  The cooperative binding of molecular agents onto a substrate is pervasive in
living systems. To study whether a system shows cooperativity, one can rely on
a fluctuation analysis of quantities such as the number of substrate-bound
units and the residence time in an occupancy state. Since the relative standard
deviation from the statistical mean monotonically decreases with the number of
binding sites, these techniques are only suitable for small enough systems,
such as those implicated in stochastic processes inside cells. Here, we present
a general-purpose grand canonical Hamiltonian description of a small
one-dimensional (1D) lattice gas with either nearest-neighbor or long-range
interactions as prototypical examples of cooperativity-influenced adsorption
processes. First, we elucidate how the strength and sign of the interaction
potential between neighboring bound particles on the lattice determine the
intensity of the fluctuations of the mean occupancy. We then employ this
relationship to compare the theoretical predictions of our model to data from
single molecule experiments on bacterial flagellar motors (BFM) of E. coli. In
this way, we find evidence that cooperativity controls the mechano-sensitive
dynamical assembly of the torque-generating units, the so-called stator units,
onto the BFM. Furthermore, in an attempt to quantify fluctuations and the
adaptability of the BFM, we estimate the stator-stator interaction potential.
Finally, we conclude that the system resides in a sweet spot of the parameter
space (phase diagram) suitable for a smoothly adaptive system while minimizing
fluctuations.
