A diffusion equation approach to spin diffusion in biomolecules

A theoretical description of $1^H-1^H$ dipolar nuclear spin relaxation in a multispin system has been worked out by forming a diffusion equation for a one-dimensional chain of equidistant spins. The spin-diffusion equation is formed from first principles by assuming nearest neighbor interactions for a molecule undergoing isotropic random reorientation. This equation describes diffusion only in the long correlation limit (for $(\omega \tau_c > 1.118)$ and is solved for driven NOE experiments, for spins in a chain of infinite length $(0 <x< \infty)$, or for spins in a chain of finite length $(0 <x< L$). The solutions are obtained using the method of the Laplace transform for specified initial and boundary conditions. The observed selectivity of the NOE transfer in driven NOE experiments on a biomolecule which has a correlation factor $\omega\tau_c \sim3$ is indeed in conformity with the predictions obtained from the spin-diffusion equation