The Aldous-Shields model revisited (with application to cellular ageing)

TL;DR

This paper reanalyzes the Aldous-Shields model for binary trees, providing explicit formulas for the profile's expectation and covariance, and applies it to model cellular aging by examining senescence in the tree structure.

Contribution

It offers new formulas for the tree profile's expectation and covariance, and applies the model to cellular aging, analyzing senescence effects in large-depth nodes.

Findings

Explicit formulas for expectation and covariance of the tree profile.

Limit results for the proportion of non-senescent vertices at large depths.

Application of the model to biological cellular aging processes.

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.