This thesis is concerned with developing a wholistic description of biological membranes: fascinating materials that make up the boundary of the cell, as well as many of the cell’s internal organelles. Our formulation of the theory of such materials relies on two well-known concepts: differential geometry and irreversible thermodynamics. The setting of differential geometry allows us to describe curves and surfaces, which in this case are embedded in the three-dimensional Euclidean space, while irreversible thermodynamics provides a theoretical framework to develop constitutive relations between the various thermodynamic forces and fluxes in a system. Both concepts are well-known, and are reviewed in Part A.
We build on these classic results in Part B, and develop the theory of irreversible thermodynamics for arbitrarily curved and deforming lipid membranes. In particular, we treat the membrane as a two-dimensional surface, in which lipids flow in-plane as a two-dimensional fluid while the membrane bends out-of-plane as an elastic shell. We then obtain the fundamental balance laws of mass, linear momentum, angular momentum, energy, and entropy, as well as the second law of thermodynamics. Finally, we apply the framework of irreversible thermodynamics to determine appropriate constitutive relations, and substitute them into the balance laws to obtain the equations governing membrane dynamics. Our main result is to present the equations of motion and appropriate boundary conditions for three systems of increasing complexity: (i) a compressible, inviscid membrane, (ii) a compressible, viscous membrane, and (iii) an incompressible, viscous membrane.
Part C of this thesis focuses on applications of the single-component theory. We begin by specializing the general governing equations to three commonly observed geometries in biological systems: planar sheets, spherical vesicles, and cylindrical tubes. A scaling analysis of the resultant equations reveals membrane dynamics are governed by two dimensionless numbers. The well-known Föppl–von Kármán number, Γ, compares tension forces to the familiar elastic bending forces, while a new dimensionless quantity—which we name the Scriven–Love number, SL—compares out-of-plane forces arising from the in-plane, intramembrane viscous stresses to bending forces. Calculations of non-negligible Scriven–Love numbers in various biological processes and in vitro experiments show in-plane intramembrane viscous flows cannot generally be ignored when analyzing lipid membrane behavior, and can never be ignored in lipid membrane tubes. Moreover, a stability analysis indicates membrane tubes are unstable above a critical value of Γ, while SL governs the spatiotemporal evolution of the deforming membrane. We close by investigating a novel hydrodynamic instability in which an initially local disturbance to an unstable tube yields propagating fronts, which leave a thin atrophied tube in their wake. Depending on the value of the Föppl–von Kármán number Γ, the thin tube is connected to the unperturbed regions via oscillatory or monotonic shape transitions—reminiscent of recent experimental observations on the retraction and atrophy of axons.