An analysis of lipid membrane morphology in the presence of coordinate-dependent non-uniformity

Abstract

A model for the mechanics of lipid membranes with non-uniform (coordinate-dependent) properties is discussed. The coordinate-dependent responses of the membranes are incorporated via the augmented non-uniform energy function and material parameters, which are dependent explicitly on the surface coordinates. We formulate the associated normal and tangential Euler equilibrium equations through which the coordinate-dependent responses of membranes are characterized. The admissible boundary conditions are taken from the existing non-linear model but reformulated and adopted to the present framework. Within the prescription of superposed incremental deformations, a compatible linear model is also formulated, from which a complete analytical solution describing the non-uniform responses of the membrane subjected to substrate–membrane interactions is obtained.

Keywords

Lipid membranes coordinate-dependent properties non-uniform energy distribution functions linear theory membrane–substrate interactions

1. Introduction

The morphological responses of lipid membranes have been the subject of intense research that has significantly enhanced our understanding of a wide range of essential cellular functions, such as fusion, budding, and vesicular transport [1 –3]. Since lipid membranes are quite fragile and negligibly thin (typically 3–5 nm), analyses of their various mechanical properties are, most often, achieved via the use of an artificial “model”. This includes the development of continuum-based models describing the mechanics of the membranes, which is also in a period of intense study (see, for example, [4 –7]). Lipid bilayer membranes are complex assemblies of lipid molecules (phospholipids), which are characterized by hydrophilic head groups and hydrophobic tails. These molecules arrange themselves into a two-layered sheet (a bilayer) with opposing orientations that form a critically important interface within biological cells, providing a selective permeability barrier for each cell [6 –9]. This occurs, driven by the hydrophobic effect, over length scales of the order of molecular dimensions. Therefore, a lipid bilayer can be regarded as a closed membrane, much like a thin-film sandwich structure, where a fluid-like substance is present between the two films. Within this context, the development of theoretical prediction models for the mechanics of lipid membranes is facilitated by the differential geometry of a surface and the theory of an elastic surface, such that the deformation energy of a thin membrane can be expressed by the mean and Gaussian curvature of a surface Gaussian curvature of a surface [10, 11]. The energy potential that accommodates the bilayer symmetry with ensuing free energy minima was proposed by Helfrich [12], from which a well-known “membrane shape equation” is formulated. The Helfrich model has been successfully implemented in a wide range of membrane problems, including budding formations [13, 14], membrane–substrate interactions [5, 15, 16], and spontaneous curvatures [17, 18].

Recent efforts emphasize the refinement of the membrane’s energy potential and the associated mathematical framework to obtain more comprehensive and efficient prediction models for the descriptions of lipid membranes. Rangamani et al. [19] developed the general non-linear model of membranes, which incorporates the effects of intramembrane viscosity from the elastic model of surfaces [20, 21], and predicted the deformations of membranes subjected to a uniformly distributed pressure. A compatible linear model was formulated by Zeidi and Kim [22], within the prescription of superposed incremental deformations. Zeidi and Kim [22] obtained a complete analytical expression to describe the deformations of a lipid membrane, subjected to intramembrane viscous flow and substrate–membrane interactions. A constitutive framework that incorporates the tilt and distention of lipid membranes in involved responses of membranes was established by Steigmann [23]. The model is further adopted in studies of variable tilt [24] and thickness distension (without tilt) [25] of the membranes. To this end, Steigmann [6] developed a comprehensive model (including shape equations and admissible boundary conditions) for the mechanics of lipid membranes, which accommodates tilt, distension, diffusion, and viscous flow, from the theory of three-dimensional liquid crystals [26 –28].

Most of the aforementioned studies presume uniform properties throughout the membrane (i.e. coordinate-independence) in an effort to obtain mathematically tractable systems and analyses. In general, though, the responses of membranes are coordinate-dependent, owing to the complex nature of such membrane systems and processes as diffusion and non-uniform protein distributions [29]. For example, drug-induced protein diffusion may be considered an energy-dissipative process [30] and therefore may further induce non-uniformity in the membrane. In addition, phase separations or local enrichments of particular lipid species arising in membrane–protein substrate interactions [31] may be considered a source of non-uniformity. Protein-induced pre-deformations may also give rise to the possible non-uniformity. For example, BIN/amphiphysin/Rvs (BAR) proteins behave as a scaffold-like structure, acting as a local tension field, which may be viewed as a non-uniformity [32]. A class of problems pertaining to the non-uniform properties of lipid membranes is discussed in [4] and [20], where the non-uniformity is accommodated via a coordinate-dependent energy function of membranes. Further, Agrawal and Steigmann [17, 29] investigated the non-uniform properties of membranes induced by surface diffusion and possible non-uniform protein distributions, and predicted the non-uniform spontaneous curvature of the membranes. However, deformation analyses of membranes accounting for explicit coordinate-dependent non-uniformities, particularly those arising in the membrane shape equation, remain lacking in the literature. In this work, we study a model that describes the non-uniform properties of lipid membranes subjected to membrane–substrate interactions and lateral pressure. The non-uniformity of the membrane is incorporated via the introduction of non-uniform energy distribution functions and material parameters, which are explicitly dependent on the surface coordinates. The corresponding normal and tangential Euler equilibrium equations are then derived, through which the coordinate-dependent responses of membranes are characterized. The admissible boundary conditions are adopted from the work of Agrawal and Steigmann [5], yet they are reformulated in the present context with the refined energy potential of the membrane. Within the prescription of superposed incremental deformations [33, 34], a compatible linear model is developed from which a complete analytical solution describing the non-uniform responses of the membrane undergoing substrate–membrane interactions is obtained.

More importantly, we found that, unlike those from the non-linear theory, the coordinate-dependent material parameters of membranes do not necessarily result in non-uniform responses of membranes. In fact, it is shown through the tangential and normal Euler equations that the terms associated with the non-uniform material parameters vanish for small deformations superposed on large deformations. This result further suggests that the descriptions of the non-uniform responses of membranes are intrinsically limited to those achieved via the augmented energy potential, which is explicitly dependent on the surface coordinates. Nevertheless, the solutions obtained from the proposed linear model demonstrate reasonable agreement with those obtained from the non-linear analysis for the small deformation regime. Both circumferential and radial non-uniform energy distribution functions are considered in order to assimilate possible non-uniformities, and the resulting deformation fields demonstrate clear signs of coordinate dependency. In particular, we have shown that the principle of superposition from linear elasticity remains valid for the present application. The result may promote the relevant morphological study of membranes, in that the solution of combined non-uniform energy distribution cases can be directly obtained via the summation of the solutions obtained, respectively, from the circumferential and radial non-uniform cases. Numerical non-linear solutions are also obtained for the purposes of comparison with those from the linear analysis and demonstration of the more general coordinate-dependent responses of membranes. The non-linear solution predicts the off-centered biconcave morphology of lipid [35, 36] and the multiple peak formations of abnormal cell membranes (burr cell), which are commonly observed in uremia and chronic kidney disease [37, 38]. Lastly, the solution presented accommodates the results in [5] and [22].

Throughout the paper, we make use of a number of well-established symbols and conventions. Thus, unless otherwise stated, Greek indices take the values in { $1, 2$ } and, when repeated, are summed over their ranges. Lastly, $(*)_{, α}$ denotes the derivative of “*” with respect to a coordinate $θ^{α}$ and $W_{K}$ stands for the derivatives of a scalar-valued function $W (K)$ with respect to the parameter K.

