Continuum Theory for the Edge of an Open Lipid Bilayer

CHAPTER ONE

Continuum Theory for the Edge of an Open Lipid Bilayer Aisa Biria, Mohsen Maleki, Eliot Fried Department of Mechanical Engineering, McGill University, Montre´al, Que´bec H3A 0C3, Canada

Contents 1. Introduction 2. Mathematical Preliminaries 2.1 Superficial fields 2.2 Differential geometry of the surface 2.3 Useful differential and integral identities 2.4 Differential geometry of the edge 2.5 Notational conventions 3. Variations of Geometric Quantities, Integrals over Surfaces and Curves, and Volume 3.1 Areal quantities 3.2 Lineal quantities 3.3 Volume 4. Variational Derivation of the Equilibrium Equations of a Lipid Vesicle 4.1 Variation of the net free-energy 4.2 Virtual volumetric work 4.3 Virtual work of the areal loads 4.4 Combined results 5. Variational Derivation of the Equilibrium Equations of an Open Lipid Bilayer with Edge Energy 5.1 Constant edge-energy density 5.2 Geometry-dependent edge-energy density 6. Force and Bending Moment Exerted by an Open Lipid Bilayer on Its Edge 6.1 Force and bending moment expressions 7. Alternative Treatment of the Edge 7.1 Edge kinematics 7.2 Balance laws 7.3 Constitutive equations and thermodynamic restrictions 7.4 Governing equations 7.5 Retrieving the Euler–Lagrange equations at the edge 8. Summary Acknowledgments References

Advances in Applied Mechanics, Volume 46 ISSN 0065-2156 http://dx.doi.org/10.1016/B978-0-12-396522-6.00001-3

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2013 Elsevier Inc. All rights reserved.

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Abstract Using a variational approach, the Euler–Lagrange equations of an open lipid bilayer subject to forces and couples distributed on its surface and edge are derived. Both constant and geometry-dependent edge-energy densities are considered. For the second of these alternatives, the edge-energy density is a general function of the normal and geodesic curvatures and geodesic torsion of the edge. Focusing on a generic segment of the edge, the global forms of the force and moment balances and the free-energy imbalance are stated and their local counterparts are derived. While the force and moment balances lead to the governing equations of the edge element under internal and external loads, the free-energy imbalance provides a mechanism for ensuring the thermodynamic compatibility of constitutive relations. Inspired by various experimental and theoretical studies showing the importance of dissipative mechanisms at the edge of an open lipid bilayer, the internal force and moment are decomposed into elastic and viscous parts. Considering the geometry-dependent edge-energy density and following the Coleman–Noll procedure, constitutive relations for the elastic contributions to the internal moment and tangential component of the internal force are derived. Additionally, the constitutive relations for the viscous contributions to the internal force and moment are restricted by a reduced dissipation inequality. In the purely elastic regime, it is shown that the governing equations for the edge arising from augmenting the force and moment balances with thermodynamically compatible constitutive relations reduce to the Euler–Lagrange equations previously obtained on variational grounds.

NOMENCLATURE1 S surface C boundary @S of S n unit normal field on S f generic scalar-valued superficial field; tangential component of the internal force f g generic vector-valued superficial field rS surface gradient f e smooth extension of f ge smooth extension of g r three-dimensional gradient P perpendicular projector onto S divS surface divergence tr trace (applied to second-order tensors) DS surface Laplacian x element of three-dimensional point space o arbitrarily chosen origin r position vector directed from o to x 1 second-order identity tensor h generic vector-valued superficial field 1

With all entries listed in order of appearance.

Continuum Theory for the Edge of an Open Lipid Bilayer

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R bounded region of three-dimensional point space V volume of R A generic subsurface of S @A boundary of A n@A unit vector on @A that is tangent to S at @A, normal to @A, and directed outward from A L curvature tensor of S H mean curvature of S I1 first principal invariant (applied to second-order tensors) K Gaussian curvature of S I2 second principal invariant (applied to second-order tensors) e unit tangent to the edge C of S n unit tangent-normal vector n of C s arclength of C kn normal curvature of C kg geodesic curvature of C tg geodesic torsion of C k curvature of C k curvature vector of C kg geodesic curvature vector of C kn binormal curvature of C e scalar variation parameter S e virtual version of S accompanying xe R generic quantity associated with S or its boundary C d first variation P arbitrary material region including S and C xe virtual deformation P e virtual version of P accompanying xe u virtual displacement ut tangential component of u U scalar normal component of u Fe virtual deformation gradient Je virtual volumetric Jacobian det determinant ne virtual version of n accompanying xe ne extension of the unit normal n dHn normal variation of the mean curvature H of S dHt tangential variation of the mean curvature H of S dKn normal variation of the Gaussian curvature K of S w shorthand for rS U dKt tangential variation of the Gaussian curvature K of S je virtual areal Jacobian of S dj first variation of the areal Jacobian of S G generic surface integral on S g generic field g defined on S ge virtual version of g accompanying xe ue component of variation u in the unit tangent e direction un component of variation u in the unit tangent-normal n direction

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le virtual lineal stretch of an infinitesimal material element on the edge C of S dl first variation of the lineal stretch of an infinitesimal material element on the edge C of S ee virtual version of e accompanying xe x generic field defined on C xe virtual version of x accompanying xe se virtual version of the arclength s accompanying xe X generic line integral on C @R boundary of R c bending-energy density per unit area of S
bending-energy density function per unit area of S c E net free-energy s areal Lagrange multiplier p unknown pressure f distributed external force on S m distributed external couple on S W S work performed by the areal external loads v virtual angular-velocity vector of the unit normal n
with respect to H
partial derivative of c c H
partial derivative of c
with respect to K c K fn scalar normal component of f ft tangential component of f E C lineal or edge contribution of E associated with C f edge-energy density, measured per unit arclength of C i distributed force exerted by the suspending solution on the edge C m distributed couple exerted by the suspending solution on the edge C W sC virtual work of the distributed force i and distributed couple m on the edge C fo constant edge-energy density per unit arclength of C ie component of i in the unit tangent e direction in component of i in the unit normal n direction in component of i in the unit tangent-normal n direction me component of m in the unit tangent e direction mn component of m in the unit normal n direction mn component of m in the unit tangent-normal n direction vC virtual angular-velocity of the Darboux frame {e, n, n} l arbitrary element of the Darboux frame {e, n, n}
geometry-dependent edge-energy density function measured per unit arclength of C f

with respect to kn fkn partial derivative of f
partial derivative of f
with respect to kg f kg
with respect to tg
partial derivative of f f tg m splay modulus
saddle-splay modulus m Ho spontaneous mean curvature kb constant coefficient in the edge-energy density f e force, per unit arclength, exerted by the edge C on S me bending moment, per unit arclength, exerted by the edge C on S WSe virtual work performed on S by f e and me

Continuum Theory for the Edge of an Open Lipid Bilayer

fn component of f e in the unit normal n direction fn component of f e in the unit tangent-normal n direction me component of me in the unit tangent e direction mn Component of me in the unit normal-tangent n direction V velocity field defined on the surface S and the edge C  stretching rate of the edge C d1, d2, d3 orthonormal directors defined on C U twist tensor W spin tensor U Darboux vector W Darboux angular-velocity G generic segment of the edge C x initial point of G xþ terminal point of G s initial arclength of G sþ terminal arclength of G x evaluation of x at the initial point x of G xþ evaluation of x at the terminal point xþ of G ½   jump (across the endpoints x and xþ) of G f  the contact force exerted by the portion C n G of C external to G, at x f þ the contact force exerted by the portion C n G of C external to G, at xþ m the contact moment exerted by the portion C n G of C external to G, at x mþ the contact moment exerted by the portion C n G of C external to G, at xþ f ext external distributed force along the segment G mext external distributed moment along the segment G f internal force of G m internal moment of G F ðGÞ net free-energy of G DðGÞ net dissipation of G WðGÞ net power expended on G by external agencies t time variable
f component of the internal force f in the plane perpendicular to the unit tangent e ^ constitutive response function for f f fel elastic part of f fvis viscous part of f mel elastic part of m mvis viscous part of m f^el constitutive response function for fel m ^ el constitutive response function for mel ^f vis constitutive response function for fvis m ^ vis constitutive response function for mvis ^ with respect to tg ^ partial derivative of f f tg ^ partial derivative of f ^ with respect to kg f kg ^ ^ with respect to kn fkn partial derivative of f a1, a2, a3 unknown coefficients b constant coefficient

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1. INTRODUCTION A phospholipid is a composite molecule in which a phosphorylated alcohol is linked by a three-carbon glycerol backbone to a pair of fatty acid chains, forming a water-soluble head group with two dangling waterinsoluble tails (Fig. 1.1). When dispersed in an aqueous solution at sufficiently high concentrations, phospholipid molecules self-assemble to form structures in which their head groups shield their tails from water. These structures include lipid bilayers, which are thin membranes composed of two adjacent leaflets in which the constituent molecules are oriented transversely and set tail to tail (Fig. 1.2). Relative misalignment of these molecules is accompanied by an energetic penalty which is manifested in the form of bending elasticity. Lipid bilayers compose the outer envelope of the plasma membrane, the walls of cellular compartments and subunits, and synthetic systems such as

water-insoluble tails water-soluble head group

Figure 1.1 Schematic of a phospholipid molecule consisting of a water-soluble head group and two water-insoluble tails.

A

B

Figure 1.2 (A) Schematic of an open lipid bilayer and two cross-sections depicting the different arrangements of lipid molecules in its interior region and near its edge. (B) Schematic of a lipid vesicle with a pore.

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giant unilamellar vesicles. The degree of order within a lipid bilayer decreases with increasing temperature, transitioning through the subgel, gel, ripple, and liquid-crystalline fluid phases (Nagle & Tristram-Nagle, 2000). It is widely accepted that many biologically relevant processes occur in the latter phase (Garcia-Manyes, Oncins, & Sanz, 2005), wherein phospholipid molecules are free to diffuse laterally within each leaflet and to flip between leaflets. When a free edge forms on a lipid bilayer, phospholipid molecules spontaneously rearrange along the edge line to form a hemicylindrical rim that shields the hydrophobic core from the surrounding solvent (Fig. 1.2). As confirmed by experiments and molecular dynamics simulations, this misalignment of lipid molecules in the immediate vicinity of the edge causes an excess energy concentrated along it (Jiang, Bouret, & Kindt, 2004; Kasson & Pande, 2004; May, 2000; Smith, Vinchurkar, Gronbech-Jensen, & Parikh, 2010). Due to this excess of stored energy, open lipid bilayers display a tendency to close up into vesicles, thereby eliminating edges. This, however, does not rule out the existence of stabilized or transient open lipid bilayers. Thermal instabilities, fluctuations of the transmembrane electrochemical potential, and mechanical stresses can lead to transient opening of lipid bilayers, allowing for the passage of water, ions, and a variety of water-soluble molecules. Transient open bilayers are also present during electroformation, which is a classical technique for creating vesicles (Lasic, 1988). Furthermore, various strategies exist for forming stable open lipid membranes. Pores on lipid membranes may be stabilized by the application of electric fields (Tsong, 1991), the addition of edge-active chemical agents (Fromherz, Ro¨cker & Ru¨ppel, 1986), and sonication (Marmottant & Hilgenfeldt, 2003). These techniques permit control over both the size and life span of pores. In contrast, irreversible electroporation may lead to the bursting of the cell membrane (Neu & Neu, 2010). With advances in sonoporation and electroporation techniques, open lipid bilayers have become the subject of increasing research interest. The presence of pores allows for the targeted uptake of macromolecules such as protein and DNA into cells, promising a revolution in gene therapy and drug delivery (Bao, Thrall, & Miller, 1997; Mehier-Humbert, Bettinger, Yan, & Guy, 2005; Newman & Bettinger, 2007). Irreversible electroporation may also artificially initiate apoptosis, making it a suitable candidate for tumor ablation and cancer therapy (Lee, Thai & Kee, 2010). This important application has driven theoretical and numerical studies on the mechanics and physics of open lipid bilayers and their edges, both at

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the molecular scale (Jiang et al., 2004; May, 2000; Wohlert, den Otter, Edholm, & Briels, 2006) and in the realm of continuum mechanics. In the latter setting, lipid bilayers are modeled as two-dimensional elastic surfaces endowed with bending energy but unable to resist tangential shear. This vision dates back to the proposition of the fluid mosaic model for cell membrane advanced by Singer & Nicolson (1972) and to the seminal works of Canham (1970) and Helfrich (1973). The continuum mechanical description of lipid bilayers has been applied extensively in various investigations of the morphology of vesicles, for which a comprehensive and far-reaching literature is available (Lipowsky, 1995; Seifert & Langer, 1993). Even though closed lipid membranes have been studied in detail for decades, the equilibrium equations for open lipid bilayers, in which the shape equations familiar from studies of closed lipid bilayers are supplemented by edge equations, were not addressed until more recently. Boal & Rao (1992) were the first to consider the problem. They augmented a Canham–Helfrich-type energy functional for the surface representing a lipid bilayer with a line energy proportional to the length of the edge of the bilayer and, restricting attention to axisymmetric configurations, used variational calculus to derive equilibrium equations for the bilayer and its edge. A decade later, Capovilla, Guven, & Santiago (2002) relaxed the assumption of axisymmetry and extracted corresponding equilibrium equations. They also provided physical interpretations of the equations they obtained. Tu & Ou-Yang (2003, 2004) used exterior differential forms to derive the governing equilibrium equations and extended the work of Capovilla, Guven, & Santiago (2002) by allowing the edge-energy density to depend also on the geodesic and normal curvatures of the boundary. In addition, Tu (2010, 2011) and Tu & Ou-Yang (2003) attempted to solve the equilibrium equations of open lipid bilayers analytically for the special case of axisymmetric configurations with uniform edge-energy density. Different numerical schemes have also been employed to obtain equilibrium configurations (Umeda, Suezaki, Takiguchi, & Hotani, 2005; Wang & Du, 2008; Yao, Sknepnek, Thomas, & de la Cruz, 2012; Yin, Yin, & Ni, 2005). Cup-shaped, gourd-shaped, and funnel-shaped configurations, as well as portions of torii have also been reported (Tu & Ou-Yang, 2003, Umeda et al., 2005; Wang & Du, 2008). Except for the work of Tu & Ou-Yang (2004), the aforementioned studies of the mechanics of open lipid membranes take the edge-energy density to be a given constant. Moreover, physical justification for the particular form of the edge-energy density used by Tu & Ou-Yang (2004) is somewhat lacking.

