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\begin{document}

\title{The discernible and hidden effects of clonality on the genotypic and
genetic states of populations: improving our estimation of clonal rates}



\author[1]{Solenn Stoeckel}%
\author[2]{Barbara Porro}%
\author[3]{Sophie Arnaud-Haond}%
\affil[1]{INRA}%
\affil[2]{Université de Nice Sophia Antipolis}%
\affil[3]{IFREMER Centre de Brest}%


\vspace{-1em}



  
  \date{March 17, 2025}


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\selectlanguage{english}
\begin{abstract}
Partial clonality is widespread across the tree of life, but most
population genetics models are designed for exclusively clonal or sexual
organisms. This gap hampers our understanding of the influence of
clonality on evolutionary trajectories and the interpretation of
population genetics data. We performed forward simulations of diploid
populations at increasing rates of clonality (c), analysed their
relationships with genotypic (clonal richness, R, and distribution of
clonal sizes, Pareto \selectlanguage{greek}β\selectlanguage{english}) and genetic (FIS and linkage disequilibrium)
indices, and tested predictions of c from population genetics data
through supervised machine learning. Two complementary behaviours
emerged from the probability distributions of genotypic and genetic
indices with increasing c. While the impact of c on R and Pareto \selectlanguage{greek}β \selectlanguage{english}was
easily described by simple mathematical equations, its effects on
genetic indices were noticeable only at the highest levels
(c\textgreater{}0.95). Consequently, genotypic indices allowed reliable
estimates of c, while genetic descriptors led to poorer performances
when c\textless{}0.95. These results provide clear baseline expectations
for genotypic and genetic diversity and dynamics under partial
clonality. Worryingly, however, the use of realistic sample sizes to
acquire empirical data systematically led to gross underestimates (often
of one to two orders of magnitude) of c, suggesting that many
interpretations hitherto proposed in the literature, mostly based on
genotypic richness, should be reappraised. We propose future avenues to
derive realistic confidence intervals for c and show that, although
still approximate, a supervised learning method would greatly improve
the estimation of c from population genetics data.%
\end{abstract}%



\sloppy


\textbf{Introduction}

Clonality occurs across the entire tree of life, including all kingdoms
of Eukaryota (Avise \& Nicholson, 2008; Sch\selectlanguage{ngerman}ön, Van Dijk, \& Martens,
2009; Tibayrenc, Avise, \& Ayala, 2015). Most, if not all, clonal
eukaryotic species alternate between clonal and sexual reproduction at
the population scale over a few generations (see Box 1 for the
definitions used in this work). This mode of reproduction, called
partial clonality (PC), has been reported in a broad range of species,
especially primary producers, ecosystem engineers, pathogens and
invasive species. Their evolutionary trajectories may thus have major
consequences for ecosystem functioning and for human health and
development (Schön et al., 2009; Yu, Roiloa, \& Alpert, 2016).
Challenging environments and the edges of species ranges may also favour
populations able to reproduce using PC and putatively populations with
higher rates of clonality, emphasising the importance of understanding
evolutionary trajectories of PC species when dealing with global changes
(Barrett, 2015, 2016; Tibayrenc \& Ayala, 2012; Yu et al., 2016).

\textbf{Box 1: relevant definitions and concepts}

\textbf{Clonal reproduction}

A precise definition of clonal reproduction has been historically used
in population genetics, corresponding to ``\emph{an individual produces
new individuals that are genetically identical to the ancestor at all
loci in the genome, except at those sites that have experienced somatic
mutations} '' (as defined in De Meeûs, Prugnolle, \& Agnew, 2007, see
also Marshall \& Weir, 1979). This definition implicitly includes
reproduction through agametic tissues and apomictic parthenogenesis.

\textbf{Partial clonality}

The term partial clonality (PC) refers to the reproductive system of
species undergoing both clonal and sexual reproduction through selfing
or outcrossing, or both.

\textbf{Comment on the use of the terms clonality and asexuality:}

In this work, we favoured the use of \emph{clonality} rather than
asexuality due to its etymology. Clone comes from the ancient Greek
\selectlanguage{greek}κλῶνος, \selectlanguage{english}referring to a regrowth, a root shoot or, lately, a growing
organic extension with new vigour (Liddell, Scott, Jones, \& McKenzie,
1940). Paradoxically, clonality, the initial definition of which
mentioned plant regrowth, seems to have been used more by biologists
working on animals, while asexuality is more common in the plant
literature.

The term asexuality is currently used beyond its initial definitions,
applying to all uniparental reproduction with incomplete meiosis
schemes, including those occurring beyond prophase I and resulting in
higher levels of recombination (Nougu\selectlanguage{ngerman}é et al., 2015). In addition, a
societal definition emerged last year: ``\emph{a}
{[}human{]}\emph{sexual orientation} {[}\ldots{}{]} \emph{not valuing
sex or sexual attraction to others enough to pursue it} '' (Decker, 2015
in Bibr, 2017).

\textbf{Rate of clonality}

In line with the definition summarised above, the rate of clonality
(here, \emph{c} ) corresponds to ``\ldots{}\emph{the probability of
clonality} versus \emph{sexual reproduction through selfing or
outcrossing} '' (Marshall \& Weir, 1979), which in this article
corresponds to the ratio of the effective number of descendants produced
by clonality to the total effective number of descendants produced in a
population (see also Balloux, Lehmann, \& de Meeus, 2003; Berg \&
Lascoux, 2000).

When inferred by genotypic and genetic indices in a population sample,
this rate is a \emph{proxy} for the idealised number of descendants
produced by clonality relative to the total idealised number of
descendants produced in the population that would result in the same
genotypic and genetic effects.

Despite the prevalence of PC and the potential extent of its
consequences at the ecosystem level, the consequences of PC for the
evolution of species and the ecological dynamics of their natural
populations have been subject to little in-depth theoretical or
empirical development (Yonezawa, Ishii, \& Nagamine, 2004). This lack of
development makes a substantial number of studies on partially clonal
species confusing when analysing population genetics data and
interpreting them in terms of demographic and evolutionary dynamics
(Avise, 2015; Fehrer, 2010; Yu et al., 2016). Nevertheless, the effects
of PC are likely to be extremely important at all spatial and temporal
scales. For example, evolutionarily speaking, the ability of a given
genotype to persist across generations adds a new target for natural
selection, namely, the genotype (Ayala, 1998).

Three main knowledge gaps are related to PC: diagnosing it in species
where its occurrence is not obviously inferred by classical naturalistic
observations (e.g\emph{.} , human pathogens, in contrast to rhizomatic
clonal plants); quantifying its extent once a given species is
determined to be partially clonal; and understanding its influence on
the ecological and evolutionary trajectories of partially clonal species
by investigating their population genetics. These gaps have been only
partly filled during the past 30 years. The use of molecular markers in
a population genetics framework paved the way for easier detection of PC
(De Meeûs, Lehmann, \& Balloux, 2006; Halkett, Simon, \& Balloux, 2005;
Tibayrenc, Kjellberg, \& Ayala, 1990) through the discrimination of
clonal lineages and detailed analysis of the genotypic and genetic
compositions of species suspected of having PC (Arnaud-Haond et al.,
2005; Bailleul, Stoeckel, \& Arnaud-Haond, 2016; Tibayrenc et al.,
1990). However, conditions allowing (or not allowing) the detection of
PC and its consequences for the trajectories of natural populations over
different time scales are likely important yet still poorly understood
(Avise, 2015; Dia et al., 2014; Fehrer, 2010; Yu et al., 2016). We still
face difficulties in inferring the rate of clonal (denoted \emph{c} )
versus sexual (1-\emph{c} ) reproduction or an approximate but
consistent\emph{proxy} for it (\emph{i.e.} , the ``level of
clonality''). These difficulties prevent access to the empirical
information necessary to compare the ecological dynamics and
evolutionary trajectories of partially clonal populations living in
different environments (McMahon et al., 2017). To understand the effect
of PC on the fate of natural populations and species, the value of
\emph{c} should first be estimated.

