Mechanics of cellular membranes 1 Lipid membranes

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Figure 1: a) Bilayer composed of lipid molecules, and b) Lipid distortion at free ... lar, variations in the direction of the surface normal yield the well known ... the projection on the tangent plane yields the equation that determines the ... In parametric form, position on the surface, relative. 3 ...... Contact lines for fluid surface.
Mechanics of cellular membranes Ashutosh Agrawal1 and David J. Steigmann2 1 2

Department of Mechanical Engineering, University of Houston

Department of Mechanical Engineering, University of California-Berkeley

Abstract: In this chapter we summarize the theory of cellular membranes required to model a diverse range of biological phenomena. A typical lipid bilayer is modeled as a two dimensional fluid shell with flexural resistance. We discuss the notion of fluidity and obtain the governing equilibrium equations for membranes with inhomogeneous properties. The theory is specialized to axisymmetric problems and employed to model protein mediated endocytosis. We obtain the contact conditions required to model the interactions of membranes with curved substrates in the presence of wetting and adhesion. Finally, we discuss the theory of membranes with coexistent phases.

1

Lipid membranes

Membranes are fundamental components of cell structures that strongly influence cell function. They constitute the plasma membrane that separates the contents of the cell from the surrounding environment. In eukaryotic cells, intracellular organelles such as nucleus, endoplasmic reticulum, golgi apparatus, mitochondria, and lysosomes are enclosed by their own internal membranes [4]. In addition to providing the basic shape, these membranes play a vital role in transporting material between various organelles and the plasma membrane and signal transduction. Biomembranes consist primarily of transversely oriented lipid molecules containing hydrophilic head groups and hydrophobic tails. The former consist of charged atoms capable of forming electrostatic bonds with the polar

1

(a)

(b)

Figure 1: a) Bilayer composed of lipid molecules, and b) Lipid distortion at free edges [2].

water molecules. The latter comprise of non-polar groups incapable of making electrostatic interactions with water molecules and are thus, averse to exposure to water. Due to these contrasting properties, lipid molecules arrange themselves in opposite orientations forming bilayers that effectively shield the tail groups from the surrounding aqueous solution (Figure 1(a)). Formation of an edge entails the reorientation of lipid molecules to prevent the exposure of tail groups (Figure 1(b)). This realignment, occurring on the length scale of molecular dimensions, is associated with an energetic cost. For this reason, membranes in general form closed structures. The lipid molecules constituting the membranes are free to diffuse on the surface. As a consequence, they possess long range orientational order but lack long range positional order. This grants membranes a fluid-like behavior, and thus they may be regarded as two-dimensional liquid crystals. A homogeneous composition and an identical environment on either side render bilayers a preferred flat configuration. However, in the presence of different species of lipids or interaction with other biological structures, a locally curved state becomes the preferential configuration.

2

A typical bilayer is characterized by a large surface area and a relatively vanishing thickness. In addition, relative misalignment of the lipids entails an energetic cost which manifests itself as flexural stiffness. Furthermore, due to the diffusivity of lipid molecules on the surface, membranes offer vanishing in-plane shear resistance. Owing to these physical characteristics, the membranes are modeled as two-dimensional fluid surfaces immersed in three-dimensional space with bending elasticity. The theory of biomembranes has been developed primarily from two approaches. First is the variational approach where the equilibrium equations governing the response of membranes emerge as the Euler-Lagrange equations associated with an energy functional [31, 45, 29]. In particular, variations in the direction of the surface normal yield the well known shape equation which, as the name suggests, determines the equilibrium configuration. Variations in the tangential plain yield an equation that determines the surface tension. In the second approach, the theory is derived directly from the balance and constitutive equations of nonlinear shells [43, 42, 21]. Since the local interactions orient the lipids in the direction of the surface normal, the appropriate framework is that of the Kirchhoff-Love shell theory. Fluidity is incorporated into the theory by imposing material symmetry restrictions on the constitutive function. Here, the equation of motion projected onto the surface normal yields the shape equation and the projection on the tangent plane yields the equation that determines the surface tension. In this Chapter, we restrict ourselves to the variational approach which is discussed in the next section.

2

Equilibrium equations

Let ω be the membrane surface with coordinates θµ ; µ = 1, 2. Here and henceforth Greek indices range over {1, 2} and, if repeated, are summed over that range.

In parametric form, position on the surface, relative

3

to a specified origin, is described by the function r(θµ ).

The surface

parametrization induces the basis aα = r,α for the tangent plane to ω at the point with coordinates θµ (Figure 2). Subscripts preceded by commas indicate partial derivatives with respect to the coordinates, while those preceded by semicolons indicate covariant derivatives. The induced metric is aαβ = aα · aβ , and is assumed to be positive definite. A dual basis on the tangent plane is then given by aα = aαβ aβ , where (aαβ ) = (aαβ )−1 . The orientation of the surface is defined locally by the unit-normal field

a2 p

ᶿ

2

a1

ᶿ

1

r

Figure 2: Parameterization of a surface.

n = a1 × a2 / |a1 × a2 | , and its local curvature by the tensor field b = bαβ aα ⊗ aβ ,

(1)

bαβ = n · r,αβ = −aα · n,β .

