The mechanism of protein-only prion replication is controversial. A detailed mathematical model of prion replication by nucleated polymerisation is developed, and its parameters are estimated from published data. PrP-res decay is around two orders of magnitude slower than PrP-sen decay, a plausible ratio of two parameters estimated from very different experiments. By varying the polymer breakage rate, we reveal that systems of short polymers grow the fastest. Drugs which break polymers could therefore accelerate disease progression. Growth in PrP-res seems slower than growth in infectious titre. This can be explained either by a novel hypothesis concerning inoculum clearance from a newly infected brain, or by the faster growth of compartments containing smaller polymers. The existence of compartments can also explain why prion growth sometimes reaches a plateau. Published kinetic data are all compatible with our mathematical model, so the nucleated polymerisation hypothesis cannot be ruled out on dynamic grounds.