g · e, it follows that
He = α1e1 + α2e2 + (h · e)m. (57)
However, by (52)2, h must satisfy
h · e = 0, (58)
which, in combination with (56), yields h = m. Thus, Hmust admit a representation of the form
H = H + m ⊗ m, (59)
where, bearing in mind (55),
H := e1 ⊗ f + e2 ⊗ g (60)
is a fully tangential tensor and, therefore, obeys
Hm = H�m = 0,H = PmH = HPm = PmHPm.
}
(61)
Notice that, granted the representation (59) for H, the condi-tion (52)3 is equivalent to the following condition on H:
I2(H) = 1. (62)
4.2 Representation theorem for the energy density of a lipidbilayer
Observe that the right Cauchy–Green tensor C can beexpressed in the form
C = C + v ⊗ m + m ⊗ v + (m · Cm)m ⊗ m, (63)
where C is a fully tangential tensor defined by
C = (FPm)�(FPm) = PmCPm, (64)
and v is a tangent vector belonging to Tξ◦ defined by
v = Cm − (m · Cm)m. (65)
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Kinematics, material symmetry, and energy densities 1005
In view of (51) and (63)–(65), the energy density W can beexpressed as a function W depending on C, v, and m · Cm:
W = W (C)
= W (C,v,m · Cm). (66)
Additionally, by (59) and (63), H�CH can be expressedas
H�CH = H�CH + H�v ⊗ m + m ⊗ H�v
+(m · Cm)m ⊗ m, (67)
and hence, the symmetry property (53) takes the form
W (H�CH,H�v,m · Cm) = W (C,v,m · Cm). (68)
Being fully tangential (see Appendix 9.1), C and H can beviewed as mappings of the tangent space Tξ◦ to itself, inwhich case the transformation rule
C −→ H�CH (69)
can be interpreted as one involving tensors that map the tan-gent space Tξ◦ to itself. Similarly, since v belongs to Tξ◦ , thetransformation rule
v −→ H�v (70)
can be interpreted as one involving the two-dimensional ref-erential tangent vectors belonging to Tξ◦ .
Consider the subgroup of the unimodular group comprisedby all rotations in the tangent plane with normal vector m.Bearing in mind that, like C and H, any element of thatsubgroup can be viewed as a mapping of the space of two-dimensional referential tangent vectors to itself, a represen-tation theorem due to Zheng (1993) can be applied to arriveat the following representation for the energy density:
W = ˜W (I1(C), I2(C),v · v,v · Cv,m · Cm). (71)
However, since, by (63),
v · Cv = det C − (m · Cm)I2(C)
+I1(C)[v · v − (m · Cm)2], (72)
(71) can be written as
W = W (I1(C), I2(C),v · v, det C,m · Cm). (73)
The five arguments of W are mutually independent. To seethis, first, notice that the tensor C has six independent com-ponents and, hence, that the decomposition in (63) definesthree independent quantities C, v, and m · Cm. Since C issymmetric, it admits a spectral decomposition
C = ω1t1 ⊗ t1 + ω2t2 ⊗ t2 (74)
involving eigenvalues ω1 and ω2 corresponding, respec-tively, to eigenvectors t1 and t2 tangent to Tξ◦ with the
property t1 × t2 = m. Additionally, in terms of the basis{t1, t2}, v can be expressed in the form
v = v1t1 + v2t2, (75)
where vα denote the component of v in the tα direction(α = 1, 2). Also, since m · Cm indicates the componentof C in the direction perpendicular to the tangent space Tξ◦ ,it can be chosen independently of the remaining arguments.Put τ = m · Cm, bearing in mind that, since C is positive-definite, τ must obey τ > 0. The arguments of (73) can thenbe expressed as
I1(C) = ω1 + ω2,
I2(C) = ω1ω2,
v · v = v21 + v2
2,
det C = ω1ω2τ − ω1v22 − ω2v
21,
m · Cm = τ.
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
(76)
Since ω1, ω2, v1, v2, and τ are independent, (76) can beused to show that the numbers I1(C), I2(C), det C, and v ·Cv, m · Cm can be varied independently.
We next show that, of the five arguments upon which Wmay depend, as indicated on the right-hand side of (73), onlyI2(C), det C, and m · Cm are invariant under any unimod-ular symmetry transformation H consistent with (61) and(62). This elucidates the distinction between the symmetryproperties of a lipid bilayer with in-plane fluidity (that is, alipid bilayer with symmetry group consisting of all fully tan-gential unimodular transformations H) with those of a solidtransversely isotropic about m. Further insight regarding thesymmetry properties of fluid films and transversely isotropicsolids due, respectively, to Steigmann (1999) and Green andAdkins (1970) are particularly relevant.
To substantiate the foregoing assertion, choose orthonor-mal unit tangent vectorse1 ande2 spanning the tangent spaceTξ◦ . A generic tangent vector v in (73) can be representedin the form v = λ1e1 + λ2e2, with λ1 and λ2 being scalars.Without loss of generality, assume that e1 is chosen in the vdirection, so that e1 = v/|v|. To show that W can depend atmost on I2(C), det C, and m · Cm, it suffices to: (i) showthat they are invariant under all unimodular symmetry trans-formations of the kind described in (52), and (ii) find specifictranformations under which I1(C) and v ·v are not properlyinvariant. Bearing in mind (52)3, (59), (61)1, and (184), it isstraightforward to show that, for any H satisfying (52),
I2(H�CH) = I2(C),
det(H�CH) = det C,m · H�CHm = m · Cm.
