The tool, called EWF 2.0, was developed by Jaromir Sant of the University of Turin and the Collegio Carlo Alberto, together with Paul A. Jenkins, Jere Koskela and Dario Spanò of the University of Warwick and Newcastle University. It extends an earlier line of work known as the EWF algorithms, which allow researchers to draw exact samples from the Wright–Fisher diffusion — that is, to generate simulated allele frequency trajectories whose statistical properties match the theoretical model perfectly, without the approximation errors that plague standard numerical schemes. The crucial innovation in version 2.0 is that the population size is no longer required to be constant. Instead, the demographic history can change over time, which mathematically turns the drift coefficient of the diffusion into a time-inhomogeneous one: a process whose random dynamics depend explicitly on when you look at it.

To appreciate why this matters, it helps to understand what exact simulation means in this context. The Wright–Fisher diffusion is a stochastic differential equation, and the most common way to simulate such equations is to discretise time into small steps and approximate the motion between them. These Euler-type schemes are easy to implement but introduce bias: the simulated trajectories are not true draws from the model, and the bias can be difficult to quantify, particularly for quantities that depend on the fine structure of the path rather than just its endpoints. Exact algorithms sidestep this problem entirely. Using clever probabilistic constructions — often based on rejection sampling ideas in which a candidate path is proposed and then accepted or rejected according to criteria that guarantee the output has exactly the right distribution — they produce trajectories that are statistically indistinguishable from the real thing, whatever the intended application.

The difficulty with time-varying demography is that these rejection-based constructions were designed for diffusions with fixed coefficients. When the population size changes, the drift term of the allele frequency dynamics acquires a time-dependent factor, and the bounds and transformations that make the earlier algorithms valid no longer hold. Previous versions of EWF could handle time-varying mutation and selection in certain settings, but the demographic component had to remain static. Since virtually every dataset from natural populations — humans, livestock, pathogens, crop species — carries the imprint of a fluctuating census history, this limitation was a serious obstacle for anyone hoping to use exact simulation in realistic inference pipelines.

EWF 2.0 removes that obstacle. According to the authors, the algorithm accommodates a specified demographic history alongside mutation parameters, a selection function and a set of sampling times, and from those inputs it generates exact draws from the law of the corresponding Wright–Fisher diffusion. Perhaps more strikingly, it can also simulate diffusion bridges: trajectories that are conditioned to start at one allele frequency and end at another. Bridge simulation is a notoriously hard problem, because the conditioning fundamentally alters the dynamics of the process, and the authors note that existing methods cannot handle time-varying mutation and selection rates in this setting. Bridges are precisely what many statistical applications need, since inferring past evolutionary forces often amounts to asking which hidden paths most plausibly connect observed genetic data points collected at different times.

The applications of such a tool extend across modern population genetics. Ancient DNA studies, for example, now routinely provide snapshots of allele frequencies at multiple time points separated by hundreds or thousands of generations. Reconstructing the trajectories between those snapshots — and quantifying the uncertainty around them — requires simulating bridges under a model that includes the demographic history inferred from archaeological and genomic evidence. Similarly, forward-time simulations used to benchmark inference methods, to test the power of genome-wide association designs, or to explore how selection responds to environmental change all benefit from trajectories that are exact rather than approximate. Any bias in the simulator propagates into the conclusions drawn from it, so the guarantee of exactness is not a mathematical luxury but a practical safeguard.

The authors validated their implementation using distributional tests, including Kolmogorov–Smirnov tests and quantile-quantile plots, which compare the empirical distribution of simulated quantities against theoretical expectations. Agreement in these tests provides strong evidence that the algorithm samples from the intended law, since even subtle biases in a simulation scheme tend to show up as systematic deviations in such comparisons. Equally important from a practical standpoint is the performance result: despite its greater generality, EWF 2.0 retains the same runtime as previous versions of the algorithm. Adding the flexibility of time-varying demography therefore comes at no computational penalty, which the authors highlight as ensuring efficiency and scalability for large-scale simulation studies.

The software is written up as an open-access methods paper and the code is freely available on GitHub, lowering the barrier for other groups to adopt it. In a field where simulation underpins everything from method validation to likelihood-free inference, a tool that combines exactness, demographic realism and competitive speed is likely to find rapid uptake. The work was supported by funding from the Italian Ministry of University and Research through a PRIN 2022 grant financed by the European Union’s Next Generation EU programme, reflecting the increasingly international and computationally oriented character of statistical genetics research.

Looking at the broader picture, the advance illustrates a familiar pattern in computational science: the gap between the models researchers want to use and the models their tools can actually handle. The Wright–Fisher diffusion with time-varying demography has been part of the theoretical literature for decades, and approximate simulators have long been able to produce plausible-looking trajectories under it. What was missing was the guarantee — the assurance that a simulated history is a genuine draw from the specified model rather than an artefact of numerical approximation. By extending exact simulation into the time-inhomogeneous regime, EWF 2.0 brings a foundational class of population genetic models fully within reach of rigorous simulation-based inference, and it does so without asking practitioners to sacrifice speed. For studies probing how genes rose and fell across the shifting demographic tides of the past, that combination may prove hard to resist.

Subject of Research: Exact simulation of allele frequency trajectories under the Wright–Fisher diffusion with time-varying demography

Article Title: EWF 2.0: exact sampling of allele trajectories using the Wright–Fisher diffusion with time-varying demography

Article References: Sant, J., Jenkins, P. A., Koskela, J., & Spanò, D. (2026). EWF 2.0: exact sampling of allele trajectories using the Wright–Fisher diffusion with time-varying demography. BMC Bioinformatics. https://doi.org/10.1186/s12859-026-06576-z

Image Credits: AI Generated

DOI: 10.1186/s12859-026-06576-z

Keywords: Wright–Fisher diffusion, exact simulation, population genetics, allele frequency, demography, genetic drift, selection, mutation, diffusion bridge, computational biology, stochastic modelling, BMC Bioinformatics