The Aldous-Shields model revisited with application to cellular ageing

Abstract

In Aldous and Shields (1988), a model for a rooted, growing random binary tree was presented. For some c > 0, an external vertex splits at rate c −i (and becomes internal) if its distance from the root (depth) is i. For c > 1, we reanalyse the tree profile, i.e. the numbers of external vertices in depth i = 1, 2, .... Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we assume that nodes in depth h + 1 are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senescent vertices for large h.

Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

“…Recently, such models have also been considered for important cellular biology questions related to ageing and cancer, where generational dependent cell division rates occur and decrease with generations; see e.g. [6,22] and other related references in the medical and biological literature. Remark 1.5.…”

“…For the percolation model [1], the event of explosion corresponds to the occurrence of a cluster of infinitely many "wet sites"connected to the root in finite time. For the biological model [6], the ageing is represented by non-explosive conditions for the cascade.…”

A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

“…Our model is similar to the Aldous-Shields model [1,6,3] for a rooted, growing random binary tree. While in their model each active vertex becomes inactive and activates its two descendant at a rate which decreases exponentially with the distance of the vertex to the root, in our model the rate decreases linearly.…”