Modelling the role of osmotic pressure for lipid vesicles in a confined space using continuum elastic theories

Fluid-lipid vesicles are ubiquitous and fundamental structures in biology, as they play a
central role in the regulation of many cellular processes. As a consequence, they have
been the subject of extensive study over the last decades, both from an experimental
and a theoretical standpoint. Mathematical models addressing lipid vesicles in osmotic
environments have been a focus, aiming to determine their shapes by minimizing a cur-
vature free energy for given volume and surface area constraints. In this thesis, a similar
approach is pursued, although in an ensemble in which the volume of the vesicle is al-
lowed to change while the surface area remains constant. In particular, our calculations
are carried out in a finite box and enable us to directly compare theoretically obtained
shapes with experimental results. The new model is first introduced for a bidimensional
vesicle and then extended to tridimensional counterparts. Stability and instability of
the solutions obtained within this framework are studied in both the bidimensional and
tridimensional cases and bifurcation diagrams are obtained, allowing the direct compar-
ison of our results with experiments. Furthermore, the minimization calculations within
the calculus of variations framework present unique mathematical challenges, offering
intriguing avenues for future exploration and theoretical development. In summary, this
research not only contributes to the understanding of lipid vesicles but also highlights
the potential for extending the model designed for lipid vesicles to the study of bacte-
ria. This suggestion serves as a stimulating prospect for further investigation into the
broader implications and applications of the proposed theoretical framework in the realm
of biological systems.