Protein refolding/misfolding to an alternative form plays an aetiologic role in many diseases in humans, including Alzheimer's disease, the systemic amyloidoses, and the prion diseases. Here we have discovered that such refolding can occur readily for a simple lattice model of proteins in a propagatable manner without designing for any particular alternative native state. The model uses a simple contact energy function for interactions between residues and does not consider the peculiarities of polypeptide geometry. In this model, under conditions where the normal (N) native state is marginally stable or unstable, two chains refold from the N native state to an alternative multimeric energetic minimum comprising a single refolded conformation that can then propagate itself to other protein chains. The only requirement for efficient propagation is that a two-faced mode of packing must be in the ground state as a dimer (a higher-energy state for this packing leads to less efficient propagation). For random sequences, these ground-state dimeric configurations tend to have more beta-sheet-like extended structure than almost any other sort of dimeric ground-state assembly. This implies that propagating states (such as for prions) are beta-sheet rich because the only likely propagating forms are beta-sheet rich. We examine the details of our simulations to see to what extent the observed properties of prion propagation can be predicted by a simple protein folding model. The formation of the alternative state in the present model shows several distinct features of amyloidogenesis and of prion propagation. For example, an analog of the phenomenon of conformationally distinct strains in prions is observed. We find a parallel between 'glassy' behavior in liquids and the formation of a propagatable state in proteins. This is the first report of simulation of conformational propagation using any heteropolymer model. The results imply that some (but not most) small protein sequences must maintain a sequence signal that resists refolding to propagatable alternative native states and that the ability to form such states is not limited to polypeptides (or reliant on regular hydrogen bonding per se) but can occur for other protein-like heteropolymers.