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arxiv: 2604.12004 · v1 · submitted 2026-04-13 · 🧬 q-bio.PE · cond-mat.stat-mech

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Fixation probabilities for multi-allele Moran dynamics with weak selection

Ian Braga, Lucas Wardil, Ricardo Martinez-Garcia

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Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.stat-mech
keywords fixation probabilityMoran processweak selectionmulti-allele dynamicsperturbative expansionFokker-Planck equationstochastic evolutionary dynamics
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The pith

Fixation probabilities for multiple competing alleles under weak selection expand systematically around the neutral solution.

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

The paper develops a perturbative framework for calculating fixation probabilities in Moran processes involving any number of alleles under weak selection. It demonstrates that these probabilities can be expanded as a series around the neutral case, where selection is absent, for arbitrary fitness functions. This works by exploiting the structure of the backward Fokker-Planck operator in the weak-selection limit. The approach is first stated generally for M alleles and then demonstrated on specific models including constant fitness, frequency-dependent coordination, and mutualistic clonal interference. If valid, the method provides analytical access to fixation outcomes in evolutionary scenarios that previously required simulation or were limited to two alleles.

Core claim

We develop a perturbative framework to compute fixation probabilities in multi-allele Moran processes under weak selection. Exploiting the general structure of the backward Fokker-Planck operator in this regime, we show that fixation probabilities admit a systematic expansion around their neutral solution for M competing alleles and arbitrary fitness functions.

What carries the argument

The backward Fokker-Planck operator of the multi-allele Moran process, whose weak-selection structure permits a perturbative expansion of fixation probabilities around the neutral solution.

If this is right

  • Fixation probabilities become analytically accessible for a simple three-allele model with constant fitness.
  • The expansion applies to coordination games where allele fitness increases with its own frequency.
  • The method handles clonal interference between mutualistic alleles.
  • Analytical results for fixation extend beyond pairwise interactions to general multi-strategy dynamics.

Where Pith is reading between the lines

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

  • The same perturbative structure might apply to other birth-death processes with weak selection if their transition rates admit a similar expansion.
  • Higher-order terms in the series could be computed explicitly to improve accuracy for moderate selection strengths.
  • The framework could be tested on frequency-dependent fitness functions beyond the three examples given.

Load-bearing premise

The backward Fokker-Planck operator in the weak-selection regime has a structure that permits a systematic perturbative expansion of fixation probabilities without breakdown or need for resummation.

What would settle it

Direct numerical simulation of the Moran process for three alleles with a chosen fitness function, followed by comparison of the computed fixation probabilities against the perturbative series truncated at low orders.

Figures

Figures reproduced from arXiv: 2604.12004 by Ian Braga, Lucas Wardil, Ricardo Martinez-Garcia.

Figure 1
Figure 1. Figure 1: Fixation probabilities in a three-allele Moran pro [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Deviation of allele-1 fixation probability from [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fixation probability perturbation ϕ (s) 1 = ϕ1 − x1 for allele 1 in the three-allele Moran process under frequency￾dependent selection. The color map represents the analytical solution over the simplex, while streamlines indicate the direction of maximal increase of the perturbation. (a) In the one-sided interaction regime, s1 = 0, gradient flows largely follow directions in which both x1 and x2 increase. … view at source ↗
read the original abstract

Fixation probabilities are essential for characterizing stochastic evolutionary dynamics, but analytical results remain limited mainly to systems with two competing types. We develop a perturbative framework to compute fixation probabilities in multi-allele Moran processes under weak selection. Exploiting the general structure of the backward Fokker-Planck operator in this regime, we show that fixation probabilities admit a systematic expansion around their neutral solution. We first introduce the framework in a general case with $M$ competing alleles and arbitrary fitness functions, and then apply it to three biologically motivated examples: a simple model of three competing alleles with a constant fitness function, a coordination game in which allele fitness increases with its frequency in the population, and a model of clonal interference between mutualistic alleles. These results extend the analytical understanding of fixation probabilities beyond pairwise interactions, establishing a framework for investigating multi-strategy stochastic evolutionary dynamics.

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

2 major / 2 minor

Summary. The manuscript develops a perturbative framework to compute fixation probabilities for multi-allele Moran processes under weak selection. It expands the fixation probability as a power series in selection strength around the neutral solution by exploiting the structure of the backward Fokker-Planck operator, first presenting the general case for M alleles with arbitrary fitness functions and then applying the method to three examples: constant fitness for three alleles, a coordination game with frequency-dependent fitness, and clonal interference between mutualistic alleles.

