Technical Summary: The Site Frequency Spectrum in an Exponentially-Growing Population with Selection

Problem Statement

This work investigates the Site Frequency Spectrum (SFS), specifically the distribution of mutant allelic frequencies, within a population undergoing exponential growth driven by driver mutations. While the SFS for neutral evolution (where mutations do not confer a fitness advantage) is well-characterized—typically following a $1/j^2$ law in expectation for exponentially growing populations—the behavior of the SFS under selection remains less understood.

The authors focus on a two-type continuous-time linear birth-death process where:

• Type-0 (Wild-type) cells grow with a net Malthusian rate $\lambda_0 > 0$.
• Type-1 (Mutant/Driver) cells arise from Type-0 cells via mutation and possess a higher net growth rate $\lambda_1 = \lambda_0 + b$, where $b > 0$ represents the selective advantage.
• The model allows for both replication-dependent mutations (occurring at cell division) and replication-independent mutations (occurring at a constant rate).

The primary objective is to characterize the driver SFS, denoted $S_j(t)$, which counts the number of distinct driver clones of size $j$ at time $t$. The study aims to derive exact moments, asymptotic behaviors (in time and frequency), and almost sure convergence results, extending previous work on neutral evolution and semi-deterministic approximations.

Methodology

The authors employ a rigorous probabilistic framework based on branching process theory.

1. Model Formulation:

   • The system is modeled as a reducible branching process where Type-0 cells form a supercritical process seeding Type-1 clones.
   • Each Type-1 clone evolves as an independent linear birth-death process with birth rate $a+b$ and death rate $d$.
   • The analysis considers two scenarios: deterministic fitness (fixed $b$) and random fitness (where the fitness advantage $B$ of each clone is drawn from a distribution $Q$).

2. Analytical Techniques:

   • Exact Moments: The authors derive integral representations for the first and second moments of $S_j(t)$ by conditioning on the initiation times of clones and the trajectory of the Type-0 population.
   • Asymptotic Analysis: Large-time limits ($t \to \infty$) and large-frequency limits ($j \to \infty$) are analyzed using properties of the limiting random variables of the branching processes (e.g., $Y = \lim e^{-\lambda_0 t} Z_0(t)$ and $Y^{(i)}_1 = \lim e^{-\lambda_1 t} Z^{(i)}_1(t)$).
   • Almost Sure Convergence: To prove strong laws of large numbers, the authors construct $L^2$-approximations. They decompose the SFS into monotone processes (counting entries and exits from size $j$) and utilize martingale arguments and Borel-Cantelli lemmas to establish convergence.
   • Fixed Detection Size: The analysis extends to random detection times $\tau_n$ (when total population size reaches $n$), utilizing the relationship between time and size in supercritical growth ($\tau_n \sim \frac{1}{\lambda_1} \ln n$).

Key Contributions and Results

1. Exact Moments and Mean Behavior

The paper provides exact integral formulas for the mean and second moment of the driver SFS.

• Selection Effect: The authors prove that increasing the selection strength $b$ decreases the frequency of small clones ($j$ near 1) in the mean SFS. This is because stronger selection causes clones to grow past small sizes more rapidly, reducing the time they spend in the "small frequency" bin.
• Asymptotic Power Laws: For large $j$ and large $t$, the mean SFS follows a power law:
  $$\lim_{t \to \infty} e^{-\lambda_0 t} E[S_j(t)] \sim C \cdot j^{-\lambda_0/\lambda_1 - 1}$$
  In the random fitness model, the tail behavior is dominated by clones with the maximal fitness advantage $b_{max}$, exhibiting a power law with a logarithmic correction.

2. Almost Sure Convergence (Strong Laws)

A major contribution is the proof of Strong Laws of Large Numbers for the driver SFS.

• Fixed Time: Conditional on non-extinction, $e^{-\lambda_0 t} S_j(t)$ converges almost surely to a random variable proportional to the limiting Type-0 population size $Y$.
• Fixed Size: Similarly, $n^{-\lambda_0/\lambda_1} S_j(\tau_n)$ converges almost surely to a random variable involving $Y$ and the limiting Type-1 population size $W$.
• Normalized SFS: The fraction of mutations at frequency $j$ relative to the total number of mutations converges to a deterministic limit that depends only on the ratio of growth rates $\lambda_0/\lambda_1$ and the extinction probability $p_1$. This suggests that the shape of the SFS can be used to infer evolutionary parameters.

3. Sparsity and Cutoff Frequencies

The authors investigate the behavior of $S_j(t)$ when the frequency $j$ varies with time, specifically $j(t)$. They identify a critical cutoff frequency scaling as:
$$j(t) \sim \exp\left( \frac{\lambda_0 \lambda_1}{\lambda_0 + \lambda_1} t \right)$$

• Trichotomy:
  • If $j(t)$ grows faster than this rate, the number of clones of that size tends to 0.
  • If $j(t)$ grows at this rate, the number of clones converges in distribution to a Poisson random variable (conditional on non-extinction).
  • If $j(t)$ grows slower, the number of clones tends to infinity.
    This threshold represents the balance between the production of new clones and their growth to size $j(t)$.

4. Relative SFS and Large Clones

The paper analyzes the relative SFS, $R_f(t)$, which counts clones comprising a fraction $> f$ of the total mutant population.

• Small Frequency Limit: As $f \to 0$, the relative SFS follows a power law $f^{-\lambda_0/\lambda_1}$.
• Connection to Semi-Deterministic Models: In the small mutation limit, the authors rigorously justify the semi-deterministic approximations used in prior literature (e.g., Durrett and Tung). They show that the stochastic model converges to the semi-deterministic model where the wild-type population grows deterministically.
• Largest Clone: The size of the largest mutant clone, normalized by the total population, converges to the supremum of a Poisson-Dirichlet distribution (specifically $PD(\alpha, 0)$ where $\alpha = \lambda_0/\lambda_1$) in the small mutation limit.

5. Tail Behavior at Fixed Detection Size

At a fixed detection size $n$, the SFS exhibits a point mass at $j=n$ (representing the single largest clone that reached size $n$). The probability of this event depends on the competition between the wild-type and mutant populations at the time of the first mutation.

Significance and Claims

The paper claims to provide a comprehensive theoretical characterization of the driver SFS in a stochastic, exponentially growing population with selection.

• Parameter Estimation: The authors assert that the derived limit theorems allow for the estimation of evolutionary parameters, specifically the fitness increase ($b$) of mutant cells relative to wild-type cells. Since the normalized SFS converges to a deterministic function of $\lambda_0/\lambda_1$, one can theoretically invert the observed SFS to estimate selection strength.
• Rigorous Justification: The work provides a rigorous stochastic foundation for previously used semi-deterministic models, confirming their validity in the small mutation limit.
• Distinction from Neutral Evolution: The results highlight how selection fundamentally alters the SFS shape compared to neutral models, particularly by suppressing small-frequency clones and creating a specific power-law tail determined by the ratio of growth rates.

The authors note that their current work focuses on a single driver mutation and the driver SFS specifically. They acknowledge that future work will extend these analyses to include "hitchhiker" (passenger) mutations and sequential driver acquisitions in multi-type populations.