Self-assembly in mechanical systems

Inspired by biological membrane shaping in the cell through means of curvature-inducing proteins, we investigate the interplay between membrane curvature and the distribution and movement of shape-inducing objects which are free to move as a consequence of the underlying shape. We initially study the self-assembly of a filament, taken as a proxy for the cross-section of a biomembrane, which is primarily driven by the chemical kinetics of attaching proteins and find that, under certain mechanical stiffness regimes of the attaching proteins, pattern formation occurs. Regions of high and low protein concentration form before spatially uniform filament shapes are obtained by means of protein adhesion and movement governed by diffusion and local curvature-seeking. However, noting that the curvature-mediated protein movement on membranes has been biologically observed to be long-range, we next study the self-assembly of embedded inclusions on a membrane as a result of the underlying geometry. We first derive an interaction law for the shape-mediated interaction of inclusions which break symmetry and find that there is a finite equilibrium distance to which the inclusions will aggregate. We derive corresponding equations of motion which describe this curvature-mediated aggregation mechanism and, using this framework, we investigate some of the properties of these self-assembled configurations, including their energy, stability, and their collective elastic behavior. Lastly, we consider the interaction energies of embedded inclusions on a periodic domain and determine that this mechanism may explain computational results of how proteins form rings to promote tubulation on cylindrical membranes.