The kinetic master-equation approach provides a powerful tool to study molecular machines operating far from equilibrium. By carefully designing models and assigning appropriate rates, one can gain relevant insight into the molecular mechanisms underlying their function. In particular, we developed kinetic models to elucidate the mechanism of proton pumping by CcO driven by redox chemistry. Experimental thermodynamic and kinetic data as well as mutagenesis results can easily be incorporated into the models. Similarly, results of electrostatics and quantum-chemical calculations, and of MD simulations can be used in the design and parameterization of the model [18
]. Studies of the simplest three-site models revealed basic principles of the redox-coupled proton pump [41
]. These master-equation models showed in particular that kinetic gating is critical to achieving high pumping efficiency. The required rate enhancement of the internal pT from Glu 242 to the pump site, mediated by the electron in heme a
, is consistent with the concept of the water-gate model [21
The thermodynamic efficiency of most three-site models lies between 20 % and 60 % well below the efficiency of CcO operating at physiological condition. As seen in the analysis of a four-site model with an additional electron site [41
], increasing the complexity of the kinetic models improved the efficiency. This result suggests that the complexity of the electron and proton pathways in CcO is crucial in its operation at such a high efficiency. We showed here that the inclusion of a Glu-valve, in which Glu 242 switches between a high-population down conformer and a low-population up conformer, indeed resulted in an increased thermodynamic efficiency of almost 80%. Thus the analysis of kinetic models with greater complexity by including additional redox centers as well as proton sites may well be justified. Although the development of such complex kinetic models requires additional efforts, not least on the experimental side, they should prove useful in identifying the general principles of CcO function and in creating blueprints for artificial redox-driven molecular machines.
Our study addresses a question central to proton pumping: how can gating of the proton flow be achieved when all microscopic processes are fully reversible. The modulation of the barriers governing the rates of microscopic transitions is often presumed to be essential for gating [78
]. However, in ref. 41
we used a simple 3-site model to show that proton pumping is possible even without any barrier modulation, as long as reactant and product concentrations are out of equilibrium. We could show that already a modest-sized network model with two proton sites and one electron site, resulting in eight microscopic states, provides sufficient complexity that an exergonic reaction (the reduction of oxygen to water) can be coupled to an endergonic reaction (the pumping of protons against an electrochemical potential). In this simplest 3-site model, gating was achieved entirely by modulating the proton and electron affinities of the microscopic states, and without state-dependent modulation of the kinetic prefactors. The analysis of kinetic master-equation models thus shows that barrier modulation is not essential for gating, as long as the microscopic system has sufficient chemical complexity.
Nevertheless, the kinetic gating associated with barrier modulation can greatly increase the efficiency of pumping [41
]. Ref. 41
showed that the pumping efficiency of a master-equation model can be improved either by increasing the number of electron and proton sites, and thus its chemical complexity, or by modulating the kinetic prefactors κij
of key microscopic transitions. Ref. 42
subsequently showed that kinetic gating of the proton transfers from Glu 242 to the pump site and into the BNC for chemistry is key to the high pumping efficiency of CcO, consistent with the water-gated mechanism of pumping [21
]. Here we showed that the Glu valve of ref. 44
further improves the efficiency by preventing proton leakage. In our model of the Glu valve, detailed balance is satisfied, i.e.
, the up-down shuttling and the pT reactions are microscopically reversible and satisfy Eq. (2)
. This analysis thus addresses a concern raised in Ref. 79
that “an isomerization of a side chain by itself cannot establish a gate (because of microscopic reversibility).” In an open system of sufficient complexity (of the underlying network of chemical reaction steps), a device such as the Glu valve can serve effectively as a gate that increases the overall efficiency. So, while proton pumping can be achieved even in a simple model without barrier modulation [41
], increased complexity, kinetic gating, and the Glu valve all help improve the efficiency of the pump.
Nevertheless, one must keep in mind that the master equation approach employed here does, by itself, not address the question how a particular element is realized atomistically. Based on their recent MD simulations of CcO [80
], Yang and Cui questioned both the water-gate and Glu valve mechanisms as gating elements. They argue that with their refined dielectric model the orientational preferences of both the active site water and of the conserved glutamic acid are substantially altered. Clearly, this is a serious concern that deserves further investigation by simulation and, ultimately, by experiment. Interestingly, though, their finding that the Glu-242 side chain (bovine numbering) is strongly biased to pointing up into the BNC (counter to what is seen in the crystal structures) could also result in a gate functioning similar to the one studied in , by breaking the connectivity between the active site region and the D channel in a way that depends on the charge state of the active site region.
So while the barrier modulation involved in kinetic gating improves the pumping efficiency dramatically within our simplified kinetic models, uncovering the underlying molecular mechanisms requires further experimental investigations. Furthermore, it is still an open question whether such barrier-modulating mechanisms are necessary to achieve the high pumping efficiency of actual oxidase. Earlier studies using a more complex four-site model showed substantially improved pumping efficiency [41
]. Preliminary studies using a five-site model that includes a more detailed rendering of the BNC () indicate that even higher efficiencies are possible in principle. Overall, by expanding the number of states in more complex kinetic networks, “gates” become regular parts of the system without the need of explicit barrier modulation.
The kinetic master-equation approach employed here can readily be applied also to other molecular machines, e.g., motor proteins. Similar approaches were already taken in studying the motion of motor proteins [59
], but our approach differs from these with respect to the explicit inclusion of the chemical reaction step that effectively breaks detailed balance in an open system with given reactant concentrations. Following a similar route, models designed to couple ATP hydrolysis and the unidirectional motion of motor proteins may reveal important substeps as well as the mechanisms of motor-protein motion.