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Universal Bound on the Fano Factor in Enzyme Kinetics

The Fano factor, an observable quantifying fluctuations of product generation by a single enzyme, can reveal information about the underlying reaction scheme. A lower bound on this Fano factor that depends on the thermodynamic affinity driving the transformation from substrate to product constrains the number of intermediate states of an enzymatic cycle. So far, this bound has been proven only for a unicyclic network of states. We show that the bound can be extended to arbitrary multicyclic networks, with the Fano factor constraining the largest value of the effective length, which is the ratio between the number of states and the number of products, among all cycles.