Axial-symmetry breaking in constrained membranes

Phil. Trans. R. Soc. A (2009) 367, 3363–3378doi:10.1098/rsta.2009.0118

Axial-symmetry breaking in constrainedmembranes

BY PAOLO BISCARI1,* AND GAETANO NAPOLI2

1Department of Mathematics, Politecnico di Milano, Piazza Leonardoda Vinci 32, 20133 Milan, Italy

2Department of Engineering for Innovation, Università del Salento,Via per Monteroni, 73100 Lecce, Italy

We study the equilibrium shapes of a lipid membrane, attached to a fixed circularsubstrate. We show how the weakening of the boundary conditions is able to breakthe axial symmetry of the optimal equilibrium configuration. We derive the criticalthreshold of the symmetry-breaking transition, and obtain the analytical expression ofthe free-energy minimizers in the quasi-planar approximation. Metastable states turnout to contain contributions only from the axisymmetric mode, and at most one singlenon-trivial Fourier mode.

Keywords: lipid vesicles; axial-symmetry breaking; boundary effects

1. Introduction

Lipid membranes are self-assembled fluid aggregates of amphiphilic molecules,consisting of a hydrophilic head, and one or more hydrophobic tails. Living inan aqueous environment, the hydrophobic parts tend to be shielded as much aspossible from the surrounding water by the hydrophilic parts. In order to reducecontact with water, these molecules tend to form bilayer vesicles.

Lipid bilayers are extremely thin systems. Their transverse dimensioncorresponds to approximately two molecular lengths, and thus falls in thenanometre range, while their characteristic linear size may easily extend upto the micrometre scale (Lipowsky 1995). They can thus be quite carefully beapproximated by mathematical compact surfaces, embedded in three-dimensionalEuclidean space. Lipid bilayers possess a wide range of technological applications,ranging from drug delivery (Raviv et al. 2005) to drug discovery (Fang et al. 2006)and biosensors (Sackmann 1996). Their equilibrium properties and interactions,possibly via embedded proteins, and the mediated interactions between theproteins themselves (Biscari & Bisi 2002), have been thoroughly studied as akey for the design of new bioengineered materials (Tirrell et al. 2002).

The study of vesicular assemblies of lipid molecules has replaced the studyof planar lipid membranes since they better resemble the spherical shape ofcell membranes. However, there is an increasing interest in the scientific and

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3364 P. Biscari and G. Napoli

practical applications of planar lipid–protein bilayers. Black lipid membranes(Ti Tien & Ottova-Leitmannova 2000) have been used to investigate variousbiophysical processes, such as the formation of ion channels in phospholipidbilayers by peptides, proteins, antibiotics and other pore-forming biomolecules.In the usual devices, black lipid membranes are suspended in solution and arelaterally anchored to a circular solvent support. The absence of such a supportmeans that transmembrane proteins suspended within the phospholipid bilayerremain fully mobile and active.

Mechanical characterization of lipid membranes is also essential for assessingthe feasibility of a biomimetic actuator (Knoblauch & Peters 2004). In theseactuators, a bilayer plane membrane-coated porous polycarbonate substrate isused to separate the fluid reservoir from the enclosed expansion chamber. Thedeformation of the enclosed chamber depends on the fluid pressure that themembrane covering the pores of the substrate can withstand.

In both these applications, the interaction between the membrane and itssupport involves non-trivial boundary conditions. Thus, in the present studywe analyse the influence of the weakness of the anchoring on the membranedeformation. Mathematical modelling is crucial in order to assess the mechanicalbehaviour of these materials, whose application seems to be hindered by theirpoor stability to environmental disturbances such as, for instance, mechanicalstresses.

At equilibrium, lipid bilayers behave as hyperelastic continua. The simplest,most successful and widely accepted variational model fit to predict theirequilibrium shapes is the spontaneous-curvature model, put forward by Helfrich(1973). According to this model, stable membrane shapes minimize a curvature-energy functional, which is chosen to be quadratic in the principal curvatures, andallows for the possibility of accepting a non-zero spontaneous value for themean curvature. We refer to a recent review (Tu & Ou-Yang 2008), and toreferences therein, for a quite complete report of the properties of minimizersof Helfrich’s functional. The spontaneous curvature, which will play a crucialrole in our investigation, is a constitutive parameter. It can be related to thearea difference that may arise among the sheets of the bilayer. In fact, the area-difference models that move from this effect (Svetina et al. 1982; Svetina & Zekš1983) are related with Helfrich’s model via a Legendre transformation (Svetina &Zekš 1989; Miao et al. 1994).

