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Protein Secondary Structure Classification

Categorizing protein structures based on their repeating, localized structural motifs: alpha-helices, beta-sheets, and random coils.
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The statement of the theorem

Let the local backbone geometry of a residue ii be defined by the dihedral angles (ϕi,ψi)(\phi_i, \psi_i). The classification of secondary structure is based on the allowed regions in the Ramachandran plot. Define the structural motif S\mathcal{S} as the set of allowed (ϕ,ψ)(\phi, \psi) pairs: \n\nS=Sα∪Sβ∪Srandom\mathcal{S} = \mathcal{S}_{\alpha} \cup \mathcal{S}_{\beta} \cup \mathcal{S}_{\text{random}} \n\nwhere Sα\mathcal{S}_{\alpha} is the region corresponding to α\alpha-helices (e.g., −60∘<ϕ<−30∘,−70∘<ψ<−30∘-60^{\circ} < \phi < -30^{\circ}, -70^{\circ} < \psi < -30^{\circ}), Sβ\mathcal{S}_{\beta} is the region corresponding to β\beta-strands (e.g., −120∘<ϕ<−60∘,10∘<ψ<140∘-120^{\circ} < \phi < -60^{\circ}, 10^{\circ} < \psi < 140^{\circ}), and Srandom\mathcal{S}_{\text{random}} encompasses the remaining, less constrained regions.