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Behind the Science: Approximating the Coalescent Under Facultative Sex

**This post is a part of the series on the 2019 AGA Presidential Symposium – Sex and Asex: the genetics of complex life cycles**


About the author: Matthew Hartfield is a NERC Independent Research Fellow at the Institute of Evolutionary Biology, University of Edinburgh UK. His research explores the interaction between genetic selection and reproduction, using mathematical and computational methods. See his lab website for more details.





I’m excited to see my paper “Approximating the Coalescent Under Facultative Sex” in print for the AGA’s special issue relating to their 2019 symposium “Sex and Asex: The genetics of complex life cycles”. It was an honour to have been invited to speak at their meeting, and especially to see the exciting evolution of sex research that was taking place around the world. It’s also fun to be writing about my paper since it has an unconventional history, spanning several years and different countries. It’s a good example of how scientific ideas can arise in unusual ways.


The initiation: 2014–2016

July 2014: the sun was shining, the football World Cup in Brazil was in full swing, and I arrived in Toronto to start a post-doc with Aneil Agrawal and Stephen Wright. I joined their labs to learn how to use my theoretical expertise to investigate genome sequences. At the time, they were researching the genetics of duckweed, an organism that is predominantly asexual but has shown previous evidence of sexual reproduction. My initial task was to develop mathematical models for interpreting the resulting genetic data.

The coalescent is a mathematical framework for predicting the evolutionary history of genetic samples, by formalising when gene copies reach their common ancestor (‘coalesce’) in the recent past. These models are used to understand what evolutionary forces (for example, mutation, selection or recombination) shape observed diversity. I threw myself head–first into developing coalescent models assuming a facultative sexual reproductive mode. The resulting paper focussed primarily on the effects of rare sex (‘rare’ being defined as the inverse of the population size). It was previously hypothesised that, if sex occurs this infrequently, then genetic variants at the same site within an individual will diverge from one another, due to a lack of gene exchange, so gene copies are essentially isolated from one another. However, this effect was seldom seen in empirical studies; we showed that this signature can be erased due to gene conversion, where one DNA strand is copied to the other one.

A reviewer for the paper suggested that we could apply what is called Möhle’s theorem to the model. This theorem shows how complex coalescent processes can be simplified, by separating out actions that occur in the recent past from long–term effects that are likely to have the biggest effect on genetic diversity. For example, if sex is frequent then when we look at both genetic variants at a site from diploid individuals, then in the recent past these variants will segregate out into individual ancestors. These genes will subsequently coalesce over longer periods of time. We presented some alternative formulations using this theorem and added them to the final paper, but at the time I didn’t think much more about the broader implications of it.


The idea takes shape: 2017

Fast forward to December 2017. I had subsequently moved to Aarhus University in Denmark to continue my post–doctoral research. My colleague Asger Hobolth was teaching a course on coalescent models and asked if I wanted to present a guest lecture. I proposed a lesson built around the ‘structured’ coalescent, which is used to model genetic histories where outcomes differ depending on how genome sequences are sampled. It is also utilised in the facultative sexual framework, since genetic histories differ depending on whether one looks at two gene copies at the same site within an individual, or instead at gene copies sampled from different individuals.

While writing the lecture, I remembered Möhle’s theorem and thought I’d read up on it to see what other examples existed. Rather fortuitously, I came across an article by John Wakeley that was also published for an AGA special issue, which illustrated the use of this theorem in many other evolutionary cases. It set off the science spark in my brain, making me realise that these facultative sex models can be simplified in more general ways than I previously thought. I made a few notes but put them to one side; I was busy enough with my current research projects, and trying to obtain funding for my first group leader position.


A paper is born: 2019

A year and a half passes. I now live in Edinburgh, having obtained my funding and am setting up my first research group. Nearly five years after making the transatlantic move to Toronto, I am crossing the Atlantic again to present these facultative sex coalescent models at the AGA annual symposium. After my talk, I spoke to the organiser Maria Orive, who asked if I had any ideas for a special issue. I mentioned there’s this one idea that was going round my head, but I needed to find a time and opportunity to write it up. There was no going back: I now had to complete this work!

I spent the summer of 2019 writing up these results and shaping them into a paper. After working through the mathematics, I demonstrated that if sex was frequent in facultative sexuals (assuming no selection and a fixed population size), then the facultative sex coalescent process can be well–approximated by simpler models assuming obligate sex. I also looked at cases where sex was extremely rare. While such a simplification was not feasible here, it was possible to shed light on different sampling strategies for disentangling the competing roles of rare sex and gene conversion in shaping neutral diversity.


The paper is published: 2021

Now, the paper is published as part of the Journal of Heredity’s special issue dedicated to the AGA symposium. This tale goes to show that scientific ideas do not necessarily follow a neat trajectory where we plan a project, work on a study, and subsequently write it up for publication. The creative scientific process ebbs and flows in different directions; one of the most exciting parts of my work is navigating these ideas and seeing where they will take me.

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