Continuum Theory for the Edge of an Open Lipid Bilayer

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In the present work, the edge-energy density is allowed to depend on the geometry of the boundary of the lipid bilayer through the normal and geodesic curvatures and the geodesic torsion. No assumptions are imposed regarding the particular form of dependence on these quantities. Since opening, closing, and stabilization of pore on a lipid bilayer are predominantly due to external stimuli, an open lipid bilayer is generally subject to external forces and couples which should be taken into account in any practical theory intended for making comparisons with experimental measurements. The external stimuli may be due, for example, to an applied electric field, mechanical stresses, or viscous forces. The theory developed here therefore accounts for external forces and couples distributed on both the surface and the edges of open lipid bilayers. Equilibrium equations that account for these forces and couples are derived using the principle of virtual work. In contrast to the works of Tu & Ou-Yang (2003, 2004), the variational method used here avoids elaborate mathematics. The principle of virtual work is also used to derive the force and bending moment which the interior part of the lipid bilayer exerts on its edge. The first stage in pore formation is characterized by an exponential growth of pore radius with time, followed by a second stage of linear shrinkage, or—in the case of irreversible pore formation—a second stage of linear growth. While the viscosity of the suspending solution is believed to be the primary source of dissipation in the linear regime, the exponential stage in the growth of a pore is dominated by lipid membrane viscosity (Brochard-Wyart, de Gennes, & Sandre, 2000; Karatekin, Sandre, & Brochard-Wyart, 2003; Neu & Neu, 2010; Sung & Park, 1997). Furthermore, Ryham, Berezovik, & Cohen (2011) and Seifert & Langer (1993) suggested that membrane viscosity has a significant dissipative role in submicron-sized lipid vesicles, such as small and large unilamellar vesicles. This observation is in agreement with the experiments of Watson & Brown (2010), continuum simulations of Brown (2011), and molecular dynamics simulations of Shkulipa, den Otter, & Briels (2005). Recently, Arroyo & DeSimone (2009) discovered that membrane viscosity is significant for pore radii in the range of tens of microns. Many eukaryotic cells, as well as giant unilamellar vesicles, fall into this range. Motivated by all the aforementioned investigations of pore dynamics, the present work also addresses dissipative effects associated with the rearrangement of phospholipid molecules as the edge of an open lipid bilayer grows or shrinks, an effect which has been neglected in most continuum mechanical descriptions of open lipid bilayers.

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To incorporate dissipative effects, the equilibrium equations for the edge are derived using an approach based on postulating balance laws for forces and moments and an imbalance that represents the second law of thermodynamics, which govern a broad spectrum of materials, along with constitutive relations, which define particular classes of materials. In this context, the edge is treated as a unidimensional continuum defined, in a fashion reminiscent of Cosserat rod theory (Cosserat & Cosserat, 1909), by a curve endowed with a triad of directors at each of its points. The balance and imbalance laws are stated for an arbitrary segment of the edge. The pointwise consequences of the force and moment balances encompass the requirement that the internal force and moment within the edge be in equilibrium with their external counterparts. Restricting attention to isothermal processes, the first and second laws of thermodynamics collapse to a free-energy imbalance. Relying on the local version of this imbalance, the Coleman–Noll procedure (Coleman & Noll, 1963) is used to derive a thermodynamically consistent constitutive theory for the edge. The constitutive expressions for the internal force and moment together with the pointwise versions of the force and moment balances provide the final equations that govern the configuration of the edge. In accord with ideas familiar from the conventional theory of viscoelasticity, the internal force and moment are decomposed into equilibrium and viscous parts. After extracting the equilibrium relations, the free-energy imbalance reduces to a residual dissipation inequality in which only the viscous contributions to the internal force and moment appear. All thermodynamically compatible expressions for the dissipative quantities must comply with this residual inequality. This requirement is then used to determine the restrictions on the constitutive expressions for the viscous force and moment. The chapter is organized as follows. Mathematical preliminaries, including the notion of a superficial field, differential and integral identities, and differential geometry of the surface and the edge, are covered in Section 2. Sections 3–6 are devoted to the variational approach. Section 3 includes derivations of various useful relations, including the first variations of different areal and lineal quantities as well as the volume enclosed by a vesicle. Section 4 is devoted to a derivation of the equilibrium equations of a lipid vesicle (or closed lipid bilayer) on the basis of the principle of the virtual work. Section 5 provides a detailed derivation of the equilibrium equations of an open lipid bilayer. To make contact and allow comparisons with the majority of the existing theoretical works, Subsection

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5.1 is devoted to the case where the edge energy is uniform. The case of an edge-energy density dependent on the normal and geodesic curvatures and the geodesic torsion of the edge is considered in Subsection 5.2. Representations for the force and bending moment exerted by a lipid bilayer on its edge are derived in Section 6. The second part of the chapter, containing Section 7, is devoted to the approach based on balance laws augmented by thermodynamically consistent constitutive relations, with a view to characterizing dissipation associated with the expansion or contraction of openings on lipid bilayers. After covering some essential kinematical preliminaries in Subsection 7.1, global statements of the force and moment balances are presented, along with derivations of their local counterparts, in Subsection 7.2. The global free-energy imbalance for the edge of the lipid bilayer is presented, along with a derivation of its local counterpart, in Subsection 7.3. This is followed by discussion of constitutive relations for the internal force and moment. Augmenting the pointwise versions of the balance laws with the thermodynamically consistent constitutive relations yields the governing equations for the edge. Finally, these are compared with the variationally-based results appearing in Section 5.

2. MATHEMATICAL PRELIMINARIES 2.1. Superficial fields Consider an open lipid bilayer identified with a smooth, orientable surface S with rectifiable boundary C ¼ @S (Fig. 1.3). Suppose that S and C are disjoint, viz.

Figure 1.3 An open lipid bilayer identified with a surface S with edge C. Also depicted are the Darboux frame {e, n, n} at a generic point on C, the curvature vector k, and its normal and tangent components kn n, with kn the normal curvature, and the geodesic-curvature vector kg.

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S \ C ¼ ∅,

ð1Þ

so that S is an open two-dimensional set that represents the interior of the bilayer and C is a closed space curve that represents the edge of the bilayer. To emphasize the latter correspondence, C is referred to as the edge of S. Let n denote a unit normal field on S. Fields like n that are defined on but not away from S are called superficial. Although the conventional threedimensional gradient operator r cannot be applied to a superficial field, it can be applied to a smooth extension of a superficial field to a neighborhood of S. However, given a point on S, there is no unique way to extend a superficial field smoothly to a three-dimensional neighborhood of that point. Among the various possibilities, a particularly convenient choice is the normally constant extension, wherein a superficial field is extended to be constant along line segments normal to S, as described for instance by Fried & Gurtin (2007). Given scalar- and vector-valued superficial fields f and g, their surface gradients rS f and rS g are defined uniquely in terms of smooth extensions f e and ge via rS f ¼ Prf e

and

rS g ¼ ðrge ÞP,

ð2Þ

where the three-dimensional gradients rf e and rge of f e and ge are evaluated on S and where P ¼ 1  n n

ð3Þ

denotes the perpendicular projector onto S (with 1 being the threedimensional identity tensor). Further, the surface divergence divS g of g is defined uniquely as divS g ¼ trðrS gÞ ¼ P  rge :

ð4Þ

Lastly, the surface Laplacian of f is defined by DS f ¼ divS ðrS f Þ:

ð5Þ

The uniqueness of the definitions (2), (4), and (5) is tantamount to the observation that the expressions for rS f , rS g, divS g, and DS f are independent of whatever strategy is adopted to extend f and g.

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2.2. Differential geometry of the surface The curvature tensor L ¼ rS n

ð6Þ

of S is a mapping of the space of three-dimensional vectors into itself with the properties Ln ¼ 0,

L ¼ L> :

ð7Þ

Given an extension ne of the unit normal n, it follows from (2) that the curvature tensor L admits a representation of the form L ¼ ðrne ÞP:

ð8Þ

The mean and Gaussian curvatures H and K of S are defined by 1 H ¼ I1 ðLÞ 2 1 ¼ trL 2 1 ¼  divS n 2

ð9Þ

and K ¼ I 2 ðL Þ
 1
¼ ðtrLÞ2  tr L2 , 2

ð10Þ

where I1(L) and I2(L) indicate first two principal invariants of L. Finally, in view of the properties (7), the Cayley–Hamilton theorem for L reads L2  I1(L)L þ I2(L)P ¼ 0, which—with the definitions (9) and (10) of H and K—becomes L2  2HL þ KP ¼ 0:

ð11Þ

2.3. Useful differential and integral identities In this section, various useful mathematical relations are introduced. For proofs of nontrivial results, the reader is referred to sources previously existing in the literature.

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Given an element x of three-dimensional point space and an arbitrarily chosen origin o, let r ¼xo

ð12Þ

denote the vector directed from o to x. Since rr ¼ 1, (2)2 and (4) yield rS r ¼ P

and

divS r ¼ 2:

ð13Þ

For a scalar-valued superficial field f and vector-valued superficial fields g and h, Gurtin & Murdoch (1975) establish the identities rS ð f gÞ ¼ f rS g þ g rS f ,

)

rS ð g hÞ ¼ ðrS gÞ> h þ ðrS hÞ> g:

ð14Þ

Taking the trace on both sides of (14)1 and invoking the definition (4) of the surface divergence yields divS ð f gÞ ¼ f divS g þ g  rS f :

ð15Þ

Consider a bounded region R with volume V. Then, as a simple consequence of the three-dimensional divergence theorem, ð r  n ¼ V 1: ð16Þ @R

Lastly, consider a subsurface A of S with boundary @A and let n@A denote the unit-vector-valued field defined on @A that is tangent to S at @A, normal to @A, and directed outward from A. Then, given a vectorvalued superficial field g, the surface-divergence theorem—as presented, for instance, by Gurtin & Murdoch (1975)—ensures that ð @A

ð g  n@A ¼

A

ðdivS g þ 2Hg nÞ,

ð17Þ

where H is the mean curvature, as defined in (9).

2.4. Differential geometry of the edge Let e denote the unit tangent to the edge C of S, selected so that the unit tangent-normal vector n of C defined by n¼en

ð18Þ

Continuum Theory for the Edge of an Open Lipid Bilayer

15

is directed outward from S. The vector triad {e, n, n} then furnishes an oriented orthonormal basis on C, often called the Darboux frame of C. The elements of this triad obey 9 e0 ¼ kn n  kg n, > = n0 ¼ kn e  tg n, ð19Þ > ; 0 n ¼ kg e þ tg n, where a prime indicates the derivative with respect to the arclength s of C, kn ¼ e 0  n ¼ n0 e

ð20Þ

and kg ¼ e0 n ¼ n0  e

ð21Þ

are the normal and geodesic curvatures of C, and tg ¼ n0  n ¼ n0 n

ð22Þ

is the geodesic torsion of C. The (scalar) curvature k ¼ je0 j of C is related to the normal and geodesic curvatures by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ k2n þ k2g : ð23Þ It is useful to define curvature and geodesic-curvature vectors k and kg for C via the relations k ¼ e0 ¼ kn n  kg n

ð24Þ

and kg ¼ k  k n n ¼ kg n:

ð25Þ

The decomposition of the curvature vector k into the normal and tangent components knn and kg is illustrated in Fig. 1.3. The normal curvature kn and geodesic torsion tg depend on the curvature tensor L of S at C. To establish the nature of this dependence, notice that, by the definition (6) of L,

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n0 ¼ ðrS nÞe ¼ Le:

ð26Þ

Thus, by (6), (20), and (24), the normal curvature kn obeys kn ¼ k n ¼ e n0 ¼ e  ðrS nÞe ¼ e  Le;

ð27Þ

similarly, by (6), (7)2, and (19)3, the geodesic torsion tg obeys tg ¼ n0 n ¼ n  n0 ¼ n ðrS nÞe ¼ n Le ¼ e  Ln:

ð28Þ

Two equivalent geometrical interpretations of the geodesic torsion tg are provided in Fig. 1.4. In contrast to the normal curvature kn and the geodesic torsion tg, the geodesic curvature kg is independent of the curvature tensor L. Less commonly encountered that the normal and geodesic curvatures kn and kg of C, but equally valuable here is the tangent-normal curvature kn of C defined by kn ¼ n  Ln:

ð29Þ

Figure 1.4 Illustration of the geodesic torsion tg at a generic point on the boundary C of S.

