  Prediction of one-dimensional protein structures such as secondary structures
and contact numbers is useful for the three-dimensional structure prediction
and important for the understanding of sequence-structure relationship. Here we
present a new machine-learning method, critical random networks (CRNs), for
predicting one-dimensional structures, and apply it, with position-specific
scoring matrices, to the prediction of secondary structures (SS), contact
numbers (CN), and residue-wise contact orders (RWCO). The present method
achieves, on average, $Q_3$ accuracy of 77.8% for SS, correlation coefficients
of 0.726 and 0.601 for CN and RWCO, respectively. The accuracy of the SS
prediction is comparable to other state-of-the-art methods, and that of the CN
prediction is a significant improvement over previous methods. We give a
detailed formulation of critical random networks-based prediction scheme, and
examine the context-dependence of prediction accuracies. In order to study the
nonlinear and multi-body effects, we compare the CRNs-based method with a
purely linear method based on position-specific scoring matrices. Although not
superior to the CRNs-based method, the surprisingly good accuracy achieved by
the linear method highlights the difficulty in extracting structural features
of higher order from amino acid sequence beyond that provided by the
position-specific scoring matrices.
