In this paper we further develop the concept of interatomic tunneling currents [A.A. Stuchebrukhov, J. Chem. Phys. 104, 8424 (1996)] for the description of long-range electron tunneling in proteins. Here we discuss a formulation of the theory for the case when nonorthogonality of the atomic basis set of the medium propagating electron is explicitly taken into account. This method provides an effective computational scheme for an exact, i.e., nonperturbative, evaluation (in one-electron approximation) of the superexchange electron tunneling matrix element, and allows one to determine which regions in the protein matrix are important for the tunneling process. The theory is applied for calculation of tunneling currents and the electronic matrix element in His126-Ru-modified blue copper protein azurin from a recent experimental work of Gray and co-workers. Analysis of interatomic currents reveals a nontrivial structure of the tunneling flow between donor and acceptor in the intervening protein medium in this system.