The rate of clonality in natural populations may be estimated by
tracking clonal spreads or determining groups of clones. In plants,
groups of clones have sometimes been identified at local scales through
extremely time-consuming and tedious mark-recapture studies of rhizomes
(Eckert, 2002; Marbà \& Duarte, 1998). However, using this method on
large spatial scales and for most species exhibiting PC through
fragmentation or multiplication at microscopic stages is unrealistic.
Therefore, tracking clonal spread or determining groups of clones
through population genetics is the only solution for the vast majority
of species. Unfortunately, although population genetics studies can
illuminate the occurrence of PC in nature, no method has been developed
thus far to reliably infer (or at least estimate) such potentially
crucial parameters in natural populations using indices gathered through
a classical one-time step sampling strategy. Two recently developed
methods allow the quantification of rates of clonality in populations
genotyped at two time steps. However, they require sampling the
population twice at an interval of at least one generation and, more
importantly, a comprehensive knowledge of major life history traits,
such as generation time, which are seldom available except for
well-known macroscopic species for which extensive field data have been
collected (Ali et al., 2016; Becheler et al., 2017).

Most empirical studies thus infer the importance of clonal reproduction
in populations using a one-time step sampling strategy to compute the
ratio of genotypes to the number of sampling units (genotypic
richness,\(Pd=G/N\)) as an estimate of clonal richness.
Genotypic richness is often implicitly assumed to have a linear
relationship with the rate of sexual reproduction \(1-c\) and
to be comparable among natural populations submitted to the same
sampling strategy. Theoretical studies have shown the strong influence
of high clonality rates (c\textgreater{}0.95) alone on parameters such
as\emph{F} \textsubscript{IS} and linkage disequilibrium (\emph{LD} )
(Balloux et al., 2003; De Meeûs et al., 2006; Navascués, Stoeckel, \&
Mariette, 2010) but no noticeable departure from expectations under
purely sexual reproduction at lower rates of clonality. However, more
recent mathematical developments have shown that the distribution
of\emph{F} \textsubscript{IS} is wider at high clonality rates but is
actually affected at all clonality rates (Stoeckel \& Masson, 2014),
depending on the strength of departure from equilibrium (Reichel,
Masson, Malrieu, Arnaud-Haond, \& Stoeckel, 2016).

This research led to a present-day paradox in the literature on PC. Many
populations exhibit average or elevated genotypic diversity, leading
several authors to conclude that these populations exhibit a high
incidence of sexual reproduction, whereas in the same studies,
consistent departure from Hardy-Weinberg equilibrium (HWE), when
reported (which is much rarer), would instead have led them to conclude
that the populations exhibit a negligible occurrence of sexual
recombination versus clonal reproduction (e.g., Orantes, Zhang, Mian, \&
Michel, 2012; Villate, Esmenjaud, Van Helden, Stoeckel, \& Plantard,
2010). This paradox is seldom obvious because \emph{F}
\textsubscript{IS}values are often not reported or, if reported, are not
interpreted in relation to clonality. In any case, part of this paradox
may lie in the pervasive effect of sampling on the estimation of
genotypic richness (Arnaud-Haond, Duarte, Alberto, \& Serrão, 2007;
Gorospe, Donahue, \& Karl, 2015). These two studies demonstrated this
worrying effect by using two empirical datasets (of seagrasses and
corals) where the true rates of clonality were unknown; assessing the
order of magnitude of these rates thus requires further investigation.

A roadmap to fill the gaps

Given the current state of knowledge, the characterisation of the
genotypic (based on groups of clones) and genetic (based on allele and
genotype frequencies at loci) compositions of populations is both the
target and the \emph{proxy} of population genetics studies aiming to
understand the influence of clonal reproduction on the dynamics and
evolution of natural populations. Reconciling the effects of PC on both
the genotypic and genetic compositions of populations in a robust
theoretical framework is thus necessary to illuminate the concomitant
changes in their respective estimators depending on the rate of
clonality.

Here, we propose a simulation-based exploratory approach both to enhance
our understanding of the consequences of clonality and to improve our
ability to reliably assess its rate within natural populations. We aim
to provide the first exploration of the effect of increasing \emph{c} on
the genotypic and genetic compositions of populations to provide
baseline expectations for the composition of natural populations
depending on the extent of clonality. We used comprehensive forward
individual-based simulations to obtain the theoretical distribution of
genotypic (genotypic richness and size distribution of lineages) and
genetic (departure from HWE and LD) parameters describing the population
composition at increasing rates of clonality from 0 to 1, including all
populations with some clonal reproduction, hereafter denoted PC for
brevity, ranging from partial (0\textless{}c\textless{}1) to strict
clonality (c=1), and populations with solely sexual reproduction (c=0),
hereafter denoted sexual populations. We explored the temporal evolution
of these populations along trajectories towards equilibrium and under
various levels of genetic drift (population sizes spanning three orders
of magnitude). To move from insights about the expected effects of PC on
natural populations towards more reproducible and formalised arguments,
we assessed the signature of PC in the genotypic and genetic index
distributions using a classical and robust Bayesian supervised learning
method. This method allowed the selection of descriptors that were more
clearly affected to in turn develop sound estimates of the extent of
clonality. Finally, we tested the robustness of the method in examining
the influence of sample size on the accuracy of estimates and proposed
further improvements based on realistic sample sizes.

\textbf{Materials and methods}

Approach

PC is empirically known to affect genotypic and genetic descriptors
commonly used in population genetics studies (Halkett et al., 2005): 1)
the number of different genotypes per population, as characterised by
the genotypic richness indices \emph{R} and Pareto
\(\beta\)(Arnaud-Haond et al., 2007), and two genetic indices,
namely, 2) the inbreeding coefficient \emph{F} \textsubscript{IS} and
its moments (Balloux et al., 2003; Stoeckel \& Masson, 2014) and 3) the
LD index (Navascués et al., 2010). To date, no analytical formalisation
has been developed to predict the theoretical probability distributions
of these descriptors under varying rates of clonality. We thus used
simulations to i) synthesise the effects of varying rates of clonality
on the ranges and dynamics of these genotypic and genetic descriptors,
ii) assess whether these descriptors actually provide the ability to
discriminate and quantify rates of clonality using a classic supervised
learning method, and iii) determine which descriptors best account for
specific ranges of rates of clonality, with the aim of providing
recommendations for future analyses and interpretations.

Simulations

Theoretical results were obtained using forward individual-based
simulations run over 10\textsuperscript{4} non-overlapping generations
to reach quasi-stationary distributions of both genotypic and genetic
diversity. In the initial generations, alleles at all neutral loci were
randomly drawn from a uniform distribution (\emph{i.e.} , maximum
genetic diversity merged at random within individuals). In these
simulations, all diploid individuals lived in constant finite-sized
populations.

Each population produced the next generation using clonal or panmictic
sexual reproduction following a fixed rate of clonality. All
hermaphrodite individuals in each generation had identical probabilities
of being parents, in both clonal and sexual events. The probability of
an individual parent being drawn i) to birth a clonal descendant and ii)
to sire half a sexual descendant followed a Bernoulli scheme, with
respective probabilities \(P(clonal\ parent)=\frac{c}{N}\) and\(P(sexual\ parent)=\frac{1-c}{2N}\), where
\emph{N} is the population size. In clonal reproduction, new independent
individuals were produced as full genetic copies of their only parent,
with somatic mutations occurring at a fixed rate of
10\textsuperscript{-6} mutations per generation per locus. This choice
was driven by the high end of estimates of DNA polymerase mutations
ranging between 10\textsuperscript{-8} and 10\textsuperscript{-9}
bp/generation (McCulloch \& Kunkel, 2008), which for a locus of 100 to
1000 base pairs would imply a mutation rate of 10\textsuperscript{-5} to
10\textsuperscript{-7.} In panmictic sexual reproduction, new
independent individuals descended from two parents chosen at random
within the previous generation, from which the individuals inherited
half their genomes and mutated at a rate of
10\textsuperscript{-3}mutations per generation per locus, following
estimated mutation rates for sexual eukaryotes ranging from
10\textsuperscript{-4} to 10\textsuperscript{-7} and
10\textsuperscript{-2} to 10\textsuperscript{-5} across generations for
single-nucleotide polymorphisms (SNPs) and microsatellites, respectively
(Payseur \& Cutter, 2006). Genomes were coded as 100 independent loci.
Alleles mutated following a K-allele mutation (KAM) model (Putman \&
Carbone, 2014; Weir \& Cockerham, 1984), which has the advantage of
simulating the behaviour of both microsatellites and SNPs well and which
best approximates the ``disturbing factor of gene frequencies'' (in the
sense of Wright, 1931) in finite-sized populations. Mutating alleles in
both clonal and sexual reproduction were drawn at random from the
respective pools of clonal and sexual offspring. In simulations, the
clonality rate, genetic drift and mutation rate were applied
homogeneously across generations and loci.