2. Preliminary: Surface geometry

In this section, we summarize the results from the differential geometry for use in the derivation of the non-uniform membrane formulation. Let $r (θ^{α})$ define the parametric position vector in three-dimensional space of a point on the evolving surface $ω$ with coordinates $θ^{α}$ . Accordingly, the tangent vectors to the surface $ω$ induced by the parameterization $r (θ^{α})$ can be computed as

r_{, α} = \partial r / \partial θ^{α} = a_{α} .

(1)

The local surface orientation $n$ is then obtained by $n (θ^{α}) = (ε^{α β} a_{α} \times a_{β}) / 2$ , where $ε^{α β} = e^{α β} / \sqrt{a}$ refers to the permutation tensor density with $a = det (a_{α β})$ and $e^{11} = e^{22} = 0, e^{12} = - e^{21} = 1 .$ Further, the associated surface metric is defined as

a_{α β} = a_{α} \cdot a_{β},

(2)

where the dot denotes the conventional Euclidean inner product. The matrix of the surface metric is positive-definite (i.e $a > 0$ ), so that the inverse of the matrix of the surface metric $a_{α β}$ exists (i.e. $a^{α β} = (a_{α β})^{- 1}$ ) and is referred to as the dual metric component of $a_{α β}$ . Hence, the surface covariant differentiation is computed as [39]

a_{α; β} = a_{α, β} - Γ_{α β}^{λ} a_{λ},

(3)

where $Γ_{α β}^{λ} = a_{α, β} \cdot a^{λ}$ are the Christoffel symbols induced by the local surface coordinate and $a^{λ}$ is the dual basis of $a_{β}$ via the relation $a^{α} = a^{α β} a_{β}$ . These results furnish the well-known Gauss and Weingarten equations:

a_{α, β} = Γ_{α β}^{γ} a_{γ} + b_{α β} n; b_{α β} = a_{α, β} \cdot n (Gauss)

(4)

and

b_{α β} = - n_{, α} \cdot a_{β} (Weingarten) .

(5)

Further, equations (1) to (5) furnish the useful relations

a_{α; β} = b_{α β} n and n_{, α} = - b_{α β} a^{β} = - b_{α}^{β} a_{β} .

(6)

Here, $b_{α β}$ are the coefficients of the second fundamental form and $b_{α}^{β}$ are the mixed components of the curvature. The symmetric property of the second fundamental from $(b_{α β})$ can be seen as

b_{α β} = - n_{, α} \cdot a_{β} = n \cdot a_{β, α} = n \cdot r_{, β α} = b_{β α},

(7)

which may also be used in later sections.

Finally, the mean and Gaussian curvatures are given by

H = \frac{1}{2} a^{α β} b_{α β} and K = \frac{1}{2} ε^{α β} ε^{λ μ} b_{α λ} b_{β μ},

(8)

which satisfy the following equalities

b^{α β} = 2 H a^{α β} - {\tilde{b}}^{α β} (Cayley - Hamilton theorem) and a^{β α} K = b_{μ}^{β} {\tilde{b}}^{μ α},

(9)

where ${\tilde{b}}^{α β}$ is the contravariant cofactor;

{\tilde{b}}^{α β} = ε^{α λ} ε^{β γ} b_{λ γ} .

(10)

3. Non-uniform membrane shape equation

The membrane shape equations are given by [5]

Δ (\frac{1}{2} W_{H}) + (W_{K})_{; β α} {\tilde{b}}^{β α} + W_{H} (2 H^{2} - K) + 2 KH W_{K} - 2 H (W + λ) = p on ω,

(11)

and

λ_{, α} = - \frac{\partial W}{\partial θ^{α}} .

(12)

If the membrane energy potential W does not depend explicitly on the coordinates (i.e. uniform membranes), this yields [17]

λ = constant, ∵ λ_{, α} = 0,

(13)

and the values of $λ$ may be configured via the membrane shape equation (see, for example, [16]). In the case of a non-uniform membrane, it is found that [5]

λ_{, α} = - \frac{\partial W}{\partial θ^{α}} \neq 0 .

(14)

Equation (14) arises as a result of the explicit coordinate dependence in the function W, which arises from the possible non-uniformity of the membrane properties. Hence, to accommodate a particular state of non-uniformity, $λ$ most be determined by solving the relevant Euler equations: two tangential equations in the coordinate directions (equation (14)) and the membrane shape equation (equation (11)).

In this study, we propose the following form of the modified Helfrich energy potential to assimilate non-uniform responses of membranes:

W (H, K; θ^{α}) = ϕ (θ^{α}) + α (θ^{α}) H^{2} + β (θ^{α}) K .

(15)

It is also noted that, the uniform membrane case (i.e. $W = k H^{2} + \bar{k} K$ [12]) can be retrieved from equation (15) by setting $ϕ (θ^{α}) = 0,$ $α (θ^{α}) = k$ and $β (θ^{α}) = \bar{k} .$

In view of equation (15), we evaluate

W_{H} = 2 α (θ^{α}) H, W_{K} = β (θ^{α}),

(16)

and thereby obtain from equation (11)

Δ (α (θ^{α}) H) + (β (θ^{α} {))}_{; β α} {\tilde{b}}^{β α} + 2 α (θ^{α}) H (H^{2} - K) - 2 H (ϕ (θ^{α}) + λ) = p,

(17)

which serves as the shape equation for non-uniform membranes. Here, since $β (θ^{α})$ is a scaler-valued function, $(β (θ^{α} {))}_{; β α}$ can be evaluated as

β (θ^{α})_{; β α} = (β (θ^{α})_{, β})_{; α} = β (θ^{α})_{, β α} - β (θ^{α})_{, λ} Γ_{β α}^{λ},

(18)

and ${\tilde{b}}^{β α}$ is the contravariant cofactor defined in equation (10).

Further, from equation (15), we compute

\begin{matrix} W_{, α} = W_{H} H_{, α} + W_{K} K_{, α} + \frac{\partial W}{\partial θ^{α}} \\ = 2 α (θ^{α}) H H_{, α} + β (θ^{α}) K_{, α} + \frac{\partial W}{\partial θ^{α}}, \end{matrix}

(19)

where $\partial W / \partial θ^{α}$ is the explicit coordinate derivative of W,

\frac{\partial W}{\partial θ^{α}} = \frac{\partial ϕ (θ^{α})}{\partial θ^{α}} + \frac{\partial α (θ^{α})}{\partial θ^{α}} H^{2} + \frac{\partial β (θ^{α})}{\partial θ^{α}} K .

(20)

Thus, the associated Euler equilibrium equations are found to be

λ_{, r} = - \frac{\partial W}{\partial θ^{r}} = - [\frac{\partial ϕ (θ^{α})}{\partial θ^{r}} + \frac{\partial α (θ^{α})}{\partial θ^{r}} H^{2} + \frac{\partial β (θ^{α})}{\partial θ^{r}} K],

(21)

and

λ_{, θ} = - \frac{\partial W}{\partial θ^{θ}} = - [\frac{\partial ϕ (θ^{α})}{\partial θ^{θ}} + \frac{\partial α (θ^{α})}{\partial θ^{θ}} H^{2} + \frac{\partial β (θ^{α})}{\partial θ^{θ}} K],

(22)

where $ϕ (θ^{α}),$ $β (θ^{α})$ , and $γ (θ^{α})$ can be chosen to achieve particular types of non-uniform distribution.

