Quenching Single-Fluorophore Systems and the Emergence of Nonlinear Stern–Volmer Plots

Reduction in fluorescence intensity upon 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, a system with 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 nonlinear with either upward or downward curvature. By maintaining the same physical mechanisms for quenching, we derive in this paper an alternative equation to describe fluorescence quenching. Here, however, we assume a local equilibrium (steady-state) between a single fluorophore and a finite number of surrounding quencher molecules, effectively partitioning the (macroscopic) system into many noninteracting small subsystems. Depending on the fluorophore's properties, the association's strength, and conditions, the resulting behavior exhibits linear dependencies, upward curvatures, or downward curvatures. More specifically, the relation reads, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>I</mml:mi> <mml:mo>°</mml:mo></mml:msub> <mml:mo>/</mml:mo> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>Z</mml:mi> <mml:mi>K</mml:mi> <mml:msub><mml:mrow><mml:mo>[</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>]</mml:mo></mml:mrow> <mml:mi>T</mml:mi></mml:msub> <mml:mo>/</mml:mo> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mi>K</mml:mi> <mml:msub><mml:mrow><mml:mo>[</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>]</mml:mo></mml:mrow> <mml:mi>T</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , where <i>K</i> is a steady-state equilibrium constant for complex formation and [Q] _{_<i>T</i>} is the total concentration of quencher in the small subsystem. The dimensionless parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> has different expressions for dynamic and static mechanisms. In the former, it is a ratio between the maximum rate of quenching and the rate of fluorophore excitation, whereas in the latter, it is a function of the fraction of excited fluorophore. Intriguingly, this relation applies also for systems with exciplex emissions. We tested the validity of this model 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.