A model of kinks appearance in the lipid bilayer membrane of erythrocytes, which are responsible for gas molecule transport, in particular, oxygen, is proposed. It was shown that the kinks arise due to the simultaneous action of transverse and tensile longitudinal mechanical stresses compressing the membrane. This model explains the membrane's permeability sharp increase for gases during an erythrocyte passage through the microcapillary network with the compressive transverse mechanical stresses sharply increasing in its membrane. It was found that the equation of kinks motion has a soliton solution, so that a kink-soliton is formed in the bilayer of the erythrocyte membrane. The developed model is consistent with the previously experimentally established fact that the native erythrocyte membranes in the bloodstream undergo a structural transition, when small changes in blood pH, hormone concentration, and temperature dramatically change the conformation of the biomembranes and its functions by changing the mechanical stress field in the biomembrane.