⎫
⎬
⎭
(77)
Consider, now, the tensor
H1 = e1 ⊗ e1 + e2 ⊗ e2 + αe2 ⊗ e1, α ∈ R (78)
123
1006 M. Maleki et al.
which satisfies (61) and (62) for all choices of the parameterα. Then, since
I1(H�1CH1) = I1(C)+ ϕ(α), (79)
with ϕ(α) = 2α(e1 · Ce2)+ α2(e2 · Ce2), and since
H�1v · H�
1v = v · v, (80)
W must obey
W (I1(C)+ ϕ(α), ·,v · v, ·, ·) = W (I1(C), ·,v · v, ·, ·)(81)
for all α. It is possible to choose α in (81) such that ϕ(α) =ν I1(C) for an arbitrary ν ≥ 0. Also, on defining x := I1(C)
and y := (1 + ν)I1(C), fixing I2(C), v · v, det C, andm · Cm, and defining f via f (x) := W (x, I2(C),v · v,det C,m ·Cm), it is evident that f (y) = f (x). Hence, sincex and y may be chosen arbitrarily, f must be constant. Itfollows that W must be independent of the argument I1(C).Next, consider the tensor
H2 = βe1 ⊗ e1 + 1
βe2 ⊗ e2, β ∈ R, β �= 0, (82)
which satisfies (61) and (62) for all choices of the parameterβ �= 0. Then, since
H�2v · H�
2v = β2v · v, (83)
W must obey
W (·, β2v · v, ·, ·) = W (·,v · v, ·, ·) (84)
for allβ �= 0. On choosingβ in (84) such thatβ2 = (v·v)−1,it follows that W must be independent of the argument v ·v.Consequently, we conclude that the energy density W of alipid bilayer must admit a representation of the form
W = Φ(I1, I2, I3), (85)
with
I1 = I2(C), I2 = det C, I3 = m · Cm. (86)
5 Kinematical discussions
With the representation (38) for the deformation gradient F,the right Cauchy–Green tensor C = F�F can be expressedas
C = Q◦C◦Q◦ − 2ξQ◦F�◦LF◦Q◦ + ξ2Q◦F�◦L2F◦Q◦+(Q◦∇S◦ξ)⊗ (Q◦∇S◦ξ)
+ ∂ξ
∂ξ◦((Q◦∇S◦ξ)⊗ m + m ⊗ (Q◦∇S◦ξ))
+( ∂ξ
∂ξ◦
)2m ⊗ m, (87)
where we have introduced
Fig. 2 Schematic depiction of changes of volume and tangent areaelements
C◦ = F�◦F◦. (88)
In addition, on comparing (64) and (87), it follows that thefully tangential tensor C can be expressed as
C = Q◦C◦Q◦ − 2ξQ◦F�◦LF◦Q◦+ξ2Q◦F�◦L2F◦Q◦ + (Q◦∇S◦ξ)⊗ (Q◦∇S◦ξ).
(89)
5.1 The invariants I1, I2, and I3
Prior to formulating specific constitutive relations, it seemsnecessary to understand the geometric properties of theinvariants I1, I2, and I3 entering the representation (85) ofthe generic energy density for a lipid bilayer.
5.1.1 The invariant I1
We now show that the invariant I1 = I2(C) controls changesin the area of infinitesimal area elements parallel to the mid-surface S◦ (Fig. 2). Toward this goal, choose linearly inde-pendent vectors e1 and e2 belonging to the tangent spaceTξ◦ that satisfy e1 × e2 = m. Further, choose infinitesimalmaterial line elements dX1
ξ◦ and dX2ξ◦ directed along e1 and
e2 such that they span the infinitesimal area element
dAξ◦ = |dX1ξ◦ × dX2
ξ◦ |. (90)
With reference to (20), define elements e1◦ and e2◦ of T◦by e1◦ = Q◦e1 and e2◦ = Q◦e2, and let the infinitesimalline elements dX1 and dX2, as described in (20), denote theimages of dX1
ξ◦ and dX2ξ◦ directed along e1◦ and e2◦ on S◦.
Obviously, dX1 and dX2 span the image
dA◦ = |dX1 × dX2| (91)
of dAξ◦ on S◦. The area element dAξ◦ corresponding to dAξ◦in the deformed body is
dAξ◦ = |Fe1 × Fe2| dAξ◦ . (92)
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Kinematics, material symmetry, and energy densities 1007
Since e1 and e2 are tangent vectors, the deformation gradi-ent F in (92) can be replaced by the superficial deformationgradient
F := FPm, (93)
giving
dAξ◦ = |Fe1 × Fe2| dAξ◦ , (94)
whereby the area ratio Jξ◦ = dAξ◦/dAξ◦ takes the form
Jξ◦ = |Fe1 × Fe2|= |Fc(e1 × e2)|= |Fcm|, (95)
with Fc being the cofactor of F. By (95),
J 2ξ◦ = |Fcm|2
= Fcm · Fcm
= m · (Fc)�Fcm
= m · (F�F)cm
= m · Ccm. (96)
By (183), m · Ccm = I1, which with (96) yields
I1 = I2(C) = J 2ξ◦ =
(dAξ◦dAξ◦
)2, (97)
confirming the assertion that I1 controls changes in the areaof infinitesimal area elements parallel to S◦. Evaluating (97)at the mid-surface S◦ gives
J 2 =(dA◦
dA◦
)2 = I2(C◦), (98)
where J is defined in (33), and dA◦ is the image, on S◦, ofarea element dA◦.
According to the definition (172) of the surface gradientand the chain rule, ∇S◦ξ in (89) can be replaced by F�◦∇S◦ξ ,where ∇S◦ξ = Pn∇xξ is the surface gradient of ξ on S◦,and
Pn = 1 − n ⊗ n (99)
is the projection tensor onto the tangent plane of S◦. SinceQ◦ is symmetric, (89) can be reorganized as
C = Q◦F�◦KF◦Q◦, (100)
where we emphasize that
K = Pn − 2ξL + ξ2L2 + (∇S◦ξ)⊗ (∇S◦ξ) (101)
is a fully tangential tensor which can be viewed as a mappingfrom the tangent space of S◦ to itself. From (184)2, it followsthat
I1 = I2(C) = I 22 (Q◦)I2(C◦)I2(K). (102)
The definition (196) of the second principal invariant I2 anda straightforward calculation lead to
I2(K) = γ 2(ξ)+ |((2Hξ − 1)Pn − ξL)∇S◦ξ |2, (103)
with
γ (ξ) := 1 − 2ξH + ξ2 K . (104)
In view of (21), (98), and (184)2, (102) can be expressed as
I1 = J 2(γ◦(ξ◦))−2(γ 2(ξ)
+|((2Hξ − 1)Pn − ξL)∇S◦ξ |2). (105)
5.1.2 The invariant I2
It is evident that the invariant I2 = det C = (det F)2 controlsvolume changes of infinitesimal material regions (Fig. 2).However, it is useful to obtain I2 in terms of relevant kine-matical quantities. According to the definition of the deter-minant and upon using (38),
det F = [(Fe1)× (Fe2)] · Fm.
