## Abstract

We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We show how the weakening of the boundary conditions is able to break the axial symmetry of the optimal equilibrium configuration. We derive the critical threshold of the symmetry-breaking transition, and obtain the analytical expression of the free-energy minimizers in the quasi-planar approximation. Metastable states turn out to contain contributions only from the axisymmetric mode, and at most one single non-trivial Fourier mode.

## 1. Introduction

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

Lipid bilayers are extremely thin systems. Their transverse dimension corresponds to approximately two molecular lengths, and thus falls in the nanometre range, while their characteristic linear size may easily extend up to the micrometre scale (Lipowsky 1995). They can thus be quite carefully be approximated by mathematical compact surfaces, embedded in three-dimensional Euclidean 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 the proteins themselves (Biscari & Bisi 2002), have been thoroughly studied as a key for the design of new bioengineered materials (Tirrell *et al.* 2002).

The study of vesicular assemblies of lipid molecules has replaced the study of planar lipid membranes since they better resemble the spherical shape of cell membranes. However, there is an increasing interest in the scientific and practical applications of planar lipid–protein bilayers. *Black lipid membranes* (Ti Tien & Ottova-Leitmannova 2000) have been used to investigate various biophysical processes, such as the formation of ion channels in phospholipid bilayers by peptides, proteins, antibiotics and other pore-forming biomolecules. In the usual devices, black lipid membranes are suspended in solution and are laterally anchored to a circular solvent support. The absence of such a support means that transmembrane proteins suspended within the phospholipid bilayer remain fully mobile and active.

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

In both these applications, the interaction between the membrane and its support involves non-trivial boundary conditions. Thus, in the present study we analyse the influence of the weakness of the anchoring on the membrane deformation. Mathematical modelling is crucial in order to assess the mechanical behaviour of these materials, whose application seems to be hindered by their poor stability to environmental disturbances such as, for instance, mechanical stresses.

At equilibrium, lipid bilayers behave as hyperelastic continua. The simplest, most successful and widely accepted variational model fit to predict their equilibrium 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, and allows for the possibility of accepting a non-zero *spontaneous* value for the mean curvature. We refer to a recent review (Tu & Ou-Yang 2008), and to references therein, for a quite complete report of the properties of minimizers of Helfrich’s functional. The spontaneous curvature, which will play a crucial role in our investigation, is a constitutive parameter. It can be related to the area 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 & Žekš 1983) are related with Helfrich’s model via a Legendre transformation (Svetina & Žekš 1989; Miao *et al.* 1994).

The optimal shapes of closed vesicles depend on the value of the spontaneous curvature, and possibly on the dimensionless ratio between the membrane area and the enclosed volume (Deuling & Helfrich 1976; Seifert *et al.* 1991; Jülicher *et al*. 1993). Special attention should be paid to the onset of classes of non-axisymmetric equilibrium shapes. In free vesicles, this phenomenon is triggered by the diminution of the enclosed volume. In this paper we analyse the possibility of 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 bilayer forced to lean on a fixed circumference. We assume that the membrane area is close to the area of the disc enclosed by the assigned circumference, so that the quasi-planar approximation holds for the membrane shape. Our geometry closely resembles the one studied in De Vita *et al*. (2007). What is new in our study is that we leave the membrane normal free to choose its direction up to the very boundary. We thus replace *strong anchoring* conditions with a *weak anchoring* energy. From the mechanical point of view, this amounts to considering the case in which an assigned external boundary torque is applied that pushes the membrane normal to approach the radial direction of the external circumference. The bulk term, measuring the elastic energy associated with the Gaussian curvature, turns into a further effective boundary torque.