(2)

where

If ν and τ are orthonormal vectors on the tangent plane, then b = κν ν ⊗ ν + κτ τ ⊗ τ + τ (ν ⊗ τ + τ ⊗ ν),

(3)

where κν and κτ are the normal curvatures on these axes and τ is the twist given respectively by κν = bαβ ν α ν β ,

κτ = bαβ τ α τ β

4

and τ = bαβ ν α τ β .

(4)

The two invariants associated with the curvature tensor play a fundamental role in the mechanics of biomembranes. These are given by H=

1 2

tr b = 12 (κν + κτ )

(5)

where is the mean curvature and K = det b = κν κτ − τ 2

(6)

is the Gaussian curvature. The strain energy for fluid films is dependent on the strain and curvature of the film defined with respect to some fixed reference configuration. Since the lipid molecules forming the membranes diffuse freely on the surface, the lipid membranes exhibit a fluid behavior. Following an approach similar to Noll’s treatment of simple fluids as special elastic materials, fluidity is incorporated into the theory by requiring the relevant strain energy density to satisfy a material symmetry restriction appropriate for two-dimensinal fluids. This leads to the requirement that [42] T ˆ ˆ Ψ(C, κ) = Ψ(RCR , ±RκRT );

det R = ±1.

(7)

ˆ is the strain energy of the fluid film, C and κ are the symmetric where Ψ surface strain tensor and the relative curvature tensor, respectively, relative to a fixed reference configuration, and R is a tensor belonging to the unimodular group. The above restriction generates a basis of invariants under unimodular transformations. These include the areal mass density ρ, mean curvature H and Gaussian curvature K ([42, 44]). Thus, for fluid films, the constitutive function takes the following canonical form ˆ Ψ(C, κ; θµ ) = W (ρ, H, K; θµ ),

(8)

in which the explicit dependence on coordinates θµ occurs if the film has non-uniform properties. Governing equations that determine equilibrium configurations emerge as the Euler-Lagrange equations associated with the potential energy functional Z E = [W (H, K; θα ) − µ(θα )]da − pV (ω), ω

5

(9)

where W (H, K) is the strain-energy per unit area of the membrane surface ω, V (ω) is the volume enclosed by ω, and µ(θα ) and p are Lagrange multipliers associated with the area and the volume constraints respectively. The first multiplier is a function defined on the surface and is associated with the local area constraint. In contrast, the second multiplier is a constant associated with a global constraint on the enclosed volume. We note that on regarding Lagrange multipliers as independent variables, variations with respect to them yield the constraint equations and thus need not be made explicit. For closed surfaces without boundary, the tangential variation of the potential energy functional yields the Euler equation [3] µ,α = ∂W/∂θα ,

(10)

The partial derivative on the right is due to the explicit coordinate dependence in the function W . This arises from the non-uniformity of the film properties in the present context. In the case of membranes with uniform properties and on identification of µ with −λ, the above equation implies that λ is constant as encountered in the literature frequently. Normal variations of the potential energy functional, for closed surfaces, lead to the Euler equation ([45, 3]) ∆( 21 WH )+(WK );αβ ˜bαβ +WH (2H 2 −K)+2H(KWK −W )−2λH = p (11) where ∆(·) = [aαβ (·);α ];β is the surface Laplacian. This is the well-known shape equation in a generalized form. In the case of vanishing bending resistance it reduces to the classical Laplace equation of capillarity which yields a constant mean curvature over the surface. The theory of membranes with a strain energy quadratic in the mean curvature (Helfrich energy) has been employed to study a wide variety of equilibrium shapes for closed vesicles and cells [15, 35, 31, 29]. These include shapes such as spheres, prolate and oblate ellipsoids, dumbbells, discocytes and stomatocytes. Transitions between these shapes have been obtained under varying surface area to volume ratio and varying constant 6

spontaneous curvature [15, 35]. One of the most widely studied shapes has been the biconcave discoid shape of a red blood cell [14, 22, 31, 29]. Here we consider a simple extension of the Helfrich model [20] to the case of non-uniform spontaneous curvature. Thus, W = k(H − C(θα ))2 ,

(12)

where k(> 0) is the (constant) bending modulus and C(θα ) is the (variable) spontaneous curvature. Here, we have suppressed the term linear in Gaussian curvature, by virtue of the Gauss-Bonnet theorem, as the surfaces considered have fixed genus. In addition, we assume the function C(θα ) to be assigned. In a more general setting, not pursued here, C(θα ) would evolve as a consequence of the interactions between the membrane curvature and the protein distribution. Proceeding with the expression (12) for the energy density and with µ = −λ, we find that (10) reduces to λ,α = 2k(H − C)∂C/∂θα ,

(13)

while the shape equation (11) becomes k∆(H − C) + 2k(H − C)(2H 2 − K) − 2kH(H − C)2 = p + 2λH.

