Deformation analysis of lipid membranes subjected to general forms of intra-membrane viscous flow and interactions with an elliptical-cross-section substrate

We study the morphology of lipid membranes subjected to intra-membrane viscous flows and interactions with elliptical cylinder substrates. From the non-linear theory of elastic surfaces, a linearized shape equation and admissible boundary conditions are formulated in elliptical coordinates via the Monge representation of a surface. In particular, the intra-membrane viscosity terms are linearized and mapped into elliptic coordinates in order to accommodate more general forms of viscous flow. The assimilated viscous flow is characterized by potential functions which satisfies the continuity condition. A complete solution in terms of Mathieu function is then obtained within the prescription of incremental deformations superposed on large. The results describe smooth morphological transitions over the domain of interest and, more importantly, predicts wrinkle formations in the presence of intra-membrane viscous flow in the surface. Lastly, the obtained solution accommodates the results from the circular cases in the limit of vanishing eccentricity and intra-membrane viscous flow.

The study of the mechanical responses of membranes under the influences of intra-membrane viscous flow are of particular mechanical interest due to its importance in the explanation of essential cellular functions including budding, fission and vesicle formations [20][21][22][23] . The theoretical frame work accounting for the effects of intra-membrane viscosity into the model of membrane deformations has been established in 24 . In there, authors reveal that the dynamics of the membrane system is notably influenced by the presence of intra-membrane viscous flow. The authors in 25 developed the comprehensive non-linear model of membranes incorporating the effects of intra-membrane viscosity from the elastic model of surfaces 26,27 . To this end, authors in 28 discussed a compatible linear model within the setting of superposed incremental deformations. However, the analysis presented in 28 is limited to certain types of problems where viscous flow is characterized as either constant or simple linear functions, and the interaction occurs through a circular contact region to obtain a mathematically tractable system. In a typical environment, a lipid membrane system is involved in more complex processes 4,5 (e.g. interactions through a non-circular domain and the influences from multi-source viscous flows). Therefore, the development of a more comprehensive model may be necessary to promote researches on the related subjects.
In the present work, we study the deformations of lipid membranes interacting with intra-membrane viscous flow and an elliptical cylinder substrate. Utilizing the Monge parameterization of a surface and general curvilinear coordinates, the expressions of linearized shape equation and associated boundary conditions are obtained from the non-liner theory 25 . The intra-membrane viscosity terms are formulated by means of 'admissible linearization' and successively transformed into elliptical coordinates to assimilate more general types of viscous flow. More importantly, we obtained a complete analytic solution by employing adapted iterative reduction and the method of eigenfunction expansion [29][30][31] , which describes the deformations of lipid membranes when interacting with intra-membrane viscous flow and an elliptical-cross-section substrate. It is found that intra-membrane viscosity induces wrinkle formations of the lipid membrane and the corresponding number of wrinkles exhibits sensitivity to both the radius of the ellipse and the intensity of viscous flow. Comparisons with phenomenologically compatible cases such as a circular substrate -membrane interactions and capillary wrinkle of polymer films, are made where the proposed model successfully reproduces the results from 28,32 in the limit of vanishing eccentricity of an ellipse. Further, we obtain solutions corresponding to the case of a lipid membrane subjected to non-uniform viscous flows and dual source flows. This is facilitated by the relaxed form of the prescribed tangential and normal force, and the condition of continuity along and within the elliptical boundaries, unlike those arising in circular boundaries where the admissible set of viscous flows are strictly uniform in one of the coordinate directions 28 . The resulting deformation fields show clear signs of dual source interference in that both the radial and circumferential wave forms are simultaneously predicted. Case study vis a vis morphologically similar results of shape memory films 33 are presented to investigate the potential applicability of the proposed model in the analysis of different types of membrane. In particular, it is found that the principles of superposition from linear elasticity remains valid, even in the presence of general forms of dual source viscous potentials. That is the solution of a dual source problem can be directly obtained by adding solutions of two single source problems. The solutions presented here are of more practical interest in that, essentially, they lead to solutions of problems in which the viscosity effects are characterized by a wide class of potential functions and so can accommodate a correspondingly large set of physically relevant problems. For example, potential applications may be expected in the study of wrinkle-caused disease (e.g. a macular epiretinal membrane 34 ) and the influences of membrane viscosity on various cellular functions such as fusion, fission and budding 35 . Further, the presented solution reproduces the existing results 17 when viscosity effects are removed, and does incorporate the solution of the classical membrane-substrate interaction problem 11 in the limit of vanishing eccentricity. In fact, the classical solution obtained directly from the proposed model produces more accurate predictions by identifying the additional Bessel functions, which is reduced from the Mathieu potentials.
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.