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

This paper is organized as follows. Section 2 is devoted to a review of the basic properties of Helfrich’s spontaneous-curvature model, and to the description of the functional to be studied below. Section 3 contains the derivation of our main result, which is summarized there in a phase diagram showing the symmetry-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
2.1
where *Σ* is a surface describing the vesicle shape, *H* and *K* denote the mean and Gaussian curvatures along *Σ*, *σ*_{0} is the spontaneous curvature, *κ* is a rigidity modulus and *μ* is a dimensionless constitutive parameter. This latter cannot be chosen at will. Indeed, if we write the free-energy density in terms of the principal curvatures, we obtain
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
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 inserted as a global, instead of a local, constraint. However, we recall that, in the absence of external loads, the two choices are equivalent (Biscari *et al*. 2002).

Let *γ*=∂*Σ*. The Gauss–Bonnet theorem states that
2.3
where *k*_{g} 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 the geodesic curvature vector. It measures the deviance of the curve from following a geodesic on *Σ*. More precisely, *k*_{g} at a point *P* is the curvature of the planar curve, 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 any symmetry breaking induced by the boundary shape. Let *D* be the disc delimited by *γ*, and *e*_{z} a unit vector orthogonal to *D*. We introduce a set of cylindrical coordinates (*r*,*θ*,*z*), centred at *O* and with symmetry axis *e*_{z}. We also let {*e*_{r},*e*_{θ}} be the unit vectors pointing in the radial and tangential directions. Then *D*={(*r*,*θ*)∣*r*∈[0,*R*],*θ*∈[0,2*π*)}. We assume that *Σ* is an explicit surface, i.e. that there exists a function such that
2.4
When this is the case, the infinitesimal area and boundary length elements, the mean and the Gaussian curvature along *Σ* are given by (e.g. Ou-Yang *et al*. 1999)
where a comma denotes partial differentiation with respect to the relative subscript.

### Lemma 2.1

*Let Σ be any regular surface, and* *an embedded curve. Then the geodesic curvature of γ is given by k*_{g}=*c* |** ν**·

**|,**

*b**where*.

**b**and c denote the binormal unit vector and the curvature along γ, and**ν**is the unit normal of Σ### Proof.

Let *γ*(*s*_{0}) be the p oint at which we aim to compute the geodesic curvature, where *γ* is assumed to be parametrized in terms of its arc length *s*. Then *γ*′(*s*_{0})=** t** is the tangent unit vector, and

*γ*′′(

*s*

_{0})=

*c*

**is proportional to the principal normal. If we denote by**

*n***the unit normal of**

*ν**Σ*at

*γ*(

*s*

_{0}), the plane tangent to

*Σ*is

*π*

_{0}={

*P*:

**·(**

*ν**P*−

*γ*(

*s*

_{0}))=0}. The condition implies

**·**

*t***=0. We introduce the binormal unit vector**

*ν***=**

*b***∧**

*t***and the angle**

*n**ψ*such that . The curve obtained by performing an orthogonal projection of

*γ*onto

*π*

_{0}is given by (Note that

*s*is not the arc-length parameter along

*ξ*.) Clearly, Thus, The geodesic curvature of

*γ*in

*γ*(

*s*

_{0}), being equal to the curvature of

*ξ*, is thus given by ▪

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

*e*_{θ},

**=**

*n*

*e*_{r},

**=**

*b*

*e*_{z}and

*c*=

*R*

^{−1}. Using the parametrization (2.4), we thus obtain

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 is applied to the membrane on the boundary itself. Such a torque acts on the unit normal to the membrane, since its physical origin is a torque exerted on the lipid molecules that form the bilayer. The constraint ** ν**·

**=0 forces the boundary unit normal**

*t***to lie in the plane orthogonal to**

*ν*

*e*_{θ}. Thus, admissible variations for the boundary normal are all of the form

*δ*

**=**

*ν**δ*

**∧**

*ϵ***, with infinitesimal rotation vector**

*ν**δ*

**=**

*ϵ**δ*

*ψ*

*e*_{θ}, where

*ψ*, as above, is the angle between

**and**

*ν*

*e*_{z}.

