Protein molecules often self-assemble by means of non-covalent physical bonds to form extended filaments, such as amyloids, F-actin, intermediate filaments, and many others. The kinetics of filament growth is limited by the disassembly rate, at which inter-protein bonds break due to the thermal motion. Existing models often assume that the thermal dissociation of subunits occurs uniformly along the filament, or even preferentially in the middle, while the well-known propensity of F-actin to depolymerize from one end is mediated by biochemical factors. Here, we show for a very general (and generic) model, using Brownian dynamics simulations and theory, that the breakup location along the filament is strongly controlled by the asymmetry of the binding force about the minimum, as well as by the bending stiffness of the filament. We provide the basic connection between the features of the interaction potential between subunits and the breakup topology. With central-force (that is, fully flexible) bonds, the breakup rate is always maximum in the middle of the chain, whereas for semiflexible or stiff filaments this rate is either a minimum in the middle or flat. The emerging framework provides a unifying understanding of biopolymer fragmentation and depolymerization and recovers earlier results in its different limits.