Viscous Lipid Membranes
In the study of the mechanics of lipid bilayer membranes, it is widely accepted that lipid membranes can be regarded as a continuous elastic surface. Within this idealization, the mechanical responses of a lipid membrane can be modeled via the theory of an elastic surface. In this section, we reformulate the results in the present context for the sake of clarity and completeness, and for the use in the derivation of the compatible linear model. The original derivation concerning the viscous elastic membranes can be found in 25 .
The equilibrium state of a purely elastic surface, in the presence of normal pressure p, is given by 26 Here, α T and n are the stress vectors and the local surface unit normal, respectively, and the semi-colon denotes the surface covariant differentiation in the sense of the Levi-Civita connection on a surface. The associated surface metric is defined as , are the tangent vectors to the surface ω induced by the parameterization θ α t r( , ), the position in  3 of a point on the surface with coordinate θ α . The unit vector field n, which serves as the local surface orientation, is then computed as θ ε = × α α β α β n( a a ) 1 ; , are the Christoffel symbols induced by the local surface coordinate. These results furnish the well-known Gauss and Weingarten equations: a a a (a n n a n a n ( ) ) ; (Gauss), and , , , , , where αβ b are the coefficients of the second fundamental form of surface and its covariant cofactor is defined by The deformation energy of an elastic surface can be expressed via the above two primary parameters: the coefficient of the first fundamental form αβ a and the second fundamental form αβ b 8,9,26 . Thus, for example, an elastic surface whose free-energy density is expressed by the mean and Gaussian curvatures through αβ a and αβ b (i.e. ρ = αβ αβ , )), α T takes the following compact form 26 : ; where which also satisfy the following equalities ( Cayley Hamilton theorem) and (11) Now, the viscous stress induced by the straining effects of the fluid is given by 37 where ν is the intra-membrane shear viscosity and is the time derivative of the evolving surface metric. Thus, in order to compute viscous stress, it is required to compute μ  a , which can be obtained via the material time derivative of a position vector r 26 : t Accordingly, it is found that a u a n a a n n a n ( ) , , , where θ = ∂ ∂ α α v t / , and = =w r r n n t t are respectively the tangential and normal velocities of a material point on the initial surface 26,37,38 .
It is now straightforward to show from Eqs. (8), (12) and (16)   ) on the surface Ω. Consequently, by projecting Eq. (1) onto the basis coordinate plane of α a , the following tangential equations of motion can be obtained: Much of literature on the mechanics of lipid membranes has revealed that a bilayer membrane can be regarded as a continuous two-dimensional elastic surface where the response functions are governed by the well-known Helfrich energy potential 1 . The model has been widely adopted in various subjects within bilayer membrane mechanics (see, for example 11,12,15 , and the references therein). Following the work of 25 , in this paper, we consider a symmetric membrane of Helfich type (i.e. ), subjected to the membrane-substrate interactions and the effects of intra-membrane viscosity. The corresponding free-energy density function is defined by 2 where k and k are empirical bending constants, which pertain to lipid membranes with uniform properties. Thus, from Eqs. (18) and (20), becomes

