The Effects of Intra-membrane Viscosity on Lipid Membrane Morphology: Complete Analytical Solution

We present a linear theory of lipid membranes which accommodates the effects of intra-membrane viscosity into the model of deformations. Within the Monge parameterization, a linearized version of the shape equation describing membrane morphology is derived. Admissible boundary conditions are taken from the existing non-linear model but reformulated and adopted to the present framework. We obtain a complete analytical expression illustrating the deformations of lipid membrane subjected to the influences of intra-membrane viscosity. The result predicts wrinkle phenomena in the event of membrane-substrate interactions. Finally, we mention that the obtained solutions reduce to those from the classical shape equation when the viscosity effects are removed.

effects when a rectangular portion of membranes is subjected to intra-membrane viscous flow. The result is also aligned with the numerical study conducted under the compatible settings 9 . In particular, we find that the viscous flows give rise to wrinkle phenomena when the membrane makes contact with a circular substrate. Quantitative comparison is made by assimilating the experimental results reported in 10 where we find that the number of wrinkles is sensitive to the thickness of membranes and the radius of interacting substrates. A phenomenologically compatible result is also reported in the work of 11,12 . We believe that the results may provide an important breakthrough in the study of relevant problems such as the effects of wrinkles on the vesicle fusion/diffusion processes 13 and a wrinkle-caused disease of human eyes (e.g. a macular pucker/epiretinal membrane) 14 . Lastly, we note that our solution also accommodates the scenarios presented in 15,16 in the limit of vanishing viscous flow.
Throughout the paper, we make use of a number of well-established symbols and conventions. Thus, unless otherwise stated, Greek indices take the values 1 and 2. Einstein summation is applied for the repeated indices.

Prerequisite
The theoretical formulation of a lipid membrane which accounts for the effects of intra-membrane viscosity is presented in 9 . There the authors obtain the constitutive relation from the theory of an elastic surface via the relation W = ρF and later by substituting viscosity terms in the resulting formulae. In this section, we reformulate the results directly from the membrane free-energy density W = W(H, K, ρ) for the sake of consistency and completeness.
Inviscid lipid membranes. The equilibrium state of a purely elastic surface, subjected to normal pressure p, is given by 17 ; where T α and n are the stress vectors and the local surface unit normal, respectively. The semi-colon denotes the surface covariant differentiation associated with the Levi-Civita contraction of the surface metric a αβ = a α · a β , where a α = r ,α (θ α , t) = ∂r/∂θ α . For instance, a α;β is defined by 18 ; , are the Christoffel symbols induced by the local surface coordinate θ ε = × α α β α β n a a ( ) 1 2 such that n is a unit-vector field and a α and a β are the tangent vectors on the deformed surface ω. ε = αβ αβ e a / refers to the permutation tensor with a = det(a αβ ). Thus, for example, we evaluate e αβ as e 11 = e 22 = 0, e 12 = −e 21 = 1. The matrix a αβ of the surface metric is a positive-definite, with a > 0, leading to the existence of dual metric a αβ which is the inverse of the surface metric (i.e. a αβ = (a αβ ) −1 ). Hence the to the dual basis is given as a α = a αβ a β . The energy induced on the membranes' deformations can be expressed via the two primary parameters: the coefficient of the first fundamental form a αβ (the surface metric); and the second fundamental form b αβ (the curvature) defined by b αβ = n · a α,β . For example, in the case of the surface with energy density W = W(a αβ , b αβ ), T α take the following compact form 17 In the above, ρ and γ are the surface mass density and the constitutively-indeterminate Lagrange-multiplier field, respectively. For lipid membranes whose free-energy density is expressed by the mean and Gaussian curvatures through a αβ and b αβ (i.e. W = W(H, K, ρ;a αβ , b αβ )), the expressions of σ βα and M βα can be obtained by using chain rules . . . To see this, we evaluate . Thus, Similarly, by using Eq. (4)  Applying Euclidean dot products in normal n and tangential a α directions and invoking Gauss and Weingarten equations 18 a β;α = b βα n and n ,α = −b αβ a β , Eq. (14) becomes ; ; and σ + + = .
βα μ ; ; Eq. (15) is often referred to as a membrane shape equation when used in conjunction with Helfrich potential 19 .
Viscous lipid membranes. Viscous stress arises due to the straining effects of the fluid and can be accommodated by the time derivative of the evolving surface metric 20 . In a typical environment, where lipid membranes are subjected to morphological transitions, the reference velocity of the system is low and therefore the corresponding Reynolds numbers are sufficiently small 21,22 . Further, it is widely accepted that lipid membranes are relatively stiff against areal dilation in comparison with bending or shear motions 23,24 . Thus, in the forthcoming derivations, we adopt the assumption of incompressible fluid and thereby find the expression of the corresponding stress as where ν is the intra-membrane shear viscosity and the superscript dot  ⁎ ( ) refers to the time derivative. Since a αβ = a α · a β , we find a a a a a a ( ) (18) In convected coordinates, λ  a is defined as 17  v w u r a n is the velocity of a material point on the initial surface. It is now trivial to show that a u a n a a n n a n ( ) , , , Now, in view of Eqs (18 and 19), we find ; ; Further, from Eqs (10), (11), (17) and (20), the expression of the viscous stress can be derived as where we also use the relations: . The equation of motion (normal direction) of the lipid membrane in the presence of intra-membrane viscosity effects is therefore obtianed from Eqs (13), (15) and (21) ; ; ; ; Utilizing the identities = αβ where Δ is the Laplace-Beltrami operator (i.e. Δφ = φ ;αβ a αβ ), defined on the surface. Similarly, by substituting Eqs (13) and (21) into Eq. (16), we obtain .
Since a βα ≠ 0, the above becomes , , ; ; ; , which serves as the tangential equations of motion.
In the case of uniform membranes of the Helfrich type, the energy density W is defined by 2 where k and k are empirical constants (the bending moduli). It is noted here that, within the framework of the forgoing model, membranes with continuously distributed proteins can be accommodated through the energy density function: , where σ(θ α , t) describes the areal concentration of proteins on the membrane surface. However, the case is excluded from the present study in an effort to obtain mathematically tractable equations. Now, Eqs (23) and (26) yield  (6) and (20), we also find − = .  and corner forces are the normal curvatures of ω in the direction of ν and τ and τ is the twist of w on the (ν, τ) axes with τ = n × ν. We note here that the normal force remains intact in the presence of intra-membrane viscosity effects.

