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\begin{document}

\title{On the Fixation or Non-Fixation of Adaptive Inversions}



\author[1]{Brian Charlesworth}%
\author[2]{Thomas Flatt}%
\affil[1]{University of Edinburgh}%
\affil[2]{Fribourg University}%


\vspace{-1em}



  
  \date{May 18, 2024}


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\selectlanguage{english}
\begin{abstract}
Several recent publications have stated that epistatic fitness
interactions cause the fixation of inversions that suppress
recombination among the loci involved. Under this model, however, the
suppression of recombination in an inversion heterozygote creates a form
of heterozygote advantage, which prevents the inversion from becoming
fixed by selection. This process has been explicitly modelled by
previous workers.%
\end{abstract}%



\sloppy


\textbf{On the Fixation or Non-Fixation of Adaptive Inversions}

\textbf{Brian Charlesworth\textsuperscript{1} and Thomas
Flatt\textsuperscript{2}}

\textsuperscript{\textbf{1}} \textbf{Institute of Evolutionary of
Evolutionary Biology, School of Biological Sciences, University of
Edinburgh, Edinburgh UK}

\textsuperscript{\textbf{2}} \textbf{Department of Biology, University
of Fribourg, Fribourg, Switzerland}

\textbf{Corresponding author: Brian Charlesworth}

\textbf{Email:}Brian.Charlesworth@ed.ac.uk

\textbf{Keywords:} balanced polymorphism, epistasis, fitness,
inversions, linkage disequilibrium recombination suppression

\textbf{Abstract}

Several recent publications have stated that epistatic fitness
interactions cause the fixation of inversions that suppress
recombination among the loci involved. Under this model, however, the
suppression of recombination in an inversion heterozygote creates a form
of heterozygote advantage, which prevents the inversion from becoming
fixed by selection. This process has been explicitly modelled by
previous workers.

There is a growing interest in the evolutionary role of inversions and
other types of chromosome rearrangements, with several recent review
papers (e.g., Wellenreuther \& Bernatchez, 2018; Faria et al., 2019;
Kapun \& Flatt, 2019; Huang and Rieseberg, 2020) and a special issue
of\emph{Molecular Ecology} devoted to this topic (Wellenreuther et al.,
2019, and references therein). As these papers have pointed out, there
are multiple ways in which natural selection can act on inversions. In
order to discriminate between different hypotheses, it is necessary to
have a clear understanding of their observable consequences.
Unfortunately, there appears to be a serious misconception about one
process that has been proposed as providing a selective advantage to an
inversion. This involves Dobzhansky's (e.g., Dobzhansky, 1949, 1950,
1951) concept of ``coadaptation'' among polymorphic loci that interact
in their effects on fitness, an idea that traces back to Fisher (1930,
pp.102-104).

In its simplest form, this model invokes two loci, A and B, each
segregating for a pair of alleles (A\textsubscript{1} vs
A\textsubscript{2}, B\textsubscript{1} vs B\textsubscript{2}) in a
diploid, randomly mating population. If the fitness effects of the two
loci are epistatic, in the sense that the fitnesses of the nine possible
diploid genotypes at the two loci deviate from those predicted by
additive combinations of the effects of the alleles at each locus
(Fisher, 1918), it is possible for linkage disequilibrium (LD) to be
maintained in the face of recombination at a stable equilibrium where
both loci are polymorphic, such that the fitter combinations of alleles
are in excess of the frequencies expected by randomly combining alleles
according to their frequencies (Fisher, 1930; Kimura, 1956; Lewontin \&
Kojima, 1960; Karlin, 1975). There is then selection for modifiers that
reduce the rate of recombination between the two loci (Fisher, 1930;
Kimura, 1956; Feldman, 1972). The same principle applies to more general
multilocus systems in randomly mating populations (Charlesworth, 1976;
Zhivotovsky et al., 1994), but not necessarily to partially inbreeding
populations (Charlesworth et al., 1979). In particular, an inversion
that arises on a haplotype that is present in excess of random
expectation (and is thus fitter than average) experiences a selective
advantage if recombination is suppressed in heterozygotes for the
inversion, simply because the inversion maintains its association with
higher than average fitness (Kimura, 1956; Charlesworth \& Charlesworth,
1973).

The misconception is that this process causes the inversion to spread to
fixation, implying that it cannot explain the balanced inversion
polymorphisms that have been the subject of so much recent attention.
This claim appears to have originated in Table 1 of Kirkpatrick \&
Barton (2006), and has been repeated in review papers by Hoffmann \&
Rieseberg (2008), Kapun \& Flatt (2019), and Huang \& Rieseberg (2020).
However, it overlooks the fact that crossing over is suppressed only in
inversion heterozygotes, so that the selective benefits of recombination
suppression confer a heterozygote advantage to the inversion, and hence
prevent its fixation. This scenario was first modelled by Kimura (1956),
and examined in more detail by Charlesworth (1974), who showed that the
exchange of alleles among inverted and standard arrangements by gene
conversion or double crossing over does not prevent the establishment
and maintenance of an inversion polymorphism when there is epistatic
selection. An example of this mechanism for maintaining inversion
polymorphisms is provided by the epistatic interactions among the
different components of segregation distorter systems, and probably
explains the frequent association of such systems with inversions
(Fuller et al., 2020). Direct effects of inversions in causing
segregation distortion should, therefore, not be assumed.

\textbf{References}

Charlesworth, B. (1974). Inversion polymorphism in a two-locus genetic
system. \emph{Genetical Research} , \emph{23,} 259-280.
https://doi.org/10.1017/S0016672300014919

Charlesworth, B. (1976). Recombination modification in a fluctuating
environment. \emph{Genetics} 83:181-195.
https://doi.org/10.1093/genetics/83.1.181

Charlesworth D, Charlesworth B \& Strobeck C. 1979. Selection for
recombination in partially self-fertilizing species.
\emph{Genetics}93:237-244. https://doi.org/10.1093/genetics/93.1.237

Dobzhansky, T. (1949). Observations and experiments on natural selection
in \emph{Drosophila} . \emph{Proceedings of the 8th International
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Dobzhansky, T. (1950). Genetics of natural populations. XIX. Origin of
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Dobzhansky, T. (1951). \emph{Genetics and the Origin of Species} , 3rd
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Faria, R., Johannesson, K., Butlin, R.K., \& Westram, A.M. (2019).
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Feldman, M.W. (1972). Selection for linkage modification: I. Random
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Fisher, R.A. (1918). The correlation between relatives on the
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Karlin, S. (1975). General two-locus selection models: Some objectives,
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Kirkpatrick, M., \& Barton, N. (2006). Chromosome inversions, local
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Wellenreuther, M., \& Bernatchez, L. (2018). Eco-evolutionary genomics
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Wellenreuther, M., M\selectlanguage{ngerman}érot, C., Berdan, E., Bernatchez, L. (2019). Going
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\textbf{Data accessibility:} No new data were generated in this study.

\textbf{Author contributions:} Both authors wrote the paper.

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