The divergence time of protein structures modelled by Markov matrices
  and its relation to the divergence of sequences
Sandun Rajapaksa Lloyd Allison Peter J. Stuckey Maria Garcia de la Banda Arun S. Konagurthu
A complete time-parameterized statistical model quantifying the divergent
evolution of protein structures in terms of the patterns of conservation of
their secondary structures is inferred from a large collection of protein 3D
structure alignments. This provides a better alternative to time-parameterized
sequence-based models of protein relatedness, that have clear limitations
dealing with twilight and midnight zones of sequence relationships. Since
protein structures are far more conserved due to the selection pressure
directly placed on their function, divergence time estimates can be more
accurate when inferred from structures. We use the Bayesian and
information-theoretic framework of Minimum Message Length to infer a
time-parameterized stochastic matrix (accounting for perturbed structural
states of related residues) and associated Dirichlet models (accounting for
insertions and deletions during the evolution of protein domains). These are
used in concert to estimate the Markov time of divergence of tertiary
structures, a task previously only possible using proxies (like RMSD). By
analyzing one million pairs of homologous structures, we yield a relationship
between the Markov divergence time of structures and of sequences. Using these
inferred models and the relationship between the divergence of sequences and
structures, we demonstrate a competitive performance in secondary structure
prediction against neural network architectures commonly employed for this
task. The source code and supplementary information are downloadable from
\url{http://lcb.infotech.monash.edu.au/sstsum}.