We consider an external couple of the form . The virtual work exerted by ** C** is given by . Thus

**is a conservative torque, with potential energy density . The boundary torque acts on the unit normal in order to align it along the minimizers of**

*C**σ*

_{w}. Then, the easy direction for the boundary normal is

*e*_{z}when

*w*>0, while it is the radial direction

*e*_{r}if

*w*<0. Clearly,

*w*measures the anchoring strength or equivalently the intensity of the boundary torque. In terms of the free-energy functional, the boundary torque adds an anchoring energy, concentrated on .

Once we neglect the topologically invariant term depending on the Euler characteristic of *Σ*, the complete free-energy functional to be minimized becomes
2.5

## 3. Torque-induced symmetry breaking

The shape of the membrane is forced to be a planar disc of radius *R*, centred at *O*, whenever the membrane area attains the value *A*_{0}=*π**R*^{2}. We assume that the area is slightly perturbed with respect to *A*_{0}, and introduce a small dimensionless parameter *ϵ* to account for the excess area: *A*=*A*_{0}(1+*ϵ*^{2}). Note that the area perturbation is positive, since *A*_{0} is the minimum area allowed by the boundary restriction. In the absence of any perturbation, the only available shape for the membrane is a planar disc. In this case, no difference is expected to arise between the two lipid layers that form the membrane, and thus the spontaneous curvature vanishes. In order to enlarge the membrane area, some extra lipid molecules must be 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, which would possibly give rise to a spontaneous curvature *σ*_{0}. Because of this physical reason, we assume that the spontaneous curvature is infinitesimal, and we write it in terms of a dimensionless parameter *ζ*_{0}: *σ*_{0}=*ϵ**ζ*_{0}/*R*.

Because of the area constraint, the equilibrium configuration of the perturbed membrane is forced to remain close to the flat solution. Consequently, we assume that *z*(*r*,*θ*)=*ϵ**u*(*r*,*θ*), where the perturbation is to be taken as linear in *ϵ*, in order to satisfy the area constraint (see below). Finally, we rescale the Lagrange multiplier as *λ*=*κ**Λ*/(2*R*^{2}) (with *Λ* a dimensionless parameter), and expand as *Λ*=*Λ*_{0}+*O*(*ϵ*). Within this approximation, we obtain
The expansion for the infinitesimal area element justifies the above choice for the order of the shape function perturbation:
where ∇ and Δ are the usual gradient and Laplacian operators in polar coordinates:

Before proceeding further, we now complete the rescaling of all our relevant variables, in order to deal with a dimensionless problem. We have already written the spontaneous curvature *σ*_{0} and the Lagrange multiplier *λ* in terms of the dimensionless parameters *ζ*_{0} and *Λ*_{0}. We now redefine the radial variable by introducing the coordinate *x*=*r*/*R*, and consequently write the shape function as *u*(*r*,*θ*)=*R*υ(*r*/*R*,*θ*), where the dimensionless shape function υ(*x*,*θ*) is defined in the domain *D*′={(*x*,*θ*):*x*∈[0,1],*θ*∈[0,2*π*)}. Finally, we set *w*=*ω**κ*/*R*, and define the dimensionless parameter *ν*=2(2*ω*−*μ*).

When looking for equilibrium configurations, we expand the shape function in Fourier series
3.1
We note that the boundary constraint *u*(*R*,*θ*)=0 for all *θ* implies *a*_{n}(1)=0 for all *n*≥0, and *b*_{n}(1)=0 for all *n*≥1.

### Remark

The axial symmetry inherent to the problem allows us to simplify the Fourier series just introduced. Clearly, it cannot be taken as guaranteed that all equilibrium configurations are axially symmetric, which would in turn mean assuming that *a*_{n}(*x*)=*b*_{n}(*x*)=0 for all *x*∈[0,1], and for all *n*≥1. However, if a non-axially symmetric equilibrium configuration is found, it is certain that all functions are stationary as well, for all values of *α*.

