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Conditional gene genealogies given the population pedigree for a diploid Moran model with selfing

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Pith's one-line read For a population that mostly self-fertilizes, the true pedigree changes genealogical predictions: at a critical outcrossing rate a random ancestral graph, not the Kingman coalescent, gives the times to common ancestry.

desk verdict New conditional coalescent limits for selfing Moran populations, with the headline limited-outcrossing result gated on an omitted tightness proof. read the letter →

arxiv 2411.13048 v2 pith:3JAEBJQD submitted 2024-11-20 q-bio.PE math.PR

Maximillian Newman , John Wakeley , Wai-Tong Louis Fan This is my paper · ORCID
classification q-bio.PEmath.PR MSC 60J2760F1792D15
keywords diploidMoranmodelselfingpopulationpedigreeconditionalcoalescentancestralgraphlimitedoutcrossingcoalescencetimeidentitydisequilibrium
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the actual pedigree of a population—who really reproduced with whom—changes the predictions of coalescent theory for a sample of two gene copies. In a diploid Moran model where each new offspring is produced by selfing with probability $\alpha_N$, the authors prove that the answer is controlled by one number: $N(1-\alpha_N)$, the expected number of outcrossing events per generation. If that number tends to infinity, the pedigree washes out and the classic Kingman prediction, an exponential time to common ancestry with rate $2/(2-\alpha)$, survives conditioning. If it tends to a finite $\lambda$, a qualitatively new limit object appears—the ancestral graph $G_\lambda$, where ancestral lineages branch at rate $\lambda$ and coagulate pairwise at rate 2—and coalescence times are the meeting times of random walks on it, so the conditional distribution is random and pedigree-dependent. If it tends to zero, coalescence is fixed by the pedigree's own times to common ancestry. Standard analyses average over pedigrees, and this paper shows that in strongly selfing populations those averages can mispredict the variation among unlinked loci.

What carries the argument

The ancestral graph $G_\lambda$ is the object that carries the limited-outcrossing argument: a continuous-time particle system that starts with two particles, in which each particle splits into two at rate $\lambda$ and each pair of particles coagulates at rate 2, equal in distribution to an ancestral recombination graph or an ancestral selection graph. It is obtained as the weak limit, on the $N^2$ time scale, of the discrete-time ancestral graph $G_N$, the subgraph of the pedigree containing all potential ancestors of the sample, which fragments at outcrossing events and coagulates at selfing events (Lemma 5.1, proved as Lemma 7.6). Coalescence times are then the first meeting times $T_\lambda$ of the two random walks on this graph, and the conditional survival probability is a step function jumping at the random overlap times of the graph. In the partial-selfing regime the argument uses a different device: a decomposition of the coalescence time into alternating overlap and splitting events, together with an $L^2$ second-moment comparison that couples two conditionally independent copies of the genealogy on the same pedigree.

What would settle it

Run the Moran model at $N=10^4$ with $\alpha_N=1-\lambda/N$ for $\lambda=1,5,20$; for each simulated pedigree compute the exact conditional survival function of the pairwise coalescence time for two sampled individuals, as in the paper's Figure 1, and compare it with $P(T_\lambda>t\mid G_\lambda)$ read off the pedigree's ancestral graph: the conditional survival function should be a step function jumping at the graph's overlap times, its across-pedigree variance should match the paper's explicit covariance formulas as a function of $\lambda$, and in the negligible-outcrossing case $N(1-\alpha_N)\to0$ the limit should be a single random jump rather than the smooth exponential $e^{-2t}$.

