How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime

arXiv:2609.14549 · q-bio.PE, cond-mat.stat-mech, physics.comp-ph · Submitted 2026-09-13 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: I'm Ines.

Marcus: And I'm Marcus, here with Yuki.

Ines: Today's paper: "How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime".

Marcus: The gist: This work presents an approximation scheme for allele-frequency statistics in strong selection,

Ines: First, who's behind it and why it matters.

Title and authors: Marcus: So, to summarize what we just talked about with "How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime", the paper proposes this approximation scheme. It ensures that both fixation and loss trajectories contribute to the statistics by conditioning on their eventual outcome.

Ines: Essentially, they take a statistic Q(tau) which is usually hard to approximate because it depends on random paths, and they express it as a weighted average of expectations based on whether the allele fixes or gets lost.

Yuki: The way they do this is by transforming those conditional expectations into unconditional ones involving these modified processes, Z(a)(tau), where a represents the outcome.

Marcus: They then use a small noise expansion in powers of sqrt epsilon, where epsilon is one over R, the measure of noise strength. They keep terms up to order epsilon to get an approximation for those processes and then approximate Q(tau) as Q0 plus one over R times Q1.

Ines: The main point for us is that they derive specific approximations for the mean allele frequency, Ey Y tau, and the variance, Vary (Y tau), showing how they approach their exact values in the limit of large R.

Marcus: They show that the deterministic contribution to the mean approaches pi(one)(y) as time goes to infinity, and for variance, V0 tends toward pi(zero)(y) pi(one)(y) as time goes on <ref:2609.14549#pg1>.

Yuki: This means even in the strong selection regime where standard approximations break down, this method gives us a structured way to estimate what the allele frequency distribution looks like over time.

Ines: It's a useful tool because it lets us calculate these coefficients of epsilon zero and epsilon one directly from the equations, which avoids having to rely on complex stochastic simulations for every scenario.

Marcus: They also provide error estimates,, which tells you how good their approximation is by measuring the difference between their result and the exact statistic over a certain time window.

Yuki: So, they are giving us a structured way to analyze allele frequency dynamics that respects the strong selection regime and incorporates both deterministic and fluctuating behavior.

The paper's summary: Ines: Moving on to what this paper suggests as improvements, it focuses on how we can use this framework better. They suggest making the mean allele frequency calculation more robust by incorporating the fluctuation contribution explicitly into that approximation.

Marcus: They propose using Q(tau) Q zero(tau) + one/R Q one(tau) for the long-time mean allele frequency, where Q zero is deterministic and Q one/R is the fluctuation part <ref:2609.14549#pg1>.

Yuki: That improvement lets us get a more accurate prediction of the long-time mean allele frequency, especially when we consider scenarios where R is large.

Ines: They also suggest using their error metrics,(mean) zero and(mean) one to assess how well approximations like Ey Y tau match the exact result when the effective population size equals the census size <ref:2609.14549#pg1>.

Marcus: That's a way to quantify accuracy, not just by looking at one number, but by seeing how close their estimate is to what you would get if you had the true population size.

Yuki: And for variance, they suggest using V(tau) V zero + one/R V one to see if the variance is dominated by the deterministic or fluctuation term over specific timescales <ref:2609.14549#pg1>.

Ines: That lets us estimate Vary (Y tau) and determine whether we should trust the deterministic part or if we need to account for those fluctuations depending on how long you are looking at.

Marcus: Basically, they're giving us tools to assess approximation accuracy based on selection strength, telling you what part of the dynamics is reliable over a certain time frame.

The paper's improvements: Ines: So wrapping up with "How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime", this paper gives us a framework to handle strong selection regimes where standard approximations break down.

Marcus: It provides explicit time-dependent results for general allele frequency statistics under strong selection, which is its main contribution. The method is designed to be computationally efficient for rapid estimation of those time-dependent dynamics without needing stochastic simulations.

Yuki: It’s a valuable tool because it lets us analyze the distribution of allele frequencies in population genetics, especially when we're looking at adapting populations or viral populations where the frequency plays a central role.

Ines: The approach is tailored for regimes where selection is substantially stronger than random genetic drift, characterized by R = 2Nes one and it works for both positive and negative selection <ref:2609.14549#pg1>.

