In this paper, the authors use the logistic temporal solution of the generalized Michaelis-Menten kinetics to provide a quantum basis for the tunnelling time and energy evaluations of Brownian enzymic reactions. The mono-substrate and mixed inhibition cases are treated and the associated quantum diagrams of the reaction mechanisms are depicted in terms of intermediate enzyme complexes. The methodology is suited for practically controlling the enzymic activity throughout absorption spectroscopy.
Top1. Introduction
Although in the first century from their discover the enzymes were mainly studied for elucidation of their kinetics (Schnell & Maini, 2003), emphasising on how their structure is changed with the chemical modifications of functional groups (Hirs, 1967), or for experiencing the “forced evolution” (Rigby, Burleigh, & Hartley, 1974), in current years the focus was on controlling them towards biotechnological roles through the knowledge based methods such as the site-directed mutagenesis (Graham et al., 1994; Tyagi et al., 2005) or gene-shuffling techniques (Stemmer, 1994). However, aiming to create a better enzyme, with improved specificity near the “catalytic perfection” (Albery & Knowles, 1976), raises the intrinsic difficulty to rationalize a general model for its activity since the relatively poor level of comprehension about the enzyme machinery (Nixon, Ostermeier, & Benkovic, 1998). As such, deviations from classical behaviour were reported for enzymes yeast alcohol dehydrogenase (Cha, Murray, & Klinman, 1989), bovine serum amine oxidase (Grant & Klinman, 1989), monoamine oxidase (Jonsson, Edmondson, & Klinman, 1994), glucose oxidation (Kohen, Jonsson, & Klinman, 1997), or for enzyme lipoxygenase (Jonsson et al., 1996), in which it was shown that H-transfer is catalysed by quantum tunnelling process. These, and other experimental (Bahnson & Klinman, 1995) and computational (Bala et al., 1996; Hwang & Warshel, 1996; Alper et al., 2001; Astumian et al., 1989; Ross et al., 2003) indications of conformational fluctuations during protein dynamics, suggested the attractive hypothesis that quantum tunnelling and the enzyme catalysis are inter-correlated (Ringe & Petsko, 1999; Sutcliffe & Scrutton, 2000).
The solvent dynamics, i.e. the in vitro and in vivo conditions, and “natural breathing”, i.e. the quantum fluctuations in the active site, of the enzyme molecule need to be counted in a more complete picture of enzymic catalysis. However, the quantum (fluctuating) nature of the enzymic reactions can be visualised by combining the relationship between the catalytic rate (kcat) and temperature (T) (DeVault & Chance, 1966) with that between the reaction rate and the turnover number or the effective time of reaction (Δt) via Heisenberg relation:
(1) were
and
kB stand for the reduced Planck and Boltzmann constants, respectively. Of course, in relation (1) the equivalence between quantum statistics and quantum mechanics was physically assumed when equating the thermal and quantum (tunnelling) energies,
and
respectively (Kleinert, Pelster, & Putz, 2002). Nevertheless, relation (1) is the basis of rethinking upon the static character of the energetic barrier, recalling the so called steady state approximation, usually assumed in describing enzymic catalysis (Laidler, 1955), within the transition state theory (TST) (Glasstone, Laidler, & Eyring, 1941).