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Extended reading notes

Core claim

The central claim is that conditioning on the random population pedigree yields three different limiting laws for the pairwise coalescence time, classified by the limit of $N(1-\alpha_N)$. For two gene copies sampled from different individuals (Theorem 4.1): when $N(1-\alpha_N)\to\infty$ the conditional survival probability $P_{\mathrm{diff}}(N^{-2}\tau^{(N)}>t\mid\mathcal{A}_N)$ converges to $e^{-2t/(2-\alpha)}$, matching the pedigree-averaged model; when $N(1-\alpha_N)\to\lambda\in(0,\infty)$ it converges to $P(T_\lambda>t\mid G_\lambda)$, the first-meeting time of two coalescing random walks on the ancestral graph; and when $N(1-\alpha_N)\to0$ it converges to the indicator $1_{\{\mathrm{Exp}(2)>t\}}$, a single exponentially distributed jump time fixed by the pedigree. For two gene copies sampled from the same individual (Theorem 4.5), the limit is $2^{-U}e^{-2t/(2-\alpha)}$, where $U$ is the individual's realized number of selfing generations with $P(U=k)=\alpha^k(1-\alpha)$, and this limit is $0$ at $\alpha=1$ and $e^{-t}$ at $\alpha=0$. Thus the pedigree-averaged theory sees one regime where the conditional theory sees three, and at the boundary $\alpha_N\to1$ the pedigree survives in the limit through the ancestral graph.

Load-bearing premise

If the size of the discrete ancestral graph is not bounded in probability on the $N^2$ time scale, the convergence to $G_\lambda$—and with it the random-walk description of the limiting coalescence time—fails, and the paper omits the details of exactly this compact-containment step.

Editorial extensions

If this is right

  • For samples of multiple unlinked loci, the pedigree-conditional model is the correct sampling structure when likelihoods are multiplied across loci, and under limited outcrossing it predicts that genealogies vary across loci through random choices at each branching of the ancestral graph.
  • The same-individual limit (Theorem 4.5) replaces the averaged inbreeding coefficient $F=\alpha/(2-\alpha)$ with each sampled individual's own realized number of selfing generations $U$, which is exactly the per-individual randomness underlying identity disequilibrium between loci.
  • The three regimes give a practical rule: if $N(1-\alpha_N)$ exceeds roughly 100, conditional genealogies are well approximated by the partial-selfing Kingman model; if it is of order 1, the limited-outcrossing model must be used; if it is much smaller than 1, genealogies across the genome are essentially identical and heterozygosity vanishes.
  • The limited-outcrossing model interpolates between the two extremes: as $\lambda\to\infty$ it converges to the $\alpha=1$ partial-selfing limit, and as $\lambda\to0$ it converges to negligible outcrossing, so it covers every way in which $\alpha_N$ can approach complete selfing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same classification by the product of population size and the rate of pedigree-breaking events should apply to other forms of close inbreeding, such as sib mating or cousin mating, because their splitting and coagulating events also scale differently; the paper's discussion anticipates this but does not prove it.
  • Beyond the paper: whole-genome data from predominantly selfing species could estimate $\lambda=N(1-\alpha)$ directly from the across-locus variance of genealogies using the covariance formulas, turning the limit theorem into a parameter-estimation procedure the paper does not develop.
  • Beyond the paper: the observable signature of the negligible-outcrossing limit, near-identical coalescence times across the whole genome, could serve as a diagnostic for populations sitting in that regime before the model's more extreme prediction of zero heterozygosity fully holds.
  • Beyond the paper: single-locus inference that assumes smooth exponential genealogies will overstate the uncertainty in coalescence times when applied locus-by-locus under limited outcrossing, because the correct pedigree-conditional distribution is a step function concentrated on a few fixed times.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper studies a discrete-time diploid Moran model with selfing probability α_N and derives the conditional distribution of the pairwise coalescence time τ^(N) of two sampled gene copies given the random population pedigree. The main results are Theorem 3.1 for the unconditional limit, Theorem 4.1 for the conditional limit under three regimes (partial selfing, limited outcrossing, and negligible outcrossing), and Theorem 4.5 for the same-individual conditional limit. In the limited-outcrossing regime N(1−α_N)→λ, the limit is described as the meeting time of coalescing random walks on an ancestral graph G_λ; in the negligible-outcrossing regime it is a random exponentially distributed constant. The paper also derives variance and covariance formulas for conditional survival probabilities and links the limited-outcrossing model to ancestral recombination and selection graphs.

