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The cooperative binding of molecular agents onto a substrate is pervasive inliving systems. To study whether a system shows cooperativity, one can rely ona fluctuation analysis of quantities such as the number of substrate-boundunits and the residence time in an occupancy state. Since the relative standarddeviation from the statistical mean monotonically decreases with the number ofbinding sites, these techniques are only suitable for small enough systems,such as those implicated in stochastic processes inside cells. Here, we presenta general-purpose grand canonical Hamiltonian description of a smallone-dimensional (1D) lattice gas with either nearest-neighbor or long-rangeinteractions as prototypical examples of cooperativity-influenced adsorptionprocesses. First, we elucidate how the strength and sign of the interactionpotential between neighboring bound particles on the lattice determine theintensity of the fluctuations of the mean occupancy. We then employ thisrelationship to compare the theoretical predictions of our model to data fromsingle molecule experiments on bacterial flagellar motors (BFM) of E. coli. Inthis way, we find evidence that cooperativity controls the mechano-sensitivedynamical assembly of the torque-generating units, the so-called stator units,onto the BFM. Finally, we estimate the stator-stator interaction potential andattempt to quantify the adaptability of the BFM.

Signature of (anti)cooperativity in the stochastic fluctuations of small systems: application to the bacterial flagellar motor