Analytical models describing the temperature dependence of the deep tunneling rate, useful for proton, hydrogen, or hydride transfer in proteins, are developed and compared. Electronically adiabatic and non-adiabatic expressions are presented where the donor-acceptor (D-A) motion is treated either as a quantized vibration or as a classical "gating" distribution. We stress the importance of fitting experimental data on an absolute scale in the electronically adiabatic limit, which normally applies to these reactions, and find that vibrationally enhanced deep tunneling takes place on sub-ns timescales at room temperature for typical H-bonding distances. As noted previously, a small room temperature kinetic isotope effect (KIE) does not eliminate deep tunneling as a major transport channel. The quantum approach focuses on the vibrational sub-space composed of the D-A and hydrogen atom motions, where hydrogen bonding and protein restoring forces quantize the D-A vibration. A Duschinsky rotation is mandated between the normal modes of the reactant and product states and the rotation angle depends on the tunneling particle mass. This tunnel-mass dependent rotation contributes substantially to the KIE and its temperature dependence. The effect of the Duschinsky rotation is solved exactly to find the rate in the electronically non-adiabatic limit and compared to the Born-Oppenheimer (B-O) approximation approach. The B-O approximation is employed to find the rate in the electronically adiabatic limit, where we explore both harmonic and quartic double-well potentials for the hydrogen atom bound states. Both the electronically adiabatic and non-adiabatic rates are found to diverge at high temperature unless the proton coupling includes the often neglected quadratic term in the D-A displacement from equilibrium. A new expression is presented for the electronically adiabatic tunnel rate in the classical limit for D-A motion that should be useful to experimentalists working near room temperature. This expression also holds when a broad protein conformational distribution of D-A equilibrium distances dominates the spread of the D-A vibrational wavefunction.

Abstract:Directional proton transport along "wires" that feed biochemical reactions in proteins is poorly understood. Amino acid residues with high pK a are seldom considered as active transport elements in such wires because of their large classical barrier for proton dissociation. Here we use the light-triggered proton wire of the green fluorescent protein (GFP) to study its ground electronic state proton transport kinetics, revealing a large temperature-dependent kinetic isotope effect. We show that "deep" proton tunneling between hydrogen-bonded oxygen atoms having a typical donor-acceptor distance of 2.7-2.8 Ǻ fully accounts for the rates at all temperatures, including the unexpectedly large value (2.5 10 9 s -1 ) found at room temperature. The rate limiting step in GFP is assigned to tunneling of the ionization-resistant serine hydroxyl proton. This suggests how high pK a residues within a proton wire can act as a "tunnel-diode" to kinetically trap protons and control the direction of proton flow.

A proper description of proton donor-acceptor (D-A) distance fluctuations is crucial for understanding tunneling in proton-coupled electron transport (PCET). The typical harmonic approximation for the D-A potential results in a Gaussian probability distribution, which does not appropriately reflect the electronic repulsion forces that increase the energetic cost of sampling shorter D-A distances. Because these shorter distances are the primary channel for thermally activated tunneling, the analysis of tunneling kinetics depends sensitively on the inherently anharmonic nature of the D-A interaction. Thus, we have used quantum chemical calculations to account for the D-A interaction and developed an improved model for the analysis of experimental tunneling kinetics. Strong internal electric fields are also considered and found to contribute significantly to the compressive forces when the D-A distance distribution is positioned below the van der Waals contact distance. This model is applied to recent experiments on the wild type (WT) and a double mutant (DM) of soybean lipoxygenase-1 (SLO). The compressive force necessary to prepare the tunneling-active distribution in WT SLO is found to fall in the ∼ nN range, which greatly exceeds the measured values of molecular motor and protein unfolding forces. This indicates that ∼60-100 MV/cm electric fields, aligned along the D-A bond axis, must be generated by an enzyme conformational interconversion that facilitates the PCET tunneling reaction. Based on the absolute value of the measured tunneling rate, and using previously calculated values of the electronic matrix element, the population of this tunneling-active conformation is found to lie in the range 10-10, indicating this is a rare structural fluctuation that falls well below the detection threshold of recent ENDOR experiments. Additional analysis of the DM tunneling kinetics leads to a proposal that a disordered (high entropy) conformation could be tunneling-active due to its broad range of sampled D-A distances.

The often-used "linear approximation" for treating the coupling of proton donor-acceptor (D-A) distance fluctuations to proton-coupled electron transfer tunneling reactions is systematically examined. The accuracy of this approximation is found to depend on the potential energy surfaces that are used to describe both the tunneling particle vibrations and the D-A coordinate probability distribution. Harmonic treatment of both the tunneling particle and the D-A coordinates results in a significant breakdown of the linear approximation when the width of the D-A distribution exceeds ∼0.05 Å. When a symmetric back-to-back Morse potential is used to describe the tunneling particle vibrations in the reactant and product states, the D-A distribution width can be expanded up to ∼0.09 Å before the rates calculated using the linear approximation exceed the exact result by an order of magnitude. Incorporation of a more realistic anharmonic D-A potential, based on quantum calculations, includes the important electronic D-A repulsion energy so that the sampling of short D-A distances is reduced. This approach improves the linear approximation as long as the D-A distribution has a width less than ∼0.1 Å. The conditions for the validity of the linear approximation are found to be more fragile when the calculated kinetic isotope effect (KIE) and its temperature dependence are also taken into account.

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