Suppose that possesses only one non-symmetric Fourier mode, i.e. , and that *b*_{N}(*x*)=η*a*_{N}(*x*), for all *x*∈[0,1], with *N*≥1. Then
3.2
provided we choose . In words, when only one non-axially symmetric mode is active, and the amplitudes relative to the sine and cosine submodes are linearly dependent, a redefinition of the *x* and *y* axes in the reference plane allows us 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 obtain , where, apart from inessential constants,
3.3
where a prime denotes differentiation with respect to *x*. The functional is well defined in a subspace of the Sobolev space *H*^{2}(0,1). More precisely, in order for the functional (3.3) to attain a finite value, the Fourier components must satisfy the condition that the combinations (*x**a*_{n}′+*x*^{2}*a*_{n}′′−*n*^{2}*a*_{n})^{2}/*x*^{3} and (*x**b*_{n}′+*x*^{2}*b*_{n}′′−*n*^{2}*b*_{n})^{2}/*x*^{3} be integrable when *x*→0.

When and , an iterated integration by parts yields the following expression for the variation of the functional : 3.4

The Euler–Lagrange equations require the bulk terms to vanish throughout the membrane:
3.5
for both *a*_{n} (*n*≥0) and *b*_{n} (*n*≥1). Equation (3.5) admits the following general solution:
3.6
where *J*_{n} and *Y*_{n} are Bessel functions of the first and second kind, respectively, and *A*_{n},…,*H*_{n} are integration constants. When the Lagrange multiplier *Λ*_{0} is negative, the stationary solutions involve the modified Bessel functions *I*_{n} and *K*_{n}, since, for any real *x*, *J*_{n}(i*x*)=i^{n}*I*_{n}(*x*) and *J*_{n}(i*x*)+i*Y*_{n}(i*x*)=(2/*π*)i^{−n−1}*K*_{n}(*x*).

The regularity conditions above require *B*_{n}=*D*_{n}=0 for all *n*≥0, and similarly *F*_{n}=*H*_{n}=0 for all *n*≥1. Furthermore, the boundary conditions *a*_{n}(1)=*b*_{n}(1)=0 allow us to write
3.7
The final boundary condition originates from the boundary terms in the variation (3.4). They yield
3.8
Equations (3.8) admit two types of solutions, depending on whether or not *Λ*_{0} attains one of the discrete set of values that allow for some of the terms in curly braces in (3.8) to vanish. In those special cases, some of the *C*_{n} and *F*_{n} may become different from zero, and non-axially symmetric modes may arise. Otherwise, only *C*_{0} will be different from zero, and the solution will necessarily be axially symmetric.

In all cases, the area constraint must be satisfied, which implies that
3.9
where *P*_{00},*P*_{01} and *P*_{11} are suitable polynomials (and *F*_{0}=0 has been introduced for the ease of notation). Once all the integration constants, along with the Lagrange multiplier *Λ*_{0}, have been determined, we need to compute the free-energy functional in order to identify the global minimizer of the elastic energy.

### (a) Axisymmetric shapes

For all values of the boundary torque, represented by *ν*, and the spontaneous curvature, identified by *ζ*_{0}, stationary axisymmetric shapes exist. They satisfy *C*_{n}=*F*_{n}=0 for all *n*≥1, while the remaining integration constant and the Lagrange multiplier are determined through the equations
3.10
and
3.11
The two equations can be combined to derive the following single equation for *Λ*_{0}:
3.12
Figure 1 shows how *F* depends on *Λ*_{0} for several different values of *ν*. Stationary values of *Λ*_{0} correspond to the values at which the function *F* intersects the value . We next collect some analytical properties of *F*, which may be useful in the following:

*F*(*ν*,·) is a non-negative, oscillating function that attains infinite times any positive value.For any fixed

*ν*,*F*(*ν*,·) vanishes a countable infinity of times. The set of zeros 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*Λ*_{0}smaller than the value at which vanishes., and also , as

*Λ*_{0}→0.*F*(*ν*,*Λ*_{0})=*ν*^{2}, if , while*F*(*ν*,*Λ*_{0})=*Λ*_{0}, if . Thus, values at which*J*_{1}vanishes may be easily recognized, since all the plots*F*(*ν*,*Λ*_{0}) intersect at .Let

*x*_{1}be the smallest non-zero root of the equation*J*_{1}(*x*)=0. (Then,*x*_{1}≈3.83.) The smallest root (in*Λ*_{0}) of equation (3.12) when*ν*≫1 is given by , as . In particular, .