Marcus: One limitation they mention is that this isn't equivalent to an asymptotic expansion in R, but rather a series expansion in the number of occurrences of the noise term.

Yuki: Also, they note that it doesn't include mutations, which would be necessary if we wanted to connect this directly with other areas like the site frequency spectrum.

Ines: So while it's great for studying allele frequency dynamics under strong selection, we still need more work to connect it with mutation effects or other genetic details.

Marcus: The paper offers a closed system of equations for the required quantities, enabling direct numerical computation of the coefficients of epsilon zero and epsilon one in all statistics.

Yuki: It's a method that has potential for broad applications in quantitative genetics and conservation genetics by helping us predict the fate of beneficial alleles in small populations under strong selective pressure.

Ines: This paper, "How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime", provides explicit time-dependent results for general allele frequency statistics under strong selection.

Conclusion: Ines: So we’ve looked at how this paper, "How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime", uses an approximation scheme to make sense of statistics when selection is very strong—that R value is much bigger than one.

Marcus: It’s about making sure we don't just look at one path, like just the deterministic one, but we condition on both fixation and loss trajectories so they both contribute to the final statistic.

Yuki: From a population genetics view, that’s important because it means you can get a better picture of the full range of possibilities for allele frequencies in those strong selection scenarios.

Ines: They use a small noise expansion in powers of the square root of epsilon, where epsilon is one over R—that's basically the measure of noise strength—and they truncate it to get an approximation for the processes Z(a)(tau).

Marcus: And then they approximate that statistic Q(tau) as Q0 plus one over R times Q1, which separates the deterministic part from the fluctuation contribution.

Yuki: That separation is telling because it lets us see if the variance is dominated by the deterministic term or if those fluctuations become more important at certain timescales.

Ines: They show that for the mean allele frequency, Ey Y tau, it approaches a value related to fixation probability as time goes on, which is a key insight for understanding long-term dynamics.

Marcus: And they quantify the error using this metric, which measures how much the approximation Q approx(tau) differs from the exact statistic Q(tau) over a certain time window.

Yuki: So, for someone just listening, it means we can use this method to estimate allele frequency statistics in strong selection without running huge stochastic simulations every single time.

Ines: It’s a computational tool that lets us compute the coefficients of epsilon zero and epsilon one directly from coupled differential equations instead of relying on complex simulations.

Marcus: They also show a relationship between the expectations of Z(zero)(tau) and Z(one)(tau), which is useful because it means knowing one helps you determine the other.

Yuki: That connection is solid, but we still need to see how this applies when we introduce mutation, because that would connect this work more deeply to things like the site frequency spectrum.

Ines: Exactly. This paper gives us a strong starting point for understanding transient dynamics under strong selection, and the next step is building on these results by incorporating those other genetic factors.

Marcus: So, they've given us a solid way to handle those complex trajectories in strong selection regimes using this approximation scheme.

David Waxman

Centre for Computational Systems Biology · Institute of Science and Technology for Brain Inspired Intelligence · Fudan University

q-bio.PE, cond-mat.stat-mech, physics.comp-ph

Submitted: 2026-09-13

Updated: 2026-10-14

Comments: 32 pages, 3 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: The gist: This work presents an approximation scheme for allele-frequency statistics in strong selection, ensuring that both fixation and loss trajectories contribute to the statistics by

Key concepts

Strong Selection Regime (R >> 1): This regime occurs when the product of effective population size and selection strength ($R = 2N_e|s|$) is much greater than one. It means natural selection is much stronger than random genetic drift, allowing for the analysis of allele frequency dynamics even with small selection coefficients in realistic population sizes.
Absorbing Classes (Fixation or Loss): In this model, every possible path an allele frequency can take eventually leads to one of two stable outcomes: it either becomes fixed (reaches 100% frequency) or is lost entirely. The analysis must account for both possibilities because ignoring the loss trajectories introduces significant errors into statistical calculations.
Small Noise Expansion: Because strong selection makes the system nearly deterministic, a small noise expansion in powers of $\sqrt{\epsilon}$ (where $\epsilon = 1/R$) is used. This allows researchers to approximate complex statistics by separating the result into a deterministic part ($Q_0$) and a fluctuation part ($1/R Q_1$), capturing transient behaviors accurately.
Conditioning on Outcome: To ensure both fixation and loss trajectories contribute correctly to the final statistic, the authors condition expectations separately on these eventual outcomes. This yields an outcome-conditioned representation of the statistic, which is essential for obtaining accurate results when selection is strong.