It is noted here that the proposed energy density function, equation (15), can be directly used in conjunction with the existing results (equations (11) and (12)) to yield the associated Euler equations (i.e. equations (17), (21), and (22)) without further modification (see Remark 1). This case, in which the refinement of the energy density function W results in non-standard forms of the shape equation, is discussed in [4, 17, 29].

Remark 1. Since the coordinate derivatives of W appear only in the tangential variations, the introduction of a non-uniform energy function, which depends on the surface coordinates $θ^{α}$ (e.g. $ϕ (θ^{α}),$ $α (θ^{α})$ , …), does not affect the structure of the shape equation (equation (11)) obtained from the induced normal variation.

To see this, we consider the virtual displacement of the equilibrium position field $r (θ^{α})$ evaluated at the particular configuration of the surface ( $ϵ = 0$ ) as

u (θ^{α}) = \frac{\partial r (θ^{α}; ϵ)}{\partial ϵ} \equiv \overset{\cdot}{r} .

(23)

The decompositions of $u (θ^{α})$ in the tangential and normal directions then yield

u (θ^{α}) = u^{α} a_{α} + w (θ^{α}) n,

(24)

where $u^{α} = \overset{\cdot}{r} \cdot a^{α}$ and $w = \overset{\cdot}{r} \cdot n$ , respectively, are the tangential and normal variations.

Now, the variations of the energy function W are evaluated using the chain rule:

\overset{\cdot}{W} = \frac{\partial W}{\partial ϵ} = \frac{d W}{d θ^{α}} \frac{\partial θ^{α}}{\partial r} \cdot \frac{\partial r}{\partial ϵ} + \frac{d W}{d n (θ^{α})} \frac{\partial n (θ^{α})}{\partial r} \cdot \frac{\partial r}{\partial ϵ} = W_{, α} a^{α} \cdot u + W_{n} n \cdot u .

(25)

Substituting equation (24) into equation (25) and using $a^{α} \cdot n = a_{α} \cdot n = 0$ , we find

\overset{\cdot}{W} = W_{, α} u^{α} + W_{n} w .

(26)

Thus, $W_{, α}$ , which includes the explicit coordinate derivative $\partial W /$ $\partial θ^{θ}$ (see equation (19)), is decoupled from the normal variation.

Further, the associated Euler equations are given by (see [5] and [25])

u^{α} [W_{H} H_{, α} + W_{K} K_{, α} - (W_{, α} + λ_{, α})] = 0,

(27)

and

w [Δ (\frac{1}{2} W_{H}) + {(W_{K})}_{; β α} {\tilde{b}}^{β α} + W_{H} (2 H^{2} - K) + 2 KH W_{K} - 2 H (W + λ)] = wp,

(28)

where $W_{, α}$ and $λ_{, α}$ in equation (27) arise as a result of the two-dimensional incompressibility condition (i.e. $J = 1) .$ In equation (27), caution needs to be taken that $W_{, α}$ should not be confused with $\partial W / \partial θ^{α}$ , unlike other terms (e.g. $H_{, α} = \partial H / \partial θ^{α}$ , …), since, in general, W is explicitly dependent on the surface coordinate $θ^{α}$ (i.e. $W_{, α} \neq \partial W / \partial θ^{α}$ ). Lastly, from equation (19), equation (27) is reduced to

λ_{, α} = - \frac{\partial W}{\partial θ^{α}} .

(29)

Therefore, it is evident from equations (28) and (29) that the resulting Euler equations remain intact even with the presence of the non-uniform distributions of $ϕ (θ^{α}),$ $α (θ^{α})$ , and $β (θ^{α})$ .

3.1. Formulations under Monge parametric representation

Using the Monge representation, the parametric position vector can be expressed as

r (θ^{α}) = θ + z (θ^{α}) k,

(30)

where $θ (θ^{α})$ is the surface coordinate of a plane, $ω,$ and $k$ is the unit normal to $ω$ . Hence, the shape of the membrane is determined by the single function $z (θ^{α})$ . In general non-axisymmetric polar coordinates, equation (30) becomes

r = r e_{r} (θ) + z (r, θ) k, e_{r} (θ) = \cos θ e_{1} + \sin θ e_{2} .

(31)

Substitution of equation (31) into equations (1) and (2) yields

a_{1} = e_{r} + z_{, r} \underline{k}, a_{2} = r e_{r} + z_{, θ} \underline{k},

(32)

a_{11} = a_{1} \cdot a_{1} = 1 + z_{, r}^{2}, a_{22} = r^{2} + z_{, θ}^{2}, a_{12} = z_{, r} z_{, θ} = a_{21},

(33)

and

a = det (a_{α β}) = r^{2} (1 + z_{, r}^{2}) + z_{, θ}^{2},

(34)

which are, respectively, the surface coordinates, the induced metric, and its determinant. Similarly, the dual basis vectors and the matrix of dual metric components ( $a^{α β} = (a_{α β})^{- 1}$ ) can be computed as

\begin{matrix} a^{1} = a^{11} a_{1} + a^{12} a_{2} = \frac{1}{a} [(r^{2} + z_{, θ}^{2}) e_{r} - r z_{, r} z_{, θ} e_{θ} + r^{2} z_{, r} k], \\ a^{2} = \frac{1}{a} [- r z_{, r} z_{, θ} e_{r} + r (1 + z_{, r}^{2}) e_{θ} + z_{, θ} k], \end{matrix}

(35)

and

a^{11} = \frac{r^{2} + z_{, θ}^{2}}{a}, a^{22} = \frac{1 + z_{, r}^{2}}{a}, a^{12} = a^{21} = - \frac{z_{, r} z_{, θ}}{a} .

(36)

Further, from equations (32) to (36), the associated surface normal and the curvature tensor are now defined by

n = \frac{a^{1} \times a^{2}}{\sqrt{a}} = \frac{1}{\sqrt{a}} (- z_{, θ} e_{θ} - r z_{, r} e_{r} + r k), b = b_{α β} a^{α} \otimes a^{β}; b_{α β} = a_{β, α} \cdot n .

(37)

Thus, the complete expression of $b$ , for general (non-axisymmetric) polar coordinates, is obtained:

\begin{matrix} b = \frac{r z_{, rr}}{a^{2} \sqrt{a}} [(r^{2} + z_{, θ}^{2}) e_{r} - r z_{, r} z_{, θ} e_{θ} + r^{2} z_{, r}^{2} k] \otimes [(r^{2} + z_{, θ}^{2}) e_{r} - r z_{, r} z_{, θ} e_{θ} + r^{2} z_{, r}^{2} k] \\ + \frac{r z_{, r θ} - z_{, θ}}{a^{2} \sqrt{a}} [(r^{2} + z_{, θ}^{2}) e_{r} - r z_{, r} z_{, θ} e_{θ} + r^{2} z_{, r}^{2} k] \otimes [- r z_{, r} z_{, θ} e_{r} + r (1 + z_{, r}^{2}) e_{θ} + z_{, θ} k] \\ + \frac{r z_{, r θ} - z_{, θ}}{a^{2} \sqrt{a}} [- r z_{, r} z_{, θ} e_{r} + r (1 + z_{, r}^{2}) e_{θ} + z_{, θ} k] \otimes [(r^{2} + z_{, θ}^{2}) e_{r} - r z_{, r} z_{, θ} e_{θ} + r^{2} z_{, r}^{2} k] \\ + \frac{r^{2} z_{, r} - r z_{, θ θ}}{a^{2} \sqrt{a}} [- r z_{, r} z_{, θ} e_{r} + r (1 + z_{, r}^{2}) e_{θ} + z_{, θ} k] \otimes [- r z_{, r} z_{, θ} e_{r} + r (1 + z_{, r}^{2}) e_{θ} + z_{, θ} k], \end{matrix}

