Sine-skewed toroidal distributions and their application in protein bioinformatics

In the bioinformatics field, there has been a growing interest in modeling dihedral angles of amino acids by viewing them as data on the torus. This has motivated, over the past years, new proposals of distributions on the torus. The main drawback of most of these models is that the related densities are (pointwise) symmetric, despite the fact that the data usually present asymmetric patterns. This motivates the need to find a new way of constructing asymmetric toroidal distributions starting from a symmetric distribution. We tackle this problem in this article by introducing the sine-skewed toroidal distributions. The general properties of the new models are derived. Based on the initial symmetric model, explicit expressions for the shape and dependence measures are obtained, a simple algorithm for generating random numbers is provided, and asymptotic results for the maximum likelihood estimators are established. An important feature of our construction is that no extra normalizing constant needs to be calculated, leading to more flexible distributions without increasing the complexity of the models. The benefit of employing these new sine-skewed toroidal distributions is shown on the basis of protein data, where, in general, the new models outperform their symmetric antecedents.