A later study revealed that geometrical constraints of α-helical HBs are likely to be the reason for the observed diagonal distributions 14. Earlier studies carried out by Scheraga and coworkers 15, 16, 17, 18, also pointed out the relevance of the diagonals when dealing with α-helical conformations. Since the α-helix basin is found near a sterically inaccessible region near the origin and is distributed along a diagonal, it was believed that the cause for the observed distributions is its location on the Ramachandran map 13. Empirical determination of α-helix structures demonstrated a diagonal distribution of the ( φ, ψ) pairs 13, 14. The Ramachandran map opened the doors for a wide variety of empirical and theoretical conformational studies of α-helical polypeptides. The understanding of the inaccessible and accessible Ramachandran regions is crucial for the understanding of the distribution of the empirically determined backbone conformations. One of the interesting conclusions drawn from Ramachandran’s pioneering study was that more than half of the regions on the Ramachandran map are sterically inaccessible. Neighbor dependent studies allow to take into account the effects of the neighbors on the ( φ, ψ) propensity of a given AA 10, 11, 12. An important variation of the Ramachandran analysis is the neighbor dependent analysis. The Ramachandran map is a plot of dihedral angles φ and ψ, where φ is the C-N-Cα-C dihedral angle and ψ is the N-Cα-C-N dihedral angle. The Ramachandran map allows for distinguishing between regions of similar backbone conformations of polypeptide chains 9. Nearly a decade after the discovery of the α-helix, a systematic tool was developed by Ramachandran and coworkers 8 for the analysis of the backbone conformation of polypeptides, namely the Ramachandran map. There are 3.7 amino acid (AA) residues per turn, with a translation of 1.47 Å per residue along the α-helical axis, and a hydrogen bond (HB) between the carbonyl group of every i th residue to the amide group of i + 4 th residue with an optimal distance of 2.72 Å between the H-bonded oxygen and nitrogen backbone atoms. According to Pauling and coworkers, α-helices have a well-defined structure with constant displacement distances between nitrogen (N) and alpha carbon (Cα) of N-Cα = 1.47 Å, between Cα and carbonyl carbon (C) of Cα-C = 1.53 Å, and C-N = 1.32 Å. This has led to a leap in our understanding of protein structure 4 and function 5, and later in folding prediction 6 and de novo design 7. During this period, Pauling and coworkers discovered the two most fundamental structures found within proteins: the α-helix and the β-sheet 1, 2, 3. The middle of the 20 th century is considered to be the genesis of structural biology. Flory’s isolated pair hypothesis is shown to be partially correct for α-helical conformations. The observed trends of helical conformations obtained from the PDB are captured by four conceptual simulations that theoretically examine the effects of residue bulkiness, external electric field, and externally applied mechanical forces. Analysis of α-helical conformations acquired from the Protein Data Bank (PDB) demonstrates that a conformational energy function of the α-helix backbone can be harmonically approximated on the ( ρ, ϑ) space, which is not applicable to the ( φ, ψ) space due to the diagonal distribution of the conformations. In this way, valuable information on the helical structure becomes directly available. We present an alternative coordinate system that describes helical conformations in terms of residues per turn ( ρ) and angle ( ϑ) between backbone carbonyls relative to the helix direction through an approximate linear transformation between the two coordinates system ( φ, ψ and ρ, ϑ). Representation of α-helical structures with the common ( φ, ψ) dihedrals, as in Ramachandran maps, does not provide informative details regarding the helical structure apart for the abstract geometric meaning of the dihedrals. Α-Helices are the most abundant structures found within proteins and play an important role in the determination of the global structure of proteins and their function.
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