To understand the effect of clonality on population genetics indices, we
ran simulations with varying population sizes (\emph{N} =
10\textsuperscript{3}, 10\textsuperscript{4} and 10\textsuperscript{5},
to be studied with arbitrarily fixed mutation rates), rates of clonality
(\emph{c} =0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99 and 1),
and numbers of generations elapsed since the initial population
(generations=10, 100, 500, 1000, 5000, and 10000). At each time step,
indices were examined for the whole population (all \emph{N} genotypes
considered) as well as for subsamples without replacement of different
sizes (\emph{n} =10, 20, 30, 50, 100, 200, 500, and 1000 and when
population sizes allowed, \emph{n} =5000, 10000, 50000, and 100000).

Each scenario was run 100 times and characterised by a set of parameters
(\emph{N} , \emph{c} ). When subsampling populations, we performed 10
independent resamplings of each generation and sample size, resulting in
1000 independent data points per set of parameters for each sample size.

Genotypic and genetic descriptors

To account for the genotypic composition and genetic state of
populations, we computed two indices describing the number and
distribution of genotypes (genotypic richness \(R\) and
slope of the size distribution of lineages Pareto \selectlanguage{greek}β,
\selectlanguage{english}Arnaud-Haond et al., 2007) and two genetic descriptors referring to
intra-individual genetic variation (as the first four moments of the
inbreeding coefficient\(F_{\text{IS}}\) distribution; Stoeckel \&
Masson, 2014) and LD (as the summarised unbiased multi-locus LD
\(\overset{\overline{}}{r_{D}}\); Agapow \& Burt, 2001).

Genotypic richness

The \emph{R} index of clonal diversity (Dorken \& Eckert, 2001) was
defined as follows:

\begin{equation}
R=(G-1)/(N-1)\nonumber \\
 \end{equation}

where \emph{G} is the number of distinct genotypes (genets) and
\emph{N}is the number of genotyped individuals.

Size distribution of lineages

The parameter Pareto \(\beta\) describes the slope of the
power-law inverse cumulative distribution of the size of lineages
(Arnaud-Haond et al., 2007):

\begin{equation}
N_{\geq X}=a.X^{-\beta}\nonumber \\
 \end{equation}

where \(N_{\geq X}\) is the number of sampled ramets belonging to
genets containing \emph{X} or more ramets in the sample of the
population studied, and the parameters \emph{a} and \selectlanguage{greek}β \selectlanguage{english}are fitted by
regression analysis.

Genetic variance apportionment

The Wright (1921, 1969) inbreeding coefficient\emph{F}
\textsubscript{IS} accounts for intra-individual genetic variation as a
departure from Hardy-Weinberg assumptions of the genotyped populations.
We computed one \emph{F} \textsubscript{IS} value per population and per
locus as\(F_{\text{IS}_{l}}=\frac{Q_{w,l}-Q_{b,l}}{1-Q_{b,l}}\), where\(Q_{w,l}\) is the population
probability that two homologous alleles taken within individuals are
identical at locus \emph{l} , and\(Q_{b,l}\) is the population
probability that two homologous alleles taken between different
individuals are identical at locus \emph{l} . We computed the first four
moments of the empirical\emph{F} \textsubscript{IS} distribution
obtained from the 10000 independent \emph{F} \textsubscript{IS} values
per scenario (100 independent loci x 100 replicated simulations),
respectively noted\(\text{Mean}\left[F_{\text{IS}}\right],Var\left[F_{\text{IS}}\right],Skew\left[F_{\text{IS}}\right]\ \)and\(\text{Kurt}\left[F_{\text{IS}}\right]\).

Linkage disequilibrium between loci

LD was studied using \({\overset{\overline{}}{r}}_{d}\) (Agapow \& Burt, 2001). The mean
correlation coefficient (\emph{r} ) of genetic distance (\emph{d} )
between unordered alleles at \(n\) loci ranged from 0 to 1.
This metric has the advantage of limiting the dependency of the
correlation coefficient on the number of alleles and loci and is well
suited to studies of partially clonal populations.

\begin{equation}
{\overset{\overline{}}{r}}_{d}=\frac{V_{D}-\sum_{j=1}^{j=n}\text{var}_{j}}{2\sum_{j=1}^{j=n}{\sum_{k>j}^{k=n}\sqrt{\text{var}_{j}.\text{var}_{k}}}}\nonumber \\
 \end{equation}

with\(V_{D}=\frac{\sum_{a,b\neq a}^{\nu}D_{a,b}^{2}-\frac{\left(\sum D_{a,b}\right)^{2}}{\nu}}{\nu}\)and\(\text{var}_{j}=\frac{\sum{d^{2}-\frac{\left(\sum d\right)^{2}}{\nu}}}{\nu}\)

where \(D\) is the number of loci at which two individuals,
\(a\) and\(b\), differ (the genetic distance
between two individuals over all their loci), \(d\) is the
number of different alleles between two individuals at locus
\(j\) (for diploids, \(d\) can be 0, 1 or 2),
and\selectlanguage{greek}\emph{ν} \selectlanguage{english}is the number of unique possible pairs of individuals
\(a\) and\(b\), where \(b\neq a\),
within a population.

Genotypic descriptors as empirical functions of the rate of clonality

To assess the relation between \emph{c} and the genotypic descriptors,
we explored the mean results of simulations as a function of \emph{c} .
Depending on the shape of the curves obtained with simulated data, we
tested the fit with basic functions (for example, simple sigmoids and
exponentially decreasing distributions) as well as with sigmoid and
parabolic curves. To assess the accuracy of our empirically inferred
formula to describe the relationships, we computed the mean absolute
error (MAE) and the root-mean-square deviation (RMSD) between
pseudo-observed simulated values and fitted formulae. These two
deviation measures aggregate the magnitudes of the errors of predictions
into a single measure of predictive accuracy. This measure represents
the mean deviation of predicted values with respect to the observed
values and has the advantage of sharing the same units as the model
variable under evaluation. Lower deviation measures indicate higher
accuracy of an analytical formula in the prediction of data. These
measures must be interpreted at the same scale as the mean value of the
studied parameter (Pi\selectlanguage{ngerman}ñeiro, Perelman, Guerschman, \& Paruelo, 2008).

\begin{equation}
MAE=\frac{1}{n}.\sum_{1}^{n}\left|y_{s}-y_{f}\right|\nonumber \\
 \end{equation}\begin{equation}
RMSD=\sqrt{\frac{1}{n}.\sum_{1}^{n}\left(y_{s}-y_{f}\right)^{2}}\nonumber \\
 \end{equation}

where \emph{n} is the number of pseudo-observed simulations per
scenario, \emph{y\textsubscript{s}} is the simulated value of the
genotypic descriptor under consideration, and
\emph{y\textsubscript{f}}is the calculated value of the genotypic
descriptor using the fitted formula.

