Four-state quantum chain as a model of sequence evolution

A variety of selection-mutation models for DNA (or RNA) sequences, well kno wn in molecular evolution, can be translated into a model of coupled Ising quantum chains, This correspondence is used to investigate the genetic vari ability and error threshold behaviour in dependence of possible fitness lan dscapes. In contrast to the two-state models treated hitherto, the model ex plicitly takes the four-state nature of the nucleotide alphabet into accoun t and allows for the distinction of mutation rates for the different base s ubstitutions, as given by standard mutation schemes of molecular phylogeny. As a consequence of this refined treatment, new phase diagrams for the err or threshold behaviour ale obtained, with appearance of a novel phase in wh ich the nucleotide ordering of the wildtype sequence is only partially cons erved. Explicit analytic;ll and numerical results are presented For evoluti on dynamics and equilibrium behaviour in a number of accessible situations, such as quadratic fitness landscapes and the Kimura 2 parameter mutation s cheme.