Origins of concentration dependence of waiting times for single-molecule fluorescence binding

Binding reactions at finite systems

A perpetual yearn exists among computational scientists to scale down the size of physical systems, a desire shared as well with experimentalists able to track single molecules. A question then arises whether averages observed at small systems are the same as those observed at large or macroscopic systems. Utilizing statistical-mechanics formulations in ensembles in which the total numbers of particles are fixed, we demonstrate that properties of binding reactions are not homogeneous functions. This means that averages of intensive parameters, such as the concentration of the bound-state, at finite systems are different than those at large systems. The discrepancy increases with decreasing temperature, volume, and to some extent, numbers of particles. As perplexing as it may sound, despite variations in average quantities, extracting the equilibrium constant from systems of different sizes does yield the same value. The reason is that correlations in reactants' concentrations ought to be accounted for in the expression of the equilibrium constant, being negligible at large-scale but significant at small-scale. Similar arguments pertain to the calculations of the reaction rate constants, more specifically, the bimolecular rate of the forward reaction is related to the average of the product (and not to the product of the averages) of the reactants' concentrations. Furthermore, we derive relations aiming to predict the composition only from the equilibrium constant and the system's size. All predictions are validated by Monte-Carlo and molecular dynamics simulations. An important consequence of these findings is that the expression of the equilibrium constant at finite systems is not dictated solely by the chemical equation of the reaction but requires knowledge of the elementary processes involved.

Quantitative Framework for Stochastic Nanopore Sensors Using Multiple Channels

Membrane protein channels employed as stochastic sensors offer large signal-to-noise ratios and high specificity in single molecule binding measurements. Stochastic events in a single ion channel system can be measured using current-time traces, which are straightforward to analyze. Signals arising from measurement using multiple ion channels are more complicated to interpret. We show that multiple independent ion channels offer improved detection sensitivity compared to single channel measurements and that increased signal complexity can be accounted for using binding event frequency. More specifically, the leading edge of binding events follows a Poisson point process, which means signals from multiple channels can be superimposed and the association times (between each binding event leading edge), allow for sensitive and quantitative measurements. We expand our calibration to high ligand concentrations and high numbers of ion channels to demonstrate that there is an upper limit of quantification, defined by the time resolution of the measurement. The upper limit is a combination of the instrumental time resolution and the dissociation time of a ligand and protein which limits the number of detectable events. This upper limit also allows us to predict, in general, the measurement requirements needed to observe any process as a Poisson point process. The nanopore-based sensing analysis has wide implications for stochastic sensing platforms that operate using multiple simultaneous superimposable signals.