Summary

The gist: This work presents an approximation scheme for allele-frequency statistics in strong selection, ensuring that both fixation and loss trajectories contribute to the statistics by conditioning on their eventual outcome.

Biological Model

The study considers a Wright-Fisher model for a finite population of diploid hermaphroditic organisms where generations are discrete, neglecting mutation over the timescales of interest. The dynamics are described under the diffusion approximation, where the frequency of allele A at time t satisfies the stochastic differential equation dXt = sXt(1 − Xt) dt + sXt(1 − Xt)2Ne dWt. This equation captures both deterministic changes due to natural selection and stochastic changes due to random genetic drift.

Representation of Allele Frequency Statistics

The paper addresses the issue that neglecting trajectories leading to loss can lead to large errors in statistics, as a direct approximation often only considers the deterministic trajectory. To ensure both types of trajectories contribute, the statistic Q(τ) is expressed as a weighted average of conditional expectations based on fixation and loss: Q(τ) = Ey [q(Yτ) loss] · π (0)(y) + Ey [q(Yτ) f ix] · π (1)(y). This representation is achieved by recasting conditioned expectations of the original random frequency Yτ as unconditional expectations involving modified processes Z(a)τ, where a=0 corresponds to loss and a=1 corresponds to fixation.

Small Noise Approximation

The approximation relies on carrying out a small noise expansion in powers of √ε, where ε = 1/R is the measure of noise strength. The random frequency Z(a)τ is expanded as Z(a)τ = Z(a)τ,0 + √εZ(a)τ,1 + εZ(a)τ,2 + O(ε3/2). By truncating the series at order ε to obtain the approximation Z(a)τ ≃ Z(a)τ,0 + √εZ(a)τ,1 + εZ(a)τ,2. The basic statistic Q(τ) is then approximated as Q(τ) ≃ Q0 (the deterministic contribution to the mean and variance) + 1/R Q1 (the fluctuation contribution).

Results for Mean Frequency and Variance

For the mean allele frequency, Ey [Yτ] ≃ ⟨Zτ,0⟩π + 1/R ⟨Mτ⟩π. The deterministic contribution, which is the weighted average of the two deterministic trajectories Z(0)τ,0 and Z(1)τ,0, approaches the fixation probability π (1)(y) as τ → ∞. For the variance, Vary (Yτ), it is approximated by Vary (Yτ) ≃ V0 + 1/R V1. The term V0 represents the contribution from deterministic trajectories, which tends to π(0)(y)π(1)(y) as τ → ∞.

Error Estimates and Accuracy

The error between the exact statistic Q(τ) and its approximation Qapprox(τ) is quantified by the metric ∆ = R κ 0 Qapprox(τ) − Q(τ) dτ / (R κ 0 Q(τ) dτ). The cut-off time κ is chosen such that the active window captures 99% of the total deviation Dtotal, ensuring that the integral in ∆ reflects only the main region where Q(τ) varies. Numerical comparisons show that while deterministic approximations are not too far from exact results, including fluctuations, Q0 + 1/R Q1, yields a much closer approximation to the exact result.

Relation between Trajectories

A crucial relationship established is that for an absolutely integrable function h(z), Ey [h(Z(0)τ)] = E1−y [h(1 − Z(1)τ)]. This means knowledge of expectations of Z(1)τ is sufficient to determine expectations of Z(0)τ.

The approximation developed effectively captures non-trivial transient phenomena displayed by some statistics, such as in Figure 2 and Figure 3. This approach offers a computationally efficient tool for analyzing the strong selection regime, allowing rapid estimation of time-dependent allele frequency statistics without resorting to stochastic simulations. The method is well-suited for analyzing adapting populations and viral populations where the distribution of allele frequencies plays a central role.

