Dynamics of the entropic insertion of a large sphere into a cylindrical vessel

Insertion of a solute into a vessel comprising biopolymers is a fundamental function in a biological system. The entropy originating from the translational displacement of solvent particles plays an essential role in the insertion. Here we study the dynamics of entropic insertion of a large spherical solute into a cylindrical vessel. The solute and the vessel are immersed in small spheres forming the solvent. We develop a theoretical method formulated using the Fokker-Planck equation. The spatial distribution of solute-vessel entropic potential, which is calculated by the three-dimensional integral equation theory combined with rigid-body models, serves as input data. The key quantity analyzed is the density of the probability of finding the solute at any position at any time. It is found that the solute is inserted along the central axis of the vessel cavity and trapped at a position where the entropic potential takes a local minimum value. The solute keeps being trapped without touching the vessel inner surface. In a significantly long time τ, the solute transfers to the position in contact with the vessel bottom possessing the global potential minimum along the central axis. As the solute size increases, τ becomes remarkably longer. We also discuss the relevance of our result to the functional expression of a chaperonin/cochaperonin in the assistance of protein folding.