The electrokinetic flow of an electrolyte solution in a rectangular microchannel is investigated theoretically. The present analysis extends previous results in that a two-dimensional problem is considered and the wall of the microchannel is covered by an ion-penetrable charged membrane layer; the latter simulates a biological system. The electroosmotic flow rate and total electric current, the electroviscous effect, and the streaming potential of the system under consideration are evaluated. We show that the variations of these quantities as a function of the aspect ratio of the microchannel may have a local minimum or a local maximum at a medium level of ionic strength, which depends on the thickness of the membrane layer. Some phenomena, which are predicted by the corresponding bare-wall model, may not be observed in the present case. For example, the variation of the electroviscous effect as a function of the electrokinetic radius may not have a local maximum.