Charge and dipole moment fluctuations around the thermal equilibrium values for a charge distribution on a sphere can be defined as δq = R 2 dΩ δσ(Ω) and δp = R 2 dΩ δσ(Ω) r Ω , respectively, where δσ(Ω) = σ(Ω) -σ(Ω) and we have taken into account that the charges are confined to the surface of a sphere . To calculate the fluctuations in the total charge of the protein around the mean-field solution, we have