Identifiable signals in genotypic and genetic descriptors, inferences
and machine learning

Our second objective was to test for the ability of genotypic and
genetic descriptors to estimate specific rates of clonality. These
descriptors were commonly used in previous studies to roughly assess the
importance of clonality in determining population reproductive modes,
but no theoretical development has demonstrated the existence of
identifiable signals allowing such descriptors to be used as key
parameters with which to estimate rates of clonality. To assess the
existence of identifiable signals in these descriptors and demonstrate
their potential usefulness in inferring rates of clonality for one
episode of genotyping, we used the results obtained from the simulations
as classifiers to train a Bayesian supervised learning algorithm. We
used the simulation results to compute the approximate nonparametric
probability distributions of the genotypic and genetic descriptors
(\emph{i.e} ., the seven \emph{features}\(\varphi_{7}=\left[R,\beta_{p},\overset{\overline{}}{r_{d}},Mean\left[F_{\text{IS}}\right],Var\left[F_{\text{IS}}\right],Skew\left[F_{\text{IS}}\right],Kurt\left[F_{\text{IS}}\right]\right]\)) with
combinations of Gaussian kernels under known rates of clonality,
resulting in a \emph{classifier} with 12 classes (one class for each
rate of clonality to be inferred: c=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6,
0.7, 0.8, 0.9, 0.99 and 1), hereafter referred to as \(C_{12}\).

\begin{equation}
L\left(\varphi_{7}\middle|C_{12}\right)=L\left(R,\beta_{p},\overset{\overline{}}{r_{d}},Mean\left[F_{\text{IS}}\right],Var\left[F_{\text{IS}}\right],Skew\left[F_{\text{IS}}\right],Kurt\left[F_{\text{IS}}\right]\middle|N,c,u\right)\nonumber \\
 \end{equation}

Provided that dependencies between the seven genotypic and genetic
descriptors are evenly distributed or cancel each other out or that
their distributions sufficiently segregate over their means per class,
we can approximate the joint probability model using the conditional
independence between features (Hand \& Yu, 2001; Webb, Boughton, \&
Wang, 2005; Zhang, 2004). The posterior probability of the\emph{i}
\textsuperscript{th} class, given that the seven measured features are
known, can be expressed as the product of the seven likelihoods of each
feature weighted by the prior probability of the class.

\begin{equation}
P\left(C_{i}\middle|\varphi_{7}\right)=p\left(C_{i}\right).\prod_{j=1}^{7}{L\left(\varphi_{j}\middle|C_{i}\right)}\nonumber \\
 \end{equation}

From this joint posterior probability, we identified the maximum \emph{a
posteriori} (\emph{MAP} ) to discern the class (``rate of clonality''
and ``population size'' pair) most likely to explain the measured
features.

\begin{equation}
MAP=\operatorname{}\left[p\left(C_{i}\right).\prod_{j=1}^{7}{L\left(\varphi_{j}\middle|C_{i}\right)}\right]\nonumber \\
 \end{equation}

We assumed a uniform distribution prior, \emph{i.e.} , equiprobability
for each class \(p\left(C_{i}\right)=1/12\), to place the algorithm in an initial
state of complete ignorance of the likely values that the two parameters
might take.

We built training and test databases of 100 and 30 replicates
per\emph{rate of clonality and population size} pair, respectively. We
explored by cross-validation whether there were enough identifiable
signals in the features of our classifier \(C_{12}\) to infer
the true rates of clonality with known values of only population
genotypic (\(R,\beta_{p}\)) and genetic (\(F_{\text{IS}},\overset{\overline{}}{r_{d}}\)) indices
alone and in combination. Posterior distributions of the thirty test
pseudo-observed datasets per \emph{rate of clonality} and
\emph{population size} pair were combined to plot the results.

\textbf{Results}

We first explored the results at equilibrium to understand the influence
of clonality on \emph{R} , Pareto\selectlanguage{greek}\emph{β} , \selectlanguage{english}LD measured as
ṝ\textsubscript{d}, and the mean, variance, skewness and kurtosis of
F\textsubscript{IS} at three population sizes (\emph{N}
=10\textsuperscript{5}, Figure 1;\emph{N} =10\textsuperscript{3}, Figure
S1a and\emph{N} =10\textsuperscript{4}, Figure S1b) and then examined
the evolutionary dynamics of the parameters over generations to
determine the effect of clonality at different time steps and quantify
the time needed to converge towards stationary values (Figures 2 and
S2). We assessed which genotypic and genetic parameters produced the
most identifiable signal, allowing accurate inferences (Figures 3, S3
and S4). Finally, we approached the issue of sampling strategy to
determine its effects on the accuracy of estimates for datasets obtained
from natural populations (Figures 4 and S2).

Genotypic richness and the distribution of clonal size at equilibrium
under an increasing rate of clonality

In terms of genotypic diversity, our results showed a clear,
progressive, and even stepwise decrease with increasing rates of
clonality (Figures 1 and S1).

When genotyping the entire population, the relationship between
\emph{R}and \emph{c} (Figure 3) does not follow a linear trend, such
as\(R=1-c\), as might have been assumed in some previous
studies. The relationship is best modelled by
\(R=\sqrt{1-c^{2}}\)(\emph{N} =100000: \emph{MAE} =0.011 and \emph{RMSD} =
0.020 for\(\overset{\overline{}}{R}=0.69\)). The same equation fits the simulation
results regardless of the population size, with slightly larger
deviations at smaller population sizes, as expected with an increasing
strength of genetic drift (\emph{N} =10000: \emph{MAE} =0.013 and
\emph{RMSD} = 0.021 for \(\overset{\overline{}}{R}=0.70\);\emph{N} =1000: \emph{MAE}
=0.029 and \emph{RMSD} = 0.041 for\(\overset{\overline{}}{R}=0.71\)), but still
providing an accurate approximation.

The curve describing the evolution of the parameter Pareto
\(\beta\) in the power-law distribution of clonal sizes
depending on the rate of clonality shows a slightly more complex
pattern. The curve has the typical shape of a sum of two sigmoid curves
with three sub-domains delimited by two inflection points (Figure 1).
Very low levels of clonality (0\textless{}\emph{c} \textless{}0.1) lead
to maximum Pareto\(\beta\)-values, which depend on the
population size (approximately 8 for \emph{N} =100 individuals to 15 for
\emph{N} =100000 individuals). For these distinct initial values, the
curves show an extremely similar shape regardless of population size,
with a marked sigmoid shape of Pareto \(\beta\)-values
declining from approximately 8 (value corresponding to high richness and
evenness; Arnaud-Haond et al., 2007) at \emph{c} =0.1 to nearly 0 for
\emph{c} =1. Interestingly, the value\(\beta\)=2 is reached for
clonal rates of approximately 0.8 to 0.9 for all population sizes.
Between clonal rates of 0.2 and 0.9, the decline in \(\beta\)
is nearly linear and flat for all population sizes. For\emph{N} =
100000, the sum of two fitted sigmoid curves produces the following
equation:

\(\beta=\frac{337335}{1+e^{16\times(c+0.65)}}+\frac{5}{1+e^{6.8\times(c-0.80)}}\)(\emph{MAE} =0.30 and \emph{RMSD} = 0.40
for\(\overset{\overline{}}{\beta}=4.66\));

for \(N=10000\),

\(\beta=\frac{506607}{1+e^{9.8\times(c+1.12)}}+\frac{4}{1+e^{8.3\times(c-0.81)}}\)(\emph{MAE} =0.27 and \emph{RMSD} = 0.36
for\(\overset{\overline{}}{\beta}=3.94\)); and

for \(N=1000\),

\(\beta=\frac{5.6}{1+e^{5\times(c-0.58)}}+\frac{3.8}{1+e^{50\times(c-0.19)}}\)(\emph{MAE} =0.32 and \emph{RMSD} = 0.40
for\(\overset{\overline{}}{\beta}=3.63\)).

Evolution of the genetic composition of populations under an increasing
rate of clonality

In contrast to the genotypic results but in agreement with previous
studies on populations at equilibrium with a realistic low mutation rate
(Balloux et al., 2003; Navascu\selectlanguage{ngerman}és et al., 2010; Stoeckel \& Masson,
2014), all mean genetic indices are nearly unaffected until the rate of
clonality reaches 0.95 (Figures 1 and S1). In fact, only a slightly
larger variance, exemplified at smaller population sizes, can be
observed at \emph{c} =0.9 for \emph{F} \textsubscript{IS} and its
moments and for \({\overset{\overline{}}{r}}_{d}\). The effects of \emph{c} on genetic
parameters are thus limited to extreme \emph{c} values.