(38)

where the coefficients of the second fundamental form $b_{α β}$ can be formulated using equations (32), (35) and (36). For example, we find that

b_{11} = n \cdot a_{1, 1} = \frac{1}{\sqrt{a}} r z_{, rr}; a_{1, 1} = \frac{\partial (e_{r} (θ) + z_{, r} (r, θ) k)}{\partial r} = z_{, rr} k,

(39)

and similarly for $b_{22},$ $b_{12}$ , and $b_{21} .$

Lastly, from equations (8) and (36) to (38), we find the following expressions for the the mean and Gaussian curvatures:

H = \frac{1}{2 a^{3 / 2}} [r^{3} z_{, rr} + r^{2} z_{, r} + r^{2} z_{, r}^{3} + r z_{, rr} z_{, θ}^{2} - 2 r z_{, r θ} z_{, r} z_{, θ} + r z_{, θ θ} + r z_{, θ θ} z_{, r}^{2} + 2 z_{, r} z_{, θ}^{2}],

(40)

K = \frac{1}{a^{2}} [r^{3} z_{, r} z_{, rr} + r^{2} z_{, rr} z_{, θ θ} - r^{2} z_{, r θ}^{2} + 2 r z_{, r θ} - z_{, θ}^{2}] .

(41)

In the actual implementation of the Monge parameterization, equation (17) can be readily reformulated by using the results in equations (32) to (41). The corresponding formulations are relatively straightforward and hence omitted for the sake of simplicity.

Our intention is to assimilate the substrate–membrane interactions in the presence of non-uniform energy distributions. For this purpose, the following interaction boundary conditions are adopted from the work of Agrawal and Steigmann [5]:

F_{v} \cos γ + F_{n} \sin γ - M n \cdot (\nabla_{Γ} γ - Bn) = σ, F_{τ} - M τ \cdot (\nabla_{Γ} γ - Bn) = 0,

(42)

F_{v} = W + λ - κ_{v} M, F_{τ} = - τ M, F_{n} = {(τ W_{K})}^{'} - {(\frac{1}{2} W_{H})}_{, v} - (W_{K})_{, β} {\tilde{b}}^{α β} v_{α},

(43)

where $F_{i}$ and M are the forces in the i direction and the moment on the boundary, respectively. Also, $B,$ $τ$ -v- $: n$ , and $γ$ are, respectively, the curvature tensor of the interacting boundary, the associated normal–tangential coordinates, and the interaction angle.

In the present case, in which the non-uniform membrane interacts with a circular cylindrical substrate of radius $r = a$ , we find

B = - a^{- 1} e_{θ} \otimes e_{θ} and γ = π / 2 .

(44)

Accordingly, equation (42) reduces to

F_{n} = σ and F_{τ} = 0 (∵ \cos γ = 0, \nabla_{Γ} γ = 0, and n \cdot e_{θ} = 0),

(45)

at the interacting boundary $Γ .$ Further, in view of equations (16) and (42), equation (43) now yields

F_{n} = (τ (s) β (θ^{α} {))}^{'} - (α (θ^{α}) H)_{, v} - (β (θ^{α} {))}_{, β} {\tilde{b}}^{α β} v_{α} = σ,

(46)

where $(τ (s) β (θ^{α} {))}^{'}$ is the arc-length derivative of $τ (s) β (θ^{α})$ , which can be evaluated as

τ (s)^{'} = \frac{\partial τ}{\partial s} = \frac{\partial τ}{\partial θ^{β}} \frac{\partial θ^{β}}{\partial s} = (τ)_{, β} τ^{β} = (b^{α β} τ_{α} v_{β})_{, γ} τ^{γ} = (b^{α β} τ^{β} a_{α β} v^{α} a_{α β})_{, γ} τ^{γ},

and

β (θ^{α})^{'} = (β (θ^{α} {))}_{, β} τ^{β} .

(47)

These expressions are then rewritten using the Monge representation for further analysis. For example, we have

(b^{α β} τ^{β} a_{α β} v^{α} a_{α β})_{, γ} τ^{γ} = {[\frac{1}{\sqrt{r^{2} [1 + {(z_{, r})}^{2}] + {(z_{, θ})}^{2}}} (r z_{, r θ} - z_{, θ}) (z_{, r} z_{, θ})^{2}]}_{, θ},

(48)

and similarly for other terms.

Detailed derivations and phenomenological implications of these boundary forces are available in [5] and [6], respectively. The interaction boundary conditions (i.e. equations (45) and (46)), together with equations (17) to (22) solve the deformed configuration of the non-uniform membrane subjected to membrane–substrate interactions. In the assimilation, we employed commercial packages (e.g. Matlab and Comsol) to solve the obtained partial differential equations (PDEs); the corresponding results are presented in later sections.

3.2. Linear model for non-uniform membranes

The formulation of the non-uniform equilibrium equations (i.e. equations (17) to (22)) in terms of equations (35) to (41) yields a highly non-linear PDE system, which often requires considerable computational resources. Alternatively, “admissible linearization” may be considered, through which one could obtain mathematically tractable systems and, more importantly, analytical expressions of solutions with minimal loss of generality. The concept has been widely and successfully implemented in studies of relevant subjects (see, for example, [5, 7, 15]). Within this setting, the derivatives of $z (θ^{α})$ of all orders are considered to be “small” (e.g. $z_{, α} << 1$ ), and thus their products can be neglected. Accordingly, using the notation “≈” to identify equations to the leading-order approximation of $z (θ^{α})$ , we find

\begin{matrix} a_{11} ≅ r^{2}, a_{22} ≅ 1, a_{12} = a_{21} ≅ 0, det | a_{α β} | = a ≅ r^{2}, \\ a^{11} ≅ 1, a^{22} ≅ \frac{1}{r^{2}}, a^{12} = a^{21} ≅ 0, \\ Γ_{12}^{1} = Γ_{21}^{1} = Γ_{11}^{2} = Γ_{11}^{1} = Γ_{22}^{2} = 0, Γ_{12}^{2} = Γ_{21}^{2} = \frac{1}{r}, Γ_{22}^{1} = - r, \\ n ≅ k - \nabla_{p} z, a^{1} ≅ e_{r} + z_{, r} k, a^{2} ≅ \frac{1}{r} e_{θ} + \frac{1}{r^{2}} k, \end{matrix}

and

b ≅ z_{, rr} (e_{r} \otimes e_{r}) + \frac{r z_{, r θ} - z_{, θ}}{r^{2}} [(e_{r} \otimes e_{θ}) + (e_{θ} \otimes e_{r})] + \frac{r z_{, r} + z_{, θ θ}}{r^{2}} (e_{θ} \otimes e_{θ}) = \nabla_{p}^{2} z,

(49)

where the subscript $(*)_{p}$ denotes the projected counterparts of (*) on the coordinate plane $ω_{p}$ , $\nabla_{p}^{2} z$ is the second gradient, and $Δ_{p} z = tr (\nabla_{p}^{2} z)$ is the associated Laplacian.