= ∂ξ
∂ξ◦
(
(g1◦ × g2◦) · n − ξ [(Lg1◦ × g2◦) · n
+(g1◦ × Lg2◦) · n] + ξ2(Lg1◦ × Lg2◦) · n)
, (106)
where g1◦ = F◦e1◦ and g2◦ = F◦e2◦ are tangent to S◦ and spandA◦.
In view of (183), (184)2, and (21)
e1◦ × e2◦ = Q◦e1 × Q◦e2
= Qc◦(e1 × e2)
= Qc◦m= I2(Q◦)m= (I2(S◦))−1m
= (γ◦(ξ◦))−1m (107)
which, with (96) and (98), implies that
(g1◦ × g2◦) · n = |F◦e1◦ × F◦e2◦|= (γ◦(ξ◦))−1|Fc◦m|= (γ◦(ξ◦))−1 J. (108)
By (108), (177), (181)–(183), and elementary properties ofthe trace and cofactor of L, (106) simplifies to
det F = J∂ξ
∂ξ◦γ (ξ)
γ◦(ξ◦), (109)
and consequently, I2 becomes
I2 = det C = J 2( ∂ξ
∂ξ◦
)2( γ (ξ)
γ◦(ξ◦)
)2. (110)
123
1008 M. Maleki et al.
5.1.3 The invariant I3
Finally, it is evident that the invariant
I3 = m · Cm
=( ∂ξ
∂ξ◦
)2(111)
controls the stretch of infinitesimal material fibers perpen-dicular to the mid-surface S◦.
In view of (105), (110), (111), we may use (196) in (105) toconclude that the energy density in (85) depends upon thereferential variables
H◦, K◦, ξ◦, (112)
and the spatial variables
H, K , ξ,∂ξ
∂ξ◦, |∇S◦ξ |2, (∇S◦ξ) · L(∇S◦ξ). (113)
Consistent with existing two-dimensional theories, H andK —which are paramount importance in the areal Canham–Helfrich energy density (1)—represent the dependence onthe curvature of the spatial mid-surface S◦ and the arealstretch J—which is present in the areal energy density ofSteigmann (1999)—embodies localized changes in the areain going from the referential mid-surface S◦ to the spatialmid-surface S◦. In addition to through-thickness dependencevia ξ , (85) accounts for the potential influence of transversenormal strain (and, consequently, thickness changes) via∂ξ/∂ξ◦ as well as both transverse shear strain and thicknessnonuniformity via |∇S◦ξ |2. Potential coupling between cur-vature and deviations in thickness is embodied by the quantity(∇S◦ξ) ·L(∇S◦ξ), which includes information regarding theorientation of ∇S◦ξ relative to the principal axes of the curva-ture tensorL. Notice that, for example, when ∇S◦ξ coincideswith one of the principal axes ofL, dependence upon the cou-pling term (∇S◦ξ) ·L(∇S◦ξ) is redundant. In particular, thisoccurs when the spatial mid-surface S◦ is spherical.
5.3 Incompressibility
Various studies suggest that lipid bilayers are very nearlyincompressible (Goldstein and Leibler 1989; Lipowsky andSackmann 1995; Safran 2003). If, to model this observation,the deformation χ is stipulated to be isochoric, then (110)has the elementary consequence
I2 = 1. (114)
Accordingly, the energy density must be independent of I2
and the representation (85) reduces to
W = Φ(I1, I3). (115)
In addition, using (114) in (110) along with (15), (21), and(104) yields the differential equation
∂ξ
∂ξ◦= 1
J
γ◦(ξ◦)γ (ξ)
= 1 − 2ξ◦ H◦ + ξ2◦ K◦(1 − 2ξH + ξ2 K )J
, (116)
which, with (27), integrates to yield ξ(1 − ξH + 13ξ
The relation (117) reveals that, for an incompressible lipidbilayer, H, K , J , and ξ are not generally independent. Forinstance, when J, H , and K at any point on the spatial mid-surface S◦ are given, (117) can be solved to determine thedistance ξ of a spatial point on S◦ corresponding to the mate-rial point at the distance ξ◦ on S◦ in the spontaneous state.In addition, the purely geometrical and kinematical result(117) suggests that the thickness of an incompressible lipidbilayer in its spatial configuration will not generally be uni-form unless its mid-surface is uniformly bent or stretchedand has uniform spontaneous curvature. Zurlo (2006) andDeseri et al. (2008) assume that ξ is linearly proportional toξ◦, or, alternatively, that ∂ξ/∂ξ◦ is the ratio of thickness ofthe deformed and referential lipid bilayer and refer to thiscondition as “quasi-incompressibility.” In such a case, theincompressibility condition is only satisfied at the spatialmid-surface S◦ and leads to
ξ
ξ◦= 1
J. (118)
However, this assumption is valid only if the spontaneousmean and Gaussian curvatures H◦ and K◦ and the mean andGaussian curvatures H and K of the spatial mid-surface arevery mild (or, more precisely, if ξH, ξ2 K , ξ◦ H◦, and ξ2◦ K◦are negligible in comparison with unity).
Notice that (117) can be viewed as a cubic equation for ξ .However, only a unique physically meaningful root of thisequation is of interest. First, the root must be real. Also, toguarantee its uniqueness, the root should be an increasingfunction of ξ◦. In addition, to satisfy the requirement thatξ |◦ = 0, the sign of ξ must match that of ξ◦. These condi-tions limit the range of the coefficients in (117). In particular,the sign of the discriminant of the cubic equation providessome information about the nature of the roots. A simplercase occurs if the lipid bilayer only has curvature in a two-dimensional space and is uniformly extended in one direc-tion. In this circumstance, K = 0 (for simplicity also assume
123
Kinematics, material symmetry, and energy densities 1009
(a)
(b)
Fig. 3 Illustrative isochoric deformations of a lipid bilayer: a Thick-ness change due to pure bending. b Thickness change due to purestretching. While the dashed lines are the spatial mid-surface S◦, thegray lines are spatial placements of few material surfaces with constantξ◦ in the reference configuration
that K◦ = 0) in which case (117) reduces to a quadratic equa-tion with the admissible root
ξ = 1 − √1 − 4ξ◦ H(1 − ξ◦ H◦)/J
2H, (119)
which is real if and only if
4ξ◦ H(1 − ξ◦ H◦) ≤ J. (120)
For there to exist an admissible root ξ satisfying the incom-pressibility condition (117), the ratio ξ◦H/J of the deformedlipid bilayer must be small enough to satisfy (120) for allξ◦ ∈ [−h−◦ , h+◦ ].
