Electrostatic interaction of ampholytic nanocolloidal particles (NPs), which mimic globular proteins, with polyelectrolyte brushes is analyzed within mean-field Poisson-Boltzmann approximation. In accordance with experimental findings, the theory predicts that an electrostatic driving force for the particle uptake by the brush may emerge when the net charge of the particle in the buffer and the charge of the brush are of the same sign. The origin of this driving force is change in the ionization state of weak cationic and anionic groups on the NP surface provoked by interaction with the brush. In experimental systems, the ionic interactions are complemented by excluded-volume, hydrophobic, and other types of interactions that all together control NP uptake by or expulsion from the brush. Here, we focus on the NP-brush ionic interactions. It is demonstrated that deviation between the buffer pH and the NP isoelectric point, considered usually as the key control parameter, does not uniquely determine the insertion free energy patterns. The latter depends also on the proportion of cationic and anionic groups in the NPs and their specific ionization constants as well as on salt concentration in the buffer. The analysis of the free energy landscape proves that a local minimum in the free energy inside the brush appears, provided the NP charge reversal occurs upon insertion into the brush. This minimum corresponds either to a thermodynamically stable or to a metastable state, depending on the pH offset from the IEP and salt concentration, and is separated from the bulk of the solution by a free energy barrier. The latter, being fairly independent of salt concentration in height, may strongly impede the NP absorption kinetically even when it is thermodynamically favorable. Hence, change reversal is a necessary but insufficient condition for the uptake of the NPs by similarly charged polyelectrolyte brushes.