Average Electron Tunneling Route of the Electron Transfer in Protein Media

We present a new theoretical method to determine and visualize the average tunneling route of the electron transfer (ET) in protein media. In this, we properly took into account the fluctuation of the tunneling currents and the quantum-interference effect. The route was correlated with the electronic factor ⟨T D A 2⟩ in the case of ET by the elastic tunneling mechanism. We expanded ⟨T D A 2⟩ by the interatomic tunneling currents J a b ’s. Incorporating the quantum-interference effect into the mean-square interatomic tunneling currents, denoted as ⟨J̃ ab 2⟩, we could express ⟨T D A 2⟩ as a sum of ℏ2⟨J̃ ab 2⟩. Drawing the distribution of ⟨J̃ ab 2⟩ on the protein structure, we obtain the ⟨J̃ ab 2⟩ map which visually represents which parts of bonds and spaces most significantly contribute to ⟨T D A 2⟩. We applied this method to the ET from the bacteriopheophytin anion to the primary quinone in the bacterial photosynthetic reaction center of Rhodobacter sphaeroides. We obtained J a b ’s by a combined method of molecular dynamics simulations and quantum chemical calculations. In calculating ⟨J̃ ab 2⟩, we found that much destructive interference works among the interatomic tunneling currents even after taking the average. We drew the ⟨J̃ ab 2⟩ map by a pipe model where atoms a and b are connected by a pipe with width proportional to the magnitude of ⟨J̃ ab 2⟩. We found that two groups of ⟨J̃ ab 2⟩’s, which are mutually coupled with high correlation in each group, have broad pipes and form the average tunneling routes, called Trp route and Met route. Each of the two average tunneling routes is composed of a few major pathways in the Pathways model which are fused at considerable part to each other. We also analyzed the average tunneling route for the ET by the inelastic tunneling mechanism.