Incremental Deformations of Lipid Membranes
The use of Monge parameterization and admissible linearization is a widely adopted methodology for lipid membrane analysis, and the associated procedures are well documented in the literature (see, for example 11,15,18 ). Here, we reformulate the results for the sake of completeness. Under the Monge parameterization, material points on the membrane surface Ω is defined by where θ θ α ( ) is position on a plane p with unit normal k. The problem of determining the membranes' deformed configuration is then reduced to solving a single function θ z t ( , ). In the cases of Cartesian coordinates, we have where α e { } is an orthonormal basis for the plane and, the subscripts of the surface coordinates are dropped and replaced by = = x y 1 , 2 , unless otherwise specified. Accordingly, we compute , , Here, ∇ = α α z z e , is surface gradient, δ αβ is Kronecker delta and b is the curvature tensor. Further, the expressions of the dual basis and the Christoffel symbols are obtained as is the corresponding Laplacian, respectively.

Linearization of the intra-membrane viscosity terms.
In the forthcoming derivations, we present the linearization procedures for the terms associated with the intra-surface viscous flow, which arise in the formulation of membrane equilibrium equations. To proceed, we express the surface gradient of the viscous flow fields and the curvature tensor as We then compute their traces to obtain Also, from the results in Eqs. (27) and (28) can be approximated, up to the leading order, to δ δ ; , , , Now, combining Eqs. (29) and (30), we find that, Thus, Eq. (29) simplifies to ; ; , , , However, since = αβ βα z z , , , the above can be re-written as , , , , , To obtain the simplified expression of incompressibility condition (22), we evaluate the surface divergence of the viscous flow field as (2020) 10:478 | https://doi.org/10.1038/s41598-019-57179-z www.nature.com/scientificreports www.nature.com/scientificreports/ .
Thus, from Eq. (27), the leading order approximation of the above can be found as where v (div ) P is the divergence of the projected coordinate plane Ω p . Consequently, substitution of these linearized expressions, (Eqs. (27), (33) and (36)), into Eqs. (19, 21 and 22) delivers the following normal and tangential equations, and incompressibility conditions: Here, θ = α P P t ( , ) is understood as a sequence of prescribed surface pressure (see 25 ) from the admissible set of boundary forces which satisfy where γ τ × = n. In particular, the compatible linear forms for the moments and normal interaction forces are given by 11 p and σ are the arc length derivative on the projected curve, and the empirical constant accounting for the wetting of the interacting boundary, respectively. Hence, the solution of Eq. (37) can be uniquely determined by imposing the admissible boundary conditions, Eqs. (38) and (39).

Formulations in the elliptical coordinates.
We consider the cases when lipid membranes interact through the elliptical contact domain of a transmembrane substrate, and are subjected a general class of intra-membrane viscous flow (see, Fig. 1). The deformations of lipid membranes defined on an elliptical domain can be examined by using the mapping, such that y c cosh( )cos( ) and sinh( )sin( ), (41) through which the rectangular Cartesian coordinates from which the admissible set of viscous flow field is found to be ). In the analysis, we assume = = C C 0 1 2 for the sake of simplicity. The cases of non-zero coefficients can be easily accommodated via the principles of superposition which will be discussed in the later section.
The membrane-substrate interaction occurs through the wall of the elliptical substrate where the corresponding domain of interest, Ω, and interacting boundary, ∂Ω, are defined respectively as Here, the repeated indices, α and β, when summed over their ranges {1, 2}, refer to ξ and η in elliptic coordinates. On the boundaries (i.e. ξ ξ = o and ξ ξ = i ), we find and thereby reduce Eq. (47) to In particular, since the membrane-substrate interaction condition (i.e. − ∇ = z n k, see 11,13 ) requires ∇ = z 0 at the inner boundary (ξ ξ = i ), the normal force (Eq. (39)) becomes We continue by rewriting ∇ H p using Eq. (42) and subsequently reduce Eq. (50) to Further, applying the similar schemes as in the above, it is not difficult to show , , Consequently, by combining the above results, we reformulate Eq. (37) and the associated boundary conditions as

