40 CHARLES TANFORD [3 



of closest approach between a charge and a water molecule is the same as 

 it would be in a low molecular weight organic ion in aqueous solution. 



Fig. 2 shows three model structures for which calculations have been made 

 for this paper. All calculations were averaged over all configurations result- 

 ing from permutation of charges over identical sites. They are approximate 

 in that the Bragg- Williams method^"^ was used. (This is equivalent to reten- 

 tion of only one term in equation 33 of Tanford and Kirkwood.^'^) For 

 each model, two isoelectric forms are likely to occur, one with protons 

 removed from every carboxyl group and from no other groups, the second 

 with protons removed from all but one of the carboxyl groups and from 

 one of the imidazolium groups. The corresponding electrostatic interaction 



-6400 

 -5200 



MODEL B 

 ELECTROSTATIC INTERACTION ENERGIES (CALS./MOLE.) - 



-10800 +6600 



-10000 +13200 



Fig. 2. Models used for calculation of the electrostatic contribution to the free energy 

 of compact configurations. The letters C, G, I, N and O represent, respectively, carboxyl, 

 guanidine, imidazole, amino and phenolic groups. The dodecahedra are inscribed in 

 spheres of radius 10 Â, with the vertices 1 Â below the surface, this being the distance 

 of closest approach of solvent to the charged sites on any organic acid or base.^° The 

 distance between nearest neighboring vertices in 6-5 Â. 



energies are: for model A, —6400 and —5200 calories/mole; for model B, 

 -10,800 and -10,000 calories/mole; for model C, +13,200 and +6600 

 calories/mole. These calculations are for isolated protein molecules at zero 

 ionic strength. The addition of inorganic salts reduces these values, but for 

 impermeable molecules, the effect is relatively small, as Table 2 shows, 

 because the salt ions cannot enter the space between charges, where they 

 would be most effective. 



These calculations clearly show that the contribution of electrostatic inter- 

 action to the free energy of isoelectric protein molecules can be very large, 

 and that the magnitude of the effect depends markedly on the positions of 

 charged sites relative to one another. The interaction energy may be nega- 

 tive or positive. However, models leading to positive interaction energies 

 (such as model C) require unlikely arrangement of the charges with most 

 of the positive charges well separated from most of the negative charges. 



