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Viral Quasispecies Evolution as a Branching Random Walk on the Hypercube

Jose Blanchet, Zhenyuan Zhang

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 $ρ=e$, where $ρ$ denotes the effective growth parameter. In the slow-branching regime $ρ\in(1,e)$ 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.

Viral Quasispecies Evolution as a Branching Random Walk on the Hypercube

Abstract

We study a continuous-time nearest-neighbor branching random walk on the -dimensional -ary hypercube 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 , corresponding to the first appearance of a prescribed collection of mutations. We derive sharp asymptotics for these first passage times, uniformly for as (where is a large constant), and identify a phase transition in first-passage scaling at , where denotes the effective growth parameter. In the slow-branching regime relevant to mutagenic treatment scenarios, the first passage time is asymptotically affine in the genome length and the target distance . 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 , providing a quantitative perspective on evolutionary safety.

Paper Structure

This paper contains 30 sections, 34 theorems, 202 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main contribution.
  3. Results by regime.
  4. Relation to prior work.
  5. Organization of the paper.
  6. Model setup and notation
  7. Notation.
  8. The constant- regime
  9. The case
  10. The case
  11. The ultra-slow branching regime
  12. Discussions and open questions
  13. The biological perspective
  14. The mathematical perspective
  15. Notation.
  16. ...and 15 more sections

Key Result

Proposition 1

Fix $b\in\mathbb{N}_2$ and $\rho\in(1,e)$. Then there exists a large constant $L_1>0$ (possibly depending on $b,\rho$) such that the following statements hold uniformly for $m\in[1, d/L_1]$.

Figures (2)

  • Figure 1: (a) Values of $x_0$ and $r$ as functions of $\rho\in[1.035,2.5]$ for $b=2$ (see definitions in \ref{['eq:x0 def']} and \ref{['eq:r def']}). If $\rho$ is close to $1$, the term $x_0d$ dominates in \ref{['eq:x0r']}; if $\rho$ is close to $e$, the term $rm$ dominates for $m$ large, where we recall that $m$ is the Hamming distance of the target from the origin. (b) FPT predictions from solving the first-moment equation \ref{['eq:1']} and from the asymptotic expansion \ref{['eq:x0r']} with $\rho=2,~b=4,$$d=10^4$, and $m\in[0,500]$.
  • Figure 2: Plots of solutions to the first-moment equation \ref{['eq:1']} for $\rho\in[2,5]$ with $b=2$. The $y$-axis is on a logarithmic scale.

Theorems & Definitions (67)

  • Proposition 1
  • Theorem 2
  • Corollary 3: Monotonicity in the mutation rate
  • Corollary 4
  • Theorem 5
  • Remark 1
  • Lemma 6
  • Proposition 7
  • Theorem 8
  • Corollary 9
  • ...and 57 more