To provide a qualitative insight regarding incompressibility-induced changes of thickness, we restrict attention to situa-tions where the spontaneous mean curvature H◦ vanishes andconsider two illustrative examples. Figure 3a shows a lipidbilayer under pure bending (in which case J = 1) with spatialmid-surfaceS◦ having a sinusoidal shape. It is evident that thethickness of the upper (lower) leaflet decreases (increases)as the curvature H of S◦ increases. Figure 3b shows a lipidbilayer with a flat spatial mid-surface (in which case H =0)subject to an areal stretch that decreases exponentiallytoward the left side. Notably, the thicknesses of lipid bilayerleaflets increase symmetrically as the areal stretch decreasesfrom right to left. Thickening phenomena of this kind are
well-known to occur in the presence of transmembrane pro-teins, in which case hydrophobic mismatch leads to stretch-ing of the lipid molecules in the through-thickness direction(here, the left side of Fig. 3b).
6 Stress relations
Consistent with the procedure pioneered by Coleman andNoll (1963), the elastic Cauchy stress tensor correspondingto an energy density of the form (85) is given by
T = 2√I2F(
3∑
i=1
αi∂Ii
∂C
)
F�, (121)
where αi = ∂Φ/∂Ii (i = 1, 2, 3). Straightforward calcula-tions based on the definitions (86) lead to
∂I1
∂C= tr(C)Pm−C,
∂I2
∂C=I2C−1,
∂I3
∂C=m ⊗ m. (122)
Using (122) in (121) results in
T= 2√I2
(
α1(I1(B)B − B2)+α2I21+α3Fm ⊗ Fm)
,
(123)
where B = FFT. Notice that, in contrast to C, B neednot be a fully tangential tensor. When the lipid bilayer isincompressible, (123) should be replaced by
T = −p1 + 2(α1(I1(B)B − B2)+ α3Fm ⊗ Fm), (124)
where αi = ∂Φ/∂Ii (i = 1, 3), and p is an unknownLagrange multiplier that penalizes the incompressibility.
7 Dimension reduction for an incompressible lipidbilayer
The derivation of two-dimensional models of shell-like struc-tures from three-dimensional elasticity has long been a sub-ject of interest. For a comprehensive review of this subject,see Ciarlet (2005). Simmonds (1985); Stumpf and Makowski(1986); Taber (1987, 1988, 1989), and Yükseler (2005) haveall used the procedure to develop hyperelastic shell theories.However, Zurlo (2006) and Deseri et al. (2008) were the firstto apply it to biomembranes. In contrast to the present work,Zurlo (2006) and Deseri et al. (2008) neglected spontaneouscurvature.
7.1 General strategy
Granted that a lipid bilayer has thickness considerablysmaller than its lateral dimensions, it is very reasonable toattribute to it an energy density, per unit area. The main goalof the dimension reduction described in this section is to
123
1010 M. Maleki et al.
Fig. 4 Schematic of the dimension reduction
obtain an areal energy densityψ◦ from the volumetric energydensity W in accord with the condition∫
D◦
ψ◦ dA◦ =∫
P
W dv, (125)
where ψ◦ is measured per unit area on the referential mid-surface S◦, dv is the referential volume element, D◦ ⊂ S◦is an arbitrary area on S◦, and P ⊂ B is the material regionassociated with D◦, the lateral faces of which are normal toS◦ and extended to the inner and outer surfaces Si and So
(see Fig. 4).In view of (90), (91), and (107), we conclude that
∫
D◦
ψ◦ dA◦ =∫
D◦
h+◦∫
−h−◦
Φ(I1, I2, I3)γ◦(ξ◦) dξ◦dA◦. (126)
Since D◦ is an arbitrary domain, (126) implies that
ψ◦ =h+◦∫
−h−◦
Φ(I1, I2, I3)γ◦(ξ◦) dξ◦. (127)
Regarding (105), (110), and (111), and bearing in mind thatΦ is an arbitrary function of the invariants I1, I2, and I3, itis evident that the integrand of (127) may, in general, dependon ξ◦ in a complex manner. This makes integrating (127)difficult. A suitable approximate approach is to expand theintegrand in (127) about ξ◦ = 0 and truncate consistent withsome desired degree of accuracy.
7.2 Expansion
Consistent with the consensus regarding the near incom-pressibility of lipid bilayers, we conduct this expansion onlyfor an energy density of the form (115), in which case (127)is replaced by
ψ◦ =h+◦∫
−h−◦
Φ(I1, I3)γ◦(ξ◦) dξ◦. (128)
Notice that, by (105), (111), and (116), the values of theinvariants I1 and I3 at S◦ are
I1|◦ = J 2 (129)
and
I3|◦ = J−2. (130)
In view of (129) and (130), the values of Φ and any of itspartial derivatives at ξ◦ = 0 may depend at most on theareal stretch J . Anticipating the need to expand Φ up tosecond order in ξ◦, it is, therefore, convenient to introduceJ -dependent quantities f0, f1, f3, f11, f13, and f33 via
f0 = Φ(I1, I3)|◦,
fk = ∂Φ(I1,I3)∂Ik
∣
∣
∣
∣◦, k = 1, 3,
fkl = ∂2Φ(I1,I3)∂Ik∂Il
∣
∣
∣
∣◦, k, l = 1, 3.
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(131)
Expanding Φ to second order in ξ◦ also requires the valuesof the first and second derivatives, with respect to ξ◦, of theinvariants I1 and I3 at ξ◦ = 0. On introducing
H = H J−1 − H◦, K = K J−2 − K◦, (132)
it follows that
I ′1|◦ = −4J 2 H ,
I ′3|◦ = 4J−2 H ,
I ′′1 |◦ = −2J 2(12H◦ H − 2K − J−4|∇S◦ J |2),
I ′′3 |◦ = 4J−2(8H2 + 6H◦ H − K ).