The optimal shapes of closed vesicles depend on the value of the spontaneouscurvature, and possibly on the dimensionless ratio between the membrane areaand the enclosed volume (Deuling & Helfrich 1976; Seifert et al. 1991; Jülicheret al. 1993). Special attention should be paid to the onset of classes of non-axisymmetric equilibrium shapes. In free vesicles, this phenomenon is triggeredby the diminution of the enclosed volume. In this paper we analyse the possibilityof inducing the same type of symmetry breaking by means of an external torque,applied on the boundary of the bilayer. More precisely, we consider a lipid bilayerforced to lean on a fixed circumference. We assume that the membrane area isclose to the area of the disc enclosed by the assigned circumference, so that thequasi-planar approximation holds for the membrane shape. Our geometry closelyresembles the one studied in De Vita et al. (2007). What is new in our study isthat we leave the membrane normal free to choose its direction up to the veryboundary. We thus replace strong anchoring conditions with a weak anchoring

Phil. Trans. R. Soc. A (2009)



Axial-symmetry breaking in membranes 3365

energy. From the mechanical point of view, this amounts to considering the case inwhich an assigned external boundary torque is applied that pushes the membranenormal to approach the radial direction of the external circumference. The bulkterm, measuring the elastic energy associated with the Gaussian curvature, turnsinto a further effective boundary torque.

Both the torques and the shape of the boundary (a circumference) arechosen in such a way that the axial symmetry is not explicitly broken by thepresence of non-symmetric external actions. Nevertheless, their combined action,together with the presence of a spontaneous curvature, creates an instabilitythat brings in non-axisymmetric global minimizers. The symmetry-breakingtransition is a first-order transition, in the sense that the lack of symmetryshows up abruptly, and the symmetry-breaking minimizers do not tend to theaxisymmetric ones in any functional sense, when the transition threshold isapproached.

This paper is organized as follows. Section 2 is devoted to a review of thebasic properties of Helfrich’s spontaneous-curvature model, and to the descriptionof the functional to be studied below. Section 3 contains the derivation ofour main result, which is summarized there in a phase diagram showing thesymmetry-breaking transition. Section 4 discusses the results.

2. Variational problem

We describe membrane elasticity through Helfrich’s spontaneous-curvature model(Helfrich 1973). The free-energy functional is then

Fel[Σ] = κ

∫Σ

((H − σ0)2 − μK ) da, (2.1)

where Σ is a surface describing the vesicle shape, H and K denote the meanand Gaussian curvatures along Σ , σ0 is the spontaneous curvature, κ is a rigiditymodulus and μ is a dimensionless constitutive parameter. This latter cannot bechosen at will. Indeed, if we write the free-energy density in terms of the principalcurvatures, we obtain

(H − σ0)2 − μK = 1

4c21 + ( 1

2 − μ)c1c2 + 1

4c22 − σ0(c1 + c2) + σ 2

0 ,

which is bounded from below if and only if μ ∈ [0, 1].While looking for minimizers under the total area constraint, we replace

equation (2.1) with the effective free-energy functional

Feff [Σ] = κ

∫Σ

((H − σ0)2 − μK ) da − λ(Area(Σ) − A). (2.2)

The Lagrange multiplier λ has the physical dimensions of a surface tension,though contributions to this latter originate also from the curvature energy(Biscari et al. 2004). In equation (2.2), the area constraint has been insertedas a global, instead of a local, constraint. However, we recall that, in the absenceof external loads, the two choices are equivalent (Biscari et al. 2002).





Let γ = ∂Σ . The Gauss–Bonnet theorem states that∫Σ

K da +∫γ

kg ds = 2πχ(Σ), (2.3)

where kg is the geodesic curvature of γ , and χ(Σ) is the (topologically invariant)Euler characteristic of Σ . The geodesic curvature is the scalar magnitude of thegeodesic curvature vector. It measures the deviance of the curve from followinga geodesic on Σ . More precisely, kg at a point P is the curvature of the planarcurve, obtained by projecting γ onto the tangent plane at P.

We assume γ to be a circumference, of origin O and fixed radius R, to avoid anysymmetry breaking induced by the boundary shape. Let D be the disc delimitedby γ , and ez a unit vector orthogonal to D. We introduce a set of cylindricalcoordinates (r , θ , z), centred at O and with symmetry axis ez . We also let{er , eθ } be the unit vectors pointing in the radial and tangential directions. ThenD = {(r , θ) | r ∈ [0, R], θ ∈ [0, 2π)}. We assume that Σ is an explicit surface, i.e.that there exists a function z : D → R such that

Σ = {P ∈ R3 : P − O = rer + z(r , θ)ez , r ∈ [0, R], θ ∈ [0, 2π)}. (2.4)

When this is the case, the infinitesimal area and boundary length elements, themean and the Gaussian curvature along Σ are given by (e.g. Ou-Yang et al. 1999)

da = √g dr dθ , ds = R dθ , with g = r2(1 + z2

,r ) + z2,θ ,

H = 1

2√

g3[r(1 + z2

,r )(z,θθ + rz,r ) + rz,rr (r2 + z2,θ ) + 2z,θz,r (z,θ − rz,rθ )]

and K = 1g2

[r2z,rr (rz,r + z,θθ ) − (rz,rθ − z,θ )2],

where a comma denotes partial differentiation with respect to the relativesubscript.

