P systems are computational models inspired by some biological features of the structure and the functioning of real cells. In this paper we introduce a new kind of communication between membranes, based upon the natural budding of vesicles in a cell. We define the operations of gemmation and fusion of mobile membranes, and we use membrane structures and rules over strings of biological inspiration only. We prove that P systems of this type can generate all recursively enumerable languages and, moreover, the Hamiltonian Path Problem can be solved in a quadratic time. Some open problems are also formulated.