The low-energy electronic band structure of bilayer graphene consists of four bands: a pair of bands split from zero energy by the interlayer coupling and a pair of bands which touch at zero energy in a nominally undoped system. The latter support massive, chiral quasiparticles with a parabolic dispersion and Berry phase 2. Asymmetry between the potential energies of the layers opens a tuneable gap between the conduction and valence bands. A self-consistent Hartree approximation is used to model the control of such an interlayer asymmetry gap induced by a transverse electric field in a graphene-based field-effect transistor.