3

(14)

Axisymmetric solutions

We consider axisymmetric surfaces parametrized by meridional arclength s and azimuthal angle θ. Let r(s) be the radius from the axis of symmetry and z(s) be the elevation above a base plane (Figure 3). Curves with constant θ and s form meridians and parallel of latitudes, respectively, on the surface. We select surface coordinates θ1 = s and θ2 = θ, and associate the vector ν with the unit tangent to a meridian. Since s measures arclength along meridians (r0 )2 + (z 0 )2 = 1,

7

(15)

where, here and henceforth, ()0 = d()/ds. This implies that r0 (s) = cos ψ

and z 0 (s) = sin ψ,

(16)

where ψ(s) is the angle made by ν with the radial direction.

n

! y

z s

r

Figure 3: Meridian of a surface of revolution.

Invoking the definitions of the normal curvatures, mean curvature and Gaussian curvature furnishes the differential equation ([2, 3]) rψ 0 = 2rH − sin ψ,

(17)

K = H 2 − (H − r−1 sin ψ)2 .

(18)

and

In the absence of lateral pressure, for the free-energy function (12), the shape equation (14) simplifies to L0 = 2r{H[k(H − C)2 + λ] − k(H − C)[H 2 + (H − r−1 sin ψ)2 ]},

(19)

where L = 21 r(WH )0 = kr(H − C)0 .

(20)

We assume the spontaneous curvature to be an assigned function C(s) of the meridional arclength. The remaining equilibrium equation, (13), yields the non-trivial equation λ0 = 2k(H − C)C 0 . 8

(21)

To solve the above differential equations governing the equilibrium configurations of the membranes, we need boundary conditions. Four of the boundary conditions which are frequently encountered are of the geometric type. These include r(0) = r0 ,

ψ(0) = ψ0 ,

z(0) = z0

and ψ(S) = ψS ,

(22)

where the right-hand sides and the value of S are to be specified. We append additional boundary conditions based on equilibrium considerations. For solutions with reflection symmetry about the equatorial plane, the limiting value of the transverse shear force per unit arc length vanishes. This is because equilibrium requires a non-zero force to be in opposite directions above and below the equatorial plane which breaks the desired reflection symmetry. In addition, the same force also vanishes at a pole, lying on the axis of revolution, in the absence of any applied load. In both cases this is equivalent to the requirement L=0

(23)

at the appropriate value(s) of arclength. The latter condition may be derived from a direct analysis of the equilibrium of a sector of the surface containing the pole, in the limit as the size of the sector shrinks to zero [2]. This is discussed in greater detail in Section 6. The system to be solved thus consists of the six equations (16)1,2 , (17), (19), (20) and (21), for the six unknowns r, z, ψ, H, λ and L. These can be integrated numerically using Matlab ODE solver ’bvp4c’ with the appropriate boundary conditions relevant for the problem.

4

Membrane-protein interactions and endocytosis

Proteins are an integral part of biological membranes and their interaction with the membranes is fundamental to comprehending the structure

9

of biomembranes and many physiological phenomena. It has been demonstrated that the transmembrane proteins, with conical geometry, induce bending in the membrane [25, 26]. Further, the deformation field generated due to embedding of the proteins leads to protein-protein interactions mediated by the surrounding membrane [25, 26]. In the case of vanishing surface tension, it has been shown that a minimum of five proteins arranged in a pentagon are necessary to form a stable equilibrium configuration [25]. In another study [19], energetics of a cylindrical protein embedded in a membrane in cubic phase has been used to study the transition from a cubic phase to a lamellar phase with an increase in protein density. Such phases have been observed to coexist in the endoplasmic reticulum [41]. A phenomenon of vital importance where membrane-protein interactions play an integral role is that of protein mediated endocytosis. It is a phenomenon that entails the creation of a bud on the membrane which facilitates encapsulation and transportation of material into the cell [4]. For budding yeast cells, it has been demonstrated that the bud formation is mediated by proteins and actin filaments [27]. In addition to inducing curvature, the protein coat filters lipid species thereby creating a phase boundary which is associated with a line tension that causes the constriction of the neck of the bud. In a recent work [28], increased insight into endocytosis is gained by employing a mechanochemical model for vesicle formation. The model couples the membrane morphology to local biochemical reactions such as hydrolysis of lipids, recruitment of coat proteins, bar domains and enzymes, and actin polymerization. Studies have also been performed on the receptor-mediated endocytosis of nanoparticles by modeling the binding interactions between the ligand molecules on the nanoparticles and the diffusive receptor molecules on the membranes [18, 11]. A similar study of the wrapping of membrane around a nanoparticle using a phenomenological approach was performed in [12]. Here we present the modeling of protein-assisted endocytosis based on the variable spontaneous curvature framework discussed in the previous

