The self-assembly process of clathrin coated pits during endocytosis has been simulated by combining and extending coarse grained models of the clathrin triskelion, the adaptor protein AP2, and a flexible network membrane. The AP2’s core, upon binding to membrane and cargo, releases a motif that can bind clathrin. In conditions where the core-membrane-cargo binding is weak, the binding of this motif to clathrin can result in a stable complex. We characterize the conditions and mechanisms resulting in the formation of clathrin lattices that curve the membrane, i.e., clathrin coated pits. The mechanical properties of the AP2 β linker appear crucial to the orientation of the curved clathrin lattice relative to the membrane, with wild-type short linkers giving rise to the inward curving buds enabling endocytosis while long linkers produce upside-down cages and outward curving bulges.