Notes on superposed incremental deformations in the mechanics of lipid membranes

2020

Sci Rep

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 intramembrane 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 mechanics of lipid membranes has consistently been the subject of intense research for its importance in the understanding of a wide variety of essential cellular processes 1-5 . Traditionally, it was believed that cells are surrounded by a thin oil-based barrier, yet the structure of this membrane was not well understood. In 1920s, E. Gorter and F. Grendel. 6 found that the cell membrane is composed of lipid molecules (phospholipids) which are generally divided into two important groups: the hydrophilic head parts and hydrophobic tail groups. When dispersed into aqueous solutions, lipid molecules are driven by hydrophobic effects to form a unique bilayer structure (a lipid bilayer) with opposing orientations, that maintain symmetry about a mid-surface. In fact, with the advances in electron microscopy, the bilayer structure was identified as a characteristic of all biological membranes (biomembranes) 7 . Since they are negligibly thin (typically 5-10 nm), and fragile, the study of various aspects of lipid bilayers is often achieved by employing mathematical models in order to overcome the formidable difficulties of experimental studies. Also, from a mechanical perspective, the responses of a lipid membrane can be idealized as a thin elastic film. Within this context, the development of theoretical models describing the behavior of lipid bilayers has greatly benefited from the differential geometry of a surface and the theory of an elastic surface such that the deformation of energy of a thin membrane can be expressed by the mean and Gaussian curvatures of a surface 8,9 . In particular, Helfrich proposed a well-known Helfrich energy potential 1 which addresses the symmetry of lipid bilayers and further ensures the resulting equilibrium state to be energy minimizing 10 . This, together with the use of variational principle, and the virtual-work statement, furnish the Euler-Lagrange equations also known as "membrane shape equations", which h...

Section: Incremental Deformations Of Lipid Membranesmentioning

confidence: 99%

“…the solutions of circular substrate interaction problems 11,18 can be accommodated by the proposed model as a special case (i.e. = e 0, see, Figs.…”

Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

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

Liu

2020

Sci Rep

“…We note that, here and henceforth, the subscript ( * ) refers to the projected counterparts of ( * ) on the projected plane . Up to the leading order approximation, the two coordinates are related by (see, for example, [19,20])…”

Section: Admissible Boundary Conditions and Linear Approximationsmentioning

confidence: 99%

“…Authors in [19] discussed 2 Mathematical Problems in Engineering the role of Lagrange-multiplier fields in the shape equations and demonstrate that there exist particular sets of Lagrange multiplier which produces a unique converged solution. Further, the mathematical perspective of shape equations and associated boundary conditions are discussed in [20].…”

Section: Introductionmentioning

confidence: 99%

Mechanics of Lipid Membranes under the Influence of Intramembrane Viscosity

Mathematical Problems in Engineering

Liu

2019

Self Cite

We discuss a continuum-based model describing the deformations of lipid membranes subjected to intramembrane viscosity. Within the frame work of the theory of an elastic surface, the membrane equilibrium equations and the expressions of viscous stress are obtained. The corresponding deformation energy of the membrane is computed via the first and second fundamental form of surface. A compatible linear model is also formulated within the prescription of superposed incremental deformations through which the deformation profiles of the membrane is obtained. It is shown that the intramembrane viscous flow gives rise to straining effects on the membranes. Further, the corresponding dynamic edge conditions reduce to purely elastic boundary conditions in the limit of vanishing viscous effects. Lastly, admissible sets of velocity fields are also examined and are used to formulate membrane shape equations and the associated dynamic boundary conditions.

Strain-Gradient Elasticity Theory for the Mechanics of Fiber Composites Subjected to Finite Plane Deformations: Comprehensive Analysis

2019

Multiscale Sci. Eng.

Self Cite

A general model for the mechanics of fiber-reinforced composites is delivered in finite-plane elastostatics. Within the framework of the nonlinear strain-gradient theory, we obtain the Euler equilibrium equations and admissible boundary conditions. In particular, we present a third gradient continuum model and associated formulations. Soft composite materials are also considered by employing the Mooney-Rivlin energy potential. The presented model can be used as an alternative two-dimensional Cosserat theory of nonlinear elasticity.