The ability of biological molecules to replicate themselves, achieved with the aid of a complex cellular machinery, is the foundation of life. However, a range of aberrant processes involve the self-replication of pathological protein structures without any additional factors. A dramatic example is the autocatalytic replication of pathological protein aggregates, including amyloid fibrils and prions, involved in neurodegenerative disorders. Here, we use computer simulations to identify the necessary requirements for the self-replication of fibrillar assemblies of proteins. We establish that a key physical determinant for this process is the affinity of proteins for the surfaces of fibrils. We find that self-replication can only take place in a very narrow regime of inter-protein interactions, implying a high level of sensitivity to system parameters and experimental conditions. We then compare our theoretical predictions with kinetic and biosensor measurements of fibrils formed from the Aβ peptide associated with Alzheimer's disease. Our results show a quantitative connection between the kinetics of self-replication and the surface coverage of fibrils by monomeric proteins. These findings reveal the fundamental physical requirements for the formation of supra-molecular structures able to replicate themselves, and shed light on mechanisms in play in the proliferation of protein aggregates in nature.