Entropic pressure on a confined biological vesicle with surface tension

Entropic forces play a critical role in the dynamics and stability of soft matter systems, particularly in biological membranes and vesicles. The origin of these forces lies in the significant thermal fluctuations of soft membranes, a subject that has intrigued the scientific community for decades. Most studies focus on a simplified version of the problem: a flat, tensionless membrane, rather than more complex non-planar surfaces with pre-existing curvature and surface tension. In this paper, we revisit this problem for confined biological vesicles using statistical mechanics analysis and coarse-grained molecular dynamics simulations, explicitly incorporating their curvature field and surface tension. The coupling between the deformation field and the non-zero curvature field leads to a renormalized surface tension, significantly altering the entropic force compared to that of a planar membrane. We demonstrate that while the entropic pressure $p$ follows a similar power-law behavior to that of a planar membrane at small distances, $p\propto 1/d^3$, it transitions to an exponential decay at larger distances. These findings provide insights into the coupled effects of surface tension, membrane configuration, and thermal fluctuations, particularly for understanding biological processes, such as vesicle fusion, endocytosis, and membrane-mediated interactions in crowded cellular environments.