Heteronuclear transverse and longitudinal relaxation in AX(4) spin systems: Application to (15)N relaxations in (15)NH(4)(+)

The equations that describe the time-evolution of transverse and longitudinal (15)N magnetisations in tetrahedral ammonium ions, (15)NH(4)(+), are derived from the Bloch-Wangsness-Redfield density operator relaxation theory. It is assumed that the relaxation of the spin-states is dominated by (1) the intra-molecular (15)N–(1)H and (1)H–(1)H dipole–dipole interactions and (2) interactions of the ammonium protons with remote spins, which also include the contribution to the relaxations that arise from the exchange of the ammonium protons with the bulk solvent. The dipole–dipole cross-correlated relaxation mechanisms between each of the (15)N–(1)H and (1)H–(1)H interactions are explicitly taken into account in the derivations. An application to (15)N-ammonium bound to a 41 kDa domain of the protein DnaK is presented, where a comparison between experiments and simulations show that the ammonium ion rotates rapidly within its binding site with a local correlation time shorter than approximately 1 ns. The theoretical framework provided here forms the basis for further investigations of dynamics of AX(4) spin systems, with ammonium ions in solution and bound to proteins of particular interest.