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

Abstract

Highlights

• •

  Group theory is used to simplify the calculations of nuclear spin relaxation rates.

• •

  Transverse and longitudinal ¹⁵N nuclear relaxation rates in ¹⁵NH ₄ ⁺ are derived.

• •

  An ammonium ion bound to a protein domain rotates rapidly within its binding site.

Abstract

The equations that describe the time-evolution of transverse and longitudinal ¹⁵N magnetisations in tetrahedral ammonium ions, ¹⁵NH 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 ¹⁵N– ¹H and ¹H– ¹H 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 ¹⁵N– ¹H and ¹H– ¹H interactions are explicitly taken into account in the derivations. An application to ¹⁵N-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.