When the rate of clonality reaches 0.99, values of\emph{F}
\textsubscript{IS} are slightly negative, a situation perceptible mostly
in a small population (\emph{N} =1000, Figure S1a). The three deeper
moments of \emph{F} \textsubscript{IS} distributions, however, show
values strongly departing from zero. At this highly elevated rate of
clonality, LD very slightly departs from 0 for large population sizes
(\emph{N} \selectlanguage{english}[?]10000) and then shows extreme values at\emph{c} =1
(\({\overset{\overline{}}{r}}_{d}\) of approximately 0.8 and 0.6 for \emph{N} =10000 and
100000, respectively). Interestingly, very small populations show a
modest (mean value of approximately 0.06) but noticeable departure from
0 as well as large variance at \emph{c} =0.99, whereas
\({\overset{\overline{}}{r}}_{d}\) returns to zero (in fact to a very slightly negative
value) at \emph{c} =1, with a more limited variance. This unexpected
behaviour occurs because clonality, by increasing the number of
generations to reach the genotype frequencies expected under
Hardy-Weinberg assumptions, allows genetic drift to control the dynamics
of genetic diversity (Reichel et al., 2016; Rouger, Reichel, Malrieu,
Masson, \& Stoeckel, 2016). In strictly clonal small populations
containing 1000 individuals, the number of genets ranges from a minimum
of 91 to a maximum of 102, with a median of 97. These populations are
dominated by one main multi-locus genotype (MLG), and the remaining MLGs
are scarcely represented (two or three copies each), appearing as
derived from the main one only through somatic mutations. These
populations thus consist of the same multi-locus lineage (MLL)
characterising the genet, \emph{i.e.} , the ensemble of ramets issued
from the same event of sexual reproduction (Arnaud-Haond et al., 2007).
MLGs mostly diverge from each other by 1 to a maximum of 16 alleles
(median=2) over a total of 200 alleles per MLG. The\({\overset{\overline{}}{r}}_{d}\)
values are thus driven by the random association of the few alleles
recently appearing by mutation in an overdominant clonal lineage fixed
by genetic drift. MLGs differing by very few loci imply that
\(V_{D}\) tends to zero and that\(\text{var}_{j}\) tends to
non-zero positive values at each locus\(j\).

Evolution of parameters when moving towards equilibrium

For \emph{c} \selectlanguage{english}[?]0.99, the maximum number of generations required to reach
the stationary mean value of the genotypic descriptor was several tens
to several hundreds, whereas for \emph{c\textgreater{}} 0.99\emph{,} the
convergence time far exceeded 300 generations (Figures 2 and S2). For
the genetic indices, a nearly stable mean value was observed beginning
in generation 100, with very small fluctuations. Strictly clonal
populations (\emph{c} =1) differed: their genotypic parameters also
reached a steady value early, but their genetic parameters continued to
evolve for 10000 or more generations. These simulation results are in
line with results obtained with mathematical assessments (Reichel et
al., 2016): the farther the population is from its equilibrium, the
faster it converges towards the equilibrium values. The trajectory then
slows down as the values approach those expected at equilibrium.
Mathematical analysis predicted that the equilibrium values would be
asymptotically reached after a maximum convergence time that depended on
the relative strength of clonality, genetic drift and mutation.

Identifiability of genotypic and genetic signals using machine learning

Supervised machine learning, as expected, delivers estimates consistent
with the description of parameter evolution with increasing \emph{c}when
all genotypes in the population are known (Figures 3 and S3). Genotypic
indices allow a reasonable estimate of \emph{c} throughout its range,
while genetic parameters allow such precision only for very high values.
Genotypic parameters evolve gradually with high accuracy of the
estimated \emph{c} based on \emph{R} and a slightly wider but still
rather precise distribution when based on an intermediate
Pareto\(\beta\). In contrast, but logically (as the mean values
of genetic parameters are nearly unaffected by increasing clonality
until extreme rates are reached; Figures 1 and S1), machine learning
produced a wide distribution of estimates around simulated values of
\emph{c} up to\emph{c} =0.6 for \emph{F} \textsubscript{IS} and \emph{c}
=0.9 for\({\overset{\overline{}}{r}}_{d}\). This distribution, however, is not entirely
flat, and although \emph{c} estimates are poor at modest rates of
clonality, they become precise near values of \emph{c} between 0.7 and
0.99.

Based on supervised machine learning, the variance in\emph{F}
\textsubscript{IS} was the most identifiable signal among the studied
genetic parameters (Figure S4). The mean and variance of\emph{F}
\textsubscript{IS} contain more identifiable signals
than\({\overset{\overline{}}{r}}_{d}\) in the range of \(0<c<0.9.\ \)The mean and
variance of \emph{F} \textsubscript{IS} values even show rather accurate
inferred rates of clonality from \emph{c} =0.7 to \emph{c} =1. The
variance in \emph{F} \textsubscript{IS} showed the best ability to
quantitatively infer \emph{c} \textless{}0.5 but produced an error
of\(\pm 0.3\). Using all moment values of \emph{F}
\textsubscript{IS} and\({\overset{\overline{}}{r}}_{d}\), the supervised learning
algorithm groups the strength of these parameters, increasing the
precision to quantitatively infer \emph{c} . Rates of clonality from
\emph{c} =0.7 to\emph{c} =1 were inferred with no error; from \emph{c}
=0.4 to\emph{c} =0.7, with low error (\(\pm 0.1\)); and from
\emph{c} =0 to\emph{c} =0.4, with larger errors (\(\pm 0.3\)).

Taken together, the genotypic and genetic parameters thus complement
each other to properly estimate \emph{c} , with the first allowing very
precise estimates of \emph{c} up to 0.95, where the latter become useful
and precise. The combination of genotypic and genetic parameters should
thus be considered to precisely estimate the whole range of possible
rates of clonality in natural populations, although genotypic
parameters\emph{a priori} appear to be the most important to retain
across the widest range of possible \emph{c} values. Theoretically, a
combination of \emph{R} and (variance in) \emph{F} \textsubscript{IS}
would be best for obtaining a good estimate of \emph{c} for any natural
population when no \emph{a priori} information on its extent is
available.

Subsampling

The inference method described above assumes that all individuals from
the \emph{in silico} population have been sampled and genotyped. When
subsampling is applied in a realistic manner (\emph{i.e.} , mimicking
the subsampling level of most studies in molecular ecology), however,
real issues emerge in terms of parameter accuracy and consequent
estimates of\emph{c} .

Genetic parameters (which proved to be less informative for assessing
the clonal rate) were nearly unaffected by realistic sample sizes,
whereas genotypic parameters (which were most informative) were
considerably overestimated when using realistic sample sizes, leading to
a gross underestimate of c from real datasets collected from natural
populations. This situation remained nearly unchanged when subsampling
was performed before the population reached equilibrium (see Figure S2).
The \emph{R} parameter is so susceptible to sampling bias that a
sampling effort of 50 units, consistent with many works studied thus
far, including ours, cannot reliably estimate an \emph{R} value lower
than 0.9, with the exception of highly clonal populations (\emph{c}
\textgreater{}0.8). A correct and unbiased estimate of R can be achieved
only by genotyping the entire population (Figures 4 and S2).
Interestingly, the variance in \emph{F} \textsubscript{IS} computed from
samples provided more identifiable signals of rates of clonality at all
populations sizes than genotyping all individuals. However, away from
equilibrium, the variance in \emph{F} \textsubscript{IS} became less
informative than that obtained at equilibrium (Figure S2).

\textbf{Discussion}

This work sheds new light on the precise influence of PC on the
genotypic and genetic composition of natural populations, allowing the
identification of the most accurate parameters that should theoretically
be used to estimate \emph{c} under a wide range of conditions.
Nevertheless, the results also clearly demonstrate that the most useful
parameters, namely, the \emph{R} and Pareto \selectlanguage{greek}\emph{β} \selectlanguage{english}describing
genotypic diversity, are seriously affected by sampling density. This
vulnerability raises questions about our ability to detect clonality by
sampling large populations only once and serious doubts as to the
possibility of quantitatively inferring rates of clonality in large
populations based on genotypic diversity alone.

