Enzyme Kinetics and Proton Transport

As an example, consider an early calculation of isotope effects on enzyme kinetics by Hwang and Warshel [31]. This study examines isotope effects on the catalytic reaction of carbonic anhydrase. The expected rate-limiting step is a proton transfer reaction from a zinc-bound water molecule to a neighboring water. The TST expression for the rate constant k is 

Warshel is to utilize a formula identical to (11.22) in this chapter to compute the free energy change. They employed an empirical valence bond (EVB, below) approach to approximately model electronic effects, and the calculations included the full experimental structure of carbonic anhydrase. An H/D isotope effect of 3.9 1.0 was obtained in the calculation, which compared favorably with the experimental value of 3.8. This benchmark calculation gives optimism that quantum effects on free energies can be realistically modeled for complex biochemical systems. 

The basic idea of the QM/MM methods [32, 33] is to partition the system into an inner quantum zone, in which the interesting chemistry happens, and an outer classical force field region. While this division is physically sensible, it can be quite tricky to handle the boundary between the two domains. If there is no covalent bond linking the QM and MM regions, the partitioning is simpler. If the boundary cuts through chemical bonds, however, the partitioning is more difficult. Several approaches have 

Classical force fields [69] have been used to model proton transport, but their accuracy has been questioned [68], 

The QM/MM and ab initio methodologies have just begun to be applied to challenging problems involving ion channels [73] and proton motion through them [74]. Reference [73] utilizes Hartree-Fock and DFT calculations on the KcsA channel to illustrate that classical force fields can fail to include polarization effects properly due to the interaction of ions with the protein, and protein residues with each other. Reference [74] employs a QM/MM technique developed in conjunction with Car-Parrinello ab initio simulations [75] to model proton and hydroxide ion motion in aquaporins. Due to the large system size, the time scale for these simulations was relatively short (lOps), but the influences of key residues and macrodipoles on the short time motions of the ions could be examined. 

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