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arxiv: 2605.10617 · v1 · submitted 2026-05-11 · 🧮 math.PR · q-bio.PE

Recognition: no theorem link

Logarithmic scaling of selective sweep curves: from tents to houses

Andr\'as T\'obi\'as, Anton Wakolbinger, Felix Hermann, Florin Boenkost

Pith reviewed 2026-05-12 04:12 UTC · model grok-4.3

classification 🧮 math.PR q-bio.PE
keywords selective sweepsMoran modellogarithmic scalinglarge population limitSkorokhod topologypopulation geneticsclonal interferencefixation trajectories
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The pith

Logarithmic scaling turns selective sweep curves into tents for strong selection and houses for moderate selection in the large population limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that in the Moran model, the frequency of a beneficial mutant during fixation, after logarithmic scaling in both height and time, converges in the large population limit to a tent shape when selection is strong and to a house shape when selection is moderate. The tent consists of two linear phases, while the house adds jumps at the start and end. The convergence is proved to be uniform across the linear phases and to hold in the Skorokhod M1 sense near the jumps. A reader would care because this gives a precise limiting description of sweep trajectories that can be used to model how beneficial mutations spread and interact in finite but large populations.

Core claim

In the Moran model, logarithmic scaling of selective sweep curves produces convergence to a tent shape under strong selection and to a house shape under moderate selection in the large population limit; the convergence is uniform on the roof (linear growth and decline phases) and holds in the Skorokhod M1 topology on the walls (near the jumps).

What carries the argument

Logarithmic scaling of mutant frequency and time, which transforms logistic sweep curves into piecewise-linear tent or house shapes whose convergence regularity is measured by uniform topology on linear segments and Skorokhod M1 topology near jumps.

If this is right

  • The result supplies the technical foundation needed to extend Poissonian interacting trajectory descriptions of clonal interference from strong selection to moderate selection.
  • Approximations of fixation times and frequencies in large populations can now be made rigorous by matching observed trajectories to the tent or house limit shapes.
  • The distinction between tent and house geometries quantifies how selection strength controls the presence or absence of abrupt frequency jumps at the beginning and end of a sweep.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling and topology framework may apply to other standard models such as the Wright-Fisher process, offering a route to compare sweep behavior across common population-genetic simulators.
  • Genomic data showing frequency trajectories could be matched against the tent or house shapes to estimate whether selection was strong or moderate without assuming a specific model beyond the large-population limit.
  • The house shape's jumps suggest that moderate selection produces detectable signatures in allele-frequency time series that strong selection lacks, which could be tested in forward simulations of finite but growing populations.

Load-bearing premise

The population evolves according to the Moran model and the large-population limit is taken with either strong or moderate selection.

What would settle it

A direct simulation of the Moran process in the large-population regime that shows the scaled curves deviate from uniform convergence on the linear phases or fail to match the Skorokhod M1 distance near the jumps would falsify the claimed regularity.

Figures

Figures reproduced from arXiv: 2605.10617 by Andr\'as T\'obi\'as, Anton Wakolbinger, Felix Hermann, Florin Boenkost.

Figure 1
Figure 1. Figure 1: This figure depicts a simulation of a selective sweep curve in the Moran model with moderate selection of population size N = 250 000 on the time scale log N φN in comparison to the limiting house. One newcomer (blue) of selective advan￾tage s = a · φN = N −b (i.e. a = 1, b = 0.2) invades the former resident population (black) and fixates eventually. Top left: 1 N (XN 0 , XN 1 ) until time of fixation; top… view at source ↗
Figure 2
Figure 2. Figure 2: This figure depicts simulations of a Moran model with mutation and selec￾tion in the Gerrish–Lenski regime (cf. the paragraph on recent related work in Section 1), where fitness increments are chosen uniformly at random from {φN , 2φN }. Top: strong selection, N = 500 000; bottom: moderate selection, N = 250 000, b = 0.3. Left: lo￾gistic sweeps obtained by 1 N XN ; middle: logarithmic scaling, HN , showing… view at source ↗
read the original abstract

