Random Generalized Cookie-Cutter Set And Applying Kullback-Leibler Divergence Distance To Construct Vertebrate Phylogeny

First, a random generalized Cookie-Cutter set K_T on [0, T] is constructed. Let K_n be the n-th level set of K_T in the process of construction. Then a memory function on K_n is defined. Based on this memory function, a random probability measure φ_n (τ) on K_n is defined. Let Ψ_n(·) = Eφ_n(·), where Ψ_n is the intensity measure of the random measure φ_n. At last it is proved that the sequence {Ψ_n} is weakly convergent and Ψ = limΨ_n.About the study of bioinformatics problem, we propose an approach to analyze the relationship of 64 vertebrates using complete mitochondrial genomes without sequence alignment. The method is based on FFT and Kullback-Leibler divergence (KLD) distance using compositional vectors of protein sequences from the complete genome. The phylogenetic tree shows that the mitochondrial genomes are separated into two major groups. One group corresponds to mammals and the other one involves two subgroups which correspond to fish and Archosauria (including birds and reptiles). The structures of the trees in topology are largely in agreement with the current known phylogenies of vertebrates.