Among the countable infinity of values of *Λ*_{0} that satisfy equation (3.12) for any given *ν* and *ζ*_{0}, the minimization process requires the one corresponding to the lowest free-energy value to be identified. If we use equations (3.10) and (3.11) to determine *ν* and *C*_{0}, the integrals in equation (3.3) can be carried out analytically and provide the free-energy excess associated with the perturbation:
3.13
Clearly, equation (3.13) is still an implicit expression for the free-energy excess. Indeed, it is meaningful only for *Λ*_{0}-values that obey condition (3.12), since the independent physical free parameter is the boundary torque *ν*.

In the special case when the spontaneous curvature vanishes (*ζ*_{0}=0), equation (3.13) shows, however, that the free energy is simply proportional to *Λ*_{0}. In general, the fact that the free energy is an increasing function of *Λ*_{0} can be easily explained by a direct inspection of the stationary shapes (3.6). Indeed, rising *Λ*_{0} amounts to enhancing the argument of the Bessel functions involved in the shape perturbations. Thus, greater values of *Λ*_{0} correspond to increasingly oscillating stationary shapes, which possess an excess of elastic energy.

When the spontaneous curvature does not vanish, , as given by equation (3.13), is not always a monotonically increasing function of *Λ*_{0}. Just to give an example, if we again denote by *x*_{1} the smallest zero of the Bessel function *J*_{1}, equation (3.13) shows that whenever . We have thus performed a numerical investigation to ascertain whether, for any value of *ζ*_{0}, the property described above holds, i.e. whether it can always be ensured that, among the countable infinity of values of *Λ*_{0} that obey equation (3.12), the smallest root corresponds to the stationary solution possessing the least free-energy excess. We have sampled values of the spontaneous curvature in the interval *ζ*_{0}∈[0,5], with *ν*∈[−50,50]. In all cases, we have found that the optimal choice for *Λ*_{0} corresponds to the smallest root of equation (3.12). Figure 2 illustrates how the excess free energy depends on *ν* for two different values of the spontaneous curvature. It is to be noted that the optimal free energy tends to a constant when . Indeed, , and thus . 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 axially symmetric shape. For any value of *ν*, we need to first identify the smallest zero of *F*(*ν*,·), which amounts to finding the smallest root of the equation . That value of *Λ*_{0} corresponds to the stationary value when *ζ*_{0}=0. From thence on, the left branch of *F*(*ν*,·) provides the optimal values of *Λ*_{0} for any *ζ*_{0}≠0.

### Remark

The optimal axially symmetric shapes share the property that, for any value of *ν* and *ζ*_{0}, they never cross the reference plane, where the external circumference lies. Figure 3 shows some of those optimal axially symmetric shapes.

To prove this property, we first note that the smallest zero (in *Λ*_{0}) of *F*(*ν*,*Λ*_{0}) does always occur before , which corresponds to the first non-trivial zero of . Thus, the argument above shows that, for any *ζ*_{0} and *ν*, the optimal value of *Λ*_{0} is always strictly smaller than the value . Suppose now that an optimal axially symmetric shape could cross the reference plane. This would imply that for some . In view of equation (3.7), this would yield . Thus, by virtue of Rolle’s theorem, an would exist such that , and this would yield a contradiction, since the function would not be allowed to vanish for any *x*∈(0,1).

### (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 modes but the zeroth one. In this section we analyse the solutions that allow for some of the *C*_{n} and *F*_{n} to become different from zero. In view of the structure of equations (3.8), those symmetry-breaking solutions may arise only when those values of *Λ*_{0} for which a term in curly braces in equations (3.8)_{2,3} vanishes for some . Once a value for *Λ*_{0} is found, equation (3.8)_{1} fixes the value of the factor *C*_{0}. Finally, the area constraint (3.9) provides the value of the corresponding . Before proceeding further, we remark that, without loss of generality, we may assume that *F*_{n}=0 for all *n*≥1. Indeed, were also different from zero, equation (3.7) would show that would simply be proportional to and, in view of the symmetry considerations presented above (see equation (3.2)), this implies that solutions with coincide with solutions with , up to an overall rotation of the reference axes.

