  Identifying protein-protein interactions is crucial for a systems-level
understanding of the cell. Recently, algorithms based on inverse statistical
physics, e.g. Direct Coupling Analysis (DCA), have allowed to use
evolutionarily related sequences to address two conceptually related inference
tasks: finding pairs of interacting proteins, and identifying pairs of residues
which form contacts between interacting proteins. Here we address two
underlying questions: How are the performances of both inference tasks related?
How does performance depend on dataset size and the quality? To this end, we
formalize both tasks using Ising models defined over stochastic block models,
with individual blocks representing single proteins, and inter-block couplings
protein-protein interactions; controlled synthetic sequence data are generated
by Monte-Carlo simulations. We show that DCA is able to address both inference
tasks accurately when sufficiently large training sets are available, and that
an iterative pairing algorithm (IPA) allows to make predictions even without a
training set. Noise in the training data deteriorates performance. In both
tasks we find a quadratic scaling relating dataset quality and size that is
consistent with noise adding in square-root fashion and signal adding linearly
when increasing the dataset. This implies that it is generally good to
incorporate more data even if its quality is imperfect, thereby shedding light
on the empirically observed performance of DCA applied to natural protein
sequences.