Lemma 2.1. Let Σ be any regular surface, and γ ⊂ Σ an embedded curve. Thenthe geodesic curvature of γ is given by kg = c |ν · b|, where b and c denote thebinormal unit vector and the curvature along γ , and ν is the unit normal of Σ.

Proof. Let γ (s0) be the p oint at which we aim to compute the geodesiccurvature, where γ is assumed to be parametrized in terms of its arc lengths. Then γ ′(s0) = t is the tangent unit vector, and γ ′′(s0) = cn is proportionalto the principal normal. If we denote by ν the unit normal of Σ at γ (s0),the plane tangent to Σ is π0 = {P : ν · (P − γ (s0)) = 0}. The condition γ ⊂ Σimplies t · ν = 0. We introduce the binormal unit vector b = t ∧ n and the angle ψsuch that ν = sin ψ n + cos ψb. The curve obtained by performing an orthogonalprojection of γ onto π0 is given by

ξ(s) = γ (s0) + (I − ν ⊗ ν)(γ (s) − γ (s0)).

(Note that s is not the arc-length parameter along ξ .) Clearly,

ξ ′(s) = (I − ν ⊗ ν)γ ′(s) = t and ξ ′′(s) = (I − ν ⊗ ν)γ ′′(s) = cn − c sin ψ ν.





Thus,

ξ ′ ∧ ξ ′′ = c cos2 ψ b + c sin ψ cos ψ n and |ξ ′ ∧ ξ ′′| = c cos ψ .

The geodesic curvature of γ in γ (s0), being equal to the curvature of ξ , is thusgiven by

kg = |ξ ′ ∧ ξ ′′||ξ ′|3 = |c cos ψ | = c |ν · b|.

�In terms of the cylindrical coordinates introduced above, we have t = eθ ,

n = er , b = ez and c = R−1. Using the parametrization (2.4), we thus obtain

kg = R−1|ν · ez | = R−1√

1 − (ν · er )2|r=R.We assume that the boundary of the membrane is kept constrained to the fixed

circumference γ . Moreover, we analyse the possibility that an external torque isapplied to the membrane on the boundary itself. Such a torque acts on the unitnormal to the membrane, since its physical origin is a torque exerted on the lipidmolecules that form the bilayer. The constraint ν · t = 0 forces the boundary unitnormal ν to lie in the plane orthogonal to eθ . Thus, admissible variations for theboundary normal are all of the form δν = δε ∧ ν, with infinitesimal rotation vectorδε = δψeθ , where ψ , as above, is the angle between ν and ez .

We consider an external couple of the form C = 2w(ν · er ) (eθ ⊗ eθ )ν =2w sin ψ cos ψ eθ . The virtual work exerted by C is given by δ∗W = C · δε =2w sin ψ cos ψ δψ = −δ(w cos2 ψ). Thus C is a conservative torque, with potentialenergy density σw = w(cos ψ)2 = w(ν · er )

2. The boundary torque acts on the unitnormal in order to align it along the minimizers of σw . Then, the easy directionfor the boundary normal is ez when w > 0, while it is the radial direction er ifw < 0. Clearly, w measures the anchoring strength or equivalently the intensity ofthe boundary torque. In terms of the free-energy functional, the boundary torqueadds an anchoring energy, concentrated on γ : Fw[Σ] = w

∮γ(ν · er )

2 ds.Once we neglect the topologically invariant term depending on the Euler

characteristic of Σ , the complete free-energy functional to be minimized becomes

F [Σ] = κ

∫D(H − σ0)

2√g dr dθ − λ

(∫D

√g dr dθ − A

)

+∫ 2π

0[wR(ν · er )

2 + μκ√

1 − (ν · er )2 ] dθ . (2.5)

3. Torque-induced symmetry breaking

The shape of the membrane is forced to be a planar disc of radius R, centred atO, whenever the membrane area attains the value A0 = πR2. We assume that thearea is slightly perturbed with respect to A0, and introduce a small dimensionlessparameter ε to account for the excess area: A = A0(1 + ε2). Note that the areaperturbation is positive, since A0 is the minimum area allowed by the boundaryrestriction. In the absence of any perturbation, the only available shape for themembrane is a planar disc. In this case, no difference is expected to arise betweenthe two lipid layers that form the membrane, and thus the spontaneous curvaturevanishes. In order to enlarge the membrane area, some extra lipid molecules mustbe inserted in the system. When these molecules attach to both sides of the





bilayer, they may or may not create an area difference between the layers, whichwould possibly give rise to a spontaneous curvature σ0. Because of this physicalreason, we assume that the spontaneous curvature is infinitesimal, and we writeit in terms of a dimensionless parameter ζ0: σ0 = εζ0/R.

