The collective λ-coordinate of the buffer particles is not restricted to a fixed interval by a wall-like potential. To avoid that the buffer charge exceeds the restricted interval, multiple buffer particles are needed in the simulation box. The optimal number of buffers can be calculated based on the analysis of charge fluctuations performed by Donnini et al. With a small charge, a buffer particle is apolar. To prevent clustering of such apolar particles in water, permeation into hydrophobic areas, such as membrane interiors, or interactions with the protein, the Lennard-Jones parameters of the buffers were chosen such that the buffers have only repulsive interactions with all other atoms, except water. After experimenting with parameters for the buffer particles, we settled on a σ of 0.25 nm and an of 4 kJ mol -1 . This choice leads to decreased clustering of buffers, low buffer concentrations in the proximity of titratable sites, and reduced penetration into hydrophobic regions (Fig. ). The resulting free energies of neutral buffer insertion into water and hydrophobic region of membrane are -2.09 ± 0.07 kJ mol -1 and 1.2 ± 0.6 kJ mol -1 , compared to 8.45 ± 0.05 kJ mol -1 and 7.8 ± 0.3 kJ mol -1 for the insertion of an uncharged sodium ion into these regions.