The signaling lipid phosphatidic acid (PA) is believed to interact specifically with membrane-bound globular proteins through a combination of electrostatic interactions and hydrogen bond formation known as the electrostatic-hydrogen bond switch. PA, which adjusts its protonation state according to the ambient pH, is able to regulate protein binding under physiological conditions in a pH-dependent manner. We investigate the question to what extent the electrostatic-hydrogen bond switch contributes to the pH-sensitivity of protein binding. To this end, we propose a theoretical model for the adsorption of a basic protein on a zwitterionic membrane that contains phosphatidic acid as a minor component. Our model is based on an extended continuum Poisson-Boltzmann approach that accounts for zwitterionic lipids, the protonation/deprotonation equilibrium of PA, and the lateral mobility of the lipids in the membrane. The electrostatic-hydrogen bond switch enters as an additional non-electrostatic attractive interaction of deprotonated PA with basic protein residues. For a generic model protein we calculate the adsorption free energy and its pH-dependence. Our results suggest that the electrostatic-hydrogen bond switch not only increases the affinity between PA and the protein but also its sensitivity with respect to changes in pH. That is, the electrostatic-hydrogen bond switch helps enabling the membrane to use physiological pH changes in order to trigger protein adsorption/desorption.