DEDICATED TO THE MEMORY OF DR. ZORAN MARIC ˇIC
The continuous probability distribution (CUPID) approach for isomerism in molecules of arbitrary size and shape in solution, analyzing the rotamer populations from NMR spin-spin couplings which avoids the traps of the approximation of discrete states and nuclear Overhauser enhancements [Z ˇ. Dz ˇakula, W. M. Westler, (2) and permits as input a variety of NMR data. Appropriate A. S. Edison, and J. L. Markley, J. Amer. Chem. Soc. 114, 6195 input data include vicinal spin-spin couplings (3), cross-relax-(1992)] can be expanded to allow computation of the rotational ation rates between fixed protons and rotating protons, chemical potential from the Fourier coefficients of the angular probability disshifts of b-CH 2 protons of cysteines in high-potential irontribution. This approach provides a general solution to the nonnegasulfur proteins (4), six-bond spin-spin couplings in benzene tivity problem, which appears when lack of data causes a serious derivatives (5), cross correlations as measured by cross relaxtruncation in the Fourier series that defines the probability distribuation involving higher spin orders (e.g., 3QF tilted frame tion. In favorable cases, this approach also allows thermodynamic characterization of internal rotation. Use of this extension of the ROESY) (6), dipole moments (7), and ESR hyperfine cou-CUPID method is illustrated by the analysis of internal rotations in pling constants (8). The basic idea behind CUPID is that the an amino acid, two peptides, and an oligosaccharide from published continuous probability distribution of dihedral angles can be experimental data. Three strategies have been devised for dealing determined by Fourier expansion. The fitting of an arbitrary with cases where the experimental input data do not provide enough probability distribution requires an infinite Fourier series; howinformation for complete reconstruction of the potential: (1) twoever, the number of experimentally derived Fourier coefficients dimensional grid search for the undetermined third-order Fourier is limited by the number and quality of experimental concoefficients of the potential, (2) transfer of these coefficients from straints. Because it is necessary to employ a truncated Fourier related model compounds, and (3) restriction of the magnitudes of expansion, this reduces CUPID to a maximum-likelihood the Fourier coefficients as required by the assumption of fast-exchange averaging of the input parameters. In addition, equations for method, where the model consists of low-frequency Fourier translating uncertainties in experimental NMR input data into errors components. While the truncated Fourier expansion offers pracin calculated continuous probability distributions of rotamers are tical advantages, such as linear independence of the Fourier presented. The dependence of errors on various features of the districoefficients, it is not without drawbacks. One is that the truncabutions has been studied systematically from simulations. The results tion results in negative values of the probability distribution, show that, typically, the confidence intervals are {30-40Њ for dihewhich have no physical meaning. We propose here the applicadral angles and {0.2 for rotamer populations. For x 1 rotamers of tion of the Boltzmann factor as a means of maintaining physical amino acids, the analysis is most sensitive to the uncertainties in reality in the truncated Fourier expansion of the angular proba-C-H b couplings. A critical reexamination of the use of Gaussian bility distribution. The present work focuses on recovering Foufunctions to reconstruct a probability distribution is presented. In particular, the simplifying assumption of identical widths for all rier components of the rotational potential from the Fourier Gaussian probability peaks has been justified by showing that it does coefficients of the distribution and on quantifying the error not lead to large errors in other CUPID parameters. Finally, the propagation in such CUPID analyses. These enhancements are angular dependences of cross-relaxation rates, their uncertainties, now included in the CUPID software package (available over and the potential for their use in studying x 1 internal rotations in WorldWideWeb from http://www.nmrfam.wisc.edu).
amino acids are discussed.