Sequence-structure relationships in proteins are highly asymmetric because many sequences fold into relatively few structures. What is the number of sequences that fold into a particular protein structure? Is it possible to switch between stable protein folds by point mutations? To address these questions, we compute a directed graph of sequences and structures of proteins, which is based on 2,060 experimentally determined protein shapes from the Protein Data Bank. The directed graph is highly connected at native energies with "sinks" that attract many sequences from other folds. The sinks are rich in beta-sheets. The number of sequences that transition between folds is significantly smaller than the number of sequences retained by their fold. The sequence flow into a particular protein shape from other proteins correlates with the number of sequences that matches this shape in empirically determined genomes. Properties of strongly connected components of the graph are correlated with protein length and secondary structure.