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Continuum Theory for the Edge of an Open Lipid Bilayer

Since n  Ln ¼ tr L  e  Le, the definition (9) of the mean curvature H of S and the relations (27) and (29) for the normal and tangent-normal curvatures kn and kn of C can be used to show that the restriction to C of H can be expressed as the average, 1 H ¼ ðkn þ kn Þ, 2

ð30Þ

of the normal and tangent-normal curvatures kn and kn of C. Toward obtaining a useful representation for the restriction to C of the Gaussian curvature K, notice that, by the alternative representations (19)2 and (26) for the arclength derivative n0 of the restriction to C of the unit normal n, the result Le of applying the restriction to C of the curvature tensor L of S to the unit tangent e of C can be expressed as Le ¼ kn e þ tg n:

ð31Þ

On computing the inner product of each term in the Cayley–Hamilton theorem (11) with e  e to yield K ¼ 2He  Le  jLej2 and using the representation (31) for Le, it follows that the restriction to C of the Gaussian curvature K is given in terms of the restriction to C of the mean curvature H and the normal curvature kn and geodesic torsion tg of C by K ¼ 2Hkn  k2n  t2g :

ð32Þ

2.5. Notational conventions Any quantity defined on the surface S that appears in a relation for the edge C must necessarily be evaluated on C. Simple examples of such relations are provided by (18) and (26)–(28), in which it is tacit that n and L denote the restrictions to C of the normal n to S and curvature L of S. Since this distinction is generally evident from context, restrictions like these are not indicated explicitly. Given a vector field g defined on S, its tangential and (scalar) normal components gt and gn are defined by gt ¼ Pg

and

gn ¼ g  n:

ð33Þ

Similarly, given a vector field h defined on C, its components he, hn, and hn relative to the Darboux frame {e, n, n} of C are defined by he ¼ h  e,

hn ¼ h n,

and

hn ¼ h n:

ð34Þ

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3. VARIATIONS OF GEOMETRIC QUANTITIES, INTEGRALS OVER SURFACES AND CURVES, AND VOLUME In the present section, an approach introduced by Fosdick & Virga (1989) and extended by Rosso & Virga (1999) is used to derive expressions for the first variations of certain essential quantities. See also Asgari, Maleki, Biria, & Fried (2013) and Maleki & Fried (2013a). To make the presentation self-contained, derivations of all variations used in this chapter are provided. Consider an infinitesimal virtual deformation which slightly alters the surface S to S e , with e  0 being a scalar parameter (Fig. 1.5). Under such a virtual deformation, a generic quantity R associated with S or its boundary C transforms to Re and the first variation dR of R is determined in accord with the limit Re  R : e!0 e

dR ¼ lim

ð35Þ

In (35), R may represent a scalar-, vector-, or tensor-valued superficial field or even a surface integral defined on S or a line integral defined on C. The conventional approach to calculating variations of superficial quantities involves considering a virtual deformation defined exclusively on S. A slightly different approach, which hinges on treating S as a material surface embedded in a three-dimensional body P undergoing a virtual deformation, is adopted here. Aside from the requirement that S must belong to its interior, P may be an arbitrarily chosen material region. Consistent with this perspective, a virtual deformation xe is an invertible mapping that takes a generic point x of P into a point

Figure 1.5 An infinitesimal virtual deformation xe mapping an open surface S with boundary C to an open surface S e with boundary Ce . For the sake of illustration, the two configurations are shown without overlap and the infinitesimal virtual distortion is exaggerated.

Continuum Theory for the Edge of an Open Lipid Bilayer

xe ¼ xe ðxÞ ¼ x þ eu þ oðeÞ

19

ð36Þ

belonging to the perturbed image P e of P. The quantity u on the second line of (36) is referred to as the variation. In accord with (33), it is advantageous to express the restriction of u to S as u ¼ ut þ Un,

ð37Þ

U ¼u  n:

ð38Þ

with ut ¼ P u and The limiting deformation x obtained by setting e to zero in (36) and various quantities associated with x are of primary interest here. In view of (36), the virtual deformation gradient Fe ¼ rxe, relative to the spatial placement P obeys F e ¼ rxe ðxÞ ¼ 1 þ eru þ oðeÞ:

ð39Þ

As a consequence of (39), the virtual volumetric Jacobian Je ¼ det Fe obeys Je ¼ det F e ¼ 1 þ etrðruÞ þ oðeÞ ¼ 1 þ edivu þ oðeÞ:

ð40Þ

Further, since Fe is assumed to be invertible, Fe1 and Fe> are defined and obey F 1 e ¼ 1  eru þ oðeÞ

ð41Þ

> > F e ¼ 1  eðruÞ þ oðeÞ:

ð42Þ

and Since lime!0 Fe ¼ 1, using the expansion (39) of Fe in the definition (35) of the first variation d determines the variation gradient Fe  1 e!0 e eru þ oðeÞ ¼ lim e!0 e ¼ ru:

dF ¼ lim

ð43Þ

20

Aisa Biria et al.

Similarly, using the expansion (40) of dJ in (35) determines the first variation Je  1 e ¼ trðruÞ ¼ div u

dJ ¼ lim e!0

ð44Þ

of volume, while using the expansion (41) and (42) of Fe1 and Fe> in (35) yields
 F 1  1 d F 1 ¼ lim e e!0 e 1  eru þ oðeÞ  1 ¼ lim e!0 e ¼ ru:

ð45Þ

and, completely analogously, dðF > Þ ¼ ðruÞ> :

ð46Þ

Notice that, by (43)–(45), dJ, d(F 1), and d(F >) are formally related to dF by
 dJ ¼ trðdF Þ, d F 1 ¼ dF, and dðF > Þ ¼ ðdF Þ> : ð47Þ

3.1. Areal quantities 3.1.1 Unit normal Under a virtual deformation xe with gradient Fe, the unit normal ne to S e is determined by mapping the unit normal n to S in accord with ne ¼ 

> F e n : >  F e n

ð48Þ

Since, by (41), > > F e n ¼ n  eðruÞ n þ oðeÞ

ð49Þ

> jF  e nj ¼ 1  en  ðruÞn þ oðeÞ,

ð50Þ

and, thus,

Continuum Theory for the Edge of an Open Lipid Bilayer

21

it follows from the relation (2)2 between the three-dimensional and surface gradients and the definition (3) of the perpendicular projector P of S that n  eðruÞ> n þ oðeÞ 1  en  ðruÞn þ oðeÞ ¼ n  ð1  n nÞðruÞ> n þ oðeÞ ¼ n  ½ðruÞP> n þ oðeÞ ¼ n  ðrS uÞ> n þ oðeÞ,

ne ¼

ð51Þ

whereby applying the definition (35) of the first variation to the unit normal n to S yields n n dn ¼ lim e e!0 e ð52Þ ¼ ðrS uÞ> n: To verify that dn as determined by (52) is orthogonal to n, notice that, by the relation (2)2 between the three-dimensional and surface gradients and since Pn ¼ 0, n dn ¼ n ðrS uÞ> n ¼ n  ðrS uÞn ¼ n  ðruÞPn ¼ 0:

ð53Þ

3.1.2 Projector Varying the definition (3) of the perpendicular projector P of S yields its first variation dP in the form dP ¼ dn n  n dn,

ð54Þ

which, on using the expression (52) for the first variation dn of the unit normal n, becomes dP ¼ ðrS uÞ> n n þ n  ðrS uÞ> n:

ð55Þ

3.1.3 Curvature tensor As a consequence of the alternative definition (8) of the curvature tensor L of S in terms of the extension ne of the unit normal n and the expression (55) determining the first variation dP of the perpendicular projector P of S, the first variation dL of L is given by

22

Aisa Biria et al.

dL ¼ dðrne ÞP  ðrne ÞdP   ¼ dðrne ÞP  ðrne Þ ðrS uÞ> n n þ ðn nÞrS u ¼ dðrne ÞP  ðrne ÞðnnÞrS u  LðrS uÞ> n n:

ð56Þ

By the definition (35) of the first variation and the expansion (39) virtual deformation gradient Fe, the factor d(rne) appearing in the first term on the far right-hand side of (56) obeys ðrne Þe  rne e!0
e e 1 rne F e  rne ¼ lim e!0
e  rnee  e rnee ru þ oðeÞ  rne ¼ lim e!0
e  e e r ne  n ¼ lim  ðrne Þru e!0 e  nee  ne  ðrne Þru ¼ r lim e!0 e

dðrne Þ ¼ lim

¼ rðdne Þ  ðrne Þru:

ð57Þ

By the relation (2)2 between the three-dimensional and surface gradients, the definition (3) of the perpendicular projector P of S, the expression (52) for the first variation dn of the unit normal n to S, and (57), dðrne ÞP þ ðrne Þn  ðrS uÞ> n ¼ rS ðdnÞ  ðrne ÞðrS u  ðnnÞrS uÞ   ¼ rS ðrS uÞ> n  ðrne ÞPrS u   ¼ rS ðrS uÞ> n þ LrS u, ð58Þ which, when inserted in (56) yields an alternative representation,  
 dL ¼ rS ðrS uÞ> n  L rS u  ðrS uÞ> n n ,

ð59Þ

for the first variation dL of the curvature tensor L of S. 3.1.4 Mean curvature By the definition (9) of the mean curvature H of S and the expression (59) for the first variation dL of the curvature tensor L of S, the first variation dH of H takes the form

Continuum Theory for the Edge of an Open Lipid Bilayer

1 dH ¼ dðtrLÞ 2 1 ¼ trðdLÞ, 2

23

ð60Þ

which, bearing in mind the properties (7) of L and the expression (59) for its variation dL, yields 
 trðdLÞ ¼ ðrS uÞ> n  L> n  trðLrS uÞ þ tr rS ðrS uÞ> n 
 ¼ trðLrS uÞ þ tr rS ðrS uÞ> n : ð61Þ In view of the decomposition (37), consider a purely normal variation, i.e. take u to be of the form u ¼ Un:

ð62Þ

Then, rS u ¼ UrS n þ n rS U ¼ UL þ n rS U

ð63Þ

and, thus,
 rS ðrS uÞ> n ¼ rS rS U: ð64Þ
 Substituting rS u and rS ðrS uÞ> n from (63) and (64) into (61) then leads to the identity
 trðdLÞ ¼ tr UL2 þ trðrS rS U Þ
 ¼ tr L2 U þ DS U
 ¼ 4H 2  2K U þ DS U,

ð65Þ

which implies that, by (65) and (60), the normal variation dHn of the mean curvature H of S can be expressed as
 1 dHn ¼ 2H 2  K U þ DS U: 2

ð66Þ

Next, consider a purely tangential variation u ¼ ut :

ð67Þ

Since, by (67), u  n ¼ ut  n ¼ 0, the definition (6) of the curvature tensor L of S and the differential identity (14)2 imply that

24

Aisa Biria et al.

ðrS ut Þ> n ¼ Lut ,

ð68Þ

which can be used in (60) to yield an expression, 1 ð69Þ dHt ¼ trðrS ðLut Þ  LrS ut Þ, 2 for the tangential variation dHt of the mean curvature H of S. On the basis of (6), Noll (1987) shows that rS ðLut Þ ¼ ðrS LÞut þ LrS ut :

ð70Þ

Using (70) in (69) gives 1 dHt ¼ trððrS LÞut Þ 2 1 ¼ ½rS ðtrLÞ ut 2 ¼ rS H ut :

ð71Þ

Finally, adding the relations for dHn and dHt provided in (66) and (71) delivers the complete expression,
 1 dH ¼ 2H 2  K U þ DS U þ rS H ut , 2 for the first variation dH of the mean curvature H of S. Lastly, using the identity ðrS uÞ> n ¼ rS U þ Lut ,

ð72Þ

ð73Þ

which follows from (63) and (68), in the representation for dn leads to the useful alternative, dn ¼ rS U  Lut ,

ð74Þ

to the representation (52) for the first variation dn of the unit normal n to S. 3.1.5 Gaussian curvature The definition (6) of the curvature tensor L of S, the definitions (9) and (10) of the mean and Gaussian curvatures H and K of S, and the expression (60) for the first variation dH of H give dK ¼ dI2 ðLÞ ¼ ðtrLÞtrðdLÞ  trðLdLÞ ¼ 4HdH  trðLdLÞ,

ð75Þ

Continuum Theory for the Edge of an Open Lipid Bilayer

25

which, with the expression (59) for the first variation dL of L, leads to
 
 trðLdLÞ ¼ L2 ðrS uÞ> n n  tr L2 rS u þ tr LrS ðrS uÞ> n
 
 ð76Þ ¼ tr L2 rS u þ tr LrS ðrS uÞ> n : As in the treatment of the mean curvature H of S, it is convenient to consider purely normal and tangential variations. For a purely normal variation, substituting the expressions (63) and (64) for rS u and dn into (76) gives
 ð77Þ trðLdLÞ ¼ tr L3 U þ trðLrS rS U Þ: As a consequence of the Cayley–Hamilton theorem (11), it follows that

 tr L3 ¼ I1 ðLÞtr L2  I2 ðLÞtrL
 ð78Þ ¼ 2H 4H 2  3K : Using the expression for tr (L3) from (78) in (77) yields
 tr ðLdLÞ ¼ 2H 4H 2  3K U þ trðLrS rS U Þ,

ð79Þ

while substituting the relations (66) and (79) for dH and tr (LdL) in (75) results in dKn ¼ 2K HU þ 2HDS U  trðLrS rS U Þ:

ð80Þ

Further, defining w ¼ rS U and invoking an argument used by Noll (1987) leads to LrS w ¼ rS ðLwÞ  ðrS LÞw:

ð81Þ

Also, from (81), it transpires that trðLrS wÞ ¼ divS ðLwÞ  ðrS ðtrLÞÞ w ¼ divS ðLwÞ  2ðrS H Þ w ¼ divS ðLwÞ  2divS ðUrS H Þ þ 2UDS H,

ð82Þ

which, when used in (80), gives dKn ¼ 2K HU þ 2HDS U  divS ðLrS U Þ þ 2divS ðUrS H Þ  2UDS H:

ð83Þ

For a purely tangential variation, from the identity (70) in the expression (75) for the first variation dK of the Gaussian curvature K of S, the

26

Aisa Biria et al.