10

section. To solve this problem we non-dimensionalize the equations with a positive constant H0 having the dimensions of curvature. The dimensionless arclength, radius and height are given by s¯ = sH0 ,

x = rH0

and y = zH0 ,

(24)

respectively, and the dimensionless mean and spontaneous curvatures are given by ¯ = H/H0 H

and C¯ = C/H0 .

(25)

Furthermore, non-dimensional counterparts of L, λ and W are given by ¯ = (kH0 )−1 L, L

¯ = (kH 2 )−1 λ λ 0

¯ = (kH02 )−1 W. and W

(26)

In this process of endocytosis, a protein coat, such as clathrin (Alberts et al 2002; Boal 2002), attaches to the membrane inducing a natural curvature over the coated region. We assume this induced curvature to be uniform and non-zero in the coated region and to be nearly zero in the uncoated region. An appropriate function to model this rapid transition is (see Figure 4) ¯ s) = C(¯

β 2 {1

− tanh[α(¯ s − s¯0 )]}.

(27)

The progression of the bud formation is simulated by increasing the parameter s¯0 , corresponding to an increasing protein coat domain. The full set of equations is integrated subject to the boundary conditions x(0) = 0,

ψ(0) = 0,

y(0) = 0

and

¯ L(0) =0

(28)

at the pole (¯ s = 0), together with ¯ =0 ψ(S)

and

¯ S) ¯ = 0. λ(

(29)

Figures 5(a), 6(a) and 7(a) show photo-micrographs of different stages of endocytosis observed experimentally by Perry and Gilbert [32]. Figures 5(b), 6(b) and 7(b) show membrane shapes computed from the present model with α = 5.5, β = π/2, S¯ = 3.0 and s¯0 = 0.3, 0.8, 1.3 respectively. The coated regions are the solid portions of the meridional curves. A value 11

2

¯ s) C(¯

1.5 1 0.5 0 0

0.5

1

1.5 s¯

2

2.5

3

Figure 4: Spontaneous curvature distribution in endocytosis (α = 20.0, β = 2.0 and s¯0 = 1.3) [3].

of βH0 = 0.0036 nm−1 yields a vesicle diameter of approximately 280 nm, which is in close agreement with the average size of the clathrin-coated vesicles observed by Perry and Gilbert [32]. 0.5 0

y

!0.5 !1 !1.5 !2 !1.5

(a)

!1

!0.5

0 x

0.5

1

1.5

(b)

Figure 5: (a) Shallow clathrin-coated vesicle observed by Perry and Gilbert (1979; reproduced by permission of the Company of Biologists), and (b) present simulation obtained with s¯0 = 0.3 [3]. Solid curve corresponds to the protein coated region; dashed curve to the uncoated region.

12

0.5 0

y

!0.5 !1 !1.5 !2 !1.5

(a)

!1

!0.5

0 x

0.5

1

1.5

(b)

Figure 6: (a) Intermediate shape of clathrin-coated vesicle observed by Perry and Gilbert (1979; reproduced by permission of the Company of Biologists), and (b) present simulation obtained with s¯0 = 0.8 [3]. Solid curve corresponds to the protein coated region; dashed curve to the uncoated region.

5

Edge conditions

The Euler-Lagrange equations associated with the tangential and the normal variations for a closed surface were presented in Section 2. We now consider a system where a membrane interacts with a substrate Γ (Figure 8) along an edge ∂ω. Various studies have presented the equilibrium conditions that hold at the edge of a membrane with a substrate [34, 9, 46, 30]. With (10) and (11) satisfied the variation of the energy reduces to [1] Z Z X E˙ = (Fν ν + Fτ τ + Fn n) · uds − M τ · ωds + fi · ui , (30) ∂ω

∂ω

where ν is the normal to the contact boundary in the tangent plane (Figure 8), τ is the tangent to the boundary in the direction of increasing arclength, n is the surface normal, u is the virtual displacement, ω is a vector associated with induced variations in n, M = 21 WH + κτ WK

(31)

is the bending couple applied to ω per unit length of ∂ω, Fν = W + λ − κν M,

13

(32)

0.5 0

y

!0.5 !1 !1.5 !2 !1.5

(a)

!1

!0.5

0 x

0.5

1

1.5

(b)

Figure 7: (a) Final stage of clathrin-coated vesicle prior to fission, observed by Perry and Gilbert (1979; reproduced by permission of the Company of Biologists), and (b) present simulation obtained with s¯0 = 1.3 [3]. Solid curve corresponds to the protein coated region; dashed curve to the uncoated region.

