Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution $g(w)$. This is the finite population version of Kingman’s house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.} \textbf{15}, 1 (1978)]. In contrast to Kingman’s work, the focus here is on unbounded distributions $g(w)$ which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size $N \to \infty$ and simulated numerically for finite $N$. When the genome-wide mutation probability $U$ is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a \textit{diluted record process} (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite $U$ the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of $1-U$ compared to the $U \to 0$ limit.

💡 Research Summary

The paper investigates the evolutionary dynamics of an asexually reproducing population moving on an uncorrelated random fitness landscape under the infinite‑sites assumption, where each mutation creates a completely new fitness value drawn independently from a distribution g(w). This setting corresponds to the finite‑population version of Kingman’s “House of Cards” model, but the authors focus on unbounded fitness distributions that allow the mean fitness to increase without limit.

Model definition
A population of size N experiences point mutations with genome‑wide probability U per generation. When a mutation occurs, the offspring’s fitness is sampled anew from g(w), independent of the parent’s fitness. Because the genome is infinite, the probability of the same mutation re‑appearing is zero, guaranteeing that each mutation contributes a fresh fitness draw.

Infinite‑population solution (N → ∞)
The authors write a master equation for the time‑dependent fitness density f(w,t). The dynamics separate naturally into a mutation operator M, which injects new fitness values at rate U g(w), and a selection operator S, which amplifies individuals proportionally to their fitness. Solving the equation analytically yields a decomposition of f(w,t) into two components:

1. Stationary mutation cloud – a time‑independent contribution proportional to U g(w) that represents the continuous influx of novel fitnesses.
2. Traveling wave – a moving front of high‑fitness individuals that advances in fitness space due to selection. The wave speed depends on the tail of g(w) and on U.

The traveling‑wave picture captures the “clonal interference” regime where multiple beneficial lineages compete, while the stationary cloud reflects the background of neutral or deleterious mutants.

Diluted Record Process (DRP) for small U
When U is very small, the population is almost always monomorphic at the current fitness record w_record. A new mutation that exceeds this record is a candidate to fix, but it succeeds only with probability

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