  Inverse statistical approaches to determine protein structure and function
from Multiple Sequence Alignments (MSA) are emerging as powerful tools in
computational biology. However the underlying assumptions of the relationship
between the inferred effective Potts Hamiltonian and real protein structure and
energetics remain untested so far. Here we use lattice protein model (LP) to
benchmark those inverse statistical approaches. We build MSA of highly stable
sequences in target LP structures, and infer the effective pairwise Potts
Hamiltonians from those MSA. We find that inferred Potts Hamiltonians reproduce
many important aspects of 'true' LP structures and energetics. Careful analysis
reveals that effective pairwise couplings in inferred Potts Hamiltonians depend
not only on the energetics of the native structure but also on competing folds;
in particular, the coupling values reflect both positive design (stabilization
of native conformation) and negative design (destabilization of competing
folds). In addition to providing detailed structural information, the inferred
Potts models used as protein Hamiltonian for design of new sequences are able
to generate with high probability completely new sequences with the desired
folds, which is not possible using independent-site models. Those are
remarkable results as the effective LP Hamiltonians used to generate MSA are
not simple pairwise models due to the competition between the folds. Our
findings elucidate the reasons for the success of inverse approaches to the
modelling of proteins from sequence data, and their limitations.
