  Proteins constitute a large group of macromolecules with a multitude of
functions for all living organisms. Proteins achieve this by adopting distinct
three-dimensional structures encoded by the sequence of their constituent amino
acids in one or more polypeptides. In this paper, the statistical modelling of
the protein backbone torsion angles is considered. Two new distributions are
proposed for toroidal data by applying the M\"obius transformation to the
bivariate von Mises distribution. Marginal and conditional distributions in
addition to sine-skewed versions of the proposed models are also developed.
Three big data sets consisting of bivariate information about protein domains
are analysed to illustrate the strength of the flexible proposed models.
Finally, a simulation study is done to evaluate the obtained maximum likelihood
estimates and also to find the best method of generating samples from the
proposed models to use as the proposal distributions in the Markov Chain Monte
Carlo sampling method for predicting the 3D structure of proteins.