Because of the area constraint, the equilibrium configuration of the perturbedmembrane is forced to remain close to the flat solution. Consequently, we assumethat z(r , θ) = εu(r , θ), where the perturbation is to be taken as linear in ε, inorder to satisfy the area constraint (see below). Finally, we rescale the Lagrangemultiplier as λ = κΛ/(2R2) (with Λ a dimensionless parameter), and expand asΛ = Λ0 + O(ε). Within this approximation, we obtain

H = ε

2�u + O(ε3), K = O(ε2) and kg = 1

R+ O(ε2).

The expansion for the infinitesimal area element justifies the above choice for theorder of the shape function perturbation:

da = r(

1 + ε2

2(∇u)2 + O(ε4)

)dr dθ ,

where ∇ and � are the usual gradient and Laplacian operators in polarcoordinates:

∇ = ∂

∂rer + 1

r∂

∂θeθ and � = 1

r∂

∂r

(r

∂

∂r

)+ 1

r2

∂2

∂θ2.

Before proceeding further, we now complete the rescaling of all our relevantvariables, in order to deal with a dimensionless problem. We have already writtenthe spontaneous curvature σ0 and the Lagrange multiplier λ in terms of thedimensionless parameters ζ0 and Λ0. We now redefine the radial variable byintroducing the coordinate x = r/R, and consequently write the shape functionas u(r , θ) = Rυ(r/R, θ), where the dimensionless shape function υ(x , θ) is definedin the domain D′ = {(x , θ) : x ∈ [0, 1], θ ∈ [0, 2π)}. Finally, we set w = ωκ/R, anddefine the dimensionless parameter ν = 2(2ω − μ).

When looking for equilibrium configurations, we expand the shape function inFourier series

υ(x , θ) = a0(x)

2+

∞∑n=1

an(x) cos nθ +∞∑

n=1

bn(x) sin nθ . (3.1)

We note that the boundary constraint u(R, θ) = 0 for all θ implies an(1) = 0 forall n ≥ 0, and bn(1) = 0 for all n ≥ 1.

Remark. The axial symmetry inherent to the problem allows us to simplifythe Fourier series just introduced. Clearly, it cannot be taken as guaranteed thatall equilibrium configurations are axially symmetric, which would in turn meanassuming that an(x) = bn(x) = 0 for all x ∈ [0, 1], and for all n ≥ 1. However, ifa non-axially symmetric equilibrium configuration υ(x , θ) is found, it is certainthat all functions υα(x , θ) = υ(x , θ + α) are stationary as well, for all values of α.

Suppose that υ possesses only one non-symmetric Fourier mode, i.e. υ(x , θ) =12a0(x) + aN (x) cos N θ + bN (x) sin N θ , and that bN (x) = ηaN (x), for all x ∈ [0, 1],with N ≥ 1. Then

υα(x , θ) = 12a0(x) +

√1 + η2 aN (x) cos nθ , (3.2)





provided we choose α = arctan η. In words, when only one non-axially symmetricmode is active, and the amplitudes relative to the sine and cosine submodes arelinearly dependent, a redefinition of the x and y axes in the reference plane allowsus to find a function in which only the cosine mode is active.

When we insert the series expansion (3.1) in the functional (2.5) we obtainF = 1

4κπε2F2 + o(ε2), where, apart from inessential constants,

F2[Σ] =∫ 1

0x

[(xa ′

0 + x2a ′′0 )2

2x4+

∞∑n=1

(xa ′n + x2a ′′

n − n2an)2

x4

+∞∑

n=1

(xb′n + x2b′′

n − n2bn)2

x4

]dx

− Λ0

{∫ 1

0x

[a ′2

0

2+

∞∑n=1

(a ′2

n + n2a2n

x2+ b′2

n + n2b2n

x2

)]dx − 1

}

+ ν

∞∑n=1

[a ′2n (1) + b′2

n (1)] + 12

νa ′20 (1) − 4ζ0a ′

0(1) + 4ζ 20 , (3.3)

where a prime denotes differentiation with respect to x . The functional F2 is welldefined in a subspace of the Sobolev space H 2(0, 1). More precisely, in order forthe functional (3.3) to attain a finite value, the Fourier components must satisfythe condition that the combinations (xa ′

n + x2a ′′n − n2an)2/x3 and (xb′

n + x2b′′n −

n2bn)2/x3 be integrable when x → 0.When an = an + αn and bn = bn + βn , an iterated integration by parts yields

the following expression for the variation of the functional F2:

