The static magnetic field dependence of chemical exchange linebroadening defines the NMR chemical shift time scale

The static magnetic field dependence of chemical exchange linebroadening in NMR spectroscopy is investigated theoretically and experimentally. Two- site exchange (A ⇆ B) is considered with site A more highly populated than site B (p(a) > p(b)), a shift difference between sites equal to Δω, and an exchange rate constant given by k(ex). The exchange contribution to the transverse relaxation rate constant for the more highly populated site is denoted R(ex). The dependence of R(ex) on the static magnetic field strength is characterized by a scaling parameter α = d ln R(ex)/d ln Δω; in which 0 ≤ α ≤ 2 for p(a) > 0.7. The value of α depends on the NMR chemical shift time scale for the exchange process: for slow exchange (k(ex)/Δω < 1), 0 ≤ α < 1; for intermediate exchange (k(ex)/Δω = 1), α = 1; and for fast exchange (k(ex)/Δω > 1), 1 < α ≤ 2. Consequently, the static magnetic field dependence of R(ex) defines the chemical shift time scale for an exchange process even if the populations are so highly skewed (p(a) >> P(b)) that the minor resonance is not observable in the slow exchange limit. The theoretical results are verified by measuring ¹⁵N transverse relaxation rate constants at static magnetic fields of 11.7 and 14.1 T and temperatures of 300 and 313 K for the protein basic pancreatic trypsin inhibitor. At each combination of static magnetic field and temperature, the rate constants were measured using Carr-Purcell-Meiboom-Gill and Hahn echo techniques with spin- echo delays ranging from 1.0 to 64.5 ms. ¹⁵N resonances for residues in the region of the Cys14-Cys38 disulfide bond are broadened due to chemical exchange. Values of α obtained from the relaxation rate constants range from 0.26 ± 0.17 for Arg39 at 300 K to 1.96 ± 0.25 for Cys38 at 313 K. For Cys38 and Arg39, the two residues most strongly affected by chemical exchange, values of k(ex) were determined to be 380 ± 70 s^-1 and 530 ± 90 s^-1 at 300 K and 1300 ± 290 s^-1 and 1370 ± 160 s^-1 at 313 K by global analysis of the relaxation rate constants. The scaling parameters α indicate that chemical exchange for most residues in basic pancreatic trypsin inhibitor does not satisfy k(ex)/Δω >> 1. Consequently, the assumption of fast-limit quadratic scaling of exchange broadening in proteins and other macromolecules may be incorrect, even if a single broadened resonance is observed for a nuclear spin. The theoretical results for the static magnetic field dependence of chemical exchange broadening in NMR spectroscopy are applicable to other nuclei and to other techniques for measuring chemical exchange linebroadening.