Figure 8: Interaction of membrane, substrate and bulk liquid [1].

Fτ = −τ M,

(33)

Fn = (τ WK )0 − ( 21 WH ),ν − (WK ),β ˜bαβ να ,

(34)

respectively, are the ν -,τ - and n - components of the force per unit length applied to ∂ω, and fi = WK [τ ]i n

(35)

is the force applied to the film at the ith corner of ∂ω. We now consider the problem of wetting of a rigid wall Γ by a bulk liquid bounded between the wall and the membrane film ω (Figure 8). The effect of wetting is

14

incorporated into the theory by appending a phenomenological energy EΓ∗ = −σAΓ∗

(36)

to the total energy E, where AΓ∗ is the surface area of the portion of the wall wetted by the volume of liquid and σ is an empirical constant. Positive values of σ promote wetting while negative values penalize it. In the presence of wetting, the variation of the potential energy reduces to [1] Z Z X E˙ = [(Fν cos γ + Fn sin γ − σ)ut + Fτ uτ ]ds − M τ · ωds + fi · ui , ∂ω

∂ω

(37)

where γ is the contact angle (cos γ = ν · t and sin γ = n · t). In the case of the interaction of a lipid membrane with a rigid wall with structural symmetry, such as a polished surface, the lipid alignment may get dictated by the microscopic patterns on the surface. Such an edge condition with a prescribed surface orientation is called hard anchoring. In other cases, it might be energetically expensive for configurations of n to deviate from a cone with a prescribed axis N at the boundary. Such an edge condition is called conical anchoring. Let N be the unit normal to the wall at a point of ∂ω then the above condition is associated with the constraint (see Figure 8) n · N = cos γ

(38)

in which γ is assigned. If the substrate is a plane, then the stationarity condition with conical anchoring yields the natural boundary conditions [1] Fν cos γ + Fn sin γ = σ

and Fτ = 0

on ∂ω,

(39)

the first of which generalizes the classical Young equation λ cos γ = σ of classical capillarity theory. If the substrate is curved, the natural boundary conditions take the form [1] Fν cos γ + Fn sin γ − M t · (∇Γ γ − Bt) = σ,

Fτ − M τ · (∇Γ γ − Bt) = 0, (40)

15

where B is the (symmetric) curvature tensor of the substrate. The first of these again reduces to the classical Young equation in the absence of bending energy and the second reduces to an identity.

6

Adhesion

Adhesion plays a significant role in biological processes such as endocytosis, exocytosis, cell growth and differentiation, cell movement and tissue formation. It is also important in several biotechnological processes related to drug discovery and delivery. Experimental studies have been conducted to study adhesion using atomic force microscope [37, 23] and magnetic tweezers [39]. These studies indicate that a vesicle adhered to a substrate undergoes a discontinuous transition from the adhered state to the detached state when pulled at the pole. Two theoretical approaches have been primarily adopted to model adhesion. The first approach employs an effective adhesion energy proportional to the contact area. This approach has been applied to study adhesion of vesicles to planar [36, 40] and curved substrates [33, 34, 8, 13, 10]. In addition, adhesion between two fluid surfaces has also been investigated [13]. The theory of adhesion differs from the basic theory of membranes by a jump condition associated with curvatures at the contact boundary. In the second approach, adhesion is modeled via interactions between the receptor molecules on the membrane and the ligand molecules on the substrate [17, 38]. Here the growth of the adhered area is associated with a diffusion of receptor molecules from the non-adhered area. Following the first approach, adhesion is incorporated into the theory by appending the total energy with a phenomenological contribution proportional to the contact area between the membrane and the substrate. This energy is of the form (36). Unlike the problem considered in the previous section, the contact here occurs over a portion of its interior as opposed to the edge of the membrane. For the augmented potential energy in the case

16

of adhesion, the stationary-energy condition reduces to a single non-trivial jump condition [1] [W ] − Mf [κν ] = σ,

(41)

For the Helfrich energy, the condition simplifies to [1] σ/k = 14 [κν ]2 = [H]2 .

