Abstract
In this work, we study the phenomenon known as cooperativity. Cooperativity is a widespread phenomenon that emerges in a variety of biological systems, ranging from the molecular scale to multicellular organization, extending to the emergence of complex behaviors in many bodily systems. Some examples of molecular-scale cooperativity include allosteric binding of oxygen to hemoglobin (Monod et al. 1965; Koshland et al. 1966), protein-protein interactions (Dill et al. 1993), membrane transport (Magleby 2003), interactions between antibodies and antigens (Sela-Culang et al. 2013), etc.In particular, we focus on processes of cooperativity involving the adsorption of ligands onto a substrate with a limited number of binding sites. There are several examples of such processes in living systems: antibody-antigen interactions (Sela-Culang et al. 2013), lectin-carbohydrate interactions (Sharon et al. 2004), virus-cell interactions (Chazal et al. 2003), bacterial adhesion, and other cell interactions (Bell 1978), etc.We have two main objectives:• To develop a general method to assess how short- and long-range interactions between ligands bound to the substrate affect the stochastic fluctuations of the number of adsorbed units.• To study the out-of-equilibrium characteristics resulting from the presence of such interactions.To achieve this, we introduce a minimal one-dimensional short-range lattice gas model (SRLG) with interactive units coupled to a thermal bath, described using a versatile grand canonical Hamiltonian. Using this model, we investigate how equilibrium fluctuations are influenced by model parameters, including the interaction potential and the chemical potential of the bulk reservoir. Furthermore, we solve the problem by determining a crucial relationship between the experimentally accessible average occupancy of the system and the corresponding standard deviation, as well as the probability distribution function. We then test the model by comparing our theoretical predictions from the fluctuation analysis with experimental data from the bacterial flagellar motor (BFM), a rotating macromolecular nanomachine. Through this analysis of fluctuations, we find evidence that cooperativity controls the mechano-sensitive dynamic assembly of torque-generating units, known as stators, on the BFM. We estimate the stator-stator interaction potential and use it as an attempt to quantify the "adaptability" of this machine. Once we observe that a certain degree of cooperativity is expected in the BFM, we continue to use the SRLG model to study the out-of-equilibrium characteristics of such a system applied to the BFM. To do so, we employ two methods: 1. Monte Carlo simulations (using multiple algorithms); 2. A Markov chain analysis of the model. We apply our results to the data from the same BFM experiments, in which we study the characteristic relaxation times of stator binding and unbinding in the BFM, as well as the residence times of stators within the motor.