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(133)
Expanding Φ then yields
ψ◦ =h+◦∫
−h−◦
(
f0 + α1ξ◦ + 1
2α2ξ
2◦ + o(ξ2◦ ))
γ◦(ξ◦) dξ◦, (134)
where α1 and α2 are given by
α1 = ( f1I ′1 + f3I ′
3)|◦ (135)
and
α2 =( f1I ′′1 + f3I ′′
3 + f11(I ′1)
2+2 f13I ′1I ′
3 + f33(I ′3)
2)|◦,(136)
respectively.
7.3 Restriction to mild areal stretch
Hereafter, we confine our attention to circumstances underwhich the areal stretch J of the referential mid-surface S◦ issufficiently mild to ensure that
h◦ J−1|∇S◦ J | � 1. (137)
123
Kinematics, material symmetry, and energy densities 1011
The gradient term on the right-hand side of (133)3 is thennegligible in comparison with other terms. With this in mind,substituting (131), (135), and (136) in (134), performingthe integration, and truncating yields an expression for theareal energy density ψ◦, measured per unit area of S◦, yieldsan expression that depends on H◦, K◦, H, K , and J . Thedimension reduction, therefore, provides an areal energy den-sity which includes the effects of spontaneous mean andGaussian curvatures, deformed mean and Gaussian curva-tures, and areal stretch.
In addition, an areal energy density ψ , measured per unitarea in the deformed state, has the form
ψ = J−1ψ◦ (138)
and, thus, depends on the same quantities upon which ψ◦depends.
7.4 Specialization to symmetric bilayers
Suppose that the leaflets of the bilayer have identical thick-ness h◦ = h+◦ = h−◦ and molecular composition, in whichcase they should be described by a single response functionΦ. The areal energy densityψ◦ determined by the dimensionreduction argument then simplifies to
ψ◦ = ψm + 1
2κ(H J−1 − H◦)2 + κ K , (139)
where ψm, H◦, κ , and κ are given by
ψm = 2h◦(1 + 16 h2◦K◦) f0
+ 43 h3◦H2◦ (J 2(1 − ν) f1 − J−2(1 + 7ν) f3),
H◦ = H◦(1 + ν),
κ = 323 h3◦ J 2μ, κ = 4
3 h3◦ J 2η.
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(140)
with
ν = η4μ, η = ( f1 − J−4 f3),
μ = 2J−4 f3 − 2J−2 f13 + J−6 f33 + J 2 f11,
}
(141)
Notice that the bending moduli κ and κ given in (140)3,4
scale with the cube of the leaflet thickness h◦. Moreover,those moduli differ from those obtained by Zurlo (2006) andDeseri et al. (2008). The difference stems from our use of theexact incompressibility condition (116) in place of their useof the quasi-incompressibility condition (118). For instance,the bending moduli in (140) include derivatives of Φ withrespect to I3, derivatives which are absent from the result ofZurlo (2006) and Deseri et al. (2008).
7.5 Alternative interpretations of the splay and saddle-splaymoduli
We now provide alternative interpretations of the splay andsaddle-splay moduli κ and κ . Consider the state of stress at
the spatial mid-surface S◦ of an incompressible lipid bilayer.Since F◦ is a tangential tensor, F�◦n = 0 and B◦ = F◦F�◦is a fully tangential tensor. Using (196), it can be shown thatI2(B◦) = I2(C◦) = J 2. As a consequence of these facts,(124) yields
T◦ := T|ξ= 0
= −p1 + 2(J 2 f1Pn + f3(Fm ⊗ Fm)|◦). (142)
Consistent with the kinematical assumption regarding theorientation of the phospholipid molecules, consider a defor-mation with the property
F|◦m = φn, φ > 0, (143)
where φ is an arbitrary constant. Also, as is customary in thetheory of thin shell-like structures, assume that the normalstress n · Tn is very small compared to all other relevantstress components. Then, in view of (143) and the assumptionn · T◦n = 0, (142) can be written as
T◦ = 2( f1 J 2 − φ2 f3)Pn. (144)
Moreover, (143) reduces C in (63) to
C|◦ = C◦ + φ2m ⊗ m. (145)
Since the deformation must be isochoric, we find that
φ2 = (I2(C◦))−1 = J−2, (146)
which allows us to reduce (144) to
T◦ = 2J 2( f1 − J−4 f3)Pn,
= : ΣPn. (147)
Notice that T◦ is an isotropic tensor on the tangent spaceof S◦. Also, Σ can be considered as the in-plane tensionor compression at the spatial mid-surface S◦. Invoking thedefinitions of Σ in (147) and η provided in (141)2, we findthat
η = 1
2J−2Σ. (148)
In addition, we may define
Λ := ∂Σ
∂ J, (149)
as the areal stiffness of the lipid bilayer at S◦, which, with(141) and (147), can be expressed as
Λ = 4J (μ+ η). (150)
Using (148), relation (150) becomes
μ = JΛ− 2Σ
4J 2 . (151)
Then, on using (148) and (151), the splay and saddle-splaymoduli defined in (140) can be expressed in terms of Σ andΛ as
123
1012 M. Maleki et al.
κ = 83 h3◦(JΛ− 2Σ), κ = 2
3 h3◦Σ. (152)
7.6 Canham–Helfrich-type energy density
In many studies, due to high in-plane resistance, the lipidbilayer is stipulated to be inextensible and this constraintis imposed by adding a suitable term to the areal energydensity. Necessarily, the introduction of such a constraint isaccompanied by the need for a Lagrange multiplier. In thiscase, the deformation of the lipid bilayer is dominated bybending. To address this limit in our setting, consider thelimiting case of J ≈ 1. Regarding the definition of ν in(141)1, and using (148) and (151), it can be concluded that
ν = Σ
2(JΛ− 2Σ). (153)
In accordance with the high in-plane resistance of the lipidbilayer, the ratio of the areal stressΣ and stiffnessΛ shouldbe very small (i.e., ΣΛ−1 ≈ 0), leading to ν ≈ 0. Thus, theareal energy density is well approximated by
ψ◦ = ψm + 1
2κ(H − H◦)2 + κ(K − K◦), (154)
with
ψm = 2h◦(
f0
(
1 + 16 h2◦K◦
)
+ 23ηh2◦ H2◦
)
, (155)
and
κ = 323 h3◦(2 f3 − 2 f13 + f33 + f11),
κ = 43 h3◦( f1 − f3).