Monge Representation and Superposed Incremental Deformations
In order to study the responses of the membrane, we use the Monge representation with space vector r(θ α , t) representing material points on the membrane surface w, which is given by where θ(θ α ) is position on a plane defined by the unit normal k and z(θ, t) is height function that describes the bilayer membrane mid-plane shape. The Monge representation is an approximation of out-of-plane deformations in which no folds of the membrane are allowed, and hence, z(θ, t) is restricted to a single-valued function. For instance, the membrane surface can be represented by orthonormal Cartesian basis θ = θ α e α and, unless otherwise specified, the subscripts of the surface coordinates are dropped and replaced by 1 = x, 2 = y for convenience. Within this setting, we compute , , Further, the normal velocity and the Christoffel symbols are computed as , , , , respectively. The evaluation of the resulting shape equation Eq. (27) in terms of Eqs (33-36) furnishes a highly nonlinear PDE system, which most often requires heavy computational resources. Instead, a means of 'admissible linearization' can be employed to make the system mathematically tractable with minimum loss of generality. The concept of the superposed incremental deformations has been widely and successfully adopted in the relevant subject of studies (see, for example 15,[26][27][28]. Within this prescription, the derivatives of z(θ, t) of all orders are considered to be 'small' (e.g. , and therefore, their products can be neglected. Accordingly, using the notation '' to identify equations to the leading order in z, we have is the corresponding Laplacian, respectively. In particular, the straining (20), viscous stress (17) and incompressibility condition (28) can be approximated as: , , , , and Further, the equations of motion in normal and tangential directions (Eqs (25) and (27)) can be approximated as In view of, Eqs (38) and (40) ), the above are equivalent to To obtain simplified edge conditions, let r(S, t) = r(θ(S), t), where θ(S) is the arc length parameterization of the projected curve ∂ω p on the plane ω p . Thus, under the Monge representation, we obtain  where ν p = τ p × k is the unit normal to the projected curve. Consequently, where (ν α ) p = e α · v p and (ν α ) p = e α · τ p and e αβ is the unit alternator defined by e 12 = −e 21 = 1 and e 11 = e 22 = 0. Also, invoking Eqs (31), (37), (47) and (48), becomes  Now, the normal force is given by where * ν ( ) , is the normal derivatives on ∂ω. By Eq. (46), the arclength derivatives satisfies the approximation Here τ τ τ p is the arclength derivative on the projected curve. In addition, H ,ν can be re-written as

