  In the bioinformatics field, there has been a growing interest in modelling
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 bivariate
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 paper 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 parameters 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 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
distributions is shown on the basis of protein data, where, in general, the new
models outperform their symmetric antecedents.