Significance. If fully supported, the result is significant for population genetics: it shows that conditioning on the pedigree can change coalescent predictions relative to the pedigree-averaged Kingman coalescent precisely in the near-selfing regime, and it provides a tractable one-parameter limiting object, the ancestral graph G_λ. The paper's strengths include a detailed overlap/splitting decomposition of the coalescence time, generator computations for the ancestral graph, and explicit L^2 convergence arguments in the partial-selfing regime. The covariance formulas in Section 5 are also useful and connect the new model to classical two-locus results. However, the central limited-outcrossing and negligible-outcrossing rows of Theorem 4.1 depend on a weak-convergence proof whose compact-containment step is explicitly omitted, and one step in the proof of Lemma 7.8 is not a valid use of conditional Markov's inequality. These are load-bearing technical gaps that should be repaired before the results can be considered fully established.

major comments (4)
  1. [Section 5 and Section 7.2.2, Lemma 7.6] The weak convergence of the discrete-time ancestral graph G^N to G_λ is not fully proved. Lemma 7.6 asserts the compact containment condition (33) by saying that 'the same argument' as for the continuous-time size process L_t applies, but the discrete-time process has one-step transition probabilities (27) with O(N^-2) and o(N^-3) terms, and no discrete-time analogue of the quadratic-variation computation in (38) is supplied. Since Lemma 7.7, Theorem 7.9, and Corollary 7.11 all rely on this convergence, the limited-outcrossing and negligible-outcrossing rows of Theorem 4.1 are not fully supported without a proof that |G^N(⌊tN^2⌋)| is tight. The manuscript itself acknowledges this gap after Lemma 5.1, but the appendix does not fill it.
  2. [Section 7.2.2, proof of Lemma 7.8] The proof of Lemma 7.8 contains an invalid inequality. The displayed 'conditional Markov inequality' reads P(N^-2(τ−τ_O^1)>ε|A_N) ≤ (1/ε) P(N^-2(τ−τ_O^1)>ε), which is not a consequence of Markov's inequality and can fail in general. The correct conditional Markov bound would involve the conditional expectation E[N^-2(τ−τ_O^1)|A_N], and then convergence of the conditional probability to zero would require a control on that conditional expectation. Because Lemma 7.8 is used to pass from the first overlap time to the coalescence time in Theorem 7.9, this proof step needs to be repaired.
  3. [Section 7.2.2, proof of Lemma 7.10] The proof of Lemma 7.10 contains a sign/conditioning error. It states that P_same(O=0)=α_N/(2−α_N) tends to 0 as α_N→1, but this quantity tends to 1. The subsequent conclusion that the right-hand side of (41) tends to 0 should be justified by conditioning on O=0 (where the rescaled coalescence time is 0) and on O≥1 (where Corollary 7.4 gives convergence to 0), not by conditioning on O=1 as written. The conclusion is correct, but the proof as written is inconsistent.
  4. [Section 7.3, Proposition 5.5] The asymptotic expansions in Proposition 5.5 are delegated to a Mathematica calculation stored in Newman (2024), and the key computation leading to equation (62) is not shown in the manuscript. This is not central to Theorems 4.1 and 4.5, but the paper should provide enough detail for the expansion to be verified without reliance on an external file, or should state more explicitly which symbolic computation is being used.
minor comments (4)
  1. [Section 7.2.2, Eq. (27)] The last displayed transition probability in (27) uses the notation 'n−M−2/n−2', which is inconsistent with the surrounding notation in terms of N and m; this should be corrected.
  2. [After Theorem 4.5] The sentence 'where is the possible for a “same” sample to enter the “diff” process' is ungrammatical; it should be 'where it is possible for a “same” sample to enter the “diff” process'.
  3. [Section 7.2.2, Lemma 7.7] The state space of the joint process in Lemma 7.7 is written as D(P ×(Z × {0, 1})^2), but Z is not introduced as a set of labels in that statement; using the notation from Definition 4.3 or defining the label set explicitly would improve clarity.
  4. [Section 1.3 and Section 6] The paper uses α_N for the selfing probability and α for its limit, but in a few places (for example in the discussion of Figure 1) α denotes a finite-N value rather than the limit; the notation should be made uniform to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the conditional coalescence limits are derived from the Moran model's transition rates rather than fitted to the target results.

