Ab initio calculations of tunneling current along a model 25-Angstrom-long polypeptide chain, and similar extended hydrocarbon molecular wires, show that in the tunneling flow there exist "topological defects,'' viz. the lines on which the wave function of the tunneling electron is zero, and the phase of the wave function is not defined. As a result, the flow of quantum mechanical probability experiences circular motions (vortices) around such nodal lines, and the overall structure of the flow becomes remarkably complex. The vortices in the tunneling flow are quantized, and are similar in their mathematical nature to those in a quantum superfluid liquid.