Stochastic Coagulation-Fragmentations Models and Maximum Likelihood Phylogenetics in Genomic Sequence Evolution

We present a theoretical study of stochastic coagulation-fragmentation models and maximum likelihood methods in phylogenetics for reconstructing evolutionary trees from genomic sequences. Moving beyond standard demographic approaches, we formalize continuous-time Markov chains on phylogenetic trees, derive master equations for allele frequencies under genetic drift and natural selection, and prove asymptotic unbiasedness theorems for maximum likelihood estimators as genome lengths increase. Our analysis examines General Time Reversible (GTR) substitution matrices, evolutionary tree spaces, population gene pool dynamics, and derives an original analytical bound for the variance of branch length estimators under site-specific heterogeneous selection. Furthermore, we outline computational optimization protocols utilizing Expectation-Maximization and Newton-Raphson algorithms on trees.