On differentiability of the membrane-mediated mechanical interaction energy of discrete-continuum membrane-particle models (Q2077421)

The authors consider a discrete-continuum model of a biomembrane represented by a continuous surface with embedded particles described by rigid discrete objects which are free to move and rotate in lateral direction. Assuming that the membrane is given by a graph \(\mathfrak{M}=\{(x,u(x))\mid x\in \Omega \}\), that it is almost flat, that is \(\left\Vert \nabla u\right\Vert \ll 1\), and that the membrane shape is described by the graph of \(u\in H^{2}\), its bending energy is approximated as: \(J(\Omega ,u)=\frac{1}{2}\int_{\Omega }\kappa (\Delta u(x))^{2}+\sigma \left\Vert \nabla u(x)\right\Vert ^{2}dx\), according to the linearized Canham-Helfrich model in Monge-gauge. Here \( \kappa >0\) denotes the bending rigidity and \(\sigma \geq 0\) the surface tension. The authors consider a hydrophobic belt which is approximated by a simple closed curve \(\mathfrak{G}\), which can be parameterized over the 2D Euclidean plane by means of a simple closed curve \(\Gamma =\partial B\subset \mathbb{R}^{2}\) and of a continuous function \(g_{0}:\Gamma \rightarrow \mathbb{R}\) such that \(\mathfrak{G}=\{(x,g_{0}(x))\mid x\in \Gamma \}\). The boundary condition \(u\mid _{\Gamma }=g_{0}\) is added, which means that the membrane is connected to each particle at the interface \(\mathfrak{G}\). The additional boundary condition \(\partial _{\nu }u\mid _{\Gamma }=g_{1}\) is further imposed with a function \(g_{1}:\Gamma \rightarrow \mathbb{R}\) describing the slope. The current position of a particle is parameterized using translations \(z_{j}\) and rotations \(\alpha _{j}\) along or around the \( x_{j}\)-axes and the authors associate a rigid body transformation. They draw computations to describe the motion of a particle, they define the feasible particle configurations for a system of \(N\) particles and the interaction energy. They finally the complete model as a minimization problem involving this interaction energy over the set of feasible membranes, which depends on the particle configuration. They prove the well-posedness of this minimization problem using Lax-Milgram's theorem. They compute the differential of this interaction energy. The paper ends with few numerical examples, the authors considering two circular particles, two peanut shaped particles, or a gradient flow. They present and discuss the results of their simulations obtained through the finite element method.