The work does not include mutations, which would be necessary to connect with other areas such as the site frequency spectrum. The approach is tailored for the regime where selection is substantially stronger than random genetic drift, characterized by R = 2Nes >> 1. The core calculations rely on conditioning to force both fixation and loss trajectories to contribute, which is essential when ε is finite. This formulation provides a closed system of equations for the required quantities, enabling direct numerical computation of the coefficients of ε 0 and ε 1 in all statistics. The results are applicable to both positive and negative selection regimes. The computational requirements are modest, involving only the numerical solution of a small set of coupled differential equations. This approach has the potential for broad applications in quantitative genetics and evolutionary biology. The ability to efficiently compute transient dynamics is valuable for inferences made from time-series allele frequency data. This work serves as a component of larger inferential frameworks where many evaluations of the forward model are required. The method can be extended to inform conservation genetics by predicting the fate of beneficial alleles in small populations under strong selective pressure. This approach is related to studies on strong selection, but differs from analyses focusing on equilibrium or steady state distributions. The results are not directly comparable to analyses considering a constant population size and mutation. The study provides explicit time-dependent results for general allele frequency statistics under strong selection, which is its primary contribution. The approximation scheme is not equivalent to an asymptotic expansion in R but rather a series expansion in the number of occurrences of the noise term. This approach avoids problematic boundary layer behaviors observed in asymptotic expansions. The results are applicable when initial frequencies and selection coefficients have definite values y and s, respectively. The method is a valuable tool for analyzing the distribution of allele frequencies in population genetics.

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How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime David Waxman Centre for Computational Systems Biology Institute of Science and Technology for Brain Inspired Intelligence Fudan University, 220 Handan Road, Shanghai 200433, PRC Abstract In many biologically important contexts selection is strong, sometimes extremely strong, relative to random genetic drift. Working under the diffusion approximation, we define R = 2Nes, where Ne is the effective population size and s is the selection coefficient associated with a focal allele. Strong selection corresponds to R ≫ 1 and can occur for relatively modest, yet typical parameter values.

Improvements for AI systems

A model that accurately predicts long-time mean allele frequency by incorporating fluctuation contributions: The improved system can calculate the long-time mean allele frequency by using the approximation Q(τ) ≃ Q0(τ) + 1/R Q1(τ) where Q0(τ) is the deterministic contribution and Q1(τ)/R is the fluctuation contribution.
A tool for assessing approximation accuracy in strong selection regimes: The system can calculate error metrics such as ∆ (mean) 0 and ∆ (mean) 1 to quantify how well approximations like ⟨Zτ,0⟩π + 1/R⟨Mτ⟩π match the exact result that applies when the effective population size equals the census size.
A predictor of variance behavior based on selection strength: The system can estimate Vary (Yτ) by using the approximation Vary (Yτ) ≃ V0 + 1/R V1 to determine if the variance is dominated by the deterministic term or fluctuation term over specific timescales.
A mechanism for robust trajectory analysis irrespective of selection sign: The system can analyze allele frequency dynamics for both positive and negative selection because Z(0)τ and Z(1)τ apply, unchanged, for both negative and positive selection in the statistic Q(τ).

Abstract

We determine statistics of allele-frequency trajectories across replicate populations. Defining R = 2N es where N e is the effective population size and s is the additive selection coefficient of a focal allele, we focus on strong selection which corresponds to R 1. Strong selection does not require large selection coefficients: in populations of realistic size, it can readily arise from small selection coefficients. Our analysis concerns a biallelic locus over timescales on which mutation can be neglected. Every allele-frequency trajectory belongs to one of two absorbing classes: fixation or loss. When R is large, individual allele-frequency trajectories can appear nearly deterministic. This does not imply that an approximation of allele-frequency statistics can be based on a single deterministic trajectory. For a beneficial allele, the deterministic trajectory approaches fixation, yet a non-negligible fraction of stochastic trajectories may be lost. Ignoring trajectories that achieve loss can produce severely erroneous statistics. We address this by conditioning separately on the eventual loss and eventual fixation of the focal allele. This yields an outcome-conditioned representation of a statistic. In the strong-selection regime, we apply a small-noise expansion that reasonably captures transient behaviours of statistics while recovering their exact long-time limits. We illustrate results with numerical examples for the mean allele frequency and its variance. The biological interpretation of this work is that, even under strong selection, both trajectory types are of importance. We provide a simple framework for calculating time-dependent allele-frequency statistics relevant to experimental evolution, longitudinal population-genomic data, and the dynamics of alleles driving selective sweeps.