Long-range electron transfer in biomolecules. Tunneling or hopping?

Two competing mechanisms are relevant for long-range electron transfer (ET) in biomolecules: direct electron tunneling between donor (D) and acceptor (A), D → A, and multistep hopping D → X → A, where an electron or an electron hole is transiently localized on intermediate sites X. Which of these mechanisms dominates the ET reaction is determined by the arrangement and electronic properties of the redox centers. For thermal ET, it is shown that single-step tunneling is overcome by hopping when the energy gap E between D and X is smaller than the crossover barrier E(C), E(C) = (ΔG/2) + (3/4)k(B)TβR(DA), where ΔG is the driving force, β the decay parameter, and R(DA) the donor-acceptor distance. In proteins at T = 300 K, hopping will dominate when E < E(C) = (ΔG/2) + (R(DA)/50) (E and ΔG are in eV, R(DA) in Å); single-step tunneling will be operative when E > E(C). Thus, one can explore the ET mechanism using three quantities E, ΔG, and R(DA). When ΔG = 0 and E = 0.5 eV (the difference in redox potentials of D and X is 0.5 V), two-step hopping D → X → A will be favored at R(DA) >25 Å. In protein ET chains, the distance between redox cofactors is often smaller than 20 Å, but the gap E between the cofactors and surrounding amino acid residues is larger than 0.5 eV. Therefore, ET in the systems should occur by single-step tunneling D → A. In the activationless regime (ΔG ≈ -λ, λ is the reorganization energy) often observed for photoinduced ET, the crossing point energy is determined by E(C) = (2λkTβR(DA))(1/2) - λ. The suggested expressions for the threshold barrier may be useful to predict the ET mechanism in natural and artificial redox systems.