Mutation and optimal search of sequences in nested Hamming spaces

A representation of evolutionary systems is defined by sequences in nested Hamming spaces, which is analogous to variable length coding. Asexual reproduction is considered as a process of finding optimal codes, and conditions are formulated under which the optimal search operator is a simple random mutation, corresponding to binomial or Poisson process. Transition probability between spheres around an optimal sequence in each Hamming space is derived and used for optimal control of mutation rate. Several control functions are discussed, including a minimal information control that has a number of interesting properties. The theory makes a number of predictions about variability of length and mutation of DNA sequences in biological organisms.