These findings stimulate new interpretations of some published data and
perspectives on improvements that are required to further understand the
dynamics and evolution of the broad range of species exhibiting PC.

Parameters most influenced by c and the consequent accuracy of
inferences based on these parameters

The index \emph{R} (clonal richness) is widely used to assess the level
of clonality within natural populations, especially in correlation with
environmental drivers, to decipher the impacts of ecological features on
PC (McMahon et al., 2017). Using entire populations, we empirically
formalised the mathematical relationship between \emph{c} and the
genotypic richness indices \emph{R} and \selectlanguage{greek}β\selectlanguage{ngerman} (Figures 1 and
S1). The relationship with \emph{R} is not linear (as sometimes
seemingly assumed in the literature) but follows\(R=\sqrt{1-c^{2}}\pm\varepsilon\) (with
\(\varepsilon\) being a small, positive, almost zero random error
depending on the strength of genetic drift). Clonal evenness,
represented by Pareto \(\beta,\) is also not a linear function
of \emph{c} . Instead, Pareto \(\beta\)follows a custom sigmoid
curve with three domains (ranging from \emph{c}\(\in\) {[}0,
0.15{]}, {[}0.15, 0.9{]} and {[}0.9, 1{]}), with the first and last
showing a strong decrease in Pareto \(\beta\) with
increasing\emph{c.} In contrast, in the smooth linear domain ranging
from\emph{c} \textasciitilde{}0.15 to \emph{c} \textasciitilde{}0.9, the
relationship is almost horizontal, suggesting limited changes in
genotypic evenness in populations with balanced amounts of sexual and
clonal events.

In contrast, and in agreement with previous findings (Balloux et al.,
2003; De Meeûs et al., 2006), the genetic parameters are, on average,
largely unaffected below extreme rates of clonality (\emph{c}
\textless{}0.95). However, the variance in\emph{F} \textsubscript{IS}
and \({\overset{\overline{}}{r}}_{d}\) hints at PC and should theoretically allow
estimation of its extent under a high prevalence of clonality (\emph{c}
\selectlanguage{english}[?]0.95; Figures and S1). Clonality acts by releasing the coercive effects
of sexuality that constrain and channel the evolutionary trajectories of
genotype frequencies towards Hardy-Weinberg proportions, which in turn
increases the range of possible values for the genetic indices. This
effect results in a broader distribution of genetic indices with a
larger variance and unusual shapes, despite nearly unaffected mean
values. The effect of clonality on the composition of natural
populations is thus expected to be much more pronounced in terms of the
genotypic structure, which strongly influences the nature of the targets
of natural selection, the vectors of migration and the long-term
retention of polymorphism, than for the genetic composition of
populations.

Logically, genetic parameters reach their equilibrium value with lower
temporal variation and faster than genotypic indices, even at small
population sizes (\emph{N} \selectlanguage{english}[?]1000 in our simulations). Although they are
poorly informative regarding \emph{c} below extreme values, accounting
for genetic indices may limit the risk of misinterpretation when
estimating \emph{c} not at equilibrium.

As a consequence, genotypic and genetic parameters appear to be
complementary in terms of the estimation of \emph{c} , with the former
being helpful at equilibrium and for all values of
\emph{c}(\textless{}0.95) and the latter being more accurate for
estimating the incidence of clonality in populations not at equilibrium
or discriminating among extreme values of \emph{c} , which often implies
a longer time needed to reach equilibrium (Reichel et al., 2016).

Unfortunately, these relationships cannot provide reliable information
for detecting PC or estimating \emph{c} due to the pervasive effect of
sampling on these parameters, which has proven particularly worrying and
raises questions regarding many conclusions reached thus far in the
literature as to the importance of sexual reproduction in a diverse
range of partially clonal species.

Detecting clonality under realistic conditions

Based on our results, clonal richness (\emph{R} ) and clonal evenness
(Pareto \(\beta\)) are highly sensitive to sampling. Even using
relatively large sample sizes (from 100 to 500 individuals) leads to
deeply biased estimates of the true \emph{R} and \(\beta\) and
thus\emph{c} values. \emph{R} is always greatly overestimated, by some
orders of magnitude more than previously demonstrated with empirical
datasets for which the rates of clonality remained unknown (Arnaud-Haond
et al., 2007; Gorospe et al., 2015), and except in nearly strictly
sexual populations, \(\beta\) was also greatly overestimated
(for\(c\geq 0.1\)). Genotypic descriptors computed from realistic
sample sizes may be informative only for rare cases of small population
sizes (\emph{N} \selectlanguage{english}[?]1000 individuals in the case of our simulations). For
most situations where population sizes are large, genotypic descriptors
computed with realistic sample sizes result in extreme underestimation
of the rates of clonality (see below) or even in overlooking the
occurrence of PC (\emph{i.e.} , considering the species as strictly
sexual). These results raise questions regarding the conclusions derived
in the literature from studies assessing the occurrence or even
sometimes the extent of clonality based only on genotypic indices.

In contrast, the distribution moments of \emph{F} \textsubscript{IS} and
mean LD for common sample sizes (more than 20 individuals) produced
values consistent with those obtained from genotyping the whole
population, yet they previously could be interpreted only for extreme
rates of clonality (c\selectlanguage{english}[?]0.95). Consequently, when analysing samples from
populations with more than 1000 individuals, most genetic descriptors
should remain informative and sometimes, together with any
\emph{R}values lower than 1, should be interpreted as a likely signature
of a high prevalence of clonality (c\selectlanguage{english}[?]0.95)

This worrying limitation recalls, for example, the results recently
reported by Dia et al. (2014) for a unicellular phytoplankton species
involved in harmful algal blooms (HABs), \emph{Alexandrium minutum} .
This species, which causes paralytic shellfish poisoning (PSP), shows an
alternation between clonal (during the bloom) and sexual phases. Dia et
al. (2004) sampled populations throughout the bloom (clonal) events,
during which they grew from being nearly undetectable to exhibiting a
concentration of 10\textsuperscript{4} to 10\textsuperscript{5} cells
per litre. Of the more than 1000 strains cultivated, 265 were fully
genotyped, among which no replicated genotypes were found, driving the
estimate of clonal diversity to \emph{R} =1. Without extensive knowledge
of the biology of this species, clonality would not have been diagnosed
on the basis of this sampling, which raises questions regarding the
occurrence of clonality. Unfortunately, no \emph{F}
\textsubscript{IS}values could be reported in this study because only
the haploid phase could be sampled, and the LD detected suggested the
occurrence of recombination. However, according to these results,
genetic descriptors allow the detection or estimation of clonality when
its prevalence is extreme: the results by Dia et al. (2014) thus mainly
suggest that the clonal rate during the bloom event did not exceed 0.95
in the few previous generations, still leaving great uncertainty as to
the prevalence of sexual or clonal reproduction in this species.

Most target species in the literature, including clonal plants and
invasive and pathogenic species, exhibit extremely large population
sizes, thus raising serious questions regarding our ability to detect
clonality based on realistic sample sizes, let alone infer its
importance. The importance of sample size is reflected in the guidelines
provided by the pioneering work of Tibayrenc et al. (1991), who listed 8
criteria to detect clonality, among which fixed heterozygosity,
deviation from HWE and LD were expected to be important in the ability
to diagnose clonality. Nevertheless, these criteria would apply only to
diploid species with extreme rates of clonality, excluding haploid
lineages and diploid species with \emph{c} \textless{}0.95.