Cayley–Hamilton theorem in the form (11), and the expression for dHt provided in (71), it follows that
 dKt ¼ 4HdHt  tr LrS ðLut Þ  L2 rS ut ¼ 4HdHt  tr½LðrS LÞut  1

  ¼ 4HdHt  tr rS L2 ut 2 1 
 ¼ 4HdHt  rS tr L2  ut 2
 ¼ 4HdHt  rS 2H 2  K  ut ¼ rS K  ut :

ð84Þ

Finally, adding the relations (83) and (84) for dKn and dKt delivers the general form, dK ¼ 2H ðKU þ DS U Þ  divS ðLrS U  2UrS H Þ 2UDS HþrS K ut ,

ð85Þ

of the first variation dK of the Gaussian curvature K of S. 3.1.6 Virtual areal Jacobian The virtual areal Jacobian je of S is related to the virtual volumetric Jacobian Je by   j ¼ J F > n, ð86Þ e

e

e

which, in view of the expansions (40) and (42) of Je and Fe> and recalling the relationship (4) between the three-dimensional divergence and the surface divergence, obeys je ¼ ð1 þ edivu þ oðeÞÞ ð1  en ðruÞn þ oðeÞÞ ¼ 1 þ eðdivu  n  ðruÞnÞ þ oðeÞ ¼ 1 þ eP  ðruÞ þ oðeÞ ¼ 1 þ edivS u þ oðeÞ:

ð87Þ

Using the expansion (87) in definition (35) of the first variation determines the first variation j 1 dj ¼ lim e e!0 e ð88Þ ¼ divS u of the areal Jacobian of S. Finally, on replacing the expression (88) for the variation u with its decomposition (37) into tangential and normal

Continuum Theory for the Edge of an Open Lipid Bilayer

27

components, invoking the differential identity (15), the definition (9) of the mean curvature H, and noticing that rS f is tangential for any scalarvalued superficial field f, the areal variation dj admits a compact representation of the form dj ¼ divS ut  2HU: 3.1.7 Surface integral of a spatial field Consider the integral ð G¼ g S

ð89Þ

ð90Þ

of a generic field g defined on S. To calculate the first variation dG of G, it is necessary to consider variations of both the domain S of integration and of the integrand g. Thus, by (35), the first variation dG of G is ð ! ð 1 dG ¼ lim ge  g , ð91Þ e!0 e Se S which, with a change of variables and the relation (89) determining the first variation dj of the areal Jacobian of S, results in ð ðg j  ge Þ þ ðge  gÞ dG ¼ lim e e e!0 e ðS ¼ ðgdj þ dgÞ ðS ¼ ðdg þ gdivS uÞ ðS ¼ ½dg þ gðdivS ut  2HU Þ: ð92Þ S

3.2. Lineal quantities At the edge C of S, it is convenient to express the tangential component ut of the variation u as a sum, ut ¼ ue e þ un n,

ð93Þ

of components normal and tangent-normal to C. If augmented by the restriction to C of decomposition (37) of u into components normal and tangential to S, (93) yields an identity,

28

Aisa Biria et al.

u ¼ ue e þ Un þ un n,

ð94Þ

which is valuable in deriving the first variations of assorted lineal quantities. Also useful is the result,



 u0 ¼ u0e  kn U þ kg un e þ kn ue þ U 0 þ tg un n

 ð95Þ  kg ue þ tg U  u0n n, of differentiating u as determined by (94) with respect to arclength and using the expressions (19) for determining the arclength derivatives of the Darboux frame {e, n, n}. 3.2.1 Virtual lineal stretch The virtual lineal stretch le of an infinitesimal material element on the edge C of S is given by le ¼ jF e ej:

ð96Þ

The associated first variation le  1 e!0 e

dl ¼ lim

ð97Þ

is needed to determine the first variations of certain useful lineal quantities. For example, much as the first variation dj of the areal Jacobian of S arises in the derivation (92) of the first variation of an integral over S, the first variation dl of the lineal stretch l arises in the derivation of the first variation of an integral over C. Using the expansion (39) of the virtual deformation gradient Fe in (96) yields le ¼ 1 þ ee  ðruÞe þ oðeÞ,

ð98Þ

which, with (97), leads to the relation le  1 e ¼ e ðruÞe:

dl ¼ lim e!0

ð99Þ

In view of (43), it follows that dl and the restriction to C of the variation gradient dF ¼ ru are linked formally by dl ¼ e  ðdF Þe,

ð100Þ

Continuum Theory for the Edge of an Open Lipid Bilayer

29

from which it follows that dl is the restriction to C of the tangential component of dF. Notice that ðruÞe ¼ u0 ,

ð101Þ

and, thus, that (ru)e is the directional derivative of u along C. As a consequence of (99), the first variation dl of the lineal stretch l can also be expressed as dl ¼ u0 e:

ð102Þ

3.2.2 Unit tangent The first variation de of the unit tangent e to C arises in determining the first variations of certain lineal quantities, the geodesic torsion tg being a prominent example. Toward determining a useful representation of de, consider the unit tangent Fe ee ¼ e , ð103Þ jF e ej to the image Ce under the virtual deformation. By the expansions (39) and (98) of the virtual deformation gradient Fe and the virtual lineal stretch le ¼ jFeej, ee ¼

e þ eðruÞe þ oðeÞ 1 þ e e ðruÞe þ oðeÞ

¼ e þ eð1  e eÞ ðruÞe þ oðeÞ ¼ e þ eð1  e eÞu0 þ oðeÞ ¼ e þ eðu0  ðu0  eÞeÞ þ oðeÞ,

ð104Þ

whereby applying the definition (35) to the unit tangent e to C yields its first variation e e de ¼ lim e e!0 e ð105Þ ¼ u0  ðu0  eÞe: Consistent with the status of e as a unit vector, (105) shows that de is orthogonal to e. 3.2.3 Tangent-normal vector The orthonormality of the Darboux frame {e, n, n} ensures that the unit tangent-normal n and its first variation dn are orthogonal and, thus, that dn can be expressed in the form

30

Aisa Biria et al.

dn ¼ ðde  nÞe  ðdn  nÞn:

ð106Þ

On using the expressions (105) and (74) for the first variations de and dn of the unit tangent e to C and the restriction to C of the unit normal n, (106) delivers dn ¼ ðu0  nÞe þ ½ðrS U þ Lut Þ  nn:

ð107Þ

3.2.4 Arclength derivative of a generic quantity To avoid repeated calculations, it is useful to derive the first variation d(x0 ) of the arclength derivative x0 of a generic quantity x defined on C. Based on the definition (35) of the first variation and the expression (102) for the first variation dl of the lineal stretch l, it follows that  1 @xe @x 0 dðx Þ ¼ lim  e!0 e @se @s  1 @xe @s @x  ¼ lim e!0 e @s @se @s  1 @xe @s @xe @xe @x þ   ¼ lim e!0 e @s @se @s @s @s    @xe 1 @s @ xe  x 1 þ lim ¼ lim lim e!0 @s e!0 e @se @s e!0 e @x
 @ ðdxÞ ¼ d l1 þ @s @s ð Þ @ dx ¼ ðdlÞx0 þ @s ð108Þ ¼ ðdxÞ0  ðu0  eÞx0 : As an illustrative application of (108), the first variation d(n0 ) of the arclength derivative of the restriction of the unit normal n to C has the form dðn0 Þ ¼ ðdnÞ0  ðu0 eÞn0 ,

ð109Þ

which, on inserting the relations (19)2 and (74) for n0 and dn, becomes dðn0 Þ ¼ ðrS U þ Lut Þ0 þ ðu0 eÞðkn e þ tg nÞ:

ð110Þ

3.2.5 Curvature vector From the representation (24) of the curvature vector k of C and the general identity (108), it follows that its first variation dk takes the form

Continuum Theory for the Edge of an Open Lipid Bilayer

31

0

ð111Þ

dk ¼ ðu0  ðu0  eÞeÞ  ðu0  eÞk:

3.2.6 Normal curvature The representation (27) for the normal curvature kn of C, the identity (52) determining the first variation dn of the unit normal n to S, and the expression (111) for the first variation dk of the curvature vector k of C lead to
 dkn ¼ ðu0  ðu0 eÞeÞ n  ðu0  eÞkn  k ðrS uÞ> n , ð112Þ which, on using the identity (73), becomes 0

dkn ¼ ðu0  ðu0  eÞeÞ  n  ðu0  eÞkn þ kg ðrS U  nÞ þ kg ut  Ln:

ð113Þ

A useful alternative,



 dkn ¼ k0n ue þ k2n  t2g U þ U 00 þ t0g  kg kn þ kg ð2H  kn Þ un þ 2tg u0n þ kg ðrS U  nÞ,

ð114Þ

to the representation (113) for the first variation dkn of the normal curvature kn of C arises on using the decompositions (93) and (95) of ut and u0 and the expressions for n0 and n 0 provided in (19), the relations (28) and (29) connecting the scalar products e  Ln and n  Ln to the geodesic torsion tg and the tangent-normal curvature kn of C, and the representation (30) for the mean curvature H on C in terms of kn and kn of C in (113). 3.2.7 Geodesic curvature With reference to the definition (24) of the curvature vector k of C, the geodesic curvature kg of C is given by kg ¼ k  n:

ð115Þ

On computing the first variation of (115), invoking once again the definition (24), and taking into consideration that, since n  n ¼ 0 and jnj ¼ 1, n  dn ¼ n  dn and n  dn ¼ 0, it follows that dkg ¼ dk n  k  dn

 1 2 ¼ dk n  kn dn n þ kg d jnj 2

¼ ðkn dn  dkÞ  n:

ð116Þ

32

Aisa Biria et al.

Toward obtaining a representation for dkg that incorporates the variation u, using the expressions (74) and (111) for the first variations dn and dk of n and k in (116) yields 0

dkg ¼ kn rS U  n  kn ðLut Þ  n  ðu0  ðu0  eÞeÞ  n  kg ðu0 eÞ:

ð117Þ

A useful alternative,



 dkg ¼ k0g ue þ t0g þ kg kn U þ 2tg U 0  K þ k2g un  u00n  kn rS U  n,

ð118Þ

to the representation (117) for the first variation dkg of the geodesic curvature kg of C arises on using the decompositions in (93) and (95) of ut and u0 , the expressions for the arclength derivatives n0 and v 0 of the normal n and tangent-normal n to C provided in (19), the identities (28) and (29) determining the geodesic torsion tg and tangent-normal curvature kn of C, the relation (30) giving the restriction to C of the mean curvature H of S in terms of the normal curvature kn and kn of C, and the identity (32) giving the restriction to C of the Gaussian curvature K of S in terms of the restriction of H to C and the normal curvature kn and geodesic torsion tg of C. 3.2.8 Geodesic torsion As with the first variation dkg of the geodesic curvature kg, it is desirable to have an expression for the first variation dtg of the geodesic torsion tg involving the variation u. According to the definition (28) of the geodesic torsion tg, its variation dtg can be expressed as dtg ¼ dn  n0  n  dðn0 Þ:

ð119Þ

Using the expressions (19)2 and (110) for the arclength derivative n0 of the restriction to C of the unit normal n to S and its variation d(n0 ) along with the expression (107) for the first variation dn of the tangent-normal n to C, (119) yields dtg ¼ ðu0  nÞkn þ ðrS U þ Lut Þ0  n  ðu0  eÞtg ,

ð120Þ

which, with the representations (93) and (95) for the tangential component ut and arclength derivative u0 of the variation u, becomes dtg ¼ kn kg ue  tg u0e þ 2kn tg U  kg tg un  kn u0n þðrS U þ ue Le þ un LnÞ0  n:

ð121Þ

33

Continuum Theory for the Edge of an Open Lipid Bilayer

3.2.9 Curve integral of a spatial field Consider the integral ð X¼ x

ð122Þ

C

of a generic field x defined on the edge C of S. Analogous to the treatment of the surface integral (90) of a generic field g defined on S, calculating the first variation dX of X necessitates considering variations of the domain C of integration and the integrand x. Thus, by (35), the first variation dX of X is 1 dX ¼ lim e!0 e

ð Ce

ð ! xe  x , C

ð123Þ

which, with a change of variables and the expression (102) for the first variation dl of the lineal stretch l can be rewritten as ð dX ¼

lim

C e!0

ð ¼

C

ð ¼

C

ðxe le  xe Þ þ ðxe  xÞ e

ðdx þ xdlÞ ðdx þ xu0 eÞ:

ð124Þ

3.3. Volume Taking the trace of (16) yields 1 V¼ 3

ð @R

r n:

ð125Þ

Since (125) involves a surface integral, (92) can be applied to its righthand side, yielding ð 1 dV ¼ ðdðr nÞ þ ðr nÞdivS uÞ 3 @R ð 1 ¼ ðU þ r dn þ ðr nÞdivS uÞ: 3 @R

ð126Þ

34

Aisa Biria et al.

Substituting the expression (52) for the first variation of dn in (126) and making use of the identities (14) and (15) yields ð
 1 dV ¼ U  ðrS uÞ> n r þ ðr  nÞdivS u 3 @R ð

 1 ¼ U  ðrS ðu nÞÞ  r þ ðrS nÞ> u r þ divS ððr nÞuÞ 3 @R ðrS ðr  nÞÞ  uÞ ð

 1 U  ðrS ðu nÞÞ  r þ ðrS nÞ> u  r ¼ 3 @R

  þ divS ððr nÞuÞ  ðrS rÞ> nu  ðrS nÞ> r u : ð127Þ In view of the identity P n ¼ 0, the identity (13)1 and the definition (6) and symmetry property (7)2 of the curvature tensor L, (127) simplifies to ð 1 dV ¼ ½U  ðrS ðu  nÞÞ r þ divS ððr  nÞuÞ: ð128Þ 3 @R Further, using the differential identity (15), (128) becomes ð 1 dV ¼ ½U þ U divS r  divS ððu  nÞrÞ þ divS ððr  nÞuÞ, 3 @R which, with the identity (13)2, leads to

ð 1 dV ¼ U þ divS ððr nÞu  ðu nÞrÞ : 3 @R

ð129Þ

ð130Þ

As it is easily verified that (r  n)u  (u  n)r is tangent to C, applying the surface-divergence theorem (17) with A ¼ @R to (130), while bearing in mind that @R is a closed surface, simplifies the variation of the volume of R to ð dV ¼ U: ð131Þ @R

4. VARIATIONAL DERIVATION OF THE EQUILIBRIUM EQUATIONS OF A LIPID VESICLE In the present section, the equilibrium equations for a lipid vesicle— or, more descriptively, a closed lipid bilayer—are derived. This is achieved

35

Continuum Theory for the Edge of an Open Lipid Bilayer

by adopting the approach of Steigmann, Baesu, Rudd, Belak, & McElfresh (2003) but with two important differences. First, convected coordinates are not used. Second, allowance is made for the presence of forces and couples distributed over the surface of the lipid vesicle. Consider a lipid vesicle represented by a closed surface S and endowed with a bending-energy density
ðH, K Þ c¼c

ð132Þ

measured per unit area of S. Since S is closed in the present setting, @S ¼ ∅:

ð133Þ

A constitutive relation of the form (132) complies not only with the principle of material frame indifference but also embodies the material symmetry requirements consistent with the in-plane fluidity and isotropy exhibited by lipid bilayers (Steigmann, 1999). Thus, the net free-energy E of a vesicle represented by a surface S is ð E ¼ c: ð134Þ S

Any shape change of a lipid bilayer is dominated by bending. Such behavior arises from the extremely large resistance to areal stretching in comparison to bending. Consistent with this observation, it is conventional to adopt the assumption of local area preservation (Steigmann, Baesu, Rudd, Belak, & McElfresh, 2003). Moreover, it is reasonable to assume that the volume of the fluid enclosed by vesicle remains fixed. These constraints are incorporated in the subsequent formulation by introducing an unknown areal Lagrange multiplier field s on S and an unknown pressure p. Consider a distributed force f and a distributed couple m defined on S. According to the principle of virtual work, the first variation dE of the net free-energy E is equal to the virtual work performed by external agencies. Thus, dE ¼ W S þ pdV ,

ð135Þ

where dV, as given by (131), is the first variation of the volume V enclosed by the lipid vesicle and ð ð W S ¼ ðf  u þ m vÞ  sdj ð136Þ S

S

36

Aisa Biria et al.

is the Ðwork performed by the areal external loads, augmented by a term  S sdj that penalizes changes in the area of S. In (136), v denotes the virtual angular-velocity, which is related to the unit normal n to S and its variation dn via v ¼ n  dn:

ð137Þ

An immediate consequence of (137) is that v obeys v  n ¼ 0 and, thus, is tangent to S; moreover, (137) implies that v  n ¼ ðn  dnÞ  n ¼ ðnÞ ðn  dnÞ ¼ ðnÞ ðnÞdn ¼ Pdn,

ð138Þ

which, since n  dn ¼ 0, delivers the useful identity dn ¼ v  n:

ð139Þ

Substituting (137) in (136) and rearranging the scalar triple-product yields ð ð W S ¼ ðf  uþðm  nÞ  dnÞ  sdj: ð140Þ S

S

4.1. Variation of the net free-energy In view of the identity (92), the first variation dE of the net free-energy E of the vesicle identified with the closed surface S is given by ð dE ¼ ½dc þ cðdivS ut  2HU Þ: ð141Þ S

Applying the chain rule to the constitutive relation (132) determining the areal free-energy density c in terms of the mean and Gaussian curvatures H and K of S leads to the relation
dK,
dH þ c dc ¼ c H K

ð142Þ


and c
denote the partial derivatives of c
with respect to H and where c H K K, respectively. In (142) and hereafter explicit dependence on (H, K) is suppressed for brevity. Substituting the expressions (72) and (85) for the first variations dH and dK of H and K in (142) and subsequently in the expression (141) for dE results in

37

Continuum Theory for the Edge of an Open Lipid Bilayer

ð dE ¼

S




2H 2  K U þ cH D U þ c
r H  u þ 2c
HKU c H H S t K 2 S


div ðLr U Þ þ 2c
div ðUr H Þ
HD U  c þ2c K S K S S K S S


r K u þ cðdiv u  2HU Þ :
UD H þ c  2c K S K S t S t

ð143Þ

On invoking the differential identity (15), (143) expands further to ð 


2H 2  K þ 1 D c


c dE ¼ H S H þ 2cK KH þ 2DS cK H 2 S




 2ðr H Þ  r c
 2c
D H  2Hc U divS LrS c K S S K K S  
r K r c u
r H þc þ c H S K S S t 
 1
1


þdivS c H rS U  UrS cH þ 2cK HrS U  2UrS cK H 2 2 


: ð144Þ cK LrS U þ ULrS cK þ 2cK UrS H þ cut On using the result
r H þc
rK rS c ¼ c H S K S

ð145Þ

of applying the surface gradient of the constitutive relation (132) in (144) and invoking the surface-divergence theorem (17), the first variation dE of the net free-energy E of S simplifies to ð


2H 2  K þ 1 D c


c dE ¼ H S H þ 2cK HK þ 2DS cK H 2

S




divS LrS cK  2ðrS H Þ  rS cK  2cK DS H  2Hc U: ð146Þ

4.2. Virtual volumetric work In view of the identity (131) and the constancy of p, the virtual volumetric work pdV in (135) is simply ð pdV ¼ p U ð S ¼ pU: ð147Þ S

38

Aisa Biria et al.

4.3. Virtual work of the areal loads Substituting the decomposition (37) of the variation u into components normal and tangential to S and the representations (74) and (89) for the first variation dn of the unit normal to S and the first variation dj of the areal Jacobian of S in the expression (158) for W S leads to ð W S ¼ ½f n U þ f t  ut  ðm  nÞ  ðrS U þ Lut Þ  sðdivS ut  2HU Þ, S

ð148Þ where f n ¼ f n

and

ft ¼Pf

ð149Þ

denote the normal and tangential components of f. Applying the differential identity (15) to (148) yields ð W S ¼ ½ðf n þ divS ðm  nÞ þ 2sH ÞU S

þðf t  Lðm  nÞ þ rS sÞ ut  divS ðU ðm  nÞ þ sut Þ,

ð150Þ

which, in view of the surface-divergence theorem (17) with A ¼ S and the assumption that S is closed, reduces to ð W S ¼ ½ðf n þ divS ðm  nÞ þ 2sH ÞU þ ðf t  Lðm  nÞ þ rS sÞ  ut : S

ð151Þ

4.4. Combined results When augmented by the expressions (146), (147), and (151) for the first variation dE of the net free-energy E of S, the virtual volumetric work p dV performed on S, and the virtual work W S performed on S by the areal loads, the work identity (135) becomes ð 


2H 2  K þ 1 D c


c H S H þ 2cK HK þ 2DS cK H 2 S



 2ðr H Þ  r c
divS LrS c K S S K  2cK DS H  2H ðc þ sÞ 

p  f n  divS ðm  nÞ U þ ðLðm  nÞ  f t  rS sÞ  ut ¼ 0: ð152Þ Since the normal and tangential components U and ut of the variation u may be chosen independently, applying the fundamental lemma of the calculus of variations to (152) yields equilibrium equations,

39

Continuum Theory for the Edge of an Open Lipid Bilayer






ð2H 2  K Þ þ 1 D c

c H S H þ 2cK HK þ 2DS cK H  divS LrS cK 2

D H  2H ðc þ sÞ
 2c 2ðrS H Þ  rS c K K S ¼ p þ f n þ divS ðm  nÞ

ð153Þ

rS s ¼ Lðm  nÞ  f t ,

ð154Þ

and

on S. Equation (153) can be viewed as the generalized shape equation governing the local geometric configuration of the lipid vesicle. A noteworthy distinction between (153) and the corresponding equilibrium equation of Steigmann, Baesu, Rudd, Belak, & McElfresh (2003) is the presence of the term f n þ divS (m  n), which accounts for the influence of the distributed force f and the distributed couple m on S. Equation (154) delivers a partial-differential equation for the unknown Lagrange multiplier field s defined on S. In the absence of external loads, s must satisfy rS s ¼ 0 and thus, as Steigmann, Baesu, Rudd, Belak, & McElfresh (2003) observe, be uniform on S.

5. VARIATIONAL DERIVATION OF THE EQUILIBRIUM EQUATIONS OF AN OPEN LIPID BILAYER WITH EDGE ENERGY In the present section, the equilibrium equations for an open lipid bilayer identified with a surface S with edge C ¼ @S (Fig. 1.3) are derived. To account for the excess energy associated with the differences between the organization and packing of phospholipid molecules at the edge of an open lipid bilayer, the edge is endowed with its own energetic structure. The net free-energy E of an open lipid bilayer therefore consists of the sum, E ¼ ES þ EC,

ð155Þ

of an areal contribution E S , as given by (134), and a lineal (or edge) contribution E C ð E C ¼ f, ð156Þ C

in which f is the edge-energy density, measured per unit arclength of C. Two classes of edge-energy densities are considered. In Section 5.1, f is chosen to be a fixed constant. In Section 5.2, f is allowed to depend on

40

Aisa Biria et al.

the geometry of the edge through the normal curvature kn, geodesic curvature kg, and the geodesic torsion tg. In addition to forces and couples distributed over S of the kind considered in Section 4, allowance is made for the presence of a distributed force i and a distributed couple m on the edge C. For an open lipid bilayer endowed with edge energy E C , the principle of virtual work requires that the first variation dE ¼ dE S þ dE C of the net freeenergy obeys dE ¼ W S þ W Cs ,

ð157Þ

where ð WS ¼

S

ð ðf  u þ ðm  nÞ  dnÞ 

sdj, S

ð158Þ

is the virtual work of the areal distributed force f and distributed couple m on the surface S, augmented by a term that embodies the constraint of areal inextensibility, and W Cs is the virtual work of the distributed force i and distributed couple m exerted by the suspending solution on the edge C. The virtual volumetric work p dV defined in (147) is not included on the right-hand side of (157) because, in contrast to the situation considered in Section 4, where S represents a lipid vesicle and, thus, is closed and therefore without boundary, in the present context S is a surface with boundary.

5.1. Constant edge-energy density In the present section, the equilibrium equations are derived for an open lipid bilayer endowed with a constant edge-energy density f ¼ fo ¼ constant,

ð159Þ

in which case the edge free-energy E C , given in (156), simplifies to ð ð E C ¼ fo ¼ fo : ð160Þ C

C

5.1.1 Variation of the net free-energy of the surface Proceeding as in the derivation (144) of the first variation dE of the net freeenergy E while invoking the surface-divergence theorem (17) with A ¼ S and the expression (145) for rS c yields

41

Continuum Theory for the Edge of an Open Lipid Bilayer

ð

dE S ¼




2H 2  K þ 1 D c


c H S H þ 2cK HK þ 2DS cK H 2 S




D H  2Hc U
 2ðr H Þ  r c
 2c div Lr c S

ð

S

K

S

S

K

K

S


 1
1
þ 2c
H
Hr U  2Ur c cH rS U  UrS c H K S S K 2 C 2




ð161Þ cK LrS U þ ULrS cK þ 2cK UrS H þ cut  n: þ



Lr U  n in (161) can be rewritten as Notice that the term  c K S

Lr U  n ¼ c
½Lððr U nÞn þ U 0 eÞ  n  c K S K S
ðn  LeÞU 0
¼ cK ðn  LnÞðrS U  nÞ  c K ð162Þ
ð2H  k Þðr U  nÞ  c
t U0 ¼c K n S K g 




ð2H  k Þðr U nÞ  c
t U 0þ c
t 0 U: ¼c K n S K g K g Substituting (162) in (161) and reorganizing the terms leads to ð


2H 2  K þ 1 D c


c dE S ¼ H S H þ 2cK HK þ 2DS cK H 2 S






 divS LrS cK  2ðrS H Þ  rS cK  2cK DS H  2Hc U ð 
 1


 rS c þ H n  2rS cK H  n þ ðLnÞ  rS cK 2 C

0

þ 2cK rS H  n þ cK tg U þ cuv

 1

þ cH þ cK kn rS U  n : ð163Þ 2 5.1.2 Virtual work of the areal loads The virtual work W S performed by the areal external loads exerted on S is ð W S ¼ ½ðf n þ divS ðm  nÞ þ 2sH ÞU þ ðf t  Lðm  nÞ þ rS sÞ ut  S ð  ðU ðm  nÞ þ sut Þ  n, ð164Þ C

which is identical to the expression (150) arising in the treatment of vesicles. However, since S is not closed in the present context, applying the surface-

42

Aisa Biria et al.

divergence theorem (17) to (164) does not lead to the simplified expression (151) relevant to vesicles. 5.1.3 Variation of the net free-energy of the edge In view of the identity (124) and the assumption (159) that the edge-energy density is constant, the first variation dE C of the net free-energy E C of the edge C, as given in (160), takes the form ð dE C ¼ fo u0 e: ð165Þ C

Using the representation (95) for the arclength derivative of the virtual velocity u0 in (165), while bearing in mind that C is closed, results in a simple relation, ð dE C ¼ fo ðkn U  kg un Þ, ð166Þ C

which exposes the respective connections between the geodesic and normal curvatures kn and kg and the normal and tangent-normal components U and un of the restriction of the virtual velocity u to the edge C of the bilayer. 5.1.4 Virtual work of the lineal loads Consider the distributed force i and the distributed couple m, both measured per unit arclength, exerted by the surrounding solution on the edge C. The virtual work W Cs performed by these external agencies takes the form ð s W C ¼ ði  u þ mvC Þ, ð167Þ C

where, in accord with (34), i and m admit the following representations in terms of components relative to the Darboux frame {e, n, n}: i ¼ ie e þ in n þ in n,

m ¼ me e þ mn n þ mn n:

ð168Þ

In addition, vC is the virtual angular-velocity defined such that for any element l of the Darboux frame for C, dl ¼ vC  l:

ð169Þ

As a first step toward determining a more explicit representation for vC , substitute the decomposition (94) of the first variation u in the expressions (105), (52), and (107) for de, dn, and dn, invoke the expressions (19) determining the arclength derivatives of the elements of

Continuum Theory for the Edge of an Open Lipid Bilayer

43

the Darboux frame, and use the relation (30) between the mean curvature H on C and the normal and tangent-normal curvatures kn and kn along with the relation (73) determining ðrS uÞ> n to give



 9 > de ¼ kn ue þ U 0 þ tg un n þ kg ue  tg U þ u0n n, > > =



 > 0 dn ¼  kn ue þ U þ tg un e  tg ue þ rS U n þ ð2H  kn Þun n, > >



 > > 0 dn ¼ kg ue þ tg U  un e þ tg ue þ rS U n þ ð2H  kn Þun n: ; ð170Þ In view of (170) and (169), vC can be expressed in the form



 vC ¼  tg ue þ rS U n þ ð2H  kn Þun e þ kg ue þ tg U  u0n n

 ð171Þ þ kn ue þ U 0 þtg un n: Substituting the representations (94), (168), and (171) for the virtual velocity u, the force i and couple m, and the virtual angular-velocity vC into the expression (167) for the virtual work W Cs performed by i and m yields ð h



 s WC ¼ ie  me tg þ mn kg þ mn kn ue þ in þ mn tg  m0n U C

 i þ in  me ð2H  kn Þ þ m0n þ mn tg un  me rS U  n : ð172Þ 5.1.5 The equilibrium equations Combining the expressions for dE S , dE C , W S , and W sC from (163), (164), (166), and (172) with (157) leads to the requirement ð 



2H 2  K þ 1 D c



c H S H þ 2cK HK þ 2DS cK H  divS LrS cK 2 S




2ðrS H Þ rS cK  2cK DS H  2H ðc þ sÞ  f n  divS ðm  nÞ U  ð  þðLðm  nÞ  f t  rS sÞ ut  ðie  me tg þ mn kg þ mn kn Þue C


 1
0



þ rS c H  n þ 2rS cK H n  ðLnÞ rS cK  2cK rS H  n  ðcK tg Þ 2

0 me þ fo kn þ in þ mn tg  mn U  ðc þ s þ fo kg  in þ me ð2H  kn Þ m0n  mn tg





 1

un  cH þ cK kn þ me rS U n ¼ 0: 2

ð173Þ

44

Aisa Biria et al.