(42)

This condition rules out the possibility of a finite-area adhesive contact of a membrane with a hydrophobic wall (σ < 0). Further, we remark that for membranes of the Helfrich type, curvature discontinuities in the interior of the free part of the membrane are not allowed. This is discussed in detail in [2]. We now simulate the interactions of a vesicle with a flat substrate under the action of point loads applied at the (north) pole. To model the point load we consider equilibrium of a subsurface ω ˜ containing the pole. This is given by Z

pnda +

ω ˜

Z

f dt + F k = 0,

(43)

∂ω ˜

where t = r(˜ s)θ measures arclength around the perimeter of the parallel, p is the osmotic pressure, f is the force per unit length exerted on ω ˜ by the neighboring membrane, and F is the point load acting at the pole along the axis of symmetry aligned with k. For the Helfrich energy this reduces to F/2π = k lim (rH 0 ), s˜→0

yielding L(0) = F/2πk,

(44)

where L(s) = rH 0 . For the case of vanishing point load at the pole this yields L(0) = 0, as mentioned in Section 3. As the vesicle changes its configuration in the presence of an applied load, we assume its total surface area to remain unchanged. A convenient way to numerically impose this constraint is to rewrite the governing equations in terms of the area as the independent variable. This is a valid operation since the area a and the arclength s are in one-to-one correspondence. The transformation is easily achieved by employing the geometric relation a0 (s) = 2πr(s). In addition we non-dimensionalize the equations 17

by the radius R of a sphere with the prescribed area of the vesicle. This yields the dimensionless variables α = a/(2πR2 ), h = RH,

x = r/R,

y = z/R,

(45)

¯ = R2 λ/k. l = RL and λ

(46)

In term of these variables and a as the independent variable, the differential equations are given by xx˙ = cos ψ,

xz˙ = sin ψ,

x2 ψ˙ = 2xh − sin ψ,

x2 h˙ = l

(47)

and ¯ − (h − x−1 sin ψ)2 ], l˙ = 2h[λ

(48)

where (·)· = d(·)/dα. These are solved in conjunction with the boundary conditions x(0) = 0,

ψ(0) = 0,

x(αf ) = xp ,

y(αf ) = yp

and ψ(αf ) = ψp , (49)

together with [h]2 = σ ¯

and l(0+ ) = F¯ ,

(50)

where σ ¯ = R2 σ/k

and F¯ = F R/(2πk).

(51)

The total area is constrained to be that of the spherical membrane, namely 4πR2 . Accordingly, the equations are integrated over the domain [0, αf ], corresponding to the free membrane, with an assigned value of αf ∈ (0, 2). A sequence of shapes is obtained by varying the value of αf thereby varying the area of contact between the membrane and the substrate. In this approach, the force acting at the pole is obtained as part of the solution. Furthermore, the mean curvature in the free membrane at the contact boundary is obtained from the jump condition (50)1 . Numerical simulations of a vesicle interacting with a concave substrate with σ ¯ = 4 is shown in Figure 9(a). For Helfrich membranes the shapes

18

obtained are independent of the bending modulus k and can thus be regarded as universal. The dashed contour represents the meridian of the membrane corresponding to vanishing point load. The solid curves correspond to membrane under compressive (downward) or tensile (upward) point loads. Figure 9(b) presents the pole force as a function of contact area which exhibits a maxima at a non-vanishing contact area in the regime with applied tensile loads. For assigned point loads this leads to solutions with non-unique contact area over some interval of tensile load. Direct comparison of the total system energy, consisting of the membrane bending energy, adhesion energy and the load potential, reveals that the solution with the larger contact area is energetically optimal. Further, as there exists no equilibrium solution for point loads exceeding the computed maxima, the contact between the substrate and the vesicle is lost in a dynamic transition at this critical value. 1 2 1.8

0.5

1.6 1.4

0

y



1.2 1

−0.5

0.8 0.6

−1

0.4 0.2 0

−1.5 −1

−0.5

0 x

0.5

1

0

0.1

(a)

0.2

!c

0.3

0.4

0.5

(b)

Figure 9: (a) Equilibrium shapes of a vesicle adhering to a flat substrate (¯ σ = 4). Dashed curve corresponds to vanishing point load, and (b) Pole force vs. contact area for a vesicle adhering to a flat substrate (¯ σ = 4) [1].

7

Coexistent phases

Membranes with multiple lipid species or cholestrol-lipid mixtures may segregate into coexistent domains with different compositions. High resolution

19

images have shown a variety of shape transformations in phase-separated giant unilammellar vesicles when subjected to varying temperatures or osmotic pressure [6]. These shape transitions have been modeled based on the quadratic free energy augmented by line tension [24, 5]. Although, in general, the Gaussian energy does not contribute to the equilibrium shapes of vesicles with fixed topology, its role in determining shape transformations in phase-separated vesicles has been illustrated in [5]. A framework for coexistent phases based on a non-convex energy density was proposed in [2] in which the phase boundary is associated with a jump in curvature and the (continuous) curvatures on either side characterize the local states of the fluid phases. Across such a boundary, the jumps in the mean curvature and the Gaussian curvature are of the form [2] [H] = 12 (n · u),

and

[K] = κτ (n · u),

(52)

where u is a 3-vector, n is the surface normal, and κτ is the normal curvature of the phase boundary which can be shown to be continuous. The locations of phase boundaries are determined in part by jump conditions which depend on the type of minimizer considered. In the given context two types of equilibria that are well known in the Calculus of Variations, namely, the weak and strong relative minimizers, can be considered. Strong relative minimizers are those configurations that minimize the energy with respect to perturbations in r and r,α that are bounded at all points of the membrane. For weak relative minimzers perturbations in r,αβ are also bounded. For both the minimizers it is necessary that the force and moment be continuous across a phase boundary [f ] = 0 and

[m] = 0.

