## 3 Results and Discussions

Table 3: Determination of IC\({}_{50}\) values of FPIA calibration curves and cross-reactivity of sulfonamides at various concentrations of antibodies and labeled hapten. A—the concentration of SA-(fluorescein derivative) conjugate = 0.16 nM, the concentration of specific antibodies = 0.2 nM. B—the concentration of SA-(fluorescein derivative) conjugate = 0.8 nM, the concentration of specific antibodies = 1 nM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sulfanilamide** & **IC\({}_{50}\)**, \(\mu\)g/mL** & **CR\({}_{\%}\)** & **IC\({}_{50}\)**, \(\mu\)g/mL** & **CR\({}_{\%}\)** \\  & **A** & **A** & **B** & **B** \\ \hline Sulfachlorpyridazine & 2.6 & 100 & 23 & 100 \\ Sulfadiazine & 16.5 & 16 & 75 & 31 \\ Sulfamethazine & 17 & 15 & 50 & 46 \\ Sulfadimethoxine & 121 & 2 & 120 & 19 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sulfonamides used in the work and their IC\({}_{50}\) and CR values relative to sulfachlorpyridazine for FPIA and ELISA techniques. ### Experimental Validation of the Model  
For non-equilibrium conditions, the cross-reactivity can also depend on the duration of the immune interaction. For the homogeneous analysis (FPIA), an equilibrium was reached faster, but the time of reagents' incubation in the heterogeneous ELISA was almost an order of magnitude longer (1 h) than in FPIA (10 min). At a kinetic association constant of 10\({}^{6}\) 1/(M \(\ast\) s) and reagent concentrations of 1 nM, the reaction approached an equilibrium by 95% in 1 h and the analysis could be considered as an equilibrium one, but at lower association constants, the analysis under a common ELISA duration (1 h) will be a non-equilibrium one. To study the effect of kinetic parameters on the specificity of the analysis, a model based on the approximation of equilibrium conditions is not suitable. Therefore, we used a numerical approach to solve the kinetic equations of the reactions, implementing the CO-PASI software for this purpose. The parameters taken for calculations are described in the Supplementary Materials section. The obtained theoretical calibration curves (Figure S1) demonstrate that as an equilibrium is approached, the differences in the detection sensitivity of analytes with different binding constants increase. As can be seen from Table 4, at a reaction time of 600 s, the IC\({}_{50}\) values for analytes with kinetic association constants of 10\({}^{6}\) and 10\({}^{7}\) M\({}^{-1}\)s\({}^{-1}\) differ by a factor of 1.5, while at a reaction time of 3600 s, the IC\({}_{50}\) values for the same analytes differ by a factor of 6.4. Thus, for non-equilibrium assays, the cross-reactivity also depends on the assay duration. To obtain broad specificity, the competitive stage of the assay should be shortened. To verify the above given theoretical assumptions, an enzyme-linked immunosorbent assay was implemented for three antibiotics from the class of fluoroquinolones: clinafloxacin, moxifloxacin, and enrofloxacin. The determined parameters of the assay specificity at different durations of immune interaction are summarized in Table 5. As expected, the differences in IC\({}_{50}\) values and, accordingly, in cross-reactivities were several times greater for the assay with an hour-long incubation of the analyte with antibodies than for the assay with incubation for 7 min. The CR values for clinafloxacin and enrofloxacin differed by 4.6 times with 7-minute incubation and by 50 times with an hour of incubation (the ratio was increased by 11 times). \begin{table}
\begin{tabular}{c c c} \hline  & \multicolumn{2}{c}{**IC\({}_{50}\), nM**} \\ \cline{2-3}
**t, s** & **k\({}_{\text{a}}\) = 10\({}^{6}\) M\({}^{-1}\)s\({}^{-1}\)** & **k\({}_{\text{a}}\) = 10\({}^{7}\) M\({}^{-1}\)s\({}^{-1}\)** \\ \hline
600 & 4.46 & 2.94 \\
3600 & 2.61 & 0.41 \\ \hline \end{tabular}
\end{table} 
Table 4: IC\({}_{50}\) values of theoretical calibration curves of non-equilibrium competitive assay for varied durations of immune interaction (t) and varied kinetic association constants (ka).