DNA Reconstruction of K80 Evolution Mode
Wenjian Liu, Mathematics and Computer Science, City University of New York, USA
BACKGROUND: Determining the reconstruction threshold of broadcast models on d-ary regular tree, as the interdisciplinary subject, has attracted more and more attention from probabilists, statistical physicists, biologists, etc. OBJECTIVE: Consider a (2q)-state symmetric transition matrix as the noisy communication channel on each edge of a regular d-ary tree. Suppose there are two categories, one of which contains exactly q states, and 3 transition probabilities: remain in the same state, mutate to other state but remain in the same category, and mutate to the other category. Consider all the symbols received at the vertices of the n-th generation. Does this leaves-configuration contain a non-vanishing information on the letter transmitted by the root, as n goes to infinity? By means of a refined analysis of moment recursion on a weighted version of the magnetization, concentration investigation, and large degree asymptotics, we construct a nonlinear second-order dynamical system and show that the Kesten–Stigum reconstruction bound is not tight when q>=4. On the other side, when q=2, that is, Kimura Model of DNA evolution, the interactions of nodes on tree become weaker as d increases. This allows us to utilize the Gaussian approximation. Therefore, we explore stability of the fixed point of Gaussian approximation function in order to verify the tightness of Kesten-Stigum reconstruction bound. APPLICATION: The reconstruction problem is concerned essentially with a tradeoff between noise and duplication in a tree communication network; phylogenetic reconstruction is a major task of systematic biology; reconstruction thresholds on trees are believed to determine the dynamic phase transitions in many constraint satisfaction problems including random K-SAT and random colorings on random graphs; the reconstruction threshold is also believed to play an important role in the efficiency of the Glauber dynamics on trees and random graphs.