(53)

Strong minimizers satisfy an additional jump condition involving the free energy [2] ± ± [W ] = WH [H] + WK [K],

(54)

in which the same superscript (+ or −) must be used in both terms on

20

the right-hand side. This is known in the Calculus of Variations as the Weierstrass-Erdmann condition. Strong minimizers satisfy the stability condition [2] W (H + ∆H, K + ∆K) − W (H, K) ≥ (∆H)WH (H, K) + (∆K)WK (H, K). (55) This is the Weierstrass-Graves inequality for biomembranes. The weak minimizers satisfy the Legendre-Hadamard condition 1 4 WHH

+ ςWHK + ς 2 WKK ≥ 0,

(56)

which follows from the linearization of (55). A specialized version of this theory, for an energy density of Ginzburg-Landau type, furnishes a model of the phenomenon of necking and budding observed in phase-separated vesicles [2].

8

Conclusion

In this chapter, we reviewed the basic theory of the mechanics of biomembranes at the continuum level. This mathematical framework has been applied to study various physical phenomena associated with biomembranes such as the study of equilibrium shapes of vesicles and cells, such as red blood cells, shapes of membranes in the presence of proteins, and biological phenomena such as endocytosis, necking and budding of phase-separated vesicles, and adhesion. It has provided considerable insight into the underlying physics behind several biological phenomena. In addition to the theory presented here, direct numerical schemes have also been developed to model the shapes of two-dimensional surfaces with bending resistance. These include a program called surface evolver [7] and a method based on the phase-field theory (see [16] and references therein). They have been successfully employed to model membrane geometries lacking axisymmetry and evolving topology.

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References [1] A. Agrawal and D.J. Steigmann. Boundary-value problems in the theory of lipid membranes. Continuum Mech. Thermodyn. (In print). [2] A. Agrawal and D.J. Steigmann. Coexistent fluid-phase equilibria in biomembranes with bending elasticity. J. Elasticity, 93:63–80, 2008. [3] A. Agrawal and D.J. Steigmann. Modeling protein-mediated morphology in biomembranes. Biomechanics and Modeling in Mechanobiology, 2008. http://dx.doi.org/10.1007/s10237-008-0143-0. [4] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. Molecular Biology of the Cell. Garland Science, New York, 2002. [5] T. Baumgart, S. Das, W.W. Webb, and J. T. Jenkins. Membrane elasticity in giant vesicles with fluid phase coexistence. Biophysical J., 89:1067–1080, 2005. [6] T. Baumgart, S.T. Hess, and W.W. Webb. Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature, 425:821–824, 2003. [7] K. Brakke. Surface evolver. http://www.susqu.edu/facstaff/b/brakke /evolver/evolver.html. [8] R. Capovilla and J. Guven. Geometry of lipid vesicle adhesion. Phys. Rev. E, 66:041604 1–6, 2002. [9] R. Capovilla, J. Guven, and J.A. Santiago. Lipid membranes with an edge. Phys. Rev. E, 66:021607 1–7, 2002. [10] S. Das and Q. Du. Adhesion of vesicles to curved substrates. Phys. Rev. E, 77:011907 1–7, 2008. [11] P. Decuzzi and M. Ferrari. The receptor-mediated endocytosis of nonspherical particles. Biophysical J., 94:3790–3797, 2008.

22

[12] M. Deserno. Elastic deformation of a fluid membrane upon colloid binding. Phys. Rev. E, 69:031903 1–14, 2004. [13] M. Deserno, M.M. M¨ uller, and J. Guven. Contact lines for fluid surface adhesion. Phys. Rev. E, 76:011605 1–10, 2007. [14] H. J. Deuling and W. Helfrich. Red blood cell shapes as explained on the basis of curvature elasticity. Biophysical J., 16:861–868, 1976. [15] H. J. Deuling and W. Helfrich. The curvature elasticity of fluid membranes: A catalogue of vesicle shapes. SIAM Journal on Applied Mathematics, 37:1335–1345, 1982. [16] Q. Du, C. Liu, and X. Wang. Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Computational Phys., 212:757–777, 2006. [17] L.B. Freund and Y. Lin. The role of binder mobility in spontaneous adhesive contact and implications for cell adhesion. J. Mech. Phys. Solids, 52:2455–2472, 2004. [18] H. Gao, W. Shi, and L.B. Freund. Mechanics of receptor-mediated endocytosis. Proc. Natl. Acad. Sci. U.S.A., 102:9469–9474, 2005. [19] M. Grabe, J. Neu, G. Oster, and P. Nollert. Protein interactions and membrane geometry. Biophysical J., 84:854–868, 2003. [20] W. Helfrich. Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch, 28c:693–703, 1973. [21] J.T. Jenkins. The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math., 32:755–764, 1977. [22] J.T. Jenkins. Static equilibrium configurations of a model red blood cell. J. Math. Biology, 4:149–169, 1977. [23] A.J. Jin, K. Prasad, P.D. Smith, E.M. Lafer, and R. Nossal. Measuring the elasticity of clathrin-coated vesicles via atomic force microscopy. Biophys. J., 90:3333–3344, 2006. 23