Thus, applying the results in equation (49), the mean and Gaussian curvatures (equations (40) and (41)) can be approximated as

H \approx \frac{1}{2} [z_{, rr} + \frac{1}{r} z_{, r} + \frac{1}{r^{2}} z_{, θ θ}] = \frac{1}{2} Δ_{p} z and K \approx 0 .

(50)

In addition, from equation (50), we reduce equations (21) and (22) to

λ_{, r} = - \frac{\partial W}{\partial θ^{r}} = - ϕ (θ^{α})_{, r}

(51)

and

λ_{, θ} = - \frac{\partial W}{\partial θ^{θ}} = - ϕ (θ^{α})_{, θ},

(52)

which serve as the linearized Euler equilibrium equations in the coordinate directions.

Remark 2. It is evident from equations (21), (22), (51), and (52) that the terms associated with $α (θ^{α})$ and $β (θ^{α})$ identically vanish after admissible linearization, regardless of the types of the distributions of $α (θ^{α})$ and $β (θ^{α})$ . For example,

\frac{\partial α (θ^{α})}{\partial θ^{r}} H^{2} ≃ 0 and \frac{\partial β (θ^{α})}{\partial θ^{r}} K ≃ 0, ∵ H^{2} ≃ 0 and K ≃ 0 .

(53)

In other words, the non-uniform potentials of $α (θ^{α})$ and $β (θ^{α})$ do not necessarily result in non-uniform responses of membranes via tangential equilibrium equations (see, equations (51) and (52). This further implies that, within the setting of superposed incremental deformations, the descriptions of non-uniform membranes are intrinsically limited to those achieved by introducing a non-uniform distribution function $ϕ (θ^{α})$ , while maintaining $α (θ^{α})$ and $β (θ^{α})$ as constants (i.e. $α (θ^{α}) = k$ and $β (θ^{α}) = \bar{k}$ ).

Based on Remark 2, the following compact form of the membrane energy potential may be proposed to accommodate the non-uniform responses of membranes:

W (H, K; θ^{α}) = k H^{2} + \bar{k} K + ϕ (θ^{α}),

(54)

where $ϕ (θ^{α})$ characterizes the particular states of non-uniformity.

Therefore, equation (17) can be approximated in accordance with equation (54) as

k Δ H - 2 H (ϕ (θ^{α}) + λ) = p,

(55)

which may serve as the compatible form of the shape equation for non-uniform membranes within the linear description.

3.3. Example1: Circumferentially non-uniform membranes

We consider the following form of the non-uniformity:

ϕ (θ) + λ (θ) = α θ^{n}; n = 1, 2, 3, \dots,

(56)

where $α$ defines the rigidity of the membrane and n controls the degrees of non-uniformity in the circumferential direction. For the purpose of demonstration, the case of $n = 2$ is presented. The solutions of other cases (e.g. $n = 3, 4, 5$ , …) can also be obtained by applying the same procedure, as will be discussed later in this section.

For $n = 2$ , equation (56) yields

λ_{, θ} = - ϕ_{, θ} + 2 α θ, and λ_{, r} = ϕ_{, r} = 0,

(57)

so that the membrane is radially uniform ( $λ_{, r} = 0$ ) but circumferentially non-uniform ( $λ_{, θ} \neq 0$ ). Now, combining equations (55) and (56), we find

H,_{rr} + \frac{1}{r} H,_{r} + \frac{1}{r^{2}} H,_{θ θ} - \frac{2 α θ^{2}}{k} H = 0,

(58)

where we set $p = 0$ from the bilayer symmetry (see [5]). The solution of equation (58) may take the following form:

H (r, θ) = \sum_{m = 0}^{\infty} R (r, θ) (C_{m} \sin (m θ) + D_{m} \cos (m θ)),

(59)

where the expression of $R (r, θ)$ can be obtained via the standard separation of variables (i.e. $\tilde{θ} (θ)_{, θ θ} / \tilde{θ} (θ) = - m^{2}$ ), as

R (r, θ) = A_{m} I_{m} (\sqrt{\frac{2 α}{k}} r θ) + B_{m} K_{m} (\sqrt{\frac{2 α}{k}} r θ),

(60)

where $I_{m}$ and $K_{m}$ are, respectively, the first and second kinds of modified Bessel function. Since the surface evolution of membranes diminishes as it approaches the boundary, equations (59) and (60) may be further reduced to

H (r, θ) = \sum_{m = 0}^{\infty} K_{m} (\sqrt{\frac{2 α}{k}} r θ) (A_{m} \sin (m θ) + B_{m} \cos (m θ)) .

(61)

We continue by combining equations (50) and (61), and thereby obtain

\sum_{m = 0}^{\infty} K_{m} (\sqrt{\frac{2 α}{k}} r θ) [A_{m} \cos (m θ) + B_{m} \sin (m θ)] = \frac{1}{2} (z_{, rr} + \frac{1}{r} z_{, r} + \frac{1}{r^{2}} z_{, θ θ}) .

(62)

The solution of this PDE can be sought in a similar form to that in equation (59):

z (r, θ) = \sum_{m = 0}^{\infty} S (r, θ) (C_{m} \sin (m θ) + D_{m} \cos (m θ)) .

(63)

Further, the substitution of equation (63) into equation (62) yields

\sum_{m = 0, 1, 2, \dots}^{\infty} 2 K_{m} (\sqrt{\frac{2 α}{k}} r θ) \frac{A_{m} \cos (m θ) + B_{m} \sin (m θ)}{C_{m} \cos (m θ) + D_{m} \sin (m θ)} = S (r, θ)_{, rr} + \frac{1}{r} S (r, θ)_{, r} - \frac{m^{2}}{r^{2}} S (r, θ),

(64)

where m is the separation variable. Thus, we obtain the following expression for $z (r, θ) :$

\begin{matrix} z (r, θ) = \sum_{m = 1, 3, 5, \dots}^{\infty} [- A_{m} \cosh (0.35 \log r) \int_{a}^{r} \frac{\sinh (0.35 \log ξ_{1}) ξ_{1} K_{\frac{m}{4}} (\sqrt{\frac{2 α}{k}} ξ_{1} θ)}{\frac{m}{4}} d ξ_{1} \\ + B_{m} \sinh (0.35 \log r) \int_{a}^{r} \frac{\cosh (0.35 \log ξ_{2}) ξ_{2} K_{\frac{m}{4}} (\sqrt{\frac{2 α}{k}} ξ_{2} θ)}{\frac{m}{4}} d ξ_{2}] [C_{m} \sin (\frac{m}{4} θ) + D_{m} \cos (\frac{m}{4} θ)], \end{matrix}

(65)

where a is the radius of the interaction boundary. The unknown constants (i.e. $A_{m},$ $B_{m,}$ $C_{m,}$ and $D_{m}$ ) can be completely determined by imposing the admissible boundary conditions (see [5]):

\nabla z (a) = 0 and H_{, r} (a) = σ / k .

