Kinetic effect of clustering: Application to systems with switching between states and influence of a drift

Assembling reacting molecules ("targets") into clusters slows down the kinetics of their reactions with mobile particles from a solution due to competition between the targets. Recently, we used a simple probabilistic approach to predict the dependence of this slowing down on the structures of clusters and on the energy barriers for reactions. Within this approach, the key parameters that define the kinetic effect of clustering are "double-visiting" probabilities: the probabilities that a random-walking "searching" particle that visited one target would visit some other target before diffusion to the bulk solution. The kinetic effect of clustering is defined by the entire set of pairwise double-visiting probabilities for all targets within the cluster that depends on the cluster geometry. The greater the double-visiting probabilities, the stronger the kinetic effect of clustering. Now we generalized our approach to systems that oscillate between active and nonactive states, such as reactants switching between steric conformations, or nanopores (channels) switching between the open and closed states. We also considered the impact of a drift of searching particles on the kinetic effect of clustering. We show that in the presence of a drift, starting from some critical distance equivalent to the ratio of the effective diffusion coefficient and the characteristic velocity of the drift, the double-visiting probabilities decay as the inverse square of the distance between the targets (or, in some cases, as the inverse of the distance to the power of 3/2), in contrast to a slower decay as the inverse of the distance in the absence of a drift. The faster decay of the double-visiting probabilities with the distance in the presence of a drift makes the kinetic effect of clustering smaller, i.e., the drift can partially suppress the kinetic effect of clustering. Our results could be applicable to natural biological systems (e.g., receptors and channels on the cellular membranes), and to technological procedures (e.g., reactions with the targets tethered together or attached to solid supports).