A multivariate circular distribution with applications to the protein structure prediction problem

The protein structure prediction problem is considered to be the holy grail of bioinformatics, and circular variables in protein structure problem are ubiquitous. For example, conformational angles appear in γ turns, α helices, and β sheets. It is well known that dihedral angles (ϕ and ψ) together with ω (torsion angle of the peptide bond) and χ (torsion angle of the side chain) are considered to be important for protein structure prediction since they define the entire conformation of a protein. In order to study k conformational angles, we need a k-variate angular distribution. In this paper, we propose a multivariate circular distribution and inferential methods, which could be useful for jointly modeling those circular variables of interest. Our proposed family of k-variate circular distributions and testing methods are applied to trivariate circular data set arising from γ turns consisting of Glycine–Phenylalanine–Threonine sequences. We have shown that there is a three-way dependent relationship between the ϕ, ψ and χ, and that the side chain angles are relevant to the relationship between dihedral angles for the given sequence. The proposed model was compared with two existing multivariate circular models using bivariate and trivariate circular data sets.