(66)

For example, we expand the applied interaction force in terms of Fourier series as

\frac{σ}{k} = \sum_{m = 1, 3, 5, \dots}^{\infty} \frac{10}{m π} \sin (\frac{m π}{4}) + \sum_{m = 1, 3, 5, \dots}^{\infty} (- 1)^{(\frac{m - 1}{2})} \frac{10}{m π} \cos (\frac{m π}{4}) .

(67)

Therefore, the unknowns $A_{m}$ and $B_{m}$ can be determined from equations (61), (66), and (67) as

A_{m} = \frac{10}{m π K_{\frac{m}{4}} {(\sqrt{\frac{2 α}{k}} a θ)}_{, r}} and B_{m} = (- 1)^{(\frac{m - 1}{4})} \frac{10}{m π K_{\frac{m}{2}} {(\sqrt{\frac{2 α}{k}} a θ)}_{, r}} .

(68)

In the assimilation, we adopt the flexural modulus of the membrane as $k = 82$ pN·nm, based on the results in [40, 41]. The value of $λ$ is dependent on the membrane systems under consideration and does not have a definite range of values. The commonly used value is $λ \propto 10^{- 4}$ pN/nm (see, for example, [13, 22, 25]). In addition, the corresponding data are obtained under the normalized setting. The dimensionless parameters used in the simulations are adopted from [5, 19, 24] as

\begin{matrix} μ = \sqrt{2 λ / k} : natural length scale, \\ σ / λ : force scale (e . g . f_{n} = σ / λ : interaction force) . \end{matrix}

(69)

Figure 1 illustrates the deflections of a non-uniform membrane predicted by the proposed model at a particular configuration of $θ$ (i.e. $θ = π / 2, π, \dots$ ). It is evident that the transverse deflection of membranes increases as the intensity of the membranes’ energy distributions ( $α$ ) decreases. This is because less energy is required to induce the membranes’ deformation for small values of $α$ . Further, when the intensity factor $α$ is set as $α = 1$ , the energy potential of the non-uniform membrane at the particular configuration of $θ = 1$ rad becomes equivalent to that of the uniform membrane so that the obtained solution recovers the results in [5] (see Figure 1). Figure 2 indicates that the disparity between the linear and non-linear solutions becomes considerable with increasing interaction forces.

Figure 1.

Transverse deflections of lipid membrane with respect to $α$ when $λ + ϕ = α θ^{672}$ .

Figure 2.

Transverse deflections of circumferentially non-uniform membranes: comparison of linear and non-linear solutions.

Lastly, the non-uniform responses of drug-treated diseased cell membranes [42] may be assimilated by using the proposed model. For this purpose, we adopt the following form of the periodic energy density distribution:

ϕ + λ = α (\cos (n θ) + A),

(70)

where n characterizes circumferentially non-uniform energy distributions, $α$ is the particular rigidity of the membrane, and A is an arbitrary positive constant. Figure 3 shows that the obtained solution assimilates the echinocyte formations of cell membranes, which is one of the common abnormalities observed in deceased cells (see, for example, [42 –44]). The exact mechanisms for the phenomena have yet to be understood or predicted by the proposed model. However, the obtained solution may provide phenomenologically compatible non-uniform energy distributions of lipid membranes leading to such morphological formations. The non-uniform responses of red blood cell membranes, which result in their distinct (off-centered) biconcave discoid structure [35], may also be simulated using the proposed energy potential (equation (70)). It is shown in Figure 4 that the proposed model reproduces the off-centered non-uniform morphology of the abnormal cell membranes.

Figure 3.

(top) Assimilation of echinocyte formation when $ϕ + λ = α (\cos (8 θ) + π)$ . (bottom) Echinocyte formation of cell membrane [42].

Figure 4.

(left) Assimilation of the off-centered non-uniform morphology when $ϕ + λ = α (\cos (θ) + π)$ . (right) Off-centered non-uniform morphology of red blood cell [35].

3.4. Example 2: Radially non-uniform membranes

The cases of radially non-uniform energy distributions are also examined, where the radial distribution function is characterized by

ϕ (r) + λ (r) = α r^{n} .

(71)

In the forgoing analysis, we demonstrated the case for $n = 1 / 2$ . Like the previous section, the solutions of arbitrary-n cases can be accommodated using the same procedures, as discussed in this section.

For $n = 1 / 2$ , equation (71) furnishes

λ_{, r} = - ϕ_{, r} + \frac{α}{2} r^{- 1 / 2}, and λ_{, θ} = ϕ_{, θ} = 0,

(72)

from which the radially non-uniform membrane ( $λ_{, r} \neq 0$ ) may be assimilated. We substitute equation (71) into equation (55) and obtain

H_{, rr} + \frac{1}{r} H_{, r} + \frac{1}{r^{2}} H_{, θ θ} - \frac{2 α \sqrt{r}}{k} H = 0 .

(73)

Hence, combining equations (50) and (73) furnishes

z_{, rrrr} + \frac{2}{r^{2}} z_{, rr θ θ} + \frac{1}{r^{4}} z_{, θ θ θ θ} + \frac{2}{r} z_{, rrr} - \frac{2}{r^{3}} z_{, r θ θ} - \frac{1}{r^{2}} z_{, rr} + \frac{4}{r^{4}} z_{, θ θ} + \frac{1}{r^{3}} z_{, r} - \frac{2 α \sqrt{r}}{k} [z_{, rr} + \frac{1}{r} z_{, r} + \frac{1}{r^{2}} z_{, θ θ}] = 0 .

(74)

In the case of axisymmetric energy density distributions (i.e. circumferentially uniform and radially non-uniform), equation (74) may further reduce to

z_{, rrrr} + \frac{2}{r} z_{, rrr} - \frac{1}{r^{2}} z_{, rr} + \frac{1}{r^{3}} z_{, r} - \frac{2 α \sqrt{r}}{k} [z_{, rr} + \frac{1}{r} z_{, r}] = 0 .

(75)

The solution of equation (75) can be found as

z = A_{1} {r^{2}}_{2} F_{3} ([\frac{4}{5}, \frac{4}{5}]; [1, \frac{9}{5}, \frac{9}{5}]; \frac{4}{25} \frac{2 α}{k} r^{\frac{5}{2}}) + A_{1} r G_{2, 4}^{2, 2} (\frac{4}{25} \frac{2 α}{k} r^{\frac{5}{2}} | \begin{matrix} \frac{3}{5}, \frac{3}{5} \\ \frac{4}{5}, \frac{4}{5}, - \frac{4}{5}, - \frac{4}{5} \end{matrix}) + A_{2} In (r) + A_{3} .

(76)

Here $G_{p, q}^{m, n}$ is the Merjer $G$ function [41], where m, n, p, and q are integers corresponding to the number of the matrix group

[\frac{3}{5}, \frac{3}{5}, \frac{4}{5}, \frac{4}{5}, - \frac{4}{5}, - \frac{4}{5}]

and ${}_{2}{F_{3}}$ is the hypergeometric function [46], such that the subscripts $2$ and $3$ refer to the length of $n \times 1$ matrices, i.e.

[\frac{4}{5}, \frac{4}{5}] = 2 \times 1 and [1, \frac{9}{5}, \frac{9}{5}] = 3 \times 1 .