The Euler–Lagrange equations on S are identical to those obtained for a vesicle in Section 4.4, except that the pressure p is absent on the right-hand side of (153). On the edge C, the Euler–Lagrange equations are ie  me tg þ mn kg þ mn kn ¼ 0, ð174Þ 

 0
 1r c


LrS c K S H  2HrS cK n þ cK tg þ mn þ m e 2 ð175Þ fo kn  in  mn tg ¼ 0, c þ s þ fo kg  in þ me ð2H  kn Þ  m0n  mn tg ¼ 0, 1

k þ m ¼ 0: c þc K n e 2 H

ð176Þ ð177Þ

The equilibrium equations (174)–(177) generalize results of Capovilla, Guven, & Santiago (2002), Tu & Ou-Yang (2003, 2004), and Yin, Yin, & Ni (2005) to include effects of loads exerted by the suspending solution on the surface and edge of the vesicle. The physical interpretation of (174), (175), (176), and (177) is facilitated by identifying their respective kinematical power conjugates, namely ue, U, un, and rS U  n. Equation (174) describes the in-plane equilibrium of the edge C in the tangential direction. In the absence of the external force ie and couple m, (174) is trivially satisfied. Also, (174) reveals the coupling between ie and components of m. Interestingly, (174) does not include any terms involving quantities associated with the surface S. This is compatible with the in-plane fluidity of lipid bilayers in the liquid phase and the concomitant inability to support in-plane shear stress. The equilibrium of the edge C in the normal n direction is imposed by (175). Among the conditions (174)– (177), only (175) incorporates the possible influence of areal loads on edge equilibrium; that influence occurs only through the tangential component m  e of the distributed couple m. Equation (176) describes the in-plane equilibrium of the edge C in the tangent-normal direction n. This condition can be viewed as a lineal counterpart of the areal shape equation (153). Lastly, (177) imposes the equilibrium of bending moments at the edge C.

5.2. Geometry-dependent edge-energy density Consider a nonuniform edge-energy density f which incorporates dependence on the geometry of the edge C and surface S at the edge through the normal and geodesic curvatures kn and kg and the geodesic torsion tg, so that

Continuum Theory for the Edge of an Open Lipid Bilayer


, k , t Þ, f ¼ fo þ fðk n g g

45

ð178Þ

where, as in the particular case treated in Section 5.1, fo is constant. Employing arguments analogous to those utilized by Seguin & Fried (2013) to derive the Canham–Helfrich free-energy density, Asgari,
HowMaleki, Biria, & Fried (2013) obtain particular expressions for f. ever, recourse to such particular expressions is not made here. To avoid unnecessary repetition, this section is focused on the effect of the geometry-dependent net free-energy E C of the edge C and the provision of the associated equilibrium equations. Consistent with (156) and (178), the net free-energy E C of the edge C is ð


,k,tÞ : EC ¼ fo þfðk ð179Þ n g g C

In view of the identity (124), the first variation dE C of E C takes the form ð

þ fu
0 e fo u0  e þ df dE C ¼ C ð

þ fu0  e : df ð180Þ ¼ C

Since, by the chain rule,
¼f
dk þ f
dk þ f
dt , df kn n kg g tg g Equation (180) can be expressed as ð


dk þ f
dk þ f
dt þ fu0  e , f dE C ¼ kn n kg g tg g C

ð181Þ

ð182Þ


, and f
indicate, respectively, the partial derivatives of f
with
,f where f kn kg tg 0 respect to kn, kg, and tg. Substituting u , dkn, dkg, and dtg from (95), (114), (118), and (121) into (182) and integrating by parts yields ð 






k2  t2 þ f
k k  t0 þf
2k t þ k0  fk f dE C ¼ kn kg n g tg n g n n g g g C





0 t þ f
t0  k k þ k 2H  k
00 þ f
0 k U þ f  2f kn g n g n g kg g kn tg g




K þ k2 þ f
k0  2k t þ fk þ 2f
0 ðk  H Þ f kg tg g g g g n tg n



 0 00 0




2fkn tg  fkg un þ fkn kg  fkg kn  ftg rS U  n , ð183Þ

46

Aisa Biria et al.

Using dE C from (183) in (157) yields the areal equilibrium equations identical to those presented in Section 5.1 along with generalizations, ie  me tg þ mn kg þ mn kn ¼ 0, 

0 1



t þ m þ m e LrS cK  rS cH  2HrS cK  n þ c K g n 2






k k  t0 þ f
2k t þ k0
k2  t 2 þ f þf kn kg n g tg n g n g g g
0 t þ f
00 þ f
0 k  i  m t ¼ 0,  fkn  2f n n g kg g kn tg g




t0  k k þ k ð2H  k Þ  f
K þ k2 cþsþf kn g n g n kg g g

 0 0
ðk  H Þ  2f
t f
00
k0  2k t þ fk þ 2f þf tg g g g n tg n kn g kg  in þ me ð2H  kn Þ  m0n  mn tg ¼ 0, 1

k þm þf
k f
k  f
0 ¼ 0: c þc K n e kn g kg n tg 2 H

ð184Þ

ð185Þ

ð186Þ ð187Þ

of the lineal equilibrium equations (174)–(177) that take into account dependence of the edge-energy density f on the geometric variables kn, kg, and tg. Analogous to the observation concerning (174), the generalization (184) of that condition to account for energetic dependence of the edge-energy density f on kn, kg, and tg is satisfied trivially in the absence of the external loads on the surface S and the edge C of the bilayer. To verify the consistency of (185)–(187), consider the special case in which no external loads are present and suppose that the areal-energy density c is in Canham–Helrfich form 1
K, c ¼ mðH  Ho Þ2 þ m 2

ð188Þ

with Ho being the spontaneous mean curvature, and that the edge-energy density f is of the particular form  1

ð189Þ f ¼ kb k2n þ k2g þ fo , 2 introduced by Tu & Ou-Yang (2004), with kb being a constant. With these choices, it is easily checked that (185)–(187) reduce to 

0 1 1 2 0 00 2 0
tg þ kb kn þ kn k  tg  tg kg  tg kg  mrS H n þ m 2 2 fo kn ¼ 0,

ð190Þ

Continuum Theory for the Edge of an Open Lipid Bilayer

 1 1 2 2 00 2
K þ s  kb kg þ kg k  tg mðH  Ho Þ þ m 2 2

0 0 þtg kn þ tg kn þ fo kg ¼ 0, 1
kn ¼ 0, mðH  Ho Þ þ m 2

47

ð191Þ ð192Þ

which coincide with the equations previously obtained by Tu & Ou-Yang (2004), with the caveat that the terms involving n and tg have opposite signs due to different defining conventions for n and the geodesic torsion tg in the present work.

6. FORCE AND BENDING MOMENT EXERTED BY AN OPEN LIPID BILAYER ON ITS EDGE In the present section, expressions for the force and bending moment that an open lipid bilayer exerts on its edge are obtained. These expressions are of prominent importance in a subsequent derivation, appearing in Section 7, of the equilibrium equations and constitutive relations for the edge C. Recall from (1) that the surface S representing the bilayer is assumed to be open, and thus, that S and the closed space curve C, which represents the edge of the bilayer, are disjoint. Figure 1.6 illustrates the interaction of the lipid bilayer surface S and the edge C as well as the arrangement of phospholipid molecules at the edge C or on the surface S in the vicinity of the edge C.

Figure 1.6 Schematic demonstration of the interaction of the lipid bilayer surface S and the edge C. Also the arrangement of phospholipid molecules at the edge C or on the surface S in the vicinity of the edge C is depicted.

48

Aisa Biria et al.

Let the force and bending moment, per unit arclength, exerted by the edge C on S be denoted by f e and me, respectively. According to Newton’s third law, the action of S on C is equal in magnitude but oppositely directed to the reaction exerted by C on S. The principal of virtual work for the surface S of the lipid bilayer reads dE S ¼ W S þ W Se ,

ð193Þ

where dE S is the first variation of net free-energy of S given in (163), W S , as presented in (164), is the virtual work done by external areal loads f and m augmented by the areal inextensibility term, and W eS , given by ð W Se ¼

C

ð f e u þ me  vC Þ,

ð194Þ

is the virtual work performed on S by f e and me. Due to different arrangements of phospholipid molecules at the edge of a lipid bilayer and in its interior region (Figs. 1.2 and 1.6), the mechanical responses of material points located on the edge and in the interior surface of a lipid bilayer are generally dissimilar. In view of in-plane fluidity of lipid bilayer in the interior region of the bilayer (i.e. within S), the components of f e and me in the n and n directions must respectively vanish, in which case f e ¼ fn n þ fn n

and

me ¼ me e þ mn n:

ð195Þ

This argument is in accord with the observation that a lipid bilayer in the fluid phase cannot support in-plane shear stress. However, it is difficult to motivate the application of analogous requirements to the force i and couple m exerted on C by the surrounding fluid; hence, these objects are allowed to take the general forms appearing in (168). In view of the decomposition (94) of the variation u, the expression (171) for the angular velocity vC , and the representations (195)1 and (195)2 for f e and me, the virtual work (194) can be expressed as W Se

¼

ð h



 mn kn  me tg ue þ fn  m0n U C

 i þ fn  me ð2H  kn Þ þ mn tg un  me rS U n :

ð196Þ

49

Continuum Theory for the Edge of an Open Lipid Bilayer

6.1. Force and bending moment expressions Using the relations (163), (164), and (196) for the first variation dE S of the net free-energy E S of S, the virtual work W S performed by the areal loads on S, and the virtual work W Se performed by the edge loads on S in the work identity (193) for S yields ð h






2H 2  K þ 1 D c
c H S H þ 2cK HK þ 2DS cK H  divS LrS cK 2 S i

 2c
D H  2H ðc þ sÞ  f  div ðm  nÞ U  2ðrS H Þ  rS c K K S n S  ð  
þðLðm  nÞ  f t  rS sÞ ut þ me tg  mn kn ue C


 1



 rS c H  n þ 2rS cK H n  ðLnÞ rS cK  2cK rS H n 2

0

 0
 cK tg  m e þ fn  mn U þ c þ s  fn þ me ð2H  kn Þ  mv tg un   1

k þ m r U v ¼ 0: þ cH þ c K n e S 2

ð197Þ

The Euler–Lagrange equations on S are identical to those presented in (153) and (154), except that the pressure p on the right-hand side of (154) is absent. Since ue, U, un, and rS U  n are independent, the following equilibrium equations must hold on C: 

me tg  mn kn ¼ 0,

0 1



t þm LrS cK  rS cH  2HrS cK n þ c K g n 2 þm e  fn ¼ 0, c þ s  fn þ me ð2H  kn Þ  mn tg ¼ 0, 1

k þ m ¼ 0: c þc K n e 2 H

The foregoing equations may be solved to yield expressions,   1
1




0, fn ¼ LrS cK  rS cH  2HrS cK  v þ me  tg k1 c H n 2 2  1

k , c þc fv ¼ c þ s  Kk1 K n n 2 H

ð198Þ

ð199Þ ð200Þ ð201Þ

ð202Þ ð203Þ

50

Aisa Biria et al.



1

me ¼  cH þ cK kn , 2  1

k , þ c c mn ¼ tg k1 K n n 2 H

ð204Þ ð205Þ

for fn, fn, me, and mn. The force and bending moment that S exerts on C are  ( fnn þ fnn) and  (mee þ mnn), respectively. The consistency of (202)–(205) may be checked by considering various simple cases that arise if the distributed couple m is absent. For example, consider the purely capillary case in which the surface supports a tension s but possesses no bending elasticity. The only nonvanishing term is then fn ¼ s, which correctly predicts the expected answer. Another example is
K a spherical surface with Canham–Helfrich energy density c ¼ 12 mH 2 þ m (from which spontaneous curvature has been neglected). It can then be shown that fn ¼ 0 and fn ¼ s. These results are consistent with previous findings of, e.g., Maleki & Fried (2013b). Lastly, it can be observed from (205) that, for tg ¼ 0, the component mn of the bending moment me vanishes, whereby me is parallel to e. Examples of surfaces with tg ¼ 0 include the sphere, for which tg ¼ 0 everywhere, and surfaces of revolution, for which tg ¼ 0 along any axisymmetric curve. For example, me must be parallel to e at the edge of an open surface of revolution.

7. ALTERNATIVE TREATMENT OF THE EDGE Being inherently limited to the description of equilibria, the variational approach used thus far is incapable of accounting for inelastic phenomena. For single-component lipid vesicles, viscous flow in the suspending solution is the most commonly noted source of dissipation. However, Seifert & Langer (1993) recognized that membrane viscosity associated with flow within and friction between leaflets may also be a significant source of dissipation, at least at sufficiently small length scales. This observation has been confirmed in molecular dynamics simulations reported by Shkulipa, den Otter, & Briels (2005), experiments reported by Watson & Brown (2010), and continuum simulations reported by Brown (2011). On the basis of an analysis of the budding of a spherical cap, Arroyo & DeSimone (2009) suggested that membrane viscosity may be relevant at significantly larger scales. Recent investigations account for edge viscosity in models for the expansion and contraction of pores on lipid bilayers (Karatekin, Sandre, & Brochard-Wyart, 2003; Neu & Neu, 2010).

