Control over interactions with biomolecules holds the key to applications of graphene in biotechnology. One such application is nanopore sequencing, where a DNA molecule is electrophoretically driven through a graphene nanopore. Here, we investigate how interactions of single-stranded DNA and a graphene membrane can be controlled by electrically biasing the membrane. The results of our molecular dynamics simulations suggest that electric charge on graphene can force a DNA homopolymer to adopt a range of strikingly different conformations. The conformational response is sensitive to even very subtle nucleotide modifications, such as DNA methylation. The speed of DNA motion through a graphene nanopore is strongly affected by the graphene charge: a positive charge accelerates the motion whereas a negative charge arrests it. As a possible application of the effect, we demonstrate stop-and-go transport of DNA controlled by the charge of graphene. Such on-demand transport of DNA is essential for realizing nanopore sequencing.