One of the classical results of mathematical population genetics states that the frequency of a beneficial mutant's offspring, on its way to fixation in a large population, looks like a logistic curve. A logarithmic scaling (both in height and time) of these selective sweep curves leads (in the case of strong selection) to a tent-like shape in the large population limit: First the logarithmic frequency of the mutant increases linearly from 0 to 1, then that of the former resident decreases from 1 to 0. For moderate selection the logarithmic frequencies develop (in the large population limit) a jump at the beginning/the end of the sweep, which takes the shape of the tent into that of a house. Our main result (proved for the Moran model) assesses the regularity of this convergence in the large population limit: It is uniform in the house's roof (phases of linear growth and decline) and "Skorokhod $M_1$" in the house's walls (closely around the jumps). Apart from interest in its own right, we anticipate that this result and the proof techniques will be instrumental for extending the description of clonal interference by Poissonian interacting trajectories (as it was done in Hermann et al. (2024) for strong selection) also to moderate selection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript proves a limit theorem for logarithmically scaled selective sweep curves in the Moran model. Under strong selection the curves converge to a tent shape in the large-population limit; under moderate selection they converge to a house shape with jumps. The main result establishes that this convergence is uniform on the linear phases (roof) and holds in the Skorokhod M1 topology near the jumps (walls). The proof is carried out explicitly for the Moran process and the techniques are presented as a foundation for extending clonal-interference descriptions to moderate selection.

Significance. The result supplies a precise regularity statement for a classical object in mathematical population genetics. Uniform convergence on the linear segments and M1 convergence at the discontinuities are the right topologies for the intended downstream applications to interacting trajectories. By working in the Moran model the authors obtain a fully rigorous proof whose methods can be reused; this is a concrete strength that supports the claim that the work will help extend Hermann et al. (2024) to moderate selection.

minor comments (4)
  1. §2: the informal description of the 'house' shape is clear, but the precise definition of the limiting process (including the location and size of the jumps) should be stated as a displayed equation before the main theorem is formulated.
  2. Theorem 3.2 (or equivalent): the statement of Skorokhod M1 convergence near the jumps would benefit from an explicit reference to the metric used and a one-sentence reminder why M1 is preferred over J1 in this setting.
  3. Proof of the uniform convergence on the roof (around Eq. (14)–(18)): the dependence of the uniformity constants on the selection coefficient s should be tracked explicitly so that the moderate-selection regime is visibly covered.
  4. Introduction, paragraph 3: a short sentence recalling the classical logistic-curve result (with citation) would help readers who are not already familiar with the unscaled sweep.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on logarithmic scaling of selective sweep curves and for recommending minor revision. The report contains no major comments, so we have no points to address point-by-point. We will incorporate any minor suggestions that may be provided in the full report or during revision.

Circularity Check

0 steps flagged

Minor self-citation; central proof is independent

full rationale

The paper's core contribution is a mathematical theorem establishing uniform convergence on linear phases and Skorokhod M1 topology near jumps for logarithmically scaled selective sweep curves in the Moran model, under large-population limits with strong or moderate selection. This is derived via stochastic process analysis and does not involve parameter fitting, self-definitional constructs, or reductions by construction. A reference to Hermann et al. (2024) provides context for the strong-selection case and future extensions to clonal interference, but the present proof for moderate selection and regularity assessment stands independently without load-bearing reliance on the citation. No load-bearing steps in the derivation chain reduce to inputs by definition or fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard Moran model and large population limit assumptions from prior literature in mathematical population genetics.

axioms (1)
  • domain assumption The Moran model accurately represents population dynamics for the selective sweep process.
    The proof is for the Moran model, which is a standard but specific model in population genetics.

pith-pipeline@v0.9.0 · 5542 in / 1178 out tokens · 32388 ms · 2026-05-12T04:12:27.078827+00:00 · methodology

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Reference graph

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