Calculation of dipole-shielding polarizabilities (sigma(alphabetagamma)I): the influence of uniform electric field effects on the shielding of backbone nuclei in proteins

A significant contribution to the chemical shielding of a nucleus can arise from uniform electric fields that act to distort the electronic charge distribution surrounding a nucleus and, hence, affect the nuclear shielding. It has been shown by Buckingham (Buckingham, A. D. Can. J. Chem. 1960, 38, 300) that the nuclear magnetic shielding tensor sigmaalphabetaI of a nucleus in the presence of an external weak static uniform electric field E may be expanded using sigmaalphabetaI(E) = sigmaalphabetaI + sigmaalphabetagammaIEgamma + 1/2sigmaalphabetagammadeltaIEgammaEdelta + sigmaalphabeta,gammadeltaIEgammadelta ... The third rank tensor sigmaalphabetagammaI is referred to as the dipole-shielding polarizability and describes the nonlinear response of the electron cloud to first order in E, muI, and B0. We report calculations of sigmaalphabetagammaI for the N, HN, and C' nuclei in N-methyl acetamide (NMA) and show that these tensors can be used to provide considerable insight into the behavior of uniform electric fields upon the shielding of backbone nuclei in proteins. The sigmaalphabetagammaI values for the N, HN, and C' of NMA were calculated using the continuous transformation of the origin of the current density (CTOCD) scheme with the diamagnetic contribution set to zero (CTOCD-DZ). Values are given for the individual tensor components of sigmaalphabetagammaI for each nucleus. To test that the calculations have provided a reasonable estimate for the sigmaalphabetagammaI of N, HN, and C' nuclei in proteins, a pH titration was performed using Hen Lysozyme (HEWL). The pH-induced isotropic shielding changes for the C', N, and HN nuclei in some peptide bonds close to E35 ( approximately <8 A) were extracted from sets of fitted titration curves. Assuming the experimental shielding changes arise solely from uniform electric field effects caused by the deprotonation of E35, without any other pH-induced structural alterations which might lead to a shielding change, the experimental shielding differences were compared to those calculated via the product Agamma(I).Egamma where Agamma(I) = (1/3)sigmaalphaalphagammaIota. The agreement with the experimental data is in many cases reasonable and suggests that, within the Buckingham formalism, the complete sigmaalphabetagammaI tensors reported here will be helpful to resolve the importance of uniform electric fields upon isotropic and anisotropic shielding in proteins and their complexes.