Fluorescent proteins undergoing green to red photoconversion have proved to be essential tools in cell biology, notably in superlocalization nanoscopy. However, the exact mechanism governing photoconversion, which overall involves irreversible cleavage of the protein backbone and elongation of the chromophore pi-conjugation, remains unclear. In this paper we present a theoretical investigation of the photoconversion reaction in the fluorescent protein EosFP, using excited-state hybrid quantum chemical and molecular mechanical potentials, in conjunction with reaction-path-finding techniques. Our results reveal a mechanism in which the hydroxybenzylidene moiety of the chromophore remains protonated and there is an excited state proton transfer from His62 to Phe61 that promotes peptide bond cleavage. Excitation of the neutral green form of EosFP to the first singlet excited state is followed by two intersystem crossing events, first to a triplet state and then back to the ground state singlet surface. From there, a number of rearrangements occur in the ground state and lead to the red form. Analyses of the structures and energies of the intermediates along the reaction path enable us to identify the critical role of the chromophore environment in promoting photoinduced backbone cleavage. Possible ways in which photoconvertible fluorescent proteins can be engineered to facilitate photoconversion are considered.