On the Theory of Stochastic Replication and Evolution of Molecular Sequences
Following the hypothesis of M. Eigen the stochastic replication of molecular sequences is one of the basic processes in molecular biology. This paper is devoted to the formal analysis of the stochastic kinetics of replicating sequences and their evolutionary trees. For the microscopic descriptiuon of systems consisting of sequences of molecules, the metric space of sequences, the occupation number space and probability distributions in the latter are introduced. The complexity of a sequence is described by Kolmogorov's algorithmic entropy which is considered as a measure of the quantity of information contained in the sequence.The time evolution of sequence systems in the occupation number space is described by a dynamic semigroup corresponding to a master equation. The realizations of the stochastic process form an evolution tree which is analyzed by automata-theoretical methods. As an example a certain model of replicating sequences with 4 units is studied. As a criterion for evolutionary processes in sequence systems the increase of the mean Kolmogorov complexity in the system is proposed.