We have studied how non-DLVO forces between molecules of the globular protein apoferritin in solution affect its osmotic second virial coefficient. A model explaining the effects of the solution ionic strength and pH on the interprotein interaction is developed, to give a physical interpretation of recently published experimental findings showing that the second virial coefficient of the protein apoferritin, supported by acetate buffer, goes through a minimum as a function of ionic strength. At low ionic strengths, the apoferritin second virial coefficient initially decreases with increasing sodium ion concentration, as DLVO theory predicts. However, non-DLVO hydration forces due to overlapping of the Stern layers of the protein molecules increase the second virial coefficient with further increase of sodium ion concentration, again as found experimentally at higher ionic strengths. The non-DLVO effect arises from ionic exchange between hydrogen and sodium ions at the protein surface. An adsorption shell of hydrated sodium ions forms around the protein molecules with increasing buffer concentration.