Continuum Theory for the Edge of an Open Lipid Bilayer

51

In the present section, an alternative strategy that allows for both elastic and inelastic forces and moments internal to the edge C is developed. This strategy involves balance laws for forces and moments, an imbalance that represents the second law of thermodynamics (for isothermal processes), and constitutive relations. A systematic distinction is maintained between the fundamental laws of balance and imbalance, which govern a broad spectrum of materials, and constitutive relations, which define particular elements belonging to that spectrum. The fundamental laws are posed for an arbitrary segment of C. When localized at a generic point on C, the force and moment balances deliver field equations that hold on C and the imbalance yields an inequality. The primarily role of that inequality in the theory is to ensure that constitutive equations are physically viable in the sense that violations of the second law of thermodynamics may not occur. For simplicity, inertia and kinetic energy are neglected and attention is restricted to isothermal processes. In formulating the balance laws, the external forces exerted on C by both the surface S and the suspending solution environment are taken into account. Associated external power expenditures are similarly important in the formulation of the imbalance that represents the second law of thermodynamics.

7.1. Edge kinematics Consider an evolving open lipid bilayer with velocity field V ¼ x, _ where a superposed dot denotes the material time derivative, defined on the surface S and the edge C. The time derivative of the relevant quantities may be derived using a strategy completely analogous to that developed for calculating first variations in Section 3. For the sake of brevity, on replacing the first variation dR of a generic quantity R with the material time derivative R_ of R and the virtual velocity u with the velocity V, the existing relations in Section 3 can be used to advantage. 7.1.1 Geometry of deformation At a point of the edge C of S, the instantaneous elongation or contraction of C is determined by the edge stretching  ¼ V0  e:

ð206Þ

Since the unit tangent e to the edge C of S obeys e ¼ r0 , computing its material time derivative e_ , invoking the commutator relation (222), and bearing in mind the relations r_ ¼ x_ and V ¼ x_ yields

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Aisa Biria et al.

e_ ¼ ðr_ Þ0  r0 ¼V0  e:

ð207Þ

Define a triple {d1, d2, d3} of orthonormal directors on C satisfying d3 ¼ d1  d2 ¼e

ð208Þ

and the relations d0i ¼ Udi , d_ i ¼ W di ,

 i ¼ 1,2,3,

ð209Þ

where U and W are known as the twist and spin tensors. Since the directors are orthonormal, U and W must be skew and there must exist vectors U and W such that  d0i ¼ U  di , i ¼ 1, 2,3; ð210Þ d_ i ¼ W  di , U and W are referred to, respectively, as the Darboux vector and the angular velocity. In view of (210) and the commutator relation (222), subtracting the arclength derivative of d_ i from the material time derivative of d0i results in the relation

 d0i ¼ ðU_  W0 Þ  di þ W  ðU  di Þ  U  ðW  di Þ ¼ ðU_  W0 Þ  di þ ðW  di ÞV  ðU di ÞW:

ð211Þ

Taking the dot product of (211) with dj6¼i and using (210)1 yields dj  ðU  di Þ ¼ dj  ððU_  W0 Þ  di Þ þ ðW  di ÞðV  dj Þ  ðU  di ÞðW dj Þ, ð212Þ which is equivalent to  U  ðdi  dj Þ ¼ ðU_  W0 Þ ðdi  dj Þ þ ðU  WÞ  ðdi  dj Þ:

ð213Þ

Since di  dj ¼ dk, for {i, j, k} an even or odd permutation of {1, 2, 3}, (213) simplifies to ðU_ þ ðV0  eÞU  W0 þU  WÞ  dk ¼ 0, k ¼ 1, 2, 3, from which it can be inferred that the Darboux vector velocity W are related through

U

ð214Þ

and the angular

Continuum Theory for the Edge of an Open Lipid Bilayer

_ þ  U ¼ W0  U  W:

U

53

ð215Þ

It is natural to choose the set {d1, d2, d3} of orthonormal directors to coincide with the Darboux frame {e, n, n} introduced in (4). On doing so, the twist and spin tensors U and W admit representations of the form  U ¼ e0 eþn0 nþn0 n, ð216Þ W ¼ e_ e þ n_ n þ n_ n: In view of (19), it can be inferred from (216)1 and (216)2 that U and W can be expressed as  U ¼ kn n þ kg n  tg e, ð217Þ W ¼ ðe _  nÞn  ðe_ nÞn þ ðn_  nÞe: Consider the cross-product W  e involving the angular velocity W and the unit tangent e of C. By (217)2, W

 e ¼½ðe_  nÞn  ðe_  nÞn þ ðnn _ Þe  e ¼ ðe_ nÞn  ðe_  nÞn ¼ e_ ,

ð218Þ

which, together with (207), results in the useful identity W

 e ¼ V0  e:

ð219Þ

Computing the arclength derivative W0 of the angular velocity W and making use of the consequences e_  n þ n_  e ¼ 0 and n_  n þ n_  n ¼ 0 of the orthonormality of Darboux frame yields 0

W

¼  t_ g e  tg ðe_  ðe_ nÞn  ðe_  nÞnÞ þ k_ g n þ kg ðn_  ðn_  nÞn _ nÞn  ðn _ eÞeÞ þ U, ðn_ eÞeÞ þ k_ n n þ kn ðn_  ðn

ð220Þ

which reduces to 0

W

¼k_ n n þ k_ g n  t_ g e þ U:

ð221Þ

7.1.2 Commutator and transport identities Bearing in mind the definition (206) of the edge stretching , it is possible to derive a commutator relation,

0 _ x0 ¼ x_  ðV0 eÞx0

0 ¼ x_  x0 , ð222Þ

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Aisa Biria et al.

involving the material time derivative and the arclength derivative on C. Notice that (222) is analogous to the expression (108) for the first variation d(x0 ) of the arclength derivative of a generic field x defined on the edge C of S. Consider a generic segment G of the edge C. Then, given a field x defined on C and bearing in mind the definition (206) of the edge stretching , a transport identity, ð_ G

ð

 _ xþx , x¼

ð223Þ

G

completely analogous to (124) follows.

7.2. Balance laws Consider a generic segment G C of the edge C, and denote the initial and terminal points of G by x and xþ, which respectively correspond to arclengths s and sþ satisfying s < sþ. Given x a smooth field defined on G, let x and xþ denote its respective values at x and xþ. Additionally, define the jump ½ x of the field x across the endpoints x and xþ as ½½x ¼ xþ  x ð ¼ x0 :

ð224Þ

G

Consider the internal force f and moment m at an arbitrary point of G. Suppose that the portion C \ G of C external to G exerts contact forces f  and f þ and contact moments m and mþ at x and xþ. Suppose further that the segment G supports an external distributed force f ext and an external distributed moment mext. Notice that, in general, the force f ext and moment mext are combinations of forces and moments imposed by the lipid bilayer surface S and the surrounding solution. The force and moment balances on the segment G therefore read ð ½ f  þ f ext ¼ 0, ð225Þ G ð ð226Þ ½½m þ r  f  þ ðmext þ r  f ext Þ ¼ 0: G

Since G is an arbitrary segment, applying (224) to (225) and (226), under the assumption that all the relevant quantities are sufficiently regular, yields equivalent local statements f 0 þ f ext ¼ 0, 0

ðm þ r  f Þ þ m

ext

þrf

ð227Þ ext

¼ 0,

ð228Þ

55

Continuum Theory for the Edge of an Open Lipid Bilayer

of force balance and moment balance, valid pointwise on C. Since, by the local force balance (227) and the identity r0 ¼ e, ðr  f Þ0 ¼ r 0  f þ r  f 0 ¼ e  f  r  f ext ,

ð229Þ

the local moment balance (228) simplifies to read m0 þe  f þmext ¼ 0:

ð230Þ

7.3. Constitutive equations and thermodynamic restrictions 7.3.1 Free-energy imbalance Let f denote the free-energy density of the edge C, so that the integral ð ð231Þ F ðGÞ ¼ f G

represents the net free-energy of a generic segment G of C. Under isothermal conditions, the net dissipation D(G) of G takes the form DðGÞ ¼ W ðGÞ 

dF ðGÞ , dt

ð232Þ

ðf ext  V þ mext  WÞ

ð233Þ

where the integral ð W ðGÞ ¼ ½ f  V þ m W þ

G

represents the net power expended on G by external agencies and t indicates the time variable. The stipulation D(G)  0 that the net dissipation D(G) must be nonnegative then leads to the free-energy imbalance W ðGÞ 

dF ðGÞ  0, dt

ð234Þ

which ensures that the temporal increase in the free energy of G may not exceed the power expended on G by external agencies. For isothermal processes, the free-energy imbalance (234) represents the first and second laws of thermodynamics for the segment G of C. Substituting the explicit form (233) of the power expenditure and the definition (231) into the free-energy imbalance (234) and invoking the transport relation (223) yields

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Aisa Biria et al.

ð ½ f  V þ m W þ

G

ðf

ext

ð ð _  V þ m  WÞ f  f  0: ext

G

G

ð235Þ

The terms on the left-hand side of the inequality (235) may be interpreted as follows: • The jump ½ f  V þ m W represents the net power expended by the contact force f and ð the contact moment m at the endpoints of G. •

The integral

G

( f ext  V þ mext  W) represents the net power expended by

the external distributed force f ext and the external distributed moment mext along G.ð •

The integral

G

f_ represents the rate at which the net free-energy of G

changes with respect to time due to the rate at which the free-energy density f changes with respect to time. ð •

The integral

G

f represents change of the net free-energy of G that

accompanies changes in its length. Granted sufficient regularity, applying (224) to (235) yields an equivalent local statement, ð f  V þ m  WÞ0 þf ext  Vþmext  W  f_  f  0,

ð236Þ

of free-energy imbalance valid pointwise on C. In view of the local versions (227) and (230) of the force and moment balances on C, it follows that ð f  V þ m WÞ0 ¼ f 0  Vþf  V0 þm0  Wþm W0 ¼ f  V  ðe  f Þ Wþm W0  f ext  V  mext  W ¼ f  ðV0  W  eÞ þ m W0  f ext  V  mext  W,

ð237Þ

whereby, bearing in mind the relations (219) and (221) for W  e and W0 , the free-energy imbalance (236) becomes ð f  fÞ þ m ðk_ n v  t_ g e þ k_ g n þ UÞ  f_  0,

ð238Þ

f ¼ f e

ð239Þ

where

denotes the tangential component of the internal force f. In view of (238), the remaining component,
f ¼ f  f e,

ð240Þ

Continuum Theory for the Edge of an Open Lipid Bilayer

57

of f is unrestricted by the free-energy inequality and must be determined to ensure satisfaction of the local force and moment balances (227) and (230). 7.3.2 Constitutive assumptions For simplicity, assume that the edge-energy density f of the edge C is an isotropic function of the Darboux vector U, as expressed relative to the triad {e, n, n} by (217)1. Then, f can depend at most on the components of u, namely on the normal and geodesic curvatures kn and kg and on the geodesic torsion tg, and thus must obey a constitutive relation of the form ^ ;k ; t Þ: f ¼ fðk n g g

ð241Þ

Since changes in the length of C arise as a consequence of the accumulation or depletion of phospholipid molecules from the adjacent surface S, the material content of C is not generally fixed. In general, changes in the length of C should therefore not be misconstrued with the process of stretching an extensible filament with fixed material content. For this reason, the free-energy density f should not depend on the local length change of the edge. During the evolution of an open lipid bilayer that involves shape changes at its edge, it seems plausible that localized dissipative processes associated with the reorganization of phospholipid molecules may be significant. To account for this possibility, it is advantageous to assume that the tangential component f of the internal force f and the internal moment m admit decompositions  f ¼ fel þ fvis , ð242Þ m ¼ mel þ mvis , into elastic components fel and mel and viscous components fvis and mvis. Assume, consistent with (221), that fel and mel obey constitutive relations of the form ) fel ¼ f^el ðtg , kg ,kn Þ, ð243Þ mel ¼ m ^ el ðtg ,kg ,kn Þ: Moreover, suppose that fvis and mvis are determined by constitutive relations ) fvis ¼ ^f vis ðk_ n , k_ g , t_ g , Þ, ð244Þ mvis ¼ m ^ vis ðk_ n , k_ g , t_ g , Þ,

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Aisa Biria et al.

that incorporate dependence on the material time derivatives k_ n , and k_ g , and t_ g of the geometric quantities kn, kg, and tg entering the relations (221) and (243) for the equilibrium quantities f, fel, and mel along with dependence on the edge stretching  of C. The dissipative response functions f vis and m ^ vis may also depend on kn, kg, and tg. However, for brevity, such dependence remains tacit here.

7.3.3 Local form of the free-energy imbalance Using the constitutive relation (241) for the edge-energy density f in (238) and invoking the chain rule yields



 ^ t_ þ m n  f ^ k_ ð f þ m  U  fÞ  me þ f tg g kg g

 ^ k_  0, þ m n  f ð245Þ kn n where, as in Section 5.2, the subscripts tg, kn, and kg denote partial derivatives with respect to the corresponding arguments and, consistent with the ^ and its practice adopted in (245), the arguments of the response function f various partial derivatives are suppressed. Further, using the decomposition (242) of the tangential internal force f and the internal moment m into the elastic and viscous parts along with the corresponding constitutive relations (243) and (244) in (245) delivers the constitutively augmented freeenergy imbalance




 ^ t_ þ m ^ k_ f^el þ m ^ el  U  f   m ^ el  e þ f ^  n  f tg g el kg g


 ^ k_ þ f^ þ m ^ vis eÞ_tg þ m ^ el  n  f ^ vis  U   ðm kn vis þ ðm ^ vis  nÞk_ g þ ðm ^ vis  nÞk_ n  0:

ð246Þ

^ f^ , m ^ ^ vis that Any choice of the response functions f, el ^ el , f vis , and m allows for a violation of the inequality (246) at a given instant of time and point on C is viewed as thermodynamically inadmissible and, thus, untenable.

