  The function of proteins arises from cooperative interactions and
rearrangements of their amino acids, which exhibit large-scale dynamical modes.
Long-range correlations have also been revealed in protein sequences, and this
has motivated the search for physical links between the observed genetic and
dynamic cooperativity. We outline here a simplified theory of protein, which
relates sequence correlations to physical interactions and to the emergence of
mechanical function. Our protein is modeled as a strongly-coupled amino acid
network whose interactions and motions are captured by the mechanical
propagator, the Green function. The propagator describes how the gene
determines the connectivity of the amino acids, and thereby the transmission of
forces. Mutations introduce localized perturbations to the propagator which
scatter the force field. The emergence of function is manifested by a
topological transition when a band of such perturbations divides the protein
into subdomains. We find that epistasis -- the interaction among mutations in
the gene -- is related to the nonlinearity of the Green function, which can be
interpreted as a sum over multiple scattering paths. We apply this mechanical
framework to simulations of protein evolution, and observe long-range epistasis
which facilitates collective functional modes.