Solutions to the Linearized Systems
Consider the case where the membrane flows over a rectangular portion of the plane In view of Eq. (37), the later implies that ∇ = z 0 and thus both z ,α and the normal derivative z ,ν = ν α z ,α vanish on the boundary. Accordingly, the edge moment acting on the boundary Eq. (52) 1 becomes It is suggested that the kinetic conditions lead to the particular set of the tangential and normal force (i.e. f τ = 0, f ν = −q), where q is prescribed surface pressure (see, for example 9 and 29 ). Thus, Eq. (52) 2-3 furnishes where the approximations have been made via Eq. (53). We note here that, up to leading order, membrane shape has negligible effects on the surface flow, whereas shape is influenced by the flow via the viscosity term in the shape equation (41). Admissible linearization thus reduces the non-linear, fully coupled equations to a system of PDEs with one-way coupling. Further, the present model incorporates the purely elastic theory of lipid membranes in the limit of vanishing viscosity ν  0; ). This observation, in turn, suggests that the decay of a dynamic solution to a purely elastic solution can be investigated by assigning q(t) on the boundary in a way that where q(θ α , t) is understood as a uniform function assigned in the interior (i.e. q(θ α , t) = q(θ α )) at each sequential step. Then the equations of motion Eq. (42) and boundary conditions Eq. (61) become Within the domain of interest (a rectangular portion), we now create a particular set of intra-surface viscous flow as Ax v Ay and (66) x y Since v x,x + v y,y = A − A = 0 and either τ x ν y or τ y ν x vanishes on the boundary (e.g. for = x a 2 boundary, ν p = −v x e 1 and τ p = −τ y e 2 so that v y = τ x = 0), Eqs (64) 3 and (65) 1 are automatically satisfied. Similarly, it follows from Eqs (65 and 66) that which agree with the elastic boundary condition in the limit as discussed in [25][26][27][28][29] (i.e. λ = −q). In addition, Eq. (64) 2 is also satisfied because both λ γ , and Δv γ vanish identically (e.g. Δv x = Δ(Ax) = 0 and λ = . 0 x , ). Consequently, the systems of coupled PDEs (64-65) now reduces to In the case of vanishing p, the solution of the above is obtained by 30  By imposing boundary conditions (68) 2-4 , the unknown constants (e.g. A n , B n etc…) can be completely determined. Here, we omit details for the sake of conciseness which can be found in 16 . The value of intramembrane surface viscosity ν = 10 −4 pN · s/nm and the bending modulus of the membrane k = 82 pN · nm are adopted from the work of 31 and 32 , respectively. We also note that the data are obtained under the normalized setting unless otherwise specified. It is clear from Fig. 1 that membrane shape is influenced by the viscous flow via the viscosity term in the shape equation (64) 1 . More precisely, the applied flow gives rise to straining effects on the membrane shape ( Fig. 1) in lateral direction. The corresponding transverse deflections (Fig. 2) decrease with the increasing velocity field of viscous flow. In addition, Fig. 2a illustrates that the obtained solutions accommodate those presented in 16 in the limit of vanishing viscous flow. This can also be seen by the reduction of Eq. (64) 1 to the classical shape equation (see, for example 15,16 ) in the absence of viscous flow (i.e. v α = 0) and so too the corresponding solutions.

Membrane-Substrate Interactions in the Presence of Viscous Flow
In light of the foregoing discussions, we now investigate a membrane-substrate interaction problem under the influence of intra-membrane viscosity. Driven by the bi-normal (transverse direction) force f n , the interaction occurs on the boundary r = a of an annular portion of the membrane, while it remains clamped (i.e. z = 0 and n = k). Under a polar-coordinate parametrization of the Monge plane, Eq. (32) is replaced by Further, the normal velocity and the curvature tensor are given by     . The solutions are the first and second kinds of the modified Bessel functions and usual Bessel functions of the order n and q with parameter μ, denoted conventionally by, I n , K n , J n , and Y n , respectively 37 . Similarly as in the rectangular case, it is found that the intra-membrane viscosity, in the case of circumferentially dominant flow, leads to straining effects on the membrane (Fig. 3) and the resulting transverse deflection is reduced with increasing velocity of viscous flow (Fig. 4a). More importantly, the viscous flow gives rise to wrinkling phenomena when ≥ ν λ − 10 A 9 . Based on our analysis of a rectangular portion of membranes and the observations from 9 where no interactions are considered, we infer that the interaction forces between the membrane and the substrate give rise to wrinkling phenomena. Further clarification of such phenomena is, however, beyond the scope of present study due to the paucity of available data. It is shown in Fig. 4(a) that both the amplitude of wrinkles and the magnitude of transverse deflections decay away as they approach to far field boundary. Also, Fig. 4(b) illustrates that the solutions from the proposed model (dot lines) reduce to those predicted by the existing model (solid lines) 15

10
A 10 and they vanish at the remote boundary. The number of radial wrinkles increases with the growing effects of viscous flow (Fig. 5) and with larger radius of an inner circle. A similar tendency can be found in the work of 10 (See, Fig. 6) where the authors measure the number of wrinkles on thin polymer films under the compatible settings as considered in the present work. In addition, we assimilate  Fig. 7. Although, the obtained model is not intended for the analysis of thin polymer films, it still provides reasonable agreement with the results in 10 (see, Fig. 7) that the number of wrinkles is sensitive to both the thickness 't' (inversely proportional) and the inner radius 'a' (proportional). The results are also align with the theoretical developments regarding to finely wrinkled states of the membrane via the minimization of the strain-energy function and by its quasi-convexification 11,12 . Potential applications to biomechanics may include monitoring vesicle thickness and enhancing vesicle fusion processes. However, prior to these applications, it would be necessary to further clarify and/or justify the obtained results by employing the aforementioned theory. Ethics statement. This work did not involve any collection of human data.
Data accessibility statement. This work does not have any experimental data.