Genetic Composition of Supercritical Branching Populations under Power Law Mutation Rates

TL;DR

This paper analyzes the genetic composition of supercritical branching cancer populations under power-law mutation rates on a finite directed trait graph. By coupling a multitype branching process with power-law-scaled mutation probabilities, it derives first-order asymptotics for mutant subpopulations on both deterministic and random time scales, with a unifying random limit $W$ arising from the wild-type lineage. A key finding is that, under a non-increasing growth-rate condition, the stochastic behavior of all mutant subpopulations is driven by $W$ alone, and only shortest admissible mutation pathways (walks) with maximal neutral mutations contribute to growth, yielding precise formulas that include $ ext{log}(n)$ factors for neutral mutations. The results extend to an infinite mono-directional graph for explicit exponents and general finite trait spaces by classifying mutational walks into finitely many equivalence classes, enabling rigorous characterization of evolutionary pathways and potential statistical inference of tumor history. These contributions provide a rigorous bridge between tumor progression models, mutational-pathway theory, and statistical inference in cancer evolution under power-law mutation regimes.

Abstract

We aim to understand the evolution of the genetic composition of cancer cell populations. To achieve this, we consider an individual-based model representing a cell population where cells divide, die and mutate along the edges of a finite directed graph $(V,E)$. The process starts with only one cell of trait $0$. Following typical parameter values in cancer cell populations we study the model under power law mutation rates, in the sense that the mutation probabilities are parametrized by negative powers of a scaling parameter $n$ and the typical sizes of the population of interest are positive powers of $n$. Under a non-increasing growth rate condition, we describe the time evolution of the first-order asymptotics of the size of each subpopulation in the $\log(n)$ time scale, as well as in the random time scale at which the wild-type population, resp. the total population, reaches the size $n^{t}$. In particular, such results allow for the perfect characterization of evolutionary pathways. Without imposing any conditions on the growth rates, we describe the time evolution of the order of magnitude of each subpopulation, whose asymptotic limits are positive non-decreasing piecewise linear continuous functions.

Figures (6)

• Figure 1: Graphical representation of the model with two traits and without backward mutation
• Figure 2: Heuristics for the first-occurrence time of mutant cells
• Figure 3: Heuristics for the size of the mutant subpopulation after time $\mathfrak{t}_{\ell(0,1)}^{(n)}$
• Figure 4: Heuristics for the contribution of walks to the size order of the plain purple mutant subpopulation: in this example, the dashed red walk has a length of $7$, while the dotted blue and plain green walks have a length of $4$. Therefore, only the two latter walks may contribute to the size order of the plain purple mutant subpopulation, making them sub-admissible walks. However, the dotted blue walk has only one neutral mutation, whereas the plain green walk has two neutral mutations. As a result, only the plain green walk will ultimately contribute to the size order of the purple mutant subpopulation. For $t \geq 4$, at time $\mathfrak{t}^{(n)}_{t}$, it will grow as $\log^{2}(n)e^{\lambda(0)\mathfrak{t}^{(n)}_{t-4}}$. Notice, in particular, that the dashed red walk has the maximal number of neutral mutations, which is $3$. However, since it is not a sub-admissible walk, the multiplicative factor of $\log(n)$ remains $2$ instead of $3$.
• Figure 5: Dynamical representation of the infinite mono-directional graph

Theorems & Definitions (46)

• Definition 2.1: Deleterious and neutral vertices
• Remark 2.2
• Definition 2.3: Walk in the graph
• Remark 2.4
• Definition 2.5: Admissible walks
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9
• Corollary 2.10: Theorem \ref{['Theorem: general finite graph']} applied to a mono-directional graph