Cremer–Pople Puckering Parameters for N-Membered Rings

The out-of-plane deviations of puckered N-membered rings can be measured by the z-coordinates of the ring atoms relative to a mean plane cutting through the ring. The z-coordinates contain information about the overall movement or the shape of the puckered ring. Translation and overall rotation of the planar reference around the x- and y-axes can be removed by imposing three constraints (see Appendix 1, eqs S1–S3).

Let R_j be the position vectors of the ring atom, j, with the origin defined as the geometrical center of the puckered rings. We denote two vectors, R′ and R″, that define the mean plane (see Appendix 1, eqs S4 and S5), where n is the unit normal vector to this mean plane; z_j is then the displacement of atom j from the mean plane and is given by the scalar products in eq 1

Using the mean plane and the full set of displacements, we can compute the Cremer–Pople ring puckering parameters¹⁹ as follows.

For odd values of N, and N > 3, the puckering amplitude, q_m, and phase angle, ϕ_m, are defined as follows:

eqs 2 and 3 apply for m = 2, 3, ..., (N – 1)/2. The amplitudes, q_m, are positive-valued, while the phase angles, ϕ_m, range from −π to π radians.

For even values of N, eqs 2 and 3 apply for m = 2, 3, ..., (N/2 – 1), but an additional puckering amplitude is required, with the following form:

Note that the q_(N)/(2) value in eq 4 can take either sign.

The Cremer–Pople representation is only applicable to monocyclic ring systems. To extend it to more complex ring systems such as fused and spiro rings, we first decompose the ring systems into smaller rings and calculate the puckering parameters for each ring. In particular, we adopt the concept of unique ring families (URFs)³⁰ for this decomposition, with the resultant Cremer–Pople parameters being calculated for all relevant cycles, i.e., minimum cycle bases.

Additionally, the Cremer–Pople representation is atom-order-dependent, and we standardize the atom ordering in the ring before calculation. Bond orders, connectivity, and element types are used to determine this standardized order (see Appendix 1). Other canonical atom numberings may also be used.³¹ In symmetric rings, such as cycloalkanes, the first atom is picked at random.

We also take the volume of the amino acid into account when ordering the backbone ring atoms in cyclic peptides. The priority increases with volume, so tryptophan, tyrosine, and phenylalanine have higher ranks, while glycine has the lowest. The rank order of amino acids can be found in Appendix 1, Table S1. Note that this ordering is only applied to the cyclic peptides.