Irreversible thermodynamics of curved lipid membranes

The theory of irreversible thermodynamics for arbitrarily curved lipid membranes is presented here. The coupling between elastic bending and irreversible processes such as intramembrane lipid flow, intramembrane phase transitions, and protein binding and diffusion is studied. The forms of the entropy production for the irreversible processes are obtained, and the corresponding thermodynamic forces and fluxes are identified. Employing the linear irreversible thermodynamic framework, the governing equations of motion along with appropriate boundary conditions are provided.

Figure 1

A schematic of a membrane patch $\mathcal{P}$. At each point $\mathit{x}$ on the membrane patch $\mathcal{P}$, we define the in-plane tangent vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$ as well as the normal vector to the plane $\mathit{n}$. The set $\{{\mathit{a}}_1,{\mathit{a}}_2\}$ constitutes a basis for the tangent plane at any location, while the set $\{{\mathit{a}}_1,{\mathit{a}}_2,\mathit{n}\}$ forms a basis of ${\mathbb{R}}^3$. At every point ${\mathit{x}}_{\mathrm{b}}$ on the patch boundary $\partial \mathcal{P}$, we define the in-plane unit tangent $\mathbf{\tau }$ and in-plane unit normal $\mathbf{\nu }$ which also form a basis of the tangent plane.

Figure 2

A director traction $\mathit{M}$ acting on the unit normal $\mathit{n}$ at the membrane boundary results in edge twisting (a) and bending (b). The director traction, which has units of couple per length, acts in the manner of a force on the dimensionless normal vector to produce a moment per length $\mathit{m}=\mathit{n}\times \mathit{M}$. The inset in (a) shows a physical representation of the director tractions acting on the membrane. In general, the couple per length $\mathit{m}$ acting on the membrane is a superposition of that shown in (a) and (b), and lies in the tangent plane.

Figure 3

A schematic of a phase transition in lipid membranes. The membrane on the left is initially in the liquid-disordered state ($L_{\text{d}}$, depicted in light gray). Under a suitable change of external conditions, such as a temperature or pressure change, the membrane undergoes a phase transition and contains both liquid-ordered domains ($L_{\text{o}}$, shown in dark gray) and liquid-disordered domains. Such phase transitions are reversible, and the membrane can be returned to its initial disordered state.

Figure 4

A schematic depicting the binding and unbinding of epsin-1 proteins to and from PI(4,5)${\mathrm{P}}_2$ lipids (black) in the membrane patch $\mathcal{P}$. The rate constants for binding and unbinding are denoted by $\vec{k}$ and $\overleftarrow{k}$, respectively. Epsin-1 proteins cannot bind to the DOPC lipids (light gray). All interactions are modeled as part of the continuum of the membrane, where changes in concentration are accounted for in terms of changes to the local species density ${\rho }_k(\mathit{x},t)$.