We seek a bounded solution within a reasonably finite domain of $a \leq r \leq r_{Finite}$ ( $r_{Finite} \propto a,$ $a :$ radius of a substrate) and therefore find from equation (76) that

z (r) = A_{1} r G_{2, 4}^{2, 2} (\frac{4}{25} \frac{2 α}{k} r^{\frac{5}{2}} | \begin{matrix} \frac{3}{5}, \frac{3}{5} \\ \frac{4}{5}, \frac{4}{5}, - \frac{4}{5}, - \frac{4}{5} \end{matrix}) + A_{2} In (r) + A_{3},

(77)

where the unknown constants $A_{1},$ $A_{2}$ , and $A_{3}$ can be uniquely determined by imposing the admissible boundary conditions (see, also [5]):

z (a) = 0, z_{, r} (a) = 0, and z_{, rrr} (a) + \frac{1}{r} z_{, rr} (a) - \frac{1}{r^{2}} z_{, r} (a) = \frac{2 σ}{k} .

(78)

Thus, we find

A_{1} = - \frac{16 r^{2} σ}{125 k G_{0, 2}^{2, 0} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} - \\ \frac{2}{5}, \frac{7}{5} \end{matrix}]}, A_{2} = \frac{8 r^{3} σ G_{1, 3}^{2, 1} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} \frac{3}{5} \\ \frac{2}{5}, \frac{2}{5}, - \frac{2}{5} \end{matrix}]}{25 k G_{0, 2}^{2, 0} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} - \\ \frac{2}{5}, \frac{7}{5} \end{matrix}]},

(79)

and

A_{3} = - \frac{8 r^{3} σ 5 \log (r) * G_{1, 3}^{2, 1} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} \frac{3}{5} \\ \frac{2}{5}, \frac{2}{5}, - \frac{2}{5} \end{matrix}] - 2 G_{2, 4}^{2, 2} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} \frac{3}{5}, \frac{3}{5} \\ \frac{2}{5}, \frac{2}{5}, - \frac{2}{5}, - \frac{2}{5} \end{matrix}]}{125 k G_{0, 2}^{2, 0} [\frac{4}{25} \frac{2 α}{k} a^{\frac{5}{2}} | \begin{matrix} - \\ \frac{2}{5}, \frac{7}{5} \end{matrix}]} .

The assimilation of the obtained solution is performed within the same normalized setting, as depicted in equation (69). Like the circumferentially non-uniform cases, the transverse deflection of membranes increases as the intensity of the membrane’s energy distribution ( $α$ ) decreases (see Figure 5). As the intensity factor approaches unity ( $α = 1$ ), the radially non-uniform energy potential becomes essentially equivalent to those from the Helfrich potential [12] within the domain of interest and thus accommodates the results in [5]. Figure 6 illustrates that the obtained linear solutions produce reasonably close predictions when compared with those from the non-linear analysis for the relatively “small” deformation regime. The disparity between the linear and non-linear solutions becomes considerable in the cases of membranes subjected to “large” transverse deformations.

Figure 5.

Transverse deflections of lipid membrane with respect to $α$ when $ϕ + λ = α \sqrt{r}$ .

Figure 6.

Transverse deflections of radially non-uniform membrane: comparison of linear and non-linear solutions.

The sequences of discocyte–stomatocyte morphology in cell membranes [47] may be mapped using the proposed energy density function (equation (71)) via $α$ and n (see Figure 7). The obtained results could have phenomenological implications by enabling estimation of the desired elastic energy to form such morphological configurations and, therefore, may further promote relevant studies (see, for example, [48, 49]). It is also noted that the principles of superposition from linear elasticity remain valid in the present case, so that the solution of the combined non-uniform energy distribution case (i.e. $ϕ + λ = α (\cos (θ) + π) + α \sqrt{r}$ ) can be directly obtained via the summation of the solutions from the circumferentially and radially non-uniform cases, respectively. That is, the results in Figures 3 and 7 can be added to yield the deformation in Figure 8.

Figure 7.

(top) Sequence of deformation mapping with respect to $α$ and n when $ϕ + λ = α r^{67 n}$ . (bottom) Discocyte–stomatocyte morphology in sequential stages [47].

Figure 8.

Deformation of radially and circumferentially non-uniform membrane when $ϕ + λ = α (\cos (θ) + π) + α \sqrt{r}$ .

Lastly, we remark that more general membrane configurations may be characterized by combining the proposed non-uniform energy density functions. For example, the deformation contour predicted by the obtained non-linear model demonstrates close similarity to the highly off-centered protrusion of red blood cell membranes that are treated under 14 days of storage in a liquid medium [36] (see Figure 9). Also, Figure 10 illustrates that the non-linear solution predicts the arrangement of peak formations of abnormal cell membranes (burr cells) commonly observed in uremia and chronic kidney disease [37, 38]. In the assimilations, the magnitude of normal deformation is controlled by the intensity parameters ( $α$ and $β)$ in the proposed energy potential (see Figure 10), and the number of peaks can be characterized by the multiplied wave terms (e.g. $\cos (θ - β π)$ ). We also note here that the non-uniform energy density distributions considered in this study may have equivalent effects on the resulting deformations of the membranes, compared with those included by prescribed non-uniform tension fields. In fact, Galan et al. [50] showed that both the surface tension and surface energy of Lennard–Jones fluids can be computed via the proposed radial distribution function. Knowing that the thickness effect has been neglected, the induced variation of membrane free energy mainly appears in the form of curvature changes of the surface through which both the surface tension and surface energy function can be altered, and vice versa [51]. Therefore, the prescription of non-uniform energy density function can lead to non-uniformity in both the surface curvature (membrane deformation) and the surface tension field.

Figure 9.

(left) Deformation contour characterized by the potential $ϕ (θ, r) = α | \cos (r - 1) θ$ . (right) Scanning electron micrograph of long-stored red blood cells [36].

Figure 10.

(left) Deformation contour characterized by the potential $ϕ (θ, r) = - α \cos (r) \cos (θ) \cos (r - 1) \cos (θ - β π)$ . (right) Scanning electron micrograph of cell membranes with different morphological configurations of stomatocytosis and echinocytosis [37].

As mentioned in earlier sections, the proposed model may not be sufficient for immediate uses in the phenomenological or clinical research of cell membrane morphology. However, it could still provide quantitative information pertaining to such membrane formation via the estimations of required elastic energy and the corresponding non-uniform energy distributions over the domain of interest.

Footnotes

Acknowledgements

Chun IL Kim thanks Professor David Steigmann for his invaluable and inspirational advice in the subject.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Sciences and Engineering Research Council of Canada (Grant Number RGPIN 04742).

ORCID iD

Chun IL Kim

References

Chernomrdik

Kozlov

Mechanics of membrane fusion. Nat Struct Mol Biol 2008; 15: 675–683.

Lenz

Morlot

Roux

Mechanical requirements for membrane fission: Common facts from various examples. FEBS Lett 2009; 583: 3839–3846.

Rothman

JE.

The machinery and principles of vesicle transport in the cell. Nat Med 2002; 8: 1059–1062.

Agrawal

Steigmann

DJ.

Coexistent fluid-phase equilibria in biomembranes with bending elasticity. J Elast 2008; 93: 63–80.

Agrawal

Steigmann

DJ.

Boundary-value problems in the theory of lipid membranes. Continuum Mech Thermodyn 2009; 21: 57–82.

Steigmann

(ed.) The role of mechanics in the study of lipid bilayers (CISM International Centre for Mechanical Sciences, vol. 577). Cham: Springer, 2017.