Figure 4 shows the phase diagram relative to the axial-symmetry breaking for the constrained membrane. The symmetry-breaking transition is triggered by both the spontaneous curvature and the boundary torque. We remark that both parameters are crucial if non-symmetric shapes are to be found. Indeed, axially symmetric shapes provide the absolute minimizer for any value of the boundary torque if the spontaneous curvature is smaller than *σ*_{0,cr}=*ϵ**R*^{−1} (which corresponds to *ζ*_{0}=1). Analogously, no instability is found if *ν*≥*ν*_{cr}≃−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 a change of sign in the spontaneous curvature can be easily taken into account by simply switching the sign of the principal component *a*_{0}.

The phase diagram in figure 4 corresponds to shape instabilities relative to the first Fourier mode. It is to be expected that, within the ‘symmetry-breaking’ domain, new transitions should appear, with the onset of absolute minimizers associated with higher Fourier modes. Figure 5 explicitly shows some of the non-symmetric stationary shapes. In particular, figure 5*c* corresponds to parameter values for which it is energetically preferred with respect to the axially symmetric minimizer.

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

## 4. Conclusions

We have studied the equilibrium shape of a lipid membrane, constrained to lean on a fixed circumference, subject to an external boundary torque. In particular, we have sought free-energy minimizers that break the axial symmetry, induced by both the boundary constraint and the external torque. We framed our study in the quasi-planar approximation, which is well justified when the membrane area is 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 either axially 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 the Lagrange multiplier *Λ*_{0}.

We have compared the elastic energy of the axially symmetric solution with the energy of the first non-trivial symmetry-breaking mode, as a function of the external torque (represented by the dimensionless parameter *ν*), and the spontaneous curvature (represented by *ζ*_{0}). As a result, we have identified a region in the (*ν*,*ζ*_{0}) 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 symmetric ones.

We remark that *ν* must be sufficiently negative for the transition to take place. It is to be noted that the dimensionless parameter *ν* depends also on the parameter *μ*, which governs the elastic constant associated with the Gaussian-curvature term in the elastic energy (see equation (2.1)). Nevertheless, and since *ν*=2(2*ω*−*μ*) and *μ*∈[0,1], in the absence of external torque (which amounts to *ω*=0) we obtain *ν*∈[−2,0]. In correspondence with these values of *ν*, no symmetry-breaking instability shows up. Thus, the effect we are envisaging is clearly linked to the external torque, and it is not an effect of the Gaussian-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 have assumed that *σ*_{0} is either null or infinitesimal: *σ*_{0}=*ϵ**ζ*_{0}/*R*. However, it would be interesting to study, at least from the mathematical point of view, the situation in which the spontaneous curvature is finite in the *ϵ*→0 limit. In the presence of a finite spontaneous curvature, such a limit becomes singular. Indeed, let us consider the expression (3.3) for , the *O*(*ϵ*^{2}) approximation to the free-energy functional in the small-*ϵ* limit. Were *σ*_{0}=*O*(1), the second-last term in , 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 first derivative of the principal component *a*_{0}, is not bounded from below. Optimal shapes are thus expected to become singular and, more precisely, to develop a boundary layer of *O*(*ϵ*) at *x*=1^{−}. This mechanism, that is, the possibility of a lipid membrane being able to relax its excess of curvature energy in a thin boundary layer, has been already identified as the optimal way in which a lipid membrane modifies its equilibrium shape in the presence of a rigid inclusion (Biscari & Napoli , 2007).

## Acknowledgements

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

## Footnotes

One contribution of 12 to a Theme Issue ‘Mechanics in biology: cells and tissues’.

- © 2009 The Royal Society