Steady-state reaction networks are here inspected from the viewpoint of individual tagged molecules jumping among their chemical states upon occurrence of the reactive events. Such an agent-based viewpoint is useful for selectively characterizing the behaviour of functional molecules, especially in the presence of bimolecular processes. We present the tools for simulating the jump dynamics, both in the macroscopic limit and in the small-volume sample where the numbers of reactive molecules are of the order of few units with an inherently stochastic kinetics. The focus is on how an ideal spatial 'compartmentalization' may affect the dynamical features of the tagged molecule. The general approach is applied to a synthetic light-driven supramolecular pump constituted by ring-like and axle-like molecules that dynamically assemble and disassemble originating an average ring-through-axle directed motion under constant irradiation. In such an example, the dynamical feature of interest is the completion time of direct/inverse cycles of tagged rings and axles. We find a surprisingly strong robustness of the average cycle times with respect to the system's size. This is explained with the presence of rate-determining unimolecular processes, which may therefore play a crucial role in stabilizing the behavior of small chemical systems against strong fluctuations in the number of molecules.
I. INTRODUCTIONSeveral key functions in biochemical contexts are regulated by networks of chemical reactions taking place in a fluid and thermostated environment under out-ofequilibrium steady state conditions. A privileged viewpoint is that of an individual tagged molecule, or even a tagged molecular fragment (a moiety), that changes its chemical state when involved in a reaction. Following the fate of a tagged molecule offers an insight on subtle features that would be otherwise shadowed if looking at the global evolution of the whole reactive system 1-6 . In particular, one can access specific statistical descriptors like the distribution of first occurrence times of certain reactive events involving the individual molecule, or the distribution of completion times of cyclical processes, and so on. Think, for example, to the statistics of the turnover time for an individual enzyme molecule involved in a catalytic scheme 7,8 , or to the statistical kinetics of processive enzymes 9 and molecular motors 10 . Such information is much more detailed than the mere average rate of the products formation at the steady state. Moreover, by adopting the agent-based viewpoint of a tagged molecule, one focuses precisely on the behaviour of the functional part of the whole machinery.The specific 'function' of a tagged molecule can be characterized by a dynamical output like, for instance, the cycle time of an enzyme molecule, or the quantitative descriptor of a much more articulated event (the concept will be further elaborated later). By viewing each chemical state as a 'site', the tagged molecule follows its path among the available sites. Because of...