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 γγ turns, αα helices, and ββ sheets. It is well known that dihedral angles (ϕϕ and ψψ) together with ωω (torsion angle of the peptide bond) and χχ (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 kk conformational angles, we need a kk-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 kk-variate circular distributions and testing methods are applied to trivariate circular data set arising from γγ turns consisting of Glycine–Phenylalanine–Threonine sequences. We have shown that there is a three-way dependent relationship between the ϕϕ, ψψ and χχ, 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.