Remark 1.
It should be noted that the restrictions on the continuity conditions ( = v div( ) 0) and the prescribed tangential ( = τ f 0) force can be relaxed along and within the elliptical boundaries unlike those arising in circular cases where the admissible set of viscous flows are required to be strictly uniform in one of the coordinate directions (i.e. either = v const r or = θ v const) to satisfy the constraints 28 . This is mainly due to the confined descriptions of the circular interaction boundary where the rate of change in the unit normal and tangent on the circular boundary remains constant so that the associated normal velocity fields v r always points to the center of a circular substrate. Thus, v r is required to be vanished by its gradient v r r , or gradient of tangential velocity θ θ v , to satisfy the continuity condition; i.e., Such restriction can be relaxed in the case of the elliptic interaction boundary, since the rate of change in local coordinate is not necessarily constant, yet they vary with respect to the coordinates ξ and η (see, Eqs. (42 and 43)). This further suggests that the normal velocity field ξ v does not necessarily points to the center of an elliptical substrate (see, Fig. 1) and therefore no restrictions are necessary for ξ v . In results, the associated flow fields ( ξ v and η v ) can accommodate more general forms such as non-uniform viscous flows and periodic wave form of viscous flows (no need to be strictly constant) without violating the aforementioned constitutive restrictions. Examples regarding these cases will be discussed in the following section.
Lastly, Eqs. (54 and 55) serve as the linearized shape equation system which describes the morphology of lipid membranes under the influences of membrane-substrate interactions and general forms of viscous flows. In the analysis, we also impose ξ η = z( , ) 0 i for the purpose of comparison with the existing literature.

Solutions to the Linearized Systems
It can be seen from Eqs. (42), (43) and (54)  In other words, the associated tangential and normal velocities are simultaneously updated as material points move over the membrane surface. Therefore, the solution of the Eq. (54), which is coupled with the viscous velocity fields, cannot be accommodated by the conventional separation variable method of modified Helmholtz equation. In this section, we combine the method of adoptive iteration and the principle of eigenfunction expansions [29][30][31] , and obtained the complete expression of the membrane's shape function ξ η z( , ). To proceed, we assume the solution of the form to accommodate the desired behavior (i.e. ∇ → z 0) as approaching the boundary. In particular, the unknown potential ξ η H q ( , , ) can be expressed as 40 The detailed procedures which can be found in [29][30][31]  where μ λ = k 2 / is the natural length scale which is commonly adopted in the membrane studies (see, for example 11,14,25 ). Details regarding the dimensionless variables adopted in the present work will be discussed in later section. The unknown constants A B C , , m m m and D m can be completely determined by imposing the admissible boundary conditions. For instance, the substrate-membrane interaction conditions (55) require        for → e 0). Lastly, we note here that the predicted wrinkle states are unique and stable, since the proposed model satisfies strict quasi-convexity via the minimization of membranes' strain-energy potentials (see, for example 45,46 ). : w(ξ, η) = A sin ξ cos η (waveform). In this section, we consider membrane systems with non-uniform viscous flow. The non-uniform cases can be observed in various cellular activities such as the transportation of the intracellular membrane and the transmembrane proteins induced by the viscous flow with tension gradient 47 . In this case, the viscose flow field becomes non-uniform due to the interactions with tension gradient field.  www.nature.com/scientificreports www.nature.com/scientificreports/ Membranes subjected to the waveform of non-uniform viscous flows can be examined by introducing the following potential function,

