Deep proton tunneling in the electronically adiabatic and non-adiabatic limits: Comparison of the quantum and classical treatment of donor-acceptor motion in a protein environment

Abstract

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.

Wide-dynamic-range kinetic investigations of deep proton tunnelling in proteins

“…We set the reaction free energy to its measured value (Δ 665cm -1 ), and the O-H mode frequency, , is fixed using its experimental correlation 39 with the tunnel distance. Table 1 shows the results using a quasi-harmonic potential surface 27 . The analysis involves no arbitrary scaling.…”

“…4) and slower time scales 5,31,32 and are often attributed to conformational (classical) distributions of the D-A distance 4,5,7,[33][34][35] . Other approaches 12,27,36 factor out and quantize the D-A vibration (~100 fs), as well as the much higher frequency (~10 fs) vibrational mode of the tunneling particle, which is a common feature in all analytical models (see Supplementary Discussion 4). Computationally based transition state theories have also been applied in the large curvature limit to account for deep tunneling 22 .…”

“…Within the class of "factorized" analytical models 12,22,27,36 In the following, we assume that a single deep proton tunneling reaction is the ratelimiting event for the overall transport process in GFP (i.e., the other two protons follow in asynchronous but concerted fashion). This assumption is based on the longer tunnel distances and larger reorganization energy that must be invoked for simultaneous (synchronous) multiple proton tunneling.…”

“…The proton wire in GFP includes amino acid components that are found in other biological systems 17 and, importantly, this wire has evolved to include a high pK a residue (Ser205) as an active transport element. Thus, the detailed experiments presented here provide insight on the structure and function of such wires as well as an opportunity to test various theoretical models that apply to PT in proteins 8,10,12,[22][23][24][25][26][27] .…”

“…1, we analyze the kinetic data using a simplified (but analytic) deep tunneling model 27 that factors out the key vibrational motions of the tunneling particle and the donor and acceptor oxygen atoms, while treating the rest of the surroundings using a generalized environmental coordinate. Because there is no electronic state change during the reaction, it is considered to be electronically adiabatic 12,22 .…”