Floquet Driving of Enzymatic Reactions: Counting Statistics and Long-Time Currents
Technologies for artificially controlling chemical reaction systems, such as optogenetics, are rapidly advancing, making it increasingly important to understand reaction dynamics under time-dependent control. When the modulation of reaction rates is periodic in time, the Floquet formalism provides a systematic framework. We develop a Floquet theory for classical stochastic processes that enables the calculation of the current and its counting statistics under such periodic modulation. In particular, we formulate the theory in terms of a counting field and derive general expressions for the first cumulant and the corresponding current. The current is expressed using the effective Floquet generator and the kicked state, and we further obtain general asymptotic expressions for the current in both the high- and low-frequency regimes. As a concrete example to test our analytical expressions, we then apply the results to discrete Floquet driving -- a non-perturbative, stepwise protocol. The setup is motivated by a biochemical system known as cyclic adenosine monophosphate (cAMP) production, which is an enzymatic reaction activated and inhibited by G-proteins. This is formulated as a discretely driven Michaelis--Menten-type reaction model, in which the catalytic activity is switched on and off abruptly in time, and we obtain analytical expressions and numerical results showing how periodic switching of reaction rates generates a long-time product current. In particular, in the high-frequency limit, we show that the effect of the periodic driving can be interpreted through an effective modification of the chemical reaction rates. These results provide a basis for Floquet analysis of periodically driven chemical reactions.