Fig. 5.

Schematic picture of protein activation and lipid membrane remodelling. A density change of active receptors drives the system towards an active configuration in which the lipid bilayer shrinks around active domains, this causing membrane deformation and thickening in the form of lipid rafts. Unlike usual description of finite deformations and multiphysics, all the three configurations are global. This is justified through a non-standard multiscale geometric framework provided by the theory of Structured Deformations (see e.g. Deseri and Owen, 2003, Deseri and Owen (2010), Deseri and Owen (2015), Deseri and Owen (2019), Palumbo et al. (2018)). Here, each material particle of the body in the virgin configuration (displayed in the top-left side of this figure) gets first mapped into an intermediate region (center of the figure) through a pair of smooth bijections $(\mathbf{i},\mathbf{K})$ (where $\mathbf{i}$ stands for the vector-valued identity and the tensor $\mathbf{K}$ accounts for all the submacroscopic changes). Indeed, it can be shown that $K_r=det\mathbf{K}$ and, hence, this invariant of $\mathbf{K}$ accounts for remodelling. Secondly, each neighborhood of each particle located in this intermediate global configuration gets classically deformed through a pair $(\mathbf{y},\nabla \mathbf{y}),$ where $\mathbf{y}$ is a smooth deformation field from the intermediate global configuration onto the current one, while the second item in the pair is the deformation gradient from the intermediate global configuration. Features of Structured Deformations in connection with the examined biological systems are discussed in more details in Appendix C in which an alternative scheme of the kinematics is given in Fig. 13.