Asymmetric Karplus curves for the protein side-chain 3J couplings

The standard Karplus equation for calculating 3J coupling constants from any given dihedral angle requires three empirical coefficients be determined that relate to the magnitudes of three modes of the angle dependency of 3J. Considering cosine modes only (bimodal, unimodal and baseline component), Karplus curves are generally symmetric with respect to the sign of the angle argument. Typically, their primary and secondary maxima differ in amplitude, whereas the two minima are of equal depth. However, chiral molecular topologies, such as those surrounding the main-chain and side-chain torsions in amino-acid residues, preclude, as regards substituent positioning, exact mirror-image conformations from being formed--for any given torsion-angle value. It is therefore unlikely that 3J couplings assume identical values for the corresponding positive and negative dihedral angles. This suggests that a better empirical fit of the torsion-angle dependency of 3J could be obtained when removing the constraint of symmetrically identical coupling constants. A sine term added to the Karplus equation allows independent modelling of both curve minima typically located near dihedral-angle values of +90 degrees and -90 degrees. Revisiting an extensive 3J coupling dataset previously recorded to determine the side-chain torsions chi1 in the protein flavodoxin, the asymmetric Karplus model accomplishes a more accurate fit to the experimental data. Asymmetries revealed in the angle dependencies exceed the experimental precision in determining 3J. Accounting for these effects helps improve molecular models.