By experimenting with a cellular automaton model of the immune system, we have reproduced affinity maturation of the antibody response, a somatic adaptation to a changing environment. The simulation allowed the isolation of a number of variables, e.g. the fraction of repertoire available, the magnitude of the change in affinity with mutation, the mutation frequency and its focus on the complementarity-determining regions (CDR) of the antibody. Multiple series of immunizations were run in machina where the contribution of each variable was evaluated against the maturation observed. We found that hypermutation is not necessary for affinity maturation if the repertoire of B cell specificities is sufficiently complete, but is essential when the B cell diversity is limited (which happens to be the case in vivo), as it fills the holes in the repertoire and allows selection by antigen. Maturation also depends on the magnitude of the change in affinity with mutation, and we supply some necessary limits on this parameter. For mutations confined to the CDR, the most efficient maturation occurs at mutation rates of 0.2 per paratope and per cell division. When mutations also affect the framework regions, the peak of the most effective CDR mutation rate moves progressively to lower values. A most sensitive parameter is the speed of maturation, which reflects the rate of expansion of mutated clones. Comparing it with biological observations can help to discriminate between alternative hypotheses on the phenomena of hypermutation and affinity.