The quantification of protein-ligand interactions is essential for systems biology, drug discovery, and bioengineering. Ligand-induced changes in protein thermal stability provide a general, quantifiable signature of binding and may be monitored with dyes such as Sypro Orange (SO), which increase their fluorescence emission intensities upon interaction with the unfolded protein. This method is an experimentally straightforward, economical, and high-throughput approach for observing thermal melts using commonly available real-time polymerase chain reaction instrumentation. However, quantitative analysis requires careful consideration of the dye-mediated reporting mechanism and the underlying thermodynamic model. We determine affinity constants by analysis of ligand-mediated shifts in melting-temperature midpoint values. Ligand affinity is determined in a ligand titration series from shifts in free energies of stability at a common reference temperature. Thermodynamic parameters are obtained by fitting the inverse first derivative of the experimental signal reporting on thermal denaturation with equations that incorporate linear or nonlinear baseline models. We apply these methods to fit protein melts monitored with SO that exhibit prominent nonlinear post-transition baselines. SO can perturb the equilibria on which it is reporting. We analyze cases in which the ligand binds to both the native and denatured state or to the native state only and cases in which protein:ligand stoichiometry needs to treated explicitly.