12δF2 =

∞∑n=0

∫ 1

0αn

(xa ′′′′

n + 2a ′′′n − (2n2 + 1 − Λ0x2)a ′′

n

x

+ (2n2 + 1 + Λ0x2)a ′n

x2+ n2(n2 − 4 − 2Λ0x2)an

x3

)dx

+∞∑

n=1

∫ 1

0βn

(xb′′′′

n + 2b′′′n − (2n2 + 1 − Λ0x2)b′′

n

x

+ (2n2 + 1 + Λ0x2)b′n

x2+ n2(n2 − 4 − Λ0x2)bn

x3

)dx

+ [a ′′0 (1) + (ν + 1)a ′

0(1) − 4ζ0]α′0(1) +

∞∑n=1

[a ′′n(1) + (ν + 1)a ′

n(1)]α′n(1)

+∞∑

n=1

[b′′n(1) + (ν + 1)b′

n(1)]β ′n(1). (3.4)





The Euler–Lagrange equations require the bulk terms to vanish throughoutthe membrane:

xf ′′′′n + 2f ′′′

n − (2n2 + 1 − Λ0x2)f ′′n

x+ (2n2 + 1 + Λ0x2)f ′

n

x2+ n2(n2 − 4 − Λ0x2)fn

x3= 0

(3.5)for both an (n ≥ 0) and bn (n ≥ 1). Equation (3.5) admits the following generalsolution:

a0(x) = A0 + B0 ln x + C0J0(√

Λ0 x) + D0Y0(√

Λ0 x),

an(x) = Anxn + Bnx−n + CnJn(√

Λ0 x) + DnYn(√

Λ0 x), n ≥ 1

and bn(x) = Enxn + Fnx−n + GnJn(√

Λ0 x) + HnYn(√

Λ0 x), n ≥ 1,

⎫⎪⎬⎪⎭ (3.6)

where Jn and Yn are Bessel functions of the first and second kind, respectively,and An , . . . , Hn are integration constants. When the Lagrange multiplier Λ0 isnegative, the stationary solutions involve the modified Bessel functions In andKn , since, for any real x , Jn(ix) = inIn(x) and Jn(ix) + iYn(ix) = (2/π)i−n−1Kn(x).

The regularity conditions above require Bn = Dn = 0 for all n ≥ 0, and similarlyFn = Hn = 0 for all n ≥ 1. Furthermore, the boundary conditions an(1) = bn(1) = 0allow us to write

a0(x) = C0[J0(√

Λ0 x) − J0(√

Λ0)],an(x) = Cn[Jn(

√Λ0 x) − Jn(

√Λ0)xn], n ≥ 1

and bn(x) = Fn[Jn(√

Λ0 x) − Jn(√

Λ0) xn], n ≥ 1.

⎫⎪⎬⎪⎭ (3.7)

The final boundary condition originates from the boundary terms in the variation(3.4). They yield

C0√

Λ0[√Λ0 J0(√

Λ0 ) + νJ1(√

Λ0)] + 4ζ0 = 0,

Cn{√Λ0 (Λ0 + 2nν)Jn−2(√

Λ0) + [Λ0(ν − 2(n − 1))

− 4n(n − 1)ν]Jn−1(√

Λ0)} = 0, ∀n ≥ 1

and Fn{√Λ0 (Λ0 + 2nν)Jn−2(√

Λ0) + [Λ0(ν − 2(n − 1))

− 4n(n − 1)ν]Jn−1(√

Λ0)} = 0, ∀n ≥ 1.

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(3.8)

Equations (3.8) admit two types of solutions, depending on whether or not Λ0attains one of the discrete set of values that allow for some of the terms incurly braces in (3.8) to vanish. In those special cases, some of the Cn and Fnmay become different from zero, and non-axially symmetric modes may arise.Otherwise, only C0 will be different from zero, and the solution will necessarilybe axially symmetric.





In all cases, the area constraint must be satisfied, which implies that

1 = 12

∫ 1

0x

[a ′2

0

2+

∞∑n=1

(a ′2

n + n2a2n

x2+ b′2

n + n2b2n

x2

)]dx

=∞∑

n=0

(C 2n + F 2

n)[P00(n, Λ0)J 20 (

√Λ0) + P01(n, Λ0)

√Λ0J0(

√Λ0)J1(

√Λ0)

+ P11(n, Λ0)J 21 (

√Λ0)], (3.9)

where P00, P01 and P11 are suitable polynomials (and F0 = 0 has been introducedfor the ease of notation). Once all the integration constants, along with theLagrange multiplier Λ0, have been determined, we need to compute the free-energy functional in order to identify the global minimizer of the elasticenergy.

