Viral Quasispecies Evolution as a Branching Random Walk on the Hypercube

TL;DR

This paper models viral evolution as a branching random walk on a hypercube, deriving asymptotics for the time to reach specific mutations, revealing a phase transition in growth behavior.

Contribution

It provides the first rigorous analysis of first passage times in a high-dimensional hypercube model for viral quasispecies evolution, identifying a phase transition at a critical growth parameter.

Findings

Sharp asymptotics for first passage times as dimension grows

Identification of a phase transition at effective growth parameter e

Delayed mutation appearance when increasing mutation rate in slow-branching regime

Abstract

We study a continuous-time nearest-neighbor branching random walk on the $d$-dimensional $b$-ary hypercube $\{0,1,\dots,b-1\}^d$ as a model for viral quasispecies evolution under mutation and replication. Motivated by mutagenic antiviral treatments and evolutionary-safety questions, we analyze the first passage time to a fixed target genotype at Hamming distance $m$, corresponding to the first appearance of a prescribed collection of mutations. We derive sharp asymptotics for these first passage times, uniformly for $m\le d/L$ as $d\to\infty$ (where $L>0$ is a large constant), and identify a phase transition in first-passage scaling at $\rho=e$, where $\rho$ denotes the effective growth parameter. In the slow-branching regime $\rho\in(1,e)$ relevant to mutagenic treatment scenarios, the first passage time is asymptotically affine in the genome length $d$ and the target distance $m$. In particular, when replication is fixed and mutation exceeds branching, increasing the mutation rate can delay the first appearance of a prescribed genotype by order $d$, providing a quantitative perspective on evolutionary safety.