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Theory and implementation

Adequate and accurate cross comparison of the NMR and MD simulation data is of crucial importance in versatile studies conformational dynamics of proteins. NMR relaxation spectroscopy has proven to be a unique approach for a site-specific investigation of both global tumbling and internal motions of proteins. The molecular motions modulate the magnetic interactions between the nuclear spins and lead for each nuclear spin to a relaxation behavior which reflects its environment. Since its first applications to the study of protein dynamics, a wide variety of experiments has been proposed to investigate backbone as well as side chain dynamics. Among them, the heteronuclear relaxation measurement of amide backbone N nuclei is one of the most widespread techniques. The relationship between microscopic motions and measured spin relaxation rates is given by Redfield's theory [70]. Under the hypothesis that N relaxation occurs through dipole-dipole interactions with the directly bonded H atom and chemical shift anisotropy (CSA), and assuming that the tensor describing the CSA is axially symmetric with its axis parallel to the N-H bond, the relaxation rates of the N nuclei are determined by a time correlation function,

| (4.195)

which describes the dynamics of a unit vector pointing along the N-H bond of the residue i in the laboratory frame. Here is the second order Legendre polynomial. The Redfield theory shows that relaxation measurements probe the relaxation dynamics of a selected nuclear spin only at a few frequencies. Moreover, only a limited number of independent observables are accessible. Hence, to relate relaxation data to protein dynamics one has to postulate either a dynamical model for molecular motions or a functional form for , yet depending on a limited number of adjustable parameters. Usually, the tumbling motion of proteins in solution is assumed isotropic and uncorrelated with the internal motions, such that:

| (4.196)

where and denote the global and the internal time correlation function, respectively. Within the so-called model free approach [71,72] the internal correlation function is modeled by an exponential,

| (4.197)

Here the asymptotic value is the so-called generalized order parameter, which indicates the degree of spatial restriction of the internal motions of a bond vector, while the characteristic time is an effective correlation time, setting the time scale of the internal relaxation processes. can adopt values ranging from 0 (completely disordered) to 1 (fully ordered). So, is the appropriate indicator of protein backbone motions in computationally feasible timescales as it describes the spatial aspects of the reorientational motion of N-H peptidic bonds vector.

When performing Order Parameter analysis, nMOLDYN computes for each residue i both and . It also computes a correlation function averaged over all the selected bondsdefined as:

| (4.198)

where is the number of selected bonds for the analysis.

Next: Parameters Up: Order Parameter Previous: Order Parameter Contents