The method of tunneling: currents developed earlier by one of the authors is applied to study electron tunneling dynamics in a model organometallic donor-bridge-acceptor system, in which the donor is a blue copper center in the reduced form, the bridge is a polypeptide (5 glycine residues), and the acceptor is the -HisRu(III)bpy(2)Im complex. This system mimics pretty closely the major amino acid tunneling path in Ru-modified azurin studied by Gray and co-workers recently. It is demonstrated that the tunneling matrix element as small as 10(-4) cm(-1) can be reliably evaluated using ab initio tunneling currents method. The method consists of an ab initio electronic structure calculation of the spatial distribution of quantum mechanical flux in the tunneling transition occurring in the system, when an electron/hole tunnels from the donor site in the molecule to the acceptor site. The analysis is based on the calculation of two diabatic nonorthogonal electronic states corresponding to localization of a tunneling electron on donor and acceptor sites, respectively, and subsequent evaluation of the matrix element of current density operator between these two states. All electrons in the system are taken into account at the Hartree-Fock level, and therefore the method allows one to examine the reaction of the background electrons to the tunneling charge in a self-consistent way. Results for this system confirm earlier reported finding that in the tunneling flow there exist quantized vortices, similar to those of a superfluid liquid, such as liquid He.