full rationale

The derivation chain is self-contained relative to the paper's own model definitions. Theorem 3.1 is proved directly from the three-state transition matrix Π_N via the overlap/splitting decomposition (Lemmas 7.1, 7.2, Corollary 7.3), not by assuming the exponential limit. The conditional limits in Theorems 4.1 and 4.5 are then obtained either by L2 second-moment estimates (Theorem 7.18, Theorem 7.21) that use the unconditional result and the joint law of two conditionally independent coalescence times, or by weak convergence of the discrete-time ancestral graph (Lemma 7.6, Lemma 7.7) to the explicitly defined ancestral graph Gλ. The parameter λ is a scaling limit N(1−α_N)→λ, not a fitted constant, and the random variable U in Theorem 4.5 arises as the limit of the geometric number of selfing generations before a splitting event, matching the model's transition probabilities. Self-citations to Diamantidis et al. (2024), Kogan et al. (2023), and Wakeley et al. (2012) are used for notation, context, comparisons, and a numerical method in Figure 1; they are not load-bearing inputs to the main convergence proofs. The omitted compact-containment detail in the proof of Lemma 5.1/Lemma 7.6 is a technical incompleteness that bears on rigor, but it is not circular: it concerns tightness of a process constructed from the Moran model, not an assumption equivalent to the conclusion. Therefore no step reduces by construction to its own inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theorems do not fit any constants to data. The only continuously varying parameters are the model inputs alpha_N and the scaling limit lambda; both are stated as assumptions of the theorems rather than estimated. The analysis leans on standard Markov-process convergence theory, on the definition of the Moran model, and on the choice of scaling N(1-alpha_N) -> lambda, which is flagged above. No new physical entity is invented; the ancestral graph G_lambda is a proven limit object, not an ad hoc postulate.

free parameters (1)
  • lambda = lim N(1-alpha_N)
    Scaling parameter of the limited-outcrossing regime; it is a model constant, not fitted to data. The main theorems hold for every lambda in [0, infinity] and the covariances in Section 5 are explicit functions of lambda.
assumptions (4)
  • standard math Weak convergence of Markov chains (Ethier and Kurtz 2009), including generator convergence and compact containment, used in Lemma 7.6.
    Standard tools in Markov process convergence; invoked without proof in the paper.
  • domain assumption The diploid Moran model with constant population size N, random mating, and Mendelian inheritance is the process of interest.
    This is the model being analyzed; it is a mathematical idealization, not a measured population.
  • domain assumption Unlinked loci are conditionally independent given the pedigree.
    Invoked when interpreting the single-locus conditional coalescence time as predicting variation among unlinked loci; it is standard Mendelian segregation.
  • ad hoc to paper Asymptotic scaling N(1-alpha_N) -> lambda in R_+ with time rescaled by N^2 Moran steps.
    This scaling is chosen to make the limited-outcrossing limit nondegenerate. Other rates of approach to alpha=1 give the other two regimes. It is a mathematical modeling choice, not an empirical constraint.

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Pith. "Pith review of Conditional gene genealogies given the population pedigree for a diploid Moran model with selfing." pith.science (2026). https://pith.science/paper/3JAEBJQD

@misc{pith2026241113048,
  author       = {Pith},
  title        = {Pith review of: Conditional gene genealogies given the population pedigree for a diploid Moran model with selfing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JAEBJQD}},
  note         = {Machine review of arXiv:2411.13048}
}
abstract

