Kinetic modeling of 2e^-/1H⁺ and 2e^-/2H⁺ bidirectional catalytic cycles

When a redox enzyme or synthetic catalyst is interfaced with an electrode, the electrochemical response depends on the details of the catalytic cycle. Here we focus on the steady-state catalytic waveshape of enzymes such as formate dehydrogenase (2e^-/1H⁺), hydrogenases (2e^-/2H⁺) and other bidirectional molecular catalysts that can be adsorbed on, and undergo direct electron transfer with an electrode. We seek to examine the relations between the dependence on pH of the waveshape, the sequence of events in the catalytic cycle, and the properties of the catalytic intermediates (their reduction potentials and pK_a's). Discussing the interpretation of the dependence on pH of the limiting currents and catalytic potentials in various simple situations leads us to introduce the concept of "catalytic pK_a". The reasoning is general and could be used in relation to any bidirectional two-electron catalytic cycle. Understanding what defines and tunes the catalytic potentials will be crucial for the design of reversible catalysts, which operate at a fast rate in either direction in response to even a small overpotential.