}
(156)
Also, using the approximations J ≈ 1 andΣΛ−1 ≈ 0 in theexpression (152)1 for the splay modulus yields
κ = 83 h3◦Λ; (157)
however, the expression (152)2 for the saddle-splay modulusremains unchanged with these approximations.
Assuming that the lipid bilayer is very thin, the membranalenergy given in (155) reduces toψm = 2h◦ f0. Since f0 onlydepends on J , the membranal energy can be represented asψm = ϕ◦(J ). Therefore, it is natural to introduce an effectivesurface tension
σ := dϕ◦dJ
(158)
and an effective areal stiffness
λ := d2ϕ◦dJ 2 , (159)
for the lipid bilayer. On using the definitions of η, Σ , andΛ in (141)2, (147), and (149), respectively, and performingstraightforward differentiation, it is possible to verify that
η = 1
2J−1 d f0
d J, Σ = J
d f0
dJ, Λ = Σ J−1 + J
d2 f0
dJ 2 ,
(160)
which, in view of the approximations J ≈ 1 andΣΛ−1 ≈ 0,and (152)2 and (157), yields
κ = 43 h2◦
d2ϕ◦dJ 2 , κ = 1
3 h2◦dϕ◦dJ
; (161)
thus, on referring to (158) and (159), the splay and saddle-splay moduli can be expressed as
κ = 43 h2◦λ, κ = 1
3 h2◦σ. (162)
From relations (162), it is evident that κ and κ are directlyproportional to the effective areal stiffness and surface ten-sion of the lipid bilayer, respectively.
When the bilayer is in a state of pure bending (i.e., J = 1),I1 and I3 are both equal to unity on the spatial mid-surfaceS◦, and thus, S◦ corresponds to the natural state—that is,
Φ(I1, I3)∣
∣
ξ◦= 0;J= 1 = 0, (163)
whereby f0 = 0. Thus, ψm = 0, and
ψ◦ = 1
2κ(H − H◦)2 + κ(K − K◦). (164)
Since the areal energy density is determinable only up toan arbitrary additive constant, (164) is equivalent to the theCanham–Helfrich energy density (1).
When the lipid bilayer is assumed inextensible, the arealenergy density (164) should be considered, while the inex-tensibility constraint should be penalized by considering aLagrange multiplier.
7.7 Effect of asymmetric chemistry of the leaflets
Now suppose that, due to possible trans-bilayer asymmetricchemistry of the leaflets (Luckey 2008; Janmey and Kin-nunen 2006; Devaux and Morris 2004; McMahon and Gal-lop 2005), the response functions Φ+ and Φ− in the upperand lower leaflets differ. This situation is very probable whenthe spontaneous curvature is induced due to asymmetric dis-tribution of lipid molecules with different molecular shapesacross the mid-surface of the lipid bilayer (McMahon andGallop 2005).
The dimension reduction then leads to an energy densityof the form
Kinematics, material symmetry, and energy densities 1013
with
Hc = 2h2◦ J 2
κ[[η]], ν = 〈η〉
4〈μ〉 , (167)
In addition, for the approximation J ≈ 1, the counterpartof (154) takes the form
ψ◦ = ψm + 1
2κ(H − (H◦ + Hc))
2 + κ(K − K◦), (168)
where ψm, Hc, κ , and κ are given by
ψm = 2h◦〈 f0〉,Hc = 2 h2◦
κ[[ f1 − f3]],
κ = 323 h3◦〈2 f3 − 2 f13 + f33 + f11〉,
κ = 43 h3◦〈 f1 − f3〉.
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
(169)
If, moreover, the lipid bilayer is in a state of pure bending,then (168) reduces to
ψ◦ = 1
2κ(H − (H◦ + Hc))
2 + κ(K − K◦). (170)
Let Hc denote the constitutively induced spontaneous meancurvature and introduce the net spontaneous curvature Hsp =H◦ + Hc. The net spontaneous mean curvature Hsp for anasymmetric lipid bilayer is then seen to incorporate two con-tributions: (i) a geometrical contribution H◦, which stemsfrom the spontaneous geometry of the lipid bilayer, due toasymmetric distribution of phospholipid molecules with dif-ferent molecular shapes or due to other possible sources(McMahon and Gallop 2005; Döbereiner et al. 1999); and(ii) a constitutive contribution Hc, which stems from differ-ences between the constitutive properties of the leaflets.
Also, notice that the alternative representations of κ andκ in (152), (157), (161), and (162) remain valid under thepresent circumstances, except that Σ, Λ, and ϕ◦ should bereplaced by 〈Σ〉, 〈Λ〉, and ϕ◦ = 2h◦〈 f0〉, respectively.
7.8 Effect of the incoherency between the leaflets
The areal energy density obtained by dimension reductionmay be generalized to include the effect of incoherencybetween the leaflets. In such case, the invariants I+
k andI−
k (k = 1, 3) and the areal stretches J+ and J− in theupper and lower leaflets differ. The integration in (128) must,therefore, be performed piecewise. Being very similar to thesteps leading to (134), the steps involved are not shown.The final form of the areal energy density depends, as before,on H◦, K◦, H , and K ; however, instead of J , it includesdependence on both J+ and J−.
7.9 Remarks
• The Canham–Helfrich energy density is an acceptableareal energy density when (i) the leaflets have identical
thickness and the same response function, (ii) the ratioof the thickness to the principal radii of curvature is verysmall, and (iii) the lipid bilayer has pure bending or it isinextensible.
• In contrast to the classical Canham–Helfrich energy den-sity (1), our theory predicts that the spontaneous Gaussiancurvature should be included in the areal energy densityin a manner analogous to the spontaneous mean curva-ture. Moreover, in contrast to (1), the energy density (164)vanishes at the spontaneous state. This issue is not impor-tant in the case of homogeneous lipid bilayers because theenergy density can be additively scaled by any constant.However, for heterogeneous lipid bilayers, such as multi-phase GUVs (e.g., see Baumgart et al. (2005)), where thesaddle-splay modulus or the spontaneous curvature arenonuniform, this distinction should be considered.
• As long as the response function Φ is known, the bendingmoduli κ and κ cannot be arbitrarily chosen. Rather, theyderive from the response function Φ. This is consistentwith the conclusions of Zurlo (2006) and Deseri et al.(2008).