One may consider the clonal mechanisms and the way clonal replicates
spatially disperse to better estimate the effect of the joint incidence
of the sampling density and scale of dispersal of clones (driving the
scale of spatial autocorrelation of genotypes compared to the grain size
of sampling) on the ability of a given strategy to detect clonal
replicates and therefore on the conclusions derived from population
genetics data as to the incidence of sexual versus clonal reproduction.
Along a continuum of dispersal from microorganisms such as unicellular
algae and flying aphids to clonal plants with strong rhizomatic
connections and ramets more often clumped than dispersed, the spatial
autocorrelation of clones increases, as does the ability of a given
sampling strategy to reveal clonal replicates at equal sampling
densities. As a consequence, at the first end of this continuum, where
spatial dispersal is not limited (as is the case for \emph{A. minutum}
), genotypic parameters alone may not be informative on the existence or
extent of clonality except for nearly strictly clonal organisms such as
the human pathogen \emph{Trypanosoma cruzi} . Such power would be gained
as the spatial distance of clonal dispersal becomes lower than the
sampling mesh size (for an example of the influence of sampling strategy
in corals, see Gorospe et al., 2015; see Riginos, 2015) for a comment),
and clonal replicates would become decreasingly randomly diluted at
large population sizes and across vast spatial scales.

Quantifying clonality or merely evaluating its extent: how wrong can we
be?

In many studies, \emph{R} may reflect the orders of magnitude separating
sample size and population size (sometimes together with the clonal size
and/or clumping of clonal replicates) rather than the prevalence of
sexual reproduction. As illustrated in this work, even moderate values
of \emph{R} under our usually very small sampling densities (several
tens of sampling units in populations bearing one hundred thousand to
millions of them) may thus suggest a high prevalence of clonal
reproduction. Some examples exist in the literature in which only a good
knowledge of species biology prevents misleading conclusions based on
values of genotypic diversity. These examples indicate the need to be
very careful in interpreting genotypic parameters alone in the numerous
cases where no such extensive knowledge of the species studied exists.
An enlightening case is the study by Orantes et al. (2012) on aphids
reproducing through cyclical parthenogenesis. Eight populations
of\emph{Aphis glycines} were sampled at two time steps corresponding to
the early season, when sexual reproduction arises at rather small
population sizes, and the late season, after a demographic explosion of
populations under full clonality. Against all expectations based on a
presumed relationship between \emph{R} and \emph{c} and ignoring the
effect of sampling density, Orantes et al. (2012) found lower genotypic
diversity during the season of sexual reproduction (average \emph{R} of
0.85, average Pareto \(\beta\) of 2.9) than during the later
season of pure clonality (average \emph{R} of 0.97, average Pareto
\(\beta\) of 4.2), \emph{i.e.} , \(R_{\text{sexual}}<R_{\text{clonal}}\)
and\(\beta_{\text{sexual}}<\beta_{\text{clonal}}\). Without knowledge of the cycle and a good
understanding of the effect of population\emph{versus} sample size, a
higher rate of sexual reproduction in the late season could have been
inferred. However, using the guidelines we aimed to develop here,
\emph{R} would mostly signal the significance of clonality and call for
careful screening of genetic parameters. In fact, departure from HWE in
this study confirms the complementarity of genotypic and genetic
parameters by supporting the prevalence of clonal reproduction across
the cycle, with mean \emph{F} \textsubscript{IS}values of -0.21 and
-0.24 in the earlier and later season, respectively, suggesting a more
important influence of clonality in the later season, with a lower mean
and larger variance. Similar patterns have been found in multiple
studies on cyclical parthenogenetic species (e.g., Gilabert, Dedryver,
Stoeckel, Plantegenest, \& Simon, 2015; Loxdale et al., 2011). Another
example is a highly clonal root-sucking nematode,\emph{Xiphinema index}
, which shows mid-range \emph{R} values (0.16 to 0.39); however,
negative mean \emph{F} \textsubscript{IS} values with large variance in
agreement with \emph{LD} values suggest a rates of clonality exceeding
0.95 in all these populations, which had better agree with a
naturalistic knowledge (Villate et al., 2010).

In fact, revising the numerous data acquired on clonal plants, including
seagrasses, in light of the present results reveals very frequent
negative \emph{F} \textsubscript{IS} values, suggesting a much higher
contribution of clonality than previously thought (Evans et al., 2014;
Sinclair, Krauss, Anthony, Hovey, \& Kendrick, 2014; Stoeckel et al.,
2006) on the basis of their average \emph{R} values (see Arnaud-Haond,
Stoeckel, \& Bailleul, 2019 for a meta-analysis). Unfortunately,\emph{F}
\textsubscript{IS} is often neglected in ecological studies, possibly
due to difficulties in disentangling the influence of technical
shortcomings such as null alleles from non-random mating such as selfing
in some studies. In the seagrass literature, for example, moderate
levels of \emph{R} have led some authors to propose that sexual
reproduction has a high incidence and may thus contribute greatly to
recombination and dispersal through seed production (McMahon et al.,
2017). The joint re-analysis of \emph{R} and \emph{F}
\textsubscript{IS}values and their correlation can illuminate likely
extreme but overlooked clonal rates (also see Arnaud-Haond et al.,
2019). Although we seldom found this type of interpretation combining
genotypic and genetic parameters in the literature (but see the examples
above), this approach has been used by some authors, such as Ali et al.
(2014) (also see the references above), to infer the importance of
long-term clonality.

Interestingly, it has been shown that CloNcaSe, a method based on
repeated genotypes alone (Ali et al., 2016), can deliver incorrect
inferences, likely due to this subsampling effect on R. For a red alga
(\emph{Gracilaria chilensis} ) maintained through strict clonality for
generations, R values of 0.2 to 0.23 lead CloNcaSe to infer a \selectlanguage{polish}ĉ\selectlanguage{english}=0.82,
while ClonEstiMate, a second method based on transition probabilities of
genotype frequencies (Becheler et al., 2017), correctly infers a \selectlanguage{polish}ĉ\selectlanguage{english}=1.
Similarly, an aphid population sampled when mostly clonal lineages can
be found (\emph{Rhopalosiphum padi,} Halkett, Kindlmann, Plantegenest,
Sunnucks, \& Simon, 2006) has an R of 0.89, leading CloNcaSe to infer a
\selectlanguage{polish}ĉ\selectlanguage{english}=0.68, while ClonEstiMate better inferred a \selectlanguage{polish}ĉ\selectlanguage{english}=0.9.

Revising estimates of clonality in natural populations is particularly
important because present-day interpretations, often mostly focusing
on\emph{R,} are likely to grossly underestimate its extent. A vast body
of literature exists on the relationship between genotypic diversity and
the resistance or resilience of populations, as demonstrated in
experimental studies (Hughes \& Stachowicz, 2004; Hughes, Inouye,
Johnson, Underwood, \& Vellend, 2008; Reusch \& Lampert, 2004; but see
Massa, Paulino, Serr\selectlanguage{ngerman}ão, Duarte, \& Arnaud-Haond, 2013). Severe
overestimation of genotypic diversities may thus have led to strongly
misleading conclusions as to the resilience of the studied populations,
enhanced by their supposedly high \emph{R} value, as well as to their
ability to rely on dispersal of seeds due to recurrent events of sexual
reproduction (Kendrick et al., 2012, 2017; McMahon et al\emph{.} ,
2017). A case-by-case re-evaluation is thus needed to determine what may
hold true for some species, depending on their life history traits
(particularly longevity and turnover), but be completely incorrect for
others.

Conclusion

To conclude, our results showed a large impact of PC on the genotypic
composition of natural populations across the whole spectrum of all
possible rates of clonality, supporting its strong influence on the
tuning of evolutionary forces acting on these populations at different
spatial and temporal scales, even at low values of \emph{c} , as
conjectured by Lewis (1987). By affecting the main path of emergence of
new variants (somatic mutations rather than recombination), the targets
of natural selection and migration (``\ldots{} \emph{the entity that
persists and evolves is the clonal lineage} \ldots{}''; Ayala, 1998),
and the influence of drift (through the potentially much longer-term
retention of polymorphism; Reichel et al., 2016; Yonezawa et al., 2004;
and the present results), PC has the potential to profoundly influence
both the short-term dynamics and the evolutionary trajectories of
natural populations, even at a modest rate of clonality. Unravelling the
occurrence of clonality and understanding its extent are thus of
paramount importance for reconstructing, understanding and forecasting
the demography, ecology and evolution of the vast number of (possibly
including some that often remain undiagnosed) partially clonal species
across the tree of life.

