Domains within fluid membranes grow by the aggregation of molecules which diffuse laterally within the membrane matrix. A simple theoretical model is introduced which predicts that a flat or weakly curved domain becomes unstable at a certain limiting size and then undergoes a budding or invagination process. This instability is driven by the competition between the bending energy of the domain and the line tension of the domain edge. For lipid bilayers, the budding domain can rupture the membrane and then it pinches off from the matrix. The same mechanism should also drive the budding of non-coated domains in biomembranes, and could even be effective when these domains are covered by a coat of clathrin molecules.
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