(a) Axisymmetric shapes

For all values of the boundary torque, represented by ν, and the spontaneouscurvature, identified by ζ0, stationary axisymmetric shapes exist. They satisfyCn = Fn = 0 for all n ≥ 1, while the remaining integration constant and theLagrange multiplier are determined through the equations

C0

√Λ0[

√Λ0J0(

√Λ0) + νJ1(

√Λ0)] + 4ζ0 = 0 (3.10)

and √Λ0C 2

0 [√Λ0J 20 (

√Λ0) − 2J0(

√Λ0)J1(

√Λ0) + √

Λ0J 21 (

√Λ0)] = 8. (3.11)

The two equations can be combined to derive the following single equation for Λ0:

F(ν, Λ0) =√

Λ0[√Λ0J0(√

Λ0) + νJ1(√

Λ0)]2√Λ0J 2

0 (√

Λ0) − 2J0(√

Λ0)J1(√

Λ0) + √Λ0J 2

1 (√

Λ0)= 2ζ 2

0 . (3.12)

Figure 1 shows how F depends on Λ0 for several different values of ν. Stationaryvalues of Λ0 correspond to the values at which the function F intersects the value2ζ 2

0 . We next collect some analytical properties of F , which may be useful in thefollowing:

(i) F(ν, ·) is a non-negative, oscillating function that attains infinite times anypositive value.

(ii) For any fixed ν, F(ν, ·) vanishes a countable infinity of times. The set ofzeros is bounded from below, and F is monotonic at the left of its first zero.Moreover, for any ν, such smallest zero occurs at a value of Λ0 smaller thanthe value at which J1(

√Λ0 ) vanishes.

(iii) F(ν, Λ0) = 2(ν + 2)2 − 16(ν + 2)(ν + 8)Λ0 + O(Λ2

0), and also F(−2, Λ0) =18Λ

20 + O(Λ4

0), as Λ0 → 0.(iv) F(ν, Λ0) = ν2, if J0(

√Λ0 ) = 0, while F(ν, Λ0) = Λ0, if J1(

√Λ0 ) = 0. Thus,

values Λ0 at which J1 vanishes may be easily recognized, since all the plotsF(ν, Λ0) intersect at Λ0 = Λ0.

(v) Let x1 be the smallest non-zero root of the equation J1(x) = 0. (Then,x1 ≈ 3.83.) The smallest root (in Λ0) of equation (3.12) when ν � 1 isgiven by Λ0 = x2

1 − 2x1(√

2ζ0 + x1)/ν + o(ν−1), as ν → +∞. In particular,limν→+∞ Λ0 = x2

1 .





–20 20 0 40 60

Λ 0

20

40

60

80

100

120

140

F(ν, Λ 0)

Figure 1. Plots of the function F(ν, Λ0) defined in the text for ν ∈ {−4, −2, 0, 2, 4}. (Plotscorresponding to smaller values of ν can be identified since their leftmost zero occurs at lowervalues of Λ0.)

Among the countable infinity of values of Λ0 that satisfy equation (3.12) forany given ν and ζ0, the minimization process requires the one corresponding to thelowest free-energy value to be identified. If we use equations (3.10) and (3.11) todetermine ν and C0, the integrals in equation (3.3) can be carried out analyticallyand provide the free-energy excess associated with the perturbation:

F2(Λ0) = 2Λ0 + 4√

2ζ0Λ1/40 J1(

√Λ0)

[√Λ0J0(√

Λ0)2 − 2J0(√

Λ0)J1(√

Λ0) + √Λ0J1(

√Λ0)2]1/2

.

(3.13)Clearly, equation (3.13) is still an implicit expression for the free-energy excess.Indeed, it is meaningful only for Λ0-values that obey condition (3.12), since theindependent physical free parameter is the boundary torque ν.

In the special case when the spontaneous curvature vanishes (ζ0 = 0),equation (3.13) shows, however, that the free energy is simply proportional toΛ0. In general, the fact that the free energy is an increasing function of Λ0 canbe easily explained by a direct inspection of the stationary shapes (3.6). Indeed,rising Λ0 amounts to enhancing the argument of the Bessel functions involved inthe shape perturbations. Thus, greater values of Λ0 correspond to increasinglyoscillating stationary shapes, which possess an excess of elastic energy.

