Continuous probability distribution (CUPID) analysis of potentials for internal rotations.
The continuous probability distribution (CUPID) approach for analyzing the rotamer populations from NMR spin-spin couplings and nuclear Overhauser enhancements [Z. Dzakula, W.M. Westler, A.S. Edison, and J.L. Markley, J. Amer. Chem. Soc. 114, 6195 (1992)] can be expanded to allow computation of the rotational potential from the Fourier coefficients of the angular probability distribution. This approach provides a general solution to the nonnegativity problem, which appears when lack of data causes a serious truncation in the Fourier series that defines the probability distribution. In favorable cases, this approach also allows thermodynamic characterization of internal rotation. Use of this extension of the CUPID method is illustrated by the analysis of internal rotations in an amino acid, two peptides, and an oligosaccharide from published experimental data. Three strategies have been devised for dealing with cases where the experimental input data do not provide enough information for complete reconstruction of the potential: (1) two-dimensional grid search for the undetermined third-order Fourier coefficients of the potential, (2) transfer of these coefficients from related model compounds, and (3) restriction of the magnitudes of the Fourier coefficients as required by the assumption of fast-exchange averaging of the input parameters. In addition, equations for translating uncertainties in experimental NMR input data into errors in calculated continuous probability distributions of rotamers are presented. The dependence of errors on various features of the distributions has been studied systematically from simulations. The results show that, typically, the confidence intervals are +/- 30-40 degrees for dihedral angles and +/- 0.2 for rotamer populations. For chi 1 rotamers of amino acids, the analysis is most sensitive to the uncertainties in C'-H beta couplings. A critical reexamination of the use of Gaussian functions to reconstruct a probability distribution is presented. In particular, the simplifying assumption of identical widths for all Gaussian probability peaks has been justified by showing that it does not lead to large errors in other CUPID parameters. finally, the angular dependencies of cross-relaxation rates, their uncertainties, and the potential for their use in studying chi 1 internal rotations in amino acids are discussed.