Transverse nuclear spin relaxation studies of viscoelastic properties in membrane vesicles. I. Theory

Transverse nuclear spin relaxation measurements employing Carr-Purcell (CP) pulse sequences can provide detailed information on the slow-motional dynamics in biomembranes. In this paper, a comprehensive relaxation model is developed for the analysis of such experiments performed on unilamellar quasi-spherical vesicles. The basis of the model is the stochastic Liouville equation in which two different relaxation processes are considered (i.e., vesicle shape fluctuations and molecular translational diffusion ). It is shown that for vesicle radii R0 greater than or equal to 200 nm, translational diffusion of the lipid molecules along the vesicle shell is too slow to contribute significantly to transverse spin relaxation in the kHz range, whereas vesicle shape fluctuations constitute the dominant transverse relaxation process. The theory is employed in model calculations for pulse frequency-dependent transverse nuclear spin relaxation rates, R_2infinity^CP(omega), from CP sequences. The analysis reveals that R_2infinity^CP(omega), induced by vesicle fluctuations, depends linearly on omega^-1 over a wide frequency range in the kHz regime. Notably, within this linear dispersion regime, the bending elastic modulus is the only relevant parameter because the magnitude of R_2infinity^CP(omega) does not depend on the size of the vesicle R0, the effective Lateral tension, or the viscosity of the surrounding fluid. On the other hand, all these parameters determine the frequency at which R_2infinity^CP(omega) levels off to a constant "plateau" value independent of omega. Thus, analysis of the R_2infinity^CP(omega) dispersion profiles is a direct way to determine the bending elastic modulus and other viscoelastic parameters of membrane vesicles.