The shapes and sizes of closed, pressurized random walks

Two-dimensional cell-like membranes acted on by osmotic pressure differentials are represented by closed, unrestricted random walks. The treatment omits excluded-volume effects, and the pressure that is imposed thus favors an oriented area, so that the shriveled configuration of a vesicle with excess external pressure is inaccessible in this model. Nevertheless, the approach has the decided advantage of yielding analytic expressions in a complete statistical analysis. Results are presented for the average square of the radius of gyration, the asphericity, and the probability distribution of the principal components of the radius of gyration tensor. The analysis is done in both the constant-pressure and constant-area ensembles.