We introduce a stochastic model of a population with overlapping generations and arbitrary levels of self-fertilization versus outcrossing. We study how the global graph of reproductive relationships, or population pedigree, influences the genealogical relationships of a sample of two gene copies at a genetic locus. Specifically, we consider a diploid Moran model with constant population size $N$ over time, in which a proportion of offspring are produced by selfing. We show that the conditional distribution of the pairwise coalescence time at a single locus given the random pedigree converges to a limit law as $N$ tends to infinity. The distribution of coalescence times obtained in this way predicts variation among unlinked loci in a sample of individuals. Traditional coalescent analyses implicitly average over pedigrees and generally make different predictions. We describe three different behaviors in the limit depending on the relative strengths, from large to small, of selfing versus outcrossing: partial selfing, limited outcrossing, and negligible outcrossing. In the case of partial selfing, coalescence times are related to the Kingman coalescent, similar to what is found in traditional analyses. In the case of limited outcrossing, the retained pedigree information forms a random graph, with coalescence times given by the meeting times of random walks on this graph. In the case of negligible outcrossing, which represents complete or nearly complete selfing, coalescence times are determined entirely by the fixed times to common ancestry of diploid individuals in the pedigree.

Figures

Figures reproduced from arXiv: 2411.13048 by the authors.

Figure 1
Figure 1. shows simulation results of pedigrees and pairwise coalescence times for the model of selfing used by Möhle (1998), which corresponds to αN = α constant in our model. Specifically, for each of three selfing probabilities, 50 independent pedigrees were simulated and for each of these two different individuals were sampled at random. Colored lines, one for each pedigree-plus-sample, depict the cumulative distribution … view at source ↗
Figure 2
Figure 2. A realization of our diploid Moran process with N = 6 individuals including the genetic transmission events at one locus (left image) and the corresponding pedigree (right image). Also on the left, two gene copies (X0, Y0) = (2, 9) are sampled in the present time-step 0, and their lineages are highlighted by the solid dots. Namely, {(Xk, Yk)} 4 k=0 = {(2, 9),(2, 9),(2, 5),(6, 5),(4, 4)}. These two lineages coalesce … view at source ↗
Figure 3
Figure 3. Theorem 4.1 says that the law of N −2 τ (N) for two lineages in different individuals is robust to the structure of the pedigree so long as the rate at which two lineages in the same individual undergo a splitting event, (1 − αN )N −1 , is much greater than the rate at which two sample lineages in distinct individuals coalesce, N −2 + O(N −3 ), specifically that the ratio of the former over the latter, approximately… view at source ↗
Figures from the paper (6 more)
Figure 3
Figure 3. Figure 3: The limiting conditional CDF t 7→ limN→∞ Pdiff(N −2 τ (N) ≤ t|AN ) for our Moran model under three assumptions about αN : partial selfing, limited outcrossing, and negligible outcrossing. (Left) Partial selfing, where limN N(1 − αN ) = ∞. The limiting CDF is determinis…
Figure 4
Figure 4. Figure 4: Two realizations of Tλ on the same ancestral graph are demonstrated in [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: A realization of GN containing potential ancestors of the two sampled individuals (ovals) at the bottom. In past time-step 3, the ancestral individual on the left undergoes an outcrossing event. In past time-step 4, one of the nodes from this event is the offspring of …
Figure 6
Figure 6. Figure 6: Two conditionally independent realizations of the pairwise process, (xλ, yλ) on the same ancestral graph Gλ. The hitting times of these random walks are marked by a purple dot. These times correspond to Tλ and T ′ λ , the coalescence times of two unlinked loci given th…
Figure 7
Figure 7. Figure 7: The two components, E[Var(T|G)] and Var(E[T|G]), of Var(T) as functions of λ under limited outcrossing. By the law of total variance, Var(T) = E[Var(T|G)] + Var(E[T|G]). Also, Var(T) = 1/4 in this case because it is the average over ancestral graphs (G), that is to say…
Figure 8
Figure 8. Figure 8: Three different realizations of τ (N) . The left realization is under Pdiff with O = 2. The middle realization is under Psame with O = 0. The right realization is under Psame with O = 2. Note that the initial state makes no contribution to O. In particular, under the P…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A conditional coalescent for diploid exchangeable population models given the pedigree

    math.PR 2025-05 conditional novelty 8.0 of 10

    Conditioned on a fixed diploid pedigree, the ancestral process converges to an inhomogeneous (Ψ,c)-coalescent, mixing fixed-time multiple mergers from large families with constant-rate binary mergers.

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