[24] F. J¨ ulicher and R. Lipowsky. Shape transformations of vesicles with intramembrane domains. Phys. Rev. E, 53:2670–2683, 1996. [25] K.S. Kim, J. Neu, and G. Oster. Curvature-mediated interactions between membrane proteins. Biophysical J., 75:2274–2291, 1998. [26] K.S. Kim, J. Neu, and G. Oster. Effect of protein shape on multibody interactions between membrane inclusions. Phys. Rev. E, 61:4281– 4285, 2000. [27] J. Liu, M. Kaksonen, D.G. Drubin, and G. Oster. Endocytic vesicle scission by lipid phase boundary forces. Proc. Natl. Acad. Sci. U.S.A., 103:10277–10282, 2006. [28] J. Liu, Y. Sun, D.G. Drubin, and G. Oster. A mechanochemical model for endocytic vesicle formation. Cell (Submitted). [29] J. C. Luke. A method for the calculation of vesicle shapes. J. Phys. (France), 42:333–345, 1976. [30] M.M. M¨ uller, M. Deserno, and J. Guven. Interface-mediated interactions between particles: A geometrical approach. Phys. Rev. E, 72:061407 1–17, 2005. [31] Z.-C. Ou-Yang, J.-X. Liu, and Y.-Z. Xie. Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases. World Scientific, Singapore, 1999. [32] M.M. Perry and A.B. Gilbert. Yolk transport in the ovarian follicle of the hen (Gallus domesticus): Lipoprotein-like particles at the periphery of the oocyte in the rapid growth phase. J. Cell Sci., 39:257–272, 1979. [33] R. Rosso and E.G. Virga. Adhesion by curvature of lipid tubules. Continuum Mech. Thermodyn., 10:359–367, 1998. [34] R. Rosso and E.G. Virga. Adhesive borders of lipid membranes. Proc. R. Soc. Lond. A, 455:4145–4168, 1999. 24

[35] U. Seifert, K. Berndl, and R. Lipowsky. Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models. Phys. Rev. A, 44:1182–1202, 1991. [36] U. Seifert and R. Lipowsky. Adhesion of vesicles. Phys. Rev. A, 42:4768–4771, 1990. [37] S. Sen, S. Subramanian, and D.E. Discher. Indentation and adhesive probing of a cell membrane with afm: Theoretical model and experiments. Biophys. J., 89:3203–3213, 2005. [38] V.B. Shenoy and L.B. Freund. Growth and shape stability of a biological membrane adhesion complex in the diffusion-mediated regime. Proc. Natl. Acad. Sci. U.S.A., 102:3213–3218, 2005. [39] A.-S. Smith, B.G. Lorz, S. Goennenwein, and E. Sackmann. Forcecontrolled equilibria of specific vesicle-substrate adhesion. Biophys. J., 90:L52–L54, 2006. [40] A.-S. Smith, E. Sackmann, and U. Seifert. Effects of a pulling force on the shape of a bound vesicle. Europhys. Letters, 64:281–287, 2003. [41] E. L. Snapp, R. S. Hegde, M. Francolini, F. Lombardo, S. Colombo, E. Pedrazzini, N. Borgese, and J. Lippincott-Schwartz. Formation of stacked ER cisternae by low affinity protein interactions. J. Cell Biol., 163:257–269, 2003. [42] D.J. Steigmann. Fluid films with curvature elasticity. Arch. Rational Mech. Anal., 150:127–152, 1999. [43] D.J. Steigmann.

On the relationship between the cosserat and

Kirchhoff-Love theories of elastic shells. Math. Mech. Solids, 4:275– 288, 1999. [44] D.J. Steigmann. Irreducible function bases for simple fluids and liquid crystal films. Z. Angew. Math. Phys., 54:462–477, 2003.

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[45] D.J. Steigmann, E. Baesu, R.E. Rudd, J. Belak, and M. McElfresh. On the variational theory of cell-membrane equilibria. Interfaces and Free Boundaries, 5:357–366, 2003. [46] Z.C. Tu and Z.C. Ou-Yang. Lipid membranes with free edges. Phys. Rev. E, 68:061915 1–7, 2003.

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