Circular clustering of protein dihedral angles by Minimum Message Length

Early work on proteins identified the existence of helices and extended sheets in protein secondary structures, a high-level classification which remains popular today. Using the Snob program for information-theoretic Minimum Message Length (MML) classification, we are able to take the protein dihedral angles as determined by X-ray crystallography, and cluster sets of dihedral angles into groups. Previous work by Hunter and States has applied a similar Bayesian classification method, AutoClass, to protein data with site position represented by 3 Cartesian co-ordinates for each of the alpha-Carbon, beta-Carbon and Nitrogen, totalling 9 co-ordinates. By using the von Mises circular distribution in the Snob program, we are instead able to represent local site properties by the two dihedral angles, phi and psi. Since each site can be modelled as having 2 degrees of freedom, this orientation-invariant dihedral angle representation of the data is more compact than that of nine highly-correlated Cartesian co-ordinates. Using the information-theoretic message length concepts discussed in the paper, such a more concise model is more likely to represent the underlying generating process from which the data came. We report on the results of our classification, plotting the classes in (phi, psi) space; and introducing a symmetric information-theoretic distance measure to build a minimum spanning tree between the classes. We also give a transition matrix between the classes and note the existence of three classes in the region phi approximately -1.09 rad and psi approximately -0.75 rad which are close on the spanning tree and have high inter-transition probabilities. This gives rise to a tight, abundant and self-perpetuating structure.