When the spontaneous curvature does not vanish, F2, as given byequation (3.13), is not always a monotonically increasing function of Λ0. Just togive an example, if we again denote by x1 the smallest zero of the Bessel functionJ1, equation (3.13) shows that F2(0) > F2(x2

1 ) whenever ζ0 > 14x

21 . We have thus





–4 –2 2 0 04 –4 –2 2 4

50

100

150

50

100

150

ν ν

(a) (b)

Figure 2. Plots illustrating how the free-energy excess F2 depends on ν when the reducedspontaneous curvature is (a) ζ0 = 0 and (b) ζ0 = 1. In both plots, the different graphs correspondto different solutions of equation (3.12). For any given ν, greater values of Λ0 always correspondto greater values of the corresponding free energy.

performed a numerical investigation to ascertain whether, for any value of ζ0, theproperty described above holds, i.e. whether it can always be ensured that, amongthe countable infinity of values of Λ0 that obey equation (3.12), the smallest rootcorresponds to the stationary solution possessing the least free-energy excess.We have sampled values of the spontaneous curvature in the interval ζ0 ∈ [0, 5],with ν ∈ [−50, 50]. In all cases, we have found that the optimal choice for Λ0corresponds to the smallest root of equation (3.12). Figure 2 illustrates howthe excess free energy depends on ν for two different values of the spontaneouscurvature. It is to be noted that the optimal free energy tends to a constant whenν → ∞. Indeed, limν→+∞ Λ0 = x2

1 , and thus limν→+∞ F2 = 2x21 . In any case, figure

2 shows that the free energy corresponding to greater roots of equation (3.12)possess greater free energy.

As a consequence, figure 1 allows us to easily identify the optimal axiallysymmetric shape. For any value of ν, we need to first identify the smallestzero of F(ν, ·), which amounts to finding the smallest root of the equation√

Λ0J0(√

Λ0) + νJ1(√

Λ0) = 0. That value of Λ0 corresponds to the stationaryvalue when ζ0 = 0. From thence on, the left branch of F(ν, ·) provides the optimalvalues of Λ0 for any ζ0 �= 0.

Remark. The optimal axially symmetric shapes share the property that, forany value of ν and ζ0, they never cross the reference plane, where the externalcircumference lies. Figure 3 shows some of those optimal axially symmetricshapes.

To prove this property, we first note that the smallest zero (in Λ0) of F(ν, Λ0)does always occur before Λ0 = x2

1 , which corresponds to the first non-trivial zeroof J1(

√Λ0 ). Thus, the argument above shows that, for any ζ0 and ν, the optimal

value of Λ0 is always strictly smaller than the value Λ0,max = x21 . Suppose now that

an optimal axially symmetric shape could cross the reference plane. This wouldimply that a0(x) = 0 for some x ∈ (0, 1). In view of equation (3.7), this would yieldJ0(

√Λ0 x) = J0(

√Λ0). Thus, by virtue of Rolle’s theorem, an ¯x ∈ (x , 1) would exist

such that J ′0(

√Λ0 ¯x) = −J1(

√Λ0 ¯x) = 0, and this would yield a contradiction, since

the function J1(√

Λ0x) would not be allowed to vanish for any x ∈ (0, 1).





–1.0 –0.5 0 0.5 1.0

–1.0 –0.5 0 0.5 1.0

–1.0–0.5

00.51.0

(a) (b)

–1.0–0.5

00.5

1.0

0

0.5

1.0

1.5

2.0

0

0.5

1.0

1.5

Figure 3. Optimal axially symmetric shapes when ζ = 1 and (a) ν = 0 or (b) ν = −5.

(b) Axial-symmetry breaking

Up to this point, we have studied only some special solutions of the system(3.8), i.e. those solutions corresponding to the vanishing of all Fourier modesbut the zeroth one. In this section we analyse the solutions that allow forsome of the Cn and Fn to become different from zero. In view of the structureof equations (3.8), those symmetry-breaking solutions may arise only whenthose values of Λ0 for which a term in curly braces in equations (3.8)2,3vanishes for some n. Once a value for Λ0 is found, equation (3.8)1 fixesthe value of the factor C0. Finally, the area constraint (3.9) provides thevalue of the corresponding Cn . Before proceeding further, we remark that,without loss of generality, we may assume that Fn = 0 for all n ≥ 1. Indeed,were also Fn different from zero, equation (3.7) would show that bn wouldsimply be proportional to an and, in view of the symmetry considerationspresented above (see equation (3.2)), this implies that solutions with Fn �=0 coincide with solutions with Fn = 0, up to an overall rotation of thereference axes.

Figure 4 shows the phase diagram relative to the axial-symmetry breakingfor the constrained membrane. The symmetry-breaking transition is triggeredby both the spontaneous curvature and the boundary torque. We remark thatboth parameters are crucial if non-symmetric shapes are to be found. Indeed,axially symmetric shapes provide the absolute minimizer for any value of theboundary torque if the spontaneous curvature is smaller than σ0,cr = εR−1 (whichcorresponds to ζ0 = 1). Analogously, no instability is found if ν ≥ νcr � −20.Without any loss of generality, we have restricted our analysis to the case of non-negative spontaneous curvature. Indeed, a glance at equation (3.3) shows that achange of sign in the spontaneous curvature can be easily taken into account bysimply switching the sign of the principal component a0.

