We describe how membrane proteins diffuse laterally in the membrane plane together with the lipids surrounding them. We find a number of intriguing phenomena. The lateral displacements of the protein and the lipids are strongly correlated, as the protein and the neighboring lipids form a dynamical protein-lipid complex, consisting of similar to 50-100 lipids. The diffusion of the lipids in the complex is much slower compared to the rest of the lipids. We also find a strong directional correlation between the movements of the protein and the lipids in its vicinity. The results imply that in crowded membrane environments there are no "free" lipids, as they are all influenced by the protein structure and dynamics. Our results indicate that, in studies of cell membranes, protein and lipid dynamics have to be considered together.