Deformation of nanoporous materials induced by gas adsorption is a ubiquitous phenomenon that plays an important role in adsorption separations, gas and energy storage, nanosensors, actuators, secondary gas recovery, and carbon dioxide sequestration in coal and shale reservoirs. One of the most prominent examples is the breathing phase transformation in metal-organic frameworks (MOF) associated with significant volume variations upon adsorption and desorption of guest molecules. Here, we present a theoretical framework for the quantitative description of the breathing transitions upon adsorption of binary mixtures, drawing on the practically important example of the displacement of methane by carbon dioxide in the MIL-53 MOF. The proposed approach, which is based on the concept of adsorption stress, reveals the mechanisms of framework deformation and breathing phase transformation between the large pore (LP) and narrow pore (NP) conformations. We show that when pure CH