7.3.4 Thermodynamic restrictions on the elastic contributions to the internal force and internal moment Since the independent scalar fields t_ g , k_ n , k_ g , and  appear linearly in the first four terms on the left-hand side of (246), their values can be chosen in a way that violates the inequality. For the inequality to hold

Continuum Theory for the Edge of an Open Lipid Bilayer

59

unconditionally, the corresponding coefficients must therefore vanish and it ^ f^ , and m follows that f, ^ el must obey el ^ ¼ m ^ ¼m ^ ¼m f ^ el e, f ^ el n, f ^ el n, tg kg kn

ð247Þ

^ ¼ f^ þ m ^ el  U: f el

ð248Þ

and

In view of (247), the elastic contribution mel of the internal moment m is ^ and the elements e, n, and n of given in terms of the partial derivatives of f the Darboux frame by ^ nf ^ eþf ^ n: mel ¼ f kn tg kg

ð249Þ

Further, in view of the expression (217)1 for the Darboux vector and the restrictions (248) and (249), the elastic contribution fel to the tangential internal force f is given by ^ k f ^ ^ fel ¼ f  tg f tg g kg  kn fkn :

ð250Þ

7.3.5 Reduced dissipation inequality. Restrictions on the viscous contributions to the internal force and internal moment In view of the elastic results (249) and (250), the decomposition (242), and the primitive constitutive relations (243) and (244), f  e and m take the form

 9 ^ k f ^ k f ^ ^ = f ¼ f  tg f þ f , tg g kg n kn vis ð251Þ ; ^ nf ^ eþf ^ nþm m¼f ^ vis , kn tg kg where f^vis and m ^ vis must satisfy the residual dissipation inequality, ð f^vis þ m ^ vis  UÞ þ m ^ vis  ðk_ n n  t_ g e þ k_ g nÞ  0,

ð252Þ

resulting from using (249) and (250) in the free-energy imbalance (246). For illustrative purposes, consider the special case in which ^f vis and m ^ vis depend linearly on their arguments. Let the viscous moment have the form mvis ¼ a1 ð_tg þ tg Þe þ a2 ðk_ g þ kg Þn þ a3 ðk_ n þ kn Þn,

ð253Þ

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Aisa Biria et al.

with a1, a2, and a3 being unknown coefficients, possibly dependent on the geometric quantities kn, kg, and tg. Using the viscous moment given in (253) and the Darboux vector (217)1 in the dissipation inequality (252) leads to a1 ð_tg þ tg Þ2 þ a2 ðk_ g þ kg Þ2 þ a3 ðk_ n þ kn Þ2 þ f^vis   0:

ð254Þ

It is obvious from (254) that the response function f^vis determining the tangential component fvis of the viscous force fvis may not depend linearly on the time rates k_ n and k_ g of the normal and geodesic curvatures kn and kg or that, t_ g , of the geodesic torsion tg. Assuming that f^vis depends on the edge stretching  in the form fvis ¼ b, with b constant, the residual dissipation inequality (254) becomes a1 ð_tg þ tg Þ2 þ a2 ðk_ g þ kg Þ2 þ a3 ðk_ n þ kn Þ2 þ b2  0, from which it follows that

ð255Þ

the coefficients a1, a2, a3, and b must obey

a1  0, a2  0, a3  0, and b  0:

ð256Þ

7.4. Governing equations The local force and moment balances (227) and (230) augmented by the thermodynamically consistent constitutive relations (251) for f and m yield the final evolution equations for the edge C. The steps leading to derivation of those equations are presented next, using the specific, previously determined, forms of the constitutive relations for the elastic contribution of tangential force and internal moment fel and mel but leaving the viscous counterparts fvis and mvis of these quantities and the external force f ext and external moment mext in general form. Computing the arclength derivative of the expression (249) for the elastic contribution mel to the internal moment m yields

0 

0  ^ þf ^ þf ^ k f ^ t n f ^ k f ^ k e m0el ¼ f tg g kg g kg n kn g tg

kn 0  ^ ^ ^ ð257Þ þ fkg  ftg kn þ fkn tg n: Taking the dot product of the local moment balance (230) with e yields e  (m0 þ mext) ¼ 0, which in view of (257) results in the relation ^0  f ^ k þf ^ k þ ðmext þ m0 Þ e ¼ 0: f tg kg n kn g vis

ð258Þ

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Continuum Theory for the Edge of an Open Lipid Bilayer

Next, computing the cross product of the local moment balance (230) with e (from the left) yields f ¼ ( f  e)e þ e  (m0 þ mext), which, on invoking the decompositions (242) of f and m, the relations (239) and (240) defining the tangential and normal components f and
f of f, the expression (250) for the elastic contribution fel to f, and (257) delivers an expression,

 ^ k f ^ k f ^ þf e f ¼ f  tg f tg g kg n kn vis

0
0   ^ þf ^ k f ^ t þ m þ mext n n  f tg g kg g kn vis

0
0   ^ f ^ k þf ^ t þ m þ mext n n, þ f kg

tg n

kn g

vis

ð259Þ

for the internal force f. Finally, taking the arclength derivative of (259), and substituting the resulting expression into the local force balance (227) yields



0

0 ^ k f ^ k f ^ þf ^ ^ ^ f  tg f þ k tg g kg n kn vis n fkn þ ftg kg  fkg tg

0 
 
  ^ f ^ k þf ^ t þ m0 þ mext  n þ f ext  e e þ m0vis þ mext  n þkg f tg n kn g vis kg 

0

0
 ^ ^ ^ k f ^ t þ m0 þ mext n þ f  t f ^ þf þ  f tg g kg g g tg  kg fkg kn vis

0 
  ^ þf k þt f ^ f ^ k þf ^ t þ m0 þ mext  n kn f kn vis n g tg n kn g kg vis

0 
 0 ^ f ^ k þf ^ t þ m0 þ mext n þf ext  n n þ f tg n kn g kg vis



0 ^ k f ^ ^ ^ ^ ^  f  tg f tg g kg  kn fkn þ fvis kg þ tg fkn þ ftg kg  fkg tg 
  ð260Þ þ m0vis þ mext v þ f ext  n n ¼ 0: Equation (260) is in vectorial form and, hence, encompasses three independent scalar equations. First, on using the identity (258), the component of (260) in the direction of the tangent e simplifies to

 tg m0vis þ mext e þ kn m0vis þ mext  n
 ð261Þ þ kg m0vis þ mext n þ f ext  e ¼ 0:

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Aisa Biria et al.

Next, the component of (260) in the direction of the normal n yields

0
 0 ^ þf ^ k f ^ t þ m0 þ mext n  f kn tg g kg g vis

 ^ k f ^ k f ^ þf k þ f  tg f tg g kg n kn vis n

0
0   ^ f ^ k þf ^ t þ m þ mext n þ f ext  n ¼ 0, þ tg f ð262Þ tg n kn g kg vis while the component in the direction of the tangent-normal n gives

0
 0 ^ f ^ k þf ^ t þ m0 þ mext  n f tg n kn g vis kg

 ^ k f ^ k f ^ þf k  f  tg f tg g kg n kn vis g

0
0   ^ þf ^ k f ^ t þ m þ mext  n þ f ext n ¼ 0: þ tg f tg g kg g vis kn ð263Þ In summary, (261), (262), and (263) together with (258) constitute the set of governing equations on the lipid bilayer edge. Note that while (261) results from the tangential component of the force balance (260), (258) follows from the tangential component of the moment balance (230). Additionally, (262) and (263) follow, respectively, from the normal and tangent-normal components of the force balance (260).

7.5. Retrieving the Euler–Lagrange equations at the edge In the present section, the governing equations (258) and (261)–(263) are specialized for an edge-energy density f of the form of (178), namely
,k ,t Þ: f ¼ fo þ fðk n g g

ð264Þ

In doing so, neglecting the viscous contributions to the internal force f and internal moment m and confining attention to the purely static case yields an independent derivation of the Euler–Lagrange equations obtained using the variational method in Section 5.2. In general, the external force f ext and moment mext may include contributions  f e and  me exerted by the lipid bilayer surface and contributions i and m exerted by the surrounding environment: f ext ¼ f e þ i, mext ¼ me þ m:

ð265Þ ð266Þ

Using the components appearing in (202)–(205), the force and moment exerted by the lipid bilayer surface may be expressed as

Continuum Theory for the Edge of an Open Lipid Bilayer

63

 1
1 1
0

n f ¼ LrS cK þ rS cH þ 2HrS cK n  m e þ tg kn cH 2 2   1 1

k  c þ s  Kkn n, ð267Þ c þc K n 2 H    1
e 1 1


ð268Þ c þ cK kn n: m ¼ cH þ cK kn e þ tg kn 2 2 H e

It is easily confirmed that in the absence of the distributed couple m on the surface, the force and moment (267) and (268) exerted by the surface on the edge identically cancel from (261). Thus, substituting the decompositions (265) and (266) of f ext and mext into the governing equation (261) reduces it to an equation, ie  me tg þ mn kg þ mn kn ¼ 0,

ð269Þ

for the force and moment exerted by agencies external to the lipid bilayer. Equation (269) is identical with (184), in agreement with the observation that (184) describes the balance of force in the direction e tangent to C. Next, on substituting from the external force and moment relations (265) and (266), the remaining governing equations yield 1

k þ m  f
0  f
k þf
k ¼ 0, ð270Þ cH þ c K n e kg n kn g tg 2 
þ1r c
þ 2Hr c
LrS c K S K  n  m e 2 S H

0
þm
0 þ f
k f
t þt c  f tg g kg g g K n kn




k f
k f
k þt f
0  f
k þf
t þm þ f  tg f tg g kg n kn n g tg n kn g n kg þ in ¼ 0, ð271Þ

0


0  f



k þf
t þm  ft f f tg n kn g n g tg  kg fkg  kn fkn kg kg


0 þ f
k f
t þ m  ðc þ sÞ þ tg f tg g kg g n kn

 1 2 1

þ K þ t g kn ð272Þ c þ cK kn þ in ¼ 0: 2 H Equations (270) and (271) are identical to the Euler–Lagrange equations (187) and (185), respectively. This is in agreement with the previous

64

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interpretation of (185) as the balance of normal force; further, it transpires that (187) represents the balance of tangential moments. Additionally, (186) can be converted to (272) upon using the geometrical relation (186) and the Euler–Lagrange equation for balance of tangential moments (187). Thus, the net information contained in (187) is the balance of force in the tangent-normal direction n, consistent with previous discussion.

8. SUMMARY In this chapter, a continuum mechanical framework for studying the equilibrium equations and constitutive theory for an open lipid bilayer was provided. After presenting necessary mathematical preliminaries, including relevant aspects of the calculus of variations, the principle of virtual work was used to derive the Euler–Lagrange equations governing a lipid vesicle subject to distributed forces and couples. An extension of this approach was then employed to derive the equilibrium equations for the edge of an open lipid bilayer endowed with a nonzero edge-energy density. Such an edge-energy density is a manifestation of the excess energy due to the particular way in which phospholipid molecules near the edge of an open bilayer must be arranged to shield the water-insoluble core of the bilayer from the surrounding solvent. Constant and geometry-dependent edge-energy densities were considered. For the latter alternative, inspired by a recent microphysical derivation (Asgari et al., 2013) of the edge-energy density of an open lipid bilayer, general dependence on the normal and geodesic curvatures and the geodesic torsion of the edge was allowed. Moreover, in addition to forces and couples distributed on the surface of the bilayer, counterparts distributed on the edge of the lipid bilayer were considered. Comparisons with existing results were provided. Subsequently, using a variational formulation, the force and bending moment exerted by an open lipid bilayer on its edge were derived. Variational approaches provide useful and efficient tools for describing the elastic behavior of the materials and structures, lipid bilayers included. However, they are poorly suited to modeling inelastic phenomena where dissipation plays an important role. Various experimental and theoretical studies suggest that viscous effects are of critical importance in dynamics of open lipid bilayers. In the final part of the chapter, an alternative approach based on augmenting balance laws for forces and moments with constitutive equations restricted to ensure satisfaction of a suitable version of the second law of thermodynamics was used to derive governing equations for the edge of a lipid bilayer, accounting for both elastic and viscous effects. For

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simplicity, only isothermal process was considered and inertial effects were neglected. Considering a generic segment of the edge, the local forms of force and moment balances and the free-energy inequality were obtained. In dealing with the constitutive theory for the edge of the lipid bilayer, the edge-energy density was assumed to depend on the normal and geodesic curvatures and the geodesic torsion of the edge. Accordingly, the tangential component of the internal force and the internal moment were decomposed into elastic and viscous contributions. Whereas the elastic contributions to these quantities were assumed to depend on the same arguments as the edgeenergy density, the viscous terms were allowed to depend on the rates of these arguments as well as stretching of the edge. Subsequently, upon implementing the force and moment decompositions into the dissipation inequality and following the Coleman–Noll procedure (Coleman & Noll, 1963), constitutive relations for the elastic contributions to the internal force and internal moment were developed. Natural elimination of the elastic terms from the dissipation inequality resulted in a reduced dissipation inequality in terms of the viscous terms dictating the thermodynamical requirement for any constitutive form of the internal viscous force and moment. Lastly, upon specializing the formulation for purely elastic case and substituting the constitutive relations into the force and moment balances, the previously derived Euler–Lagrange equations using variational methods were recovered. Aside from special geometries such as axisymmetric shapes, solving the equations governing open lipid bilayers appears to be very challenging. The inherent complexity is intensified when new features such as geometry-dependent edge-energy densities are taken into account. It therefore seems likely that further progress will hinge on the development of accurate and robust numerical methods. To this end, formulations based on the principle of virtual work, such as that presented here, provide a natural platform for formulating finite-element discretizations and numerical integration schemes. Additionally, the alternative formulation presented here provides an avenue for analyzing the influence of dissipative edge forces and moments of the dynamics of open lipid bilayers.

ACKNOWLEDGMENTS The authors thank Brian Seguin for helpful discussions and suggestions. This work was supported by the National Institute of Health under grant GM084200 and the Canada Research Chairs Program.

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