Significance. If the expansion is rigorously justified, the work would extend analytical results on fixation probabilities beyond the two-allele limit and supply a systematic tool for multi-strategy stochastic evolutionary dynamics. The approach builds on the neutral operator's kernel properties to generate recursive corrections, which is a strength when the solvability conditions hold.

major comments (2)
  1. [§2 (general framework)] §2 (general framework): The central claim that the expansion is systematic for arbitrary fitness functions rests on the assumption that solvability conditions for the neutral backward operator L_0 are automatically satisfied at each order. The source term at order n must be orthogonal to the adjoint kernel (constants or neutral fixation probabilities on the simplex). The manuscript does not derive or verify this orthogonality for general fitness; it holds in the three examples by symmetry or construction but is not guaranteed a priori for arbitrary fitness, risking non-existence of solutions or secular terms.
  2. [Abstract and §3–5 (examples)] Abstract and §3–5 (examples): The three applications are presented as support for the general framework, but without explicit checks (e.g., verification that the source terms satisfy the Fredholm alternative at each perturbative order or comparison to exact neutral limits), it is unclear whether the hierarchy closes without resummation. This weakens the assertion that the method works for arbitrary fitness.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'arbitrary fitness functions' is used without qualification; a brief caveat that the expansion assumes solvability conditions would improve precision.
  2. [Notation] Notation: Define the selection-strength parameter and the fitness perturbation operator consistently between the general derivation and the example sections to avoid ambiguity in the perturbative hierarchy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points about the rigor of the general perturbative framework and the supporting examples. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [§2 (general framework)] The central claim that the expansion is systematic for arbitrary fitness functions rests on the assumption that solvability conditions for the neutral backward operator L_0 are automatically satisfied at each order. The source term at order n must be orthogonal to the adjoint kernel (constants or neutral fixation probabilities on the simplex). The manuscript does not derive or verify this orthogonality for general fitness; it holds in the three examples by symmetry or construction but is not guaranteed a priori for arbitrary fitness, risking non-existence of solutions or secular terms.

    Authors: We thank the referee for identifying this key aspect of the justification. The manuscript presents the general framework in §2 by exploiting the kernel structure of the neutral backward operator, but it is correct that an explicit derivation of the orthogonality (Fredholm alternative) condition for arbitrary fitness functions is not provided. While the construction ensures that fitness perturbations vanish in the neutral limit, a general proof that source terms remain orthogonal to the adjoint kernel at every order was omitted. We will revise §2 to include this derivation, showing that the inner product of each source term with the neutral fixation probabilities integrates to zero over the simplex by the properties of the Moran transition rates and the definition of the selection perturbation. This addition will confirm that solutions exist order by order without secular terms for arbitrary fitness. revision: yes

  2. Referee: [Abstract and §3–5 (examples)] The three applications are presented as support for the general framework, but without explicit checks (e.g., verification that the source terms satisfy the Fredholm alternative at each perturbative order or comparison to exact neutral limits), it is unclear whether the hierarchy closes without resummation. This weakens the assertion that the method works for arbitrary fitness.

    Authors: We agree that the examples would benefit from explicit verification to demonstrate that the perturbative hierarchy is well-defined. In the revised manuscript, we will augment §§3–5 with direct checks for each example: (i) confirmation that the source terms at each order are orthogonal to the adjoint kernel, (ii) explicit reduction to the exact neutral fixation probabilities when the selection coefficient is set to zero, and (iii) verification that no resummation is required because the solvability condition holds at every finite order. These additions will be concise and will reinforce that the framework applies beyond the specific cases shown. revision: yes

Circularity Check

0 steps flagged

No circularity: expansion follows from standard backward Fokker-Planck structure

full rationale

The derivation begins from the established backward Fokker-Planck operator for the multi-allele Moran process under weak selection, a standard stochastic-process tool independent of the paper. The perturbative hierarchy is constructed order-by-order by applying this operator to successive corrections, with the neutral solution serving as the zeroth-order input. No step defines the target fixation probabilities in terms of themselves, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the present work. The three examples are downstream applications, not inputs that force the general result. The framework is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard stochastic process approximations for the Moran model; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The Moran process dynamics can be approximated by a backward Fokker-Planck equation under weak selection.
    Standard modeling assumption in stochastic evolutionary dynamics.
  • standard math Neutral (zero-selection) fixation probabilities are known exactly and form a valid base for perturbation.
    In the neutral Moran model, fixation probability equals initial frequency.

pith-pipeline@v0.9.0 · 5445 in / 1172 out tokens · 41595 ms · 2026-05-10T15:55:35.971568+00:00 · methodology

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Reference graph

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