  Elastic netwok models coarse grain proteins into a network of residue beads
connected by springs. We add dissipative dynamics to this mechanical system by
applying overdamped Langevin equations of motion to normal-mode vibrations of
the network. In addition, the network is made heterogeneous and softened at the
protein surface by accounting for hydration of the ionized residues. Solvation
changes the network Hessian in two ways. Diagonal solvation terms soften the
spring constants and off-diagonal dipole-dipole terms correlate displacements
of the ionized residues. The model is used to formulate the response functions
of the electrostatic potential and electric field appearing in theories of
redox reactions and spectroscopy. We also formulate the dielectric response of
the protein and find that solvation of the surface ionized residues leads to a
slow relaxation peak in the dielectric loss spectrum, about two orders of
magnitude slower than the main peak of protein relaxation. Finally, the
solvated network is used to formulate the allosteric response of the protein to
ion binding. The global thermodynamics of ion binding is not strongly affected
by the network solvation, but it dramatically enhances conformational changes
in response to placing a charge at the active site of the protein.