Unfortunately, given the present state of knowledge and existing
analytical tools, the possibilities of inferring the rates of clonality
using one episode of population genotyping are remote. These results
also clarify the paradox of the often reported (but also often
overlooked) combination of high genotypic diversities, suggesting both
significant rates of sexual reproduction and significant heterozygote
excess, supporting nearly strict clonality (Dia et al., 2014; Orantes et
al., 2012). Many partially clonal organisms studied to date may rely on
a much higher prevalence of clonal reproduction than initially thought,
but clonal richness in these organisms may be overestimated due to the
limited sampling power at hand. This work thus calls for a reappraisal
of previously published data and conclusions on a broad range of clonal
organisms. Perspectives on how to infer the importance of clonality
using one episode of genotyping may, however, exist and can be
summarised with the following guidelines:

1) PC can be detected or quantified with the usual sampling power and
existing methods, mostly when the rate of clonality exceeds 95\%.

2) Departure from HWE towards heterozygote excess, particularly together
with a large variance in \emph{F\textsubscript{IS}} across loci,
indicates the occurrence and likely prevalence of clonality.

3) The joint examination of genotypic and genetic descriptors is often
necessary when PC detection is still needed (a recommendation
reminiscent of the ones formulated a long time ago for human pathogens
(Tibayrenc et al., 1991; see also Tibayrenc \& Ayala, 2012 but seldom
followed in ecological studies).

4) Considering both families of parameters may help better estimate the
extent of clonal reproduction but may require accepting a large
uncertainty, particularly when the rate of clonal reproduction is not
very high.

5) As such departures are expected due to clonality,\emph{F}
\textsubscript{IS} should not be used

a- for the estimation of \emph{p\textsubscript{sex}} (as initially
offered by Douhovnikoff \& Dodd, 2003 and relayed by Arnaud-Haond et
al., 2007), as it may be in most cases due to clonality rather than
non-random pairing of gametes. b- (perhaps not as strictly) when
filtering next-generation sequencing (NGS) data based on possible PC.
Such filters, failing to fit in the case of partial PC, would lead to at
best a very large number of informative loci being discarded and at
worst complete ignorance of the occurrence of PC in the dataset. c- to
detect technical artefacts such as null alleles and correct data or
select loci using models based on pure sexuality, including those
implemented in software, such as Micro-Checker (Van Oosterhout,
Hutchinson, Wills, \& Shipley, 2004).

6) Finally, due to the observed but faint signature of \emph{c} slightly
below 95\% in the second and further moments
of\emph{F\textsubscript{IS}} and to a lesser
extent\emph{r\textsubscript{d}} , which remains visually undetectable
but can be detected by machine learning methods, improvement is expected
to result from using machine learning based on informed databases
corresponding to the broadest possible range of scenarios. Such
development represents a promising avenue and will require large and
versatile databases to accommodate the diversity of life history traits
associated with clonality and subsampling to account for sampling
effects.

\textbf{Acknowledgements}

This study greatly profited from the exchange of ideas within the CLONIX
consortium. We warmly thank Fabien Halkett for his valuable comments
that improved the clarity of the manuscript and Myriam Valero, the
CLONIX consortium and particularly Ronan Becheler and Diane Bailleul for
useful discussions.

This work was funded by the French National Research Agency (projects
CLONIX: ANR-11-BSV7-007 and Clonix2D ANR-18-CE32-0001) and the INVAMAT
project (Plant Health and Environment Division of the French National
Institute of Agricultural Research).

This preprint has been reviewed and recommended by Peer Community In
Evolutionary Biology (https://doi.org/10.24072/pci.evolbiol.100078). We
thank Myriam Heuertz, Marcela Van Loo and David Macaya-Sanz for their
constructive comments that enhanced the manuscript.

\textbf{Data accessibility}

All data used for figures are fully numerically available at
doi:10.5281/zenodo.3666344

Code of simulation accessible
at\emph{https://www6.rennes.inrae.fr/igepp\_eng/Productions/Software}(temporary
link: http://gofile.me/4rKuR/DqPzrEh7y)

\textbf{Conflict of interest}

The authors of this preprint declare that they have no financial
conflicts of interest based on the content of this article.

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\par\null

\textbf{Author contributions}

SAH and SS laid the foundation of this work, conceived the study and
wrote the manuscript. SS built the simulator and formalised the
mathematical analyses and equations. SAH supervised BP's master
training. BP, SAH and SS contributed to code enhancement, computation,
data exploration and interpretation. BP tracked and managed the
bibliography. All authors contributed to editing. SAH and SS were
responsible for funding applications. All authors read and approved the
final manuscript.



\textbf{Figures legends}

\textbf{Figure 1.}Distribution of parameters explored at equilibrium
(10\textsuperscript{4} generations of quantitatively homogeneous
evolution since the initial random population) as a function of rates of
clonality, \emph{c} , at population size N=10\textsuperscript{5} (for
N=10\textsuperscript{3} and N=10\textsuperscript{4}, see Figure S1):
genotypic parameters: (a) R and (b) Pareto \selectlanguage{greek}β; \selectlanguage{english}and genetic parameters:
(c) F\textsubscript{IS} mean, (d) F\textsubscript{IS} variance, (e)
F\textsubscript{IS} skewness, (f) F\textsubscript{IS} kurtosis and (g)
linkage disequilibrium. measured as ṝ\textsubscript{d}. The X-axis is
linear from \emph{c} =0 to \emph{c} =0.9 and then non-linear for the
last two boxes at \emph{c} =0.99 and\emph{c} =1.

\textbf{Figure 2.} Temporal evolution of each parameter at a population
size of 10\textsuperscript{5} individuals per generation, as a function
of the number of generations elapsed from a fully random population at
generation 0. Genotypic parameters: (a) R and (b) Pareto \selectlanguage{greek}β; \selectlanguage{english}genetic
parameters: (c) FIS mean, (d) FIS variance, (e) linkage disequilibrium
measured as ṝd. For smaller population sizes (N=10\textsuperscript{3}and
N=10\textsuperscript{4}), see Figure S2 and S2b. Caution regarding
interpretation: all x-axes are non-linear, and the y-axis
for\({\overset{\overline{}}{r}}_{d}\) and the mean and variance of the\emph{F}
\textsubscript{IS} distributions present one to two changes in scaling.

\textbf{Figure 3.}Machine learning inferences of\emph{c} at
N=10\textsuperscript{5} and for each parameter used for inference:
genotypic parameters (a) R and (b) Pareto \selectlanguage{greek}β \selectlanguage{english}and genetic parameters (c)
F\textsubscript{IS} and (d) ṝ\textsubscript{d}, as well as (e) the
combination of all four parameters. The inferred values are plotted
against the simulated values, with the density gradient from black to
light grey indicating the most to least likely/probable.

\textbf{Figure 4.} Subsampling effects on the distributions of genotypic
indices (R and Pareto \selectlanguage{greek}β\selectlanguage{english}) and genetic indices (mean and variance of the
F\textsubscript{IS} distribution and LD measured as ṝ\textsubscript{d}),
depending on the sample sizes applied to the dataset, with
N=10\textsuperscript{5} at equilibrium (generation g=10000). For smaller
population sizes (N=10\textsuperscript{3} and N=10\textsuperscript{4})
and for the combined effect of subsampling and a non-equilibrium state,
see supplementary Figure S2a and S2b. Caution regarding interpretation:
all x-axes are non-linear, and the y-axis for\({\overset{\overline{}}{r}}_{d}\) and the
mean and variance of the\emph{F} \textsubscript{IS} distributions
presents one change in scaling.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2020-02-Figure-1/2020-02-Figure-1}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2020-02-Figure-2/2020-02-Figure-2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2020-02-Figure-3/2020-02-Figure-3}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2020-02-Figure-4/2020-02-Figure-4}

\end{center}
\end{figure}

\selectlanguage{english}
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