[Paper Review] The waiting time for m mutations

Schweinsberg, Jason|arXiv (Cornell University)|Feb 24, 2008

Cancer Genomics and DiagnosticsBiochemistry, Genetics and Molecular Biology32 references17 citations

TL;DR

This paper analyzes the waiting time for m mutations to accumulate in a finite population under the Moran process, deriving the asymptotic distribution of the time until an individual acquires m mutations. The key result shows that the distribution depends critically on how the mutation rate μ scales with population size N, revealing distinct behaviors for different scaling regimes, with applications to cancer evolution and regulatory sequence emergence.

ABSTRACT

We consider a model of a population of fixed size N in which each individual gets replaced at rate one and each individual experiences a mutation at rate μ. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with m mutations. Several different behaviors are possible, depending on how μchanges with N. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development.

Motivation & Objective

• To determine the asymptotic distribution of the time until an individual in a population of size N acquires m mutations.
• To understand how the mutation rate μ scaling with population size N affects the waiting time distribution for m mutations.
• To model the stochastic process of mutation accumulation in a population under the Moran process, accounting for fixation and stochastic tunneling.
• To provide a mathematical foundation for multi-stage cancer models and regulatory sequence evolution by analyzing mutation waiting times.
• To compare different stochastic models of mutation accumulation and establish conditions under which they converge asymptotically.

Proposed method

• Models a population of fixed size N evolving via the Moran process, with each individual replacing itself at rate 1 and acquiring mutations at rate μ.
• Tracks the number of individuals of each mutation type (X_j(t)) and defines τ_m as the first time X_m(t) > 0.
• Analyzes the process through four distinct models (Models 4 and 5) that differ in how mutations are suppressed or fixed.
• Uses coupling techniques to compare Model 4 (with suppression and fixation) to Model 5 (simplified, non-suppressed process) by modifying mutation dynamics.
• Applies limit theorems and concentration inequalities (e.g., Lemma 22) to show that the probability of large numbers of high-type individuals is negligible.
• Establishes asymptotic equivalence between modified models (Model 4') and Model 5 by showing |r_4(T) - r_5(T)| → 0 as N → ∞.

Experimental results

Research questions

• RQ1 How does the waiting time for m mutations depend on the mutation rate μ when μ scales with population size N?
• RQ2 What are the different asymptotic behaviors of the waiting time distribution for m mutations under varying scaling of μ with N?
• RQ3 In what scenarios does stochastic tunneling (accumulating m mutations before fixation) dominate over sequential fixation?
• RQ4 How do suppression of mutations and fixation dynamics affect the time to m mutations in finite populations?
• RQ5 Under what conditions do simplified models (e.g., Model 5) accurately approximate the full stochastic process (Model 4) in the limit of large N?

Key findings

• The asymptotic distribution of the waiting time τ_m for m mutations depends critically on the scaling of μ with N, leading to different limiting behaviors.
• When μ is small (e.g., μ = c/N), the waiting time τ_m converges in distribution to a gamma-type limit, with the rate depending on c and m.
• For larger μ (e.g., μ = c/N^α with α < 1), the waiting time distribution shifts, reflecting faster mutation accumulation and reduced fixation delay.
• Stochastic tunneling becomes significant when μ is large enough that multiple mutations can accumulate before fixation, altering the waiting time distribution.
• The limit lim_{N→∞} |r_4(T) - r_5(T)| = 0 is established, showing that Model 5 provides a valid asymptotic approximation to the full process.
• The probability that a type m-j individual experiences a mutation leading to type m is asymptotically proportional to μ q_j (N−ℓ)/N, where ℓ is the number of higher-type individuals.