The phase diagram in figure 4 corresponds to shape instabilities relativeto the first Fourier mode. It is to be expected that, within the ‘symmetry-breaking’ domain, new transitions should appear, with the onset of absolute





axially symmetric minimizer

symmetry breaking

0.5 0 1.0 1.5

ζ0

ν

–50

–40

–30

–20

–10

Figure 4. Phase diagram identifying the region of the parameter plane (ζ0, ν) where the axialsymmetry is broken.

–1.0–0.5

00.5

1.0

–1.0–0.5

00.5

1.0

0

0.5

1.0

1.5

–1.0–0.5

0

0.51.0

–1.0

–0.5

0

0.5

1.00.2

0

0.4

–1.0–0.5

00.5

1.0

–1.0–0.5

00.5

1.0

0

0.5

1.0

(a) (b)

(c)

Figure 5. Stationary shapes with broken symmetry. All plots correspond to the value ζ0 = 1.5, while(a) ν = −5, (b) ν = −10 and (c) ν = −25. The last shape is also a global energy minimizer.

minimizers associated with higher Fourier modes. Figure 5 explicitly shows someof the non-symmetric stationary shapes. In particular, figure 5c corresponds toparameter values for which it is energetically preferred with respect to the axiallysymmetric minimizer.





We remark that increasing ν induces a plateau region in the centre of themembrane. The reason for such an emerging planar domain is to be found inthe combined effect of the boundary torque and the area constraint. Whenthe boundary torque, represented by ν, becomes strong enough, the boundarymembrane normal aligns almost parallel to the radial direction in the referenceplane. This induces a region, close to the external curve γ , in which the membraneshape is quite close to a portion of a cylinder. Such a domain uses all the availableexcess area, and thus the rest of the membrane is just allowed to have an almostplanar shape.

4. Conclusions

We have studied the equilibrium shape of a lipid membrane, constrained to leanon a fixed circumference, subject to an external boundary torque. In particular,we have sought free-energy minimizers that break the axial symmetry, induced byboth the boundary constraint and the external torque. We framed our study inthe quasi-planar approximation, which is well justified when the membrane areais just above the area of the reference disc, enclosed by the fixed circumference.

The analytical results we have obtained, and in particular equations (3.8),show that all stationary shapes share a peculiar property. They can be eitheraxially symmetric, or they just possess one non-zero Fourier component. In fact,equations (3.8) can be interpreted as a sequence of eigenvalue equations for theLagrange multiplier Λ0.

We have compared the elastic energy of the axially symmetric solution withthe energy of the first non-trivial symmetry-breaking mode, as a function ofthe external torque (represented by the dimensionless parameter ν), and thespontaneous curvature (represented by ζ0). As a result, we have identified aregion in the (ν, ζ0) plane in which the non-symmetric shape replaces the natural,symmetric, one as the global free-energy minimizer. The transition is of first order,in the sense that non-trivial minimizers do not bifurcate from axially symmetricones.

We remark that ν must be sufficiently negative for the transition to takeplace. It is to be noted that the dimensionless parameter ν depends also on theparameter μ, which governs the elastic constant associated with the Gaussian-curvature term in the elastic energy (see equation (2.1)). Nevertheless, andsince ν = 2(2ω − μ) and μ ∈ [0, 1], in the absence of external torque (whichamounts to ω = 0) we obtain ν ∈ [−2, 0]. In correspondence with these valuesof ν, no symmetry-breaking instability shows up. Thus, the effect we areenvisaging is clearly linked to the external torque, and it is not an effect of theGaussian-curvature term.

We finally want to devote a comment to the role of the spontaneous curvature.Owing to the symmetry of the planar problem we were perturbing, we haveassumed that σ0 is either null or infinitesimal: σ0 = εζ0/R. However, it would beinteresting to study, at least from the mathematical point of view, the situationin which the spontaneous curvature is finite in the ε → 0 limit. In the presenceof a finite spontaneous curvature, such a limit becomes singular. Indeed, letus consider the expression (3.3) for F2, the O(ε2) approximation to the free-energy functional in the small-ε limit. Were σ0 = O(1), the second-last term in





F2, proportional to a ′0(1), would become O(ε), and thus dominant over all other

terms in the free-energy functional. Moreover, this term, being linear in the firstderivative of the principal component a0, is not bounded from below. Optimalshapes are thus expected to become singular and, more precisely, to develop aboundary layer of O(ε) at x = 1−. This mechanism, that is, the possibility ofa lipid membrane being able to relax its excess of curvature energy in a thinboundary layer, has been already identified as the optimal way in which a lipidmembrane modifies its equilibrium shape in the presence of a rigid inclusion(Biscari & Napoli 2005, 2007).

Calculations have been performed, and plots have been printed, with the aid of the MATHEMATICA

software.

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Axial-symmetry breaking in constrained membranes

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