The simulation of 31P NMR line shapes of lipid bilayers using an analytically soluble model

Abstract

Van Faassen's method for obtaining an explicit solution to a first order stochastic differential equation is applied to the simulation of 31P NMR line shapes of unoriented phospholipid bilayers in the L α phase and of oriented bilayers in both the L α and L β′ phases. The effects of the two slowest motions on the density matrix are described by the stochastic Liouville equation (SLE) which is solved analytically using the method of van Faassen. These two slowest motions are assumed to be a rotational re-orientation about the long molecular axis and a uniform wobble of this axis within a conical volume with re-orientation rates characterised by correlation times τ ∥ and τ ⊥ respectively. In the present work the Hamiltonian contains the intramolecular dipole–dipole interaction between the phosphorous nucleus and the four closest methylene protons of the choline headgroup, as well as the anisotropic chemical shielding interaction. Hence the contribution to relaxation from cross correlation between the dipole–dipole and anisotropic chemical shielding interactions is included. The reorientation of the headgroup is assumed to be a rotation sufficiently fast to lead to complete axially symmetric averaging of the Hamiltonian about the rotational axis (the P–O 11 bond axis). Evaluation of the line shape in the present work involves only numerical integration and is therefore less computationally demanding than the large matrix inversions involved in the approaches of Campbell, Freed et al. The present theory also uses fewer parameters than that of Dufourc et al. but nevertheless results in good agreement with these authors’ measurements on DMPC bilayers, using a fixed value of 10 for the ratio τ ⊥/ τ ∥ in the case of the L α phase. However, in contrast to Dufourc et al., we find that these correlation times are equal for the L β′ phase. Finally, we have simulated the decoupled powder line shapes obtained from the L β′ phase of DPPC by Campbell and Meirovitch. Again, we get good agreement providing τ ⊥= τ ∥.