  Diffusion-based network models are widely used for protein function
prediction using protein network data and have been shown to outperform
neighborhood- and module-based methods. Recent studies have shown that
integrating the hierarchical structure of the Gene Ontology (GO) data
dramatically improves prediction accuracy. However, previous methods usually
either used the GO hierarchy to refine the prediction results of multiple
classifiers, or flattened the hierarchy into a function-function similarity
kernel. No study has taken the GO hierarchy into account together with the
protein network as a two-layer network model.
  We first construct a Bi-relational graph (Birg) model comprised of both
protein-protein association and function-function hierarchical networks. We
then propose two diffusion-based methods, BirgRank and AptRank, both of which
use PageRank to diffuse information on this two-layer graph model. BirgRank is
an application of traditional PageRank with fixed decay parameters. In
contrast, AptRank uses an adaptive mechanism to improve the performance of
BirgRank. We evaluate both methods in predicting protein function on yeast,
fly, and human datasets, and compare with four previous methods: GeneMANIA,
TMC, ProteinRank and clusDCA. We design three validation strategies: missing
function prediction, de novo function prediction, and guided function
prediction to comprehensively evaluate all six methods. We find that both
BirgRank and AptRank outperform the others, especially in missing function
prediction when using only 10% of the data for training.
  AptRank combines protein-protein associations and the GO function-function
hierarchy into a two-layer network model without flattening the hierarchy into
a similarity kernel. Introducing an adaptive mechanism to the traditional,
fixed-parameter model of PageRank greatly improves the accuracy of protein
function prediction.
