Quenching Single-Fluorophore Systems and the Emergence of Non-linear Stern-Volmer Plots

Reduction in fluorescence's intensity upon the addition of quencher molecules is often quantified by the Stern-Volmer equation. Central to the underlying model is the formation of an adduct between quencher and excited-state (dynamic-quenching), or ground-state (static-quenching), fluorophore at steady-state conditions. Assuming a thermodynamic behavior, that is, large numbers of fluorophore and quencher molecules, the resulting dependency of the ratio between fluorescence intensities, with and without quencher, on quencher's concentration is linear. Yet, alongside abundance reports confirming this linear behavior, numerous observations indicate the dependency can also be non-linear with either upward, or downward, curvature. By maintaining the same physical mechanisms for quenching, in this paper we derive an alternative equation to describe fluorescence quenching. Here however, we assume a local equilibrium (steady-state) between a single-fluorophore and surrounding quencher molecules, effectively, partitioning the (macroscopic) system into many non-interacting small subsystems. Depending on fluorophore's properties, association's strength, and conditions, the resulting behavior exhibits linear dependencies, upward curvatures, or downward curvatures. More specifically, the relation reads, $ I_\circ/I = 1 + ZK[Q]_T/(1 + (1-Z)K[Q]_T) $, where $K$ is a steady-state equilibrium constant for complex formation and $[Q]_T$ is total concentration of quencher in the small subsystem. The dimensionless parameter $Z$ has different interpretations for dynamic and static mechanisms, though, it is related to the fraction of excited fluorophore. Partitioning the system into small subsystems naturally represents fluorophore-quencher interactions physically confined to restricted volumes. Nonetheless, this model accounts successfully also for deviations from linearity observed in homogeneous solutions. This is plausibly due to limited diffusion, triggering repetitive binding of the fluorophore with the same set of surrounding quenchers. We tested the validity of this equation (except for downward curvatures induced by a static mechanism, in which case a related equation was applied) on 151 experimental fluorescence quenching plots, taken from the literature, operated by dynamic, static, and combined mechanisms. The results of the fitting are excellent with an average correlation coefficient of 0.9985.