Abstract
The equations that describe the time-evolution of transverse and longitudinal 15N magnetisations in tetrahedral ammonium ions, 15NH4+, 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 41kDa 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 1ns. The theoretical framework provided here forms the basis for further investigations of dynamics of AX4 spin systems, with ammonium ions in solution and bound to proteins of particular interest.
Highlights
The transverse and longitudinal nuclear spin-relaxation rates, which can be obtained from NMR spectra, are accurate reporters on the interactions and dynamics of molecules ranging from small organic molecules and ions [1,2,3,4] to large macromolecular complexes [5,6,7,8]
It was assumed that the geometric structure of the ammonium ion is that of a tetrahedron, which in turn means that symmetries of the energy eigenstates fall within the symmetries of the Td point group
We presented the equations that describe the transverse nitrogen relaxations of the ammonium ion in two basis sets, the Zeeman-derived basis and the Cartesian basis, as well as the relaxation rates of the longitudinal spin-density operators in the Cartesian basis
Summary
The transverse and longitudinal nuclear spin-relaxation rates, which can be obtained from NMR spectra, are accurate reporters on the interactions and dynamics of molecules ranging from small organic molecules and ions [1,2,3,4] to large macromolecular complexes [5,6,7,8]. Is bound to proteins [16] or nucleic acid complexes [17,18,19], the exchange rate of the ammonium protons becomes sufficiently slow to allow for both detection of the ammonium protons and acquisition of 15N–1H correlation spectra. As was shown recently [16], 15NH4+ can be observed even when bound to proteins with molecular weights in excess of 40 kDa, but it is currently not clear whether it is fast reorientation of the ammonium ion within the binding site or favourable cross-correlated relaxation mechanisms that allow for such measurements. Given the development of techniques to probe ammonium ions in proteins and nucleic acids and considering the interest in probing the regulations of enzymes by monovalent cations in general, it is of interest to derive equations that describe the transverse and longitudinal relaxations of ammonium ions under various conditions.
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