Belay

Kim

Schiavone

Analytical solution of lipid membrane morphology subjected to boundary forces on the edges of rectangular membranes. Continuum Mech Thermodyn 2016; 28: 305–315.

Gorter

Grendel

On bimolecular layers of lipoids on the chromocytes of the blood. J Exp Med 1925; 41: 439–443.

Robertson

JD.

The ultrastructure of cell membranes and their derivatives. Biochem Soc Symp 1959; 16: 3–43.

10.

Naghdi

PM.

On a variational theorem in elasticity and its application to shell theory. J Appl Mech 1964; 31: 647–653.

11.

Truesdell

(ed.) Linear theories of elasticity and thermoelasticity: Linear and nonlinear theories of rods, plates, and shells. Berlin: Springer-Verlag, 1973.

12.

Helfrich

Elastic properties of lipid bilayers: Theory and possible experiments. Z Naturforsch, C: Biosci 1973; 28: 693–703.

13.

Belay

Kim

Schiavone

Mechanics of a lipid bilayer subjected to thickness distension and membrane budding. Math Mech Solids 2018; 23: 67–84.

14.

Belay

Kim

Schiavone

Budding formation of lipid membranes in response to the surface diffusion of transmembrane proteins and line tension. Math Mech Solids 2016; 22: 2091–2107.

15.

Zeidi

Kim

CI.

Notes on superposed incremental deformations in the mechanics of lipid membranes. Math Mech Solids 2017; 24: 181–194.

16.

Kim

CI.

A discussion on the mechanics of lipid membranes: Lagrange multipliers and a singular substrate. Z Angew Math Phys 2017; 68: 84.

17.

Agrawal

Steigmann

DJ.

Modeling protein-mediated morphology in biomembranes. Biomech Model Mechanobiol 2008; 8: 371.

18.

Walani

Torres

Agrawal

Anisotropic spontaneous curvatures in lipid membranes. Phys Rev E 2014; 89: 062715.

19.

Rangamani

Agrawal

Mandadapu

, et al. Interaction between surface shape and intra-surface viscous flow on lipid membranes. Biomech Model Mechanobiol 2013; 12: 833–845.

20.

Steigmann

DJ.

Fluid films with curvature elasticity. Arch Ration Mech Anal 1999; 150: 127–152.

21.

Steigmann

DJ.

On the relationship between the Cosserat and Kirchhoff–Love theories of elastic shells. Math Mech Solids 1999; 4: 275–288.

22.

Zeidi

Kim

CI.

The effects of intra-membrane viscosity on lipid membrane morphology: Complete analytical solution. Sci Rep 2018; 8: 12845.

23.

Steigmann

DJ.

A model for lipid membranes with tilt and distension based on three-dimensional liquid crystal theory. Int J Non Linear Mech 2013; 56: 61–70.

24.

Rangamani

Steigmann

DJ.

Variable tilt on lipid membranes. Proc Math Phys Eng Sci 2014; 470: 20140463.

25.

Kim

Steigmann

DJ.

Distension-induced gradient capillarity in lipid membranes. Continuum Mech Thermodyn 2015; 27: 609–621.

26.

Ericksen

JL.

Conservation laws for liquid crystals. Trans Soc Rheol 1961; 5: 23–34.

27.

Ericksen

JL.

Hydrostatic theory of liquid crystals. Arch Ration Mech Anal 1962; 9: 371–378.

28.

Ericksen

JL.

Equilibrium theory of liquid crystals. Adv Liq Cryst 1976; 233–298.

29.

Agrawal

Steigmann

DJ.

A model for surface diffusion of trans-membrane proteins on lipid bilayers. Z Angew Math Phys 2011; 62: 549.

30.

Mahapatra

Saintillan

Rangamani

Transport phenomena in fluid films with curvature elasticity. bioRxiv 2020; DOI: 10.1101/2020.01.14.906917.

31.

Capeta

Poveda

Loura

LM.

Non-uniform membrane probe distribution in resonance energy transfer: Application to protein–lipid selectivity. J Fluoresc 2006; 16(2): 161–172.

32.

Walani

Torres

Agrawal

Anisotropic spontaneous curvatures in lipid membranes. Phys Rev E 2014; 89(6): 062715.

33.

Gwinner

Non-linear elastic deformations. Acta Appl Math 1988; 11: 191–193.

34.

Steigmann

DJ.

Finite elasticity theory. Oxford: Oxford University Press, 2017.

35.

Tsubota

Wada

Liu

Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: Equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion. Biomech Model Mechanobiol 2014; 13: 735–746.

36.

Blasi

D’Alessandro

Ramundo

, et al. Red blood cell storage and cell morphology. Transfus Med 2012; 22: 90–96.

37.

Geekiyanage

Balanant

Sauret

, et al. A coarse-grained red blood cell membrane model to study stomatocyte-discocyte-echinocyte morphologies. PLoS One 2019; 14: e0215447.

38.

Weksler

Wintrobe’s atlas of clinical hematology. Philadelphia, PA: Lippincott Williams & Wilkins, 2007.

39.

Sokolnikoff

IS.

Tensor analysis: Theory and applications. New York, NY: John Wiley & Sons, Inc., 1951.

40.

Hochmuth

Waugh

RE.

Erythrocyte membrane elasticity and viscosity. Annu Rev Physiol 1987; 49: 209–219.

41.

Derényi

Jülicher

Prost

Formation and interaction of membrane tubes. Phys Rev Lett 2002; 88: 238101.

42.

Van Vliet

Hinterdorfer

Probing drug–cell interactions. Nano Today 2006; 1: 18–25.

43.

Gallagher

. The red blood cell membrane and its disorders: Hereditary spherocytosis, elliptocytosis, and related diseases. In: Lichtman

Kipps

Seligsohn

, et al. (eds.) Williams hematology. 8th ed. New York, NY: McGraw-Hill, 2010, 617–646.

44.

Wolfe

. Hemolytic anemia due to membrane and enzyme defects. In: Lanzkowsky

(ed.) Manual of pediatric hematology and oncology. 5th ed. Amsterdam: Academic Press, 2010, Chapter 7.

45.

Mathai

AM.

A handbook of generalized special functions for statistical and physical sciences. Oxford: Clarendon, 1993.

46.

Seaborn

JB.

Hypergeometric functions and their applications. New York, NY: Springer-Verlag, 1991.

47.

Escudero

Montilla

Garca

, et al. Effect of dietary (n-9), (n-6) and (n-3) fatty acids on membrane lipid composition and morphology of rat erythrocytes. Biochim Biophys Acta, Lipids Lipid Metab 1998; 1394: 65–73.

48.

Lim

HWG

Wortis

Mukhopadhyay

Stomatocyte-discocyte-echinocyte sequence of the human red blood cell: Evidence for the bilayer-couple hypothesis from membrane mechanics. Proc Natl Acad Sci USA 2002; 99: 16766–16769.

49.

Muñoz

Sebastián

Sancho

, et al. Elastic energy of the discocyte-stomatocyte transformation. Biochim Biophys Acta 2014; 1838: 950–956.

50.

Galan

Mulero

Cuadros

Calculation of the surface tension and the surface energy of Lennard–Jones fluids from the radial distribution function in the liquid phase. Mol Phys 2005; 103(4): 527–535.

51.

Gurkov

Kralchevsky

PA.

Surface tension and surface energy of curved interfaces and membranes. Colloids Surf 1990; 41: 45–68.