Non-uniform viscous flow potential
where the intensity of wavy flow can be controlled by the parameters E and F. In the assimilation, we set = = E F 1 for simplicity. Accordingly, from Eq. (62), we obtain the following expression of ξ η T ( , ) mn , addressing the viscous effects, Combining (63), (65) and (72), the complete solution describing the membranes' morphology can then be found as where σ 1 is defined in Eq. (69). Similar to the constant viscous cases, the resulting deformation fields (radial wave deformations) are sensitive to both the dimension of an inner ellipse and the intensity of viscous flow; i.e., the number of waves reduces as A decreases from − 10 5 to − 10 7 (See. Fig. 5). But, more importantly, the transverse wave deformations of the membrane and the corresponding vertical deflections die out as they approach the remote boundary. As a result, the corresponding boundary remains intact and stable (See. Fig. 6). In the event of vanishing A, the wave deformations are completely removed from the entire domain of interest so that the vertical deformation profile reduces to the results in 17 , where the authors present the analysis of elliptical substrate-membrane interaction problems without the considerations of viscosity effects (See. Fig. 6). Also, Fig. 7. shows that the obtained solution accommodates the results of circular substrate-membrane interaction problems in 28 when the eccentricity converges to zero (i.e. → e 0). In fact, the solutions in Figs. 6 and 7 become essentially identical for sufficiently small value of A; i.e., ≤ − A 10 16 for case in Fig. 6 and ≤ − A 10 8 and ≤ − A 10 10 for the cases in Fig. 7. In the assimilations, the classical solutions obtained from the proposed model are intentionally reproduced at = − A 10 15 , = − A 10 7 and = − A 10 9 for the purpose of visual demonstration.
Dual source problems: w(ξ, η) = A + A sin ξ cos η. The proposed model is sufficiently general in that the viscous effects from both radial and circumferential directions can be simultaneously considered. To demonstrate this, we introduce the following dual source potential and subsequently obtain from Eq. (62) that Thus, from Eqs. (63) and (65), the deformation mapping function ξ η z( , ) can be obtained in the same manner as in the single source cases. Figure 8 illustrates the deformation configuration of the membranes under the influence of dual source viscous flow. It is shown that both the radial and circumferential wave patterns are simultaneously observed. www.nature.com/scientificreports www.nature.com/scientificreports/ Morphologically similar cases are reported in the work of 33 where the authors examined the wrinkle phenomena of a thin gold layer (10 nm in thickness) when subjected to thermal stresses from the adjoined polymer substrate. In cases of thin membranes, thermal stresses may be understood as a particular type of the surface stress 48,49 . Therefore, the results may bear close resemblance with the present case where the membrane's deformations are induced by the surface interaction forces which are transmitted from the acting viscous flows. The obtained solution assimilates the experimental results in 33 when compatible conditions are applied (see, Figs. 9 and 10). This further suggests that the proposed model may be of practical interest in the morphological study of thin film structures. Such investigations are, however, limited in the present study due to the lack of available data. Fig. 8 further indicates that the principle of superposition remains valid in the present cases. The principle is widely adopted in various engineering problems with simple initial and/or boundary value problems of either first (Dirichlet) or second (Neumann) type [50][51][52] . However, such practices are largely absent in the membrane studies due the complexity of mixed boundary conditions (i.e. both the Dirichlet and Neumann boundary conditions are prescribed on the boundaries), and the limited access for the solutions of membrane systems subjected to coupled-physics environment. In the present case, the solutions of single source problems (i.e. = w A and ξ η = w A sin cos ) can be obtained from Eqs. (67), (68), (72) and (73) that  The above is the same as the solution obtained from the dual source problem (Eq. (75) and Fig. 8). This, in turn, suggests that the solutions of dual source problems can be obtained directly from the solutions of single source problems (see, also, Figs. 8 and 11) via simple summations. In other words, the principle of superposition remains valid even with the presence of intra-membrane viscous flows and interaction forces. The result may further promote the study of various different influences of viscous flows onto membrane-substrate systems by minimizing computational complexities and resources.  www.nature.com/scientificreports www.nature.com/scientificreports/ resulting impaired vision (a macular epiretinal membrane) 34,57 . Such wrinkle formations are most often induced by the interactions between the posterior vitreous cortex and the retina 58 . Since nearly all the emmetropic retinas are oblate in shape in both transverse axial and sagittal sections 59 , the wrinkle formations on the retina may share close similarity to the elliptical membrane-substrate systems examined by the proposed model. In addition, wrinkle involved deformations and the directional elongation of the vesicle are often caused by the viscous shear flows and/or directional viscous flows 54,55 . Therefore, the proposed model may be employed to study the morphological transitions of cell membranes associated with those cellular activities.

Remark 2. The results in
Ethics statement. This work did not involve any collection of human data.