• The bending moduli κ and κ scale with the cube of theleaflet thickness h◦. This is in harmony with the defor-mation of a thin elastic sheet, for example, as describedby the classical Föppl–von Kármán theory, where thebending rigidity is proportional to the cube of the sheetthickness. In addition, just as the bending rigidity of anisotropic homogeneous elastic sheet is linearly propor-tional to its Young modulus, the splay modulus κ is lin-early proportional to the in-plane stiffness of the lipidbilayer. In contrast, the saddle-splay modulus κ is lin-early scaled with the surface tension in the lipid bilayer.
• The membranal energy ψm includes not only a term pro-portional to h◦ but also a secondary term proportionalto h3◦. This term also contains the spontaneous mean andGaussian curvatures of the lipid bilayer. However, as longas the lipid bilayer is very thin, the contribution of thissecondary term is negligible.
• Ostensibly, the areal energy densityψ◦ obtained from thedimension reduction argument should provide a basis forformulating variational problems to determine minimumenergy configurations of lipid bilayers. However, to gen-erate well-posed variational problems, the areal energydensity ψ◦ should at very least satisfy the Legendre–Hadamard condition and thereby guarantee that thesecond (weak) variation of the underlying functionalis positive. For example, based on the establishedLegendre–Hadamard condition for elastic surfaces ofsecond-grade (see, for instance, Hilgers and Pipkin(1993)), Steigmann (1999) and Agrawal and Steigmann(2008) derived the Legendre–Hadamard condition rel-evant to a lipid bilayer with an areal energy den-sity depending generically on H, K , and J—which
123
1014 M. Maleki et al.
encompasses the result of the dimension reductionobtained here. If, in the present context, the assumption ofmild areal stretch embodied by (137) does not hold, then,as is the case in the work of Zurlo (2006) and Deseri et al.(2008), the areal energy density ψ◦ will include an extracontribution proportional to |∇S◦ J |2. However, there isno reason to expect that the Legendre–Hadamard condi-tions for an areal energy density depending on H, K , J ,and |∇S◦ J | should always be satisfied by the areal energydensity arising from the dimensional reduction argument,even if the three-dimensional energy density Φ leading toψ◦ satisfies the appropriate three-dimensional Legendre–Hadamard condition. To see this consider, for example,a flat lipid bilayer, in which the energetic contribution ofcurvature is absent, so that
ψ◦ = ϕ◦(J )+ 2
3h3◦ J−2 f1|∇S◦ J |2. (171)
The approach of Hilgers and Pipkin (1993) can then beused to show that the Legendre–Hadamard condition issatisfied only if f1 > 0. A precise understanding ofthe sign of f1 depends on the specific structure of theresponse function Φ and the value of areal stretch J onthe mid-surface. Nevertheless, in view of (147), it can beobserved that f1 may depend on the in-plane stress at themid-surface of the lipid bilayer. As long as f1 is positive,the second variation of ψ◦ is positive and the equilib-rium configuration of the lipid bilayer is stable (locally,at least). However, if f1 is negative, due to a possible con-traction in the lipid bilayer, then the Legendre–Hadamardcondition is violated, implying that the configuration ofthe lipid bilayer is unstable. One way to cure this problemis to add a term toψ◦, as Hilgers and Pipkin (1996) did intheir study of the equilibrium of elastic membranes withstrain-gradient energies.
8 Summary
A continuum approach to modeling the deformation of lipidbilayers with spontaneous curvature was provided. In adeparture from prevailing tradition, a lipid bilayer was mod-eled by a three-dimensional body. Apart from a kinemat-ical constraint incorporating natural aspects of the behav-ior of lipid molecules, no further restrictions were imposedon the deformation. In this context, a general representationfor the deformation gradient was derived. That representa-tion involves the curvature tensor of the mid-surface in thespontaneous (or reference) state, the curvature tensor of themid-surface in the deformed state,the deformation gradient
of the mid-surface, and changes in transverse thickness. Thecoherency of the leaflets that comprise a lipid bilayer, whichentails considering local coupling or sliding of those leaflets,was explored, as was the topic of area compatibility.
Geometry and kinematics aside, the material symmetryof lipid bilayers that exhibit in-plane fluidity and transverseisotropy was studied. Moreover, modeling the bilayer as ahyperelastic material, a representation theorem for the energydensity was developed. Three invariants were found to besufficient to describe the constitutive behavior of a lipidbilayer. It was shown that these invariants describe localstretch of area elements parallel to the mid-surface, volumechange, and through-thickness stretching. Explicit expres-sions for these invariants were determined and presented interms of fundamental kinematical quantities. Among thesekinematical quantities are the referential and spatial mid-surface curvatures, the areal stretch of the mid-surface, thetransverse normal and shear strains (which, in the presentsetting, control thickness change and nonuniformity, respec-tively), and a coupling term between the surface gradientof the transverse deformation and the mid-surface curvaturetensor.
The special case of a lipid bilayer that—in accord withexperimental observations—is incompressible was consid-ered. Under this constraint, transverse deformation (withrespect to the mid-surface) of the lipid bilayer is coupled tomid-surface deformation. This purely geometrical and kine-matical consideration suggests that when the mid-surfaceof an incompressible lipid bilayer has nonuniform bend-ing and/or stretching or when the spontaneous curvature isnonuniform, its thickness in the deformed state is, in general,nonuniform.
Granted that the lipid bilayer is sufficiently thin, adimension reduction argument was used to extract anareal (two-dimensional) energy density from the volumet-ric energy density. The conditions under which a Canham–Helfrich-type energy density is derivable were discussed. Anenergy density for lipid bilayer with asymmetric leaflets wasalso obtained, and it was shown that, for such a medium,the spontaneous curvature can be interpreted as combi-nation of two contributions. The first contribution repre-sents the preferred geometry (i.e., the spontaneous shape)of the lipid bilayer. The second contribution—which may becalled the constitutively induced spontaneous curvature—arises due to differences between the response functions ofthe leaflets. Lastly, the general form of the areal energy den-sity for a lipid bilayer with incoherent leaflets was consid-ered.
Acknowledgments This work was supported by NIH grant GM084200and by the Canada Research Chairs program.
