NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

In this paper we study the evolution of sequential dynamical systems $(\mathsf{SDS})$ as a result of the erroneous replication of the SDS words. An $\mathsf{SDS}$ consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The $\mathsf{SDS}$ over the word w and Y is the composed map: $[\mathfrak{F}_Y,w]=\prod_{i=1}^{k} F_{w_i}$. The word w represents the genotype of the $\mathsf{SDS}$ in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the $\mathsf{SDS}$. We introduce combinatorial properties of $\mathsf{SDS}$ which allow us to construct a new distance measure $\mathsf{D}$ for words. We show that $\mathsf{D}$ captures the similarity of corresponding $\mathsf{SDS}$. We will use the distance measure $\mathsf{D}$ to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.