Kinematic Reconstruction of Cyclic Peptides and Protein Backbones from Partial Data

We present an algorithm, QBKR (Quaternary Backbone Kinematic Reconstruction), a fast analytical method for an all-atom backbone reconstruction of proteins and linear or cyclic peptide chains from C α coordinate traces. Unlike previous analytical methods for deriving all-atom representations from coarse-grained models that rely on canonical geometry with planar peptides in the trans conformation, our de novo kinematic model incorporates noncanonical, cis–trans, geometry naturally. Perturbations to this geometry can be effected with ease in our formulation, for example, to account for a continuous change from cis to trans geometry. A simple optimization of a spring-based objective function is employed for C α–C α distance variations that extend beyond the cis–trans limit. The kinematic construction produces a linked chain of peptide units, C α–C–N–C α, hinged at the C α atoms spanning all possible planar and nonplanar peptide conformations. We have combined our method with a ring closure algorithm for the case of ring peptides and missing loops in a protein structure. Here, the reconstruction proceeding from both the N and C termini of the protein backbone (or in both directions from a starting position for rings) requires freedom in the position of one C α atom (a capstone) to achieve a successful loop or ring closure. A salient feature of our reconstruction method is the ability to enrich conformational ensembles to produce alternative feasible conformations in which H-bond forming C–O or N–H pairs in the backbone can reverse orientations, thus addressing a well-known shortcoming in C α-based RMSD structure comparison, wherein very close structures may lead to significantly different overall H-bond behavior. We apply the fixed C α-based design to the reverse reconstruction from noisy Cryo-EM data, a posteriori to the optimization. Our method can be applied to speed up the process of an all-atom description from voluminous experimental data or subpar electron density maps.