123
Kinematics, material symmetry, and energy densities 1015
9 Appendices
9.1 Superficial fields
A field defined only on a surface is called superficial. Forinstance, m is a superficial unit vector field defined on themid-surface S◦. The three-dimensional gradient of such afield is undefined. However, a smooth extension of a super-ficial field to a three-dimensional neighborhood of the sur-face on which it is defined provides a means for defining itsthree-dimensional gradient (on the relevant neighborhood).The normally constant extension, in which a superficial fieldis stipulated to be constant along lines perpendicular to thesurface on which it is defined, provides the simplest suchextension (Fried and Gurtin 2007). For example, considera scalar-valued superficial field f defined on S◦, and let f e
denote a smooth extension of f to a neighborhood of S◦. Thisextension can be used to define the surface gradient ∇S◦ f off on S◦ in terms of the three-dimensional gradient ∇ f e off e by
∇S◦ f = Pm∇ f e, (172)
where
Pm = 1 − m ⊗ m (173)
is the projection tensor onto the tangent space T◦ of So. Noticethat ∇ f e in (172) must be evaluated at points on the surfaceS◦. It should be mentioned that ∇S◦ f as determined by (172)is independent of the particular features of the extension f e.It is easily shown that ∇S◦ f is tangent to the surface S◦.Similarly, the surface gradient ∇S◦g and surface divergencedivS◦g of vector-valued superficial field g are defined as
∇S◦g = (∇ge)Pm,
divS◦g = tr(∇S◦g) = Pm · ∇ge,
}
(174)
where ge is a smooth extension of g. Here, as with ∇ f e in(172), ∇ge is evaluated at the point on S◦. Additionally, ina suitably determined neighborhood of S◦, the gradient ∇gand divergence divg of a vector field g defined on a three-dimensional region containing S◦ decompose according to
∇g = ∇S◦g + ∂g∂m ⊗ m,
divg = divS◦g + ∂g∂m · m,
}
(175)
where g = g|S◦ is the restriction of g to S◦, and
∂g∂m
= (∇g)m (176)
is the normal derivative of g.A superficial tensor field G, besides being defined only on
a surface, must satisfy
Gm = 0. (177)
For example, the surface gradient ∇S◦g of superficial vectorfield g is a superficial tensor field. If G also obeys
G�m = 0, (178)
then G is said to be a fully tangential tensor field. For exam-ple, the projection tensor Pm is fully tangential.
Other examples of fully tangential tensor fields are thecurvature tensors L◦ and L of the surfaces S◦ and S◦, asdefined by
L◦ = −∇S◦m, L = −∇S◦n. (179)
L◦ and L each possess at most two nontrivial scalar invari-ants. Convenient choices for these are the mean and Gaussiancurvatures. Specifically, while
H◦ = 12 I1(L◦) = 1
2 tr(L◦),K◦ = I2(L◦) = 1
2 [(tr(L◦))2 − tr(L2◦)],
}
(180)
define the mean and Gaussian curvatures H◦ and K◦ of thesurface S◦,
H = 12 I1(L) = 1
2 tr(L),
K = I2(L) = 12 [(tr(L))2 − tr(L2)],
}
(181)
define the analogous quantities for S◦.A useful property of any fully tangential tensor A is the
relation
Ac = A2 − I1(A)A + I2(A)1 (182)
determining its cofactor Ac, where I1(A) and I2(A) are firsttwo principal invariants of A. A simple, but useful conse-quence of (182) is that
Acm = I2(A)m. (183)
Notice that I2(A) can be viewed as the determinant of a two-dimensional matrix representation ofA. Having this in mind,other useful identities can be established, including
I2(A�) = I2(A), I2(AB) = I2(A)I2(B), (184)
with B also being a fully tangential tensor.
9.2 Pseudoinverse of a fully tangential tensor
Let A be a fully tangential tensor. Then,
Am = A�m = 0 (185)
and
A = PmA = APm. (186)
Since the determinant of Pm vanishes, (186) implies thatthe determinant of A must also vanish. Consequently, as amapping from three-dimensional vector space to itself, A isnot invertible. However, from (186),Amaps any vector fromthe tangent space T◦ to a vector in T◦. If A, considered as a
123
1016 M. Maleki et al.
mapping from T◦ to T◦, is one-to-one and, thus, invertible,then there exists a tensor A† satisfying
AA† = A†A = Pm. (187)
In view of (187), A† provides an inverse for A as a mappingfrom T◦ to T◦. However, A† does not provide an inverse ofA considered as a mapping from three-dimensional vectorspace to itself. Thus,A† may be thought of as a pseudoinverseof A.
To obtain the pseudoinverseA† of a fully tangential tensorA that is a one-to-one mapping from T◦ to T◦, consider theCayley–Hamilton equation
A3 − I1(A)A2 + I2(A)A = 0, (188)
for A. (Notice that, since A is fully tangential, its third prin-cipal invariant I3(A) obeys I3(A) = det A = 0. Hence, aterm proportional to I3(A) is absent from (188).) On apply-ing the left-hand side of (188) to an arbitrary vector t andintroducing the vector u = At, it follows that
(A2 − I1(A)A + I2(A)Pm)u = 0. (189)
Since u is a tangent vector and, thus, Pmu = u, keeping185 and (186) in mind, (189) yields
u = A[(I2(A))−1(I1(A)Pm − A)]u
= [(I2(A))−1(I1(A)Pm − A)]Au. (190)
Since, for any invertible tensor T,
TT−1u = T−1Tu, (191)
it follows from (190) that
A† = (I2(A))−1(I1(A)Pm − A) (192)
provides a pseudoinverse of the fully tangential tensorA thatis one-to-one as a mapping from T◦ to T◦.
In view of (192), a fully tangential tensor A is pseudoin-vertible if and only if
I2(A) �= 0. (193)
Granted that A is fully tangential, A† defined by (192) isalso fully tangential—that is, A† defined by (192) obeys
A†m = (A†)�m = 0, (194)
and
A† = PmA† = A†Pm. (195)
Also multiplying (192) byA and using (187), while invoking(193), yields the relation
A2 − I1(A)A + I2(A)Pm = 0, (196)
which can be viewed as the Cayley–Hamilton equation for afully tangential tensor (see also Simmonds 1985 for a discus-sion of the Cayley–Hamilton equation for a linear mapping
of two-dimensional vector space into itself). Finally, using(196) in (182) leads to
Ac = I2(A)m ⊗ m. (197)
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