Molecular dynamics simulation has been used to study the heteroepitaxy of Cu/Au(001) and Au/Cu(001) with the embedded atom potentials. It is found that Cu/Au(001) has a different heteroepitaxial behavior from Au/Cu(001). The epi-islands of Cu grow on Au (001) in the lattice of Au (001) without any defect. For the epitaxy of Au/Cu(001), however, mismatching dislocations can be observed in the island larger than 7×7. The asymmetricall epitaxial behavior can be attributed to the difference in the strain state and elastic properties between Cu and Au islands. With the increase of heteroepitaxial islands, the micro-misfit of Cu/Au(001) decreases monotonically, but the micro-misfit of Au/Cu(001) has an oscillating increase. The Cu epi-islands enable the substrate to be deformed at the edge of the islands while the substrate in Au/Cu(001) is deformed at the region under the islands.